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Duplij, Wess. (eds.) Noncommutative Structures in Mathematics and Physics (Proc.Kiev, 2000)(491s)

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Conference proceedings edited by Steven Duplij and Julius Wess from the NATO Advanced Research Workshop held in Kiev, Ukraine, September 24-28, 2000. It collects roughly 40 contributed papers on supergravity and branes, Lie superalgebras, quantum groups and Hopf algebras, q-deformed and noncommutative space-time, p-adic strings and related topics. This is a downloaded book by other authors, kept as reference material in the archive.

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Editors StevenDuplijandJuliusWess Noncommutative Structures in Mathematics andPhysics Proceedings ofthe NATO Advanced ResearchWorkshop “NONCOMMUTATIVE STRUCTURES IN MATHEMATICSAND PHYSICS” Kiev, Ukraine September 24-28,2000 kievarwe.tex; 12/03/2001; 3:49; p.1 v Editors StevenDuplij TheoryGroup Nuclear PhysicsLaboratory KharkovNational University Kharkov61077 Ukraine JuliusWess SektionPhysik Ludwig-Maximilians-Universit ¨at Theresienstr.37 D-80333M ¨unchen Germany Compilingand making-up byStevenDuplij . kievarwe.tex; 12/03/2001; 3:49; p.2 CONTENTS P REFACE viii J.WessGaugeTheories Beyond Gauge Theory 1 D.Leites,V.Serganova SymmetriesWider Than Supersymmetry 13 K. Stelle Tensions inSupergravity Braneworlds 31 P.Grozman,D.Leites AnUnconventionalSupergravity 41 E. Bergshoeff, R. Kallosh, A. Van Proeyen Supersymmetry Of RS BulkAnd Brane 49 D.Galtsov,V. Dyadichev D-branesAnd Vacuum Periodicity 61 P. Kosi´nski, J. Lukierski, P. Ma ´slankaQuantum Deformations Of Space-TimeSUSYAndNoncommutativeSuperﬁeld Theory 79 D.Leites,I.Shchepochkina TheHoweDualityAnd Lie Superalgebras 93 A.Sergeev EnvelopingAlgebraOfGL(3)AndOrthogonal Polynomials 113 S. Duplij, W. Marcinek Noninvertibility, Semisupermanifolds And CategoriesRegularization 125 F. Brandt An Overview Of New Supersymmetric Gauge Theories With 2-Form Gauge Potentials 141 K. Peeters, P. Vanhove, A. Westerberg Supersymmetric R 4 Actions AndQuantumCorrections To SuperspaceTorsion Constraints 153 S. Fedoruk, V. G. Zima Massive Superparticle With Spinorial Central Charges 161 A.Burinskii Rotating SuperBlack Holeas SpinningParticle 181 F.Toppan ClassifyingN-extended 1-dimensionalSuper Systems 195 C. Quesne Para, Pseudo, And Orthosupersymmetric Quantum MechanicsAnd Their Bosonization 203 A. Frydryszak Supersymmetric Odd Mechanical Systems And Hilbert Q-moduleQuantization 215 S. Vacaru, I. Chiosa, N. Vicol Locally Anisotropic Supergravity And GaugeGravity OnNoncommutative Spaces 229 T. Kobayashi, J. Kubo, M. Mondrag ´on, G. Zoupanos Finiteness In Conventional N= 1GUTs 245 kievarwe.tex; 12/03/2001; 3:49; p.3 CONTENTS vii J.Simon WorldVolume RealizationOf Automorphisms 259 G. Fiore, M. Maceda,J. Madore Some MetricsOn The Manin Plane 271 V.Lyubashenko Coherence IsomorphismsForA Hopf Category 283 A.Ganchev FusionRings And Tensor Categories 295 V.Mazorchuk On Categories Of Gelfand-ZetlinModules 299 D. Shklyarov, S. Sinel’shchikov, L. Vaksman Hidden Symmetry Of Some AlgebrasOf q-differential Operators 309 P. Jorgensen, D. Proskurin, Y. Samoilenko A Family Of∗-Algebras Allowing Wick Ordering: Fock Representations And Universal EnvelopingC ∗ -Algebras 321 A. U. Klimyk Nonstandard Quantization Of The Enevloping Algebra U(so(n))AndItsApplications 331 A. Gavrilik Can the Cabibbo mixing originate from noncommutative extradimensions? 343 N. Iorgov Nonclassical Type Representations Of Nonstandard Quantization Of Enveloping Algebras U(so(n)), U(so(n,1)) and U(iso(n)) 357 K. Landsteiner QuasiparticlesIn Non-commutativeFieldTheory 369 A. Sergyeyev Time Dependence And (Non)Commutativity Of Symmetries OfEvolution Equations 379 B.Dragovich,I.V. Volovich p-AdicStrings AndNoncommutativity 391 G. Djordjevi ´c, B. Dragovich, L. Ne ˇsi´cAdelic Quantum Mechanics: NonarchimedeanAnd NoncommutativeAspects 401 Y.Kozitsky GibbsStatesOfALatticeSystemOfQuantumAnharmonic Oscillators 415 D.Vassiliev A Metric-AfﬁneFieldModelFor TheNeutrino 427 M. Visinescu Generalized Taub-NUT Metrics And Killing-Yano Tensors 441 V.Dzhunushaliev AnEffectiveModel OfThe Spacetime Foam 453 A. Higuchi Possible Constraints On String Theory In Closed Space WithSymmetries 465 A.Alscher, H.Grabert SemiclassicalDynamics Of SU(2)Models 475 L IST OF SPEAKERS AND THEIR E- PRINTS 481 kievarwe.tex; 12/03/2001; 3:49; p.4 P REFACE Theconceptsofnoncommutativespace-timeandquantumgroupshavefound growing attention in quantum ﬁeld theory and string theory. The mathematical concepts of quantum groups have been far developed by mathematicians and physicists of the Eastern European countries. Especially, V. G. Drinfeld from Ukraine, S. Woronowicz from Poland and L. D. Faddeev from Russia have been pioneering the ﬁeld. It seems to be natural to bring together these scientists with researchers in string theory and quantum ﬁeld theory of the Western European countries. From another side, supersymmetry, as one of examples of noncom- mutative structure, was discovered in early 70’s in the West by J. Wess (one of the co-Directors) and B. Zumino and in the East by physicists from Ukraine V. P. Akulov and D. V. Volkov. Therefore, Ukraine seems to be a natural place to meet. Supersymmetry is a very important and intriguing mathematical concept which has become a basic ingredient in many branches of modern theoretical physics. In spite of its still lacking physical evidence, its far-reaching theoret- ical implications uphold the belief that supersymmetry plays a prominent role in the fundamental laws of nature. At present the most promising hope for a truly supersymmetric uniﬁed and ﬁnite description of quantum ﬁeld theory and general relativity is superstring theory and its latest formulation, Witten’s M- theory.Superstringspossessbyfarthelargestsetofgaugesymmetrieseverfound in physics, perhaps even large enough to eliminate all divergences in quantum gravity.Notonlydoessuperstring’ssymmetryincludethatofEinstein’stheoryof general relativity and the Yang-Mills theory, it also includes supergravity and the GrandUniﬁed Theories. One of the exciting new approaches to nonperturbative string theory involves M-theoryandduality,which,infact,forcetheoreticalphysiciststoreconsiderthe central role played by strings in supersymmetry. In this revised new picture all ﬁve superstring theories, which on ﬁrst glance have entirely different properties and spectra, are now seen as different vacua of a same theory, M-theory. This uniﬁcationcannot,however,occurattheperturbativelevel,becauseitisprecisely the perturbative analysis which singles out the ﬁve different string theories. The hope is that when one goes beyond this perturbative limit, and takes into account all non-perturbative effects, the ﬁve string theories turn out to be ﬁve different descriptions of the same physics. In this context a duality is a particular relation applying to string theories, which can map for instance the strong coupling re- gion of a theory to the weak coupling region of the same theory or of another kievarwe.tex; 12/03/2001; 3:49; p.5 PREFACE ix one, and vice versa, thus being an intrinsically non-perturbative relation. In the recent years, the structure of M-theory has begun to be uncovered, with the es- sential tool provided by supersymmetry. Its most striking characteristic is that it indicates that space-time should be eleven dimensional. Because of the intrinsic non-perturbative nature of any approach to M-theory, the study of the p-brane solitons,ormoresimply‘branes’,isanaturalsteptotake.Thebranesareextended objects present in M-theory or in string theories, generally associated to classical solutionsofthe respectivesupergravities. Quantum groups arise as the abstract structure underlying the symmetries of integrable systems. Then the theory of quantum inverse scattering gives rise to some deformed algebraic structures which were ﬁrst explained by Drinfeld as deformations of the envelopping algebras of the classical Lie algebras. An analo- gous structure was obtained by Woronowicz in the context of noncommutative C ∗ -algebras. There is a third approach, due to Yu. I. Manin, where quantum groupsareinterpretedastheendomorphismsofcertainnoncommutativealgebraic varieties deﬁned by quadratic algebras, called quantum linear spaces. L. D. Fad- deevandhiscollaboratorshadalsointerpretedthequantumgroupsfromthepoint of view of corepresentations and quantum spaces, furnishing a connection with the quantum deformations of the universal enveloping algebras and the quantum double of Hopf algebras. From the algebraic point of view, quantum groups are Hopf algebras and the relation with the endomorphism algebra of quantum linear spaces comes from their corepresentations on tensor product spaces. The usual constructionofthecoactiononthetensorproductspaceinvolvestheﬂipoperator interchanging factors of the tensor product of the quantum linear spaces with the bialgebra. This fact implies the commutativity between the matrix elements of a representation of the endomorphism and the coordinates of the quantum lin- ear spaces. Moreover, the ﬂip operator for the tensor product is also involved in many steps of the construction of quantum groups. In the braided approach toq-deformations the ﬂip operator is replaced with a braiding giving rise to the quasi-tensorcategoryof k-modules,where anatural braided coaction appears. The study of differential geometry and differential calculus on quantum groups that Woronowicz initiated is also very important and worthwile to investi- gate.Nextstepinthisdirectionisconsiderationofnoncommutativespace-timeas a possible realistic picture of how space-time behaves at short distances. Starting from such a noncommutative space as conﬁguration space, one can generalize it to a phase space where noncommutativity is already intrinsic for a quantum mechanicalsystem.Thedeﬁnitionofthisnoncommutativephasespaceisderived from the noncommutative differential structure on the conﬁguration space. The noncommutativephasespaceisa q-deformationofthequantummechanicalphase space and one can apply all the machinery learned from quantum mechanics. If one demands that space-time variables are modules or co-modules of the q- deformed Lorentz group, then they satisfy commutation relations that make them kievarwe.tex; 12/03/2001; 3:49; p.6 x PREFACE elements of a non-commutative space. The action of momenta on this space is non-commutative as well. The full structure is determined by the (co-)module property. It can serve as an explicit example of a non-commutative structure for space-time. This has the advantages that the q-deformed Lorentz group plays the role of a kinematical group and thus determines many of the properties of this space and allows explicit calculations. One can explicitly construct Hilbert space representations of the algebra and ﬁnd that the vectors in the Hilbert space can be determined by measuring the time, the three-dimensional distance, the q- deformed angular momentum and its third component. The eigenvalues of these observables form a q-lattice with accumulation points on the light-cone. In a way physics on the light-cone is best approximated by this q-deformation. One can consider the simplest version of a q-deformed Heisenberg algebra as an example ofanoncommutativestructure,ﬁrstderiveacalculusentirelybasedonthealgebra and thenformulatelaws ofphysicsbasedon thiscalculus. Bringingtogetherscientistsfromquantumﬁeldtheory,stringtheoryandquan- tum gravity with researchers in noncommutative geometry, Hopf algebras and quantum groups as well as experts on representation theory of these algebras had a stimulating effect on each side and will lead to new developments. In each ﬁeld there is a highly developed knowledge by experts which can only be transformed to another ﬁeld only by having close personal contact through dis- cussions,talksandreports.Wehopethatcommonprojectscanbefoundsuchthat working in these projects the detailed techniques can be learned from each other. The Workshop has promoted the development of new directions in the ﬁeld of modern theoretical and mathematical physics combining the efforts of scientists fromNATO,East Europeancountries andNIS. WearegreatlyindebtedtotheNATODivisionofScientiﬁcAffairsforfunding ofourmeetingandtotheNationalAcademyofSciencesofUkraineforhelpinits localorganizing.Itisalsoagreatpleasuretothankallthepeoplewhocontributed to the successful organization of the Workshop, especially members of the Local Organizing Committee Profs. N. Chashchyn and P. Smalko. Finally, we would liketothankalltheparticipantsforcreatinganexcellentworkingatmosphereand for outstanding contributions tothisvolume. Editors kievarwe.tex; 12/03/2001; 3:49; p.7 GAUGETHEORIESBEYOND GAUGE THEORY JULIUSWESS Sektion Physik der Ludwig-Maximilians-Universit ¨at Theresienstr. 37, D-80333M ¨unchen,Germany and Max-Planck-Institut f ¨ur Physik (Werner-Heisenberg-Institut) F¨ohringerRing 6,D-80805M ¨unchen,Germany 1. Algebraicpreliminaries In gauge theories we consider differentiable manifolds as base manifolds and ﬁ- bresthatcarryarepresentationofaLiegroup.Inthefollowingweshallshowthat itispossibletoreplacethedifferentiablemanifoldbyanon-commutativealgebra, ref. [1]. For this purpose we ﬁrst focus our attention on algebraic properties. The coordinatesx i x 1 ,...,x n ∈R, (1) areconsideredas elementsof an algebra over Csubject to the relations: R:x i x j −x j x i = 0. (2) This characterizes R n as a commutative space. The relations generate a 2-sided idealI R . From the algebraic point of view, we deal with the algebra freely generatedbytheelements x i and dividedby the ideal I R : A x =C /bracketleftbig [x 1 ,...,x n ] /bracketrightbig I R . (3) Formal power series are accepted, this is indicated by the double bracket. The elements of the algebra are the functions in R n that have a formal power series expansionattheorigin: f(x 1 ,...,x n )∈ A x , (4) f(x 1 ,...,x n ) = ∞ /summationdisplay r i =0 f r 1 ...r n (x 1 ) r 1 ····· (x n ) r n . kievarwe.tex; 12/03/2001; 3:49; p.8 2 J.WESS Multiplication is the pointwisemultiplication of thesefunctions. Themonomialsofﬁxeddegreeformaﬁnite-dimensionalsubspaceofthealge- bra.Thisalgebraicconceptcanbeeasilygeneralizedtonon-commutativespaces. We consider algebras freely generated by elements ˆx 1 ,...ˆx n , again calling them coordinates.Butnowwechangetherelationstoarriveatnon-commutativespaces: R ˆx,ˆx : [ˆx i ,ˆx j ] =iθ ij (ˆx). (5) Following L.Landau, non-commutativity carries a hat. Now we deal with the algebra: A ˆx =C<<ˆx 1 ,..., ˆx n >> I R ˆx,ˆx , (6) ˆf∈ A ˆx . Inthefollowingweimposeonemoreconditiononthealgebra:thedimension of the subspace of homogeneous polynomials should be the same as for com- muting coordinates. This is the so called Poincare-Birkhof-Witt property (PBW). Onlyalgebraswiththispropertywillbeconsidered,amongthemarethealgebras whereθ ij isa constant: Canonical structure, ref.[2]: [ˆx i ,ˆx j ] =iθ ij , (7) whereθ ij islinearin ˆx: Lie structure,ref. [3]: [ˆx i ,ˆx j ] =iθ ij k ˆx k , (8) whereθ ij isquadraticin ˆx: Quantumspace structure,ref.[4]: [ˆx i ,ˆx j ] =iθ ij kl ˆx k ˆx l , (9) The constants θ ij k andθ ij kl are subject to conditions to guarantee PBW. For Lie structures this will be the Jacobi identity, for the quantum space structure the Yang-Baxter equation. There is a natural vector space isomorphism between A x andA ˆx . It is based on the isomorphism of the vector spaces of homogeneous polynomialsthathavethesame degreedueto thePBWproperty. In order to establish the isomorphism we choose a particular basis in the vec- tor space of homogeneous polynomials in the non-commuting variables ˆxand characterize the elements of A ˆx by the coefﬁcient functions in this basis. The corresponding element in the algebra A x of commuting variables is supposed kievarwe.tex; 12/03/2001; 3:49; p.9 GAUGETHEORIES BEYONDGAUGETHEORY 3 to have the same coefﬁcient function. The particular form of this isomorphism depends on the basis chosen. The vector space isomorphism can be extended to analgebraisomorphism.Toestablishitwecomputethecoefﬁcientfunctionofthe productoftwoelementsin A ˆx andmapittoA x .Thisdeﬁnesaproductin A § that we denote as diamond product ( ♦product). The algebra with this ♦product we call ♦ A x .There is anaturalisomorphism: A ˆx ←→ ♦ A x . (10) The three structures that we have mentioned above have an even stronger property than PBW. It turns out that monomials in any well-deﬁned ordering of the coordinates form a basis. Among them is an ordering as we have used it before or the completely symmetrized ordering of monomials as well. For such structures we shall denote the ♦product as * product (star product), ref. [5]. For the canonical structure we obtain the Moyal-Weyl * product, ref. [6], if we start fromthe basis of completelysymmetrizedmonomials: (f∗g)(x) =e i 2 ∂ ∂xi θ ij ∂ ∂yj f(x)g(y) /vextendsingle/vextendsingle/vextendsingle y⇒x (11) = /integraldisplay d n y δ n (x−y)e i 2 ∂ ∂xi θ ij ∂ ∂yj f(x)g(y). For the Lie structure we can usetheBaker-Campbell-Hausdorf formula: e ik·ˆx e ip·ˆx =e i(k+p+ 1 2 g(k,p))·ˆx . (12) This deﬁnes g(k,p). (f∗g)(x) =e i 2 x·g(i ∂ ∂y ,i ∂ ∂z ) f(y)g(z) /vextendsingle/vextendsingle/vextendsingle y→xz→x . (13) For the quantum plane weconsiderthe exampleoftheManin plane ˆxˆy=qˆyˆx, (14) (f∗g)(x) =q −x /prime ∂ ∂x/prime y ∂ ∂y f(x,y)g(x /prime ,y /prime ) /vextendsingle/vextendsingle/vextendsingle x /prime →x y /prime →y . It is natural to use the elements of ♦ A x as objects in physics. Fields of a ﬁeld theory willbe suchobjects. φ(x)∈ ♦ A x . (15) The product of ﬁelds will always be the * product. To formulate ﬁeld equations we introduce derivatives. On the algebra A ˆx this can be done on purely algebraic grounds. We have to extend the algebra A ˆx by algebraic elements ˆ∂ i , ref. [7]. A kievarwe.tex; 12/03/2001; 3:49; p.10 4 J.WESS generalized Leibnizrulewill playtherole ofalgebraicrelations. Leibnizrule: (ˆ∂ i ˆfˆg) = ( ˆ∂ i ˆf)ˆg+O l i (ˆf)ˆ∂ l ˆg:R ˆx,ˆ∂ . (16) From the law of associativity in A ˆx follows that the operation Ohas to be an algebrahomomorphism: O i j (ˆfˆg) =O i l (ˆf)O l j (ˆg). (17) But we shall restrict the Leibniz rule by an even stronger requirement. The ideal generated by theR ˆx,ˆx relations has to remain a two-sided ideal in the larger algebragenerated by ˆxandˆ∂.Thisleadstoso calledconsistencyrelations. FinallyR ˆ∂,ˆ∂ relations have to be deﬁned. As conditions we consider the ˆ∂ subalgebra, demand PBW and derive consistency relations from R ˆ∂,ˆ∂ and the Leibnizruleasbefore.Derivativesdeﬁnedthatwayinduceamapfrom A ˆx toA ˆx : ˆf∈A ˆx ,(ˆ∂ i ˆf)∈A ˆx , (18) (ˆ∂ i ˆf) = ˆ∂ i ˆf−O l i (ˆf)ˆ∂ l . This algebraic concept of derivatives has been explained in ref[] and applied toquantumplanes.Followingthesamestrategyderivativescanbedeﬁnedforthe canonicalstructure aswell. For the rest of this talk we will restrict ourselves to the canonical case only. The Leibniz rule forthecanonical caseistheusualone: ˆ∂ i ˆx j =δ j i + ˆx j ˆ∂ i . (19) Itsatisﬁesalltheconsistencyrelations.Asexplainedabove,thederivativesinduce amapon the algebra A ˆx : ˆf∈A ˆx :ˆf→[ˆ∂ i ,ˆf]∈A ˆx . (20) Thisistherelationthatweshallusetodeﬁnederivativesonﬁelds.Forthispurpose wemap ˆ∂to ♦ A x .From(20)followsthatitbecomestheusualderivativein ♦ A x : f(x)→∂ i f(x). (21) Fromthedeﬁnitionofthe*product follows: ∂ i (f∗g) =∂ i f∗g+f∗∂ i g. (22) This is the Leibniz rule (20) when mapped to the ♦ A x algebra. As a consequence of(20)we ﬁnd that ˆx i −iθ ij ˆ∂ j (23) kievarwe.tex; 12/03/2001; 3:49; p.11 GAUGETHEORIES BEYONDGAUGETHEORY 5 commutes with all coordinates. For invertible θ ij this can be used to deﬁne the action of the derivative entirelyin A ˆx ˆ∂ i =−iθ −1 ij ˆx j . (24) Translatedto the ♦ A x algebrathisimplies: ∂ i f(x) =−iθ −1 ij [x j∗ , f]. (25) As a consequencewederive ˆ∂ j ˆ∂ k −ˆ∂ k ˆ∂ j =−iθ −1 jk :R ˆ∂,ˆ∂ . (26) ThisR ˆ∂,ˆ∂ relation satisﬁesall therequirementsof (ref7). To formulate a Lagrangian ﬁeld theory we have to learn how to integrate. Whereas it was easier to formulate derivatives on objects of A ˆx it is easier to formulateintegrationonobjectsof ♦ A x .Forthe canonicalstructurewe deﬁne: /integraldisplay ˆf= /integraldisplay d n x f(x), ˆf∈A ˆx ,f∈ ♦ A x . (27) This is alinear map ofthealgebra A ˆx intoC S:A ˆx →C, (28) S(c 1 ˆf+c 2 ˆg) =c 1 /integraldisplay ˆf+c 2 /integraldisplay ˆg, and ithasthe traceproperty: /integraldisplay ˆfˆg= /integraldisplay ˆgˆf. (29) This can be veriﬁedexplicitelyusing thedeﬁnition ofthe * product: /integraldisplay f∗g= /integraldisplay g∗f= /integraldisplay d n x f(x)g(x). (30) For the quantum space structure the deﬁnition (30) for the integral does not have thetraceproperty.Thereis,however,ameasurefortheintegrationthatleadstoan integralwith the traceproperty. /integraldisplay ˆf≡ /integraldisplay d n x µ(x)f(x) (31) For the Manin planewe canverifyexplicitelythat themeasure µ(x,y) = 1 xy(32) kievarwe.tex; 12/03/2001; 3:49; p.12 6 J.WESS hasthisproperty. In general we can construct Hilbert space representations of the algebra and deﬁne the integral as the trace. This will lead to inﬁnite sums that can be inter- preted as Riemannian sums for an integral and lead to the respective measure for theintegration. 2. Gauge theories Our aim is to formulate gaugetheories.Theywill bebased ona Lie algebra: [T a ,T b ] =if ab c T c . (33) In a usual gauge theorie on R n the ﬁelds will span a representation of the Lie algebraandtransform underaninﬁnitesimal gauge transformation: δ α 0 ψ(x) =iα 0 (x)ψ(x). (34) The transformation parameters areLiealgebravalued: α 0 (x) =α 0 a T a (35) and consequently: (δ α 0 δ β 0 −δ β 0 δ α 0 )ψ=−(β 0 α 0 −α 0 β 0 )ψ =i(α 0 ×β 0 )ψ=δ α 0 ×β 0 ψ, (36) α 0 ×β 0 ≡α 0 a β 0 b f ab c T c . covariant derivatives are deﬁned with the help of a Lie algebra valued gauge ﬁeld a: D i ψ= (∂ i −ia i )ψ, (37) a i =a a i T a . Toobtain: δ α 0 D i ψ=iα 0 D i ψ (38) we havetodemand: δa i =∂ i α 0 +i[α 0 ,a i ], (39) δa i,a =∂ i α 0 a −α 0 b f bc a a i,c . Toformulateagaugetheoryonanon-commutativespacewestartwithﬁelds ψ(x) that are elements of ♦ A ˆx and again span a representation of the Lie algebra (33). We demand thetransformation law: δ α ψ(x) =iα(x)∗ψ(x) (40) kievarwe.tex; 12/03/2001; 3:49; p.13 GAUGETHEORIES BEYONDGAUGETHEORY 7 in analogy to (34). But now we cannot demand αto be Lie algebra valued, we shallassumeittobe envelopingalgebra valued: α(x) =α 0 a (x)T a +α 1 ab (x) :T a T b : +···+α n−1 a 1 ...a n (x) :T a 1 ·····T a n : +··· (41) This is in analogy to (35). We have adopted the ::notation for the basis elements oftheenvelopingalgebra.We shall usethesymmetrized polynomials asa basis: :T a : =T a , (42) :T a T b : = 1 2(T a T b +T b T a )etc. Inanalogy to(36) weﬁnd (δ α δ β −δ β δ α )ψ= [α ∗ , β]∗ψ. (43) Naturally, [α ∗ , β]willbe anenveloping algebravalued element of ♦ A x . The unpleasant fact of the deﬁnition (41) of an enveloping algebra valued transformation parameter is that it depends on an inﬁnite set of parameter ﬁelds α n (x). In physics we would have to deal with an inﬁnite set of ﬁelds when deﬁning a covariant derivative, something we try to avoid. However, a gauge transformation can be realized by transformation parameters that depend on x viatheparameterﬁeld α 0 (x),thegaugeﬁeld a i,a (x)andtheirderivativesonly.In thenotationofeqn (41)we have α n a 1 ...a n+1 (x) =α n a 1 ...a n+1 (α 0 a (x),a 0 i,a (x),∂ i α 0 a (x),...).(44) Transformation parameters that are restricted that way we shall denote Λ α 0 (x). Theseparameterscan beconstructed insucha waythat eqn (36)holds: δ α 0 ψ(x) =iΛ α 0 (x) (x)∗ψ(x), (δ α 0 δ β 0 −δ β 0 δ α 0 )ψ=δ α 0 ×β 0 ψ, (45) (α 0 ×β 0 ) a =α 0 b β 0 c f bc a . This together with the * product is the deﬁning equations for the gauge transfor- mations. That such parameters Λ α 0 (x)can be found is not obvious, it’s rather a miracle in our present understanding of such gauge theories. Their existence is a consequenceof theSeiberg-Witten map[2]. In the second variation of ψwe also have to account for the variation of Λ α 0 as itdependson a i,a : (δ α 0 δ β 0 −δ β 0 δ α 0 )ψ=i(δ α 0 Λ β 0 −δ β 0 Λ α 0 )∗ψ+ [Λ α 0 ∗ ,Λ β 0 ]∗ψ,(46) =δ α 0 ×β 0 ψ=iΛ α 0 ×β 0 ∗ψ kievarwe.tex; 12/03/2001; 3:49; p.14 8 J.WESS We shall construct Λ α 0 in a power series expansion in θ. To illustrate the method we expand Λ α 0 to ﬁrst orderin θ Λ α 0 =α 0 a T a +θ ij Λ 1 α 0 ,ij +..., (47) To be consistent we expand the * product in (46) also to ﬁrst order in θand compare powers of θ. theθ-independent term deﬁnes α 0 ×β 0 as we have used it in (45). This had to be expected, this order is exactly the commutative case. To ﬁrstorder weobtaintheequation: θ ij /parenleftbig (δ α 0 Λ 1 β 0 ,ij −δ β 0 Λ 1 α 0 ,ij )−i([α 0 ,Λ 1 β 0 ,ij ]− (48) −[β 0 ,Λ 1 α 0 ,ij ]) /parenrightbig + 1 2∂ i α 0 a ∂ j β 0 b :T a T b :=θ ij Λ 1 α 0 ×β 0 ,ij . This equation has thesolution: θ ij Λ 1 α 0 ,ij = 1 2θ ij (∂ i α 0 a )a j,b :T a T b :. (49) We see that Λ 1 is of second order in the generators Tof the Lie algebra. The structure of eqn (46) allows a solution where Λ n , the term in (47) of order n−1 inθ, isa polynomialof order ninT. Λ α 0 =α 0 a T a + 1 2θ ij (∂ i α 0 a )a j,b :T a T b : +... (50) In a next step in the formulation of a gauge theory we introduce covariant derivatives. Eqn (24) shows that we can relate this problem to the construction of covariant coordinates. We try to deﬁne such coordinates with the help of a gauge ﬁeld,inthe samewayas wedidit forderivatives in eqn(37): X i =x i +A i (x), (51) δ α 0 X i ∗ψ=iΛ α 0 ∗X i ∗ψ. (52) This leads toatransformationlawforthegauge ﬁeld A i (x): δA i =−i[x i∗ ,Λ α 0 ] +i[Λ α 0 ∗ , A i ]. (53) We have to assume that A i is enveloping algebra valued but we try to make an ansatzwhere allthe coefﬁcient functionsonly dependon a i,a and its derivatives: A i (x) =A i,0 a (x)T a +A i,1 ab (x) :T a T b : +... (54) +A i,n−1 a 1 ...a n (x) :T a 1 ·····T a n : +..., A i,n =A i,n (a i,a ,∂a i,a ,...). kievarwe.tex; 12/03/2001; 3:49; p.15 GAUGETHEORIES BEYONDGAUGETHEORY 9 Now we expand (53) in θ, demandA i,n to be a polynomial of order ninθand solveeqn(53), A i (x) =θ ij V j , V j (x) =a j,a T a − 1 2θ ln a l,a (∂ n a j,b +F nj,b :T a T b : +..., (55) F nj,b =∂ n a j,b −∂ j a n,b +f cd b a n,c a j,d . This together with (41) is known as Seiberg-Witten map for an abelian gauge group. We have constructed it for an arbitrary non-abelian gauge group as well. Covariant derivatives follow from (37) D i ∗ψ= (∂ i −iV i )∗ψ, (56) δ α 0 D i ∗ψ=iΛ α 0 ∗D i ∗ψ. Wenowproceedwiththedeﬁnitionoftensorsasinausualgaugetheory,keeping inmind (27) ˜F ij =D i ∗D j −D j ∗D i −iθ −1 ij . (57) The transformation law of thetensor is δ α 0 ˜F ij =i[Λ α 0 ∗ ,˜F ij ]. (58) This can be veriﬁedfrom(53) and thedeﬁnitionof ˜F. Toﬁrstorderin θweﬁnd: ˜F ij =F ij,a T a +θ ln (F il,a F jn,l −(59) 1 2a l,a (2∂ n F ij,b +a n,c F ij,d f cd e )) :T a T b : +.... (60) We seethat new“contact”termsappear intheﬁeld strength ˜F. AgoodcandidateforaLagrangian is L= 1 4TrF ij ∗F ij . (61) Thetraceistakenintherepresentationspaceofthegenerators T.TheLagrangian (61) isnotinvariantbecause the * productisnot commutative: δL= 1 4Tri[Λ α 0 ∗ , L]. (62) We know, however, that the integral has the trace property (31). This allows us to deﬁne theinvariant action: W= 1 4 /integraldisplay TrF ij ∗F ij (63) = 1 4 /integraldisplay TrF ij F ij . kievarwe.tex; 12/03/2001; 3:49; p.16 10 J.WESS This action depends on the gauge ﬁeld a i,a and its derivatives only. It can be considered as a gauge-invariant object if a i,a transforms according to (39). this impliesthat WsatisﬁestheWardidentities. δ α 0 (x) W= 0, (64) δ α 0 (x) =−α 0 a (x) /parenleftbigg ∂ ∂x i δ a d +a i,b (x)f ab d /parenrightbigg δ δa i,d (x). TheLagrangianexpandedtoallordersin θ,isanon-localobject.Itremainstobe seen if it is acceptable for a quantum ﬁeld theory or if it has to be viewed as an effective Lagrangian,ref.[8]. References 1. B. Jur ˇco, S. Schraml, P. Schupp and J. Wess, Enveloping algebra-valued gauge transforma- tions for non-abelian gauge groups on non-commutative spaces , Eur. Phys. J. C 17, (2000) 521, hep-th/0006246. J.Madore,S.Schraml,P.SchuppandJ.Wess, Gaugetheoryonnoncommutativespaces ,Eur. Phys. J.C 16, (2000), 161, hep-th/0001203. B.Jurˇco,P.Schupp, NoncommutativeYang-Millsfromequivalenceofstarproducts ,Eur.Phys. J.C 14, 367 (2000),hep-th/0001032. B. Jurˇco, P. Schupp and J. Wess, Noncommutative gauge theory for Poisson manifolds , Nucl. Phys.B 584, (2000),784, hep-th/0005005. B. Jurˇco, P. Schupp and J. Wess, Nonabelian noncommutative gauge theory and Seiberg- Witten map , in preparation. 2. N. Seiberg and E. Witten, String theory and noncommutative geometry , JHEP9909(1999) 032, hep-th/9908142. 3. A. Dimakis, J. Madore, Differential Calculi and Linear Connections , J.Math.Phys. 37(1996) 4647. M. Dubois-Violette, R. Kerner, J. Madore, Gauge bosons in a noncommutative geometry , Phys.Lett. B217(1989)485. J. Hoppe, Diffeomorphism groups, Quantization and SU(∞), Int.J.Mod.Phys. A 4(1989) 5235. J. Madore, An Introduction to Noncommutative Differential Geometry and it Physical Applications , 2nd Edition, Cambridge University Press, 1999. B. deWit, J. Hoppe,H. Nicolai, Nucl.Phys. B305[FS23] (1988) 545. D. Kabat, W. Taylor IV, Spherical membranes in Matrix theory , Adv.Theor.Phys. 2 (1998) 181-206,(hep-th 9711078). 4. J.Wess, q-deformedHeisenbergAlgebras ,inH.Gausterer,H.GrosseandL.Pittner,eds.,Pro- ceedings of the 38. Internationale Universit ¨atswochen f ¨ur Kern- und Teilchenphysik, no. 543 inLect.NotesinPhys.,Springer-Verlag,2000,Schladming,January1999,math-ph/9910013. 5. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory and quantization. I. Deformationsof symplectic structures , Ann. Physics 111, 61 (1978). M. Kontsevitch, Deformationquantizationof Poissonmanifolds, I, q-alg/9709040. D.Sternheimer, DeformationQuantization: Twenty YearsAfter ,math/9809056. 6. H. Weyl, Quantenmechanik und Gruppentheorie , Z. Physik 46, 1 (1927); The theory of groups and quantum mechanics , Dover, New-York (1931), translated from Gruppentheorie undQuantenmechanik , Hirzel Verlag, Leipzig (1928). kievarwe.tex; 12/03/2001; 3:49; p.17 GAUGETHEORIES BEYONDGAUGETHEORY 11 J. E. Moyal, Quantum mechanics as a statistical theory , Proc. Cambridge Phil. Soc. 45, 99 (1949). 7. J. Wess and B. Zumino, Covariant differential calculus on the quantum hyperplane , Nucl. Phys. Proc. Suppl. 18B(1991) 302. 8. L.Bonora,M.Schnabl,M.M.Sheikh-JabbariandA.Tomasiello, NoncommutativeSO(n)and Sp(n)gaugetheories , hep-th/0006091. I. Chepelev, R. Roiban, Convergence Theorem for Non-commutative Feynman Graphs and Renormalization , hep-th/0008090. A. Bichl, J.M. Grimstrup, V. Putz, M. Schweda, Perturbative Chern-Simons Theory on non- commutativeR 3 , hep-th/0004071. A. Bichl, J.M. Grimstrup, H. Grosse, L. Popp, M. Schweda, R. Wulkenhaar, The Superﬁeld Formalism Applied to the Non-commutative Wess-Zumino Model , hep-th/0007050. kievarwe.tex; 12/03/2001; 3:49; p.18 kievarwe.tex; 12/03/2001; 3:49; p.19 SYMMETRIESWIDERTHAN SUPERSYMMETRY ∗ DIMITRYLEITES †‡ Department of Mathematics, University of Stockholm, Roslags- v¨agen.101,Kr ¨aftriket hus 6,S-10691, Stockholm, Sweden VERA SERGANOVA § Department of Mathematics, University of California at Berkeley, Berkeley,CA94720,USA Abstract. We observe that supersymmetries do not exhaust all the symmetries of the super- manifolds. On a generalization of supermanifolds (called metamanifolds ), the “functions” form a metaabelean algebra, i.e., the one for which [[x,y],z] = 0with respect to the usual commutator. Thesuperspacesconsideredasmetaspacesadmitsymmetrieswiderthansupersymmetries.Conjec- turally, inﬁnitesimal transformations of these metaspaces constitute Volichenko algebras which we introduce as inhomogeneous subalgebras of Lie superalgebras. The Volichenko algebras are natu- ral generalizations of Lie superalgebras being 2-step ﬁltered algebras. They are non-conventional deformations of Lie algebras bridging themwith Lie superalgebras. 1. Introduction:Towardsnoncommutativegeometry This is an elucidation of our paper [31]. In 1990 we were unaware of [42] to which we now would like to add later papers [14], and [2], and papers cited thereinpertainingtothistopic.ObservealsoanobviousconnectionofVolichenko algebras with structures that become more and more fashionable lately, see [22]; Volichenko algebras are one of the ingredients in the construction of simple Lie algebras over ﬁeldsofcharacteristic2, cf.[23] 1.1. The gist of idea . To describe physical models, the least one needs is a triple (X,F (X),L), consisting of the “phase space” X, the sheaf of functions on it, locally represented by the algebra F(X)of sections of this sheaf, and a Lie subalgebra Lof the Lie algebra of of differentiations of F(X) considered ∗ Instead of J. Naudts contribution by theeditor S. Duplij’s request † D.L. is thankful to an NFR grant for partial ﬁnancial support, to V. Molotkov, A. Premet and S. Majid for help. ‡ [email protected] § [email protected] kievarwe.tex; 12/03/2001; 3:49; p.20 14 D. LEITES, V.SERGANOVA as vector ﬁelds on X. HereXcan be recovered from F(X)as the collection Spec(F(X)), called the spectrum and consisting of maximal or prime ideals of F(X). Usually,Xis endowed witha suitabletopology. Afterthediscoveryofquantummechanicstheattemptstoreplace F(X)with thenoncommutative(“quantum”)algebra Abecamemoreandmorepopular.The ﬁrst successful attempt was superization [25], [5] the road to which was prepared intheworksofA.Weil,Leray,GrothendiekandBerezin,see[11].Itturnsoutthat having suitably generalized the notion of the tensor product and differentiation (by inserting certain signs in the conventional formulas) we can reproduce on supermanifolds all the characters of differential geometry and actually obtain a much reacher and interesting plot than on manifolds. This picture proved to be a great success in theoretical physics since the language of supermanifolds and supergroups is a “natural” for a uniform description of bose and fermi particles. TodaythereisnodoubtthatthisisthelanguageoftheGrandUniﬁedTheoriesof all knownfundamental forces. Observe that physicists who, being unaware of [25], rediscovered super- groups and superspaces (Golfand–Likhtman, Volkov–Akulov, Neveu–Schwartz, Stavraki)werestudyingpossibilitiestoenlargethegroupofsymmetries(orrather the Lie algebra of inﬁnitesimal symmetries) of the known objects (in particular, objects described by Maxwell and Dirac equations). Their efforts did not draw much attention (like our [25] and [31]) until Wess and Zumino [43] understood and showed to others some of the whole series of wonders one can obtain by meansofsupersymmetries. Here we show that the supergroups are not the largest possible symmetries of superspaces;therearetransformationsthatpreservemorenoncommutativitythan justa“mere”supercommutativity.Tobeabletoobservethattherearesymmetries that unify bose and fermi particles we had to admit a broader point of view on our Universe and postulate that we live on a supermanifold. Here (and in [31]) we suggest to consider our supermanifolds as paticular case of metamanifolds , introducedinwhatfollows. How noncommutative should F(X)be? To deﬁne the space correspond- ing to an arbitrary algebra is very hard, see Manin’s gloomy remarks in [33], where he studies quadratic algebras as functions on “perhaps, nonexisting” noncommutativeprojectivespaces. Manin’s idea that there hardly exists one uniform deﬁnition suitable for any noncommutative algebra (because there are several quite distinct types of them) was supported by A. Rosenberg’s studies; he managed to deﬁne several types of spectrainordertointerpretANYalgebraasthealgebraoffunctionsonasuitable spectrum,seepreprintsofhistwobooks[27],no.25,andnos.26,31(thelatterbe- ingexpandedas[35]).Inparticular,thereISaspacecorrespondingtoaquadratic (or“quadraticizable”)algebrasuchastheso-called“quantum”deformation U q ( g) ofU( g), see [12]. kievarwe.tex; 12/03/2001; 3:49; p.21 SYMMETRIES WIDER THANSUPERSYMMETRY 15 Observe that in [33] Manin also introduced and studied symmetries of super- commutative superalgebras wider than supersymmetries, but he only considered them in the context of quadratic algebras (not all relations of a supercommu- tative suepralgebra are quadratic or quadraticizable). Regrettably, nobody, as far as we know, investigated consequences of Manin’s approach to enlarging supersymmetries. Unlike numerous previous attempts, Rosenberg’s theory is more natural; still, it is algebraic, without any real geometry (no differential equations, integration, etc.). For some noncommutative algebras certain notions of differential geome- try can be generalized: such is, now well-known, A. Connes geometry, see [10], and [34]. Arbitrary algebras seem to be too noncommutative to allow to do any physics. In contrast, the experience with the simplest non-commutative spaces, the su- perspaces,tellsusthatallconstructionsexpressibleinthelanguageofdifferential geometry (these are particularly often used in physics) can be carried over to the super case. Still, supersymmetry has, as we will show, certain shortcomings, which disappearinthe theorywe propose. Speciﬁcally, we continue the study started under Berezin’s inﬂuence in [25] (later suppressed under the same inﬂuence in [5], [26]), of algebras just slightly more general than supercommutative superalgebras, namely their arbitrary, not necessarilyhomogeneous,subalgebrasandquotients.ThankstoVolichenko’sthe- orem F (F is for “functions”, see [27], no. 17 and Appendix below) such algebras areprecisely metaabelean ones, i.e.,those thatsatisfy the identity [x,[y,z]] = 0 (here [·,·] is the usual commutator) . (1.1) As in noncommutative geometries, we think of metaabelean algebras as “func- tions”onawhatwewillcall metaspace . Observe that the conventional superspaces considered as metaspaces and La- grangiansonthemhave additional symmetriesascomparedwithsupersymmetry. 1.2. The notion of Volichenko algebras . Volichenko’s Theorem F gives us a natural generalization of the supercommutativity. It remains to deﬁne the analogs of the tensor product and study differentiation (e.g., Volichenko’s ap- proach, see§3). We conjecture that the analogs of Lie algebras in the new setting areVolichenko algebras deﬁned here as nonhomogeneous subalgebras of Lie superalgebras. Supersymmetry had been already justiﬁed for physicists when mathemati- cians’ attention was drawn to it by the list of simple ﬁnite dimensional Lie superalgebras: bar one exception it was discrete and looked miraculously like the list of simple Lie algebras. Our list of simple Volichenko algebras is similar. Our main mathematical result is the classiﬁcation (under a technical hypothesis) of simple ﬁnitedimensional (and vectorial) Volichenko algebras, see[40],[31]. kievarwe.tex; 12/03/2001; 3:49; p.22 16 D. LEITES, V.SERGANOVA Remarkably,VolichenkoalgebrasarejustdeformationsofLiealgebrasthough in an entirely new sense: in a category broader than that of Lie algebras or Lie superalgebras. This feature of Volichenko algebras could be signiﬁcant for paras- tatisticsbecauseonceweabandonbose-fermistatistics,thereseemtobetoomany adhocwaystogeneralize.Ourclassiﬁcationassertsthatwithinthenaturalcontext of simple Volichenko algebras the set of possibilities is discrete or has at most 1- parameter (hence, anyway, describable!). It is important because it suggests the possibility of associating distinct types of particles to representations of these structures. Our generalization of supersymmetry and its implications for parastatistics appear to be complementary to works on braid statistics in two dimensions [15] inthecontext of [13], seealso [19].Weexpectthem to tie upat some stage. ExamplesofwhatlookslikenonsimpleVolichenkoalgebrasrecentlyappeared inanothercontext in[2], [36],[42]and [14]. 1.3.Anintriguingexample:thegeneralVolichenkoalgebra vgl µ (p|q).Let the space hof vgl µ (p|q)be the space of (p+q)×(p+q)-matrices divided into thetwosubspacesas follows: h ˆ0 = /braceleftbigg A0 0D /bracerightbigg ; h ˆ1 = /braceleftbigg 0B C0 /bracerightbigg . (1.3.1) Here h ˆ1 isanatural h ˆ0 -modulewithrespecttothebracketofmatrices;ﬁx a,b∈C such thata:b=µ∈CP 1 and deﬁne the multiplication h ˆ1 × h ˆ1 −→ h ˆ0 by the formula [X,Y ] =a[X,Y ] − +b[X,Y ] + for anyX,Y∈ h ˆ1 . (1.3.2) (The subscript−or+indicates the commutator and the anticommutator, respec- tively.) As we sill see, his a simple Volichenko algebra for any a,bexcept for ab= 0when it becomes isomorphic to either the Lie algebra gl(p+q)or the Lie superalgebra gl(p|q). To show that vgl µ (p|q)is indeed a Volichenko algebra, we have to realize it as a subalgebra of a Lie superalgebra. This is done in heading 2 ofTheorem2.7. 2. Metaabeleanalgebraasthealgebraof“functions”.Volichenkoalgebraas an analog ofLie algebra 2.1. Symmetries broader than supersymmetries . It was the desire to broaden the notion of a group that lead physicists to supersymmetry. However, in viewing supergroupsastransformationsofsuperspacesweconsideronlyeven,“statistics- preserving”, maps: nonhomogeneous “statistics-mixing” maps between super- algebras are explicitly excluded and this is why and how odd parameters of supergroupsappear,cf. [3],[11]. On the one hand, this is justiﬁed: since we consider graded objects why should we consider transformations that preserve these objects as abstract ones kievarwe.tex; 12/03/2001; 3:49; p.23 SYMMETRIES WIDER THANSUPERSYMMETRY 17 but destroy the grading? It would be inconsistent on our part, unless we decide to considerthegradingor“parity”asoneconsiderstheelectricchargeofanucleon: incertain problemswe ignoreit. On the other hand, if such parity violating transformations exist, they deserve to be studied, to disregard them is physically and mathematically an artiﬁcial restriction. Wewouldliketobroadenthenotionofsupergroupsandsuperalgebrastoallow for the possibility of statistics-changing maps. Soon after Berezin published his description of automorphisms of the Grassmann algebra [4] it became clear that Berezinmissednonhomogeneousautomorphisms,butthecompletedescriptionof automorphisms was unknown for a while. In 1977, L. Makar-Limanov gave us a correct description of such automorphisms (private communication). A. Kirillov rediscovered it while editing [3], Ch.1; for automorphisms in presence of even variables see [28]. Recall the answer: the generic ﬁnite transformation of a supercommutative superalgebraFof functions in neven generators x 1 ,...,x n andmodd ones θ 1 ,...,θ m is oftheform(here p m istheparityof m, i.e., either 0or1) x i /mapsto→[(f i + /summationtext k f i 1 ...i 2k i θ i 1 ...θ i 2k ) + /summationtext k f i 1 ...i 2k+1 i θ i 1 ...θ i 2k +1 ](1 +F i θ 1 ...θ m p m ) θ j /mapsto→[( /summationtext k g i 1 ...i 2k+1 j θ i 1 ...θ i 2k+1 ) +g j + /summationtext k g i 1 ...i 2k j θ i 1 ...θ i 2 k ](1 +g ) (2.1) wheref i ,F i andf i 1 ...i 2k i , and also g i 1 ...i 2k+1 j are even superﬁelds, whereas f i 1 ...i 2k+1 i ,g j andg i 1 ...i 2k j and alsog,F i are odd superﬁelds. (A mathematician, see [11], would say that the odd superﬁelds (underlined once) represent the pa- rameters corresponding to Λ-points with nonzero odd part of the background supercommutative superalgebra Λ.) Notice that one gserves all the θ j . The twice underlined factors account for the extra symmetry of Fas compared with supersymmetry. Comment . When the number of odd variables is even, as is usually the case in modern models of Minkowski superspace, there is only one extra func- tional parameter, g. Therefore, on such supermanifolds, the notion of a boson is coordinate-free, whereasthat ofafermiondependsoncoordinates . Summingup,(this isour main messageto the reader) supersymmetryis notthemostbroad symmetryof supercommutative superalgebras 2.2. Two complexiﬁcations . Another quite unexpected ﬂaw of supersymme- try is that the category of supercommutative superalgebras is notclosed with respect to complexiﬁcation. It certainly is if Cis understood naively, as a purely even space. Declaring √−1to be odd, we make Cinto a nonsupercommutative kievarwe.tex; 12/03/2001; 3:49; p.24 18 D. LEITES, V.SERGANOVA superalgebra. This associative superalgebra over Ris denoted by Q(1;R), see [26],[6]. The complex structure given by an odd operator gives rise to a “queer” su- peranalogue of the matrix algebra, Q(n;K)over any ﬁeldK. Its Lie version, the projectivization of its queertraceless subalgebra (ﬁrst discovered by Gell-Mann, Mitchel and Radicatti, cf. [9]) is one of main examples of simple Lie superal- gebras, whereas Q(1)corresponds to one of the two cases of Schur’s Lemma for superalgebras. An inﬁnite dimensional representation of Q(1)is crucial in A. Connes’ noncommutative differential geometry. In short, the odd complex structure onsuperspacesis animportant one. How to modify deﬁnition of supermanifold to incorporate the above struc- tures? Conjecturally, the answer is to consider arbitrary, not necessarily homo- geneous subalgebras and quotients of supercommutative superalgebras. These algebras are, clearly, metaabelean algebras. But how to describe arbitrary metaa- beleanalgebras?In1975D.L.discussedthiswithV.KacandKacconjectured(see [26])thatconsideringmetaabeleanalgebraswedonotdigressfarfromsupercom- mutative superalgebras, namely, every metaabelean algebra is a subalgebra of a supercommutative superalgebra. Therefore, the most broad notion of morphisms of supercommutative superalgebras should only preserve their metaabeleanness but not parity. (Since C, however understood, is metaabelean, we get a category ofalgebrasclosed withrespectto all algebramorphisms andcomplexiﬁcations.) VolichenkoprovedmorethanKac’conjecture(Appendix).Namely,heproved that any ﬁnitely generated metaabelean algebra admits an embedding into a uni- versalsupercommutativesuperalgebraanddevelopedananalogueofTaylorseries expansion. UntilVolichenko’sresults,itwasunclearhowtoworkwithmetaabeleanalge- bras:arethereanyanaloguesofdifferentialequations,orintegral,inotherwords, is there any “real life” on metaspaces [26]? Thanks to Volichenko, we can now consider pairs (ametaabelean algebra,its ambientsupercommutative superalgebra) and corresponding projections “superspace −→metaspace” when we consider these algebrasas algebrasof functions. It is interesting to characterize metaabelean algebras which are quotients of supercommutative superalgebras: in this case the corresponding metaspace can be embedded into the superspace and we can consider the induced structures (Lagrangeans,various differentialequations, etc.). Butevenifwewouldhavebeentotallyunabletoworkwithmetaspaceswhich arenotsuperspaces,itismanifestlyusefultoconsidersuperspacesasmetaspaces. In so doing, we retain all the paraphernalia of the differential geometry for sure, and inaddition get moretransformations ofthe sameentities. kievarwe.tex; 12/03/2001; 3:49; p.25 SYMMETRIES WIDER THANSUPERSYMMETRY 19 For example, it is desirable to make use of the formula (ﬁrst applied by Arnowitt,Coleman andNath) BerX= exp str log X which extends the domain of the berezinian (superdeterminant) to nonhomo- geneous matrices X. Then we can consider the additional nonhomogeneous transformations,liketheonesdescribedin(2.1).AllsupersymmetricLagrangeans admitmetasymmetry widerthan supersymmetry. Remark. In mathematics and physics, spaces are needed almost exclusively to integrate over them or consider limits in analytic questions. In problems where integration is not involved we need sheaves of sections of various bundles over the spaces rather than the spaces themselves. Gauge ﬁelds, Lagrangeans, etc. are all sections of coherent sheaves, corresponding to sections of vector bundles. Now, almost 30 years after the deﬁnition of the scheme of a metaabelean algebra (metavariety or metaspace) had been delivered at A. Kirillov’s seminar ([25]), there is still no accepted deﬁnition of nice (“morally coherent” as Manin says) sheaves over such a scheme even forsuperspaces(foradiscussionsee[8]).AstocandidatesforsuchsheavesseeRosenberg’sbooks onnoncommutativegeometry[27],nos.25,26,31and[35])and §9in[8].This§9is,besidesall,a possible steptowards “compactiﬁcationin odd directions”. 2.3. A description of Volichenko algebras . It seemed natural [26] to get for Lie superalgebras a result similar to Volichenko’s theorem F, i.e., to describe arbitrary subalgebras of Lie superalgebras. Shortly before his untimely death I. Volichenko (1955-88) announced such a description (Theorem A, here A is for (Lie)“algebra”).Inhismemorythen,a Volichenkoalgebra isanonhomogeneous subalgebra hof a Lie superalgebra g. The adjective “Lie” before a (super)algebra indicates that the algebra is not associative, likewise the adjective “Volichenko” remindsthatthealgebraisneitherassociativenorshoulditsatisfyJacobiorsuper- Jacobiidentities.Thus,aVolichenkoalgebra hisanon-homogeneoussubspaceof aLiesuperalgebra gclosedwithrespecttothesuperbracketof g.Howtodescribe hbyidentities, i.e., ininner terms, withoutappealing to any ambient? Theorem . A (I. Volichenko, 1987) Let A be an algebra with multiplication denotedbyjuxtaposition.DeﬁnetheJordanelements a◦b:=ab+ba andJacobi elements J(a,b,c ) :=a(bc) +c(ab) +b(ca) .Supposethat (a)A isLieadmissible,i.e., A isaLiealgebrawithrespecttothenewproduct deﬁned bythebracket (notsuperbracket) [a,b] =ab−ba ; (b) thesubalgebra A (JJ) generatedbyallJordanandJacobielementsbelongs totheanticenter of A ,in otherwords ax+xa= 0for anya∈A (JJ) , x∈A; (c)a(xy) = (ax)y+x(ay)foranya∈A (JJ) , x,y∈A . Then (1) Any (not necessarily homogeneous) subalgebra hof a Lie superalgebra g satisﬁesthe aboveconditions (a) —(c) . (2) If A satisﬁes (a)—(c) ,thenthereexistsaLiesuperalgebra SLie (A) such that A isasubsuperalgebra(closedwithrespecttothesuperbracket)of SLie (A) . kievarwe.tex; 12/03/2001; 3:49; p.26 20 D. LEITES, V.SERGANOVA Heading(1) issubjectto a directveriﬁcation. Clearly,thepartsofconditions(b)and(c)whichinvolveJordan(resp.Jacobi) elements replace the superskew-commutativity (resp. Jacobi identity). Condition (a)ensuresthat Aisclosedin SLie(A)withrespecttothebracketintheambient. Discussion . If true, Volichenko’s theorem A would have disproved a pes- simistic conjecture of V. Markov cited in [26]: the minimal set of polynomial identities that single out nonhomogeneous subalgebras of Lie superalgebras is inﬁnite . I. Volichenko did not investigate under which conditions a ﬁnite di- mensional Volichenko algebra Acan be embedded into a ﬁnite dimensional Lie superalgebra g; whichis,perhaps,thequotient of SLie(A)moduloan ideal. Volichenko’s scrap papers were destroyed after his death and no hint of his ideasremains.SeveralresearcherstriedtorefuteitandA.Baranovsucceeded.He showed[1]thatVolichenko’stheoremViswrongasstated:oneshouldaddatleast one more relation of degree 5. First, following Volichenko, Baranov introduced insteadofJ(a,b,c )moreconvenientlinearcombinationsofthe Jacobi elements j(a,b,c ) = [a,b◦c] + [b,c◦a] + [c,a◦b] fora,b,c∈A. Then Baranov rewrote identities (a)–(c) in the following equivalent but more transparent form (i)–(v): (i)[a,[b,c]] + [b,[c,a]] + [c,[a,b]] = 0; (ii)a◦b◦c= 0; (iii)j(a,b,c )◦d= 0; (iv)[a◦b,c◦d] = [a◦b,c]◦d+ [a◦b,d]◦c; (v)[j(a,b,c ),c◦d] = [j(a,b,c ),c]◦d+ [j(a,b,c ),d]◦c. Baranov’snewidentityindependentof(i)–(v)isofdegree5andissomewhat implicit;itinvolves49monomialsandnolucidexpression for it isfoundyet. Trueorfalse,Volichenko’stheoremAdoesnotaffectourresults,sincewedo notappealto anintrinsic deﬁnitionofVolichenkoalgebras. 2.4. On simplicity of Volichenko algebras . As we will see, the notion of Volichenko algebra is a totally new type of deformation of the usual Lie algebra. It also generalizes the notion of a Lie superalgebra in a sence that the Lie super- algebras areZ/2-graded algebras (i.e., they are of the form g=⊕ i=¯0,¯1 g i such that [ g i , g j ]⊂ g i+j ) whereas Volichenko algebras are only 2-step ﬁltered ones (i.e., they are of the form h=⊕ i=ˆ0,ˆ1 h i asspacesand h ˆ0 is a subalgebra. There are, however, several series of examples when Volichenko algebras are Z/2-graded (e.g., vgl µ (p|q)). Hereafter gis a Lie superalgebra over Cand h⊂ ga subspace which is not a subsuperspaceclosedwithrespecttothesuperbracketin g.Fornotationsofsimple complex ﬁnite dimensional Lie superalgebras, the list of known simple Z-graded inﬁnite dimensional Lie superalgebras of polynomial growth over CandR, and kievarwe.tex; 12/03/2001; 3:49; p.27 SYMMETRIES WIDER THANSUPERSYMMETRY 21 their gradings see [20], [38], [27], [37], [30]. A Volichenko algebra is said to be simpleif ithasnotwo-sided idealsand itsdimensionis /negationslash= 1. Remark. P. Deligne argued that for an algebra such as a Volichenko one, modules over which have no natural two-sided structure, the above deﬁnition seems to be too restrictive: one should deﬁne simplicity by requiring the absence of one-sided ideals. As it turns out, none of the simple Volichenko algebras we list in what follows has one-sided ideals, so we will stick to the above (at ﬁrst glance, preliminary) deﬁnition: it iseasier to work with. Lemma. For any simple Volichenko algebra h , h⊂ g /prime , there exists a simple Liesubsuperalgebra g⊂ g /prime thatcontains h . So, we can (and will) assume that the ambient gof a simple Volichenko al- gebra is simple. In what follows we will see that under a certain condition for a simpleVolichenkoalgebra hitssimpleambientLiesuperalgebra gisunique.here isthiscondition: 2.5. The “epimorphy” condition . Denote by p i : g−→ g i , wherei=¯0,¯1, theprojectionstohomogeneouscomponents.AVolichenkoalgebra h⊂ gwillbe calledepimorphic ifp 0 ( h) = g ¯0 .NoteveryVolichenkosubalgebraisepimorphic: forexample,thetwoextremes,Volichenkoalgebraswiththezerobracketandfree Volichenko algebras, are not epimorphic, generally. All simple ﬁnite dimensional Volichenkoalgebras known to us are,however,epimorphic. Hypothesis . Everysimple Volichenko algebrais epimorphic . A case study of various simple Lie superalgebras of low dimensions reveals that they do not contain non-epimorphic simple Volichenko algebra. Still, we can notprovethishypothesisbutwilladoptitforitlooksverynaturalatthemoment. Lemma. Let h⊂ g beanepimorphicVolichenkoalgebraand f: g ¯0 −→ g ¯1 a linearmapthatdetermines h ,i.e., h= h f :={a+f(a)|aruns over g ¯0 }. Then 1)f isa 1 -cocyclefrom C 1 ( g ¯0 ; g ¯1 ) ; 2)f can be uniquely extended to a derivation of g (also denoted by f ) such that f(f( g ¯0 )) = 0 . Example . Recall, that the odd element xof any Lie superalgebra is called a homologic one if [x,x] = 0,cf. [41].Let x∈ g ¯1 be such that [x,x]∈C( g), (2.5.1) whereC( g)isthecenterof g.Clearly,themap f= ad (x)satisﬁesLemma2.4if xsatisﬁes (2.5.1), i.e.,ishomologic modulocenter. A homologic modulo center element xwill be said to ensure nontriviality (of thealgebra h x ={a+ [a,x]|aruns over g ¯0 }) (2 .5.2) if [[ g ¯0 ,x],[ g ¯0 ,x]]/negationslash= 0, kievarwe.tex; 12/03/2001; 3:49; p.28 22 D. LEITES, V.SERGANOVA i.e.,ifthereexistelements a,b∈ g ¯0 such that [[a,x],[b,x]]/negationslash= 0. (2.5.3) The meaning of this notion is as follows. Let a,b∈ h,a=a 0 +a 1 ,b=b 0 +b 1 , wherea 1 = [a 0 ,x],b 1 = [b 0 ,x]for somex∈ g ¯1 . Notice that for xsatisfying (2.5.1)wehave [[a 1 ,b 1 ],x] = 0. (2.5.4) If (2.5.3)holds,we have [a,b] = [a 0 ,b 0 ] + [a 1 ,b 1 ] + [a 0 ,b 1 ] + [a 1 ,b 0 ] = ([a 0 ,b 0 ] + [a 1 ,b 1 ]) + [[a 0 ,b 0 ],x]. (2.5.5) It follows from (2.5.4) and (2.5.5) that if xis homologic modulo center, then h x is closed under the bracket of g; if thisxdoes not ensure nontriviality, then h x is justisomorphic to g ¯0 . In other words, an epimorphic Volichenko algebra is a deformation of the Lie algebra g ¯0 inatotallynewsence:notintheclassofLiealgebras,norinthatofLie superalgebras but in the class of Volichenko algebras whose intrinsic description is to be given. (To see that an epimorphic Volichenko algebra h x is a result of a deformationofsorts,multiply xbyanevenparameter, t.Iftwereodd,wewould haveobtaineda deformationof g ¯0 in theclass of Liesuperalgebras.) Remark.Itiseasytoshowmakinguseofformula(2.5.5)whyitisimpossible to consider any other (inconsistent with parity) Z/2-grading (call it deg) of gand deform in a similar way the Lie subsuperalgebra of elements of degree 0 with respectto deg. AnyepimorphicVolichenkoalgebra h x ⊂ gisnaturallyﬁltered:itcontainsas as subalgebrathe Liealgebra ann(x) ={a∈ g ¯0 |[x,a] = 0}. Problems . 1) We have a sandwich: between Hopf (super)algebras, U( h x ) andU( g), a non-Hopf algebra, U( h)(the subalgebra of U( g)generated by h), is squeezed. How to measure its “non-Hopfness”? This invariant seems to be of interest. 2) It is primarily real algebras and their representations that arise in applica- tions. So whatarethese notionsfor Volichenkoalgebras? We do not know at the moment the deﬁnition of a representation of a Volichenko algebra even for epimorphic ones. To say “a representation of a Volichenko algebra is a through map: the composition of an embedding h⊂ g into a minimal ambient and a representation g−→ gl(V)” is too restrictive: the adjointrepresentation and homomorphismsof Volichenkoalgebras areruled out. 3) If we abandon the technical hypothesis on epimorphy, do we obtain any simple Volichenkoalgebras?(Conjecture: wedonot.) 4) Describe Volichenko algebras intrinsically, via polynomial identities. This seems tobe adifﬁcultproblem. kievarwe.tex; 12/03/2001; 3:49; p.29 SYMMETRIES WIDER THANSUPERSYMMETRY 23 5) Classify simple Volichenko subalgebras of the other known simple Lie superalgebrasofinterest, e.g.,of polynomialgrowth,cf. [16],[17]. 2.6. Vectorial Volichenko superalgebras . For a vector ﬁeld D= /summationtext f r ∂ r from vect(m|n) = derC[x,θ],deﬁneits inverseorder withrespecttothenonstan- dard (ifm/negationslash= 0) grading induced by the grading of C[x,θ](for which degx i = 0 anddegθ j = 1for alliandj) and inv.ord(f r )is the least of the degrees of monomialsinthe power seriesexpansionof f r . There are two major types of Lie (super)algebras and their subalgebras: the ones realized by matrices and the ones realized by vector ﬁelds. The former ones will berefered toas matrix ones, the latterones asvectorial algebras. 2.6.1. Lemma . Let h⊂ g be a simple epimorphic vectorial Volichenko algebra, i.e., a subalgebra of a simple vectorial Lie superalgebra. Then in the representation h= h f we have f(·) = [·,x] , where x is homologic and inv.ord(x) =−1 . 2.6.2. Lemma . Let G be the Lie group with the Lie algebra g ¯0 , let G 0 be the Lie group with the Lie algebra g 0 of linear vector ﬁelds with respect to the stan- dard (see [37]) grading; let AutG 0 be the group of automorphisms of G 0 . Table 2.7.2 contains all, up to (AutG 0 ) -action, homologic elements of the minimal inverseorderinthevectorialLiesuperalgebras.Inparticular,for svect /prime (2n) there arenone. 2.7. Theorem . A simple epimorphic ﬁnite dimensional Volichenko algebra h⊂ g canbeonlyoneof thefollowing h= h x , where: 1)x is an element from Table 2.7.2 or an element from Table 2.7.1 satisfying theconditionensuringnon-trivialityif g/negationslash= psq(n) ; 2) if g= psq(n) , then either x is an element from Table 2.7.1 satisfying the conditionensuringnon-trivialityor x= antidiag ( X,X ) , where X= diag (a1 p ,b1 n−p )withap +b(n−p) = 0. Now, the ﬁnaltouch: Proposition . Simple epimorphic Volichenko algebras from Tables 1 , 2 have no one-sided ideals. kievarwe.tex; 12/03/2001; 3:49; p.30 24 D. LEITES, V.SERGANOVA Table 2.7.1. Homologic elements xand the condition when xensures nontriviality of hfor matrixLie superalgebras g g x (whendoesxensure nontri viality) sl(m|n),m≤ n x p q = antidiag( B,C), whereB= diag(1 p , 0) C= diag(0,1 q )(p,q> 0,p+q≤m ) psl(n|n )same as for sl(n|n)and also antidiag(1 n ,1 n )(as abov e) osp(2m|2n )the image of theabove x p p ∈ sl(m|n)⊂ osp(2m|2n ) 2p≤min (m,n)(p>0 ) osp(2m+ 1|2n ) theimage of theabove x under the embedding osp(2m|2n)⊂ osp(2m+ 1|2n ) spe(n ) antidiag( B,C), whereB= diag(1 p ,0 n−p ) , C= diag(0 n−2q ,J 2q ), p+ 2q≤n(p,q> 0 ) psq(n ) antidiag( X,X ) , whereX= diag(J 2 (0),...,J 2 (0),0,..., 0) withk-manyJ 2 (0)’s, where J 2 (0) = antidiag (1 ,0),2k≤n(k>0 ) ag 2 , ab 3 , theroot vector corresponding to osp(4|2;α ) an isotropic (odd)simple root (nev er) In Table 2.7.2 we have listed not only homologic elements — that is to say Volichenkosubalgebras—ofﬁnitedimensionalsimpleLiesuperalgebrasofvec- torﬁeldsbutalsosimpleVolichenkosubalgebrasofallnonexceptionalsimpleLie superalgebrasofvector ﬁelds, fortheirlist see[30]. Table 2.7.2. Homologic elements xof minimal inverse order in simple Lie superalgebras gof vector ﬁelds vect(m|n), wheremn/negationslash= 0,n> 1orm= 0,n> 2; ∂ ∂θ 1 svect(m|n), le(n), sle o (n)forn> 1 k(2m+ 1|n), wheren> 1 K θ 1 h(2m|n), wheremn/negationslash= 0,n> 1and sh(n),n> 3 ∂ ∂θ 1 and ∂ ∂θ 1 + √−1 ∂ ∂θ 2 m(n),n>1, and sm λ (n),λ/negationslash= 0,n> 1M 1 andM 1+θ 1 ...θ 2k for sm λ (2k ) svect(0|2n+ 1),n> 1 ∂ ∂θ 1 and(1 +tθ 2 ...θ 2n+1 ) ∂ ∂θ 1 ,t∈ C kievarwe.tex; 12/03/2001; 3:49; p.31 SYMMETRIES WIDER THANSUPERSYMMETRY 25 3. Appendix. Volichenko’s theorem F and elements of Calculus on matamenifolds 3.1. In what follows all the algebras are associative with unit over a ﬁeld K, charK/negationslash= 2. We will deal with two important PI-varieties of algebras (the varietiessingled outbypolynomialidentities): – thevarietyCofsupercommutativesuperalgebras; –thevarietyGgenerated(bytensoringandpassingtoquotients)bytheGrass- mann algebra Λ(∞)of countably many indeterminates (its natural Z/2-grading ignored). The varietyGplays a signiﬁcant role in the theory of varieties of associative algebras ([21]). It is known that if charK= 0it is distinguished by the identity (1.1).If charK/negationslash= 0, theidentity X p = 0shouldbe added. I. Volochenko wrote: “As pointed out by D. Leites [26], in the conventional supermanifold theory it seems too restrictive that not all subalgebras or quotients of superalgebras are considered as algebras of functions on supermanifolds but onlythegraded(homogenous)ones.ItistemptingtoconstructavariantofCalcu- lus which enables one to operate with arbitrary subalgebras, ideals and quotients. ...Deﬁnitionofthecategoryoftopologicalspacesringedbysuchgeneralalgebras isobvious,cf.[25], wherethealgebraiccaseis considered . It remained unclear, however, how to uniformly describe such algebras. For instance, do they constitute a variety? Leites recalls a conjecture of Kac (1975) that such algebras are metaabelean , i.e., satisfy the identity (1.1). The conjecture is a well-known fact of the theory of varieties of associative algebras, cf. [24]. From the context of [25], however, it is clear that the actual problem is, ﬁrst of all, how to describe a variety of not necessarily homogeneous subalgebras which apriori canbe less than G. Actually, I will not only prove that any algebra G∈Gcan be embedded into a commutative superalgebra but will also prove the existence of a universal (in a naturalsence)envelopingalgebra U C (G)fromtheclassCofallthesupercommu- tativesuperalgebrasandgiveanexplicitrealizationof U C (G).Therefore,wecan, in principle, reduce the study of homomorphisms of algebras from Gto that of theirenveloping superalgebrasfrom C. I hope that this is (at least partly) an answer to Leites’ question how to work withalgebrasfrom G and thecorresponding ‘supermanifolds’ ”. 3.2. LetK C [X,Y ]be the algebra determined by the system of indeterminates X∪Y= (X i ) i∈I ∪(Y j ) j∈J andrelations X i 1 X i 2 −X i 2 X i 1 = 0, X i Y j −Y j X i = 0, Y j 1 Y j 2 +Y j 2 Y j 1 = 0 fori,i 1 ,i 2 ∈I,andj,j 1 ,j 2 ∈J.Thisalgebrapossessesanaturalparity: p(X i ) = ¯0,p(Y j ) =¯1fori∈I,j∈J. kievarwe.tex; 12/03/2001; 3:49; p.32 26 D. LEITES, V.SERGANOVA LetI=J; letK G [Z]be a non-graded subalgebra in K C [X,Y ]generated by all theelements Z i =X i +Y i (i∈I). Statement .K G [Z] is a free algebra in the variety G and the elements Z i ( i∈I ) are its free generators. In other words, let K A [T] be a free associative algebrawithfreegenerators T 1 ,T 2 ,... .If f(Z 1 ,...,Z n ) = 0 in K G [Z] forsome f(T 1 ,...,T n )∈K A [T] ,then f(a 1 ,...,a n ) = 0 forany a 1 ,...,a n ∈K C [X,Y ] . 3.3.Setd= /summationtext i∈I Y i ∂ ∂X i . Statement . The polynomial f(X,Y )∈K C [X,Y ] belongs to K G [Z] if and onlyif df=f ¯1 ,or,equivalently, df ¯0 =f ¯1 . 3.4.Arelationbetween idealsof K G [Z]andK C [X,Y ]. Statement . Let A beanidealof K G [Z] and ¯A theidealof K C [X,Y ] generated by A ¯0 ∪A ¯1 ={f ¯0 ,f ¯1 :f∈A} .Then ¯A∩K G [Z] =A . Now,let ˜G=G ¯0 ⊕G ¯1 bealinearsuperspace,whereeach G i isacopyofour algebraGfromG. Consider the subalgebra K G [G]⊂K C [˜G]generated by all the elementsg ¯0 +g ¯1 , whereg∈G. Clearly,K C [˜G]/similarequalK C [X,Y ], whereXandY arebasesinG ¯0 andG ¯1 ,respectively,and K G [G]/similarequalK G [Z].ThenGisisomorphic to the quotient of K G [Z]modulo the ideal Agenerated by all the elements of the form (g ¯0 +g ¯1 )(h ¯0 +h ¯1 )−((gh) ¯0 + (gh) ¯1 ). TheuniversalC-enveloping of Gis the quotient U C [G]ofK C [˜G]modulo the ideal ¯Ageneratedby theelementsoftheform g ¯0 h ¯0 +g ¯1 h ¯1 −(gh) ¯0 , g ¯0 h ¯1 +g ¯1 h ¯0 −(gh) ¯1 . Any element g∈Gis identiﬁed with the image of g ¯0 +g ¯1 under the canonical epimorphism K C [˜G]→U C [G]. InK C [˜G], same as in K C [X,Y ], there is deﬁned the derivation: d(g ¯0 ) =g ¯1 , d(g ¯1 = 0for anyg∈G. Since ¯Aisd-invariant, it follows that dinduces a canonicalderivationof U C [G]which wewill alsodenoteby d. Proposition . The element f of U C [G] belongsto G ifand onlyif df ¯0 =f ¯1 . 3.5. An explicit description of the supercommutative envelope: Theorem F. The universal C -enveloping U C (G) of the algebra G of G is isomorphic to thesupercommutativesuperalgebra S=G (+) ⊕Ω 1 G (+) /C whoseevencomponent G (+) is G consideredwiththeJordanproduct x◦y= 1 2 (xy+yx) andtheoddcom- ponent Ω 1 G (+) /C considered as a G (+) -module is the module of differentials, i.e., the quotient of the free G (+) -module with basis (dx) x∈G modulo the submodule generatedby d(x+y)−dx−dy, d (x◦y)−xdy−ydforx,y∈G (+) ,anddcforc∈C, where C bethesubalgebra(withunit)in G andin G (+) generatedbytheelements of the form [x,y] for x,y∈G .The product of odd elements is determined by the kievarwe.tex; 12/03/2001; 3:49; p.33 SYMMETRIES WIDER THANSUPERSYMMETRY 27 formula dx·dy= 1 2[x,y] (x,y∈G). 3.6.TheTaylorformula .Hereafter charK= 0,thesetofindices Iiseither Nor{1,2,...,n}. Forarbitrary c 1 ,...,c p ∈K G [Z](p∈N) set symm(c 1 ,... ,c p ) = 1 p! /summationdisplay σ∈S p c σ(1) ...c σ(p) . The expressions of this form will be called an s-monomial (inc 1 ,... ,c p ). Determine alsoan a-monomial inc 1 ,... ,c 2q by setting alt(c 1 ,... ,c 2q ) = 1 (2q)! /summationdisplay τ∈S 2q (−1) signτ c τ(1) ...c τ(2q) = 2 −q [c 1 ,c 2 ]...[c 2q−1 ,c 2q ]. (Thelastequalityisanontrivialstatement.)Let Mbethesetofallthepairsofthe formm= (α,β), where α= (α 1 ,... ,α p ), α 1 ≤...≤α p , α ν ∈Ifor 1≤ν≤p β={β 1 ,... ,β 2q }, β 1 <...<β 2q , β µ ∈Ifor 1≤µ≤2q. Inthesenotationsfor an arbitraryfamily c= (c i ) i∈I of elements from K G [Z]set c m = symm(c α 1 ,... ,c α p )alt(c β 1 ,... ,c β 2q ). Theelementsoftheform c m (m∈M)willbecalled sa-monomials inc i (i∈I). Proposition The sa -monomials Z m ( m∈M ) constituteabasisof K G [Z] . Set ∂ ∂Z i = ∂ ∂X i + ∂ ∂Y i (i∈I) and foranarbitrary m∈Mset ∂ m ∂Z m = symm /parenleftbigg ∂ ∂Z α 1 ,... , ∂ ∂Z α p /parenrightbigg alt /parenleftbigg ∂ ∂Z β 1 ,... , ∂ ∂Z β 2q /parenrightbigg . Hereafter we assume that I={1,2,... ,n}. Form= (α,β)setδ(m) =q andletm! = (−1) δ(m) d 1 !...d n !,whered i isthedegreeof symm(Z α 1 ,... ,Z α p ) inZ i (i∈I). Theorem (The Taylor series expansion.) For an arbitrary f(Z)∈K G [Z] and anarbitrary a= (a 1 ,... ,a n )∈K n we have f(Z) = /summationdisplay m∈M 1 m!∂ m f ¯0 (a ) ∂Z m (Z−a) m . kievarwe.tex; 12/03/2001; 3:49; p.34 28 D. LEITES, V.SERGANOVA References 1. Baranov,A.A., VolichenkoalgebrasandnonhomogeneoussubalgebrasofLiesuperalgebras . (Russian) Sibirsk. Mat. Zh. 36(1995), no. 5, 998–1009; translation in Siberian Math. J. 36 (1995), no. 5, 859–868 2. Quesne C., Vansteenkiste N. C λ -extended oscillator algebra and parasupersymmetric quantum mechanics . Czechoslovak J. Phys. 48 (1998), no. 11, 1477–1482; Beckers J., De- bergh N., Quesne C., Parasupersymmetric quantum mechanics with generalized deformed parafermions , hep-th/9604132; Helv. Phys. 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B 66(1977), no. 4, 361–364; Wess J.,Supersymmetry/supergravity.Conceptsandtrendsinparticlephysics (Schladming,1986), 29–58,Springer,Berlin,1987; WessJ.,BaggerJ., Supersymmetryandsupergravity .Princeton Seriesin Physics.PrincetonUniversity Press,Princeton, N.J., 1983.i+180 pp kievarwe.tex; 12/03/2001; 3:49; p.37 TENSIONSINSUPERGRAVITY BRANEWORLDS KELLOGGSTELLE ∗ Theoretical Physics Group, Imperial College. London SW7 2BW, UK Abstract. We show how the Randall-Sundrum geometry, which has been proposed as a scenario fortheuniverserealizedasa3-braneembeddedina5-dimensionalspacetime,arisesnaturallyasan S 5 dimensional reduction of a supersymmetric 3-brane of type IIB supergravity. However, a closer inspectionofthe D= 10delta-functionsourcesforthissolutionrevealsamorecomplexsituation: inadditiontotheanticipatedpositiveandnegativeshellsof3-branesource,thereisalsoanon-brane stress-tensor delta-function. The latter singularity may be interpreted as arising from a patching of two discs of D= 10spacetime coincident with the innerand outerbrane locations. The idea that our universe might be realized as a 3-brane embedded in a higher-dimensional spacetime has been considered at various times in recent years [1–5]. In the context of string duality, it was speciﬁcally the construction ofHoˇravaandWitten[6,7]realizingheterotictoM-theorydualityviaanorbifold compactiﬁcation that set a pattern for this scenario. In particular, one may obtain a 3-brane solution to M-theory reduced on a Calabi-Yau manifold down to ﬁve spacetime dimensions [8–10]. This solution has parallel 3-brane universes facing each other across a transverse ﬁfth dimension, located at the ﬁxed planes of the Hoˇrava-Witten S 1 / Z 2 orbifold. The 3-branes are magnetically charged and satu- rate a BPS bound, so are supersymmetric. The solution is supported by a D= 5 scalarﬁeldwhichhasahigher-dimensionalinterpretationasthevolumemodulus, or “breathing mode” of the compactifying space. This scalar ﬁeld acquires a po- tentialasaresultof4-formﬂuxesbeingturnedoninthecompactiﬁeddimensions. The dimensional reduction is thus an example of a generalized (aka Scherk- Schwarz)reductionwithnon-trivialﬁeldstrengthsturnedoninthecompactifying space. InterestinsuchpicturesbecameverymuchheightenedwhenRandallandSun- drum showed [11, 12] that in such a brane-world universe, gravity could behave as if it were effectively 4-dimensional even though the distance between the two 3-branesmightbetakentoinﬁnity,providedthebulkgeometrynearthebranewe live on is aD= 5anti de Sitter space. Speciﬁcally, a model was considered that ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.38 32 K.STELLE involvedtwosegmentsofapure AdS 5 spacetimepatched together, ds 2 5 =e −2|z | L dx µ dx µ +dz 2 (1) witha“kink”at z= 0correspondingtoapositive-tension δ-functionstress-tensor source.InRef.[12]itwasshownthatthisgivesrisetoa“binding”ofgravitytothe D= 5spacetimeregionnearthe(3+1)dimensionalbraneworld,withaneffective Newtoniangravitationalpotentialplus eventualmeasurable corrections, V grav ∼ 1 r+L 2 r 3 . (2) It was the potential measurability of these corrections to Newtonian gravity that attractedsuch strongattention withinthescientiﬁccommunity. Nospeciﬁcsupergravityrealizationofsuchaconstructionwasgiven,although clearly it seems natural to try to embed the RS braneworld into a D= 5dimen- sionalreductionoftypeIIBsupergravity.RealizingtheRSbraneinasupergravity contextranintocertaindifﬁculties,however,primarilyconcerningthebehaviorof the scalar ﬁeld that would need to be used to support the 3-brane solution. No known scalar ﬁeld in any of the dimensionally-reduced versions of D= 5su- pergravity has the properties needed to ﬂow correctly to a ﬁxed point at locations far from the RS brane, and this was encoded in a “no-go theorem” [13, 14]. As is frequently the case with no-go theorems, however, the main result may be to direct attention towards the underlying assumptions that need to be relaxed. The keyone inthiscaseconcerns the natureofthe supportingscalar. Even before the Randall-Sundrum work on our universe as a braneworld em- beddedinaD= 5spacetime,ageneralstudyhadbeenmade[15]ofthespherical dimensional reductions of various supergravity theories and of the branes and domain walls that exist in these reduced theories. For the speciﬁc case of the S 5 reduction of type IIB supergravity down to D= 5, it was shown that the familiar D3-brane geometry of D= 10type IIB theory dimensionally reduces to a 3-brane in D= 5, supported by the “breathing mode” scalar modulus that determines the volume of the compactifying S 5 . This works in a very similar way to the breathing-mode supported 3-brane in the Calabi-Yau reduction of M- theory [8–10]. It should be noted, however, that the breathing mode for an S 5 reductiondoesnotitselfbelongtothemasslesssupergravitymultiplet.Instead,the breathing mode belongs to a massive spin-two multiplet, as is appropriate, since the dimensional reduction turns on a ﬂux in the internal S 5 directions, and this gives the breathing mode ϕa scalar potential that allows this mode to support a 3-branesolution.Withoutthispotential,thebreathingmodecouldnotsupporta3- branesolution.Butthemassivecharacterofthismodeplacesitoutsidetheclassof modes normally considered in D= 5compactiﬁcations of supergravity theories. The importance of this mode for realizing the Randall-Sundrum braneworld as a supergravity construction was recognized in Refs [16, 17], although a main focus kievarwe.tex; 12/03/2001; 3:49; p.39 TENSIONSIN SUPERGRAVITY BRANEWORLDS 33 was still on the difﬁculty of realizing RS geometries as a fully “smoothed-out” solitonic solution. To see how a construction analogous to the M-theory 3-brane solution can be madeinS 5 reducedtypeIIBtheory,considerasimpliﬁedtheoryjustretainingthe D= 10metric and the self-dual5-form ﬁeld strength H [5] , R µν = 1 96(H [5] ) 2 µν H [5] = ∗ H [5] dH [5] = 0, (3) wheretheequationsofmotionfortheﬁve-formareimpliedbytheBianchiidentity dH≡0taken together with the H [5] = ∗ H [5] duality relation. Dimensionally reducingon S 5 ,onemakes theKaluza-Kleinansatz ds 2 10 =e 2αϕ ds 2 5 +e 2βϕ ds 2 (S 5 ) (4) H [5] = 4me 8αϕ ε [5] + 4mε [5] (S 5 ) (5) α= 1 4 /radicalbigg 5 3β=−3 α 5. This reductionyields the D= 5bosonic theory L 5 =eR− 1 2 e∂ µ ϕ∂ µ ϕ−8m 2 ee 8αϕ +R 5 ee 16 αϕ 5 + more (6) wherethetermsrepresentedby“more”includebosonicﬁeldsthatarenotrelevant for the 3-brane solution, plus all the fermions. Note that there are two potential termsin(6):theonewithcoefﬁcient −8m 2 comesfromthe H [5] ﬂuxesturnedon in the reduction ansatz (5), while the one with the coefﬁcient R 5 comes from the Einstein-Hilbert action in the ﬁve compactiﬁed directions, since S 5 is not Ricci- ﬂat.Thecoefﬁcient R 5 isequaltotheconstant Ricci scalarof theinternal S 5 . Thepresenceoftwopotentialtermsin(6)withoppositesignsenablesapartic- ularly simple and maximally symmetric solution to the D= 5reduced theory. In this case, one can ﬁnd a solution with a constant breathing-mode scalar ϕ=ϕ ∗ , with e 24αϕ ∗ 5 =R 5 20m 2 R µν =−4m 2 e 8αϕ ∗ g µν . (7) Solving this D= 5Einstein equation with a cosmological term, one ﬁnds the AdS 5 ×S 5 “vacuum”ofthe S 5 compactiﬁedtheory.Theexistenceofthisvacuum makes this a simpler situation than the one obtained in M-theory reduced on a Calabi-Yau manifold, where only a single potential term is obtained, and where no maximally-symmetric solutionin D= 5is found. In addition to the AdS 5 ×S 5 solution (7), one can also search for brane so- lutions with less symmetry, but which tend asymptotically in appropriate regions kievarwe.tex; 12/03/2001; 3:49; p.40 34 K.STELLE to the above solution. Before pursuing this search, let us make a small change to the reduction ansatz (5) which is frequently made when considering domain- wall solutions ( i.e.for codimension-one branes). The original type IIB theory in D= 10has a Z 2 symmetry (actually, it is just a discrete D= 10proper Lorentz transformation) that couples an orientation-reversing transformation on the S 5 coordinates together with a sign ﬂip on one of the lower D= 5coordinates, say y→−y. This symmetry is broken by the original ansatz (5), but will be restored if one generalizes the ansatz by inclusion of θfunctions (θ(y) = 1fory >0and θ(y) =−1fory<0): H [5] = 4mθ(y)e 8αϕ ε [5] + 4mθ(y)ε [5] (S 5 ). (8) Notethatbothtermsin(8)needtohave θfunctionsinordertosatisfythe H [5] self- duality condition in (3). With this modiﬁed ansatz, one has traded in translation invariance in the ycoordinate for this preserved Z 2 symmetry. Although the ﬁeld strengthH [5] isdiscontinuousin(8),theunderlyingfour-formgaugepotential A [4] canstillbecontinuous.Weshalladoptabasicboundaryconditionrequirementof continuityfor themetric andthegauge potentialsatsuch“kink” locations. Adopting the ansatz (8) and searching for a domain-wall solution, one ﬁnds thefollowing[15]: ds 2 5 =e 2A dx µ dx ν η µν +e 2B dy 2 e 4A =e −B =˜b 1 H 2 7 +˜b 2 H 5 7 e −7 ϕ √ 15 =H=e −7ϕ 0 √ 15 +k|y| ˜b 1 =±28 m 3k˜b 2 =± 14 15k /radicalbig 5R 5 . (9) Of the sign choices allowed in (9), we shall pick ˜b 2 >0,˜b 1 <0in order to ensure reality of the metric and to permit a k→0limit so as to recover the pure AdS Randall-Sundrum bulk spacetime [18]. We shall also choose the integration constantϕ 0 so thatH(0)> H ∗ =e −7ϕ ∗ √ 15 and we shall take the slope parameter kto be negative. Then the “kink” at y= 0faces downward, so that the function H(y)reachesH ∗ atsomeﬁnitevalue y ∗ .Thesolution(9)maythenbeconsidered to be a “semi-interpolating soliton” in the sense that, although the point y= 0at which the domain-wall kink is located is not null, i.e.not a horizon, the solution evolves as one moves away from y= 0through either positive or negative y values towards the AdS 5 ×S 5 vacuum solution (7) at y=y ∗ . With the “kink- down” structure selected here, the surface at y= 0corresponds to an extended object of positive tension. Thesolution(9)isafullysupersymmetricsolution,despiteitskinksingularity. The bulk geometry admits a 16-component Killing spinor, since it is none other thantheregularD3-branegeometryoftypeIIBtheory.Moreover,the Z 2 invariant kievarwe.tex; 12/03/2001; 3:49; p.41 TENSIONSIN SUPERGRAVITY BRANEWORLDS 35 structure of (9) is precisely what is needed for the Killing spinor equation to be valid at all points, including at the kink location y= 0, with a continuous Killing spinor.Theﬂipofsigninthe5-formﬂuxvalue masgiveninthemodiﬁedKaluza- Klein ansatz (8) is essential for the Killing spinor equations to be solved in this way. The positive-tension nature of the y= 0surface and the approach to the AdS 5 ×S 5 vacuum solution at y ∗ suggests that one should be able to take a limit of the solution (9) and obtain the Randall-Sundrum spacetime [18]. This limit needs to be taken conjointly in both the integration constants ϕ 0 andk. We let H 0 =e −7ϕ 0 √ 15 =H ∗ +β|k|and then take the limit k→0 − . In this conjoint limit, factorsofk −1 cancelagainstfactorsof k,and thelimiting metric becomes ds 2 = 2 √ L(β−|y|) 1 2 dx µ dx ν η µν +L 2 16dy 2 (β−|y|) 2 , L =m −1 /parenleftBigg 20m 2 R 5 /parenrightBigg 5 6 .. (10) Thissolutionisapatched D= 5antideSitterspacewiththehorizonat y=y ∗ = ±β. To recognize it in a more standard form, make a ﬁnal coordinate change: β−|y|=βe −4|˜y | L , thusobtaining AdS 5 spacetime inPoincar ´e coordinates: ds 2 =e −2|˜y | L dx µ dx ν η µν +d˜y 2 . (11) Let us now consider how this Randall-Sundrum metric has been successfully obtained as a solution of type IIB supergravity theory, despite the apparent impli- cations of the various “no-go” theorems for the necessary scalar ﬂows [13, 14]. Consideratheoryconsistingofgravitycoupledtoascalarﬁeld φwithapotential V(φ): L=e[R− 1 2 ∇ µ φ∇ µ φ−V(φ)], (12) where the minimum of the potential is taken to occur at φ=φ 0 . Then expand V(φ)nearφ 0 :V(φ) =−12g 2 + 1 2 µ 2 (φ−φ 0 ) 2 +...(theconstant gischosento make the AdS curvature equal to −g 2 (g MP g NQ −g MQ g NP ). Writing the metric asds 2 =e 2A(y) dx µ dx ν η µν +e 2B(y) dy 2 and solving the Einstein equations up to linear order in φ, one ﬁndsA(y) =±gy. Then one ﬁnds for static φ(y)nearφ 0 theapproximate ﬁeld equation φ /prime/prime ±4gφ /prime −µ 2 φ≈0. (13) This equation has twosolutions: φ≈φ 0 +ce −E 0 A(y) (14) φ≈φ 0 +ce −(4−E 0 )A(y) (15) E 0 = 2 + /radicalBigg/parenleftbigg µ g /parenrightbigg 2 + 4≥2 (16) kievarwe.tex; 12/03/2001; 3:49; p.42 36 K.STELLE whereE 0 istheAdSenergy.Requiringastableinfraredﬂowtothevacuumvalue φ 0 asA→−∞(i.e.e 2A →0, so one moves in to the horizon), one must take the second solution (15) and also impose a restriction that the scalar ﬁeld’s AdS energy be bounded below by 4: E 0 >4. Now, the AdS energy is a ﬁxed constant for a given ﬁeld, determined by the Lagrangian. General ﬁelds in D= 5AdS spacetimecarryAdSrepresentations D(E 0 ,j 1 ,j 2 ),wherej 1 andj 2 arespins.For “standard” supergravities in D= 5(i.e.supergravities containing the massless graviton and vector multiplets, plus hypermultiplets and tensor multiplets), one ﬁnds scalars D(E 0 ,0,0)withE 0 = 2,3,4only, so an infrared stable ﬂow of the abovetypeisnotpossible.However,thesolution(9)issupportedbythebreathing- mode scalar φ, obtained from the S 5 dimensional reduction down from D= 10. This mode belongs to a short massive multiplet of D= 5,N= 4supergravity, which contains a massive spin-two mode, so it does not belong to one of the supermultiplets customarily considered in D= 5massless supergravity models. Comparison of the breathing-mode potential V(φ) = 8m 2 e 8αϕ −e 16 αϕ 5 R 5 ,α= 1 4 /radicalBig 5 3 , with the formula (16) for E 0 givesE 0 = 8, clearly satisfying the required bound forastableﬂow to ϕ 0 =ϕ ∗ . Since the Kaluza-Klein ansatz (4,8) constitutes a consistent truncation of the D= 10theory down to D= 5, one may automatically oxidize the solution (9) back up toD= 10and consider its structure there. In this case, it becomes a patched set of domains of a standard type IIB D3-brane geometry. Each patch runs from a horizon at isotropic-coordinate radius r= 0↔y=y ∗ out to an outer radius r=r RS ↔y= 0, wherer RS = /radicalBig 20 R 5 [e − /radicalbig 3 5 ϕ 0 −e − /radicalbig 3 5 ϕ ∗ ]. At this outer radius r=r RS , the solution is patched onto a Z 2 mirror solution on another sheet of spacetime, corresponding to the D= 5region with y < 0. The Randall-Sundrum limit k→0 − ,ϕ 0 →ϕ ∗ corresponds to shrinking down to zero the radius r RS at which the patch to the second sheet is made. Alternately, one could take a limit m→∞for the ﬂux parameter in the reduction ansatze (4,8). In either case, one obtains a spacetime that has a uniform AdS structure: in the ﬁrst case, because one is restricting the spacetime ever more narrowly down to a solid annulus around the horizon, which is asymptotically AdS 5 ×S 5 ; in the secondcasebecausethisasymptoticregionspreadsouttoﬁllthewholespacetime. Regardless of the perspective one takes on this limit, the proper length running from a given radius 0< r < r RS down to the horizon at r= 0diverges. So, in this sense, the horizon is an inﬁnite proper distance away along a radial ( i.e. spacelike) geodesic. However, as is generally the case with extremal geometry horizons, one may also reach the horizon along a timelike or lightlike geodesic within a ﬁnite afﬁne parameter interval. So the question of whether this Randall- Sundrum spacetimeisreally inﬁniteor notrequires carefulinterpretation. At the horizon itself, one has a choice of interpretations for the structure of the solution (9) when oxidized back up to D= 10. The D3 brane geometry is kievarwe.tex; 12/03/2001; 3:49; p.43 TENSIONSIN SUPERGRAVITY BRANEWORLDS 37 actually non-singular and Z 2 symmetric at the r= 0horizon [19]. If one takes the horizons in the two sheets patched together at r=r RS to be distinct, then oneconsidersapatched-branerealizationofRSIIgeometry[12],whichin D= 5 consists of a single kinked warp-factor AdS metric as in (11), extending out then to inﬁnite proper distances in the y >0andy <0regions. On the other hand, if one decides to exploit the Z 2 symmetry of the D3 brane solution at the horizon, one may alternatively make a second patch of the horizon at y=y ∗ onto the secondsheethorizonat y=−y ∗ .Thisproducesasecond,upwards-facingkinkin theD= 5geometry,correspondingtoanextendedobjectofnegativetension,re- producingtheRSIgeometry[11]withtwobranesofoppositetension,facingeach otheracrossacompactdimension.ThissituationisclearlyatypeIIBanalogueof the M-theory 3-brane solution obtained in a Calabi-Yau compactiﬁcation [8–10]. The second patching surface can equally well be moved off from the horizon by moving the inner patching radius away from r= 0, corresponding to moving the secondD= 5brane intoa ﬁniteproperdistancefromthe y= 0surface. Whatever the interpretation given to the horizon region, the kink surface at y= 0↔r=r RS possesses the essential properties of the Randall-Sundrum solution. This surface has a positive tension σ RS >0, as can be veriﬁed using the Israelmatchingconditions ∆K µν =K + µν −K − µν =−8π G 3σ RS g µν , (17) forthediscontinuityintheextrinsiccurvature K µν = 1 2 n λ ∂ λ g µν ,wheren λ isthe outward-pointingsurfacenormal.Consequently,inaccordancewiththeresultsof Ref. [12] this surface has the property of “binding” gravity to it: matter on this 3+1 dimensional surfacegravitationallyinteracts asif the theory were in D= 4. TheabovepictureoftheRandall-SundrumspacetimeasapatchingoftypeIIB 3-brane geometries leaves some important questions unaddressed. The principal one of these is the nature of the singular sources that must be present as a result of the curvature delta-functions arising from the patching process. An immediate appreciationofthismaybehadbyconsideringthesignsofthesourcebranedelta functions. The bulk geometry between the inner and outer patching radii in the D= 10perspective is a normal D3-brane geometry with a positive energy. At the same time, if the outermost source is of positive tension, as it must be in ordertoagreewiththeRandall-Sundrumtensionasobtainedfrom(17)in D= 5, then the inner source would have to be of opposite, i.e.negative, tension. This is clearly inconsistent with the positive-energy D3-brane geometry in the solid annulusbetweentheinnerandoutersources.Arelatedproblemisthatnotonlythe sign, but also the magnitude of the tensions do not agree with D3-brane tensions: the D3-brane tension is only 2 3 of the Randall-Sundrum value as determined by (17) [20]. Both of the above problems are resolved by a recognition that the sources at the inner and outer radii in D= 10cannot simply be D3-brane sources alone kievarwe.tex; 12/03/2001; 3:49; p.44 38 K.STELLE [21]. A brane stress tensor in D= 10would have nonzero components onlyin thebraneworldvolumedirections, ˆT µν =−σg µν δ(z).However,the D= 5stress tensorforthelimiting solution(10)oxidizesupto D= 10in theform ˆT µν =−56m 2 β /parenleftBigg 20m 2 R 5 /parenrightBigg − 25 12 δ(y)g µν + Reg. ˆT 55 = 0 + Reg. ˆT ab =− 224 3m 2 β /parenleftBigg 20m 2 R 5 /parenrightBigg − 25 12 δ(y)g ab + Reg., (18) wherethea,bindiceslieinthecompact S 5 directions.Itisimmediatelyapparent thatthisisnotoftheformofabranestresstensor,notwithstandingthefactthatthe surrounding spacetime is a limit of a normal type IIB 3-brane solution. One may understand what is going on by taking the difference between the stress tensor (18) and that expected from the 3-brane bulk geometry. Alternatively (and this is much simpler in practice), one may ﬁnd the structure of the difference stress tensor by keeping the general domain-wall form of the D= 5solution (9, 10) withthe|y|modulus,butturningoffthemagneticﬂuxparameter m.Theresultof thisanalysis isastress tensoroftheform ˆT Diﬀ. µν = 3κδ(y)g µν ˆT Diﬀ. 55 = 0 ˆT Diﬀ. ab = 12 5κδ(y)g ab , (19) whereκisaconstant.Thissingularstresstensoroccursevenintheabsenceofthe 3-brane,i.e.itisasingularityoccurringbetweenpatchesofﬂatspace.The D= 5 interval−β < y < β∼−∞<˜y <∞lifts to two copies of a disc in the ﬂat D= 10spacetime, with an outermost patch corresponding to y= 0, and another patch atthe horizon, y=y ∗ =±β. Although the stress-tensor (19) is not of the form of a brane stress tensor, one can still compare its ˆT 00 component to that of the 3-brane. Comparing the value ofκobtained with that of the D3 brane source for the bulk geometry shows that thestress-tensor(19) has an effective “tension”related tothat ofa3-brane by σ ﬂatpatch =− 5 2σ D3 . (20) This explains what is happening in the relationship between the type IIB 3-brane solution and the Randall-Sundrum solution. The D= 5Randall-Sundrum solu- tion(priortotakingthepureAdSlimit)liftstoa D= 10solutionthatiscomposed of two copies of the 3-brane geometry, patched together at a radius r RS and at the kievarwe.tex; 12/03/2001; 3:49; p.45 TENSIONSIN SUPERGRAVITY BRANEWORLDS 39 horizon. The extra stress-tensor component (19), related to that of the 3-brane by (20),combineswiththe3-branestresstensortomakeacompositesingularstress- tensor which when viewed from a D= 5viewpoint appears to be a brane stress tensor of sign oppositeto that of the 3-brane in D= 10, and with a magnitude 3 2 that ofthe3-brane, explaining thediscrepancynoted in Ref.[20]. Theoverallsolutionliftedto D= 10isstill Z 2 symmetric,andifonedemands thatthisdiscretesymmetryberespected,togetherwiththe S 5 sphericalsymmetry required for a spherical dimensional reduction down to D= 5, then the location of the “patch” stress-tensor singularity (19) is ﬁxed by the symmetry. This is not the case, however, with the 3-brane itself. There is no symmetry principle that restricts this to be superposed on the patch singularity (19) – it may freely move inwards from the patch. For static solutions, this has the effect of joing the D3 brane spacetime continuously onto an outermost solid annulus of ﬂat space. In generalized solutions, however, this boundary may also become dynamical. Owing to the sign ﬂip inherent in (20), it is clear that what looks like a positive tension brane from the D= 5perspective actually contains a negativetension 3- brane from the D= 10perspective. Establishing the stability or otherwise of this conﬁguration clearly remains an essential task for future analysis of braneworld scenarioslike thatof RandallandSundrum. References 1. K.Akama, Pregeometry , Lect. Notes Phys. 176, 267 (1982)[hep-th/0001113]. 2. V. A. Rubakov and M. E. Shaposhnikov, Do We Live Inside A Domain Wall? , Phys. Lett. B125, 136(1983). 3. V. A. Rubakov and M. E. Shaposhnikov, Extra Space-Time Dimensions: Towards A Solution To TheCosmological ConstantProblem ,Phys. Lett. B125,139 (1983). 4. G.W.GibbonsandK.Maeda, BlackHolesAndMembranesInHigherDimensionalTheories With DilatonFields , Nucl. Phys. B298, 741(1988). 5. G.T.HorowitzandA.Strominger, BlackstringsandP-branes ,Nucl.Phys. B360,197(1991). 6. P. Ho ˇrava and E. Witten, Heterotic and type I string dynamics from eleven dimensions , Nucl. Phys.B460, 506(1996) [hep-th/9510209]. 7. P. Hora ˇva and E. Witten, Eleven-Dimensional Supergravity on a Manifold with Boundary , Nucl. Phys. B475, 94 (1996)[hep-th/9603142]. 8. A.Lukas,B.Ovrut,K.S.StelleandD.Waldram, TheUniverseasaDomainWall ,Phys.Rev. D59, 086001 (1999) [hep-th/9803235]. 9. A. Lukas, B. Ovrut, K. S. Stelle and D. Waldram, Heterotic M–theory in Five Dimensions , Nucl. Phys. B552, 246 (1999)[hep-th/9806051]. 10. A.LukasandK.S.Stelle, Heteroticanomalycancellationinﬁvedimensions ,JHEP0001,010 (2000)[hep-th/9911156]. 11. L.RandallandR.Sundrum, Alargemasshierarchyfromasmallextradimension ,Phys.Rev. Lett.83, 3370 (1999) [hep-ph/9905221]. 12. L. Randall and R. Sundrum, An alternative to compactiﬁcation , Phys. Rev. Lett. 83, 4690 (1999)[hep-th/9906064]. 13. R. Kallosh and A. Linde, Supersymmetry and the brane world , JHEP0002, 005 (2000) [hep- th/0001071]. kievarwe.tex; 12/03/2001; 3:49; p.46 40 K.STELLE 14. K.BehrndtandM.Cveti ˇc, Anti-deSittervacuaofgaugedsupergravitieswith8supercharges , Phys. Rev. D61, 101901 (2000)[hep-th/0001159]. 15. M.S.Bremer,M.J.Duff,H.L ¨u,C.N.PopeandK.S.Stelle, Instantoncosmologyanddomain walls fromM-theory andstring theory , Nucl. Phys. B543,321 (1999)[hep-th/9807051]. 16. M. Cveti ˇc, H. L¨u and C. N. Pope, Domain walls and massive gauged supergravity potentials, hep-th/0001002. 17. M. Cveti ˇc, H. L¨u and C. N. Pope, Localised gravity in the singular domain wall background? hep-th/0002054. 18. M.J. Duff, J.T. Liu and K.S. Stelle, A supersymmetric type IIB Randall-Sundrum realization, hep-th/0007120. 19. G.W.Gibbons,G.T.HorowitzandP.K.Townsend,Class.QuantumGrav. 12,297(1995)[hep- th/9410073]. 20. P.Kraus, Dynamicsofanti-deSitterdomainwalls ,JHEP9912,011(1999)[hep-th/9910149]. 21. M. Cveti ˇc, M.J. Duff, J.T. Liu, C.N. Pope and K.S. Stelle, Randall-Sundrum Brane Tensions, hep-th/0011167. kievarwe.tex; 12/03/2001; 3:49; p.47 ANUNCONVENTIONALSUPERGRAVITY PAVELGROZMANand DIMITRYLEITES †‡ Department of Mathematics, University of Stockholm, Roslagsv. 101,Kr¨aftrikethus 6,S-10691,Stockholm, Sweden Abstract. WeintroduceandcompletelydescribetheanaloguesoftheRiemanncurvaturetensorfor the curved supergrassmannian of the passing through the origin (0|2)-dimensional subsuperman- ifolds in the (0|4)-dimensional supermanifold with the preserved volume form. The underlying manifold of this supergrassmannian is the conventional Penrose’s complexiﬁed and compacti- ﬁed version of the Minkowski space, i.e. the Grassmannian of 2-dimensional subspaces in the 4-dimensional space. The result provides with yet another counterexample to Coleman–Mandula theorem. 1.Newsupertwistors .Penrosesuggestedanunusualdescriptionofourspace- time, namely to compactify the Minkowski space-time model of the Universe (nontrivially: with a light cone at the inﬁnity) and complexify this compactiﬁ- cation. The ﬁnal result is Gr 4 2 , the Grassmanian of 2-dimensional subspaces in the4-dimensional(complex)space(ofso-calledtwistors).Therearemanypapers and several monographs on advantages of this interpretation of the space-time in various problems of mathematical physics; we refer the reader to Manin’s book [5], where an original Witten’s idea to incorporate supervarieties and con- siderinﬁnitesimalneighborhoodsforinterpretationofthe“usual”,i.e.,non-super, Yang-Mills equations is thouroghly investigated together with several ways to superize Minkowski space. Oursisonemore,distinct, way. Observe that the supermanifold of (0|2)-dimensional subsuperspaces in the (0|4)-dimensional superspace is identical with Gr 4 2 , only the tautological bundle is different: the ﬁber is purely odd. In this work we consider not subsuper spaces butsubsuper manifolds . Weconsideredthestructurefunctions—analogsoftheRiemanntensor—for thecurvedsupergrassmannian CGr 0|4 0|2 of(0|2)-dimensionalsubsuper manifolds in the(0|4)-dimensional supermanifold. Recall that the “usual” grassmannian con- sists of linear subspaces of the linear space passing through the origin whereas ‡ We gratefully acknowledge partial ﬁnancial support of The Swedish Institute and an NFR grant, respectively. † [email protected] kievarwe.tex; 12/03/2001; 3:49; p.48 42 P. GROZMAN,D.LEITES the curved one consists of submanifolds, in other words, nonlinear embeddings are allowed and the submanifolds do not haveto pass through a ﬁxed point. Obviously, the curved Grassmannian is inﬁnite dimensional, but the curved su- pergrassmannianCGr 0,n 0,k is of ﬁnite superdimension: it is a quotient of the supergroup of superdiffeomorphisms of the linear supermanifold C 0,n (the Lie superalgebra of this Lie supergroup is vect(0|n) = derC[θ 1 ,...,θ n ]). For the list of classical superspaces including curved supergrassmannians see [4].) The underlying manifold of CGr 0|4 0|2 is the conventional Gr 4 2 butCGr 0,4 0,2 has also odd coordinates. OnCGr 0|4 0|2 , we have expanded the curvature supertensor in components with respect to the (complexiﬁcation of the) Lorentz group and saw that it does not contain the components used for the ordinary Einstein equations (EE), namely, there is no Ricci curvature Ricand no scalar curvature Scalar(in what follows R(22)andR(00),respectively). So we decided to amend the initial model and consider the supergrassmanian CGr 0|4 0|2 (0)of subsupermanifolds through the origin. It turns out that this does not help: no RicandScalar,either. We decided not to give up, and took for the model of Minkowski superspace thesupergrassmannian SCGr 0|4 0|2 (0)ofsubsupermanifoldsthroughtheoriginwith the volume element of the ambient and the subsupermanifolds preserved. On SCGr 0|4 0|2 (0), the expansion of the curvature supertensor does contain R(22)and R(00)! There areno analogsofconformal(offshell) structurefunctions. Ourmodelanditssupergroupofmotion—ananalogueofthePoincar ´egroup — do not contradict the restrictions of the famous no-go theorems by Haag– Łopuszanski–Sohnius and Coleman–Mandula (for further discussions see [1]) and provides us with a new, missed so far, version of the Poincar ´e supergroup. The analogues of Einstein equations we suggest are a totally new version of SUGRA. Equating to zero other conformally non-invariant components we get extra conditions;we donot knowhowtointerpretethem. Wedonotseeanyreasonfordiscardingthisandsimilarmodels.Inparticular, we suggest to analyze the structurre functions (deﬁnition below) on CGr 0|4 0|2 and CGr 0|4 0|2 (0)which we have abandoned above. The conventional reading of Coleman–Mandula’s theorem (cf. [6]) assumes that the complexiﬁed Lorentz Lie algebra L= sl(2) L ⊕ sl(2) R commutes with the Lie algebra of internal symmetries i(for us iis equal to sl(2) L ⊗Cξ 1 ξ 2 , see sec.4). In our case Lacts on iand forms a semidirect sum with it; the bracket on i is identically zero. This possibility does not contradict assumptions of Coleman– Mandula’s theorem butwas notconsidered. kievarwe.tex; 12/03/2001; 3:49; p.49 ANUNCONVENTIONAL SUPERGRAVITY 43 The odd parameters have a correct statistics with respect to the Lorentz Lie algebra. We represent Einstein’s equations as conditions on conformally noninvariant components of the analog of the Riemann tensor, and represent the Riemann tensor as a section of the bundle on the (locally) Minkowski space whose ﬁber is certain Lie algebra cohomology . This is a more user-friendly description of the Riemannian tensor than the classical treatment of obstructions to nonﬂatness in differential geometry. We have in mind Spencer homology , cf. [7], where the case of any G-structure, not only G=O(n)is considered. Superization of the deﬁnitionsfrom[7]is the routinestraightforward application of theSign Rule. Remark. It is interesting to test the whole list of curved supergrassmannians with the simple Lie supergroup of motion (see Tables in [4]) and similarly to the above sacrify the simplicity of the supergroup of motion in order to get EE. Grozman’s package SuperLie (see [2]) is a useful tool in this research problem: withouta computer(and agoodcode)thistask is hardlyfeasible. 2. Structure functions: recapitulation ([7]) . LetF(M)be the frame bundle over a manifold M, i.e., the principal GL(n)-bundle. Let G⊂GL(n)be a Lie group. AG-structure on Mis a reduction of the principal GL(n)-bundle to the principalG-bundle. The simplest G-structure is the ﬂatG-structure deﬁned as follows. Let Vbe R n (orC n )withaﬁxedframe.Theﬂatstructureisthebundleover Vwhoseﬁber overv∈Vconsistsofallframesobtainedfromtheﬁxedoneunderthe G-action, Vbeingidentiﬁedwith T v Vbymeans oftheparalleltranslation by v. Examples of ﬂat structures . The classical spaces, e.g., compact Hermi- tian symmetric spaces, provide us with examples of manifolds with nontrivial topologybutﬂat G-structure. In [7] the obstructions to identiﬁcation of the kth inﬁnitesimal neighbour- hood of a point mon a manifold MwithG-structure with the kth inﬁnitesimal neighbourhood of a point of the ﬂat manifold Vwith the above described ﬂat G- structurearecalled structurefunctionsoforder k.In[7]itisshownfurtherthatthe tensors that constitute these obstructions are well-deﬁned provided the structure functions of all orders < kvanish. (In supergravity the conditions that structure functions of lesserorders vanish are called Wess-Zumino constraints .) The classical description of the structure functions uses the notion of the Spencer cochain complex . Let us recall it. Let S i denote the operator of the i- th symmetric power. Set g −1 =T m M, let g 0 be the Lie algebra of G; fori >0 set: g i ={X∈Hom ( g −1 , g i−1 )|X(v 0 )(v 1 ,...,v i ) =X(v 1 )(v 0 ,...,v i ) for anyv 0 ,v 1 ,...,v i ∈ g −1 }.(2.1) Finally, set ( g −1 , g 0 ) ∗ =⊕ i≥−1 g i . This is the Lie algebra of alltransforma- tions that preserve on g −1 the same structure which is preserved by the linear kievarwe.tex; 12/03/2001; 3:49; p.50 44 P. GROZMAN,D.LEITES transformationsfrom g 0 . Suppose that the g 0 -module g −1 is faithful, i.e., each nonzero element from g 0 actsnontrivially.Then, clearly, ( g −1 , g 0 ) ∗ ⊂ vect(n) = derR[x 1 ,...,x n ], wheren= dim g −1 ,with g i ={X∈ vect(n) i : [X,D ]∈ g i−1 for anyD∈ g −1 } fori≥1.Itis easytocheck that ( g −1 , g 0 ) ∗ is aLiesubalgebraof vect(n). The Lie algebra ( g −1 , g 0 ) ∗ will be called the Cartan’s prolong (the result of Cartan’sprolongation )of thepair ( g −1 , g 0 ). LetE i be the operator of the i-th exterior power; set (prime denotes dualization) C k,s ( g −1 , g 0 ) = g k−s ⊗E s ( g /prime −1 ). Deﬁne the differential ∂ s :C k,s ( g −1 , g 0 ) −→C k,s+1 ( g −1 , g 0 ) by setting for any v 1 ,...,v s+1 ∈V(asusual, theslotwiththe hattedvariableis tobe ignored): (∂ s f)(v 1 ,...,v s+1 ) = /summationdisplay (−1) i [f(v 1 ,..., ˆv s+1−i ,...,v s+1 ),v s+1−i ].(2.2) As expected, ∂ s ∂ s+1 = 0, and the homology H k,s ( g −1 , g 0 ) of the bicomplex ⊕ k,s C k,s ( g −1 , g 0 ) is called the (k,s)-thSpencer cohomology of( g −1 , g 0 ) ∗ . (Observe that we use a grading of the Spencer complex different form that in [7]. Ours is a more natural one.) Proposition ([7]) Thestructurefunctionsoforder k constitutethespaceofthe (k,2) -thSpencercohomology ofthe ( g −1 , g 0 ) ∗ . 3. Spencer cohomology in terms of Lie algebra cohomology . We observe that ⊕ k H k,2 ( g −1 , g 0 ) =H 2 ( g −1 ; ( g −1 , g 0 ) ∗ ). (3) The advantage of this reformulation: the Lie algebra cohomology (the right hand sideof(3))iseasiertocompute(e.g.,bymeansofthepackageSupeLiewhenthe general theory fails, or with the help of various theorem). At the same time the ﬁne grading of Spencer homology is not lost: the Z-grading of ( g −1 , g 0 ) ∗ which induces the grading (3)ofH 2 ( g −1 ; ( g −1 , g 0 ) ∗ )coincides (up to a shift) with the oderofthestructure functions. AnalogsofWeylandRiemanntensors .Suppose g 0 containsacenter(likein the case when a metric is preserved up to a conformal factor). Then the elements ofH 2 ( g −1 ; ( g −1 , g 0 ) ∗ )areanalogs oftheWeyltensor. Letˆ g 0 be the semisimple part of g 0 and let ˆ g ∗ be a shorthand for ( g −1 ,ˆ g 0 ) ∗ . The elementsof H 2 ( g −1 ;ˆ g ∗ )are analogsof the Riemanntensor. kievarwe.tex; 12/03/2001; 3:49; p.51 ANUNCONVENTIONAL SUPERGRAVITY 45 The relation between ˆH=H 2 ( g −1 ;ˆ g ∗ )andH=H 2 ( g −1 ; ( g −1 , g 0 ) ∗ )is more intricate for the general ˆ g 0 than in the Riemannian case ( ˆ g 0 = o(n)) when ˆHstrictly contains H. In general, these spaces have common components (con- formally invariant, “on shell” ones) and have other components, analogs of “off shell”components,cf.[3]. In the Riemann case, there are two “off shell” components: with the highest weights (2,2)(the traceless Ricci tensor) and (0,0)(the scalar curvature). Here the highest weights are given with respect to the complexiﬁcation L= sl(2) L ⊕ sl(2) R ofthe o(1,3).TheEinsteinequaton(invacum)isavanishingconditionof thesecomponents.Remarkably,therearenostructurefunctionsoflesserorder.If they had existed, we would have to impose analogs of Wess-Zumino constraints tobe abletodeﬁne theusualRiemanncurvature tensor. 4. The description of ( g −1 , g 0 ) ∗ for the curved supergrassmannians . For the general curved supergrassmannian of (0,k)-dimensional subsupermanifolds Sinthe (0,n)-dimensionalsupermanifold Tletξ 1 ,...,ξ k bethecoordinatesof S andθ 1 ,...,θ n−k the remaining coordinates of T. Then setting degxi i = 0for all ianddegθ j = 1for alljwe getaZ-gradingof vect(0|n)oftheform g 0 = ( gl(V)⊗C[ξ])⊃ + vect(ξ); g −1 =V⊗C[ξ]; (4) whereV= Span( ∂ ∂θ 1 ,..., ∂ ∂θ n−k )is the identity gl(V)-module, and⊃ + is the sign ofasemidirect sumofalgebras: a⊃ + bwiththeideal a. Forn= 4wecomputed H 2 ( g −1 ; ( g −1 , g 0 ) ∗ )inthe following cases: (a) thegeneral curvedsupergrassmannians; (b)thesupergrassmanniansofsubspacesthrough 0,i.e.,weremovedfrom vect all partial derivatives (since this is not an invariant formulation, it is better to say: we onlyconsidered thevectorﬁeldsthatvanish attheorigin); (c)incase(b)weonlyconsideredvolume-preservingtransformations,i.e.,we diminished g 0 as well: g 0 = ( sl(V)⊗C[ξ])⊃ + sl(Span(ξ)); g −1 =V⊗Cξ. Inparticular,since g −1 isisomorphictothetangentspaceatapointofthecurved supergrassmannian, we see that its even part in cases (a) – (c) is the same Gr 4 2 whilethetangentspacetothewholesupermanifoldatthe“origin”is Span(ξ i ∂ ∂θ j : 1≤i,j≤2). So the number of odd coordinates of our model varies from 4 in case (a) to2in cases(b) and(c). Table. In the ﬁrst line there are indicated the degrees, i.e., orders, of all nonzero structure functions and the rest of the table lists their the weights (with respect to L) (superscript denotes the multiplicity of the weight the subscript the degree of the corresponding structure function). The g 0 -action is nontrivial and glues distinct irreducible ( g 0 ) ¯0 -modules. (We did not show the action though we havecomputedit.) kievarwe.tex; 12/03/2001; 3:49; p.52 46 P. GROZMAN,D. LEITES Oddstructure functions Evenstructure functions −2−1 0 (11) (01) 2 (11) (13) (23) (03) (21)0 1 2 (00) 2 (10) (00) (02) (12) (02) (04) 2 (14) (04) (22) (32) (22) (24) (40) The( g 0 ) ¯0 -moduleswhosehighestweightsaregiveninthetablearegluedinto g 0 -modulesasfollows(an arrowindicatesasubmodule).The even tensors: (00) 2 0 −→ (02) 2 ; /arrowsoutheast (12) 1 /arrownortheast;(04) 0 −→ (04) 2 ; /arrowsoutheast (14) 1 /arrownortheast; (22) 0 −→(14) 1 −→(22) 2 ; (22) 0 −→(32) 1 −→(22) 2 ; (24) 0 −→(32) 1 ; (12) 1 −→(04) 2 ; (40) 0 −→(32) 1 ; (12) 1 −→(22) 2 . The odd tensors: (11) −2 −→(23) −1 ; (01) 2 −1 −→(11) 0 ; (13) −2 −→(23) −1 . 5. The Einstein equations . The conventional EE in vacum are the conditions on thetwo tensorsofdegree 2andweight (00)and(22),namely, R(22) = 0 and R(00) =λg, (5) whereλ∈Cis interepreted in terms of the cosmological constant and gis the metric preserved. For an analog of the Einstein equations on the curved supergrassmannian we may take the same vanishing conditions of the 2-nd order structure functions of weights (00)and(22)with respect to L. However, unlike the Einstein’s case, we have to vanish the constraints, the structure functions of lesser orders, both even and odd. The meaning of these analogs of Wess-Zumino constraints is unclear to us. References 1. Deligne P. et al (eds.) Quantum ﬁelds and strings: a course for mathematicians . Vol. 1, 2. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. AMS, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ,1999. Vol. 1: xxii+723 pp.;Vol. 2:pp. i–xxiv and 727–1501 kievarwe.tex; 12/03/2001; 3:49; p.53 ANUNCONVENTIONAL SUPERGRAVITY 47 2. Grozman P., Leites D., Mathematica -aided study of Lie algebras and their cohomology. From supergravity to ballbearings and magnetic hydrodynamics In: Ker¨anen V. (ed.) The second International Mathematica symposium , Rovaniemi, 1997, 185–192 3. Grozman P., Leites D., Supergravities and N-extended Minkowski superspaces for any N. In: Wess J., Ivanov E. (eds.) Supersymmetries and quantum symmetries . Proc. International Conference in memory of V. Ogievetsky, June 1997, Lecture Notes in Physics 524, Springer, 1999, 58–67 4. Leites D., Serganova V., Vinel G., Classical superspaces and related structures . In: Bartocci C. et al. (eds) Differential Geometric Methods in Theoretical Physics . Proc. DGM-XIX, 1990, Springer, LN Phys. 375, 1991, 286–297 5. Manin, Yu. Gauge ﬁeld theory and complex geometry , Springer-Verlag,Berlin, 1997. 6. SalamA., SezginE., Supergravities in diversedimensions , v.v.1, 2, WorldScientiﬁc, 1989 7. Sternberg S., Lectureson differential geometry , Chelsey,2nd edition, 1985 kievarwe.tex; 12/03/2001; 3:49; p.54 kievarwe.tex; 12/03/2001; 3:49; p.55 SUPERSYMMETRYOFRS BULK ANDBRANE ERICBERGSHOEFF InstituteforTheoreticalPhysics,Nijenborgh4,9747AGGroningen, TheNetherlands RENATA KALLOSH Department of Physics, Stanford University, Stanford, CA 94305, USA ANTOINE VANPROEYEN Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan200D B-3001Leuven,Belgium Abstract. We review the construction of actions with supersymmetry on spaces with a domain wall.Thelatterobjectsactassourcesinducingajumpinthegaugecouplingconstant.Despitethese singularities,supersymmetrycanbeformulated,maintainingitsroleasasquarerootoftranslations in this singular space. The setup is designed for the application in ﬁve dimensions related to the Randall–Sundrum (RS) scenario. The space has two domain walls. We discuss the solutions of the theory with ﬁxed scalars and full preserved supersymmetry, in which case one of the branes can be pushed to inﬁnity, and solutions where half of the supersymmetries are preserved. 1. Introduction Itisnotobvioushowsupersymmetrycanbeimplementedinaspacewithdomain walls. The wall is at a ﬁxed place and its presence seems to lead to a breaking of translations orthogonal to the plane. Supersymmetry, being the square root of translations, seems rather difﬁcult to realize in this context. It is interesting to see howthisobstacle hasbeen avoidedin [1],which wesummarize here. The work is mostly motivated by the Randall–Sundrum (RS) scenarios [2]. Thesimplestformofthesituationthatisunderinvestigationconsistsofa3-brane in a 5-dimensional bulk. The solution can be generalized e.g. to 8-branes in D= 10,butthefullimplementationof thatsituationisstill underinvestigation. WhentheRSscenariosappeared,supersymmetrisationwassooninvestigated. After initial attempts, it was found that no smooth supersymmetric RS single- branescenarioispossible[3].Thisscenariowithonebranewasputforwardasan alternative tocompactiﬁcation. kievarwe.tex; 12/03/2001; 3:49; p.56 50 E. BERGSHOEFF, R. KALLOSH, A. VAN PROEYEN Figure 1. Two-brane scenario. The ﬁfth dimension is a circle with branes at opposite ends and a Z 2 identiﬁcation ofpoints symmetric w.r.t. x 5 = 0. ThisleadustotheoriginalRSsetupwithtwobranes.The2-branescenariohas a compactiﬁed ﬁfth dimension, x 5 /similarequalx 5 + 2˜x 5 , with two branes ﬁxed at x 5 = 0 andx 5 = ˜x 5 .Thereismoreoveranorbifoldconditionrelatingpoints x 5 and−x 5 . Thus,theﬁve-dimensionalmanifoldhastheform M=M 4 × S 1 Z 2 .Thisissimilarto theHoˇrava–Witten [4]scenario.Thelatteroneembeds10-dimensionalmanifolds in an 11-dimensional space. They obtain the supersymmetry by a cancellation between anomalies of the bulk theory and a non-invariance of the classical brane action. Lukas, Ovrut, Stelle and Waldram [5] reduced this on a Calabi–Yau man- ifold to ﬁve dimensions, and further developed this setup in ﬁve dimensions. Further steps have been taken by [6–9]. In [7, 9] the gauge coupling constant does not change when crossing the branes, while in [6, 8] this coupling constant changes sign. In that respect, our approach is most close to the latter. In these papers, the action in the bulk is modiﬁed, such that it is not supersymmetric any morebyitself,butthenon-invarianceiscompensatedbythebraneactiontoobtain invarianceofthetotalaction.We[1]obtainseparateinvarianceofbulkandbrane action. The ﬁrst part of this report will treat the construction of the action with local supersymmetry on the singular space. In that part, we will show how the bulk and brane action are separately invariant under supersymmetry. The supersym- metry that we are considering is the one with 8 real components, i.e. minimal (N= 2) supersymmetry in 5 dimensions. The algebra is preserved despite the discontinuity. The second part treats background solutions. The Killing spinors are discussed. There are solutions with ﬁxed scalars and 8 Killing spinors, and solutions of 1/2supersymmetry, i.e. with 4 Killing spinors. Finally a summary is given,discussingopenissues. 2. Theactionforbulk andbrane The construction of the action involves three steps. First, we consider the bulk action.Thatistheactionofsupergravityin D= 5withmattercouplings.Aquite general action has been given in [10] based on the general methods developed in 4 dimensions in [11]. But it may not be excluded that further generalizations are kievarwe.tex; 12/03/2001; 3:49; p.57 SUPERSYMMETRY OF RSBULK 51 possible [12]. We will restrict ourselves to the couplings of vector multiplets, for which the general couplings were found in [13]. One can separate the ungauged part,andthepartdependentonagaugecouplingconstant g.Wewillconsideronly thegaugingofa U(1)R-symmetry group. In the second step, the gauge coupling constant gis replaced by a ﬁeld G(x). ALagrangemultiplierﬁeld,a (D−1)-form(4-formforourapplication),isintro- duced, whose ﬁeld equation imposes the constancy of G(x)such that effectively it isstillaconstant. The third step introduces the brane action. That action has extra terms for the Lagrangemultiplier (D−1)-form,whichallows G(x)tovarycrossingthebrane. We willshowhow every steppreservesthesupersymmetry! Before embarking on that programme, we want to repeat the fundamental algebraic relation between the cosmological constant and the gauge coupling constant ofR-symmetry. The super-anti-de Sitter algebra for N= 2inD= 5 isSU(2,2|1). It involves the anti-de Sitter algebra SO(4,2)/similarequalSU(2,2)with translationsP a andLorentzrotations M ab ,thesupersymmetries Q i ,withi= 1,2, a symplectic Majorana spinor, and a U(1)generator as R-symmetry. The most characteristic (anti)commutatorrelationsare /braceleftBig Q i ,Q j /bracerightBig = 1 2 ε ij γ a P a + igQ ij γ ab M ab + iε ij U, /bracketleftBig U,Q i /bracketrightBig =gQ ij Q j , [P a ,P b ] =g 2 Q ij Q ji M ab , /bracketleftBig P a ,Q i /bracketrightBig = iγ a gQ ij Q j . (1) Q ij satisﬁes Q ij =Q ji , Q ij ≡ε ik Q kj = i (q 1 σ 1 +q 2 σ 2 +q 3 σ 3 ), q 1 , q 2 ,q 3 ∈R, (q 1 ) 2 + (q 2 ) 2 + (q 3 ) 2 = 1. (2) This matrix determines the embedding of U(1)in the automorphism group of the supersymmetries SU(2). This choice is not physically relevant in itself. The second of the commutators in (1) implies that gis the coupling constant of R-symmetry. But the third equation says that g 2 determines the curvature of spacetime, i.e. it determines the cosmological constant. This fact is the corner- stone of the situation that we describe. The gauge coupling and the cosmological constant are related. However, one can change the coupling constant from +g to−g, not affecting the cosmological constant. That is what will happen going through the branes. This jump in the sign of gwill thus occur together with the action of theZ 2 . ThisZ 2 acts on the ﬁelds, which therefore live on an orbifold. Onecandistinguishoddandevenﬁelds.Thecircleconditionontheﬁeldsandthe kievarwe.tex; 12/03/2001; 3:49; p.58 52 E. BERGSHOEFF, R. KALLOSH, A. VAN PROEYEN orbifold condition are then Φ(x 5 ) = Φ(x 5 + 2˜x 5 ), Φ even (−x 5 ) = Φ even (x 5 ), Φ odd (−x 5 ) =−Φ odd (x 5 ).(3) These conditions imply that odd ﬁelds vanish on the branes: at x 5 = 0and at x 5 = ˜x 5 . Alsothesupersymmetriessplit.Halfofthemareeven,andhalfareodd.There- fore, on the brane one has 4 supersymmetries, i.e. N= 1in 4 dimensions. This splitting of the fermions requires a projection matrix in SU(2)space. Now the relativechoiceofthisprojectionmatrixand Qin(2)matters.Iftheyanticommute, the choice that has been taken in [7, 9], then gdoes not change when one crosses the brane. If they commute, as in [6, 8], then gjumps over the brane. And the latter iswhatwe willtakefurther. After these general remarks, we come to step 1. We thus consider the action ofsupergravitycoupled to nvectormultiplets [13].The ﬁeldsare e a µ , ψ i µ , A I µ , ϕ x , λ ix , (4) i.e. the graviton, gravitini, n+ 1gauge ﬁelds ( I= 0,1,... ,n), including the graviphoton, nscalars (x= 1,... ,n), andndoublets of spinors. The scalars describe a manifold structure that has been called very special geometry [14]. That geometry, and the complete action, is determined by a symmetric tensor C IJK .Thescalarsarebestdescribedaslivinginan n-dimensionalscalarmanifold embedded in an (n+ 1)-dimensional space. h I are the coordinates of this larger space. The submanifold is deﬁned by an embedding condition such that the h I as functions of the independentcoordinates ϕ x shouldsatisfy h I (ϕ)h J (ϕ)h K (ϕ)C IJK = 1. (5) The metric and all relevant quantities of this bulk theory is thus so far only dependenton C IJK . Then we add the gauging of a U(1)group. That means that we take a lin- ear combination of the vectors as gauge ﬁeld for this R-symmetry. The linear combination isdeﬁned byreal constants V I : A (R) µ ≡V I A I µ . (6) Theactionandthetransformationlawsarethenmodiﬁedbytermsthatalldepend ongQ ij . Instep 2, the coupling constant gis replaced by a coupling ﬁeld G(x). In the G¨unaydin–Sierra–Townsend (GST) action, the coupling constant appears up toterms ing 2 .We thusreplace S GST (g) =S 0 +gS 1 +g 2 S 2 ⇒S GST (G(x)) =S 0 +G(x)S 1 +G(x) 2 S 2 . (7) kievarwe.tex; 12/03/2001; 3:49; p.59 SUPERSYMMETRY OF RSBULK 53 Another term is added to the bulk action that forces G(x)to be a constant, using aLagrange-multiplier4-form A µνρσ : S bulk =S GST (G(x)) + /integraldisplay d 5 xe 1 4!ε µνρστ A µνρσ ∂ τ G(x) =S 0 − /integraldisplay d 5 xeV− /integraldisplay d 5 xeˆF(x)G(x) +fermionic terms. (8) Inthesecondline,thetermshavebeenreordered.Thepotential Voriginatesfrom S 2 in(7), andleadstothepotential V=−6G 2 /bracketleftBigg W 2 − 3 4 /parenleftbigg ∂ W ∂ϕ x /parenrightbigg 2 /bracketrightBigg , W≡ /radicalBig 2 3 h I V I , (9) where the linear combination Wappears, analogous to (6). The third term in (8) appearsfromintegratingbypartthetermwiththeLagrangemultiplier,leadingto theﬂux ˆF≡ 1 4! e −1 ε µνρστ ∂ µ A νρστ +covariantization. (10) The covariantization terms come from S 1 in (7). This method of describing a constant using a (D−1)-form is in fact an old method that was already used in[15]. It is easy to understand how supersymmetry is preserved. Indeed, the GST action is knownto be invariant: δ(/epsilon1)S GST (g) = 0. (11) Therefore,theonlynon-invariancefor S GST (G(x))appears,ifwedeﬁne δ(/epsilon1)G= 0, from the x-dependence of G(x). It is thus proportional to its spacetime derivative δ(/epsilon1)S GST (G(x)) =B µ ∂ µ G(x), (12) whereB µ issomeexpressionoftheotherﬁeldsandparameters,whoseexactform isnotimportantfortheargumenthere.Oneimmediatelyseesthenthatinvariance of(8)isobtainedby deﬁning thetransformationlaw of the 4-form as δ(/epsilon1) 1 4! ε µνρστ A µνρσ =B τ = e /bracketleftBig −i 3 2 ψ i µ γ µτ /epsilon1 j W −ψ i µ γ µτρ /epsilon1 j A (R) ρ + 3 2 λ i x W ,x γ τ /epsilon1 j /bracketrightBig Q ij ,(13) where we gave also the explicit form for our case. However, it is clear that the methodis alsovalid in othertheories. Step3introduces thebraneaction, suchthat thetotalaction is S new =S bulk +S brane . (14) kievarwe.tex; 12/03/2001; 3:49; p.60 54 E. BERGSHOEFF, R. KALLOSH, A. VAN PROEYEN Figure 2. The coupling constant gjumps atx 5 = 0and atx 5 = ˜x 5 . The braneaction has theform S brane =−2g /integraldisplay d 5 x /parenleftBig δ(x 5 )−δ(x 5 −˜x 5 ) /parenrightBig/parenleftBig e (4) 3W+ 1 4! ε µ ν ρ σ A µ ν ρ σ /parenrightBig =S brane, 1 −S brane, 2 . (15) Underlined indices refer to the values in the brane directions: µ= 0,1,2,3. The actionispresentedasanintegralover5dimensions,butthedeltafunctionsimply that it is a four-dimensional action for each brane separately. The action of each brane consists of a Dirac–Born–Infeld (DBI) term and a Wess–Zumino (WZ) term. However, both parts depend only on the pullback of the bulk ﬁelds to the branes.Therearenoﬁeldslivingonthebrane.Thefunction WappearsintheDBI term, and plays the role of the central charge of the brane. But most importantly, the4-formLagrangemultiplierappearsintheWZterm,andthisthusmodiﬁesits ﬁeldequation.The newﬁeld equation is ∂ 5 G(x 5 ) = 2g /parenleftBig δ(x 5 )−δ(x 5 −˜x 5 ) /parenrightBig , (16) and leads tothesolution (taking intoaccountthe cyclicity condition) G(x) =gε(x 5 ). (17) The function ε(x 5 )jumps as well at x 5 = 0as atx 5 = ˜x 5 , see ﬁgure 2. It is clear from this picture that we need the second brane. Indeed, one has to come back to theoriginalvalueof g,inorderthattotalderivativesin x 5 donotcontributetothe action.The ﬂux,whichisdetermined bythe ﬁeldequation of G(x), is ˆF= 12G /bracketleftBigg W 2 − 3 4 /parenleftbigg ∂ W ∂ϕ x /parenrightbigg 2 /bracketrightBigg +fermionic terms. (18) kievarwe.tex; 12/03/2001; 3:49; p.61 SUPERSYMMETRY OF RSBULK 55 The overall factor changes when crossing each brane due to (17). These jumps imply that the wallacts asa sinkfortheﬂuxes . That supersymmetry is still preserved by the addition of the brane is less obvious and is the non-trivial part of the construction. It turns out that the super- symmetryispreservedthankstotheprojections.Oneﬁnds(indices maretangent space indicesin branedirections) δS brane =−3g /integraltext d 5 x /parenleftbig δ(x 5 )−δ(x 5 −˜x 5 ) /parenrightbig e (4) /bracketleftBig W¯/epsilon1 i γ m e µ m /parenleftBig ψ µi −iγ 5 Q ij ψ j µ /parenrightBig + +W ,x ¯/epsilon1 i /parenleftBig iλ x i −γ 5 Q ij λ xj /parenrightBig/bracketrightBig .(19) The combinations of the gravitino and the gauginos that are in brackets are the components that are odd under the Z 2 projection, and thus vanish on the brane. Thisleadstotheinvariance.Remarkthatineachcaseoneofthetwotermscomes fromtheDBI(mass)termandtheotherfromtheWZ(charge)term.Thistherefore determinestherelativeweightofthetwoterms,andisthemass =chargerelation, that says that the brane is BPS. We thus see, indeed, that the brane action is separately invariant. Note, that if we would not use (or eliminate) the Lagrange multiplier, then this would relate bulk and brane, and only the sum would be invariant. 3. Thebackground:BPSsolutions We consider solutionswith awarpedmetric, i.e. ds 2 =a 2 (x 5 ) dx µ dx ν η µ ν + (dx 5 ) 2 . (20) The energydensity for solutionsthatdepend only on x 5 is E(x 5 ) =−6a 2 a /prime2 + 1 2 a 4 (ϕ x/prime ) 2 +a 4 V− 1 4! ε µνρσ 5 A µνρσ G /prime + + 2g /parenleftBig δ(x 5 )−δ(x 5 −˜x 5 ) /parenrightBig/parenleftBig 3a 4 W+ 1 4! ε µ ν ρ σ A µ ν ρ σ /parenrightBig ,(21) where the prime denotes a derivative w.r.t. x 5 . The ﬁrst three terms come from the GST action, the last one on the ﬁrst line from the term that we added with the Lagrange multiplier. The second line comes from the brane action. For this type ofbraneactions, onecan rewriteit usingsquaresandtotal derivatives: E= 1 2a 4 /braceleftbigg/bracketleftbig ϕ x/prime −3GW ,x /bracketrightbig 2 −12[a /prime a+GW] 2 /bracerightbigg + 3[a 4 GW] /prime + + /bracketleftBig 2g /parenleftBig δ(x 5 )−δ(x 5 −˜x 5 ) /parenrightBig −G /prime /bracketrightBig/parenleftBig 3a 4 W+ 1 4! ε µ ν ρ σ A µ ν ρ σ /parenrightBig .(22) The expression in square brackets in the second line is the ﬁeld equation of the Lagrange multiplier, and this line can thus be omitted. The last term of the ﬁrst kievarwe.tex; 12/03/2001; 3:49; p.62 56 E. BERGSHOEFF, R. KALLOSH, A. VAN PROEYEN line is a total derivative in x 5 and thus also does not contribute to the energy due to the continuity of the ﬁelds. The vanishing of the squared terms gives thus the minimum of the energy, and this minimum is even zero, as the zero energy of a closeduniverse.The BPSconditionsarethus ϕ x/prime = 3GW ,x ,a /prime a=−GW. (23) These equations are also called stabilization equations. These equations are im- portant to investigate the preserved supersymmetries. The transformations of the fermionsare δ(/epsilon1)λ x i =−i 1 2 γ 5 ϕ x/prime /epsilon1 i − 3 2 GQ ij W ,x /epsilon1 j , δ(/epsilon1)ψ µi =∂ µ /epsilon1 i + 1 2 δ m µ γ m /parenleftBig a /prime γ 5 /epsilon1 i + iaGQ ij W/epsilon1 j /parenrightBig , δ(/epsilon1)ψ 5i =/epsilon1 /prime i + 1 2 iGQ ij Wγ 5 /epsilon1 j . (24) Tosolve these,we splitthe supersymmetriesin theirevenand oddparts: /epsilon1 i =/epsilon1 + i +/epsilon1 − i , /epsilon1 ± i = 1 2 /parenleftBig /epsilon1 i ±iγ 5 Q ij /epsilon1 j /parenrightBig =±iγ 5 Q ij /epsilon1 ±j .(25) The vanishing of the last transformation of (24) determines the x 5 dependence of both parts. We have /epsilon1 ± i =a ±1/2 /epsilon1 ± i (x µ ). The transformations of the other components of the gravitino then determines the dependence on the other four spacetimevariables. Thisgives the generalsolution, /epsilon1 i =a 1/2 /epsilon1 +(0) i +a −1/2 /parenleftbigg 1−a /prime ax µ γ µ γ 5 /parenrightbigg /epsilon1 −(0) i , (26) asfunctionof /epsilon1 ±(0) i ,whichareconstantspinorswitheachonly4realcomponents. Thereremainsthetransformationsofthegaugino,whichlead to ϕ x/prime /epsilon1 −(0) i = 0. (27) This leaves two possibilities. The ﬁrst factor can be zero, which implies that we have constant scalars. In that case 8 Killing spinors survive. The other possibility allows non-constant scalars. Then the second factor should be zero, and this thus eliminates 4 supersymmetries. There remain 4 Killing spinors, /epsilon1 +(0) i , which are the4 that are non-vanishing also on thebrane. We consider both possibilities. First, let us look at the situation with ﬁxed scalars.TheBPS equationsarethen (ϕ y ) /prime = 0, /parenleftbigg ∂ W ∂ϕ x /parenrightbigg crit = 0,a /prime a=−gε(x 5 )W. (28) kievarwe.tex; 12/03/2001; 3:49; p.63 SUPERSYMMETRY OF RSBULK 57 The constancy of Wis translated by formulae of very special geometry in a ‘supersymmetricattractor equation’ C IJK ¯h J ¯h K =q I , ¯h K ≡ /radicalbig W crit h K , q I ≡ /radicalBig 2 3 V I .(29) This equation is well-known from black-hole physics [16]. A solution gives rise toa metric ofthe form ds 2 =e −2gW crit |x 5 | dx µ dx ν η µ ν + (dx 5 ) 2 ,ora=e −2gW crit |x 5 | .(30) In this case, the negative-tension brane can be pushed to inﬁnity. Indeed, there is no obstructionas anever vanishes. To consider supersymmetric domain walls with non-constant scalars , we use another coordinate, y,suchthat ∂ ∂x 5 =a 2 ∂ ∂y .The metric isthen ds 2 =a 2 (y)dx µ dx ν η µ ν +a −4 (y)dy 2 . (31) The stabilizationequationstaketheform a 2 d dyϕ x = 3G(y)W ,x , a d dya=−G(y)W. (32) Thesen+ 1equationsarecombined, usingrelations ofvery specialgeometry,to d dy(C IJK ˜h J ˜h K ) =−2G(y)q I where ˜h I ≡a(y)h I ,(33) whosesolutionsaregiven intermsofharmonicfunctions H I (y): C IJK ˜h J ˜h K =H I (y) =c I −2gq I |y|, (34) wherec I areintegrationconstants,while q I aretheconstantsthatwereintroduced inthegauging ( V I upto anormalization).Theyare harmonic inthe sensethat d dy d dyH I =−4gq I [δ(y)−δ(y−˜y)]. (35) The warpfactoris a 2 (y) =h I H I . (36) In this case the distance between the branes is restricted. There can be two types ofrestrictions: 1. There can be fundamental restrictions due to the origin of the functions h I . E.g. these are in various applications related to integrals over Calabi–Yau cycles.Theirvanishing canputa restriction onthedistance. kievarwe.tex; 12/03/2001; 3:49; p.64 58 E. BERGSHOEFF, R. KALLOSH, A. VAN PROEYEN 2. The vanishing of the harmonic functions also puts a restriction. Indeed, these harmonic functions enterinthewarpfactor, whichshouldbenon-vanishing. Ineachcase this restricts thedistancetobesmaller than acritical distance |˜y|<|y| sing . (37) 4. Summaryandoutlook The RS scenario in 5 dimensions can be made supersymmetric despite the singu- larities of the space. The action and transformation laws can be obtained using a 4-form,suchthatbulkandbraneareseparatelysupersymmetric.Supersymmetric solutionsexistwithﬁxed scalars or 1/2 supersymmetry. Half of the supersymmetries vanish on the branes. Also the translation gen- erator in the ﬁfth direction vanishes on the brane. That is how the algebra can be realized.Thesealgebraicaspectscouldstillbeclariﬁedfurther.Alsotheextension to hypermultiplets deserves further study. The same mechanism could be applied to study 8-branes in D= 10and other similar situations. It is furthermore an intriguing questionhow supersymmetricmatter can live on the branes. Acknowledgments. This work was supported by the European Commission RTN programme HPRN- CT-2000-00131,inwhichE.B.isassociatedwithUtrechtUniversity.Theworkof R.K.wassupportedbyNSF grantPHY-9870115. References 1. E. Bergshoeff, R. Kallosh and A. Van Proeyen, Supersymmetry in singular spaces , JHEP10 (2000)033 [hep-th/0007044]. 2. L.RandallandR.Sundrum, Alargemasshierarchyfromasmallextradimension ,Phys.Rev. Lett.83, 3370 (1999) [hep-ph/9905221]; An alternative to compactiﬁcation , Phys. Rev. Lett. 83, 4690(1999) [hep-th/9906064]. 3. R. Kallosh and A. Linde, Supersymmetry and the brane world , JHEP0002, 005 (2000) [hep- th/0001071]; K.BehrndtandM.Cveti ˇc,Anti-deSittervacuaofgaugedsupergravitieswith8supercharges , Phys. Rev. D61, 101901 (2000)[hep-th/0001159]. 4. P.Ho ˇravaandE.Witten, Eleven-dimensionalsupergravityonamanifoldwithboundary ,Nucl. Phys.B475(1996) 94[hep-th/9603142]. 5. A.Lukas,B.A.Ovrut,K.S.StelleandD.Waldram, Theuniverseasadomainwall ,Phys.Rev. 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Kallosh, Supersymmetry and attractors , Phys. Rev. D54, 1514 (1996) [hep- th/9602136]; Universality of supersymmetric attractors , Phys. Rev. D54, 1525 (1996) [hep- th/9603090]. kievarwe.tex; 12/03/2001; 3:49; p.66 kievarwe.tex; 12/03/2001; 3:49; p.67 D-BRANES AND VACUUMPERIODICITY DMITRIGALTSOV ∗ Laboratoire de Physique Th ´eorique LAPTH (CNRS), B.P.110, F- 74941Annecy-le-Vieuxcedex, France, and Department of Theoretical Physics, Moscow State University, 119899,Moscow,Russia VLADIMIR DYADICHEV † Department of Theoretical Physics, Moscow State University, 119899,Moscow,Russia Abstract. The superstring/M-theory suggests the Born-Infeld type modiﬁcation of the classical gaugeﬁeldlagrangian.Wediscusshowthischangestopologicalissuesrelatedtovacuumperiodic- ityintheSU(2)theoryinfourspacetimedimensions.Anewfeature,whichisduetothebreakingof scale invariance by the non-Abelian Born-Infeld (NBI) action, is that the potential barrier between the neighboring vacua is lowered to a ﬁnite height. At the top of the barrier one ﬁnds an inﬁnite family of sphaleron-like solutions mediating transitions between different topological sectors. We review these solutions for two versions of the NBI action: with the ordinary and symmetrized trace. Then we show the existence of sphaleron excitations of monopoles in the NBI theory with the triplet Higgs. Soliton solutions in the constant external Kalb-Ramond ﬁeld are also discussed whichcorrespondtomonopolesinthegaugetheoryonnon-commutativespace.Anon-perturbative monopole solutionfor thenon-commutative U(1)theory is presented. 1. Introduction Recent development in the superstring theory [1, 2] suggests that the low-energy dynamics of a Dp-brane moving in a ﬂat D-dimensional spacetime z M = z M (x µ ), M = 0,...,D−1, µ= 0,...pis governed by the Dirac-Born-Infeld (DBI)action S p = /integraldisplay/parenleftbigg 1− /radicalBig −det(g µν +F µν ) /parenrightbigg d p+1 x, (1) ∗ [email protected] † [email protected] kievarwe.tex; 12/03/2001; 3:49; p.68 62 D. GALTSOV,V.DYADICHEV where g µν =∂ µ z M ∂ ν z N η MN , (2) is an induced metric on the brane and F µν is aU(1)gauge ﬁeld strength. Using the gauge freedom under diffeomorphisms of the world-volume, one can choose coordinatesz M = (x µ , X m ), whereX m are transverse to the brane, and rewrite theactionas S p = /integraldisplay/parenleftbigg 1− /radicalBig −det(η µν +∂ µ X m ∂ ν X m +F µν ) /parenrightbigg d p+1 x. (3) A trivial solution to this action is X m = 0, F µν = 0, what means that the p- braneisﬂatandthereisnoelectromagneticﬁeld.Becauseofthesymmetry X m → −X m , the planar solution remains true when F µν does not vanish, in which case theelectromagneticﬁeldisgovernedbytheBorn-Infeld(BI)action.Moreover,as was noticed by Gibbons [3], the only regular static source-free solution of the BI electrodynamics which fallsoffat spatial inﬁnityis atrivialone. This is no longer true in the case of NcoincidentDp-branes whose low- energydynamicsisdescribedbythenon-AbeliangeneralizationoftheDBIaction involvingthe SU(N)Yang-Mills(YM)ﬁeld.Namely,forﬂat D3-branesthereg- ular sourceless ﬁnite energy conﬁgurations of the YM ﬁeld were found to exist [4,5].Thetopologicalreasonforthisliesinthevacuumperiodicityofthe SU(2) gaugeﬁeldinfourdimensions.NeighboringYMvacuaareseparatedbypotential barriers which in the case of the BI action are lowered down to a ﬁnite height due to the breaking of the scale invariance in the BI theory. This removes the well-known obstruction for classical glueballs [6–8], which can be summarized as follows. Scale invariance of the usual quadratic Yang-Mills action implies that the YM ﬁeld stress–energy tensor is traceless: T µ µ = 0 =−T 00 +T ii , where µ= 0,...,3, i= 1,2,3.Sincetheenergydensityispositive, T 00 >0,thesumof the principal pressures T ii is also everywhere positive, i.e.the Yang–Mills matter isrepulsive.Consequently,mechanicalequilibriumwithinthelocalizedstaticYM ﬁeld conﬁguration is impossible [9]. In the spontaneously broken gauge theories scale invariance is broken by scalar ﬁelds, what opens the possibility of particle- like solutions: magnetic monopoles (in the theory with the real triplet Higgs) and sphalerons(in thetheory with thecomplexdoubletHiggs). The role of the Higgs ﬁeld in these two cases is somewhat different. For monopoles the topological signiﬁcance of the Higgs ﬁeld is essential: indeed, monopolesinterpolatebetweentheunbrokenandbrokenHiggsphases.Inthecase of sphalerons, the Higgs ﬁeld plays mostly a role of an attractive agent which is able to glue the repulsive YM matter. Historically, topological signiﬁcance of the Dashen-Hasslacher-Neveu (DHN) solution in the SU(2)theory with the doublet Higgs [10] was ﬁrst explained by Manton [11] as a consequence of non–triviality of thethirdhomotopy group of the Higgs broken phase manifold π 3 (G/H ). This kievarwe.tex; 12/03/2001; 3:49; p.69 D-BRANES AND VACUUM PERIODICITY 63 isequivalenttoexistenceofnon-contractibleloopsinthespaceofﬁeldconﬁgura- tionspassingthroughthevacuum.Thenbytheminimaxargumentoneﬁndsthata saddlepointexistsontheenergysurfacewhichisaproperplaceforthesphaleron. Later it became clear that similar solutions arise in some models without Higgs, suchasEinstein-Yang-Mills[12]orYang-Millswithdilaton[13](forareviewand furtherreferencessee[14]).Themaincommonfeatureofthesetheoriesisthatthe conformal invariance of the classical YM equations is broken, what removes the ”mechanical” obstruction for existence of particle-like conﬁgurations. As far as thetopologicalargumentisconcerned,itisworthnotingthat H= 1fortheDHN solution, so the same third homotopy group argument applies to the gauge group Gitself, thatis, it works equally inthetheorieswithout Higgs. BreakingofthescaleinvarianceintheNBItheoryalsogivesrisetosphaleron glueballs which mediate transitions between different topological sectors of the theory. Their mass is related to the BI ﬁeld-strength parameter which for the D- branes is 2πα /prime . We will discuss here the difference between glueball solutions in twoversionsoftheNBItheory:withtheordinaryandsymmetrizedtrace.Wealso show that, when the triplet Higgs ﬁeld is added, the theory admits, apart from the usual magnetic monopoles, the hybrid solutions which can be interpreted as sphaleron excitations of monopoles. At the end we brieﬂy discuss monopole so- lutionsingaugetheoriesonnon-commutativespacesandgiveanexplicitsolution for theU(1)monopole with Higgs in the D-brane picture with the Kalb-Ramond ﬁeld. 2. NBIaction with ordinary andsymmetrized trace A precise deﬁnition of the NBI action was actively discussed during past few years [15–20], for an earlier discussion see [21]. An ambiguity is encoded in specifyingthetraceoperationoverthegaugegroupgenerators.Formallyanumber of possibilities can be envisaged. Starting with the determinant form of the U(1) Dirac-Born-Infeldaction S= 1 4π /integraldisplay/braceleftbigg 1− /radicalBig −det(g µν +F µν ) /bracerightbigg d 4 x, (4) one can use the usual trace, the symmetrized or antisymmetrized [15] ones, or evaluatethedeterminantbothwithrespecttoLorentzandthegaugematrixindices [19]. Alternatively one can start with the ’square root’ form, which is most easily derivedfrom(4)usingtheidentities det(g µν +F µν ) = det(g µν −F µν ) = det(g µν +i˜F µν ) = = det(g µν −i˜F µν ) = /bracketleftBig det(g µν −F 2 µν )(g µν +˜F 2 µν ) /bracketrightBig 1/4 ,(5) kievarwe.tex; 12/03/2001; 3:49; p.70 64 D. GALTSOV,V.DYADICHEV whereF 2 µν =F µα F αν (similarlyfor ˜F µν ),and F µα F αν −˜F µα ˜F αν = 1 2g µν F αβ F αβ , F µα ˜F αν =− 1 4g µν F αβ ˜F αβ . (6) This givesthe relation /radicalBig −det(g µν +F µν ) = /radicalBig −det(g) /radicalbigg 1 + 1 2F 2 − 1 16(F˜F) 2 ,(7) withF 2 =F µν F µν , F˜F=F µν ˜F µν . Foranon-Abeliangaugegrouptherelations(6)arenolongervalid,sothereis no direct connection between the ’determinant’ and the ’square root’ form of the lagrangian.Thereforethelattercanbechosenasanindependentstartingpointfor anon-Abelian generalization. There is, however, a particular trace operation – symmetrized trace – under which generators commute, so both forms of the lagrangian remain equivalent. This deﬁnition is favored by the no-derivative argument, as was clariﬁed by Tseytlin [15]. Restricting the validity of the non-Abelian effective action by the constant ﬁeld approximation, one has to drop commutators of the matrix- valuedF µν since these can be reexpressed through the derivatives of F µν . This corresponds to the following deﬁnition S= 1 4πStr /integraldisplay/braceleftbigg 1− /radicalBig −det(g µν +F µν ) /bracerightbigg d 4 x, (8) where symmetrization applies to the ﬁeld strength (not to potentials). This action reproduces an exact string theory result for non-Abelian ﬁelds up to α /prime2 order. Although there is no reason to believe that this will be true in higher orders in α /prime , theStraction is an interesting model providing minimal generalization of the Abelianaction[15]. AnexplicitformoftheSU(2)NBIactionwiththesymmetrizedtraceforstatic SO(3)-symmetricmagnetictypeconﬁgurationswasfoundonlyrecently[5].One startswiththedeﬁnition L NBI =β 2 4πStr /parenleftBigg 1− /radicalBigg −det /parenleftBig g µν + 1 βF µν /parenrightBig/parenrightBigg =kβ 2 4πStr(1−R),(9) where R= /radicalBigg 1 + 1 2β 2 F µν F µν − 1 16β 4 (F µν ˜F µν ) 2 , (10) kievarwe.tex; 12/03/2001; 3:49; p.71 D-BRANES AND VACUUM PERIODICITY 65 andβof the dimension of length −2 is the BI ’critical ﬁeld’. The normalization of thegaugegroup generators is unusual and ischosen as follows F µν =F a µν t a ,trt a t b =δ ab . (11) The symmetrizedtrace oftheproduct of pmatrices is deﬁned as Str(t a 1 ...t a p )≡ 1 p!tr /parenleftbig t a 1 ...t a p +all permutations /parenrightbig , (12) and it is understood that the general matrix function like (9) has to be series expanded. It has to be noted that under the Stroperation the generators can be treated as commuting objects, and the gauge algebra should not be applied, (e.g. the square of the Pauli matrix τ 2 x /negationslash= 1) until the symmetrization in the series expansionis completed. A generalSO(3)symmetricSU(2)gauge ﬁeld is described by the Witten’s ansatz √ 2A=a 0 t 1 dt+a 1 t 1 dr+{w 2 t 2 −(1−w)t 3 }dθ+ (13) {(1−w)t 2 + ˜wt 3 }sinθdφ, wherethefunctions a 0 , a 1 , w, ˜wdependonr,tand √ 2isintroducedtomaintain the standard normalization. Here we use a rotating basis t i , i= 1,2,3for the SU(2)generators deﬁnedas t 1 =n a τ a / √ 2, t 2 =∂ θ t 1 ,sinθt 3 =∂ ϕ t 1 , (14) wheren a = (sinθcosϕ,sinθsinϕ,cosθ), withτ a being the Pauli matrices. Thesegeneratorsobeythecommutationrelations [t i ,t j ] = 1 √ 2 /epsilon1 ijk t k . Fromfourfunctionsenteringthisansatzonecanbegaugedaway.Inthestatic case we can further reduce the number of independent functions to two, while the static purely magnetic conﬁgurations are fully described by a single function w(r): √ 2A θ =−(1−w)t 3 , √ 2A ϕ = sinθ(1−w)t 2 . A t =A r = 0.(15) The ﬁeldstrength tensorhas thefollowingnon-zerocomponents √ 2F rθ =w /prime t 3 , √ 2F rϕ =−sinθw /prime t 2 , √ 2F θϕ = sinθ(w 2 −1)t 1 ,(16) whereprime denotesderivativeswithrespect to r. For purely magnetic conﬁgurations the second term under the square root is zero,and the substitution of(16) gives R 2 = 1 +(1−w 2 ) 2 β 2 r 4 t 2 1 +w /prime 2 β 2 r 2 (t 2 2 +t 2 3 ). (17) kievarwe.tex; 12/03/2001; 3:49; p.72 66 D. GALTSOV,V.DYADICHEV To ﬁnd an explicit expression for the lagrangian one has to expand the square root in a triple series in terms of the even powers of generators t 1 ,t 2 ,t 3 , then to calculate the symmetrized trace of the powers of generators in all orders, and ﬁnallyto make aresummationof theseries.The resultreads L NBI =β 2 4π /parenleftBigg 1−1 +V 2 +K 2 A √ 1 +V 2 /parenrightBigg , (18) where V 2 =(1−w 2 (r)) 2 2β 2 r 4 , K 2 =w /prime2 (r ) 2β 2 r 2 , A= /radicalBigg 1 +V 2 V 2 −K 2 arctanh /radicalBigg V 2 −K 2 1 +V 2 . (19) Here we assumed that V 2 > K 2 , otherwise an arctanform is more appropriate. Note that when the difference V 2 =K 2 changes sign, the k function Aremains real valued. It can be checked that when β→∞, the standard Yang-Mills la- grangian (restricted to monopole ansatz) is recovered. In the strong ﬁeld region ourexpression differsessentially fromthesquareroot/ordinary trace lagrangian. The corresponding explicit action deﬁned in a square root form with an ordinarytrace reads: L NBI =β 2 4π /parenleftBig 1− /radicalbig 1 +V 2 + 2K 2 /parenrightBig (20) 3. Topologicalvacuaand sphalerons As is well-known, vacuum in the SU(2)YM theory in the four-dimensional spacetime splits into an inﬁnite number of disjoint classes which can not be deformed into each other by ’small’ (contractible to a point) gauge transforma- tions. Writing the pure gauge vacuum YM potentials as A=iUdU −1 , where U∈SU(2)andimposing anasymptoticcondition lim r→∞ U(x i ) =1, (21) we can interpret U(x i )as mappings S 3 →SU(2). All sets of such U’s falls into thesequenceof homotopyclassescharacterized bythewinding number k[U] = 1 24π 2 tr /integraldisplay R 3 UdU −1 ∧UdU −1 ∧UdU −1 . (22) kievarwe.tex; 12/03/2001; 3:49; p.73 D-BRANES AND VACUUM PERIODICITY 67 Arepresentativeof the k-thclass can be chosenas U k = exp{iα(r)t 1 / √ 2},whereα(0) = 0,α(∞) =−2πk.(23) The corresponding potential will be given by the Witten ansatz with a= 0,w=exp(iα(r)). The asymptotic condition (21) leads to the following fall-off requirements. A a =o(r −1 )forr→∞. (24) The representatives of different vacuum classes with different kcannot be continuously deformed into each other within the class of the purely vacuum ﬁelds. But there exists an interpolating sequence of nonvacuum ﬁeld conﬁgu- rations of ﬁnite energy (the latter can be deﬁned on shell and then continued off-shell) satisfying the required boundary conditions (24) that connects different vacuumclasses.Finiteenergysolutionsfortheactions(18)or(20)shouldsatisfy thefollowingboundaryconditionsnearthe origin w= 1 +br 2 +O(r 4 ), (25) and atthe inﬁnity w=±1 + c r+O( 1 r 2 ), (26) wherebandcare free parameters. (The value w(∞) = 0together with ﬁniteness of the energy implies that w≡0.) The leading terms are the same as required for the vacuum conﬁgurations. These solutions, if exists, can be shown to lie on the pathinthesolutionspaceconnectingtwotopologicallydistinctvacua.Considera one-parametersequenceofﬁeldconﬁgurations(offshellgenerally)dependingon acontinuousparameter λ∈[0, π][22] A[λ] =i1− w 2U + dU −1 + +i1 + w 2U − dU −1 − , (27) where U ± = exp /braceleftBig iλ(w±1)t 1 / √ 2 /bracerightBig . (28) This ﬁeld vanishesfor λ= 0,whereas for λ=πit can berepresented as A[π] =iUdU −1 ,with U = exp{iπ(w−1)t 1 / √ 2}.(29) In view of the above boundary condition for w, in the case w(∞) =−1one has thek= 1vacuum. Now, the crucial thing is that for λ=π/2we come back to the conﬁguration (15). So if the solution to the classical ﬁeld equations with the kievarwe.tex; 12/03/2001; 3:49; p.74 68 D. GALTSOV,V.DYADICHEV required asymptotics exists indeed, this can be interpreted as a manifestation of theﬁniteness ofthe potential barrierbetweendistinctvacua. Note that the same reasoning holds for the ordinary Yang–Mills system. But due to the scale invariance of this theory there is no function wwhich minimizes theenergy functional. Both the analysis of the equations following from NBI lagrangians (18,20) usingthemethodsofdynamicalsystems[4]andnumericalexperiments[5]shows that such solutions exist in both NBI models — with ordinary and symmetrized trace. They form a discrete sequence labeled by the number of nodes of the functionw(r), and the lower one-node solution is similar to the sphaleron of the Weinberg-Salamtheory. In the NBI theory βis the only dimensionful parameter giving a natural scale of length, i.e. theories with different values of βare equivalent up to rescaling. Settingβ= 1we obtain the equations of motion for the symmetrized trace NBI model d dr /braceleftBigg w /prime 2(V 2 −K 2 ) /parenleftBigg K 2 √ 1 +V 2 1 +K 2 −(2V 2 −K 2 ) A √ V 2 −K 2 /parenrightBigg/bracerightBigg (30) =wV(K 2 A−V 2 ) (V 2 −K 2 ) √ 1 +V 2 . Forthe ordinary tracemodel one has d dr /braceleftbigg w /prime √ 1 +V 2 + 2K 2 /bracerightbigg =−w V √ 1 +V 2 + 2K 2 , (31) We are looking for the solutions satisfying the boundary conditions (25,26). For largerboth equations reduce to that of the usual YM theory, so the solutions are not much different in the far zone. Near the origin the equations are different, more careful analysis reveals that the nature of stationary points associated with theoriginisdifferentfortwo versionsof thetheory. A trivial solution to these equations (valid for both models) is an embedded abelian monopole w= 0. In the BI theory it has the ﬁnite energy. From the general analysis, as discussed in [14] for the ordinary trace, one ﬁnds that wcan not have local minima for 0< w < 1, w <−1and can not have local maxima for−1<w< 0, w> 1. The same remains true for the symmetrized trace. Thus any solution which starts at the origin on the interval −1< w < 1must remain within the strip−1< w < 1. Oncewleaves the strip, it diverges in a ﬁnite distance. Regular solutions exist for a discrete sequence of bshown in the table I togetherwithcorrespondingmasses M n fortheﬁrstsix nwhichisthenumberof zeroes ofw(r). Then= 1solution is analogous to the sphaleron known in the Weinberg-Salam theory [10, 11], it is expected to have one decay mode. Higher odd-nsolutions may be interpreted as excited sphalerons, they are expected to kievarwe.tex; 12/03/2001; 3:49; p.75 D-BRANES AND VACUUM PERIODICITY 69 –1–0.500.51 w(t) –6 –4 –2 0 2 4 6 8 t=ln(r) Figure 3. Sphaleron glueball solutions w n forn= 1,2,3in the symmetrized trace (solid line) and ordinary trace (dashed line) models havendecay directions. Even- nsolutions are topologically trivial, they can be regardedassphaleronicexcitationofthevacuum.Qualitativelypictureisthesame as fortheordinary trace[4], butthe discretevaluesof barerather different. Numericalsolutionsforbothmodelsareshownintheﬁgure3.Itissurprising that the solutions with the ordinary and the symmetrized trace are rather similar inspiteofthesubstantialdifferenceofthelagrangians.Theyhavehoweversome- what different behavior near the origin: those with the symmetrized trace leave the vacuum value w= 1faster and stay longer in the intermediate region where w(r)is close to zero. In this region the magnetic charge is almost unscreened, so thisistheparticlecore.Thusforall nsolutionsaremorecompactintheordinary tracecase.Forbothmodelstheparameters b n growinﬁnitelywithincreasingnode numbern.Thismeansthatthereisnolimitingsolutionas n→∞contrarytothe EYM casewhere such solutionsdo exist. 4. Magnetic monopolesand hybrid solutions Magnetic monopoles are associated with the deformed D3-branes with non zero transverse coordinates X m interpreted as Higgs scalars. The deformation can be thought of as caused by an open string attached to the brane. In the BPS limit the solutions are the same as for the quadratic YM theory [17, 18] Monopoles fortheordinarytracemodelwereconstructedbyGrandi,MorenoandSchaposnik kievarwe.tex; 12/03/2001; 3:49; p.76 70 D. GALTSOV,V.DYADICHEV TABLE I. Values of bandMfor ﬁrst six glueball solutions in NBI models with ordinary and symmetrized traces Ordinary trace Symmetrized trace n b tr M tr b Str M S tr 1 1.27463×10 1 1.13559 1 .23736×10 2 1.20240 2 8.87397×10 2 1.21424 5 .05665×10 3 1.234583 3 1.87079×10 4 1.23281 1 .67739×10 5 1.235979 4 1.27455×10 6 1.23572 7 .11885×10 6 1.236046 5 2.65030×10 7 1.23603 4 .94499×10 8 1.2360497 6 1.80475×10 9 1.23604 4 .52769×10 10 1. 2360497 [23]. For monopoles the function wmonotoneously varies from the value w= 1 at the origin to the asymptotic value w= 0at inﬁnity. Note, that assuming the asymptotic value w= 0for pure gauge NBI theory we will get only embedded abeliansolution w≡0.Ouraimhereistoshowthat,inaddition,therearehybrid NBI-Higgs solutions for which the function w(r)oscillates in the core region. In otherwords,startingfromthevacuum w= 1attheoriginthefunction w(r)tries tofollowthesphaleronicbehavior,butﬁnallyturnsbacktothemonopoleregime. AddingtotheNBIactiontheHiggsterm S=S NBI +S H whereS H istaken intheusualform S H = 1 8π /integraldisplay/parenleftbigg D µ φ a D µ φ a − λ 2 /parenleftBig φ a φ a −v 2 /parenrightBig/parenrightbigg , (32) oneobtainstheNBI-Higgstheory,containing,apartfrom β,thesecondparameter λ(without loss of generality we put the gauge coupling constant equal to unity). For spherically symmetric static purely magnetic conﬁgurations the YM ansatz remains thesame , whilefor theHiggs ﬁeld φ a =H(r ) rn a . (33) For simplicity we consider here the square root form of the NBI action (20). Performinganintegrationoversphericalanglesoneobtainstheenergyfunctional (equal to minusaction for staticconﬁgurations) E= 4π /integraldisplay drr 2 /braceleftbigg 2β 2 (R−1) + 1 2r 2 /parenleftbigg (H /prime − H r) 2 + 2 r 2 H 2 w 2 /parenrightbigg +V /bracerightbigg ,(34) kievarwe.tex; 12/03/2001; 3:49; p.77 D-BRANES AND VACUUM PERIODICITY 71 where R= /radicalBigg 1 + 1 β 2 r 4 (r 2 w /prime2 + 1 2(w 2 −1) 2 ), V = λ 4 /parenleftBigg H 2 r 2 −1 /parenrightBigg 2 .(35) Varyingthisfunctionalone ﬁndstheequationsof motion r 2 w /prime/prime =w(RH 2 +w 2 −1) +r 2 R /prime Rw /prime , (36) r 2 H /prime/prime = 2Hw 2 −λH(r 2 −H 2 ). (37) Boundaryconditionsatinﬁnityforasolutionwithaunitmagneticchargeread lim r→∞ w(r) = 0, lim r→∞ H(r ) r= 1, (38) whileat theorigin w(0) = 1, H (0) = 0. (39) Starting with (39) one can construct the following power series solution converginginanon-zero domain aroundtheorigin: w= 1−br 2 +βb 2 /parenleftbig 22b 2 +β 2 /parenrightbig +d 2 /parenleftbig 6b 2 +β 2 /parenrightbig 3 2 10β(2b 2 +β 2 )r 4 +O(r 6 )(40) H=dr 2 − /parenleftbigg 1 10λd+ 2 5db /parenrightbigg r 4 +O(r 6 ),(41) wherebanddarefreeparameters.For β→∞thetheoryreducestothestandard YMH-theory,admittingmonopoles.In[23]itwasshownthatmonopolesolutions totheEqs.(36,37) continue toexistupto somelimitingvalue β cr . Nowwehavetoexplainwhyonecanexpecttohavealsothehybridsolutions. NeartheorigintheHiggsﬁeldisclosetozero,sotheinﬂuenceoftheterm H 2 KR is negligible, and the YM ﬁeld behaves like in the pure NBI case. As was argued in [4], NBI theories with different βare equivalent up to rescaling, and so for β large enough the solution starts forming just near the origin. But for larger rthe role of Higgs is increased, so one can expect that some solutions can be trapped to the monopole asymptotic regime. More precisely, in the region of r≈1/ √β, thefunction w(r)issimilartothesphaleronsolutionof[4]:startingwith w= 1it passesthrough w= 0andthentendstothevalue w= 1.Afterleavingthisregion the solution enters the region where it has properties of the NBI monopole and atr→∞both ﬁeld functions tend to their asymptotical values (38). The Higgs ﬁeldH(r)forthesehybridsolutionsbehavesqualitativelyinthesamewayasfor themonopoles. kievarwe.tex; 12/03/2001; 3:49; p.78 72 D. GALTSOV,V.DYADICHEV N=0 N=2 N=1 –0.200.20.40.60.81 –10 –8 –6 –4 –2 0 2 t=ln(r) Figure 4. Magnetic monopole and ﬁrst two hybrid solutions in the ordinary trace model for β= 30,λ= 1/2. Solid line — w, dashed line— H/r To obtain hybrid solutions numerically we introduce the logarithmic variable t= ln(r)and apply a shooting strategy to ﬁnd the values of parameters bandd ensuringthemonopoleasymptoticconditions(40-41)afterseveraloscillationsof w. As an initial guess for bone can take the (appropriately rescaled for given β) glueballvaluesfoundin[4].Anotherparameter dturnsouttobeweaklysensitive onβforβlarge enough. The resulting solutions for n= 1,2andλ= 1/2are shown on Fig. 1,2 together with the ground state monopole ( n= 0). The masses increasewith nandconvergerapidlytothemassofanembeddedAbeliansolution withfrozenHiggs: w(r)≡0, H (r)≡r. (42) Although this singular solution does not satisfy the boundary conditions (39) it has ﬁnite energy within the NBI-Higgs theory, which can be obtained by substituting theEq. (42)into theEq.(34): E lim = 2 /integraldisplay β 2 (R−1)r 2 dr= /radicalbig β /integraldisplay/parenleftBigg /radicalbigg 4 + 2 x 4 −2 /parenrightBigg x 2 dx= 1.467338 /radicalbig β. (43) With decreasing β, the discrete values of the parameter b n also decrease until relativelysmallvaluesof β.Then,with βfurtherdecreasingbothparameters band dstartgrowinguntilsomecriticalvalueof β crn isreachednearwhichparameters kievarwe.tex; 12/03/2001; 3:49; p.79 D-BRANES AND VACUUM PERIODICITY 73 b n andd n tend to inﬁnity and monopole solutions with given number of zeroes cease to exist. The lowest of these critical values is β cr0 ≈0.45for unexcited monopole solution. The excited solutions disappear at greater values of β. The mass of excited monopoles is well described by the formula (43), even for the lowest excited solution the difference with the exact numerical value is less then 4% for all values of β. The ﬁgure 4 shows the behavior of functions w, Hfor some intermediate value of β. Note, that at critical βall branches of monopole solutions(includingunexcitedbranches)convergetothelimitingAbeliansolution (42) (withdifferentrate). The excited monopole solutions also exist in the Einstein-Yang-Mills-Higgs theory [24]. There the role of non-linear excitations is played by Bartnick- McKinnon gravitating sphalerons of EYM theory [12]. The phase diagram (regions of existence in parameter space) is somewhat different in our case, the detailswill begivenelsewhere. 5. Non-commutativemonopoles HerewediscussanotheraspectoftheD-branepictureofgaugetheories,whichis the direct subject of the present workshop. Recently it was discovered that gauge theories onnoncommutativemanifolds [x µ ,x ν ] =iθ µν (44) are connected with the gauge theories on D-branes with the constant background Kalb-Ramondﬁeld Bturnedon [25] B µν =−θ µν (2πα /prime ) 2 . (45) Therelationbetweenthesetwoversionsisnon-localandisdeﬁnedperturbatively through the Seiberg-Witten map [26] (for a more recent discussion see [27, 28]). Namely,theYMtheoryon anoncommutativefour-dimensionalspace ˆS= Tr /integraldisplay/parenleftbigg 1 4ˆg 2 ˆF µν ∗ˆF µν +... /parenrightbigg d 4 x, (46) deﬁned usingthe star-product F(x)∗G(x) = exp /parenleftbigg iθ µν 2∂ µ ∂ /prime ν /parenrightbigg F(x)G(x /prime )| x /prime =x , (47) and theD-brane theory with A µ , F µν are relatedperturbatively via ˆA µ =A µ −θ αβ 4{A α , ∂ β A µ +F βµ } + +O(θ 2 ). (48) kievarwe.tex; 12/03/2001; 3:49; p.80 74 D. GALTSOV,V.DYADICHEV The issue of magnetic monopoles in both treatments of the non-commutative YM was discussed recently in a number of papers [29–32, 30]. It was argued that BPS-saturated monopoles exist in the non-commutative case as well. Apart from the BPS bound most of the previous discussion was perturbative in terms of the non-commutativityparameter θ µν . Adding the constant B-ﬁeld spoils the spherical symmetry of monopoles and therefore their non-perturbative treatment in the D-brane picture becomes rather complicated. At best one can construct an axially symmetric model using B µν as aKalb-Ramondanalogofthehomogeneousmagnetic(electric)ﬁeld.Eveninthis case the NBI model is still too complicated both for Tr and Str versions. Here we giveanon-perturbativemonopolesolutioninthesimplestcaseofthe U(1)gauge ﬁeld with Abelian Higgs. As was shown by Gibbons [3], the system of BI U(1) andHiggsﬁeldspossessestheboostsymmetry(inthemixedspaceofcoordinates andtheﬁeldvariables)whichcanbeusedasasolutiongeneratingtechniquetoadd a constant magnetic ﬁeld to the pointlike magnetic monopole (resp. electric ﬁeld to the electric BIon). Reinterpreted as the Kalb-Ramond ﬁeld, this homogeneous ﬁeldmaybe accountedfortheparameterofnon-commutativity. WestartwiththeDBI action S DBI =− /integraldisplay d 4 x /radicalBig −det (η µν +∂ µ y∂ ν y+F µν ) (49) with one external coordinate y(playing the role of the Higgs ﬁeld) and introduce themagneticpotential χ H=−∇χ, (50) where His the magnetic ﬁeld strength — canonical conjugate to the magnetic induction B: H=−∂ L ∂B. (51) Performing the corresponding Legendre transformation we obtain the follow- inghamiltonian functional H= /integraldisplay d 3 x /radicalBig 1−(∇χ) 2 + (∇y) 2 + (∇χ) 2 (∇y) 2 −(∇χ·∇y) 2 ,(52) which can be interpreted as the volume of the three-dimensional hypersurface parametrized by coordinates x i in the ﬁve-dimensional pseudoeuclidean space {x i ,y,χ}with the metric diag(+,+,+,+,−)(minus corresponds to χ). We use the symmetries of this functional to generate ﬁrst the scalar charge from the monopole charge and then to generate a constant background ﬁeld which will be then interpreted as the Bﬁeld. So we start with the spherically symmetric conﬁgurations.The ﬁeldequationsare then reducedto y /prime/prime = 2y /prime /parenleftbig χ /prime2 −y /prime2 −1 /parenrightbig r, χ /prime/prime = 2χ /prime /parenleftbig χ /prime2 −y /prime2 −1 /parenrightbig r,(53) kievarwe.tex; 12/03/2001; 3:49; p.81 D-BRANES AND VACUUM PERIODICITY 75 where prime denotes the derivative with respect to the radial variable r. It is easy to see that two potentials should be proportional. Depending on which potential dominates,onecanﬁnd three different typesof behaviour: 1. The spacelike vector in the {y,χ}plane. By some rotation the magnetic ﬁeld canberemoved.Thisisthecatenoidalsolution[3].Sinceitdoesnotexistfor allr, wewillnotconsider it further. 2. The timelike vector in the {y,χ}plane. By a rotation it can be reduced to a U(1)monopolewithoutexcitationsofthetransversedegreesoffreedom.The potential for thisparticularsolution(withunit charge) is χ 0 (r) = r /integraldisplay dr √ 1 +r 4 , (54) andcould bewritten explicitlyintermsofelliptic integrals. 3. The lightlikevector y=±χ.Thisis the BPSmonopole: χ BPS (r) =±y BPS (r) = 1 r. (55) To obtain the non-BPS monopole solution that also has a nonzero Higgs counterparty(r)one can simply perform aboostin the {χ,y}plane: χ(r) = coshψχ 0 (r)y(r) = sinhψχ 0 (r). (56) The next step is to perform a boost in the {χ,z}plane to generate the constant background magnetic ﬁeld. To understand why this ﬁeld may be equally inter- preted as aBﬁeld one should notice that the ﬁeld equations do not change if we replaceF µν byF µν +B µν with constant B. So,ifwedenote χ=g(ρ,z),thenafterthesecond boost we obtain: coshφg+ sinhφz= coshψχ 0 /parenleftbigg /radicalBig ρ 2 + (coshφz+ sinhφg) 2 /parenrightbigg ,(57) whereρ= /radicalbig x 2 +y 2 andχ 0 is deﬁnedby theEq.(54). This nonlinear equation cannot be solved explicitly but it is simple to explore it numerically. The key point is to note that for a given g,ρ,z, using equations (54),(57), one can ﬁnd the vector F+B(magnetic induction plusB-ﬁeld). Then the monopole ﬁeld is obtained by subtracting the constant background. Note that, depending on the values of the boosts parameters φandψ, the so- lution can become double-valued. Let us consider this feature in more detail. For magnetic monopole without excitations of the transversal component the three-dimensionalhypersurface χ 0 (x,y,z )isspacelikeeverywhereexceptforthe originwhereittouchesthelightcone.Whenweboostinthe {χ,y}directions,the surfaceχ(x,y,z )acquires the timelike piece which can cause multivaluedness kievarwe.tex; 12/03/2001; 3:49; p.82 76 D. GALTSOV,V.DYADICHEV Figure5. Non-commutative U(1)monopole: constant |F|curves Figure 6. Non-commutative U(1)monopole: constant ycurves after boosting in the {χ,z}directions. (When treated as a hypersurface in the ﬁve-dimensional space {r,χ,y}it remains of course spacelike). This effect is interpreted from the string theory point of view as tilting the D-brane, but from the point of view of 3-dimensional ﬁeld theory this multivaluedness should be interpreted as a signal that no well deﬁned solution exists. It is worth noting that for BPS solution such multivaluedness emerges for any value of the background ﬁeld. In the ﬁgures 5,6 the sections of level surfaces of constant yand constant|B| are shown. The full solution is axially symmetric and is obtained by rotating the pictures alongthe symmetryaxis. kievarwe.tex; 12/03/2001; 3:49; p.83 D-BRANES AND VACUUM PERIODICITY 77 6. Discussion WehavediscussedsomenewissuesassociatedwiththeD-branepictureofgauge theories.Apartfromgivinganicegeometricframework,D-branessuggestamodi- ﬁcationofdynamicsoftheYMﬁeldintroducingtheBorn-Infeldtypelagrangian. This latter breaks the conformal invariance of the YM equations removing the obstruction for existence of classical glueballs in the SU(2) theory in four di- mensions. Topological reason for existence of such glueballs lies in the vacuum periodicitywhichholdsequallyintheordinaryYMtheoryandintheNBItheory, with an important difference that in the latter case the potential barriers between neighboring vacua have ﬁnite heights. Classical NBI glueballs (more precisely, halfofthem)aresphaleronsmediatingthetopologicaltransitions.Wehavefound thattheyexistbothfortheordinarytraceandthesymmetrizedtraceversionsofthe NBItheorywithsomewhatdifferentcorestructure.Wehavealsoshownthatinthe NBI theory with the triplet Higgs one encounters, apart from the usual magnetic monopoles,thehybridsolutionswhichcanberegardedassphaleronicexcitations of monopoles. Finally, adding the constant Kalb-Ramond ﬁeld, one is able to account for non-commutative monopoles. We presented a new nonperturbative axisymmetric solutionfortheU(1) non-commutative monopole withHiggs. Acknowledgements Oneoftheauthors(DG)isgratefultotheorganizersoftheWorkshopforinvitation and support and especially to Steven and Diana Duplij for a stimulating atmo- sphereduringthismeeting.HeisalsogratefultoLAPTH(Annecy)forhospitality while the ﬁnal version of this paper was completed. This work was supported in partbythe RFBR grant00-02-16306. References 1. 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S.Moriyama, Noncommutativemonopolefromnonlinearmonopole , Phys.Lett. B485(2000), 278–284, hep-th/0003231 . kievarwe.tex; 12/03/2001; 3:49; p.85 QUANTUMDEFORMATIONSOF SPACE-TIMESUSY AND NONCOMMUTATIVESUPERFIELD THEORY P.KOSI´NSKI Instituteof Physics,University ofL ´od´z, ul.Pomorska149/5390–236 L ´od´z,Poland JERZYLUKIERSKI ∗ Instituteof TheoreticalPhysics, University of Wroclaw pl.M. Borna9, 50-205Wrocław,Poland P.MA´SLANKA Instituteof Physics,University ofL ´od´z, ul.Pomorska149/5390–236 L ´od´z,Poland Abstract. We review shortly present status of quantum deformations of Poincar ´e and conformal supersymmetries.Afterrecallingthe κ–deformationofD=4Poincar ´esupersymmetrieswedescribe the corresponding star product multiplication for chiral superﬁelds. In order to describe the de- formation of chiral vertices in momentum space the integration formula over κ–deformed chiral superspace isproposed. 1. Introduction The noncommutative space–time coordinates were introduced as describing al- gebraically the quantum gravity corrections to commutative ﬂat (Minkowski) background (see e.g. [1, 2]) as well as the modiﬁcation of D–brane coordinates in the presence of external background tensor ﬁelds (e.g. B µν inD= 10string theory;see[3]–[5]).Weknowwellthatbothgravityandstringtheoryhavebetter properties (e.g. less divergent quantum perturbative expansions) after their super- symmetrization.Itappearsthereforereasonable,ifnotcompelling,toconsiderthe supersymmetric extensionsofthe noncommutativeframework. The generic relationfor the noncommutative space–timegenerators /hatwide x µ [ /hatwide x µ , /hatwide x ν ] =iΘ µν ( /hatwide x) =i /parenleftBig Θ µν + Θ ρ µν /hatwide x ρ +... /parenrightBig (1) ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.86 80 P.KOSI´NSKI, J.LUKIERSKI,P. MA ´SLANKA has been usually considered for constant value of the commutator (1), i.e. for Θ µν ( /hatwide x) = Θ µν . In such a case the multiplication of the ﬁelds φ k ( /hatwide x)de- pending on the noncommutative (Minkowski) space–time coordinates can be represented by noncommutative Moyal ∗–product of classical ﬁelds φ k (x)on standard Minkowskispace φ k ( /hatwide x)φ l ( /hatwide x)←→φ k (x)∗φ l (x) =φ k (y)e i 2 Θ µν ∂ ∂yµ ∂ yν φ l (z)| x=y (2) It appears that the relation (1) with constant Θ µν can be consistently supersym- metrized (see e.g. [6]–[9]) by supplementing the standard relations for the odd Grassmannsuperspacecoordinates(furtherwechoose D= 4N= 1SUSYand α,β= 1,2). {θ α ,θ β }={θ α ,θ ˙β }= {θ ˙α ,θ ˙β }= 0 [ /hatwide x µ ,θ α ] = [ /hatwide x µ ,θ ˙α ] = 0 (3) Such a choice of superspace coordinates ( /hatwide x µ ,θ α ,θ ˙α ) implies that the supersym- metrytransformationsremainclassical: /hatwide x /prime µ = /hatwide x µ −i /parenleftBig /epsilon1σ k θ α −θσ k /epsilon1 /parenrightBig θ /prime α =θ α +/epsilon1 α θ /prime ˙α =θ ˙α +/epsilon1 ˙α (4) i.e.thecovariancerequirementsofdeformedsuperspaceformalismdonotrequire thedeformationof classicalPoincar ´e supersymmetries 1 . OuraimhereistoconsiderthecasewhenthestandardPoincar ´esupersymme- tries can not be preserved. For this purpose we shall consider the case with linear Lie–algebraic commutator (1). Its supersymmetrization leads to the deformed superspace coordinates /hatwide z A = ( /hatwide x µ , /hatwide θ α , /hatwide θ ˙β )satisfyingLie superalgebra relation: [ /hatwide z A , /hatwide z B ] =iΘ C AB /hatwide z C (5) where Θ C AB satisﬁes graded Jacobiidentity: Θ D AB Θ E CD +gradedcycl. (A,B,C ) = 0 (6) It appears that in such a case for some choices of the “structure constants” Θ C AB one can ﬁnd the deformed quantum D= 4Poincar´e supergroup, which provide the relations (5) as describing the deformed translations and deformed supertranslations. 1 It should be stressted, however, that the introduction of constant tensor Θ µν in (1) leads to breaking (O(3,1)→O(2)×O(1,1))) ofD= 4Lorentz symmetry. The way out is to consider Θ µν asaconstantﬁeld,withgeneratorofLorentzsubalgebracontainingcontributionwhichrotates theΘ µν components(seee.g.[10]).Therelation(1)canbemadecovariantonlyfor D= 2(Θ µν ≡ /epsilon1 µν forD= 2); for 2+1 Euclidean case see [11] kievarwe.tex; 12/03/2001; 3:49; p.87 QUANTUMDEFORMATIONS OFSPACE-TIMESUSY 81 The plan of the paper is following: In Sect. 2 we shall brieﬂy review the considered in literature quantum deformations of Poincar ´e and conformal su- persymmetries. The list of these deformations written in explicit form as Hopf algebras is quite short, and only the knowledge of large class of classical r– matrices shows that many quantum deformations should be still discovered. As the only nontrivial quantum deformation of D= 4supersymmetry given in the literature is the so–called κ–deformation,obtained in1993 [12]–[14]. In Sect. 3 we consider the Fourier supertransform of superﬁelds in classical (undeformed)and κ–deformedform.Wepresentalsotheintegrationformulaover κ–deformedsuperspace,whichprovidesthedescriptioninsupermomentumspace leadingtothe κ–deformed Feynmann superdiagrams. In Sect. 4 we consider the κ–deformed superﬁeld theory in chiral superspace. We introduce the∗–product multiplication of κ–deformed superﬁelds. It appears that there are two distinguished ∗–products, which both can be written in closed form: one described by standard supersymmetric extenion of CBA formula and other physical, providing the addition of fourmomenta and Grassmann momenta in terms of the coproduct formulae. In such a way we obtain the supersymmetric extensionoftwo∗–products, consideredrecentlyin [15]. In Sect. 5 we shall present some remarks and general diagram describing the deformationschemeof superﬁeldtheory. 2. Quantum Deformations of Space–TimeSupersymmetries Therearetwobasic space–timesymmetriesinD dimensions: - Conformal symmetries O(D,2), having another interpretation as anti–de– Sittersymmetries in D+ 1dimensions -Poincar´e symmetries T D−1,1 + ⊃O(D−1,1). i)Quantumdeformationsofconformalsupersymmetries. The conformal symmetries can be supersymmetrized without introducing tensorialcentralcharges in D= 1,2,3,4and6. One gets: D= 1 :O(2,1)−→OSp(N; 2|R)orSU(1,1 :N) D= 2 :O(2,2) =O(1,2)⊗O(1,2)−→OSp(M; 2|R)⊗OSp(N; 2|R) D= 3O(3,2)−→OSp(N; 4|R) D= 4O(4,2)−→SU(2,2;N) D= 6O(6,2)−→U α U(4;N|H) kievarwe.tex; 12/03/2001; 3:49; p.88 82 P.KOSI´NSKI, J.LUKIERSKI,P. MA ´SLANKA All conformal supersymmetries listed above are described by simple Lie su- peralgebras. It is well–known that for every simple Lie superalgebra one can introduce the q–deformed Cartan–Chevaley basis describing quantum (Hopf– algebraic)Drinfeld–Jimbodeformation[16,17].These q–deformedrelationshave beenexplicitlywritteninphysicalbasisofconformalsuperalgebraindifferentdi- mensions(seee.g.[18]).Itiseasytoseethatthedeformationparameter qappears as dimensionless. It follows, however, that there is another class of deformations of conformal andsuperconformalsymmetries,withdimensionfullparameter κ,playingtherole ofgeometricfundamentalmass.For D= 1onecanshowthattheJordaniandefor- mation ofSL(2;R)/similarequalO(2,1)describes the κ–deformation of D= 1conformal algebra [22]. This result can be extended supersymmetrically, with the following classical /hatwide r–matrixdescribing Jordanian deformation U κ (OSp(1; 2|R)[23] r= 1 κh∧e SUSY =⇒r= 1 κ /parenleftbig h∧e+Q + ∧Q + /parenrightbig Jordaniandeformation Jordanian deformation ofSp(2;R)/similarequalO(2; 1;R)ofOSp(1,2;R) (D= 1conformal ) (D= 1superconformal )(7) TheOSp(1; 2;R)Jordanian classical /hatwide r–matrix can be quantized by the twist method.Semi–closedformforthetwist functionhas beenobtained in[24]. It appears that one can extend the Jordanian deformations of D= 1con- formal algebra to D > 1; forD= 3andD= 4the extended Jordanian classicalr–matrices were given in [22]. It should be also mentioned that the generalized Jordanian deformation of D= 3conformalO(3,2)algebra has been obtained in full Hopf–algebraic form [25]. The extension of Jordanian deforma- tion ofOSp(1,2;R)forD> 1superconformal algebras is not known even in its inﬁnitesimal form givenbyclassical r–matrices. ii) Quantum deformations ofPoincar ´esupersymmetries. Contrary to DJ scheme for simple Lie (super)algebras it does not exist a systematic way of obtaining quantum deformations of non–semisimple Lie (su- per)algebras. A natural framework for the description of deformed semi–direct products, like quantum Poincar ´e algebra, are the noncocommutative bicrossprod- uctHopfalgebras(seee.g.[26]).Itappearshoweverthatintheliteratureithasnot beenformulatedanyeffective schemedescribingthese quantum bicrossproducts. One explicit example of quantum deformation of D= 4Poincar´e super- algebra and its dual D= 4Poincar´e group in form of graded bicrossproduct Hopfalgebrawasgivenin[14].Bymeansofquantumcontractionof q–deformed N= 1anti–de–Sitter superalgebra U q (OSp(1|4))there was obtained in [12] the κ–deformedD= 4Poincar´e subalgebra U κ (P 4;1 ). Subsequently by nonlinear change of generators the quantum superalgebra U κ (P 4;1 )was written in chiral bicrossproduct basis [13]. The κ–deformed Poincar ´e subalgebra is given by the kievarwe.tex; 12/03/2001; 3:49; p.89 QUANTUMDEFORMATIONS OFSPACE-TIMESUSY 83 deformationof the following gradedcross–product 2 p 4;1 = /parenleftBig SL(2;C)⊕ (SL(2;C) + ⊃T 0;2 /parenrightBig /multicloseleftT 4;2 (8) where the generators of SL(2;C)are given by two–spinor generators M αβ = 1 8 (σ µν ) αβ M µν , the generators of (SL(2;C)byM ˙α˙β =M ∗ αβ = 1 8 σ µν ˙α˙β M µν , T 0;2 describes two antichiral superchar gesQ ˙α , andT 4;2 the graded Abelian superalgebra T 4;2 : [P µ ,P ν ] = [P µ ,Q α ] ={Q α ,Q β }= 0 (9) Therelations(9)describethealgebraofgeneratorsoftranslationsandsupertrans- lations in chiral superspace. The algebra (SL(2;c)⊕ (SL(2;c) + ⊃T 0;2 )has the form sl(2;c) : [M αβ ,M γδ ] =/epsilon1 αγ M βδ −/epsilon1 βγ M αδ (10a) +c βδ M αγ −/epsilon1 αδ M β γ sl(2;c) + ⊃T 0;2 : [M ˙α˙β ,M ˙γ˙δ ] =/epsilon1 ˙α˙γ M ˙β˙δ −/epsilon1 ˙β˙γ M ˙α˙δ +/epsilon1 ˙β˙δ M ˙α˙γ −/epsilon1 ˙α˙δ M ˙β˙γ [M ˙α˙β ,Q ˙γ ] =/epsilon1 ˙αγ Q ˙β −/epsilon1 ˙β˙γ Q ˙α {Q ˙α ,Q ˙β }= 0 (10b) It should be observed that in the cross-product (8) the basic supersymmetry algebra{Q α ,Q ˙β }= 2(σ µ p µ ) α˙β istheonebelongingto thecross–relations. Theκ–deformed bicrossproductis given bythe formula U κ (p 4;2 ) = (SL(2;c) ⊕SL(2;c) + ⊃T 0;2 )⊿/triangleleftsldT κ 4;2 (11) The relations (9) and (10a) remain valid but T κ 4;2 describes now the Hopf algebra withdeformed coproducts: ∆P 0 =P 0 ⊗1 + 1⊗P 0 ∆P i =P i ⊗e − P 0 κ + 1⊗P i ∆Q α =Q α ⊗e − P 0 2κ + 1⊗Q α (12) The cross–relationsare thefollowing (M i = 1 2 /epsilon1 ijk M jk ,N i =M i0 ) : 2 In [13] for the crossproduct formula describing D= 4superPoincar ´e algebra the following notation was used: p 4;1 =O(1,3; 2)/multicloseleftT 4,2 . In the notation (8) proposed in present paper the extensionof Lorentz algebra by odd generators is described more accurately. kievarwe.tex; 12/03/2001; 3:49; p.90 84 P.KOSI´NSKI, J.LUKIERSKI,P. MA ´SLANKA [M i ,P j ] =i/epsilon1 ijk P k [M i ,P 0 ] = 0 [N i ,P j ] =iδ ij [ κ 2(1−e − 2P 0 κ + 1 2κ /arrowrighttophalf P 2 ) + 1 κP i P j ] [N i ,P 0 ] =iP i (13) and [M i ,Q α ] =− 1 2(σ i ) β α Q β [N i ,Q α ] = 1 2ie − P 0 κ (σ i ) β α Q β + 1 2κ/epsilon1 ijk P j (σ k ) β α Q β {Q α ,Q ˙β }= 4κδ α˙β sinhP 0 2κ−2e P 0 2κ p i (σ i ) α˙β (14) The notion of bicrossproduct (11) implies also the modiﬁcation of primitive coproducts for SL(2;c) ⊕SL(2;c) + ⊃T 0;2 generators.Onegets: ∆M i =M i ⊗1 + 1⊗M i ∆N i =N l ⊗1 +e − P 0 κ ⊗N i + 1 κ/epsilon1 ijk P j ⊗M k − i 4κ(σ i ) α˙β Q α ⊗e P 0 κ Q ˙β ∆Q j =Q ˙α ⊗1 +e P 0 2κ ⊗Q ˙α (15) It appears that the classical N= 1D= 4Poincar´e superalgebra can be put as well intheform p 4;1 = (SL(2;c) + ⊃T 0;2 )⊕ (SL(2;c) /multicloseleftT 4;2 (16) whereT 0 4;2 describethetranslationandsupertranslationgenerators( P κ ,Q ˙α ).Sub- sequently the κ–deformation of D= 4N= 1Poincar´e superalgebra can be obtained bydeforming(16) intogradedbicrossproduct Hopfsuperalgebra U κ (p 4;1 ) = (SL(2;c) + ⊃T 0;2 )⊕ (SL(2;c)⊿ /triangleleftsldT κ 4;2 (17) Inordertodescribethe κ–deformedchiralsuperspaceoneshouldconsiderthe Hopf superalgebra /tildewide T κ 4;2 obtained by dualization of the relations (9) and (12), and describing by functions C( /hatwide z A )onκ–deformed chiral superspace /hatwide z A = ( /hatwide z µ , /hatwide θ α ), kievarwe.tex; 12/03/2001; 3:49; p.91 QUANTUMDEFORMATIONS OFSPACE-TIMESUSY 85 where /hatwide z µ denotesthecomplexspace–timecoordinates.Oneobtainsthefollowing set ofrelations: [ /hatwide z 0 , /hatwide z i ] = i κ /hatwide z i [ /hatwide z i , /hatwide z j ] = 0 [ /hatwide z 0 , /hatwide θ α ] = i 2κ /hatwide θ α [ /hatwide z i , /hatwide θ α ] = 0 { /hatwide θ α , /hatwide θ β }= 0 (18a) and theprimitive coproducts: ∆ /hatwide z µ = /hatwide z µ ⊗1 + 1⊗ /hatwide z µ ∆ /hatwide θ α = /hatwide θ α ⊗1 + 1⊗ /hatwide θ α (18b) Theκ–deformed chiral superﬁeld theory is obtained by cosidering suitably or- dered superﬁelds. In the following Section we shall consider the superFourier transform of deformed superﬁelds and consider the κ–deformed chiral superﬁeld theory. 3. FourierSupertransformsand κ–deformedBerezin Integration i) Fouriersupertransformonclassicalsuperspace. The superﬁelds are deﬁned as functions on superspace. Here we shall restrict ourselvesto D= 4chiralsuperspace z A = (z µ ,θ α ) (µ= 0,1,2,3;α= 1,2)and tochiral superﬁelds Φ(z,θ). The Fourier supertransform of the chiral superﬁeld and its inverse take the form: Φ(x,θ) = 1 (2π) 2 /integraldisplay d 4 pd 2 η /tildewide Φ(p,η)e i(px+ηθ) (19a) /tildewide Φ(p,η) = 1 (2π) 2 /integraldisplay d 4 xd 2 θΦ(x,θ)e −i(px+ηθ) (19b) The Fourier supertransforms were considered ﬁrstly in [29, 30]. It appears that the set of even and odd variables ( z µ ,θ α ;p µ ,η α ) describes the superphase space, withGrassmannvariables η α describing“oddmomenta”.TheBerezinintegration rulesare validinbothoddpositionandmomentumsectors: /integraldisplay d 2 θ= /integraldisplay d 2 θθ α = 0 1 2 /integraldisplay d 2 θθ α θ α = 1 (20a) /integraldisplay d 2 η= /integraldisplay d 2 ηη α = 0 1 2 /integraldisplay d 2 θη α η α = 1 (20b) kievarwe.tex; 12/03/2001; 3:49; p.92 86 P.KOSI´NSKI, J.LUKIERSKI,P. MA ´SLANKA whereη α =/epsilon1 αβ η β andη α η α = 2η 1 η 2 . It is easy to see that θ 2 = 1 2 θ α θ α η 2 = 1 2 η α η α playthe role ofDiracdeltas, because /integraldisplay d 2 θθ 2 Φ(z,θ) = Φ(z,θ)| θ=0 (21a) /integraldisplay d 2 ηη 2 /tildewide Φ(p,η) = /tildewide Φ(p,η)| η=0 (21b) The formulae(19a)–(19b) in componentformalism Φ(z,θ) = Φ(z) + Ψ α (z)θ α +F(z)θ 2 (22a) leadto /tildewide Φ(p,η) = /tildewide F(p)− /tildewide Ψ ν (p)η ν − /tildewide Φ(p)η 2 (22b) Let us consider for example the chiral vertex Φ 3 (z,θ), present in Wess–Zumino model.This vertexcan be writteninmomentum superspaceas follows: /integraldisplay d 4 zd 2 θΦ 3 (z,θ) = /integraldisplay d 4 p 1 ...d 4 p 3 d 2 η 1 ...d 2 η 3 ·Φ(p 1 ,η 1 ) Φ(p 2 ,η 2 ) Φ(p 3 ,η 3 )δ 4 (p 1 +p 2 +p 3 )(η 1 +η 2 +η 3 ) 2 (23) We see therefore that in Feynmann superdiagrams the chiral vertex (23) will be representedbytheproductofDiracdeltasdescribingtheconservationatthevertex ofthefourmomentaas wellastheGrassmann oddmomenta. ii) Fouriersupertransformon κ–deformed superspace. Following the formulae (18a)–(18b) we obtain the supersymmetric extension of ofκ–deformed Minkowski space to κ–deformed superspace /hatwide x µ −→( /hatwide x µ , /hatwide θ α ). The orderedsuperexponential isdeﬁnedas follows: :e i(p µ /hatwide z µ +η α /hatwide θ α ) :=e −ip 0 /hatwide z 0 e i(/vector p/vector z+η α /hatwide θ α ) (24) where (p µ ,θ α )satisfy the Abeliangraded algebra )9), i.e. [p µ ,p ν ] = [p µ ,η α ] ={η α ,η β }= 0 (25) Fromtheformulae (8)and(24)–(25) followsthat: :e i(p µ /hatwide z µ +η α /hatwide θ α ) : :e i(p /prime µ /hatwide z µ +η /prime α /hatwide θ α ) :=:e i∆ (2) µ (p,p /prime ) /hatwide z µ +∆ (2) α (η,η /prime ) /hatwide θ α :(26) where ∆ 0 (p,p /prime ) =p 0 +p /prime 0 ∆ i (p,p /prime ) =p i +e − p 0 κ p /prime i kievarwe.tex; 12/03/2001; 3:49; p.93 QUANTUMDEFORMATIONS OFSPACE-TIMESUSY 87 ∆ α (η,η /prime ) =η α +e − p 0 2κ η /prime α (27) Theκ–deformedFourier supertransformcanbedeﬁned as follows: Φ( /hatwide z, /hatwide θ) := 1 (2π) 2 /integraldisplay d 4 pd 2 η /tildewide Φ κ (p,η) :e i(p /hatwide z+η /hatwide θ) : (28) If wedeﬁneinverse Fouriersupertransform /hatwide Φ(p,η) = 1 (2π) 2 /integraldisplay d 4 /hatwide zd 2 /hatwide θΦ( /hatwide z, /hatwide θ) :e −i(p /hatwide z+η /hatwide θ) : (29) undertheassumption that ( /hatwide θ 2 = 1 2 /hatwide θ α /hatwide θ α ) /integraldisplay d 2 /tildewide θ /tildewide θ 2 = 1 (30a) orequivalently( η 2 ≡ 1 2 η α η α ) 1 (2π) 4 /integraldisplay/integraldisplay d 4 /hatwide zd 2 /hatwide θ:e i(p /hatwide z+η /hatwide θ) :=δ 4 (p)·η 2 (30b) one gets /hatwide Φ κ (p,η) =e − 4p 0 κ /tildewide Φ /parenleftBig e p 0 κ /vector p,p 0 ,e p 0 2κ η α /parenrightBig (31) Forκ–deformedchiralﬁeldsonecanconsidertheirlocalpowers,andperform theκ–deformed superspace integrals. Onegets /integraldisplay/integraldisplay d 4 /hatwide zd 2 /hatwide θ: Φ( /hatwide z, /hatwide θ) = /hatwide Φ(0,0) /integraldisplay/integraldisplay d 4 /hatwide zd 2 /hatwide θΦ 2 ( /hatwide z, /hatwide θ) = /integraldisplay d 4 p 1 d 4 p 2 d 2 η 1 d 2 η 2 (32a) /tildewide Φ κ (p 1 ,η 1 ) /tildewide Φ κ (p 2 ,η 2 )δ(p 01 +p 02 )δ (3) /parenleftBig /vector p 1 +e p 01 κ /vector p 2 /parenrightBig (η 1 +e p 01 2κ η 2 ) 2 /integraldisplay/integraldisplay d 4 /hatwide zd 2 /hatwide θΦ 3 ( /hatwide z, /hatwide θ) = /integraldisplay 3 /productdisplay i=1 d 4 p i d 2 η i · /tildewide Φ κ (p i ,η i ) ·δ(p 01 +p 02 +p 03 )·δ (3) /parenleftbigg /vector p 1 +e p 01 κ /vector p 2 +p 0 +p 02 κ/vector p 3 /parenrightbigg · /parenleftbigg η 1 +e p 02 2κ η 2 +e p01+p 02 κ η 3 /parenrightbigg 2 (32b) The formulae (32a) can be used for the description of κ–deformed vertices in Wess–Zumino model for chiral superﬁelds. kievarwe.tex; 12/03/2001; 3:49; p.94 88 P.KOSI´NSKI, J.LUKIERSKI,P. MA ´SLANKA 4. Star Productfor κ–deformedSuperﬁeld Theory In this section we shall extend the star product for the functions on κ–deformed Minkowski space given in [15] to the case of functions on κ–deformed chiral superspace,describedby therelations(18a)–(18b). The CBH⋆–productformulafor unordered exponentials takesthe form e ip µ z µ +η ˙ α θ ˙α ·e ip /prime ν z ν +η /prime ˙ β θ ˙β =e iγ µ (p,p /prime )z µ +σ ˙α (p,p /prime ,η ,η /prime )θ ˙α (33) where γ 0 =p 0 +p /prime 0 (34a) γ k =p k e p/prime 0 κ f /parenleftbig p 0 κ /parenrightbig +p /prime k f /parenleftBig p /prime 0 κ /parenrightBig f /parenleftBig P 0 +p /prime 0 κ /parenrightBig (34b) σ ˙α =η ˙α e p/prime 0 2κ f /parenleftbig p 0 2κ /parenrightbig +η /prime ˙α f /parenleftBig p /prime 0 2κ /parenrightBig f /parenleftBig P 0 +p /prime 0 2κ /parenrightBig (34c) andf(x)≡ e x − 1 x .The starproduct multiplicationreproduces theformula (33). e ip µ z µ +η ˙ α θ ˙α ⋆e ip /prime ν z ν +η /prime ˙ β θ ˙β =e iγ µ (p,p /prime )z µ +σ ˙α (p,p /prime ,η ,η /prime )θ ˙α (35) For arbitrarysuperﬁelds φ(z,θ)andχ(z,θ)onegets φ(z,θ)⋆χ(z,θ) = =φ /parenleftBigg 1 i ∂ ∂p µ , ∂ ∂η ˙α /parenrightBigg χ /parenleftBigg 1 i ∂ ∂p /primeµ , ∂ ∂η /prime ˙α /parenrightBigg e iγ µ (p,p /prime )z µ +σ ˙α (p,p /prime ,η ,η /prime )θ ˙α /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle p=0 p /prime =0 η =0 η /prime =0 (36) orequivalently φ(z,θ)⋆χ(z,θ) =e iz µ /parenleftBig γ µ /parenleftBig ∂ ∂y , ∂ ∂y/prime /parenrightBig − ∂ ∂yµ − ∂ ∂y/primeµ /parenrightBig −θ ˙α /parenleftBig σ ˙α /parenleftBig ∂ ∂y , ∂ ∂y/prime , ∂ ∂ω , ∂ ∂ω/prime /parenrightBig − ∂ ∂ω˙α − ∂ ∂ω/prime ˙α /parenrightBig ·φ(y,ω)χ(y /prime ,ω /prime ) /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle y=y /prime =z ω=ω /prime =θ (37) Inparticularwe get kievarwe.tex; 12/03/2001; 3:49; p.95 QUANTUMDEFORMATIONS OFSPACE-TIMESUSY 89 z i ⋆z j =z i z j z 0 ⋆z i =z 0 z i + i 2κz i z i ⋆z 0 =z 0 z i − i 2κz i z i ⋆θ ˙α =z i θ ˙ α θ ˙α ⋆z i =z i θ ˙α z 0 ⋆θ ˙α =z 0 θ ˙α + i 4 κθ ˙ α θ ˙α ⋆z 0 =z 0 θ ˙α − i 4 κθ ˙ α θ ˙α ⋆θ ˙β =θ ˙ α θ ˙β (38) Star product/circleasteriskcorresponding to the multiplication of ordered exponentials (24) takestheform: e ip µ z µ +η ˙ α θ ˙α /circleasteriske ip /prime µ z µ +η /prime ˙ α θ ˙α =e i(p 0 +p /prime 0 z 0 +i(e p/prime 0κ p κ +p /prime κ )z κ +(e p/prime 0 2 κ η ˙α +η /prime ˙α )θ ˙α (39) The superalgebra (18a) of κ–deformed superspace is obtained from the followingrelations: z k /circleasteriskθ ˙α =θ ˙α /circleasteriskz k =z k θ ˙ α θ ˙α /circleasteriskθ ˙β =θ ˙ α θ ˙β z k /circleasteriskz i =z k z i z 0 /circleasteriskz i =z 0 z i z i /circleasteriskz 0 =z 0 z i − i κz i z 0 /circleasteriskθ ˙α =z 0 θ ˙ α θ ˙α /circleasteriskz 0 =z 0 θ ˙α − i 2 κθ ˙α (40) Similarly like in nonsupersymmetric case the star–product (39) is more phys- icalbecausereproducesthecompositionlawofevenandoddmomentaconsistent withcoalgebrastructure. kievarwe.tex; 12/03/2001; 3:49; p.96 90 P.KOSI´NSKI, J.LUKIERSKI,P. MA ´SLANKA 5. FinalRemarks In this lecture we outlined present status of quantum deformations of space–time supersymmetries 3 , and for the case of κ–deformation of D= 4supersymmetries proposed the corresponding deformation of chiral superﬁeld theory. It appears that only the κ–deformed chiral superspace generators describe a closed sub- algebra ofκ–deformedD= 4Poincar´e group. At present it can be obtained theκ–deformation of superﬁeld theory on real superspace can be obtained. The deformationofchiralsuperﬁeldtheorycanbedescribedbythefollowingdiagram: Classical local superfield theory on standard superspacedeformationκ− κ-deformation Fouriersupertransform Classical nonlocal κ-deformed superfield theory on standard superspace standard inverse Fourier supertransform κ-deformed theory on graded commutative momentum superspacelocal κ-deformed superfield theory on κ-deformed Minkowski superspace∗ 14 3 2 -multiplication κ-deformed superfield Figure 7.κ–deformationof local superﬁeld theory The star product/circleasteriskgiven by formula (39) (see 4 /circlecopyrton Fig. 1) is selected by the choice of superFourier transform (28), with ordered Fourier exponential de- scribed by (24). Equivalently, the /circleasterisk–product multiplication can be obtained by thefollowingthree consecutive steps: i)Deformationoflocal superﬁeldtheory(see 1 /circlecopyrton Fig. 1) ii)κ–deformed superﬁeld transform(28)(see 2 /circlecopyrton Fig.1) iii)inverse classical Fouriertransform (see 3 /circlecopyrtonFig. 1) Φ(z,θ) = 1 (2π) 2 /integraldisplay d 4 pd 2 θe −i(p µ z µ +η α θ α ) /tildewide Φ(p,η) (41) obtained in thelimit κ→∞from theinverse Fourier transform (29). 3 We did not consider here however, the quantum deformations of inﬁnite – parameter super- conformal symmetries in 1 + 1dimensions, described by superVirasoro algebras as well as afﬁne OSp(N; 2)–superalgebras kievarwe.tex; 12/03/2001; 3:49; p.97 QUANTUMDEFORMATIONS OFSPACE-TIMESUSY 91 Finally it should be observed that for the deformation (1) with constant /hatwide θ µν therewerecalculatedsomeexplicitcorrectionstophysicalprocesses,inparticular forD= 4QED [29]–[31]. We would like to stress that these calculations should be repeated for Lie algebraic deformations of space–time and superspace, in par- ticularinthe κ–deformedframework.Thepreliminaryresultsinthisdirectionhas beenobtainedin [32,33]. References 1. S. Dopplicher, K. Fredenhagen, J. Roberts, Phys. Lett. B331, 39 (1994); Comm. Math. Phys. 172, 187(1995) 2. L.J. Garay, Int. Journ. Mod. Phys. A10, 145 (1995) 3. Chong-SunChu, Pei-Ming Ho, hep–th/9812219; hep–th/9906192 4. N.Seiberg, E. Witten, JHEP 9909:032 (1999) 5. J.Madore, S. Schraml, P. Schupp, J. Wess,hep—th/0001203 6. C. Chu,F. Zamorra, hep–th/9912153 7. S. Ferrara,M. Lledo, hep–th/0002084 8. S. Terashima, hep–th/0002119 9. A.A. Bichl, J.M. Grimstrup, H. Grosse, L. Popp, M. Schweda, R. Wulkenhaar, hep– th/0007050 10. S. Dopplicher, Ann. Inst. HenriPoinc. 64, 543 (1996) 11. J.Lukierski, P. Stichel,W.J. Zakrzewski, Ann,. Phys. 261, 224(1997) 12. J.Lukierski, A.Nowicki,J. Sobczyk, J. Phys. 26A, L1109 (1993) 13. P. Kosi ´nski, J. Lukierski,P. Ma ´slanka,J.Sobczyk, J. Phys. 27A, 6827 (1994); ibid.28A, 2255 (1995) 14. P. Kosi ´nski, J. Lukierski,P. Ma ´slanka,J.Sobczyk, J. Math. Phys. 37,3041 (1996) 15. P. Kosi ´nski, J. Lukierski,P. Ma ´slanka,hep–th/0009120 16. M. Chaichian, P.P. Kulish, Phys. Lett. B234, 72 (199) 17. S.M. Khoroshkin, V.N. Tolstoy, Comm. Math. Phys. 141, 599 (1991) 18. J.Lukierski, A.Nowicki,Phys. Lett.B279, 299 (1992) 19. V. Dobrev, Journ.Phys. A26, 1317 (1993) 20. L. Dabrawski, V.K. Dobrev, R. Floreanini, V. Husain, Phys. Lett. B302,215 (1993) 21. C. Juszczak, Journ. Phys. A27, 385(1994) 22. J.Lukierski, P. Minnaert, M.Mozrzymas, Phys. Lett. B371, 215 (1996) 23. C. Juszczak, J. Sobczyk, math.QA/9809006 24. R. Celeghini, P. Kulish,q–alg/9712006, 25. F. Herranz, Journ. Phys. A30, 6123 (1997) 26. S. Majid, “Foundationsof QuantumGroup Theory” Cambridge Univ. Press,1995 27. D. Leites, B.M. Zupnik “Multiple Processes at High Energies” (in Russian), Tashkent 1976, p. 3–26 28. F.A.Berezin, M.S. Marinov, Ann. Phys. 104, 336 (1977) 29. I.Mocioiu, M. Pospelov, R. Roiban, Phys. Lett. B489, 390(2000) 30. N.Chair, M.M. Sheikh–Jabbari, hep–th/0009037 31. M. Chaichian, M.M. Sheikh–Jabbari,A. Tureanuhep–th/0010175 32. J.Lukierski, H.Ruegg, W. Ruehl, Phys. Lett. 313, 357 (1993) 33. L.C. Biedenharn, B. Mueller, M. Tarlini,Phys. Lett. 318, 613 (1993) kievarwe.tex; 12/03/2001; 3:49; p.98 kievarwe.tex; 12/03/2001; 3:49; p.99 THEHOWE DUALITYANDLIESUPERALGEBRAS DIMITRYLEITES †‡ Department of Mathematics, University of Stockholm, Roslagsv. 101,Kr¨aftrikethus 6,S-10691,Stockholm, Sweden IRINA SHCHEPOCHKINA § Independent Univ. of Moscow, Bolshoj Vlasievsky per., 121002 Moscow, Russia Abstract. Howe’s duality is considered from a unifying point of view based on Lie superalgebras. New examples are offered. In particular, we construct several simplest spinor-oscillator representa- tions and compute their highest weights for the “stringy” Lie superalgebras (i.e., Lie superalgebras of complex vector ﬁelds (or their nontrivial central extensions) on the supercircle S 1|n and its two-sheeted cover associated with the M ¨obiusbundle). In our two lectures we brieﬂy review, on the most elementary level, several resultsandproblemsuniﬁedby“Howe’sduality”.Detailswillbegivenelsewhere. The groundﬁeld inthe lecturesis C. 1. Introduction In his famous preprint [24] R. Howe gave an inspiring explanation of what can be “dug out” from H. Weyl’s “wonderful and terrible” book [55], at least as far as invariant theory is concerned, from a certain unifying viewpoint. According to Howe, much is based on a remarkable correspondence between certain ir- reducible representations of Lie subalgebras ΓandΓ /prime of the Lie algebra o(V) or sp(V)provided ΓandΓ /prime are each other’s “commutants”, i.e., centraliz- ers. This correspondence is known ever since as Howe’s correspondence or Howe’s duality . In [24] and subsequent papers Howe gave several examples of such a correspondence previously known, mostly, inadvertently. Let us remind ‡ We gratefully acknowledge ﬁnancial support of an NFR grant and RFBR grant 99-01-00245, respectively. D.L is thankful to B. Feigin, E. Poletaeva, V. Serganova and Xuan Peiqi for helpful discussions. † [email protected] § [email protected] kievarwe.tex; 12/03/2001; 3:49; p.100 94 D.LEITES, I.SHCHEPOCHKINA some of them (omitting important Jacquet-Langlands-Shimizu correspondence, S.Gelbart’s contributions, etc.): 1)decomposition of o(V)-moduleS.(V)into spherical harmonics; 2)Lefschetz decomposition of sp(V)-module Λ.(V)into primitive forms (sometimes this iscalled Hodge–L ´epage decomposition); 3)astrikingresemblancebetween spinorrepresentation of o(n)andoscillator (Shale–Segal–Weil–metaplectic –... )representationof sp(2n). As an aside Howe gives the “shortest possible” proof of the Poincar´e lemma. (Recallthatthislemmastatesthatinanysufﬁcientlysmallopenstar-shapedneigh- borhoodofanypointonanymanifoldanycloseddifferentialformisexact.)Inthis proof, Lie superalgebras, that lingered somewhere in the background in the pre- vious discussion but were treated rather as a nuisance than help, are instrumental to reach the goal. This example shows also that the requirement of reductivity ofΓandΓ /prime to form a “dual pair” is extra. Elsewhere we will investigate what are the actual minimal restrictions on ΓandΓ /prime needed to reach one of the other problems usually solved by means of Howe duality: decompose the symmetric or exterior algebra of a module over Γ⊕Γ /prime . Howe’s manuscript was written at the time when supersymmetry theory was being conceived. By the time [24] was typed, the deﬁnition of what is nowadays called superschemes ([34]) was not yet rewritten in terms to match physical papers (language of points was needed; now we can recommend [5]) nor translated into English and, therefore, was unknown; the classiﬁcation of simple ﬁnite dimensional Lie superalgebras over Chad just been announced. This was, perhaps, the reason for a cautious tone with which Howe used Lie superalgebras, although he made transparent how important they might be foralucid presentationof hisideas andexplicitlystatedso. Since[25],thepublishedversionof[24],thoughputasidetostewfor12years, underwent only censorial changes, we believe it is of interest to explore what do we gain by using Lie superalgebras from the very beginning (an elaboration of other aspects of this idea [4] are not published yet). Here we brieﬂy elucidate some of Howe’s results and notions and give several new examples of Howe’s dual pairs. In the lectures we will review the known examples 1) – 3) mentioned abovebutconsiderthem in anappropriate “super” setting, andadd to them: 4) a reﬁnement of the Lefschetz decomposition — J. Bernstein’s decomposi- tion([2])ofthespace Ω. /planckover2pi1 of“twisted”differentialformsonasymplecticmanifold with values in a line bundle with connection whose curvature form differs by a factor/planckover2pi1fromthe canonical symplecticform; 5) a decomposition of the space of differential forms on a hyper-K ¨ahlerian manifold similar to the Lefschetz one ([53]) but with sp(4)instead of sp(2) = sl(2)anditsreﬁnementassociatedwiththe osp(1|4). 6) Apart from general clariﬁcation of the scenery and new examples even in the old setting, i.e., on manifolds, the superalgebras introduced ab ovomake it manifest that there are at least two types of Howe’s correspondence: the conven- kievarwe.tex; 12/03/2001; 3:49; p.101 HOWEDUALITY ANDLIE SUPERALGEBRAS 95 tionaloneandseveral“ghost”onesassociatedwithquantizationoftheantibracket [40]. 7) Obviously, if Γ⊕Γ /prime is a maximal subalgebra of osp, then (Γ,Γ /prime )is an ex- ampleofHowedualpair.Section6givessomefurtherexamples,partlyborrowed from[49],where more examplescanbe found. We consider here only ﬁnite dimensional Lie superalgebras with the invariant theory in view. In another lecture ( §§3,4) we consider spinor-oscillator represen- tations in more detail. In these elementary talks we do not touch other interesting applications such as Capelli identities ([30],[43]), or prime characteristic ([47]). OfdozensofpaperswithexamplesofHowe’sdualityininﬁnitedimensionalcases andstillotherexamples,wedrawattentionofthereadertothefollowingselected ones:[12],andvariousinstancesofbose-fermicorrespondence,cf.[13]and[26]. ObservealsothattheHowedualityoftenmanifestsitselffor q-deformedalgebras, e.g., in Klimyk’s talk at our conference, or [6]. To treat this q-Howe duality in a similarway,weﬁrsthavetoexplicitly q-quantizePoissonsuperalgebras po(2n|m) (formn= 0thisisstraightforwardreplacementof(super)commutatorsfrom[39] withq-(super)commutators. 2. ThePoissonsuperalgebra g= po(2n|m) 2.1. CertainZ-gradings of g. Recall that gis the Lie superalgebra whose super- space isC[q,p,Θ]and the bracket is the Poisson bracket{·,·} P.b. is given by the formula {f,g} P.b. = /summationtext i≤n /parenleftbigg ∂ f ∂p i ∂ g ∂q i − ∂ f ∂q i ∂ g ∂p i /parenrightbigg − (−1) p(f) /summationtext j≤m∂ f ∂θ j ∂ g ∂θ j forf,g∈C[p,q,Θ].(2.1) Sometimesit ismoreconvenient toredenotethe Θ’s andset ξ j = 1 √ 2 (Θ j −iΘ r+j );η j = 1 √ 2 (Θ j +iΘ r+j ) forj≤r= [m/2] (herei 2 =−1), θ = Θ 2r+1 and accordingly modifythe bracket (if m= 2r, thereis no term with θ): {f,g} P.b. = /summationtext i≤n /parenleftbigg ∂ f ∂p i ∂ g ∂q i − ∂ f ∂q i ∂ g ∂p i /parenrightbigg − (−1) p(f) /bracketleftbigg/summationtext j≤m ( ∂ f ∂ξ j ∂ g ∂η j + ∂ f ∂η j ∂ g ∂ξ j ) + ∂ f ∂θ∂ g ∂θ /bracketrightbigg . Setting deg Lie f= degf−2for any monomial f∈C[p,q,Θ], where degp i = degq i = deg Θ j = 1foralli,j,weobtain the standardZ-grading of g: kievarwe.tex; 12/03/2001; 3:49; p.102 96 D.LEITES, I.SHCHEPOCHKIN A degree of f− 2− 1 0 1.. . f 1p, q, θ f : degf= 2f: degf= 3.. . Clearly, g=⊕ i≥−2 g i with g 0 /similarequal osp(m|2n). Consider now another, “ rough”, grading of g.To thisend,introduce: Q= (q,ξ),P= (p,η)andset degQ i = 0,degθ= 1,degP i = /braceleftbigg 1 ifm= 2k 2 ifm= 2k+ 1.(∗) Remark. Physicists prefer to use half-integer values of degform= 2k+ 1by setting degθ= 1 2 anddegP i = 1at alltimes. Theabovegrading (∗)ofthepolynomialalgebrainducesthefollowing rough gradingof the Lie superalgebra g. Form= 2kjust delete the columns of odd degreesand deletethe degreesby2: m= 2k+ 1 :de gree.. . 2 1 0− 1− 2 elements.. .C[Q]P 2 C[Q]P θC[Q] PC[Q] θC[Q ] 2.2. Quantization . We call the nontrivial deformation Qof the Lie super- algebra po(2n|m)quantization (for details see [40]). There are many ways to quantize g,butallofthemareequivalent.Recallthatweonlyconsider gwhoseel- ementsarerepresentedbypolynomials;forfunctionsofothertypes(say,Laurent polynomials)the uniquenessof quantizationmaybeviolated. Consider the following quantization, so-called QP-quantization, given on lineartermsby theformulas: Q:Q/mapsto→ˆQ, P/mapsto→/planckover2pi1 ∂ ∂Q, (∗) where ˆQistheoperatorofleftmultiplicationby Q;anarbitrarymonomialshould be ﬁrst rearranged so that the Q’s stand ﬁrst (normal form) and then apply (∗) term-wise. The deformed Lie superalgebra Q( po(2n|2k))is the Lie superalgebra of dif- ferential operators with polynomial coefﬁcients on R n|k . Actually, it is an analog of gl(V).This ismostclearlyseenfor n= 0. Indeed, Q( po(0|2k)) = gl(Λ.(ξ)) = gl(2 k−1 |2 k−1 ). Ingeneral, for n/negationslash= 0,wehave Q( po(2n|2k)) = “ gl”(F(Q)) = diff(R n|k ). Form= 2k−1we consider po(0|2k−1)as a subalgebra of po(0|2k); the quantization sends po(0|2k−1)into q(2 k−1 ). Forn/negationslash= 0the image ofQis an kievarwe.tex; 12/03/2001; 3:49; p.103 HOWEDUALITY ANDLIE SUPERALGEBRAS 97 inﬁnitedimensional analogof q,indeed (for J=i(θ+ ∂ ∂θ )withi 2 =−1): Q( po(2n|2k−1)) = qdiff(R n|k ) ={D∈ diff(R n|k ) : [d,J] = 0}. 2.3. Fock spaces and spinor-oscillator representations . The Lie superalge- bras diff(R n|k )and qdiff(R n|k )have indescribably many irreducible representa- tions even for n= 0. But one of the representations, the identity one, in the superspace of functions on R n|k , is the “smallest” one. Moreover, if we con- sider the superspace of diff(R n|k )or qdiff(R n|k )as theassociative superalgebra (denoted Diﬀ(R n|k )orQDiﬀ(R n|k )), this associative superalgebra has only one irreducible representation — the same identity one. This representation is called theFock space . As is known, the Lie superalgebras osp(m|2n)are rigid for (m|2n)/negationslash= (4|2). Therefore, thethroughmap h−→ g 0 = osp(m|2n)⊂ g= po(2n|m) Q −→ diff(R n|k ) sends any subsuperalgebra hof osp(m|2n)(for(m|2n)/negationslash= (4|2)) into its iso- morphic image. (One can also embed hinto diff(R n|k )directly.) The irreducible subspace of the Fock space which contains the constants is called the spinor- oscillator representation of h. In particular cases, for m= 0orn= 0this subspace turns into the usual spinororoscillator representation , respectively. We have just given a uniﬁed description of them. (A more detailed description follows.) 2.4. Primitive alias harmonic elements . The elements of osp(m|2n)(or its subalgebra h) act in the space of the spinor-oscillator representation by inhomo- geneous differential operators of order ≤2(order is just the ﬁltration associated withthe “rough” grading): m= 2k : degree− 1 0 1 elemen ts ˆP 2 ˆPˆ Q ˆQ 2 m= 2k+ 1 : degree− 2− 1 0 1 2 elemen ts ˆP 2 ˆPˆ θˆPˆ Q ˆQˆ θˆQ 2 The elements from (C[Q]) ˆP 2 form= 2kor(C[Q,θ]) ˆPˆθ form= 2k+ 1 are called primitive orharmonic ones. More generally, let h⊂ osp(m|2n)be aZ-graded Lie superalgebra embedded consistently with the rough grading of osp(m|2n). Then the elements from (C[Q]) h −1 form= 2kor(C[Q,θ]) h −1 for m= 2k+ 1willbe called h-primitive or h-harmonic . 2.4.1. Nonstandard Z-gradings of osp(m|2n). It is well known that one simple Lie superalgebra can have several nonequivalent Cartan matrices and systems of Chevalley generators, cf. [20]. Accordingly, the corresponding divi- sions into positiveandnegativeroot vectors are distinct. The following problem kievarwe.tex; 12/03/2001; 3:49; p.104 98 D.LEITES, I.SHCHEPOCHKINA arises: How the passage to nonstandard gradings affects the highest weight of the spinor-oscillatorrepresentation deﬁned insec.3? (Cf. [44]) 2.5. Examples of dual pairs . Two subalgebras Γ,Γ /prime of g 0 = osp(m|2n)will becalleda dualpair ifone ofthem isthe centralizer ofthe other in g 0 . IfΓ⊕Γ /prime is a maximal subalgebra in g 0 , then, clearly, Γ,Γ /prime is a dual pair. A generalization: consider a pair of mutual centralizers Γ,Γ /prime in gl(V)and embed gl(V)into osp(V⊕V ∗ ).Then Γ,Γ /prime isadualpair(in osp(V⊕V ∗ )).Foranumber ofsuch examplessee[49]. Let usconsider severalof these examples in detail. 2.5.1.Γ = sp(2n) = sp(W)andΓ /prime = sp(2) = sl(2) = sp(V⊕V ∗ ).Clearly, h= Γ⊕Γ /prime isamaximalsubalgebrain o(W⊗(V⊕V ∗ )).TheFockspaceisjust Λ.(W). Thefollowingclassicaltheoremanditsanalog5.2illustratetheimportanceof theabovenotionsand constructions. Theorem .TheΓ /prime -primitive elements of Λ.(W)of each degree iconstitute an irreducible Γ-moduleP i sp ,0≤i≤n. This action of Γ /prime in the superspace of differential forms on any symplectic manifoldiswellknown: Γ /prime isgenerated(asaLiealgebra)byoperators X + ofleft multiplication by the symplectic form ωandX − , application of the bivector dual toω. 2.5.2.Γ = o(2n) = o(W)andΓ /prime = sp(2) = sl(2) = sp(V⊕V ∗ ). Clearly, h= Γ⊕Γ /prime is a maximal subalgebra in sp(W⊗(V⊕V ∗ )). The Fock space is justS.(W). Theorem .TheΓ /prime -primitive elements of S.(W)of each degree iconstitute an irreducible Γ-moduleP i o ,i= 0,1,.... This action of Γ /prime in the space of polynomial functions on any Riemann mani- foldisalsowellknown: Γ /prime isgenerated(asaLiealgebra)byoperators X + ofleft multiplication by the quadratic polynomial representing the metric gandX − is thecorrespondingLaplace operator. Clearly, a mixture of Examples 2.5.1 and 2.5.2 corresponding to symmetric or skew-symmetric forms on a supermanifold is also possible: the space of Γ /prime - primitiveelementsof S.(W) ofeachdegree i isanirreducible Γ -module ,cf.[44] and Sergeev’s papers [51], [52]. In [24], [25] the dual pairs had to satisfy one more condition: the through actionofboth ΓandΓ /prime ontheidentity g 0 -moduleshouldbecompletelyreducible. Even for the needs of the First Theorem of Invariant Theory this is too strong a requirement, cf. examples with complete irreducibility in [51, 52] with our last example, in which the complete reducibility of pe(n)is violated. Investigation of therequiremetson ΓandΓ /prime neededfortheFirstTheoremofInvariantTheorywill begivenelsewhere. 2.5.3.Bernstein’ssquarerootoftheLefschetzdecomposition .LetLbethe spaceofa(complex)linebundleoveraconnectedsymplecticmanifold (M 2n ,ω) with connection∇such that the curvature form of ∇is equal to/planckover2pi1ωfor some kievarwe.tex; 12/03/2001; 3:49; p.105 HOWEDUALITY ANDLIE SUPERALGEBRAS 99 /planckover2pi1∈C. This/planckover2pi1will be called a twist; the space of tensor ﬁelds of type ρ(hereρ: sp(2n)−→ gl(U)is a representation which deﬁnes the space Γ(M,U )of tensor ﬁelds with values in U), and twist/planckover2pi1will be denoted by T /planckover2pi1 (ρ). Let us naturally extendtheactionof X + ,X − fromthespace Ωofdifferentialformson Montothe space Ω /planckover2pi1 of twisted differential forms using the isomorphism of spacesT /planckover2pi1 (ρ)/similarequal T(ρ)⊗Γ(L),where Γ(L) = Ω 0 /planckover2pi1 isthespaceofsectionsofthelinebundle L,i.e., thespace oftwisted functions. Namely, set X + /mapsto→X + ⊗1, etc. LetD + =d+αbe the connection∇itself andD − = [X − ,D + ].OnΩ /planckover2pi1 ,introduceasuperspacestructuresetting p(ϕ⊗s) = degϕ(mod 2),forϕ∈Ω,s∈Ω 0 /planckover2pi1 . Theorem . ([2])OnΩ /planckover2pi1 , the operators D + andD − generate an action of the Lie superalgebra osp(1|2)commuting with the action of the group ˆGof ∇-preservingautomorphisms of thebundle L. Bernstein studied the ˆG-action, more exactly, the action of the Lie algebra po(2n|0)corresponding to ˆG; we are interested in the part of this action only: in sp(2n) = po(2n|0) 0 -action. In Example 2.5.1 the space P i consisted of differential forms with constant coefﬁcients. Denote by P i =P i ⊗S.(V)the space of primitive forms with polynomial coefﬁcients. The elements of the space √ P i /planckover2pi1 =KerD − ∩P i /planckover2pi1 will becalled∇-primitiveforms of degreei(andtwist/planckover2pi1). Bernstein showed that √ P i /planckover2pi1 is an irreducible g= po(2n|0)-module. It could be that over subalgebra g 0 the module √ P i /planckover2pi1 becomes reducible but the general theorem of Howe (which is true for osp(1|2n)) states that this is not the case, it remains irreducible. Shapovalov and Shmelev literally generalized Bernstein’s result for (2n|m)-dimensional supermanifolds, see review [37]. In particular, Shapovalov, who considered n= 0, “took a square root of Laplacian and the metric”. 2.5.4. Inspired by Bernstein’s construction, let us similarly deﬁne a “square root” of the hyper-K ¨ahler structure. Namely, on a hyper-K ¨ahlerean manifold (M,ω 1 ,ω 2 )consider a line bundle Lwith two connections: ∇ 1 and∇ 2 , whose curvature forms are equal to /planckover2pi1 1 ω 1 and/planckover2pi1 2 ω 2 for some/planckover2pi1 1 ,/planckover2pi1 2 ∈C. The pair /planckover2pi1= (/planckover2pi1 1 ,/planckover2pi1 2 )will be called a twist; the space of tensor ﬁelds of type ρand twist/planckover2pi1 will be denoted by T /planckover2pi1 (ρ). Verbitsky [53] deﬁned the action of sp(4)in the space Ωof differential forms on M. Let us naturally extend the action of the generators X ± j forj= 1,2ofof sp(4)from Ωontothespace Ω /planckover2pi1 oftwisteddifferentialforms using the isomorphism T /planckover2pi1 (ρ)/similarequalT(ρ)⊗Γ(L), where Γ(L) = Ω 0 /planckover2pi1 is the space of sectionsofthelinebundle L;hereX + j istheoperatorofmultiplicationby ω j and X − j istheoperatorof convolutionwith thedual bivector. Deﬁnethespaceofprimitive i-forms(withconstantcoefﬁcients)onthehyper- K¨ahlerean manifold (M,ω 1 ,ω 2 )by setting P i =KerX − 1 ∩KerX − 2 ∩Ω i . (HK) kievarwe.tex; 12/03/2001; 3:49; p.106 100 D.LEITES, I.SHCHEPOCHKINA According to the general theorem [25] this space is an irreducible sp(2n;H)- module. SetD − i = [X − i ,D + i ].Thepromisedsquarerootofthedecomposition(HK)is thespace P i /planckover2pi1 =KerD − 1 ∩KerD − 2 ∩Ω i /planckover2pi1 . ( √ HK) The operators D ± i , whereD + i =∇ i ,generate osp(1|4). 2.6. Further examples of dual pairs . The following subalgebras g 1 (V 1 )⊕ g 2 (V 2 )aremaximalin g(V 1 ⊗V 2 ), hence, aredual pairs: g 1 g 2 g osp(n 1 |2m 1 ) osp(n 2 |2m 2 ) osp(n 1 n 2 + 4m 1 m 2 |2n 1 m 2 + 2n 2 m 1 ) o(n ) osp(n 2 |2m 2 ) osp(nn 2 |2nm 2 ),n/negationslash= 2, 4 sp(2n ) osp(n 2 |2m 2 ) osp(2mn 2 |4nm 2 ) pe(n 1 ) pe(n 2 ) osp(2n 1 n 2 |2n 1 n 2 ),n 1 ,n 2 > 2 osp(n 1 |2m 1 ) pe(n 2 ) pe(n 1 n 2 + 2m 1 n 2 ) ifn 1 /negationslash= 2m 1 spe(n 1 n 2 + 2m 1 n 2 ) ifn 1 = 2m 1 o(n ) pe(m ) pe(nm ) sp(2n ) pe(m ) pe(2nm ) In particular, on the superspace of polyvector ﬁelds, there is a natural pe(n)- module structure, and pe(1), its dual partner in osp(2n|2n), is spanned by the divergence operator ∆(“odd Laplacian”), called the BRST operator ([1]), the even operator of pe(1)being deg x −deg θ , whereθ i =π( ∂ ∂x i ),πbeing the shift ofparity operator. Forfurtherexamplesofmaximalsubalgebrasin gland qsee[49].Thesesubal- gebrasgiverisetoothernewexamplesofHowedualpairs.Forthedecomposition ofthetensoralgebracorrespondingtosomeoftheseexamplessee[51,52],some ofthelatterarefurtherelucidatedin[3].SomefurtherexamplesofHowe’sduality, considered in a detailed version of our lectures, are: (1) over reals; (2) dual pairs in simple subalgebras of po(2n|m)distinct from osp(m|2n); in particular, (3) embeddings into po(2n|m;r), the nonstandard regradings of the Poisson super- algebra, cf. [50]; (4) a “projective” version of the Howe duality associated with embeddingsintotheLiesuperalgebraofHamiltonianvectorﬁelds,thequotientof the Poisson superalgebra, in particular, the exceptional cases in dimension (2|2), cf. [40]. It is also interesting to consider the prime characteristic and an “odd” Howe’s duality obtained from quantization of the antibracket (the main objective of[4]),to saynothingof q-quantized versionsofthe above. kievarwe.tex; 12/03/2001; 3:49; p.107 HOWEDUALITY ANDLIE SUPERALGEBRAS 101 3. Generalitieson spinor and spinor-likerepresentations 3.1.ThespinorandoscillatorrepresentationsofLiealgebras .Theimportance ofthespinorrepresentationbecameclearveryearly.Oneofthereasonsisthefol- lowing.Asisknownfromanytextbookonrepresentationtheory,thefundamental representations R(ϕ 1 ) =W,R(ϕ 2 ) = Λ 2 (W),...,R(ϕ n−1 ) = Λ n−1 (W)of sl(W),where dimW=nandϕ i isthehighestweightof Λ i (W),areirreducible. Any ﬁnite dimensional irreducible sl(n)-moduleL λ is completely determined by its highest weight λ= /summationtext λ i ϕ i withλ i ∈Z + . The module L λ can be realized as asubmodule(or quotient)of ⊗ /parenleftBig R(ϕ i ) ⊗λ i /parenrightBig . Similarly, every irreducible gl(n)-moduleL λ , whereλ= (λ 1 ,...λ n−1 ;c) andcis the eigenvalue of the unit matrix, is realized in the space of tensors, perhaps,twistedwiththehelpof c-densities,namelyinthespace ⊗ i /parenleftBig R(ϕ i ) ⊗λ i /parenrightBig ⊗ tr c ,where tr c istheLiealgebraicversionofthe cthpowerofthedeterminant,i.e., inﬁnitesimally, trace, given for any c∈Cby the formula X/mapsto→c·tr(X)for any matrixX∈ gl(W). Thus, all the irreducible ﬁnite dimensional representations of sl(W)are naturally realized in the space of tensors, i.e., in the subspaces or quotient spaces of the space T p q =W⊗···⊗W /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright p ⊗W ∗ ⊗···⊗W ∗ /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright q , whereWis thespaceoftheidentityrepresentation.For gl(W),wehavetoconsiderthespace T p q ⊗tr c . For sp(W),theconstructionissimilar,exceptthefundamentalmodule R(ϕ i ) isnow apartof themodule Λ i (id)consistingof the primitive forms . For o(W),thesituationistotallydifferent:notallfundamentalrepresentations can be realised as (parts of) the modules Λ i (id). The exceptional one (or two, for o(2n))ofthemiscalledthe spinorrepresentation ;for o(W),where dimW= 2n, itisrealizedintheGrassmannalgebra E.(V)ofa“half”of W,whereW=V⊕ V ∗ is a decomposition into the direct sum of subspaces isotropic with respect to theformpreservedby o(W).FordimW= 2n+1,itisrealizedintheGrassmann algebraE.(V⊕W 0 ), whereW=V⊕V ∗ ⊕W 0 andW 0 is the 1-dimensional space onwhich theorthogonal formisnondegenerate. The quantization of the harmonic oscillator leads to an inﬁnite dimensional analog of the spinor representation which after Howe we call oscillator repre- sentation of sp(W). It is realized in S.(V), where as above, Vis a maximal isotropicsubspaceof W(withrespecttotheskewformpreservedby sp(W)).The remarkablelikenessofthespinorandoscillatorrepresentationswasunderlinedin atheoryof dualHowe’spairs , [23]. The importance of spinor-oscillator representations is different for distinct classes of Lie algebras and their representations. In the description of irreducible ﬁnitedimensionalrepresentationsofclassicalmatrixLiealgebras gl(n), sl(n)and sp(2n)we can do without either spinor or oscillator representations. We can not kievarwe.tex; 12/03/2001; 3:49; p.108 102 D.LEITES, I.SHCHEPOCHKINA do without spinor representation for o(n), but a pessimist might say that spinor representation constitutes only 1 n th of the building bricks. Our, optimistic, point of view identiﬁes the spinor representations as one of the two possible types of thebuildingbricks. For the Witt algebra wittand its central extension, the Virasoro algebra vir, everyirreducible highest weight module is realized as a quotient of a spinor or, equivalently, oscillator representation, see [8], [10]. This miraculous equivalence isknowninphysicsunderthenameof bose-fermicorrespondence ,see[18],[26]. For the list of generalizations of wittand vir, i.e., simple (or close to simple) stringy Lie superalgebras or Lie superalgebras of vector ﬁelds on N-extended supercircles, often called by an unfortunate (as explained in [21]) name “super- conformalalgebras”,see[21].Theimportanceofspinor-oscillatorrepresentations diminishes as Ngrows, but for the most interesting — distinguished ([21]) — stringy superalgebrasit is high,cf. [11],[46]. 3.2. Semi-inﬁnite cohomology . An example of applications of spinor- oscillator representations: semi-inﬁnite (or BRST) cohomology of Lie superalge- bras. These cohomology were introduced by Feigin ﬁrst for Lie algebras ([9]); then he extended the deﬁnition to Lie superalgebras via another construction, equivalent to the ﬁrst one for Lie algebras ([7]). For an elucidation of Feigin’s construction see [14], [31] and [54]. Feigin rewrote in mathematical terms and generalizedtheconstructionsphysicistsusedtodeterminethe criticaldimensions ofstringtheories,i.e.,thedimensionsinwhichthequantizationofthesuperstring is possible, see [42], [18]. These critical dimensions are the values of the cen- tral element (central charges) on the spinor-oscillator representation constructed from the adjoint representation; to this day not for every central element of all distinguished simple stringy superalgebras their values are computed on every spinor-oscillatorrepresentation,notevenontheonesconstructedfromtheadjoint representations. 4. Thespinor-oscillatorrepresentationsandLiesuperalgebras 4.1. Spinor (Clifford–Weil–wedge– ...) and oscillator representations . As we saw in [40], po(2n|m) 0 ∼= osp(m|2n), the superspace of elements of degree 0 in the standard Z-grading of po(2n|m)or, which is the same, the superspace of quadratic elements in the representation by generating functions. At our ﬁrst lecture we deﬁned the spinor-oscillator representation as the through map (here k= [ m 2 ]andQis thequantization) g−→ po(2n|m) Q −→ /braceleftbigg diff(n|k) ifm= 2k qdiff(n|k) ifm= 2k−1, where Im( g)⊂ po(2n|m) 0 = osp(m|2n). Actually, such requirement is too restrictive, we only need that the image of gunder embedding into po(2n|m) kievarwe.tex; 12/03/2001; 3:49; p.109 HOWEDUALITY ANDLIE SUPERALGEBRAS 103 remainsrigidunderquantization.Sovarioussimplesubalgebrasof po(2n|m)will do asambientsof g. This spinor-oscillator representation is called the spinor representation of g ifn= 0, or theoscillator representation ifm= 0. We will denote this rep- resentation Spin(V)and set Osc(V) = Spin(Π( V)), whereVis the standard representation of osp(m|2n). In other words, if Spin(V)is a representation of osp(m|2n),then Osc(V)isarepresentationof osp(2n|m),soOsc(V)onlyexists formeven. IfVis a g-module without any bilinear form, but we still want to construct a spinor-oscillatorrepresentationof g,considerthemodule W=V⊕V ∗ (wherein theinﬁnitedimensionalcasewereplace V ∗ withtherestricted dualofV;roughly speaking, if V=C[x], thenV ∗ =C[[ ∂ ∂x ]], whereas the restricted dual is C[ ∂ ∂x ]) endowedwiththeform(for v 1 ,w 1 ∈V,v 2 ,w 2 ∈V ∗ )symmetricfortheplussign and skew-symmetricotherwise: B((v 1 ,v 2 ),(w 1 ,w 2 )) =v 2 (w 1 )±(−1) p(v 1 )p(w 2 ) w 2 (v 1 ). Now,inW,selectamaximalisotropicsubspace U(notnecessarily VorV ∗ )and realize thespinor-oscillatorrepresentationof gin the exterior algebraof U. Observe that the classical descriptions of spinor representations differ from ours, see, e.g., [17], where the embedding of g(in their case g= o(n)) into the quantized algebra (namely into Q( po(0|n−1))) is considered, not into po(0|m). The existence of this embedding is not so easy to see unless told, whereas our constructionsare manifestandbring aboutthesameresult. To illustrate our deﬁnitions and constructions, we realize the orthogonal Lie algebra o(n)asthe subalgebra intheLie superalgebra po(0|n). Case o(2k).Basis: X + 1 =ξ 2 η 1 , ..., X + k−1 =ξ k η k−1 , X + k =η k η k−1 ; X − 1 =ξ 1 η 2 , ..., X − k−1 =ξ k−1 η k , X − k =ξ k−1 ξ k ; H 1 =ξ 1 η 1 −ξ 2 η 2 , ..., H k−1 =ξ k−1 η k−1 −ξ k η k , H k =ξ k−1 η k−1 +ξ k η k . ForR(ϕ k )take the subspacespace functions C[ξ] ev which contains the constants C·ˆ1,where ˆ1isjust the constantfunction 1;clearly, ˆ1is thevacuum vector. Quantization (see above) sends: ξ i intoˆξ i , andη i into/planckover2pi1 ∂ ∂ξ i , soX ± i ˆ1 = 0for i<k, hence,H i ˆ1 = [X + i ,X − i ]ˆ1 = 0fori<k.Contrariwise, H k ˆ1 = [X + k ,X − k ]ˆ1 = [∂ k ∂ k−1 ,ˆξ k−1 ˆξ k ]ˆ1 =∂ k (−ˆξ k−1 ∂ k−1 + 1)ˆξ k ˆ1 =ˆ1. So weseethat thespinorrepresentation is indeedafundamental one. kievarwe.tex; 12/03/2001; 3:49; p.110 104 D.LEITES, I.SHCHEPOCHKINA Case o(2k+ 1).Basis: X + 1 =ξ 2 η 1 , ..., X + k−1 =ξ k η k−1 , X + k = √ 2η k θ; X − 1 =ξ 1 η 2 , ..., X − k−1 =ξ k−1 η k , X − k = √ 2θξ k ; H 1 =ξ 1 η 1 −ξ 2 η 2 , ..., H k−1 =ξ k−1 η k−1 −ξ k η k , H k = 2ξ k η k . ForR(ϕ k )consider the space of even functions C[ξ 1 ,...,ξ k ,θ] ev and realize o(2k+ 1)so thatξ i /mapsto→ˆξ i ,η i /mapsto→/planckover2pi1 ∂ ∂ξ i ,θ/mapsto→/planckover2pi1(ˆθ+ ∂ ∂θ ). As above for o(2k), set/planckover2pi1= 1. Then, asabove, H i v= [X + i ,X − i ]ˆ1 = 0fori<k, whereas H k ˆ1 = [X + k ,X − k ]ˆ1 = 2 2 /parenleftbigg ∂ k (ˆθ+ ∂ ∂θ) 2 ˆξ k +ˆξ k (ˆθ+ ∂ ∂θ) 2 ∂ k /parenrightbigg ˆ1 =ˆ1. Soˆ1isindeed the highestweightvectorof the kthfundamental representation. 4.2. Stringy superalgebras. Case vir. For the basis of virtakee i =t i+1 d dt , i∈Z,andthe centralelement z;letthe bracketbe [e i ,e j ] = (j−i)e i+j − 1 12δ ij (i 3 −i)z. (∗) We advise the reader to refresh deﬁnitions of stringy superalgebras and various modules over them, see [21], where we also try to convince physicists not to use theterm“superconformalalgebra”(except,perhaps,for k L (1|1)and k M (1|1)).In particular,recallthat F λ,µ = Span(ϕ i =t µ+i (dt) λ |i∈Z). Statement .The only instances when F λ,µ possesses an invariant symmetric nondegenerate bilinear form are the space of half-densities, √ Vol =F 1/2,0 , and itstwistedversion, F 1/2,1/2 and inbothcasestheform is: (f √ dt,g √ dt) = /integraldisplay fg·dt; the only instances when F λ,µ possesses an invariant skew-symmetric forms are thequotientspaceoffunctionsmoduloconstants, dF=F 0,0 /C·1,and 1 2 -twisted functions, √ tF=F 0,1/2 andinbothcasesthe formis: (f,g) = /integraldisplay f·dg. Let∂ i = ∂ ∂ϕ i (whereϕ i =t µ+i (dt) λ ).Let osc( √ Vol)bethe vir-submoduleof theexterioralgebraonϕ i fori<0containingtheconstant ˆ1.Sincethegenerators kievarwe.tex; 12/03/2001; 3:49; p.111 HOWEDUALITY ANDLIE SUPERALGEBRAS 105 e i of viracts onF λ,µ as (sums over i∈Z) e 1 = /summationtext (µ+i+ 2λ)ϕ i+1 ∂ i = /summationtext iϕ i+1 ∂ i , e −1 = /summationtext (µ+i+ 1)ϕ i ∂ i+1 = /summationtext (i+ 1)ϕ i ∂ i+1 ; e 2 = /summationtext (µ+i−λ)ϕ i+1 ∂ i = /summationtext iϕ i+1 ∂ i , e −2 = /summationtext (µ+i+ 3λ)ϕ i ∂ i+1 = /summationtext (i+ 1)ϕ i ∂ i+1 , and representing e 0 andzas brackets of e ±1 ande ±2 from (∗)we immediately deducethat thehighest weights (c,h)ofosc( √ Vol)is(− 1 3 ,0). Forthespinorrepresentations spin( √ tF)andspin(dF)(realizedonthe sym- metricalgebra ofϕ i fori<0) we similarly obtain that the highest weights (c,h) are( 1 6 , 1 2 )forspin( √ tF)and(− 1 6 ,0)forspin(dF). Observe that the representations spin( √ tF),spin(dF)andosc( √ Vol)are constructed on ahalfof the generatorsusedto construct Spin(F λ;µ ). 4.3. The highest weights of the spinor representations of k L (1|n)and k M (1|n). In the following theorem we give the coordinates (c,h;H 1 ,...)of the highest weight of the spinor representations Spin(F λ;µ )of the contact superal- gebra k L (1|n)with respect to z(the central element), K t , and, after semicolon, on the elements of Cartan subalgebra, respectively. For k M (1|n)we write ˜h;˜H i . (Observe that for n > 4the Cartan subalgebra has more generators than just H 1 =K ξ 1 η 1 ,...,H k =K ξ k η k which generate the Cartan subalgebra of k(1|2k), thealgebraof contact vector ﬁeldswithpolynomial coef ﬁcients.) n 0 1 2≥ 3 c12λ 2 −12λ+ 2−12λ+ 3 6 0 h(µ+ 2λ)(µ+ 1)µ+ 2 λ 2µ+ 2λ+ ν2 n−1 (µ+λ) + 2 n− 3 ˜ h – 2µ+ 3λ− 1 4 2µ+ 2λ− 1 2 2 n−1 (µ+λ ) Theorem . Let (c,h;H 1 ,...) be the highest weight of the spinor representa- tion Spin(F λ;µ ) of k L (1|n) . The highest weight of the oscillator representation Osc(F λ;µ ) = Spin(Π(F λ;µ )) is (−c,h;H 1 ,...) andsimilarlyfor k M (1|n) . For n/negationslash= 2 ,allthecoordinatesofthehighestweightotherthan c , h vanish.For n= 2 the value of H on the highest weight vector from Spin(F λ,ν;µ ) is equal to ν . The values of c and h (or ˜h ) on modules Spin(F λ;µ ) are given in the above table. Up torescaling,these resultsareknownforsmall n,see [29],[28] andrefs. Remark. For the contact superalgebras gon the 1|n-dimensional supercircle our choice of g-modulesV=F λ;µ from which we constructed Spin(V⊕V ∗ )is natural for small n: there are no other modules! For larger nit is only justiﬁed if kievarwe.tex; 12/03/2001; 3:49; p.112 106 D.LEITES, I.SHCHEPOCHKINA weareinterestedinsemi-inﬁnitecohomologyof gandnotinrepresentationtheory per se. For the superalgebras gof series vectand svectthe adjoint module gis of the formT(id ∗ ), i.e, it is either coinduced from multidimensional representation ( vect), or is a submodule of such a coinduced module ( svect). Spinor-oscillator representations of this typewerenot studiedyet,cf. sec.5. 4.4. Other spinor representations . 1) Among various Lie superalgebras for which it is interesting to study spinor-oscillator representations, the simple (or close to them) maximal subsuperalgebras of poare most interesting. The list of suchmaximalsubalgebrasisbeingcompleted;variousmaximalsubalgebraslisted in [48] distinct from the sums of mutual centralizers also provide with spinor representations. As an interesting example consider A. Sergeev’s Lie superalgebra as, the nontrivial central extension of the Lie superalgebra spe(4)preserving the odd bilinear form and the volume on the (4|4)-dimensional superspace, see [49, 50]. Namely, consider po(0|6), the Lie superalgebra whose superspace is the Grass- mann superalgebra Λ(ξ,η)generated by ξ 1 ,ξ 2 ,ξ 3 ,η 1 ,η 2 ,η 3 and the bracket is the Poisson bracket. Recall also that the quotient of po(0|6)modulo center is h(0|6) = Span(H f |f∈Λ(ξ,η)),where H f = (−1) p(f) /summationdisplay (∂ f ∂ξ j ∂ ∂η j +∂ f ∂η j ∂ ∂ξ j ). Now, observe that spe(4)can be embedded into h(0|6). Indeed, setting degξ i = degη i = 1for alliwe introduce a Z-grading on Λ(ξ,η)which, in turn, induces aZ-grading on h(0|6)of the form h(0|6) =⊕ i≥−1 h(0|6) i . Since sl(4)∼= o(6), we canidentify spe(4) 0 with h(0|6) 0 . Itisnotdifﬁculttoseethattheelementsofdegree −1inthestandardgradings of spe(4)and h(0|6)constitute isomorphic sl(4)∼= o(6)-modules. It is subject to adirectveriﬁcation that itis reallypossibleto embed spe(4) 1 into h(0|6) 1 . A. Sergeev’s extension asis the result of the restriction onto spe(4)⊂ h(0|6) of the cocycle that turns h(0|6)into po(0|6). The quantization (with parameter λ) deforms po(0|6)into gl(Λ(ξ)); the through maps T λ : as−→ po(0|6)−→ gl(Λ(ξ))are representations of asin the 4|4-dimensional modules Spin λ . The explicitformof T λ is asfollows: T λ : /parenleftbigg a b c−a t /parenrightbigg +d·z/mapsto→ /parenleftbigg a b−λ˜c c−a t /parenrightbigg +λd·1 4|4 , where 1 4|4 is the unit matrix and ˜c ij =c kl for any skew-symmetric matrix c ij = E ij −E ji andanyevenpermutation (1234)/mapsto→(ijkl).Clearly,T λ isanirreducible representationfor any λandT λ /negationslash/similarequalT µ forλ/negationslash=µ. 2) Maximal subalgebras (for further examples see [48]) and a conjecture. LetV 1 be a linear superspace of dimension (r|s); letΛ(n)be the Grassmann kievarwe.tex; 12/03/2001; 3:49; p.113 HOWEDUALITY ANDLIE SUPERALGEBRAS 107 superalgebra with nodd generators ξ 1 ,...,ξ n and vect(0|n) = derΛ(n)the Lie superalgebra ofvector ﬁeldson the (0|n)-dimensional supermanifold. Let g= gl(V 1 )⊗Λ(n)⊃ + vect(0|n)bethesemidirectsum(theidealattheopen partof⊃ + )withthenaturalactionof vect(0|n)ontheideal gl(V 1 )⊗Λ(n).TheLie superalgebra ghasanaturalfaithfulrepresentation ρinthespace V=V 1 ⊗Λ(n) deﬁned by theformulas ρ(X⊗ϕ)(v⊗ψ) = (−1) p(ϕ)p(ψ) Xv⊗ϕψ, ρ(D)(v⊗ψ) =−(−1) p(D)p(v) v⊗Dψ for anyX∈ gl(V 1 ),ϕ,ψ∈Λ(n),v∈V 1 ,D∈ vect(0|n). Let us identify the elementsfrom gwiththeirimagesunder ρ,soweconsider gembeddedinto gl(V). Theorem ([48]) 1) The Lie superalgebra gl(V 1 )⊗Λ(n)⊃ + vect(0|n) is max- imal irreducible in sl(V 1 ⊗Λ(n)) unless a)dimV 1 = (1,1) or b)n= 1 and dimV 1 = (1,0) or (0,1) or (r|s) for r/negationslash=s . 2) If dimV 1 = (1,1) ,then gl(1|1)∼=Λ(1)⊃ + vect(0|1) , so gl(V 1 )⊗Λ(n)⊃ + vect(0|n)⊂Λ(n+ 1)⊃ + vect(0|n+ 1) anditisthe biggersuperalgebrawhich is maximalirreduciblein sl(V) . 3) If n= 1 and dimV 1 = (r|s) for r >s> 0 , then g is maximal irreducible in gl(V) . Conjecture .Supposer+s= 2 N .Then, dimVcoincideswith dim Λ(W)for somespaceW.Wesuspectthatthiscoincidenceisnotaccidentalbutisoccasioned by the spinor representations of the maximal subalgebras described above. The same applies to q(V 1 )⊗Λ(n)⊃ + vect(0|n), a maximal irreducible subalgebra in q(V 1 ⊗Λ(n)). 4.5. Selected problems . 1) The spinor and oscillator representations are real- izedinthesymmetric(perhaps,supersymmetric)algebraofthemaximalisotropic (at least for g= sp(2k)and o(2k)) subspace Vof the identity g-module id =V⊕V ∗ . But one could have equally well started from another g-module. For an interesting study of spinor representations constructed from W/negationslash= id, see [45]. To consider in a way similar to sec. 2 contact stringy superalgebras g= k L (1|n)and k M (1|n), as well as other stringy superalgebras from the list [21], we have to replace F λ,µ with modulesT µ (W)of (twisted) tensor ﬁelds on the supercircle and investigate how does the highest weight of ˆ1∈Osc(T µ (W)) orˆ1∈Spin(T µ (W))constructed from an arbitrary irreducible co(n)-module W=V⊕V ∗ depend on the highest weight of W. (It seems that the new and absolutely remarkable spinor-like representation Poletaeva recently constructed [46] isobtained inthis way.) To give the reader a feel of calculations, we consider here the simplest non- trivial case o(3) = sl(2). The results may (and will) be used in calculations of kievarwe.tex; 12/03/2001; 3:49; p.114 108 D.LEITES, I.SHCHEPOCHKINA Spin(T µ (W))for g= k L (1|n)and k M (1|n)forn= 3,4. As is known, for every N∈Z + there exists an irreducible (N+ 1)-dimensional g-module with highest weightN. This module possesses a natural nondegenerate g-invariant bilinear form which is skew-symmetric for N= 2k+ 1and symmetric for N= 2k. The corresponding embeddings g−→ o(2k+ 1)and g−→ sp(2k)are called principal, see [19] and references therein. Explicitly, the images of the Chevalley generatorsX ± of sl(2)areasfollows: X − /mapsto→ /summationtext X − i , X + /mapsto→    N(N+ 1)X + N + /summationtext 1≤i≤N−1 i(N+ 1−i)X + i forN= 2k+ 1 N 2 X + N + /summationtext 1≤i≤N−1 i(2N−i)X + i forN= 2k. From the commutation relations between X + andX − we derive that only X ± N give a nontrivial contribution to the highest weight HWof the sl(2)-module Spin(L N );we have: HW =    N(N+ 1) ifN= 2k+ 1 − 1 2 N 2 ifN= 2k. 2) Observe, that the notion of spinor-oscillator representation can be broad- ened to embrace the subalgebras of the Lie superalgebra hof Hamiltonian vector ﬁeldsandtheirimagesunderquantization;wecallthethroughmapthe projective spinor-oscillator representation . 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FunctionalAnal. Appl. 24(1990), no.3, 229–230 54. Voronov A., Semi-inﬁnite homologicalalgebra , Invent. math. 113,1993, 103–146 55. Weyl H., The classical groups. Their invariants and representations . Fifteenth printing. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ,1997. xiv+320 pp. kievarwe.tex; 12/03/2001; 3:49; p.118 kievarwe.tex; 12/03/2001; 3:49; p.119 ENVELOPING ALGEBRAOF GL(3)AND ORTHOGONALPOLYNOMIALS ALEXANDER SERGEEV †‡ (Correspondence)DepartmentofMathematics,UniversityofStock- holm,Roslagsv.101,Kr ¨aftrikethus6,S-10691,Stockholm,Sweden; [email protected] (On leave of absence from Balakovo In- stitute of Technique of Technology and Control, Branch of Saratov TechnicalUniversity,Balakovo, SaratovRegion, Russia) Abstract. LetAbeanassociativealgebraover CandLaninvariantlinearfunctionalonit(trace). Letωbe an involutive antiautomorphism of Asuch thatL(ω(a)) =L(a)for anya∈A. Then Aadmits a symmetric invariant bilinear form /angbracketlefta,b/angbracketright=L(aω(b)). ForA=U( sl(2))/ m, where m is any maximal ideal of U( sl(2)), Leites and I have constructed orthogonal basis whose elements turned out tobe, essentially, Chebyshev and Hahn polynomials inone discretevariable. Here I takeA=U( gl(3))/ mfor the maximal ideals mwhich annihilate irreducible highest weight gl(3)-modules of particular form (generalizations of symmetric powers of the identity rep- resentation). In whis way we obtain multivariable analogs of Hahn polynomials. Clearly, one can similarly consider gl(n)and gl(m|n)instead of gl(3)but the amount of calculations isappalling. §1.Background 1.1. Lemma .LetAbe an associative algebra generated by a set X. Denote by [X,A]thesetoflinearcombinationsoftheform /summationtext [x i ,a i ],wherex i ∈X,a i ∈A. Then [A,A] = [X,A]. Proof. Letus applythe identity ([3],p.561) [ab,c] = [a,bc] + [b,ca]. (1.1.1) Namely, let a=x 1 ...x n ; let us induct on nto prove that [a,A]⊂[X,A]. Forn= 1the statement is obvious. If n > 1, thena=xa 1 , wherex∈Xand due to(1.1.1) wehave [a,c] = [xa 1 ,c] = [x,a 1 c] + [a 1 ,cx] . ‡ I am thankful to D. Leites for encouragement and help and to ESI, Vienna, for hospitality and support. † [email protected] kievarwe.tex; 12/03/2001; 3:49; p.120 114 A.SERGEEV 1.2. Lemma .LetAbe an associative algebra and a/mapsto→ω(a)be its involutive antiautomorphism (transposition for A= Mat (n)). LetLbe an invariant func- tional onA(like trace, i.e., L([A,A]) = 0) such thatL(ω(a)) =L(a)for any a∈A.Deﬁne thebilinearform on Aby setting /angbracketleftu,v/angbracketright=L(uω(v)) foranyu,v∈A. (1.2.1) Then i)/angbracketleftu,v/angbracketright=/angbracketleftv,u/angbracketright; ii)/angbracketleftxu,v/angbracketright=/angbracketleftu,ω(x)v/angbracketright; iii)/angbracketleftux,v/angbracketright=/angbracketleftu,vω (x)/angbracketright; iv)/angbracketleft[x,u],v/angbracketright=/angbracketleftu,[ω(x),v]/angbracketright. Proof. (Clearly,iii)issimilarto ii)). i)/angbracketleftu,v/angbracketright=L(uω(v)) =L(ω(uω(v))) =L(vω(u)) =/angbracketleftv,u/angbracketright. ii)/angbracketleftxu,v/angbracketright=L(xuω(v)) =L(uω(v)x) =L(uω(ω(x)v)) =/angbracketleftu,ω(x)v/angbracketright. iv)/angbracketleft[x,u],v/angbracketright=/angbracketleftxu,v/angbracketright−/angbracketleftux,v/angbracketright /angbracketleftu,ω(x)v/angbracketright−/angbracketleftu,vω (x)/angbracketright=/angbracketleftu,[ω(x),v]/angbracketright. 1.3. Traces and forms on U( g). Let gbe a ﬁnite dimensional Lie algebra, Z( g)thecenterof (U( g),WtheWeylgroupof gand haCartansubalgebraof g. The following statements are provedin[1]. 1.3.1. Proposition .i)U( g) =Z( g)⊕[U( g),U( g)]. ii)Let/sharp:Z( g)⊕[U( g),U( g)]−→Z( g)be thenaturalprojection.Then (uv) /sharp = (vu) /sharp and (zv) /sharp =z(v) /sharp for anyu,v∈U( g) andz∈Z( g). iii)U( g) =S( h) W ⊕[U( g),U( g)]. iv)Letλbe the highest weight of the irreducible ﬁnite dimensional g-module L λ andϕtheHarish-Chandrahomomorphism. Then ϕ(u /sharp )(λ) =tr(u| L λ ) dimL λ . 1.3.2. OnU( g),deﬁneaformwith valuesin Z( g)by setting /angbracketleftu,v/angbracketright= (uω(v)) /sharp , (∗) whereωistheChevalley involution in U( g). LemmaTheform (∗)is nondegenerate on U( g). Proof. Let/angbracketleftu,v/angbracketright= 0for anyv∈U( g).ByProposition 1.3.1 tr(uω(v)) =ϕ((uω(v)) /sharp )(λ)·dimL(λ) =ϕ(/angbracketleftu,v/angbracketright)(λ)·dimL(λ) = 0; kievarwe.tex; 12/03/2001; 3:49; p.121 ALGEBRAOF GL(3)AND ORTHOGONAL POLYNOMIALS 115 hence,u= 0onL(λ)foranyirreducibleﬁnitedimensional L(λ),and,therefore, u= 0inU( g). 1.3.3. Lemma .For anyλ∈ h ∗ deﬁne aC-valuedform on U( g)by setting /angbracketleftu,v/angbracketright λ =ϕ(/angbracketleftu,v/angbracketright)(λ). The kernelof thisform isamaximal ideal in U( g). Proof. The form/angbracketleft·,·/angbracketright λ arises from a linear functional L(u) =ϕ(u /sharp )(λ); hence, by Lemma 1.2 its kernel is a twosided ideal IinU( g). OnA=U( g)/I, theforminducedisnondegenerate.If z∈Z( g),then /angbracketleftz,v/angbracketright λ =L(zω(v)) =L(z)L(ω(v)); hence,z−L(z)∈I.Therefore,theonly g-invariantelementsin Aarethosefrom Span (1). LetJbe a twosided nontrivial ( /negationslash=A,0) ideal inAandJ=⊕ µ J µ be the decomposition into irreducible ﬁnite dimensional g-modules (with respect to the adjoint representation). Since J/negationslash=A, it follows that J 0 = 0. Hence,L(J) = 0 and/angbracketleftJ,A/angbracketright.Thus,J= 0. 1.4.Gelfand–Tsetlinbasisandtransvectoralgebras .(Forrecapitulationon transvectoralgebrassee [7].) LetE ij be the matrix units. In gl(3), we ﬁx the subalgebra gl(2)embedded into the left upper corner and let hdenote the Cartan subalgebra of gl(3) = Span (E ii :i= 1,2,3). There is a one-to-one correspondence between ﬁnite dimensional irreducible representations of gl(3)and thesets (λ 1 ,λ 2 ,λ 3 ) such that λ 1 −λ 2 ,λ 2 −λ 3 ∈Z + . Such sets are called highest weights of the corresponding irreducible representa- tionwhosespaceisdenoted L λ .Witheachsuch λweassociateaGelfand–Tsetlin diagram Λ: λ 31 λ 32 λ 33 λ 21 λ 22 λ 11 (1.4.1) wherethe upperline coincideswith λandwhere “betweenness”conditions hold: λ k,i −λ k−1,i ∈Z + ;λ k−1,i −λ k,i+1 ∈Z + foranyi= 1,2;k= 2,3.(1.4.2) Set z 21 =E 21 , z 12 =E 12 ;z 13 =E 13 , z 32 =E 32 ; z 31 = (E 11 −E 22 + 2)E 31 +E 21 E 32 , z 23 = (E 11 −E 22 + 2)E 23 −E 21 E 13 .(1.4.3) kievarwe.tex; 12/03/2001; 3:49; p.122 116 A.SERGEEV Set(L λ ) + = Span (u:u∈L λ ,E 12 u= 0). 1.4.1.Theorem .(see[4]) Letvbeanonzerohighestweightvectorin L λ ,and ΛaGelfand–Tsetlindiagram. Set v Λ =z λ 21 −λ 11 21 z λ 31 −λ 21 31 z λ 32 −λ 22 32 v andletl ki =λ ki −i+ 1.Then i)The vectors v Λ parametrized by Gelfand–Tsetlin diagrams form a basis in L λ . ii)The gl(3)-action on vectors v Λ is given bythefollowing formulas E 11 v Λ =λ 11 v Λ ; E 22 v Λ = (λ 21 +λ 22 −λ 11 )v Λ ; E 33 v Λ = ( 3 /summationtext i=1 λ 3i − 2 /summationtext j=1 λ 2j )v Λ ; E 12 v Λ =−(l 11 −l 21 )(l 11 −l 22 )v Λ+δ 11 ; E 21 v Λ =v Λ−δ 11 ; E 23 v Λ =−(l 21 −l 31 )(l 21 −l 32 )(l 21 −l 33 ) (l 21 −l 22 )v Λ+δ 11 − (l 22 −l 31 )(l 22 −l 32 )(l 22 −l 33 ) (l 22 −l 21 )v Λ+δ 22 ; E 32 v Λ =(l 21 −l 11 ) (l 21 −l 22 )v Λ−δ 11 +(l 22 −l 11 ) (l 22 −l 21 )v Λ−δ 22 , where Λ±δ ki is obtained from Λby replacing λ ki withλ ki ±1and we assume thatv Λ = 0ifΛdoesnotsatisfyconditions onGTs-diagrams. iii)Thevectorsv Λ correspondingtotheGTs-diagramswith λ 21 =λ 11 forma basisof (L λ ) + . §2.Formulationsofmainresults 2.1. Modules S α (V). Let g= gl(3)be the Lie algebra of 3×3matrices overC. For anyα∈Cdenote byS α (V)the irreducible g-module with highest weight (α,0,0). Ifα∈Z + , thenS α (V)is the usual α-th symmetric power of the identity g-moduleV.Namely: S α (V) = Span (x k 1 1 x k 2 2 x k 3 3 :k 1 +k 2 +k 3 =α;k 1 ,k 2 ,k 3 ∈Z + ). Forα/negationslash∈Z + wehave(likeinsemi-inﬁnitecohomology ofLie superalgebras) S α (V) = Span (x k 1 1 x k 2 2 x k 3 3 :k 1 +k 2 +k 3 =α;k 2 ,k 3 ∈Z + ). kievarwe.tex; 12/03/2001; 3:49; p.123 ALGEBRAOF GL(3)AND ORTHOGONAL POLYNOMIALS 117 Remark. The expression x k fork∈Cis understood as a formal one, satisfying ∂x k ∂x =kx k−1 . OnS α (V)the g= gl(3)-actionisgivenby E ij /mapsto→x i ∂ ∂x j . 2.2.Theorem . i)S α (V)isanirreducible g-module forany α. ii)The kernelJ α of the corresponding to S α (V)representation of U( g)is a maximalidealif α/negationslash∈Z <0 . Set A α =U( g)/J α and letθbe the highest weight of the adjoint rep- resentation of g. Now consider A α as g-module with respect to the adjoint representation. iii) A α = ∞ ⊕ k=0 L kθ ifα/negationslash∈Z ≥0 . iv) A α = α ⊕ k=0 L kθ ifα∈Z ≥0 . v)The form/angbracketleftu,v/angbracketright α =ϕ(uω(v) /sharp )(α,0,0)is nondegenerate on A α forα/negationslash∈ Z <0 . 2.3. Let h= Span (E 11 ,E 22 ,E 33 )be Cartan subalgebra in gandε 1 ,ε 2 ,ε 3 the dual basis of h ∗ . LetQ={ /summationtext k i ε i : /summationtext k i = 0}be the root lattice of g. For anyµ∈Qdeﬁne ( A α ) µ ={u∈ A α : [h,u] =µ(h)ufor anyh∈ h}. (2.3.1) Clearly, A α isQ-graded: A α =⊕ µ∈Q ( A α ) µ . Theorem2.5belowshowsthat ( A α ) µ =Ru µ ,whereu µ ∈ A α isdeﬁneduniquely up toaconstantfactorand R=C[E 11 ,E 22 ,E 33 ]/(E 11 +E 22 +E 33 −α). Denote by ( A α ) + the subalgebra of giconsisting of vectors highest with respecttothe ﬁxed gl(2): ( A α ) + ={u∈ A α : [E 12 ,u] = 0}. (2.3.2) The algebra ( A α ) + also admitsQ-grading: ( A α ) + =⊕ ν∈Q ( A α ) + ν . (2.3.3) Denote:Q + ={ν∈Q: ( A α ) + ν /negationslash= 0}. Theorem 2.4 below shows that ( A α ) + ν =C[E 33 ]u + ν , whereν∈Q + . For f,g∈C[E 33 ]andν∈Q + set /angbracketleftf,g/angbracketright + ν =/angbracketleftfu + ν ,gu + ν /angbracketright α . (2.3.4) Forf,g∈Randµ∈Qset /angbracketleftf,g/angbracketright µ =/angbracketleftfu µ ,gu µ /angbracketright α . (2.3.5) kievarwe.tex; 12/03/2001; 3:49; p.124 118 A.SERGEEV Fork≥0andν∈Q + set f k,ν (E 33 )u ν = /braceleftBigg (adz 31 ) k (u ν+k(ε 1 −ε 3 ) ) forν(E 33 )≤0 (adz 23 ) k (u ν+k(ε 3 −ε 2 ) ) forν(E 33 )≥0 /parenleftbig (2.3.6) (2.3.7) /parenrightbig Fork,l≥0andν∈Q + set f ν l,k (E 11 ,E 22 ,E 33 )u ν = /braceleftBigg (adz 21 ) l (adz 31 ) k (u ν+k(ε 1 −ε 3 )+l(ε 1 −ε 2 ) ) forν(E 33 )≤0 (adz 21 ) l (adz 23 ) k (u ν+k(ε 3 −ε 2 )+l(ε 1 −ε 2 ) forν(E 33 )≥0 /parenleftbig (2.3.8) (2.3.9) /parenrightbig 2.4.Theorem .0)( A α ) + ν =C[E 33 ]u + ν ,whereu ν isdetermineduniquelyupto aconstantfactor. 1)/angbracketleft( A α ) + ν , A α ) + ν /angbracketright α = 0forν/negationslash=µ. 2)The polynomials f k,ν (E 33 )areorthogonalrelative /angbracketleft·,·/angbracketright + ν . 3)The polynomials f k,ν (E 33 )satisfythedifferenceequation (E 33 −ν(E 33 ) + 1)(E 33 +ν(E 11 )−α)∆f−E 33 (E 33 +ν(E 22 )−α−2)∇f= k(k+ 2ν(E 11 ) + 2)fifν(E 33 )<0; (E 33 + 1)(E 33 +ν(E 11 )−α)∆f−(E 33 −ν(E 33 ))(E 33 +ν(E 22 )−α−2)∇f= k(k−2ν(E 11 ) + 2)fifν(E 33 )≥0. 4)Explilcitely, f k,ν (E 33 )is of theform f k,ν (E 33 ) = const× 3 F 2   −k, k + 2ν(E 11 ) + 2,−E 33 1−ν(E 33 ), ν(E 11 )−α|1   , where 3 F 2 /parenleftbigg α 1 ,α 2 ,α 3 β 1 ,β 2 |z /parenrightbigg = ∞ /summationdisplay i=0 (α 1 ) i (α 2 ) i (α 3 ) i (β 1 ) i (β 2 ) i z i i! isageneralizedhypergeometricfunction, (α) 0 = 1and(α) i =α(α+ 1)...(α+ i−1)fori>0. 2.5. Theorem . 0)( A α ) ν =C[E 11 ,E 22 ,E 33 ]u ν , whereu ν is determined uniquely uptoaconstantfactor. 1)/angbracketleft( A α ) ν , A α ) ν /angbracketright α = 0forν/negationslash=µ. 2)Thepolynomials f ν l,k (E 11 ,E 22 ,E 33 )formanorthogonalbasisof Rrelative /angbracketleft·,·/angbracketright ν . 3)The polynomials w(f l,k )(E 11 ,E 22 ,E 33 )forw∈Wform an orthogonal basis ofRrelative/angbracketleft·,·/angbracketright w(ν) provided polynomials f l,k (E 11 ,E 22 ,E 33 )form an orthogonalbasis of Rrelative/angbracketleft·,·/angbracketright ν . kievarwe.tex; 12/03/2001; 3:49; p.125 ALGEBRAOF GL(3)AND ORTHOGONAL POLYNOMIALS 119 4)The polynomials f ν l,k (E 11 ,E 22 ,E 33 )forν∈Q + andν(E 33 )≤0satisfy thesystem oftwodifference equations (where H 1 =E 11 −E 22 ,H 2 =E 33 ) [f(H 1 + 2,H 2 )−f(H 1 ,H 2 )]· 1 4 (H 1 −H 2 +α+ 1)(H 1 +H 2 −α)− [f(H 1 ,H 2 )−f(H 1 −2,H 2 )]· 1 4 (H 1 −H 2 +α−ν(E 11 ))(H 1 +H 2 −α−1+ ν(E 22 )) = [l 2 +l(ν(E 11 ) +ν(E 22 ) + 1)) +ν(E 22 )−ν(E 11 )]f; [2α−ν(H 2 )(α+ 2 +ν(H 2 )) +H 2 (2α+ 1 + 2ν(H 2 ))−2H 2 2 ]f(H 1 ,H 2 )− 1 2 (H 2 + 1−ν(H 2 ))(H 1 −H 2 +α−2ν(E 11 ))f(H 1 −1,H 2 + 1)− 1 2 H 2 (H 1 −H 2 +α+ 2)f(H 1 + 1,H 2 + 1)− 1 2 (H 2 + 1−ν(H 2 ))(α−H 1 −H 2 )f(H 1 + 1,H 2 + 1)− 1 2 H 2 (α−H 1 −H 2 + 2−2ν(E 22 ))f(H 1 −1,H 2 −1) = [2k 2 + 4kl+ 4k(1 +ν(E 11 )) + 2l(1 +ν(E 11 )−ν(E 22 ))+ ν(E 11 ) 2 −ν(E 22 ) 2 + 4ν(E 11 )]f(H 1 ,H 2 ). §3.Proof ofTheorem 2.2 i) The module S α (V)is irreducible if and only if it has no vacum vectors (i.e, vectorsannihilatedby E 12 andE 23 .This is subjectto a direct veriﬁcation. ii) Follows from Exercise858 ofCh.8of [1]. iii) LetA 3 be the Weyl algebra (i.e., it is generated by the p i andq i fori= 1,2,3satisfying p i p j −p j p i =q i q j −q j q i = 0;p i q j −q j p i =−δ ij . (3.1) SettingE ij /mapsto→p i q j we see that the homomorphism ϕ:U( g)−→End (S α (V)) factors through A 3 andA 3 acts onS α (V)so thatp i /mapsto→x i andq i /mapsto→ ∂ ∂x i . Let us describe the image of ϕ. Tothisend, on A 3 ,introcude a grading by setting degp i = 1 degq i =−1 fori= 1,2,3. (3.2) Now itisclear that Imϕis thealgebra B 3 of elementsof degree0. To describe highest weight elements in B 3 , it sufﬁces to describe same in S k (V)⊗S k (V ∗ ). Let us identify S k (V)⊗S k (V ∗ )withEnd (S k (V)), letu∈ End (S k (V))commutes with the action of E 12 andE 23 onS k (V). But thenu is uniquely determined by its value on the lowest weight vector x k 3 ∈S k (V); moreover,E 12 x k 3 = 0. Hence, u(x k 3 ) =a 0 x k 3 + k /summationdisplay i=0 a i x i 1 x k−i 3 , kievarwe.tex; 12/03/2001; 3:49; p.126 120 A.SERGEEV so u(x k 3 ) = 1 ka 0 ( k /summationdisplay i=0 x i ∂ ∂x i )x k 3 + k /summationdisplay i=0 (k−i )! k!a i (x 1 ∂ ∂x 3 ) i x k 3 . This shows that the algebra of highest weight vectors in B 3 is generated by p 1 q 3 andz=p 1 q 1 +p 2 q 2 +p 3 q 3 . Ifα/negationslash∈Z ≥0 , then A α is the quotient of B 3 modulo (z−α).This proves iii). iv)Inthiscase A α = End (S k (V))andtheprooffollowsfromthearguments attheend of the aboveparagraph. v) By 1.3.3 the kernel of /angbracketleft·,·/angbracketright α inU( g)is a maximal ideal. But A α = U( g)/J α , whereJ α is maximal due to i). So J α coincides with the kernel of /angbracketleft·,·/angbracketright α inU( g)andthe formisnondegenerate on A α . §4.Proof ofTheorem 2.4 0) Direct computations show that the set of elements from A 3 commuting with E 12 is a subalgebra generated by p 1 ,q 2 ,p 3 ,q 3 andz=p 1 q 1 +p 2 q 2 +p 3 q 3 . So thisalgebrais the linearspan of theelements oftheform u=p k 1 1 q k 2 2 p k 3 3 q k 4 3 z k 5 . Ifu∈B 3 ,thenk 1 +k 3 =k 2 +k 4 ,so u= /braceleftBigg p k 1 1 q k 2 2 p k 3 −k 4 3 p k 4 3 q k 4 3 z k 5 ifk 3 ≥k 4 p k 1 1 q k 2 2 p k 3 3 q k 3 3 q k 4 −k 3 3 z k 5 ifk 3 ≤k 4 .(4.1) Hence,settingfor ν= /summationtext k i ε i suchthat /summationtext k i = 0,k 1 ≥0andk 2 ≤0 u + ν = /braceleftBigg p k 1 1 q −k 2 2 p k 3 3 ifk 3 ≥0 p k 1 1 q −k 2 2 q −k 3 3 ifk 3 ≤0(4.2) we obtainthestatement desired. 1) Letu∈( A + α ) µ ,v∈( A + α ) ν , andh∈ h. Then by heading iv) of Lemma 1.2 we obtain: /angbracketleft[h,u],v/angbracketright=µ(h)/angbracketleftu,v/angbracketright=/angbracketleftu,[h,v]/angbracketright=ν(h)/angbracketleftu,v/angbracketright. So/angbracketleftu,v/angbracketright= 0ifµ/negationslash=ν. 2)Letν(E 33 )≤0.We have: f k,ν u + ν = (adz 31 ) k (u + ν+k(ε 1 −ε 3 ) ) = adz 31 (adz 31 ) k−1 (u + ν+(k−1)(ε 1 −ε 3 )+(ε 1 −ε 3 ) ) = (adz 31 )f k−1,ν+(ε 1 −ε 3 ) u + ν+(ε 1 −ε 3 ) . kievarwe.tex; 12/03/2001; 3:49; p.127 ALGEBRAOF GL(3)AND ORTHOGONAL POLYNOMIALS 121 Directveriﬁcation showsthat (here h=E 33 ) (adz 31 )(fu + ν ) ={(E 33 −ν(E 33 ))[(E 33 −α)ν(E 11 ) + (ν(E 11 )−1)ν(E 11 )]f(h)− E 33 [(E 33 −α)ν(E 11 )−(ν(E 22 ) + 2)ν(E 11 )]f(h−1)}u ν−(ε 1 −ε 3 ) . (4.3) It easilyfollowsfromLemma1.2thatfor any z∈U( g)wehave /angbracketleft(adz)(u),v/angbracketright=/angbracketleftu,(adω(z))(v)/angbracketright, but ω(z 31 ) =ω((E 11 −E 22 + 2)E 31 +E 21 E 32 ) =E 13 (E 11 −E 22 + 2) +E 23 E 12 . Sincefu ν isahighestweightvectorwithrespecttotheﬁxed gl(2),itfollows that (adω(z 31 ))(fu ν ) = (ad (E 13 (E 11 −E 22 + 2))(fu ν ) = (ν(E 11 )−ν(E 22 ) + 2)∆f·u ν+(ε 1 −ε 3 ) . Now, let us induct on k. Fork= 0the statement is obvious. For k > 0and degg<kwe have /angbracketleftf k,ν ,g/angbracketright ν =/angbracketleftf k,ν u + ν ,gu + ν /angbracketright= /angbracketleftf k−1,ν+(ε 1 −ε 3 ) u + ν+(ε 1 −ε 3 ) ,(ad (ω(z 31 )))(gu + ν )/angbracketright= /angbracketleftf k−1,ν+(ε 1 −ε 3 ) ,(ν(E 11 )−ν(E 22 ) + 2)∆g/angbracketright ν+(ε 1 −ε 3 ) = 0 by inductivehypothesis. The caseν(E 33 )≥0is similar. 3) Observe that z=E 13 E 31 +E 23 E 32 belongs to the centralizer of gl(2)in U( g). Letν(E 33 )≤0. Thenu + ν =p k 1 1 q k 2 2 q k 3 3 as in (4.1.2). Having applied adz kievarwe.tex; 12/03/2001; 3:49; p.128 122 A.SERGEEV tofu + ν we obtain: (adz)(fu + ν ) =E 13 E 31 fu + ν +fu + ν E 31 E 13 −E 13 fu + ν E 31 −E 31 fu + ν E 13 + E 23 E 32 fu + ν +fu + ν E 32 E 23 −E 23 fu + ν E 32 −E 32 fu + ν E 23 = E 11 (E 33 + 1)fu + ν +fu + ν E 33 (E 11 + 1)− f(E 33 + 1)u + ν E 11 (E 33 + 1)−f(E 33 −1)E 33 (E 11 + 1)u + ν +E 22 (E 33 + 1)fu + ν + fu + ν E 33 (E 22 + 1)−f(E 33 + 1)E 22 u + ν (E 33 + 1)−f(E 33 −1)E 33 u + ν (E 22 + 1) = (E 11 +E 22 )(E 33 + 1)fu + ν + (E 33 −ν(E 33 ))(E 11 + 1−ν(E 11 ) +E 22 + 1−ν(E 22 ))fu + ν − f(E 33 + 1)·(E 33 + 1−ν(E 33 ))(E 11 +E 22 −ν(E 11 ))u + ν − f(E 33 −1)E 33 (E 11 +E 22 −ν(E 22 ) + 2)u + ν = [f(E 33 + 1)·(E 33 + 1−ν(E 33 ))(E 33 −α+ν(E 11 ))+ f(E 33 −1)E 33 (E 33 +ν(E 22 )−α−2)− (E 33 −α)(E 33 + 1)f−(E 33 −ν(E 33 ))(E 33 +ν(E 11 ) +ν(E 22 )−α−2)f]u + ν . This givesustheright handsideof theﬁrstequation ofheading3). Since adzcommutes with the gl(2)-action and preserves the degree of poly- nomialf, it follows that (adz)(fu ν ) =c·(fu ν ). Counting the constant factor, we arrivetotheﬁrst equationof heading3). The proofof thesecondequation issimilar. §5.Proof ofTheorem 2.5 0) Recall that B 3 is the subalgebra of A 3 of the elements of degree 0 relative grading (3.2). Fork∈Zsetr k i = /braceleftBigg =p k i ifk≥0 ,q −k i ifk≤0.Forγ= /summationtext k i ε i , where /summationtext k i = 0,set u γ =r k 1 1 r k 2 2 r k 3 3 . Clearly,B 3 isthelinear spanoftheelementsof theform p m 1 1 q l 1 1 p m 2 2 q l 2 2 p m 3 3 q l 3 3 ,wherem 1 +m 2 +m 3 =l 1 +l 2 +l 3 . It is alsoclearthat eachsuch element can berepresented in the form f(E 11 ,E 22 ,E 33 )r k 1 1 r k 2 2 r k 3 3 . This completestheproofofheading0). kievarwe.tex; 12/03/2001; 3:49; p.129 ALGEBRAOF GL(3)AND ORTHOGONAL POLYNOMIALS 123 1)Proofis similar tothatfromsec. 4.2. 2) Letν(E 33 )≤0. By setting H 1 =E 11 −E 22 ,H 2 =E 22 −E 33 we identifyR=C[E 11 ,E 22 ,E 33 ]/(E 11 +E 22 ,+E 33 −α)withC[H 1 ,H 2 ].LetΛis aGelfand–Tsetlindiagramofthe followingform: ν(E 11 ) +k+l 0 −(ν(E 11 ) +k+l) ν(E 11 ) +l−(ν(E 22 ) +l) ν(E 11 ) Fromtheexplicitformula for f ν k,l we derivethat f ν k,l u ν =v Λ . (5.1) Now, consider the following operators from the maximal commutative subal- gebra ofU( g): E 11 ,E 22 , Ω 2 =E 2 11 +E 2 22 +E 11 −E 22 + 2E 21 E 12 , Ω 3 =E 2 11 +E 2 22 +E 2 33 +E 11 −E 22 +E 11 −E 33 +E 22 −E 33 + 2E 21 E 12 + 2E 31 E 13 + 2E 32 E 23 .(5.2) Then we have: E 11 v Λ =ν(E 11 )v Λ ;E 22 v Λ =−ν(E 22 )v Λ ; Ω 2 v Λ = [2l 2 + 2l(ν(E 11 ) +ν(E 22 ) + 1)+ ν(E 11 ) 2 +ν(E 22 ) 2 ]v Λ ; Ω 3 v Λ = 2(ν(E 11 ) +k+l)(ν(E 11 ) +k+l+ 2]v Λ .(5.3) It is easy tocheckthat the operators (5.2)satisfy ω(E 11 ) =E 11 ;ω(E 22 ) =E 22 ;ω(Ω 2 ) = Ω 2 ;ω(Ω 3 ) = Ω 3 and, therefore, they are selfadjoint relative the form /angbracketleft·,·/angbracketright. Formula (5.3) makes it manifest that operators (5.2) separate the vectors v Λ , hence, these vectors are pairwise orthogonal.Moreover, itiseasytoseethat f ν k,l isof the form f ν k,l =H l 1 H k 2 +..., where the dots designate the summands of degrees ≤k+lof the form H a 1 H b 2 , where (a,b)<(l,k)with respect to the lexicographic ordering. Thus, the f ν l,k constituteabasisof C[H 1 ,H 2 ]. kievarwe.tex; 12/03/2001; 3:49; p.130 124 A.SERGEEV 3) The statement follows from the fact that the Weyl group acts on A α and preservestheform /angbracketleft·,·/angbracketright. 4) Since the polynomials f ν l,k u ν are elements of a Gelfand–Tsetlin basis, they are eigenvectors for Ω 2 andΩ 3 with respect to the adjoint action of g= gl(3)on A α . As wehave showninsec5.2,wehave Ω 2 f ν l,k u ν = [2l 2 + 2l(ν(E 11 ) +ν(E 22 ) + 1)+ ν(E 11 ) 2 +ν(E 22 ) 2 ]f ν l,k u ν ; Ω 3 f ν l,k u ν = 2(ν(E 11 ) +k+l)(ν(E 11 ) +k+l+ 2]f ν l,k u ν . To derive the corresponding equations, we have to explicitly compute the actions ofΩ 2 andΩ 3 onfu ν . These straightforward computations imply the second equation. References 1. Dixmier J. Alg`ebres envellopentes , Gautier-Villars, Paris, 1974; Enveloping algebras , AMS, 1996 2. Leites D., Sergeev A., Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices . Theor. and Math. Physics, 1232000,no.2, 582–609 3. Montgomery S., Constructing simple Lie superalgebras from associative graded algebras . J. Algebra195(1997), no. 2, 558–579 4. Molev A.I., Yangiansand transvectoralgebras . Math.RT/9902060 5. Nikiforov A. F., Suslov S. K., Uvarov V. B. Classical orthogonal polynomials of a discrete variable. Translated from the Russian. Springer Series in Computational Physics. Springer- Verlag, Berlin, 1991. xvi+374 pp. 6. PinczonG., TheenvelopingalgebraoftheLiesuperalgebra osp(1|2) . J.Algebra, 132(1990), 219–242 7. Zhelobenko D., Predstavleniya reduktivnykh algebr Li . (Russian) [Representations of reduc- tive Lie algebras] Nauka, Moscow, 1994. 352 pp.; Zhelobenko D., Shtern A., Predstavleniya grupp Li. (Russian) [Representations of Lie groups] Spravochnaya Matematicheskaya Bib- lioteka.[Mathematical Reference Library], Nauka, Moscow, 1983.360 pp. kievarwe.tex; 12/03/2001; 3:49; p.131 NONINVERTIBILITY, SEMISUPERMANIFOLDSAND CATEGORIESREGULARIZATION STEVENDUPLIJ ∗† Kharkov NationalUniversity, Kharkov 61001, Ukraine WłADYSłAW MARCINEK Institute of Theoretical Physics, University of Wrocław, Pl. Maxa Borna 9,50-204Wrocław,Poland Abstract. The categories with noninvertible morphisms are studied analogously to the semisuper- manifoldswithnoninvertibletransitionfunctions.Theconceptsofregular n-cycles,obstructionand the regularization procedure are introduced and investigated. It is shown that the regularization of a category with nonivertible morphisms and obstruction form a 2-category. The generalization of some related structures to the regular case is given. 1. Introduction In the supermanifold noninvertible generalization approach [1–3] we study here the obstructed cocycle conditions in the category theory framework and extend themtosuchstructuresascategories,functions,(co-)algebras,(co-)modulesetc. This approach is connected with the higher regularity concept [4] and reconsid- ering the role of identities [5]. The introduced category regularization together withobstructionforma 2-category.Similarabstractstructuregeneralizationswere considered in topological QFT [6, 7], for n-categories [8–10], near-group cate- gories [11, 12] (with noninvertible elements) and weak Hopf algebras [13, 14] in which the counit does not satisfy ε(ab) =ε(a)ε(b)or satisfy ﬁrst order (in our classiﬁcation) regularity conditions [15, 16]. We ﬁrst show how to deal with noninvertibilityinthesupermanifoldtheory[17,18]andthenapplythisapproach tomoregeneral structures. ∗ [email protected] † WWW:http://gluon.physik.uni-kl.de/˜duplij kievarwe.tex; 12/03/2001; 3:49; p.132 126 S. DUPLIJ,W.MARCINEK 2. Supermanifolds and semisupermanifolds In the supermanifold theory [17–19] the phenomenon of noninvertibility obvi- ouslyarisesfromoddnilpotentelementsandzerodivisorsofGrassmannalgebras (also in the inﬁnite dimensional case [20]). Despite the invertibility question is quite natural, the answer is not so simple and in some cases can be nontrivial, e.g. in some superalgebras one can introduce invertible analog of an odd symbol [21], or construct elements without number part which are not nilpotent even topologically [22]. Several guesses concerning inner noninvertibility inherent in the supermanifold theory were made before, e.g. “...there may be no inverse pro- jection 0 atall”[23],“...ageneralSRSneedsnothaveabody 0 ”[24],or“...abody 0 may not even exist in the most extreme examples” [25]. It were also considered pure odd supermanifolds [26, 27] which give an important counterexample to the Coleman-Mandula theorem “...and provides us with a new, missed so far, version of the Poincar ´e supergroup” [28], exotic supermanifolds with nilpotent even co- ordinates [29] and supergravity with noninvertible vierbein [30]. Some problems with odd directions and therefore connected with noninvertibility in either event are described in [31, 32], and a perspective list of supermanifold problems was statedbyD. Leitesin [33]. The patch deﬁnition of a supermanifold M 0 in most cases differs from the patch deﬁnition of an ordinary manifold [34, 35] by “super-” terminology only and is well-known [36]. Let /uniontext α {U α ,ϕ α }is an atlas of a supermanifold M 0 , then itsgluingtransitionfunctions Φ αβ =ϕ α ◦ϕ −1 β satisfy the cocycle conditions Φ −1 αβ = Φ βα ,Φ αβ ◦Φ βγ ◦Φ γα = 1 αα (1) on overlaps U α ∩U β and on triple overlaps U α ∩U β ∩U γ respectively, where 1 ααdef =id(U α ).Toobtainapatchdeﬁnitionofanobjectanalogoustosuperman- ifold we try to weaken demand of invertibility of coordinate maps ϕ α . Consider a generalized superspace Mcovered by open sets U α as M= /uniontext α U α . We assume here that the maps ϕ α :U α →V α ⊂R n|m are not all homeomorphisms, i.e. amongthemtherearenoninvertible maps 1 . Deﬁnition 1. A semisupermanifold is a noninvertibly generalized superspace M represented as a semiatlas M= /uniontext α {U α ,ϕ α }with invertible and noninvertible coordinatemaps ϕ α :U α →V α ⊂R n|m . We do not concretize here the details, how the invertibility appears here, but instead we will describe it by some general relations between semitransition 0 number part. 1 UnderR n|m we imply some its noninvertible generalization[3]. kievarwe.tex; 12/03/2001; 3:49; p.133 SEMISUPERMANIFOLDSAND CATEGORIES 127 functionsandotherobjects.WeThenoninvertiblyextendedgluing semitransition functions of asemisupermanifoldaredeﬁned bythe equations Φ αβ ◦ϕ β =ϕ α , Φ βα ◦ϕ α =ϕ β (2) insteadof Φ αβ =ϕ α ◦ϕ −1 β ,whichobviouslyextendstheclassoffunctionstonon- invertible ones. Then we assume that instead of (1) the semitransition functions Φ αβ of asemisupermanifold Msatisfythe following relations Φ αβ ◦Φ βα ◦Φ αβ = Φ αβ (3) onU α ∩U β overlaps (invertibility isextendedto regularity) and Φ αβ ◦Φ βγ ◦Φ γα ◦Φ αβ = Φ αβ , (4) Φ βγ ◦Φ γα ◦Φ αβ ◦Φ βγ = Φ βγ , (5) Φ γα ◦Φ αβ ◦Φ βγ ◦Φ γα = Φ γα (6) on tripleoverlaps U α ∩U β ∩U γ and Φ αβ ◦Φ βγ ◦Φ γρ ◦Φ ρα ◦Φ αβ = Φ αβ , (7) Φ βγ ◦Φ γρ ◦Φ ρα ◦Φ αβ ◦Φ βγ = Φ βγ , (8) Φ γρ ◦Φ ρα ◦Φ αβ ◦Φ βγ ◦Φ γρ = Φ γρ , (9) Φ ρα ◦Φ αβ ◦Φ βγ ◦Φ γρ ◦Φ ρα = Φ ρα (10) onU α ∩U β ∩U γ ∩U ρ . We can write similar cycle relations to inﬁnity and call them towerrelations whichsatisfyidenticallyinthestandardinvertiblecase [36]. R EMARK 1. In any actions with noninvertible functions Φ αβ we are not allowed to cancel by them, because the semigroup of Φ αβ ’s is a semigroup without can- cellation, and we are forced to exploit the corresponding semigroup methods [37, 38]. Conjecture 2. The functions Φ αβ satisfying the relations (3)–(10) can be viewed assomenoninvertiblegeneralizationofthetransitionfunctionsascocyclesinthe corresponding ˇCech cohomologyofcoverings[39,40]. 3. Obstructedness and additional orientationonsemisupermanifolds The semisupermanifolds deﬁned above belong to a class of so called obstructed semisupermanifolds [1, 3] in the following sense. Let us rewrite relations (1) as theinﬁniteseries n= 1 : Φ αα = 1 αα , (11) kievarwe.tex; 12/03/2001; 3:49; p.134 128 S. DUPLIJ,W.MARCINEK n= 2 : Φ αβ ◦Φ βα = 1 αα , (12) n= 3 : Φ αβ ◦Φ βγ ◦Φ γα = 1 αα , (13) n= 4 : Φ αβ ◦Φ βγ ◦Φ γδ ◦Φ δα = 1 αα (14) ··· ··· Deﬁnition 3. A semisupermanifold is called obstructed , if some of the cocycle conditions (11)–(14)are broken. It can happen that starting from some n=n m all higher cocycle conditions holdvalid. Deﬁnition4. Obstructednessdegree ofasemisupermanifoldisamaximal n m for which the cocycle conditions (11)–(14) are broken. If all of them hold valid, then n mdef = 0. Obviously, that ordinary manifolds [35] (with invertible transition functions) havevanishingobstructedness,andtheobstructednessdegreeforthemisequalto zero,i.e.n m = 0. R EMARK 2. Theobstructedsemisupermanifoldsmayhavenonvanishingordinary obstruction which can be calculated extending the standard methods [17] to the noninvertible case. Therefore, using the obstructedness degree n m , we have possibility to clas- sify semisupermanifolds properly. Moreover, the pure soul supernumbers do not containunity.Obviouslythatobstructedsemisupermanifoldscannothaveidentity semitransition functions. The orientation of ordinary manifolds is determined by the Jacobian sign of transitionfunctions Φ αβ writtenintermsoflocalcoordinateson U α ∩U β overlaps [34, 35]. Since this sign belong to Z 2 , there exist two orientations on U α . Two overlapping charts are consistently oriented (or orientation preserving ) ifΦ αβ has positive Jacobian, and a manifold is orientable if it can be covered by such charts, thus there are two kinds of manifolds: orientable and nonorientable [35]. In supersymmetric case the role of Jacobian plays Berezinian [17] which has a “sign” belonging to Z 2 ⊕Z 2 , and so there are four orientations on U α and ﬁve corresponding kindsof supermanifold orientability[41,42]. Deﬁnition 5. In case a nonvanishing Berezinian of Φ αβ is nilpotent (and so has nodeﬁnitesignintheprevioussense)thereexistsadditional nilpotentorientation onU α ofasemisupermanifold. kievarwe.tex; 12/03/2001; 3:49; p.135 SEMISUPERMANIFOLDSAND CATEGORIES 129 AdegreeofnilpotencyofBerezinianallowsustoclassifysemisupermanifolds havingnilpotent orientability (see e.g. [43,44]). 4. Higherregularityand obstruction The above constructions have the general importance for anyset of noninvert- ible mappings. The extension of n= 2cocycle given by (3) can be viewed as some analogy with regular [45] or pseudoinverse [46] elements in semigroups or generalized inverses in matrix theory [47], category theory [48] and theory of generalized inverses of morphisms [49]. The relations (4)–(10) and with other ncan be considered as noninvertible analogue of regularity for higher cocycles. Therefore, byanalogywith(3)–(10)it is naturaltoformulate thegeneral Deﬁnition 6. An noninvertible mapping Φ αβ isn -regular , if it satisﬁes on overlaps n /bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright U α ∩U β ∩...∩U ρ tothe following conditions n+1 /bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright Φ αβ ◦Φ βγ ◦...◦Φ ρα ◦Φ αβ = Φ αβ +perm. (15) The formula (3) describes 3-regular mappings, the relations (4)–(6) corre- spond to 4-regular ones, and (7)–(10) give 5-regular mappings. Obviously that 3-regularitycoincideswith the ordinaryregularity. Let us consider a series of the selfmaps e (n) αα :U α →U α of a semisupermani- fold deﬁned as e (1) αα = Φ αα , (16) e (2) αα = Φ αβ ◦Φ βα , (17) e (3) αα = Φ αβ ◦Φ βγ ◦Φ γα , (18) e (4) αα = Φ αβ ◦Φ βγ ◦Φ γδ ◦Φ δα (19) ··· ··· We will call e (n) αα ’s tower identities (or obstruction of U α ). From (11)–(14) it follows that for ordinary supermanifolds obstruction coincide with the usual identitymap e (n),ordinary αα = 1 αα . (20) kievarwe.tex; 12/03/2001; 3:49; p.136 130 S. DUPLIJ,W.MARCINEK So the obstructedness degree can be treated as a maximal n=n m for which tower identities differ from the identity, i.e. (20) is broken. The obstruction gives the numerical measure of distinction of a semisupermanifold from an ordinary supermanifold. When morphisms are noninvertible (a semisupermanifold has a nonvanishing obstructedness), we cannot “return to the same point”, because in general e (n) αα /negationslash= 1 αα ,andwehavetoconsider“nonclosed”diagramsduetothefact that therelation e (n) αα ◦Φ αβ = Φ αβ isnoncancellative now (see R EMARK 1). Summarizing the above statements we propose the following intuitively con- sistent changing of the standard diagram technique as applied to noninvertible morphisms.Ineverycasewegetanewarrowwhichcorrespondstotheadditional multiplier, and so for n= 2weobtain Invertiblemorphisms Φ αβ Φ β α =⇒ Noninvertiblemorphisms Φ βα Φ αβ n=2 whichdescribesthetransitionfrom(12)to(3)andpresentstheordinaryregularity conditionformorphisms[48,49].Themostintriguingsemicommutativediagram isthetriangle one Invertiblemorphisms Noninvertiblemorphisms Φ αβ Φ γ α =⇒ +perm. Φ βγ Φ γα Φ αβ Φ βγ n=3 which generalizesthe cocycle condition(1). The higher n-regular semicommutative diagrams can be considered in the frameworkof generalizedcategories[9, 12,50]in the following way. 5. Categoriesand 2-categories There is an algebraic approach to the formalism considered in previous sections based on the category theory [5, 4]. A category Ccontains a collection C 0 of objects and a collection hom (C)of arrows (morphisms) (see e.g. [51]). The kievarwe.tex; 12/03/2001; 3:49; p.137 SEMISUPERMANIFOLDSAND CATEGORIES 131 collection hom (C)is the union of mutually disjoint sets hom C (X,Y )of arrows X f −→YfromXtoYdeﬁned for every pair of objects X,Y∈ C. It may happen that for a pair X,Y∈Cthe set hom C (X,Y )is empty. The associative composition of morphisms is also deﬁned. By an equivalence in Cwe mean a classofmorphisms hom /prime (C) = /uniontext X,Y∈(C 0 ) hom /prime C (X,Y )where hom /prime C (X,Y )isa subset of hom C (X,Y ). Two objects X,Yof the categoryCis equivalent if and onlyifthereis an morphism X s −→Yinhom /prime C (X,Y )suchthat s −1 ◦s=id X , s◦s −1 =id Y (21) LetX= (X 1 ,···,X n )be a sequence of objects of C. Our category can con- tainsaclassof noninvertible morphisms[48,4].A(strict) 2-categoryCconsistsof a collectionC 0 of objects as 0-cells and two collections of morphisms: C 1 andC 2 called 1-cells and 2-cells, respectively [52]. For every pair of objects X,Y∈C 0 thereisacategory C(X,Y )whoseobjectsare 1-cellf:X→YinC 1 andwhose morphisms are 2-cells. For a pair of 1-cellsf,g∈C 1 there is a 2-cells:f→g inC 2 .Foreverythree objects X,Y,Z∈C 0 thereis a bifunctor c:{C(X,Y )×C(Y,Z)−→C (X,Z)} (22) which is called a composition of 1-cells. There is an identity 1-cellid X ∈ C(X,X )which acts trivially on C(X,Y )orC(Y,X). There is also 2-cellid id X which actstriviallyon 2-cells. LetCbe a category with equivalence. Then one can see that collection of all equivalenceclassesofobjectsof Cformsa 2-category C(C).Theseclassesare 0- cellsof C(C),1-cellsareclassesofmorphismsof C.and 2-cellsaremapsbetween these classes. Observe that 1-cells of C(C)can be represented by morphisms of the underlying category C, but such representation is not unique. One equiva- lence class can be represented by several equivalent morphisms. One can deﬁne 2-morphismsonequivalenceclasses,and C(C)becomesa2-category.Ifthecate- goryCisequippedwithcertainadditionalstructures,thenonecantransformthem intoC(C). If for instanceCis monoidal category with product ⊗:C×C−→C , thenC(C)becomestheso-calledsemistrictmonoidal 2-category.Thismeansthat the product⊗(under some natural conditions) is deﬁned for all cells of the 2- category C(C). In the case of braided categories one can obtain the semistrict braided monoidal category [52]. Algebras, coalgebras, modules and comodules can be also included in this procedure. We apply such method to regularize categorieswith noninvertiblemorphisms andobstruction [5, 4]. 6. Categoriesand regularization LetCbe a category with invertible and noninvertible morphisms [5] and equiva- lence.Theequivalencein Cisheredeﬁnedastheclassofinvertiblemorphismsin thecategoryC. kievarwe.tex; 12/03/2001; 3:49; p.138 132 S. DUPLIJ,W.MARCINEK Deﬁnition7. A sequenceofmorphisms X 1f 1 −→X 2f 2 −→··· f n−1 −→X nf n −→X 1 (23) such that there is an (endo-)morphism e (3) X 1 :X 1 −→X 1 deﬁned uniquely by the followingequation e (n) X 1 :=f n ◦···◦f 2 ◦f 1 (24) and subjects to the relation f 1 ◦f n ◦···◦f 2 ◦f 1 =f 1 is said to be a regular n-cycleonCanditisdenoted by f= (f 1 ,...f n ). The (endo-)morphisms e (n) X i :X i −→X i corresponding for i= 2,... ,nare deﬁned by asuitablecyclicpermutationof abovesequence. Deﬁnition 8. The morphism e (n) X is said to be an obstruction of X. The mapping e (n) :X∈C 0 →e (n) X ∈hom(X,X )is called a regular n-cycle obstruction structure onC. If X 1g 1 −→X /prime 2g 2 −→··· g n−1 −→X /prime ng n −→X 1 is an another n-tuple of morphisms such that e (n) X 1 :g n ◦···◦g 2 ◦g 1 , then we assume that X /prime i is equivalent to X i ,fori= 2,... ,n. Deﬁnition 9. A maps:f⇒gwhich sends the object X i into equivalent object X /prime i andmorphism f i intog i is said to be obstruction n-cycle equivalence. Wehavethe diagram X 2f 2 −→··· f n−1 −→X n f 1 /arrownortheast f n /arrowsoutheast X 1 ⇓s X 1 g 1 /arrowsoutheast g n /arrownortheast X /prime 2g 2 −→··· g n−1 −→X /prime n (25) Lemma 10. There is a one to one correspondence between equivalence classes ofregularn-cycles andregular n-cycleobstruction structures. Iff= (f 1 ,...f n )is a class of regular n-cycles, then there is the correspond- ing regularn-cycle obstruction structure e:X∈C 0 →e X ∈hom(X,X )such that the relation (24) holds true. Let e (n) :X∈C 0 →e (n) X ∈hom(X,X )be a regularn-cycleobstruction in C. kievarwe.tex; 12/03/2001; 3:49; p.139 SEMISUPERMANIFOLDSAND CATEGORIES 133 Deﬁnition11. Amorphism α:X−→Yof thecategoryCsuchthat α◦e (n) X =e (n) Y ◦α (26) issaidto bearegular n-cycleobstructionmorphism from XtoY. It follows from (23) that the morphism αis in fact a sequence of morphism α:= (α 1 ,... ,α n )suchthatthediagram X 1f 1 −→X 2f 2 −→··· f n−1 −→X nf n −→X 1 α 1 ↓ ↓ ↓ ↓ α 1 Y 1g 1 −→Y 2g 2 −→··· g n−1 −→Y ng n −→Y 1 (27) iscommutative. Deﬁnition 12. A collection of all equivalence classes of objects C 0 with obstruc- tion structures e (n) :X∈C 0 →e (n) X ∈hom(X,X )is denoted by/Rfractureg n (C)and called an obstruction n-cycle regularization of C. The class of all regular n-cycle morphisms from XtoYisdenoteby/Rfractureg n (C)(X,Y ). Corollary 13. It follows from the Lemma 10 that the map s:α−→βwhich sends an arbitrary regular n-cycle morphisms α∈/Rfractureg n (C)(X,X /prime )into a reg- ularn-cycle morphisms β∈/Rfractureg n (C)(X,X /prime )is a regular obstruction n-cycle equivalence. One can deﬁne 2-morphisms and an associative composition of 2-morphisms suchthat/Rfractureg n (C)(X,Y )becomesacategoryforeverytwoobjects X,Y∈C 0 .If α:X−→Yandβ:Y−→Zaretwon-cyclemorphisms,thenthecomposition β◦α:X→Zisalsoan-cyclemorphism.Inthiswayweobtainthecomposition as bifunctors c /Rfractureg n :={/Rfractureg n (C)(X,Y )×/Rfractureg n (C)(Y,Z)−→/Rfractureg n (C)(X,Z)}(28) We summarizeourconsiderationsinthe following lemma: Lemma 14. The class/Rfractureg n (C)forms a (strict) 2-category whose 0-cells are equivalence classes of objects of Cwith obstructions, whose 1-cells are regular n-cycleobstructionmorphisms,andwhose 2-cellsareregularobstruction n-cycle 2-morphisms. kievarwe.tex; 12/03/2001; 3:49; p.140 134 S. DUPLIJ,W.MARCINEK 7. Regularization of monoidal categories functions and Yang-Baxter equation LetC=C(I,⊗)be a monoidal category, where Iis the unit object and ⊗: C×C−→C isthemonoidalproduct [53, 54]. If the following relation e (n) X ⊗e (n) Y =e (n) X⊗Y . (29) holdstrue,thenwe have Proposition 15. The monoidal product of two regular n-cyclesX 1 ,... ,X n and Y 1 ,... ,Y n withobstruction e (n) X 1 ,ande (n) Y ,respectively,is theregular n-cycle X 1 ⊗Y 1 ,⊗···⊗X n ⊗Y n with theobstruction e (n) X⊗Y . One can see that in this case /Rfractureg n (C)is the so-called semistrict monoidal category[52]. LetCandDbe two monoidal categories and let /Rfractureg n (C),/Rfractureg n (D)be their regularization 2-categories. We can introduce the notion of regular 2-functions, pseudonatural transformations and modiﬁcations. All deﬁnitions do not changed, butthepreservationoftheidentitycanbereplacedbytherequirementofpreserva- tion of obstruction morphisms e (n) X and the invertibility is replaced by regularity. If, for instance, there is a regular 2-functorF:/Rfractureg n (C)−→/Rfractureg n (C), then in additiontothestandarddeﬁnition[51] we havethefollowingrelation F(e X ) =e F(X) . (30) In the same manner we can “regularize” pseudo-natural transformations and modiﬁcations [50]. Let /Rfractureg n (C)be a semistrict monoidal 2-category. A pseudo- natural transformations B={B X,X /prime :X⊗X /prime →X /prime ⊗X}and two regular modiﬁcations B X⊗Y,Z ,B X,Y⊗Z such that B X⊗Y,Z X⊗Y⊗Z−→Y⊗Z⊗X B X,Y ⊗e Z /arrowsoutheast /arrownortheast e Y ⊗B X,Z Y⊗X⊗Z(31) and B X,Y⊗Z X⊗Y⊗Z−→Z⊗X⊗Y e X ⊗B Y,Z /arrowsoutheast /arrownortheast B X,Z ⊗e Y X⊗Z⊗Y(32) kievarwe.tex; 12/03/2001; 3:49; p.141 SEMISUPERMANIFOLDSAND CATEGORIES 135 and B X,X /prime ◦e X⊗X /prime =e X /prime ⊗X ◦B X,X /prime , (33) are said to be a regular n-cycle braiding. Obviously, these operations must sat- isfying all conditions of [52] with two changes indicated at the beginning of this section. Then the 2-category/Rfractureg n (C)is called a semistrict regular n-cycle braided monoidal category. This allows us to obtain here the following regular n-cycle Yang–Baxter equation[5, 4] B (1) Y,Z,X ◦B (2) Y,X,Z ◦B (1) X,Y,Z =B (2) Z,X,Y ◦B (1) X,Z,Y ◦B (2) X,Y,Z ,(34) wherethe notation B (1) X,Y,Z =B X,Y ⊗e Z ,B (2) X,Y,Z =e X ⊗B Y,Z has been used and the obstruction e X is exploited instead of the identity Id X . Solutions of the regular n-cycle Yang–Baxter equation (34) can be found by application ofthe endomorphism semigroupmethods used in [55, 16]. 8. Regularizationofalgebras,coalgebras,modulesandcomodules Let(C)beamonoidalcategoryand /Rfractureg n (C)beitsregularization.Itisknownthat an associative algebra in the category Cis an objectAof this category such that there is an associative multiplication m:A⊗A→A which is also a morphism of this category. If the multiplication is in addition a regular n-cycle morphism, thenthealgebraAissaidtobearegular n-cyclealgebra.Thismeansthatwehave therelation m◦(e A ⊗e A ) =e A ◦m. (35) Obviouslysuchmultiplicationnotneedtobeunique.Denoteby /Rfractureg n (C)(A⊗ A,A)a class of all such multiplications. We can see that a regular n-cycle 2- morphismss:m⇒nwhich send the multiplication minto a new one nshould be an algebra homomorphism. One can deﬁne regular n-cycle coalgebra or bial- gebra in a similar way. A comultiplication /triangle:A−→A⊗A can be regularized accordingto therelation /triangle◦e A = (e A ⊗e A )◦/triangle. (36) Inthiscase weobtain aclass /Rfractureg n (C)(A,A⊗A )of comultiplications. Let A Cbe a category of all left A-modules, whereAis a bialgebra. For the regularization/Rfractureg n ( A C)of theA–module action ρ M :A⊗M−→Mwe use thefollowingformula ρ M ◦(e A ⊗e M ) =e M ◦ρ M , (37) kievarwe.tex; 12/03/2001; 3:49; p.142 136 S. DUPLIJ,W.MARCINEK whereρ M :A⊗M−→Mis the left module action of AonM. The class of all such module actions is denoted by /Rfractureg n ( A C)(A⊗M,M). The monoidal operationin this categoryisgiven as the followingtensorproductof A-modules ρ M⊗N := (id M ⊗τ⊗id N )◦(ρ M ⊗ρ N )◦(/triangle⊗id M⊗N ),(38) whereτ:A⊗M→M⊗Ais the twist, i. e. τ(a⊗m) :=m⊗afor every a∈A,m∈M. Lemma 16. For the tensor product of module actions we have the following formula ρ M⊗N ◦(e A ⊗e M⊗N ) =e M⊗N ◦ρ M⊗N . (39) This lemma means that the tensor product of two module actions satisfy our regularity condition if and only if these two actions also satisfy the regularity condition(37). Observethatthereisalsoacategory C A ofrightA-comodules,where Aisan algebra.Wecanregularizethiscategoryinthefollowingway.Forthecoactionwe have ρ◦e A = (e M ⊗e A )◦ρ M , (40) and ρ M⊗N := (id M ⊗m A )◦(id M ⊗τ⊗id N )◦(ρ M ⊗ρ N ),(41) whereτ:M⊗N→N⊗Misthetwist,m A :A⊗A→A isthemultiplication inA. Conclusions Thus noninvertible extension of many abstract structures can be done in common generalway:byintroductionoftheobstructions(or n-cycles) ewhichareanalogs of units of the invertible case. In search of possible analogies we observe that “lne” can play the role of ﬁrst “fundamental group” for “space” of categories and vanishes for invertible morphisms, while its difference from “zero” can be treated as nontrivial “noninvertible topology” of such “space”. We also note that “nil-” extension of supermanifolds – semisupermanifolds [56, 3] – can be com- paredwiththe“meta-”extensionofsupermanifolds–metamanifolds[57–59]–to ﬁnd their complementarity or additivity and possibly for further generalizations simultaneously inbothways. Acknowledgments . 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Serganova, Symmetries wider than supersymmetry , Talk at the Kiev NATO ARW, September 2000, thisvolume . kievarwe.tex; 12/03/2001; 3:49; p.146 kievarwe.tex; 12/03/2001; 3:49; p.147 ANOVERVIEW OF NEWSUPERSYMMETRICGAUGE THEORIES WITH 2-FORM GAUGE POTENTIALS FRIEDEMANN BRANDT ∗ Max-Planck-Institut f ¨ur Mathematik in den Naturwissenschaften, Inselstraße 22-26, D-04103 Leipzig,Germany After October 1,2000: Max-Planck-Institut f ¨ur Gravitationsphysik, Albert-Einstein-Insti- tut, Am M ¨uhlenberg 1,D-14476 Golm, Germany Abstract. An overview of new 4d supersymmetric gauge theories with 2-form gauge potentials constructed by various authors during the past ﬁve years is given. The key r ˆole of three particular types of interaction vertices is emphasized. These vertices are used to develop a connecting per- spective on the new models and to distinguish between them. One example is presented in detail to illustratecharacteristicfeaturesofthemodels.Anewresultoncouplingsof2-formgaugepotentials to Chern-Simons forms is presented. 1. Introduction During the past ﬁve years, several new 4d supersymmetric gauge theories have been constructed by various authors [1]–[13]. Common to all these models is the presenceof2-formgaugepotentialsandacomplicated(nonpolynomial)structure of interactions and symmetry transformations (gauge symmetries, supersymme- try). The initial motivation to construct such models came from string theory and focused the attention ﬁrst on the vector-tensor (VT) multiplet [14, 15] of N=2 supersymmetry. Namely, in N=2 supersymmetric 4d heterotic string vacua, the dilaton is believed to reside in a VT multiplet (see, e.g., section 3 of the review [16]). In order to couple this multiplet to N=2 supergravity, its so-called central charge must be gauged and this leads inevitably to the structures characteristic of the new models (cf. remarks at the end of section 3). Only two of the works [1]–[13] are not devoted to the VT multiplet: in [11] a rather general class of new supersymmetric gauge theories with 2-form gauge ﬁelds is constructed, and [13] deals with the double tensor (TT) multiplet of N=2 supersymmetry and its ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.148 142 F.BRANDT couplings to vector and hyper multiplets. The TT multiplet is believed to be the dilaton-multipletofN=2supersymmetrictypeIIBsuperstringvacua[16]andthus it shouldplaytherea r ˆoleanalogoustothe VT multipletin heterotic vacua. The purpose of this contribution is to give an overview of the new models and to emphasize the key r ˆole of three types of cubic interaction vertices in these models. To this end, ﬁrst a brief excursion to consistent interactions of p-form gauge potentials in general is made in section 2. This will also show how the newmodelsﬁtintherecentclassiﬁcation[17–19]ofinteractionsbetween p-form gauge potentials. The three particular types of interaction vertices are identiﬁed and discussed in some detail in section 3, including a new result on couplings of 2-form gauge potentials to Chern-Simons forms. Then these vertices and the supersymmetrymultipletstructureareusedtocharacterizethevariousmodelsand to distinguish between them. In section 4, an explicit example is treated in detail to illustrate characteristic features of the new models. The example is an N=2 supersymmetricmodelfoundin[13],couplingtheTTmultipletmentionedabove totwoN=2vectormultiplets.Section5containsaselectionofopenproblemsand possible futuredevelopments. 2. Interactions of p-formgauge potentials Gauge invariance restricts the possible interactions of p-form gauge ﬁelds quite severely. In the simplest case, the gauge transformation of a p-form gauge poten- tialA= (1/p!)dx µ 1 ∧···∧dx µ p A µ 1 ...µ p is a natural generalization of the gauge transformationof theelectromagnetic gaugeﬁeld: δ (0) gauge A=dω⇔δ (0) gauge A µ 1 ...µ p =p∂ [µ 1 ω µ 2 ...µ p ] , (1) whereω µ 1 ...µ p−1 are arbitrary gauge parameter ﬁelds. Analogously to the electro- magneticcase,correspondinggaugeinvariantﬁeldstrengths arethus F=dA⇔F µ 0 ...µ p = (p+ 1)∂ [µ 0 A µ 1 ...µ p ] , (2) and the standard Lagrangian for a set of free p-form gauge ﬁelds is a linear combination ofMaxwell-type kinetic terms F µ 0 ...µ p F µ 0 ...µ p . A systematic investigation of the possible interaction vertices which can be added consistently to such a free Lagrangian L (0) was carried out by Hen- neaux and Knaepen [17–19]. They studied consistent deformations of the free LagrangianL (0) and ofthe gaugetransformations δ (0) gauge , L=L (0) +g α V (1) α +g α g β V (2) αβ +... (3) δ gauge =δ (0) gauge +g α δ (1) gaugeα +g α g β δ (2) gaugeαβ +... , (4) whereg α are continuous coupling constants (deformation parameters), such that thedeformedLagrangian Lisinvariantunderthedeformedgaugetransformations kievarwe.tex; 12/03/2001; 3:49; p.149 NEW SUPERSYMMETRIC GAUGE THEORIES 143 δ gauge moduloa total derivative, δ gauge L=∂ µ K µ . (5) To ﬁrst order in the coupling constants, (5) requires that the V (1) α beδ (0) gauge - invariant on-shell in the free theory modulo a total derivative. Furthermore, without loss of generality, one may neglect all V (1) α which vanish on-shell in the free theory modulo a total derivative because they can be removed by ﬁeld redef- initions (such vertices are therefore called trivial ones). Henneaux and Knaepen found thefollowing resultforthe remainingﬁrst-order vertices: Category 1: Vertices that are δ (0) gauge -invariant off-shell modulo a total deriva- tive and therefore do not modify the gauge transformations to ﬁrst order. There are two types of such vertices (modulo total derivatives). Those of the ﬁrst type dependonp-formgaugeﬁeldsonlyviatheﬁeldstrengths F µ 0 ...µ p andtheirderiva- tives.Ofcourse,thereareinﬁnitelymanyverticesofthistype.Thoseofthesecond typeareverticesoftheChern-Simons type A∧F∧···∧F (6) where theF’s may have different form-degrees and all form-degrees must sum up to the spacetime dimension. These vertices are δ (0) gauge -invariant only modulo a total derivative. Category 2: Vertices that are δ (0) gauge -invariant only on-shell in the free theory moduloatotalderivative.Theseverticesareofparticularinterestbecausetheyare accompanied by deformations of the gauge transformations. A remarkable result is that, when ordinary gauge ﬁelds (1-form gauge potentials) are absent, all these verticescanbebroughttothefollowingform(modulotrivialverticesandvertices ofcategory 1): A∧F∧···∧F∧ ∗ F∧···∧ ∗ F /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright atleastone ∗ F(7) where ∗ FdenotestheHodgedualof Fandtheremustbeatleastone ∗ Fbecause otherwise the vertex would be of the Chern-Simons type (6). Again, the F’s may have different form-degrees and all form-degrees must sum up to the spacetime dimension. Therefore there are only ﬁnitely many vertices (7) for a ﬁnite number ofp-form gauge ﬁelds. The ﬁrst order deformations of the gauge transformations which correspondto avertex(7) takethe form δ (1) gauge A=ω∧F∧···∧F∧ ∗ F∧···∧ ∗ F (8) where one of the ∗ F’s that occurs in (7) is omitted (for instance, when (7) con- tains only one ∗ F, then (8) contains no ∗ F). When 1-form gauge potentials are present, (7) still gives nontrivial ﬁrst-order vertices of category 2, but then there kievarwe.tex; 12/03/2001; 3:49; p.150 144 F.BRANDT maybeadditionalverticesofcategory2whichcannotbebroughttotheform(7). In particular, when at least three 1-form gauge potentials are present, there are Yang-Millscubicverticeswhichdifferfrom(7)becausetheycontaintwo‘naked’ gaugepotentialsinsteadofonlyone(thestructureofYang-Millscubicverticesis A∧A∧ ∗ Fwhere theA’s are 1-form gauge potentials and Fis a 2-form ﬁeld strength). Infour-dimensionalspacetimetherearethreedifferenttypesofcubicvertices (7)involving1-formgaugepotentials A 1 ,2-formpotentials A 2 andcorresponding ﬁeldstrengths F 2 =dA 1 andF 3 =dA 2 : A 2 ∧ ∗ F 3 ∧ ∗ F 3 (9) A 1 ∧ ∗ F 2 ∧ ∗ F 3 (10) A 1 ∧F 2 ∧ ∗ F 3 . (11) Thesearetheverticesmentionedin theintroduction. 3. Overviewofthenew models In accordance with commonly used nomenclature (which is actually somewhat unfair, see remarks at the end of this section), the vertices (9), (10) and (11) will be referred to as “Freedman-Townsend” (FT), “Henneaux-Knaepen” (HK) and “Chapline-Manton”(CM)vertices,respectively.Eachofthenewsupersymmetric models reviewed here contains at least one of these vertices. We label 1-form potentials and 2-form potentials by indices a= 1,2,...andi= 1,2,...re- spectively, and denote their component ﬁelds by A a µ andB i µν =−B i νµ . The ﬁeld strengths of A a µ are denoted by F a µν =∂ µ A a ν −∂ ν A a µ , the Hodge-dualized ﬁeld strengths of B i µν byH iµ = 1 2 ε µνρσ ∂ ν B i ρσ . The vertices (9), (10) and (11) read explicitly,usinga suitablenormalization, FT vertices: 1 4f ijk H i µ H j ν B k ρσ ε µνρσ (12) HKvertices: T iab H i µ F aµν A b ν (13) CM vertices: 1 2S iab H i µ F a νρ A b σ ε µνρσ (14) wherethef ijk ,T iab andS iab are constant coefﬁcients, with f ijk =−f jik , S iab =S iba . [S iab =S iba can be imposed without loss of generality because S i[ab] can be removed from the vertices (14) by subtracting trivial vertices.] These coefﬁcients aresubjecttoconditionsimposedby(5)atsecondorderinthecouplingconstants kievarwe.tex; 12/03/2001; 3:49; p.151 NEW SUPERSYMMETRIC GAUGE THEORIES 145 (deformationparameters).Viewing T iab andS iab astheentriesofmatrices T i and S i , these conditions read f ijl f klm +f jkl f ilm +f kil f jlm = 0 (15) [T i ,T j ] =f ijk T k (16) (S i T j −S j T i ) + (S i T j −S j T i ) /latticetop =f ijk S k . (17) To derive these conditions, it was assumed that the zeroth order Lagrangian is L (0) =−(1/2)H µi H i µ −(1/4)F a µν F µνa , and that (9), (10) and (11) are the only vertices of category 2 with non-vanishing coefﬁcients (vertices of category 1 do not modify these conditions, but switching on other vertices of category 2 might cause modiﬁcations orlead toadditionalconditions). (15) and (16) were already found in [17] and require that the f ijk be structure constants of a Lie algebra and that the T i be representation matrices of that Lie algebra, respectively. (17) was not derived in a previous work, to my knowledge. It requires that the symmetric parts of the matrices 2(S i T j −S j T i )be equal to f ijk S k .Thisisfulﬁlled,forinstance,if S i =NT i +T /latticetop i NwhereNisanarbitrary symmetricmatrix(i.e., S iab =N ac T icb +N bc T ica withN ab =N ba ),butthereare other solutionsas well. The correspondingﬁrstorderdeformations of the gauge transformationsare δ (1) gauge B i µν =−f ijk (H j µ ω k ν −H j ν ω k µ )− 1 2ε µνρσ T iab F ρσa ω b +S iab F a µν ω b δ (1) gauge A a µ =−T iab H i µ ω b . (18) The following table gives an overview of the new supersymmetric models. The vertices discussed above are used to distinguish between the various models. Inadditionthenumberofsupersymmetries(N=1orN=2supersymmetry)andthe supersymmetry multiplets are given. In the case of N=1 supersymmetry, T and V stand for tensor multiplets (also called linear multiplets) and vector multiplets respectively. In the case of N=2 supersymmetry, VT, TT and V stand for vector- tensormultiplets, double-tensor multipletsand vector multipletsrespectively. kievarwe.tex; 12/03/2001; 3:49; p.152 146 F.BRANDT susy multiplets interactions papers N=2 VT, V HK, CM [1,2, 7,8, 10] N=2 VT, V CM [3–5, 9] N=2 VT CM [6] N=1 T, V FT,HK, CM [11] N=2 VT HK [12] N=2 TT, V FT, HK [13] Of course, this table characterizes the various models only very roughly. The example in the next section is to illustrate characteristic features of these models. It is beyond the scope of this paper to review the various models in greater detail but I would like to add at least a few remarks: (a) Among all these models only those in [7] are locally supersymmetric, the other ones are globally supersym- metric. (b) The works on the VT multiplet overlap in part because some of these works rederive models which had already been found by means of other methods in previous works. (c) Models in the same row of the table may of course still differ. For instance, CM vertices in two models with the same multiplet content may contain different Chern-Simons forms (in the literature, this has led to a distinction between “linear” and “nonlinear” VT multiplets [2]). Different CM couplingscorrespondtodifferentsolutionstoEq.(17).Ofcourse,analogousstate- ments apply to the FT and HK vertices. (d) Some of the models in [11] possess extended (N≥2) supersymmetry. For instance, it has been pointed out in [12] that the model constructed there can be obtained from [11]. However, it is not clear how to sieve out systematically those models in [11] which have extended supersymmetry. Finallyafewcommentsonthehistorymaybeinorder.ModelswithFTinter- actionswereconstructedalreadybyOgievetskyandPolubarinov[20]alongtime before the work by Freedman and Townsend [21]. CM interactions have a long history too. It seems that they appeared ﬁrst in the early 80’s [22–24] and, again, the work by Chapline and Manton was not the ﬁrst one with such interactions. CM interactions attracted particular attention because of their crucial r ˆole in the Green-Schwarz anomaly cancellation mechanism [25] (the anomaly cancellation ismadepossiblebythedeformationofthegaugetransformationsassociatedwith CM vertices,seesection 2). HK interactions (in four-dimensional spacetime) were discovered much later. However, the ﬁrst models with such interactions were not found by Henneaux and Knaepen. Rather, it seems that HK interactions occurred for the ﬁrst time in [1] where the central charge of the VT multiplet was gauged. The connection of kievarwe.tex; 12/03/2001; 3:49; p.153 NEW SUPERSYMMETRIC GAUGE THEORIES 147 that gauging to HK vertices is the following. Gauging the central charge (e.g., via the Noether method) gives rise to a vertex V µ j µ whereV µ is a 1-form gauge ﬁeld andj µ is the Noether current corresponding to the central charge symmetry. That Noether current is j µ =H ν F νµ , and thus the vertex V µ j µ is a HK vertex. Combined FT and HK interactions, and the relation to Lie algebras, were found afterwards by Henneaux and Knaepen [17]. It seems that the ﬁrst and so far only workwith modelscontaining simultaneouslyFT,HK and CM vertices is [11]. 4. Example The example is an N=2 supersymmetric model coupling one TT multiplet to two V multiplets and involves HK vertices but no FT or CM vertices. A TT multiplet contains two 2-form gauge potentials B i µν (i= 1,2), two real scalar ﬁelds a i and two Weyl fermions χandψ. Each V multiplet contains a 1-form gauge potential A µ , a complex scalar ﬁeld φand two Weyl fermions λ i . The V multiplets are labeled by the index a= 1,2. This ﬁeld content is supplemented with auxiliary ﬁeldsh i µ which are embedded in the TT multiplet. These auxiliary ﬁelds allow one to construct the model in a compact polynomial form. In fact, it would be verycumbersometoconstructthemodelwithouttheseauxiliaryﬁeldsbecauseof the complicated nonpolynomial structure which arises then, see below. Note that, in contrast to other supersymmetric models, the auxiliary ﬁelds do not lead to an off-shellclosedsupersymmetryalgebra.Onthecontrary,theauxiliaryﬁeldsmake the supersymmetry algebra even “more open” (a formulation of the TT multiplet withanoff-shellclosed supersymmetry algebraisnot kno wn). bosons We yl-fermions TTB i µν a i (h i µ )χ ψ V a A a µ φ a λ ai Thanks to the inclusion of the auxiliary ﬁelds, the Lagrangian takes the fol- lowing simple form (using conventions as [26] adapted to the Minkowski metric diag(1,−1,−1,−1)), L=∂ µ a i ∂ µ a i +h i µ h µi + 2h i µ H µi −iχ∂¯χ−iψ∂¯ψ − 1 4ˆF a µν ˆF aµν + 1 2ˆD µ φ a ˆD µ ¯φ a −2iλ ia ˆD¯λ ia (19) where ˆF a µν =ˆD µ A a ν −ˆD ν A a µ =∂ µ A a ν +g i h i µ ε ab A b ν −(µ↔ν) ˆD µ φ a =∂ µ φ a +g i h i µ ε ab φ b ˆD¯λ ia =σ µ (∂ µ ¯λ ia +g i h i µ ε ab ¯λ ib ). kievarwe.tex; 12/03/2001; 3:49; p.154 148 F.BRANDT Theg i arerealcouplingconstants(deformationparameters).Notethat ˆD µ hasthe formofacovariantderivativeeventhoughtheauxiliaryﬁeldscannotbeviewedas gaugeﬁelds(infact,theysubstituteforﬁeldstrengths,astheequationsofmotion giveh i µ =−H i µ +...).Theauxiliaryﬁeldsalsosimplifythestructureofthegauge andsupersymmetrytransformationsconsiderably.Thegaugetransformationsread δ gauge A a µ =ˆD µ ω a =∂ µ ω a +g i h i µ ε ab ω b δ gauge B i µν = 1 4g i ω a ε ab ε µνρσ ˆF bρσ +∂ µ ω i ν −∂ ν ω i µ δ gauge = 0onotherﬁelds whereω a andω i µ are the gauge parameter ﬁelds associated with A a µ andB i µν respectively. The supersymmetry transformations read, with constant anticom- mutingWeyl-spinors ξ i astransformationparameters, δ susy A a µ =ε ij ξ i σ µ ¯λ ja −ξ i Γ i ε ab A b µ +c.c. δ susy φ a = 2ξ i λ ia −(ξ i Γ i +¯ξ i ¯Γ i )ε ab φ b δ susy λ ia = i 2(ε ij ξ j σ µν ˆF a µν −¯ξ i ¯σ µ ˆD µ φ a )−(ξ j Γ j +¯ξ j ¯Γ j )ε ab λ ib δ susy B i µν =−ε ij ξ j σ µν χ+ξ i σ µν ψ +ig i ε ab (¯φ a ξ j σ µν λ jb +ε jk A a [µ ξ j σ ν] ¯λ kb ) +c.c. δ susy a i = 1 2(ξ i χ−ε ij ξ j ψ) +c.c. δ susy χ=−¯ξ i ¯σ µ (ε ij h j µ + i∂ µ a i ) δ susy ψ=−¯ξ i ¯σ µ (h i µ + iε ij ∂ µ a j ) δ susy h i µ = i 2∂ µ (ξ i ψ−ε ij ξ j χ) +c.c. where Γ i = i 2g j (ε ij χ+δ ij ψ). The commutator algebra of the supersymmetry and gauge transformations is rathercomplicatedoff-shell buton-shellit isquite simple, [δ susy ,δ /prime susy ]≈δ translation +δ gauge (20) [δ susy ,δ gauge ]≈δ /prime gauge (21) [δ gauge ,δ /prime gauge ]≈0, (22) where≈is equality on-shell. (20) is the standard N=2 supersymmetry algebra on-shell(modulogaugetransformations),withvanishingcentralcharge.Iremark kievarwe.tex; 12/03/2001; 3:49; p.155 NEW SUPERSYMMETRIC GAUGE THEORIES 149 thatthegaugetransformationswhichappearontherighthandsideof(20)involve explicitly the spacetime coordinates, see [13] and [27] for details and comments onthispoint.(21)illustratesafeaturetypicalofmanyofthenewmodels,namely that gauge and supersymmetry transformations do not commute (not even on- shell). Explicitly, the gauge parameter ﬁelds ω a/prime andω i/prime µ ofδ /prime gauge on the right handsideof(21)read ω a/prime = (ξ i Γ i +¯ξ i ¯Γ i )ε ab ω b ω i/prime µ =− i 2g i ε ab ε jk ω a (ξ j σ µ ¯λ kb −λ kb σ µ ¯ξ j ) where theξ’s andω’s are supersymmetry parameters and gauge parameter ﬁelds ofδ susy andδ gauge onthelefthandsideof(21).Accordingto(22),thegaugetrans- formations commute on-shell which is also typical of the new models [note: the algebra of the gauge transformations is not related to the Lie algebra underlying Eqs.(15)through (17)!]. Let me ﬁnally discuss the nonpolynomial structure which arises when one eliminates the auxiliary ﬁelds. The Lagrangian (19) contains the auxiliary ﬁelds atmostquadratically, L=− 1 4F a µν F aµν +∂ µ a i ∂ µ a i + 1 2∂ µ φ a ∂ µ ¯φ a −iχ∂¯χ−iψ∂¯ψ−2iλ ia ∂¯λ ia + 2h i µ H µi +h i µ K µi,νj h j ν where H µi =H µi −g i ε ab ( 1 2 F aµν A b ν + 1 4 φ a↔ ∂ µ ¯φ b + iλ ja σ µ ¯λ jb ) K µi,νj =η µν δ ij + 1 2 g i g j [η µν (φ a ¯φ a −A a ρ A aρ ) +A aµ A aν ] The auxiliary ﬁelds can be eliminated by solving their algebraic equations of motion. Thesolution is h i µ =−(K −1 ) µi,νj H νj , (23) whereK −1 is the inverse of the ﬁeld dependent matrix K,(K −1 ) µi,ρk K ρk,νj = δ ν µ δ j i . Note thatKdoes not involve derivatives of the ﬁelds and therefore K −1 is nonpolynomial in the ﬁelds but still local. Hence, using (23), the Lagrangian, gauge and supersymmetry transformations become nonpolynomial but remain strictly local. Expanding the resulting Lagrangian in the coupling constants, one ﬁnds at ﬁrst order HK vertices as well as vertices of category 1 which complete the HK vertices such that the sum is supersymmetric on-shell in the free theory kievarwe.tex; 12/03/2001; 3:49; p.156 150 F.BRANDT moduloatotal derivative, L=− 1 4F a µν F aµν +∂ µ a i ∂ µ a i + 1 2∂ µ φ a ∂ µ ¯φ a −iχ∂¯χ−iψ∂¯ψ−2iλ ia ∂¯λ ia −H µi (K −1 ) µi,νj H νj =L (0) +g i ε ab H i µ F aµν A b ν /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright HK vertices+g i ε ab H i µ ( 1 2φ a↔ ∂ µ ¯φ b + 2iλ ja σ µ ¯λ jb ) /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright category 1vertices (susy completion ofHKvertices)+...(24) Itwasmentionedalreadythatnonpolynomialstructuresasinthisexampleare typical of the new gauge theories. They cannot be avoided in models with FT or HK vertices because they are necessary consequences of these vertices, already in the non-supersymmetric case. The use of appropriate auxiliary ﬁelds that sim- plifytheconstructionisanalmostindispensabletoolforconstructingcomplicated modelsofthistype,especiallysupersymmetricones.Theﬁndingofsuchauxiliary ﬁelds and their embedding in supersymmetry multiplets is in general a nontrivial and subtle ingredient of the construction. In contrast, models which contain CM vertices but no FT or HK vertices are simpler and the issue of auxiliary ﬁelds is less involved. In particular, such models are not necessarily nonpolynomial although supersymmetry often enforces a nonpolynomial dependence on scalar ﬁeldseveninsuchmodels. 5. Comments Thefollowingisaselectionofopenproblemswhichmaypointtopossiblefurther developmentsintheﬁeld: (i) In my opinion, the r ˆole of the matter ﬁelds (scalar ﬁelds, fermions) in the new supersymmetric models has not been fully understood yet. In particular, the relation of scalar ﬁelds to the underlying geometry (Lie algebra) is somewhat mysterious.Abetterunderstandingofthisissuemightbeakeytoadeeperunder- standing of the supersymmetry structure of the models and to a more systematic constructionof such models. (ii) Systematic classiﬁcations of the possible consistent and supersymmetric interactions involving p-form gauge potentials, analogous to the classiﬁcation [17–19] of non-supersymmetric interactions, are largely missing. An exception is the classiﬁcation of the lowest dimensional interaction vertices involving a TT multipletin[13].Supersymmetrysupplements(5)withtheadditionalrequirement δ susy L=∂ µ M µ whereδ susy are the deformed supersymmetry transformations. This restricts the possible interactions as compared to the non-supersymmetric case, and relates coefﬁcients of various interaction terms. A typical example is (24) where the coefﬁcients of the HK vertices are related to coefﬁcients of inter- action vertices of category 1. In fact, supersymmetry can even completely forbid kievarwe.tex; 12/03/2001; 3:49; p.157 NEW SUPERSYMMETRIC GAUGE THEORIES 151 interactions which would be allowed if supersymmetry were not imposed. An exampleistheabsenceofN=2supersymmetricCMcouplingsoftheTTmultiplet [13]. Furthermore, it depends on the supersymmetry multiplet structure which interactions are possible. For instance, it was just mentioned that there are no N=2 supersymmetric CM couplings involving the TT multiplet, whereas such couplings do exist for the VT multiplet (cf. table in section 3). Such results could be relevant in the context of string theory when comparing properties of different superstring vacua. (iii) Locally supersymmetric models with FT or HK couplings are almost completely missing so far. In fact, the only exception is the work [7] where N=2 supergravitymodelswithVTmultipletswereconstructed.Theconstructionoflo- cally supersymmetric extensions of some of the other models could be of interest in the string theory context. In particular this applies to supergravity models with the TT multiplet because of the conjectured importance of this multiplet to type IIBsuperstring vacua(cf. introduction). (iv)RecallthatFT,HKandCMverticesarespecialcasesofvertices(7).Non- supersymmetric models in spacetime dimensions >4with such vertices have been constructed already [17, 28]. Analogous globally or locally supersymmetric modelsinhigherspacetimedimensionshavenotbeenconstructedsofar.Infactit seems that the only vertices (7) which have been used in supersymmetric models inspacetimedimensions >4sofararethefamiliarCMvertices(14).Forinstance, theseverticesoccurin10-dimensionalsupergravityinconnectionwiththeGreen- Schwarz anomalycancellationmechanism(cf.remarks atthe end ofsection 3). 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Bagger, Supersymmetry and supergravity , Princeton Series in Physics (Prince- ton Univ. Press, Princeton1992). 27. F. Brandt, Hidden symmetries of supersymmetric p-form gauge theories , Phys. Lett. B (to appear), hep-th/0009133. 28. F. Brandt, J. Sim ´on and U. Theis, Exotic gauge theories from tensor calculus , Class. Quant. Grav. 17(2000) 1627-1636,hep-th/9910177. kievarwe.tex; 12/03/2001; 3:49; p.159 SUPERSYMMETRIC R 4 ACTIONS AND QUANTUM CORRECTIONS TOSUPERSPACETORSION CONSTRAINTS KASPER PEETERS 1 , PIERRE VANHOVE 1 and ANDERS WESTERBERG 2∗ 1 CMS/DAMTP, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 2 NORDITA,Blegdamsvej 17,DK-2100 Copenhagen Ø,Denmark Abstract. We present the supersymmetrisation of the anomaly-related R 4 term in eleven dimen- sions and show that it induces no non-trivial modiﬁcations to the on-shell supertranslation algebra and the superspace torsion constraints before inclusion ofgauge-ﬁeldterms. 1 1. Higher-derivativecorrectionsandsupersymmetry The low-energy supergravity limits of superstring theory and D-brane effective actions receive inﬁnite sets of correction terms, proportional to increasing pow- ers ofα /prime =l 2 s and induced by superstring theory massless and massive modes. At present, eleven-dimensional supergravity lacks a corresponding microscopic underpinning that could similarly justify the presence of higher-derivative cor- rections to the classical Cremmer-Julia-Scherk action [1]. Nevertheless, some corrections of this kind are calculable from unitarity arguments and super-Ward identities in the massless sector of the theory [2] or by anomaly cancellation arguments[3,4]. Supersymmetry puts severe constraints on higher-derivative corrections. For example,itforbidstheappearanceofcertaincorrections(like,e.g, R 3 corrections tosupergravityeffectiveactions[5]),andgroupstermsintovariousinvariants[6– 9].Thestructureoftheinvariantsthatcontainanomaly-cancellingtermsisof great 1 Based on talks given by K.P. at the SPG meeting, Cambridge, February 2000, by A.W. at the NordicNetworkMeeting,Copenhagen,May2000,andbyP.V.attheFradkinMemorialConference, Moscow, June2000, and at the ARW Conference, Kiev, September 2000. ∗ k.peeters,[email protected], [email protected] kievarwe.tex; 12/03/2001; 3:49; p.160 154 K.PEETERS,P.VANHOVE,A.WESTERBERG importance due to the quantum nature of the anomaly-cancellation mechanism and is themainconcernofthis note. Higher-derivativeadditionstothesupergravityactionsareingeneralcompati- blewithsupersymmetryonlyifthetransformationrulesfortheﬁeldsalsoreceive higher-derivativecorrections: /parenleftbigg δ 0 + /summationdisplay n (α /prime ) n δ n /parenrightbigg/parenleftbigg S 0 + /summationdisplay n (α /prime ) n S n /parenrightbigg = 0. (1) Asaconsequence,theﬁeld-dependentstructurecoefﬁcientsontheright-handside ofthesupersymmetryalgebra, [δ susy 1 ,δ susy 2 ] =δ translation +δ susy +δ gauge +δ Lorentz , (2) willbemodiﬁedaswell.Whenthetheoryisformulatedinsuperspacethestructure of the algebra is related to the structure of the tangent bundle, the link being provided by the constraints on the superspace torsion. In particular, corrections to the parameters modify the superspace constraints. However, since some cor- rections are reabsorbable by suitable rotations of the tangent bundle basis, not all corrections arephysical. We report here on the supersymmetrization of the anomaly-related terms (α /prime ) 2 B∧F 4 for super-Maxwell theory coupled to N=1 supergravity in ten di- mensions and (α /prime M ) 3 C∧t 8 R 4 (where (α /prime M ) 3 = 4π(l P ) 6 ) in eleven dimensions performedin[10].Inbothcases,thesesuperinvariantsdonotimplyanymodiﬁca- tions to the superspace constraints. We present here only the more salients aspect of the analysis and refer to the article [10] for computational and bibliographical details. Our main motivation to look for non-trivial corrections to superspace con- straints comes from the link between these constraints and the kappa symmetry of M-branes [11–13] and D-branes [14–16]. Classical kappa invariance of the M- and D-brane world-volume actions — a key requirement for these objects to be supersymmetric — imposes the on-shell constraints on the background super- spacesupergravityﬁelds,amongthemthesuperspacetorsion.Forthisreason,any non-trivial modiﬁcation to the constraints is expected to require new terms in the world-volumeactionsforthebranesinorderforkappasymmetrytobepreserved. 2. Construction of anabelian F 4 superinvariant in D=10 As a ﬁrst step in our analysis of the implications of higher-derivative correc- tions to the supersymmetry algebra, we discuss the construction of the abelian (α /prime ) 2 (t 8 F 4 −B∧F 4 )forN=1 super-Maxwell theory coupled to gravity in ten dimensions. The ﬁeld content of the on-shell super-Maxwell theory comprises an abelian vectorA µ and a negative-chirality Majorana-Weyl spinor χ. Since we are inter- ested in local supersymmetry invariance we have to take into account also the kievarwe.tex; 12/03/2001; 3:49; p.161 QUANTUM-CORRECTEDSUPERSPACE 155 interactions with the zehnbein e µr , the negative-chirality Majorana-Weyl grav- itinoψ µ and the two-form B µν from the supergravity multiplet. The classical action (leaving outthe gravitationalsector) S F 2 = /integraldisplay d 10 xe /bracketleftbig − 1 4F µν F µν −8 ¯χ/negationslashD(ω)χ+ 2 ¯χΓ µ Γ νρ ψ µ F νρ /bracketrightbig (3) isinvariant underthe local supersymmetry transformations δA µ =−4 ¯/epsilon1Γ µ χ, δχ = 1 8Γ µν /epsilon1F µν , (4) For local supersymmetry we have to consider the transformations of the super- gravity multiplet ﬁelds as well (neglecting terms proportional to the two-form B µν andthecorresponding ﬁeld strength, H µνρ ): δe µr = 2¯/epsilon1Γ r ψ µ , δψ µ =D µ (ω)/epsilon1+···, δB µν = 1 √ 2¯/epsilon1Γ [µ ψ ν] .(5) TheF 4 action invariant under the local supersymmetry transformations listed aboveis [17,18, 10]: S F 4 = (α /prime ) 2 32 /integraltext d 10 x /bracketleftBig 1 6 et (r) 8 F r 1 r 2 ···F r 7 r 8 + 1 3 √ 2 ε (r) 10 B r 1 r 2 F r 3 r 4 ···F r 9 r 10 − 32 5 et (r) 8 η r 2 r 3 (¯χΓ r 1 D r 4 (ω)χ)F r 5 r 6 F r 7 r 8 + 12· 32 5 e(¯χΓ r 1 D r 2 (ω)χ)F r 1 m F mr 2 − 16 5! ε (r) 10 (¯χΓ r 1 ···r 4 Γ r 5 D r 6 (ω)χ)F r 7 r 8 F r 9 r 10 + 16 3 et (r) 8 (¯ψ r 1 Γ r 2 χ)F r 3 r 4 F r 5 r 6 F r 7 r 8 + 8 3 e(¯ψ m Γ mr 1 ···r 6 χ)F r 1 r 2 ···F r 5 r 6 /bracketrightBig . (6) Note that our string-amplitude based analysis has allowed us to group also the fermionic terms using the well-known t 8 tensor. The local supersymmetry in- variance of the combined action S F 2 +S F 4 requires that the supersymmetry transformationsbemodiﬁedaccordingto ( F 2 :=F mn F nm ) δA µ =−4 ¯/epsilon1Γ µ χ−(α /prime ) 2 /bracketleftBig 1 4(¯/epsilon1Γ µ χ)F 2 −(¯/epsilon1Γ m χ)F 2 mµ − 1 8(¯/epsilon1Γ r 1 ···r 4 µ χ)F r 1 r 2 F r 3 r 4 /bracketrightBig , δχ= 1 8Γ µν /epsilon1F µν + 1 768(α /prime ) 2 /bracketleftBig t (r) 8 Γ r 7 r 8 /epsilon1−Γ r 1 ···r 6 /epsilon1 /bracketrightBig F r 1 r 2 F r 3 r 4 F r 5 r 6 .(7) It can be veriﬁed that the structure of the supersymmetry algebra is not modiﬁed by theorder- (α /prime ) 2 corrections[17,18,10]: /bracketleftbig δ (α /prime ) 0 /epsilon1 1 +δ (α /prime ) 2 /epsilon1 1 ,δ (α /prime ) 0 /epsilon1 2 +δ (α /prime ) 2 /epsilon1 2 /bracketrightbig A µ = /bracketleftbig δ (α /prime ) 0 /epsilon1 1 ,δ (α /prime ) 0 /epsilon1 2 /bracketrightbig A µ +O /parenleftbig (α /prime ) 4 /parenrightbig .(8) kievarwe.tex; 12/03/2001; 3:49; p.162 156 K.PEETERS,P.VANHOVE,A.WESTERBERG Consequently,thestructureofthesuperspacetorsionconstraintswillbethesame as for the classical theory to this order. This observation is related to the fact that it is possible to supersymmetrise the Dirac-Born-Infeld actions while imposing onlytheclassicalconstraints[19]. 3. Construction of the C∧R 4 superinvariantin D=11 Noticing the close parallel between the classical supersymmetry transformations for thesuper-Maxwellandthesupergravityﬁelds δχ = 1 8 Γ µν /epsilon1F µν , δψ rs = 1 8Γ µν /epsilon1R µνrs +···, (9) δF µν =−8D [µ (¯/epsilon1Γ ν] χ), δR µνrs =−8D [µ (¯/epsilon1Γ ν] ψ rs ) +4D [µ (¯/epsilon1Γ ν] ψ rs + 2 ¯/epsilon1Γ [r ψ s] ν] ) +···, it istempting tomakethefollowingsubstitution inthe super-Maxwell action: F r 1 r 2 →R r 1 r 2 s 1 s 2 , χ→ψ s 1 s 2 , D r χ→D r ψ s 1 s 2 . (10) Unfortunately,thedifferenceinstructurebetweentheequationsofmotionforthe gauge potential and the spin connection implies that the previous mapping does not commute with supersymmetry, as can be seen by the presence of the second line in the supersymmetry transformation of the Riemann tensor above. Another crucialdifferencebetweenthesuper-Maxwellandsupergravitycasesisthat,when subtracting all the lowest-order equations of motions, it is necessary to make the followingsubstitution fortheRiemanntensor: R mnpq →W mnpq − 16 d−2δ [m[p (¯ψ |r| Γ |r| ψ n]q] −¯ψ |r| Γ q] ψ n]r ).(11) Taking all these facts into account, as well as the information from string- amplitude analysis that the extra s-type indices in (10) should be contracted with an additional t (s) 8 tensor, we arrive at the following M-theory C∧R 4 invariant kievarwe.tex; 12/03/2001; 3:49; p.163 QUANTUM-CORRECTEDSUPERSPACE 157 after liftingto elevendimensions[10]: (α /prime M ) −3 L Γ [0] = + 1 192et (r) 8 t (s) 8 W r 1 r 2 s 1 s 2 ···W r 7 r 8 s 7 s 8 (12) + 1 (48) 2 ε t 1 t 2 t 3 r 1 ···r 8 t (s) 8 C t 1 t 2 t 3 W r 1 r 2 s 1 s 2 ···W r 7 r 8 s 7 s 8 , (α /prime M ) −3 L Γ [1] =−4et (s) 8 (¯ψ s 1 s 2 Γ r 1 D r 2 ψ s 3 s 4 )W r 1 r 3 s 5 s 6 W r 3 r 2 s 7 s 8 − 1 4et (s) 8 (¯ψ r 1 Γ r 2 ψ s 7 s 8 )W r 1 r 2 s 1 s 2 W mns 3 s 4 W nms 5 s 6 −et (s) 8 (¯ψ r 1 Γ r 2 ψ s 7 s 8 )W r 1 ms 1 s 2 W mns 3 s 4 W nr 2 s 5 s 6 +et (s) 8 (¯ψ r 1 Γ s 7 ψ r 2 s 8 )W r 1 r 2 s 1 s 2 W mns 3 s 4 W nms 5 s 6 −4et (s) 8 (¯ψ r 1 Γ s 7 ψ r 2 s 8 )W r 1 ms 1 s 2 W mns 3 s 4 W nr 2 s 5 s 6 + 2 9et (s) 8 (¯ψ m Γ n ψ ms 8 )W pqs 1 s 2 W qps 3 s 4 W ns 7 s 5 s 6 − 8 9et (s) 8 (¯ψ m Γ n ψ ms 8 )W nps 1 s 2 W pqs 3 s 4 W qs 7 s 5 s 6 , (α /prime M ) −3 L Γ [3] = + 2et (s) 8 (¯ψ s 5 s 6 Γ r 1 r 2 r 3 D r 4 ψ s 7 s 8 )W r 1 r 2 s 1 s 2 W r 3 r 4 s 3 s 4 − 1 8et (s) 8 (¯ψ m Γ mr 1 r 2 ψ s 7 s 8 )W r 1 r 2 s 1 s 2 W pns 3 s 4 W nps 5 s 6 + 1 2et (s) 8 (¯ψ m Γ mr 1 r 2 ψ s 7 s 8 )W r 1 ps 1 s 2 W pns 3 s 4 W nr 2 s 5 s 6 +et (s) 8 (¯ψ m Γ r 1 r 2 r 3 ψ s 7 s 8 )W r 1 r 2 s 1 s 2 W mns 3 s 4 W nr 3 s 5 s 6 , (α /prime M ) −3 L Γ [5] = + 1 8et (s) 8 (¯ψ r 6 Γ r 1 ···r 5 ψ s 7 s 8 )W r 1 r 2 s 1 s 2 W r 3 r 4 s 3 s 4 W r 5 r 6 s 5 s 6 , (α /prime M ) −3 L Γ [7] = + 1 48et (s) 8 (¯ψ m Γ mr 1 ···r 6 ψ s 7 s 8 )W r 1 r 2 s 1 s 2 W r 3 r 4 s 3 s 4 W r 5 r 6 s 5 s 6 . Even if the elfbein supersymmetry transformation rule receives (α /prime M ) 3 modiﬁ- cations, by computing the closure of the supersymmetry algebra (2), we ﬁnd [10] thatthetranslationparameterdoes notreceivecorrectionsthatcannotbeabsorbed by ﬁeldredeﬁnitions. kievarwe.tex; 12/03/2001; 3:49; p.164 158 K.PEETERS,P.VANHOVE,A.WESTERBERG 4. Superspace approach It can be argued that in the completely general Ansatz for the dimension zero torsion constraint T abr = (CΓ r 1 ) ab X rr 1 + 1 2!(CΓ r 1 r 2 ) ab X rr 1 r 2 + 1 5!(CΓ r 1 ···r 5 ) ab X rr 1 ···r 5 , (13) the coefﬁcient X rr 1 can be set equal to δ rr 1 , and all fully antisymmetric tensors containedin X rr 1 r 2 andX rr 1 ···r 5 tozerobyachoiceoftangentbundlebasis(see, e.g.,[21]).Thisleavesastheonlycandidatesfornon-trivialM-theorycorrections the SO(1,10) representations 429and4290of the Γ [2] andΓ [5] coefﬁcients, re- spectively. Therefore, from the component analysis of the previous section we concludethatthehigher-orderinvariant(12)doesnotinduceanymodiﬁcationsto thetorsionconstraint(13). Howeshowedin [20], thatimposing onlythe constraint T abr = (CΓ r ) ab (14) on the dimension-zero component of the superspace torsion, the classical, on- shell, eleven-dimension supergravity theory of [1] follows withouthaving to introduce a four-form superﬁeld. An analysis of the superspace Bianchi iden- tities for this superﬁeld would necessitate a more complete analysis of the R 4 invariant(12) withthe inclusion ofhigherpowersof thefour-formﬁeld strength. In this context, let us also mention that in parallel with our component-space basedapproachtouncoverthesuperspaceunderlyingM-theory,acomplementary line of attack based on an analysis of the superspace Bianchi identities has been initiatedby Cederwallet al.in[21]. Acknowledgements K.P. and P.V. are supported by PPARC grant PPA/G/S/1998/00613. 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Nilsson, Manifestly Supersymmetric M- Theory , hep-th/0007035. kievarwe.tex; 12/03/2001; 3:49; p.166 kievarwe.tex; 12/03/2001; 3:49; p.167 MASSIVESUPERPARTICLEWITH SPINORIALCENTRAL CHARGES S.FEDORUK Ukrainian Engineering-Pedagogical Academy, 61003 Kharkiv, 16 UniversitetskaStr.,Ukraine V.G. ZIMA Kharkiv National University, 61077 Kharkiv, 4 Svobody Sq., Ukraine Abstract. WeconstructthemanifestlyLorenz-invariantformulationofthe N= 1D= 4massive superparticle with spinorial central charges. The model possesses from one to three κ-symmetries. Thelocaltransformationsof κ-symmetryarewrittenout.Theusingofindexspinorforconstruction of the tensorial central charges is considered. The equivalence at the classical level between the massiveD= 4superparticle with one κ-symmetry and the massive D= 4spinning particle is obtained. 1. Introduction Recently it became clear that some interesting supersymmetric theories admit besides scalar central charges which are presented in conventional D= 4 Poincare supersymmetry [1, 2] also nonscalar central charges: tensorial [3]-[7] orspinor[8,9]ones.Althoughthetensorialcentralchargesinthesupersymmetry algebra are usually associated with topological contributions of the extended ob- jectsitisattractivetoconsiderthepuresuperparticlemodelshavingsymmetryof thiskind.Suchmodelswereﬁrstlyobtainedinmasslesscase[10]for D= 4with twoorthreelocal κ-symmetries. CentralchargeisaquantitywhichisinertwithrespecttoSUSYbuttransforms underinternalorLorentz groups. Weconstructthemodelofthemassive D= 4nonextendedsuperparticlewith spinorialcentralchargespossessingoneortwolocal κ–symmetries 1 .Inparticular in such a way we obtain the superparticle with a single κ-symmetry which is equivalenttotheusualspinning(spin 1/2)particle [12,13]inthespinorialcentral charge background. 1 The Lagrangian of the massive superparticle with vector central charge and with two κ–symmetries has been presented alreadyin [11] kievarwe.tex; 12/03/2001; 3:49; p.168 162 S.FEDORUK,V.G.ZIMA Generalized central extension of N= 14–dimensional supersymmetry algebra /braceleftBig Q,Q /bracerightBig = 2(γ µ )P µ (1) with Majorana supercharges Q + =Q, energy–momentum vector Pandγ– matrices in Majorana representation, so that C=γ 0 =C −1 and as in any representationwehave C T =−C,canbewritten in theform {Q,Q}= 2Z (2) whereZ T =Zis the most general symmetric matrix of Abelian generalized central charges with a total of ten real entries. In decomposition of this matrix on the basis deﬁned by products of γ–matrices we have tensorial central charges as coefﬁcients ZC= (γ µ )P µ + i 2(γ µν )Z µν (3) wherePis (in general) a linear combination of the energy–momentum vector P and a “string charge”. Six real charges Z µν =−Z νµ are related to the symmetric complexWeylspin–tensor Z αβ =Z βα bythe relation Z µν = 1 2(¯Z ˙α˙β ˜σ ˙α˙β µν −Z αβ σ αβ µν ). (4) The spin–tensors Z αβ and¯Z ˙α˙β = (Z αβ )represent the self–dual and anti–self– dual parts of the central charge matrix. The tensorial central charges commute withfour–momentumandtransformascomponentsofatensorundertheLorentz group transformations. There are two types of model with central charges. Some of them have ex- act SUSY due to presence of special tensorial central charge coordinates which transform together with Grassmannian spinor θand space–time vector x. Their derivative with respect to development parameter τabsorbs the tensorial part in SUSY variation of product θ˙θin complete analogy with absorption of the vector partinthevariationbythespace–timevector xunderordinarySUSYtransforma- tions.OthermodelshavenosimilarcoordinatesandtheirSUSYisreachedonlyon themassshell.Hereweexaminenamelythesecondtypemodelwhichisobtained byaddingtocoordinatesofnonextendedmassivesuperparticlecertaindynamical even spinor ζ. This spinor parameterizes [15] in the rest frame of particle the compact groupmanifold ofquantum–theoryrotationgroup SU(2). In this paper we use the D= 4spinor conventions of [2]. Majorana and Weyl odd spinors are denoted by the same literal. One can easy identiﬁes the meaning of a denotation viewing its nearest encirclement. Bispinor expressions with Majorana spinors are written, as a rule, in conventional form which makes obvious atransition toWeylspinors. kievarwe.tex; 12/03/2001; 3:49; p.169 LORENTZ SPINORIAL CENTRAL CHARGES 163 2. Actionanditssymmetries 2.1. SUPERPARTICLE LAGRANGIAN Letustakefor superparticleLagrangianthe expression L=L super +L SCC ≡p˙x+i¯θZC˙θ− e 2(p 2 +m 2 ) +L SCC ≡p˙x+iP α˙β (θ α ˙¯θ ˙β −˙θ α ¯θ ˙β ) +iZ αβ θ α ˙θ β +i¯Z ˙α˙β ¯θ ˙α ˙¯θ ˙β − e 2(p 2 +m 2 ) +L SCC . (5) Hereeis Lagrange multiplier for mass constraint p 2 +m 2 ≈0. Let us take spinorialcentralchargeLagrangian L SCC ,i.e.apartofLagrangian(5)containing kinetic term for commuting spinor coordinates ζ α ,¯ζ ˙α = (ζ α )and generating constraintonthese variables,intheform L SCC =˙ζv+ ¯v˙¯ζ−λ(ζˆp¯ζ−j). (6) wherevis canonical conjugate momentum for ζandλis Lagrange multiplier for ”spin constraint” r−j≡ζˆp¯ζ−j≈0 (r≡ζˆp¯ζ). (7) This constraintgivesusat j/negationslash= 0thecompletenesscondition rδ β α =ζ α (¯ζ˜p) β + (ˆp¯ζ) α ζ β and c.c. (8) for spinorsζ,ˆp¯ζ. Here matrices ˆpand˜pare the contractions of the space–time momentumpandσ–matrices with lower and upper spinor indices, respectively. Similar tojnumerical constant plays the role of “classical spin” in the index spinorformalism[14,11,15]whichisattractiveintaskofparticlespindescription with commuting spinors. In what follows it is important that some equations of motionfollowingfromtheLagrangian(5)read as ˙ζ= 0,˙p= 0. (9) We can construct the vector and tensor central charges Zin the Lagrangian (5) with the spinors ζ, ˆp¯ζwithout derivatives in τ. So on shell, due to (9), these quantities will be constants. Such constructions for central charges Zin terms of spinorialonesdonot modifytheequations(9) and are speciﬁedbelow. kievarwe.tex; 12/03/2001; 3:49; p.170 164 S.FEDORUK,V.G.ZIMA 2.2. SUPERSYMMETRY SUSY transformations can be viewed as global translations in odd spinor coor- dinatesθaccompanied translations in space–time vector xand possibly some scalar or tensorial even central charge coordinates ywhich leave invariant certain differential1–forms.MentionedformscovariantlytransformunderglobalLorentz and internal groups and are invariant under space–time translations. These forms are sums of corresponding even coordinate differentials and terms which are bilinear in odd spinor coordinates and their differentials. Using such forms one can construct theories with exact off–shell SUSY but in absence of some central charge coordinates and corresponding differential forms it is possibly to reach SUSYonlyonshell.Inthecaseunderconsiderationwehaveuniquefundamental vectorialsuperform ω µ ≡˙ω µ dτ=dx+id¯θγ µ θ=dx+iθσ µ d¯θ−idθσ µ ¯θ (10) and usualSUSY transformations are δx µ =−i¯θγ µ δθ= +iθσ µ δ¯θ−iδθσ µ ¯θ (11) with constant δθ. Supercharges can be obtained as coefﬁcients at the derivative (δθ) . intheintegrandofthelocalvariationoftheactioninHamiltonianform.The variationof theLagrangian (5) is δL=iP(θσ(δ¯θ) . −(δθ) . σ¯θ)−2i(Z αβ (δθ α ) . θ β +¯Z ˙α˙β (δ¯θ ˙α ) . ¯θ ˙β )− i(˙Z αβ δθ α θ β +˙¯Z ˙α˙β δ¯θ ˙α ¯θ ˙β ) + (iZ αβ δθ α θ β +i¯Z ˙α˙β δ¯θ ˙α ¯θ ˙β ) . + P(δω) . −(p−P) . δx+ ((p−P)δx) . . (12) WeseethatδL= 0if(δθ) . = 0uptosurfacetermsinabsenceoftensorialcentral charge coordinates. Equations of motion (9) are twice used for this conclusion. In the ﬁrst place we use these equations to change multiplier pat˙xbyPand as consequence to collect variations of xandθat vectorial part of Zin variation of superform (10). In the second place we use equations (9) to represent variation with tensorial part of Zas total derivative. Constancy of δθis used as well. The price for the presence of the supersymmetry is the inﬁnite number of the spin states in the spectrum. At the restriction of the bosonic spinor sector to the index spinor one [14, 11, 15] the number of the states in spectrum becomes ﬁnite but the supersymmetry disappears. But in both cases the models possess local κ-symmetries. In coordinate representation for odd variables one obtains as generators of SUSY transformations Q= ∂ ∂θ+Zθ. (13) kievarwe.tex; 12/03/2001; 3:49; p.171 LORENTZ SPINORIAL CENTRAL CHARGES 165 IntermsofWeyl spinorwe have Q α = ∂ ∂θ α + (ˆP¯θ) α +θ β Z βα , (14) ¯Q ˙α = ∂ ∂¯θ ˙α + (θˆP) ˙α +¯Z ˙α˙β ¯θ ˙β . (15) So generators of SUSY contain “anomalous” extra pieces with central charges. The algebra(2)of SUSYgenerators {Q α ,Q β }= 2Z αβ , /braceleftBig Q α ,¯Q ˙β /bracerightBig = 2P α˙β (16) istheN= 1D= 4SUSY algebraextended by tensorial central charges. One can introduce terms with derivatives of central charge coordinates yto the multipliers at central charges in the Lagrangian (5). Then the model becomes SUSY invariantnot only quasi-invariant. 2.3.κ–SYMMETRY Grassmannianconstraintsofthe model (5) are d θ ≡−ip θ −Zθ≈0. (17) IntermsofWeyl spinorwe have d θα ≡−ip θα −(ˆP¯θ) α −θ β Z βα ≈0, (18) ¯d θ˙α ≡−i¯p θ˙α −(θˆP) ˙α −¯Z ˙α˙β ¯θ ˙β ≈0. (19) Poissonbrackets algebraofconstraints(17) is {d θ ,d θ }= 2iZ. (20) IntermsofWeyl spinorsitis {d θα ,d θβ }= 2iZ αβ , /braceleftBig ¯d θ˙α ,¯d θ˙β /bracerightBig = 2i¯Z ˙α˙β , /braceleftBig d θα ,¯d θ˙β /bracerightBig = 2iP α˙β . (21) Let us analyze all possibilities of different numbers of κ–symmetries. The number of κ–symmetries is deﬁned rank of Poisson bracket matrix for the fermionicconstraints.In consideredcase detZ= (P 2 ) 2 −P α˙α P β˙β Z αβ ¯Z ˙α˙β + 1 4Z αβ Z αβ ¯Z ˙α˙β ¯Z ˙α˙β .(22) kievarwe.tex; 12/03/2001; 3:49; p.172 166 S.FEDORUK,V.G.ZIMA The characteristic polynomial, which is obtained by substitution p 0 →p 0 −λ in(22), has theform λ 2 (λ 2 −4p 0 λ+A) + 2Bλ+ detZ where A= 4(p 0 ) 2 −2P 2 −σ 0 α˙α σ 0 β˙β Z αβ ¯Z ˙α˙β , B= 2p 0 P 2 −σ 0 α˙α P β˙β Z αβ ¯Z ˙α˙β . Thus ifA= 0,B= 0,detZ= 0we have three fermionic ﬁrst class constraints and superparticle model with 3/4conserved SUSY. In case B= 0,detZ= 0 butA/negationslash= 0two eigenvalues λamong four ones are zero and superparticle model conserves 1/2SUSY. Only in case detZ= 0butA/negationslash= 0,B/negationslash= 0we have system with1/4conservedSUSY. Notedthatsomesuperparticlemodelassociatedwithsuperalgebrawithtenso- rialcentralchargeswasconsideredin[16].Bosonicconstraintsofthemodel[16] aregeneralizedmass shellcondition ZCZ= 0 (23) which in Weylspinor notationreads Z αβ Z βγ =P 2 δ αγ ,¯Z ˙α˙β ¯Z ˙β˙γ =P 2 δ ˙α˙γ , Z αβ P β˙α +P α˙β ¯Z ˙β˙α = 0.(24) It is easy to see that the model [16] preserves two supersymmetries or more. Preserving of one supersymmetry is not possible in that model. From (24) we haveB= 0,detP= 0and thus necessarily two eigenvalues λamong four ones are zero. Thus the condition (24) are too much strong to have system with 1/4 conserved SUSY. 3. Equivalence between massive spinning particle and superparticle with oneκ–symmetry 3.1. SPINNINGPARTICLE INTHEPSEUDOCLASSICAL APPROACH In the pseudoclassical approach the Lagrangian of spinning particle has the followingform[12,13] L 1/2 =p µ ˙x µ + i 2(ψ µ ˙ψ µ +ψ 5 ˙ψ 5 )− e 2(p 2 +m 2 )−iχ(pψ+mψ 5 ).(25) ThespinvariablesinthisdescriptionaretheGrassmannian(pseudo)vector ψ µ and the Grassmannian (pseudo)scalar ψ 5 . Besides mass constraint T≡p 2 +m 2 ≈0 kievarwe.tex; 12/03/2001; 3:49; p.173 LORENTZ SPINORIAL CENTRAL CHARGES 167 in Hamiltonian formalism the physical sector of the model is subjected to the GrassmannianconstraintsfromwhichoneDirac constraint D≡p µ ψ µ +mψ 5 ≈0 (26) playstheroleoftheﬁrstclassconstraintandﬁveself-conjugacyconditionforthe Grassmannianvariables g µ ≡p µ ψ − i 2ψ µ ≈0, g 5 ≡p ψ5 − i 2ψ 5 ≈0 (27) are the second class constraints. Thus the number of physical odd degrees of freedom in the model (25) is [number of ( ψ µ ,ψ 5 ,p ψµ ,p ψ5 )] – [number of the secondclassconstraints( g µ ,g 5 )]–2[numberoftheﬁrstclassconstraint( D)]=3. The usual model of the massive CBS superparticle [22] with Grassmannian spinorcoordinates θ α ,¯θ ˙α hasonly thefermionicspinorconstraints d θα ≡−ip θα −(ˆp¯θ) α ≈0, ¯d θ˙α ≡−i¯p θ˙α −(θˆp) ˙α ≈0 which all are the second class constraints. Here the number of the physical odd degrees of freedom is [number of ( θ α ,¯θ ˙α ,p θα ,¯p θ˙α )] – [number of ( d θ ,¯d θ )] = 4. In order to obtain desired three physical fermionic degrees of freedom it is necessary that from fermionic four spinor constraints three constraints are of the second class whereas one constraint should be of the ﬁrst class. Such situation with nonsymmetric separation of the fermionic constraints into the ones of ﬁrst andsecondclasshasbeenproposedinmasslesssuperparticlemodels[10]aswell as in the massive particle case [23]. Precisely the situation with one ﬁrst class fermionic constraint has been presented in [23] in the construction of N= 4→ N= 1PBGS ind= 1. The relation between that model and our one will be given below. Thus in the massive case the equivalence of spinning particle and superparticlewithtensorialcentralchargeswithone κ-symmetryisexpected.Let us note that in massless case [24, 25] the spinning particle is equivalent, at least on classical level, to the usual CBS superparticle without any central charges. This fact of identifying the local fermionic invariances of spinning particle and κ-symmetries of superparticle is essential for superﬁeld formulation of massless superparticletheory[24,25]andconsequentgeneralizationsonsuperbranes[26]. Accounting above mentioned preliminary arguments for the possible relation betweenmassivespinningparticleandmassivesuperparticlewithtensorialcentral chargeswetakethefollowingwayforconstructionofthesuperparticlemodel.We shall realize the covariant transition, under preservation of the physical content, fromthemodelofthemassivespinningparticletothesystemwithGrassmannian spinor variables. As result of this procedure we arrive at model of the N= 1 D= 4massive superparticle with tensorial central charges possessing one gauge fermionicinvariance ( κ-symmetry). kievarwe.tex; 12/03/2001; 3:49; p.174 168 S.FEDORUK,V.G.ZIMA Covariant transition from the Grassmannian vector ψ µ and scalarψ 5 to the Grassmannian spinors θ α ,¯θ ˙α requires using of the commuting spinor variables ζ α ,¯ζ ˙α . The total system which we consider as initial under transition to Grassman- nian spinors is in fact the sum of the two sectors coupled through the space-time momentum. One of these sectors is the usual massive spinning particle with Lagrangian (25) whereas the second is the sector of the bosonic spinor with Lagrangian (6).Thus the Lagrangianoftheinitial systemhas the followingform L=L 1/2 +L SCC =p˙x+ i 2(ψ˙ψ+ψ 5 ˙ψ 5 )− e 2(p 2 +m 2 )−iχ(pψ+mψ 5 ) +˙ζv+ ¯v˙¯ζ−λ(ζˆp¯ζ−j). (28) As result of the constraint ζˆp¯ζ=jthe sign of the constant jdeﬁnes the sign of theenergy.Infollowing we consider thepositive energy sector where j >0. 3.2. CONVERSION OF SPINNING PARTICLETO SUPERPARTICLE WITH TENSORIAL CENTRALCHARGES The conversion of spinning particle model described by the Grassmannian vari- ablesψ µ ,ψ 5 to the model with the Grassmannian spinor variables θ α ,¯θ ˙α is realized bythegeneral resolution[11]oftheform ψ µ =r −1/2 (θσ µ ˜pζ+¯ζ˜pσ µ ¯θ)−mρζσ µ ¯ζ, (29) ψ 5 =r −1/2 m(ζθ+¯θ¯ζ) +rρ+˜ψ 5 . (30) The initial Grassmannian variables ψ µ ,ψ 5 (5 variables) are expressed in terms of two Grassmannian scalars ρ,˜ψ 5 and three components of spinor θ. Just for projections of ψ µ ≡− 1 2 ˜σ µ˙αα ˆψ α˙α in the basis formed by spinors ζ α ,(¯ζ˜p) α we have ζˆψ¯ζ= 2r 1/2 (ζθ+¯θ¯ζ),¯ζ˜pˆψ˜pζ= 2mr 2 ρ, (31) ζˆψ˜pζ= 2r 1/2 (ζˆp¯θ),¯ζ˜pˆψ¯ζ= 2r 1/2 (θˆp¯ζ), (32) where ˆψ=ψ µ σ µ . Thefourth componentof thespinor φ=i(θζ−¯ζ¯θ) (33) does not participate in the expression for ψ-variables. The inversion of (29), (30) and (33) looksasfollows θ α = 1 4r −3/2 /bracketleftBig (ζˆψ¯ζ)(ˆp¯ζ) α + 2(¯ζ˜pˆψ¯ζ)ζ α /bracketrightBig + i 2r −1 φ(ˆp¯ζ) α , kievarwe.tex; 12/03/2001; 3:49; p.175 LORENTZ SPINORIAL CENTRAL CHARGES 169 ¯θ ˙α = 1 4r −3/2 /bracketleftBig (ζˆψ¯ζ)(ζˆp) ˙α + 2(ζ˜ψˆpζ)¯ζ ˙α /bracketrightBig − i 2r −1 φ(ζˆp) ˙α , ρ= 1 2mr −2 (¯ζ˜pˆψ˜pζ), ˜ψ 5 = 1 m(p µ ψ µ +mψ 5 )−(2mr) −1 (ζˆψ¯ζ)(p 2 +m 2 ). In the new variables the Dirac constraint takes a simple form. On mass shell p 2 +m 2 = 0wehave D=pψ+mψ 5 =m˜ψ 5 ≈0. (34) Moreover, we can extract from the new variables a pure gauge degree of free- dom for fermionic local symmetry of the spinning particle [12, 13] (world-line supersymmetry) δχ= ˙/epsilon1, δe =−2i/epsilon1χ, δψ µ =−/epsilon1p µ , δψ 5 =−/epsilon1m, δx µ =i/epsilon1ψ µ . Inthenewvariablesthis transformationtakes theform δθ α =− 1 4/epsilon1r −1/2 (ˆp¯ζ) α , δ¯θ ˙α =− 1 4/epsilon1r −1/2 (ζˆp) ˙α , δρ=− 1 2/epsilon1mr −1 , δ ˜ψ 5 =− 1 2m/epsilon1(p 2 +m 2 )≈0. Thus, theonlytransformedarethevariable ρand onecomponent ofspinor θ δ(θζ+¯ζ¯θ) = 1 2/epsilon1r 1/2 . Subsequentlythecombination ρ+mr −3/2 (θζ+¯ζ¯θ)ofthiscomponent θandρis invariantunderthegaugetransformations, δ[ρ+mr −3/2 (θζ+¯ζ¯θ)] = 0,whereas thevariable ρ−mr −3/2 (θζ+¯ζ¯θ) (35) isthepuregauge degreeof freedom, δ[ρ−mr −3/2 (θζ+¯ζ¯θ)] =−mr −1 /epsilon1. Accounting the equation of motion for bosonic spinor ˙ζ= 0and substituting theresolvingexpressions(29),(30)for ψ µ ,ψ 5 intheLagrangian(28)wearriveat theLagrangian L=p( ˙x−i˙θσ¯θ+i˙θσ˙¯θ)−im 2 r −1 (θζ¯ζ˙¯θ−˙θζ¯ζ¯θ) + i 2r 2 /bracketleftBig ρ+mr −3/2 (θζ+¯ζ¯θ) /bracketrightBig/bracketleftBig ˙ρ+mr −3/2 (˙θζ+¯ζ˙¯θ) /bracketrightBig + i 2r /bracketleftBig ρ−mr −3/2 (θζ+¯ζ¯θ) /bracketrightBig ˙˜ψ 5 + i 2r˜ψ 5 /bracketleftBig ˙ρ−mr −3/2 (˙θζ+¯ζ˙¯θ) /bracketrightBig + i 2˜ψ 5 ˙˜ψ 5 −imχ˜ψ 5 − e 2(p 2 +m 2 ) +˙ζv+ ¯v˙¯ζ−λ(ζˆp¯ζ−j). (36) kievarwe.tex; 12/03/2001; 3:49; p.176 170 S.FEDORUK,V.G.ZIMA It should be stressed that the equation ˙ζ= 0for bosonic spinor, which has been used for derivation of the Lagrangian (36), is reproduced by the same La- grangian (36). As we see from the Lagrangian, the gauge variable (35) is the cor- responding conjugate variable for ˜ψ 5 which generates the local transformations. The simplergauge ﬁxingconditionfor it ρ−mr −3/2 (θζ+¯ζ¯θ) = 0 gives us the possibility to resolve the scalar ρin term of spinor projection (θζ+ ¯ζ¯θ). Wetake themoregeneralcondition ofthistype ρ−mr −3/2 (θζ+¯ζ¯θ) = 2(k−1)mr −3/2 (θζ+¯ζ¯θ) (37) whichisthegaugeﬁxingconditionatall kexceptk= 0.Atk= 0(37)isreduced totheconditionon gauge invariant variable ρ+mr −3/2 (θζ+¯ζ¯θ) = 0 and ofcourseitisnot a gauge ﬁxing. Substituting in the Lagrangian (36) the constraint condition ˜ψ 5 = 0(the equation ofmotion fortheLagrange multiplier χ) and theexpression ρ= (2k−1)mr −3/2 (θζ+¯ζ¯θ) (38) (following from thegaugeﬁxing condition (37)) we obtain the Lagrangian L=p˙ω θ +iZ αβ θ α ˙θ β +i¯Z ˙α˙β ¯θ ˙α ˙¯θ ˙β +iZ α˙β (θ α ˙¯θ ˙β −˙θ α ¯θ ˙β )− e 2(p 2 +m 2 ) +˙ζv+ ¯v˙¯ζ−λ(ζˆp¯ζ−j). (39) In this expression ω θ ≡˙ω θ dτ=dx−idθσ¯θ+iθσd¯θis the usual N= 1 superinvariant ω-form. The quantities Z αβ =Z βα ,¯Z ˙α˙β = (Z αβ )andZ α˙β = (Z β˙α )areexpressed in terms of bosonicspinor ζ(for similarformulasee[10]) Z αβ = 2k 2 m 2 j −1 ζ α ζ β , Z α˙β = (2k 2 −1)m 2 j −1 ζ α ¯ζ ˙β .(40) Z αβ and¯Z ˙α˙β aretensorcentralcharges(types (1,0)and(0,1))andZ α˙β isvector one (type (1/2,1/2)) fortheD= 4N= 1supersymmetry algebra [17]-[21]. The same result is obtained if we consider the connection of the systems (28) and (5) in the Hamiltonian formalism. Precisely there is the canonical transfor- mation which connect the models with each other. Now in order to make equal the number of Grassmannian variables in the models we introduce pure gauge variableφin the initial model of the spinning particle. Its pure gauge nature is achievedbythe presence oftheﬁrst classconstraint p φ ≈0 (41) kievarwe.tex; 12/03/2001; 3:49; p.177 LORENTZ SPINORIAL CENTRAL CHARGES 171 in the initial model. So in the canonical transformation we imply that the term p φ ˙φ−µp φ is added to the Lagrangian (28). Here µis Lagrange multiplier. The resolutionof φin terms ofthe spinorsisgivenbytheexpression(33). As the generating function of the canonical transformation from system with coordinates ψ µ ,ψ 5 ,φ,x µ ,ζ α ,¯ζ ˙α to the system with coordinates θ α ,¯θ ˙α ,ρ,˜ψ 5 , x /primeµ ,ζ /primeα ,¯ζ /prime˙α wetake F=−p µ ψ ψ µ (p µ ,ζ,θ,ρ )−p ψ5 ψ 5 (ζ,θ,ρ, ˜ψ 5 )−p φ φ(ζ,θ) +ζ α v /prime α + ¯v /prime ˙α ¯ζ ˙α −p µ x /prime µ . (42) Here the expressions for old variables in term of new ones from the right hand side of the equations (29), (30), (33) have been used. That construction of the generatingfunction(42)reproduces,bydeﬁnitionofthecanonicaltransformation, the resolution (29), (30), (33) of the initial Grassmannian coordinates in spinors ψ µ =−∂ l F/∂p µ ψ ,ψ 5 =−∂ l F/∂p ψ5 ,φ=−∂ l F/∂p φ and leaves invariable bosonic spinor coordinates ζ /primeα =∂F/∂v /prime α =ζ α ,¯ζ /prime˙α =∂F/∂ ¯v /prime ˙α =¯ζ ˙α and the momentum vector p /prime µ =−∂F/∂x /primeµ =p µ . The expression of new Grassmannian momentaintermsof initialonesare p θα =−∂ r F/∂θ α =r −1/2 (σ µ ˜pζ) α p µ ψ −mr −1/2 ζ α p ψ5 +iζ α p φ , ¯p θ˙α =−∂ r F/∂¯θ ˙α =r −1/2 (¯ζ˜pσ µ ) ˙α p µ ψ −mr −1/2 ¯ζ ˙α p ψ5 −i¯ζ ˙α p φ , p ρ =−∂ r F/∂ρ =−m(ζσ µ ¯ζ)p µ ψ +rp ψ5 , p ˜ψ5 =−∂ r F/∂˜ψ 5 =p ψ5 . The expressions of the initial bosonic spinor momenta v α =∂F/∂ζ α ,¯v ˙α = ∂F/∂ ¯ζ ˙α and space-time coordinate x µ =−∂F/∂p µ in terms of the new phase space coordinates contain besides corresponding new phase variables the addi- tional terms depending on the new Grassmannian phase space variables. These terms arise because of the dependence of the resolution expressions (29), (30), (33) onζ,¯ζandp. Here we do not need the expressions for v /prime ,¯v /prime andx /prime in the explicitformdue toindependenceof allconstraints onthese phase variables. Nowweeliminatethevariables ˜ψ 5 ,p ˜ψ5 bymeansoftheDiracconstraint(26) and gaugeﬁxing conditionforDirac constraint p ˜ψ5 −i(k−1)mr −1/2 /bracketleftbig θζ+¯ζ¯θ /bracketrightbig ≈0 (43) atk/negationslash= 0 2 . After fulﬁllment of the additional canonical transformation p ρ → p ρ /prime =p ρ −ikmr 1/2 /bracketleftbig θζ+¯ζ¯θ /bracketrightbig , which leads to resolving form p ρ /prime ≈0of one 2 ThediagonalizedDiracconstraint D /prime ≡D−ip µ g µ −img 5 =−i[p µ (p µ ψ + i 2 ψ µ )+m(p ψ5 + i 2 ψ 5 )]≈0hasinnewvariablestheform D /prime = i 4 r −1/2 /bracketleftbig ¯ζ˜pp θ + ¯p θ ˜pζ /bracketrightbig − i 2 mr −1 p ρ + 1 2 m˜ψ 5 ≈0. ThePoissonbracketofthecondition(43)and D /prime isequalto (km)/2,i.e.atk= 0thecondition(43) does not ﬁxthe gauge for the Dirac constraint. kievarwe.tex; 12/03/2001; 3:49; p.178 172 S.FEDORUK,V.G.ZIMA Fermi-constraint from (27), we eliminate the variables ρ,p ρ with the help of two from ﬁve second class Fermi-constraints (27). Because of the resolving form of theconstraintswithrespecttoeliminatedvariables, ˜ψ 5 ≈0andp ρ /prime ≈0,theDirac bracketsforremainingvariablesarethesameastheirPoissonbrackets.Afterthat theremainingGrassmannian constraints take thefollowing form ¯ζ˜pp θ −¯p θ ˜pζ≈0, (44) /bracketleftbig ¯ζ˜pp θ + ¯p θ ˜pζ /bracketrightbig −4ik 2 m 2 /bracketleftbig θζ+¯ζ¯θ /bracketrightbig ≈0, (45) ζ /bracketleftbig −ip θ −ˆp¯θ /bracketrightbig ≈0, [−i¯p θ −θˆp]¯ζ≈0 (46) whicharethesameastheprojectionsonspinors ζ,ˆp¯ζoftheGrassmannianspinor constraints d θα ≡−ip θα −(ˆp¯θ) α −θ β Z βα −Z α˙β ¯θ ˙β ≈0, (47) ¯d θ˙α ≡−i¯p θ˙α −(θˆp) ˙α −¯Z ˙α˙β ¯θ ˙β −θ β Z β˙α ≈0 (48) withquantities Z αβ ,Z α˙β deﬁnedin(40).Frominvarianceofthevariables ζ α ,¯ζ ˙α , p µ under the canonical transformation, all bosonic constraints, i.e. p 2 +m 2 ≈0 andζˆp¯ζ−j≈0, are not changed. The system with remaining variables and the constraints is described by the above mentioned Lagrangian (5). The Lagrangian (5)reproducesaccuratelythissetof the constraintsand nothing else. ThusweestablishthatthemodeldescribedbyLagrangian L=L 1/2 +L b.s. is equivalentphysicallytothemodelwithLagrangian L=L super +L b.s. atclassical level.HereL 1/2 istheLagrangian(25)ofthemassivespinningparticle(spin 1/2) whereasL super is Lagrangian of the massive N= 1superparticle with tensorial centralcharges (40) L super =p˙ω θ +iZ αβ θ α ˙θ β +i¯Z ˙α˙β ¯θ ˙α ˙¯θ ˙β +iZ α˙β (θ α ˙¯θ ˙β −˙θ α ¯θ ˙β )− e 2(p 2 +m 2 ). (49) Lagrangians L b.s. of the bosonic spinor in the both equivalent models are quite identical. It should be noted that the value of constant kin the formula (40) for central charges of the superparticle is nonzero, k/negationslash= 0, in the case of its equivalence to the spinning particle. But in general the value k= 0is not forbidden in model of superparticle with central charges. Next we consider the cases both with k/negationslash= 0 andk= 0. As we see below at k/negationslash= 0andk= 0we have superparticle models withone andtwo κ-symmetries respectively. kievarwe.tex; 12/03/2001; 3:49; p.179 LORENTZ SPINORIAL CENTRAL CHARGES 173 3.3. ANALYSIS ON LEVEL OF PHYSICALDEGREES OFFREEDOM Alternative way for a proof of classical equivalence of the massive spin 1/2 particle (25) and the massive superparticle with central charges (49), at k/negationslash= 0, possessing one κ-symmetry is the reduction of both models to physical degrees of freedom [27]. In the examining positive energy sector after choice of gauge ψ − =ψ 0 −ψ 5 = 0forDiracconstraintandexclusionof ψ + =ψ 0 +ψ 5 bymeans of the constraint condition we obtain for the physical odd degrees of freedom of spinning particle [28, 27] the Lagrangian in the form of L (ph) 1/2,Gr = i 2 /vectorψ˙/vectorψ. On the otherhandtheGrassmannianpartofthesuperparticleLagrangian L super takesthe form L (ph) super,Gr =i¯q˙q−iq˙¯q+ 2k 2 iη˙η after usingofthevariables η=mr −1/2 (θζ+¯ζ¯θ), σ =−imr −1/2 (θζ−¯ζ¯θ), (50) q=r −1/2 (θˆp¯ζ),¯q=r −1/2 (ζˆp¯θ). (51) Setting q= (ψ 1 +iψ 2 )/2,¯q= (ψ 1 −iψ 2 )/2, η =ψ 3 /2k we obtainexactlythe same Grassmannian partof the Lagrangian L (ph) super,Gr =L (ph) 1/2,Gr = i 2/vectorψ˙/vectorψ. (52) SuchLagrangianforthephysicaloddvariablescomesoutalsofromwork[23] in non-Lorentz covariant Grassmannian sector N= 4→N= 1PBGS. In ﬁrst order formalism the target space actionof this workhas the Lagrangian L=/vectorP/vectorΠ−P 0 Π 0 + e 2(P 02 −/vectorP 2 −1)−Θ˙Θ−/vectorΨ˙/vectorΨ (53) where Π 0 =˙X 0 + Θ˙Θ +/vectorΨ˙/vectorΨ,/vectorΠ =˙/vectorY−˙Θ/vectorΨ + Θ˙/vectorΨ(weremainherethenotations of[23]).Inaccountingthelastexpressions, the Lagrangian (53) takes the form L=/vectorP˙/vectorY−P 0 ˙X 0 + e 2(P 02 −/vectorP 2 −1) −(P 0 + 1) /bracketleftbigg /vectorΨ− 1 P 0 + 1/vectorPΘ /bracketrightbigg/bracketleftbigg /vectorΨ− 1 P 0 + 1/vectorPΘ /bracketrightbigg · . Afterusingof thevariables /vectorψ= √ 2(P 0 + 1) 1/2 /bracketleftbigg /vectorΨ− 1 P 0 + 1/vectorPΘ /bracketrightbigg kievarwe.tex; 12/03/2001; 3:49; p.180 174 S.FEDORUK,V.G.ZIMA we obtainexactlythe Lagrangian (52) forGrassmannian variables. 3.4. SUPERPARTICLE WITH INDEX SPINOR In order to analyze the properties of the obtained massive superparticle with ten- sorial central charges let us consider the model of spinning particle with index spinor [14, 11, 15] as additional bosonic coordinates. It is naturally because we have used for bosonic spinor the relation ζˆp¯ζ−j≈0which is inherent in the index spinor approach. In the Hamiltonian formalism the index spinor sector is restrictedbythe spinorself-conjugacy conditions d ζ ≡ip ζ −ˆp¯ζ≈0, ¯d ζ ≡−iˆp ζ −ζˆp≈0 (54) whicharethesecondclassconstraintsinthemassivecase.Itisachievedinabove model (28) by the substitution v=−iˆp¯ζ,¯v=iζˆp. ThenL b.s. (6) takes the form oftheindexspinorLagrangian [14] L index =−i˙ζˆp¯ζ+iζˆp˙¯ζ−λ(ζˆp¯ζ−j). (55) The constraint ζˆp¯ζ−j≈0included in the Lagrangian generates in Hamiltonian formalismthe spinconstraint i 2(ζp ζ −¯p ζ ¯ζ)−j≈0 (56) which together with second class constraints (54) leads [14] to the particle state ofthesinglespinassociatedwithgivensectorofindexspinor.Spinoftheparticle in the quantum spectrum is the value of the constant jrenormalized by ordering constants (thus jcan benamed“classicalspin”). The realization of the previously considered canonical transformation to the modelwithLagrangian L /prime =L 1/2 +L index ,i.e.L index insteadL b.s. in(28),leads totheLagrangian L /prime =p˙ω+iZ αβ θ α ˙θ β +i¯Z ˙α˙β ¯θ ˙α ˙¯θ ˙β +iZ α˙β (θ α ˙¯θ ˙β −˙θ α ¯θ ˙β ) +iY αβ ζ α ˙ζ β +i¯Y ˙α˙β ¯ζ ˙α ˙¯ζ ˙β +iY α˙β (ζ α ˙¯ζ ˙β +˙ζ α ¯ζ ˙β ) −iN(˙ζˆp¯ζ−ζˆp˙¯ζ) − e 2(p 2 +m 2 )−λ(ζˆp¯ζ−j). (57) Heretheform ω≡˙ωdτ =dx−idζσ¯ζ+iζσd¯ζ−idθσ¯θ+iθσd¯θisinvariantwith respect to the transformations of the usual N= 1supersymmetry with Grass- mannian spinor parameter and “bosonic supersymmetry” with c-number spinor parameter [14, 11, 15]. The central charges Z αβ ,Z α˙β have the same form (40). kievarwe.tex; 12/03/2001; 3:49; p.181 LORENTZ SPINORIAL CENTRAL CHARGES 175 So the kinetic terms of the space-time coordinate and Grassmannian spinor in L /prime (57) are identical to the corresponding terms in L(5) and hence the algebras of thefermionicconstraintsinbothmodelsareidentical.Butthekinetictermsofthe index spinor in Lagrangian L /prime are different from the kinetic terms of the bosonic spinorinLagrangian Lbyadditional terms with quantities Y αβ = 2k(k−2)m 2 j −1 θ α θ β , ¯Y ˙α˙β = −(Y αβ ), Y α˙β =−(2k 2 −4k+ 1)m 2 j −1 θ α ¯θ ˙β (58) which canberegarded asthecentralchargesofthe“bosonic SUSY” aswell as N≡j −1 /bracketleftBig (θˆp¯θ) + 2(2k−1)m 2 j −1 (θζ)(¯ζ¯θ) /bracketrightBig . (59) The appearance of these extra terms is the result of modiﬁcation of index spinor momentap ζ ,¯p ζ underthecanonicaltransformationand,asconsequence,themod- iﬁcation of the spin constraint (56) and bosonic spinor constraints (54) expressed by newvariables. Speciﬁc peculiarity of the model (57) with index spinor is an interconnec- tion between usual fermionic supersymmetry and “bosonic one” and at present its meaning is not yet quite clear. Some duality appears in the invariance under permutation of Grassmannian and bosonic spinors both ω-form and certain terms withcentralcharges ofdifferenttypes. 4. Gaugesymmetriesofmassivesuperparticlewithtensorialcentralcharges For local transformation ofthe Grassmannianspinor δθ α =iκ(¯ζ˜p) α , δ ¯θ ˙α =−i¯κ(˜pζ) ˙α (60) and standardSiegeltransformation[29,30]of thespace-timecoordinate δx µ =−iθσ µ δ¯θ+iδθσ µ ¯θ (61) withlocalcomplexGrassmannianparameter κ(τ)thevariationoftheLagrangians up toatotal derivativeis δL=−2k 2 m 2 (θζ+¯ζ¯θ)(κ−¯κ) · + 2k 2 m 2 (θζ+¯ζ¯θ) · (κ−¯κ) −4km 2 j −1 [(θˆp¯ζ)ζ˙ζ+ (ζˆp¯θ)˙¯ζ¯ζ](κ−¯κ). (62) As we see,δL= 0for realκ= ¯κat arbitrary values of constant k. But atk= 0 we haveδL= 0for arbitrary complex parameter κ. Thus atk/negationslash= 0when the tensor central charge Z αβ is present the models have one κ-symmetry with real kievarwe.tex; 12/03/2001; 3:49; p.182 176 S.FEDORUK,V.G.ZIMA Grassmannianparameter κ= ¯κ.Butatk= 0whenthereisonlythevectorcentral chargeZ α˙β wehavetwo κ-symmetrieswithcomplexGrassmannianparameter κ. A ﬁrst class constraint is associated to each gauge symmetry in Hamilto- nian formalism. As is already noted our systems are described by the fermionic constraints(covariantderivatives)(47),(48). TheirPoissonbracketsalgebra is {d θα ,d θβ }= 2iZ αβ , /braceleftBig ¯d θ˙α ,¯d θ˙β /bracerightBig = 2i¯Z ˙α˙β , /braceleftBig d θα ,¯d θ˙β /bracerightBig = 2i /parenleftBig p α˙β +Z α˙β /parenrightBig (63) with central charges (40). Covariant separation of the fermionic ﬁrst and second classconstraintsisachievedbytheprojectiononthespinors ζ α ,(ˆp¯ζ) α .Letusput χ θ ≡ζd θ =−iζp θ −ζˆp¯θ≈0,¯χ θ ≡¯d θ ¯ζ=−i¯p θ ¯ζ−θˆp¯ζ≈0,(64) g θ ≡¯ζ˜pd θ +¯d θ ˜pζ=−i(¯ζ˜pp θ + ¯p θ ˜pζ)−4k 2 m 2 (θζ+¯ζ¯θ)≈0,(65) f θ ≡i(¯ζ˜pd θ −¯d θ ˜pζ) =¯ζ˜pp θ −¯p θ ˜pζ≈0. (66) The nonzeroPoisson bracketsof theseprojections are {χ θ ,¯χ θ }= 2ij,{g θ ,g θ }= 16k 2 m 2 ij. (67) Thus the constraints χ θ ,¯χ θ are always the second class constraints whereas the constraintf θ is always the ﬁrst class constraint generating one κ-symmetry with localparameter (κ+ ¯κ)onvariable (θζ−¯ζ¯θ), /braceleftbig f θ ,θζ−¯ζ¯θ /bracerightbig = 2r,δ(θζ−¯ζ¯θ) = ir(κ+ ¯κ). The constraint g θ is the second class constraint at k/negationslash= 0. But atk= 0 the constraint g θ becomes the ﬁrst class constraint and generates additional κ- symmetry with local parameter i(κ−¯κ)on variable (θζ+¯ζ¯θ), /braceleftbig g θ ,θζ+¯ζ¯θ /bracerightbig = −2ir,δ(θζ+¯ζ¯θ) =ir(κ−¯κ). Thus we obtain the models of the D= 4N= 1massive superparticle with tensorial central charges possessing one or two Siegel κ-symmetries. In the language of the brane theories these models correspond to the BPS superbrane conﬁgurations preserving 1/4or1/2of supersymmetry (see [21] and references therein). It should be noted that constant kin the construction of the superparticle appears in the gauge ﬁxing condition under transition from the spinning particle. Therefore at all k/negationslash= 0the superparticle has quite similar systems of the con- straints and the same number of physical degrees of freedom. The models at all k/negationslash= 0areequivalent.Undertransformationswhichcanbeconsideredascanonical transformations θ α →θ α +br −1 (θζ+¯ζ¯θ)(¯ζ˜p) α ,¯θ ˙α →¯θ ˙α +br −1 (θζ+¯ζ¯θ)(˜pζ) ˙α (68) kievarwe.tex; 12/03/2001; 3:49; p.183 LORENTZ SPINORIAL CENTRAL CHARGES 177 wherebis real number the Lagrangian L(orL /prime ) transforms into the same La- grangian with akin place ofkwherea≡1 + 2b. As ﬁnal result at level of the free superparticle we have two substantially different models of the massive superparticle with tensorial central charges. First of them at k= 1/ √ 2has only tensorcentralcharge Z αβ andpossessesone κ-symmetry.Secondmodelat k= 0 hasonlyvectorcentralcharge Z α˙β and possesses two κ-symmetries. 5. Quantum spectrumofthe models In process of the construction it is established the equivalence at classical level between the massive D= 4N= 1superparticle with one κ-symmetry and the massiveD= 4n= 1spinning particle. But they may lead to distinct quantum theories[27].Belowweestablishthatthespinningparticleandsuperparticlewith tensorial central charges, which have index spinor as additional one, have iden- tical state spectrum. By analogy with results in paper [12–14] the ﬁrst operator quantization of the spinning particle with index spinor described by Lagrangian L 1/2 +L index isimmediate.WavefunctioninthemodelisdeﬁnedbyDiracspinor with (anti)holomorphic dependence in index spinor of homogeneity degree 2J whereJis the classical spin jrenormalized by the ordering constant. Writing Dirac spinor in terms of Weyl spinors as /parenleftBig ψ χ /parenrightBig , in according to analysis carried out in [14] we have in holomorphic case two multispinor ﬁelds ψ α 1 ...α 2J β and χ α 1 ...α 2J ˙β which are symmetrical in 2Jindicesαs. Hereβand˙βcorrespond to bispinor index. Theseﬁelds are connectedwith eachother by Dirac equation /parenleftbigg 0 ˜p ˆp0 /parenrightbigg/parenleftbigg ψ χ /parenrightbigg =m /parenleftbigg ψ χ /parenrightbigg (69) (quantumcounterpartoftheDiracconstraint(26)).Comparisonwithsuperparticle modelismoreimmediateifwetaketheﬁeld χ α 1 ...α 2J ˙β asbasicone.Buttheﬁeld ψ α 1 ...α 2J−1 α 2J β =φ (α 1 ...α 2J β) +φ (α 1 ...α 2J−1 /epsilon1 α 2J )β exhibitssimplythattwospins J± 1 2 are presented in spectrum at ﬁxed Jas it should be when one adds spin J whichisgivenbyindexspinorandspin 1 2 whichcorrespondstotheGrassmannian variablesψ µ ,ψ 5 of the pseudoclassical mechanicsunder quantization. The quantization of the superparticle (57) is suitable to carry out in vari- ables (50), (51) in term of which the fermionic constraints (64)-(66) take the extremelysimple forms ip q + ¯q≈0, i¯p q +q≈0, ip η + 2k 2 η≈0, (70) p σ ≈0. kievarwe.tex; 12/03/2001; 3:49; p.184 178 S.FEDORUK,V.G.ZIMA We gauging out the variable σ, the introduce the Dirac brackets for taking into account of the fermionic second class constraints and the represent the remaining fermionic variables q,¯q,η(in fact/vectorψ) by means of the usual Pauli σ-matrices. Thusthewavefunctionofthisproblemhastwocomponentsdependingappropri- ately on index spinor and space-time variables. The quantization of the bosonic spinor sector shows certain difference with [14]. Additional term of the form q¯q in spin constraint (56) arising due to interaction of bosonic and fermionic sectors leads to different homogeneity degrees (which correspond to different represen- tations of Lorentz group) for two components of wave function. Bosonic spinor constraints (54) ((anti)homogeneity conditions) acquire the additional terms both withq¯qand alsoqη(or¯qη). These last terms, which are proportional σ + (orσ − ), σ ± ≡(σ 1 ±iσ 2 )/2in matrix realization of odd variables, connect two compo- nentsofwavefunction.Asresulttheirreducible (2J+ 1)-componentspinorﬁeld φ α 1 ...α 2J+1 , in term of which one component of wave function is determined, is expressedbyDirac equation p γ˙β χ α 1 ...α 2J ˙β =mφ α 1 ...α 2J γ (71) via ﬁeldχ α 1 ...α 2J ˙β which determines second component of wave function. This last ﬁeldχ α 1 ...α 2J ˙β can be identiﬁed with basic ﬁeld of the spinning particle spectrum. In case of models (28) and (5), when there is not present the truncation of bosonic spinor sector to the index one because of absence of bosonic spinor con- straints, the quantum equivalence apparently remains too. One can expect it from the quite identity of bosonic sectors of the models (28) and (5) and identifying of physical fermionicdegrees offreedom whichhasbeen demonstrated in Sec. 2. In case of the Lagrangian (5) one can include vector central charge Z µ into vector of space-time momentum by the shift p µ →p µ +Z µ after taking into account the bosonic spinor equation of motion ˙ζ= 0. Therefore at k= 0, when there is vector central charge only, it disappears completely from the action and superparticle model reduces in fact to massless case. Unlike this in the particle model(57)withindexbosonicspinorat k= 0theredeﬁnitionofmomentumdoes not exclude vector central charge due to accompanying modiﬁcation of bosonic spinor and spin constraints. In this case the wave function contains two usual spin-tensor ﬁelds φ α 1 ...α 2J±1 , satisfying massive Klein-Gordon equation and dis- connected with each other because of missing terms with qηin bosonic spinor constraints. 6. Conclusion In this work we presented the manifestly Lorentz-invariant formulation of the D= 4N= 1free massive superparticle with tensorial central charges. The tensorial central charges are construct by commuting bosonic spinor and also by kievarwe.tex; 12/03/2001; 3:49; p.185 LORENTZ SPINORIAL CENTRAL CHARGES 179 indexspinor.Themodelpossessesingeneraloneortwo κ-symmetries.Inparticu- larcasethemodelcontainsarealparameter kandatk/negationslash= 0ithasoneκ-symmetry while atk= 0the number of κ-symmetries is two. The local transformations ofκ-symmetry are written out. It is obtained the equivalence at classical level between the massive D= 4superparticle with one κ-symmetry and the massive D= 4spinningparticle. Acknowledgments. We would like to thank I.A.Bandos, E.A.Ivanov, S.O.Krivo- nos,J.Lukierski,A.Yu.Nurmagambetov,D.P.Sorokin,A.A.Zheltukhinforinterest to the work and for many useful discussions. The authors are grateful to I.A.Ban- dos, V.P.Berezovoj, A.Yu.Nurmagambetov, D.P.Sorokin for the hospitality at the NSC Kharkov Institute of Physics and Technology. This work was partially supportedbyresearchgrantoftheMinistryofEducationandScienceofUkraine. References 1. R.Haag,J.Lopuszanski andM.Sonius, Nucl. Phys. B78(1995)257 2. J.WessandJ.Bagger, SupersymmetryandSupergravity ,1983(Princeton:PrincetonUniversity Press) 3. J.van Holten andA. van Proyen, J. Phys. A15(1982)3763 4. P.K.Townsend, P-brane democracy , inParticles, Strings and Cosmology , eds. J.Bagger, G.Domokos, A.Falk and S.Kovesi-Domokos (World Scientiﬁc 1996), hep-th/9507048; Four lectures on M-theory , hep-th/9612121; F-theory fromits superalgebra ,hep-th/9712004 5. I.Bars, Phys. Lett. B 373(1996)68;Phys.Rev. D 54(1996)2503 6. Y.Eisenberg and S.Solomon, Phys.Lett. B220(1989)562 7. S.F.Hewson and M.J.Perry, Nucl. Phys. B492(1997)249 S.F.Hewson, Nucl. Phys. B 501(1997)445; Anapproach to F-theory , hep-th/9712017 8. 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B446(1995)79; I.A.Bandos, D.P.Sorokin and D.V.Volkov, Phys. Lett. B 352(1995)269 D.P.Sorokin, Phys. Rep. 329(2000)1 (and refs. therein) 27. P.K.Townsend, Phys. Lett. B 261(1991)65 28. J.P.Gauntlett, J.Gomis andP.K.Townsend, Phys. Lett. B 248(1990)288 29. J.A.de Azcarraga and J.Lukierski, Phys. Lett. B 113(1982)170 30. W.Siegel, Phys. Lett. B 128(1983)397 kievarwe.tex; 12/03/2001; 3:49; p.187 ROTATING SUPERBLACK HOLEAS SPINNING PARTICLE ALEXANDER BURINSKII ∗ NSI,Russian AcademyofSciences, Moscow, Russia 1. Introduction The Kerr rotating black hole solution displays some remarkable features indicat- ingarelationtothestructureofthespinningelementaryparticles.Inparticular,in the1969Carter[1]observed,thatifthreeparametersoftheKerr-Newmanmetric are adopted to be ( /planckover2pi1=c=1 )e 2 ≈1/137, m≈10 −22 , a≈10 22 , ma = 1/2,then one obtains a model for the four parameters of the electron: charge, mass,spinandmagneticmoment,andthegyromagneticratioisautomaticallythe same as that of the Dirac electron. Investigations along this line [2–6] allowed to ﬁnd out stringy structures in the real and complex Kerr geometry and to put forward a conjecture on the baglike structure of the source of the Kerr-Newman solution. The earlier investigations [2, 13, 5] showed that this source represents a rigid rotator ( a relativistic disk ) built of an exotic matter with superconducting properties. Since 1992 black holes have paid attention of string theory. In 1992 the Kerr solution was generalized by Sen to low energy string theory [7], and it was shown [17] that near the Kerr singular ring the Kerr-Sen solution acquires a metricsimilartotheﬁeldaroundaheteroticstring.Thepointofviewhasappeared that black holes can be treated as elementary particles [8]. On the other hand, a description of a spinning particle based only on the bosonic ﬁelds cannot be complete, and involving fermionic degrees of freedom is required. Therefore, the spinningparticlemustbebasedonasuper-Kerr-Newmanblackholesolution[18] representinganaturalcombinationoftheKerrspinningparticleandsuperparticle model.Angularmomentum Lofspinningparticlesisveryhigh |a|=L/m≥m, and the horizons of the Kerr metric disappear. There appears a naked ring-like singularitywhichhastoberegularizedbeingreplacedbyasmoothmattersource. In this review we consider a source representing a rotating superconducting bag with a smooth domain wall boundary described by a supersymmetric version of ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.188 182 A.BURINSKII theU(I)×U /prime (I)ﬁeldmodel[23].Infact,thismodeloftheKerr-Newmansource represents a generalization of the Witten superconducting string model [16] for thesuperconductingbaglikesources[6]. 2. Complex sourceof Kerrgeometry and itsstringy interpretation The Kerr-Newmansolutioncanberepresented inthe Kerr-Schild form g µν =η µν + 2he 3 µ e 3 ν , (1) whereη µν is metric of an auxiliary Minkowski space η µν =diag(−1,1,1,1), andhis a scalar function. Vector ﬁeld e 3 is null,e 3 µ e 3µ = 0,and tangent to PNC ( principal null congruence ) of the Kerr geometry. The Kerr PNC is twisting i.e. correspondingtoavortexofanullradiation. 1 Oneofthemainpeculiaritiesofthe KerrgeometryissingularringrepresentingabranchlineoftheKerrspaceonthe ‘positive’(r>0)and‘negative’( r<0)sheetswhicharedividedbythedisk r= 0spannedbythisring.TheKerrsingularringisexhibitedasapoleofthefunction h(r,θ) = mr−e 2 / 2 r 2 +a 2 cos 2 θ ,whererandθare the oblate spheroidal coordinates. The KerrPNCisin-goingonthe‘negative’sheetofspace,itcrossesthedisk r= 0and turns into out-going one on the ‘positive’ sheet. Appearance of the Kerr singular ringontherealspace-timecanalsobeobservedintheCoulombsolution f=e/˜r when its point-like source is shifted in complex region (x 0 ,y 0 ,z 0 )→(0,0,ia). Radial distance ˜rbecomes complex in this case and can be expressed as ˜r= r+iacosθ(Appel,1987!).Similarly,thesourceofKerr-Newmansolutioncan be considered from complex point of view as a ”particle” propagating along a complexworld-line [9,12]parametrized bycomplextime. The objects described by the complex world-lines occupy an intermediate position between particles and strings. Like the strings they form the two- dimensional surfaces or the world-sheets in the space-time. It was shown that thecomplexKerrsourcemaybeconsideredasacomplexhyperbolicstringwhich requires an orbifold-like structure of the world-sheet. In many respects this string is similar to the ‘mysterious’ N= 2string of superstring theory shedding a light onthepuzzleofitsphysicalinterpretation.Aswehavealreadymentioned,thereis one more stringy structure in the Kerr geometry connected with the Kerr singular ring. In fact the both these stringy structures are different exhibitions of some membrane-like source. This source has a complex interpretation alongside with some realimage intheform of a rotating bubblewhich will bediscussed further. The Kerr PNC may be obtained from the complex source by a retarded-time construction. The rays of PNC are the tracks of null planes of the complex light cones emanated from the complex world line [11, 12]. The complex light cone 1 Besides,theKerrPNCisgeodesicandshearfree,itrepresentsabundleoftwistorsandcanbe described by the Kerrtheorem [10,11, 9, 12]. kievarwe.tex; 12/03/2001; 3:49; p.189 ROTATINGSUPERBLACKHOLE 183 withthevertexatsomepoint x 0 ofthecomplexworldline x µ 0 (τ):(x µ −x 0µ )(x µ − x µ 0 ) = 0,can be split into two families of null planes: ”left” planes spanned by null vectors e 1 ande 3 , and”right”planes spanned by null vectors e 2 ande 3 . The Kerr PNC arises as the real slice of the family of the ”left” null planes of the complexlightcones whichvertices lieon thestraight complex worldline x 0 (τ). Onlytheconeslyingonthestrip |Imτ|≤|a|havearealslice.Therefore,the ends of the resulting complex string are open. To satisfy the complex boundary conditions,anorbifold-likestructureoftheworldsheetmustbeintroduced[9,12], which iscloselyconnected with theabovementioned Kerr’s twosheetedness. 3. Super-Kerr-Newman geometry A supergeneralization of the Kerr-Newman solution can be obtained as a natural combinationoftheKerrspinningparticleandsuperparticle[18].Infact,thecom- plex structureof theKerr geometrysuggests the way ofits supergeneralization. Note,thatanyexactsolutionoftheEinsteingravityisindeedatrivialsolution of supergravity ﬁeld equations. The supergauge freedom allows one to turn any gravitysolutionintoaformcontainingspin-3/2ﬁeld ψ i satisfyingthesupergravity ﬁeld equations. However, since this spin-3/2 ﬁeld can be gauged away by the reverse transformation, such supersolutions have to be considered as trivial. The hint how to avoid this triviality problem follows from the complex structure of theKerrgeometry.Infact,fromthecomplexpointofviewtheSchwarzschildand Kerr geometries are equivalentandconnectedby a trivialcomplex shift. Thenon-trivial twisting structure of the Kerr geometry arises as a result of the complex shift of the real slice concerning the center of the solution [11, 9]. Similarly,itispossibletoturna trivialsuperblackholesolutionintoa non-trivial . Thetrivial supershift can be represented as a replacement of the complex world line by a superworldline X µ 0 (τ) =x µ 0 (τ)−iθσ µ ¯ζ+iζσ µ ¯θ,parametrized by Grassmann coordinates ζ,¯ζ, or as a corresponding coordinate replacement in theKerrsolution x /primeµ =x µ +iθσ µ ¯ζ−iζσ µ ¯θ;θ /prime =θ+ζ,¯θ /prime =¯θ+¯ζ, (2) Assuming that coordinates x i before the supershift were the usual c-number coordinates one sees that coordinates acquire nilpotent Grassmann contributions after supertranslations. Therefore, there appears a natural splitting of the space- time coordinates on the c-number ‘body’-part and a nilpotent part - the so called ‘soul’.The‘body’subspaceofsuperspace,orB-slice,isasubmanifoldwherethe nilpotent part is equal to zero, and it is a natural analogue to the real slice of the complexcase. Reproducing the real slice procedure of the Kerr geometry in superspace one hasto usethereplacements: a/ complex world line →superworldline, kievarwe.tex; 12/03/2001; 3:49; p.190 184 A.BURINSKII b/complex lightcone →superlightcone, c/ realslice→bodyslice. Performingthebody-sliceprocedureto superlightcone constraints s 2 = [x µ −X 0µ (τ)][x µ −X µ 0 (τ)] = 0, (3) one selects the body and nilpotent parts of this equation and obtains three equa- tions.TheﬁrstoneisthediscussedaboverealsliceconditionofthecomplexKerr geometryclaimingthatcomplexlightconescanreachtherealslice.Thenilpotent partof(3)yields twoB-sliceconditions [x µ −x µ 0 (τ)](θσ µ ¯ζ−ζσ µ ¯θ) = 0; (4) (θσ¯ζ−ζσ¯θ) 2 = 0. (5) These equations can be resolved by representing the complex light cone equa- tion via the commuting two-component spinors Ψand˜Ψ:x µ =x 0µ + Ψσ µ ˜Ψ. ”Right” (or ”left”) null planes of the complex light cone can be obtained keeping Ψconstant and varying ˜Ψ(or keeping ˜Ψconstant and varying Ψ.) As a result we obtain the equations ¯Ψ¯θ= 0, ¯Ψ¯ζ= 0,which in turn are conditions of proportionality of the commuting spinors ¯Ψ(x)determining the PNC of the Kerr geometry and anticommuting spinors ¯θand¯ζ, these conditions providing the left null superplanes of the supercones to reach B-slice. It also leads to ¯θ¯θ=¯ζ¯ζ= 0, and equation(5)is satisﬁedautomatically. Thus,asaconsequenceoftheB-sliceandsuperlightconeconstraintsweobtain anon-linearsubmanifoldofsuperspace θ=θ(x),¯θ=¯θ(x).Theoriginalfour- dimensional supersymmetry is broken, and the initial supergauge freedom which allowed to turn the super geometry into trivial one is lost. Nevertheless, there is a residual supersymmetrybasedonfree Grassmannparameters θ 1 ,¯θ 1 . TheaboveB-sliceconstraintsyieldinfactthenon-linearrealizationofbroken supersymmetryintroducedbyVolkovandAkulov[20,21]andconsideredinN=1 supergravity by Deser and Zumino [19]. It is assumed that this construction is similar to the Higgs mechanism of the usual gauge theories and ζ α (x),¯ζ ˙α (x) represent Goldstone fermion which can be eaten by appropriate local supertrans- formation/epsilon1(x)with a corresponding redeﬁnition of the tetrad and spin-3/2 ﬁeld. Complex character of supertranslations in the Kerr case demands to use in this scheme the N=2 supergravity. We omit here details referring to [18] and mention only that in the resulting exact solution the torsion and Grassmann contributions to tetrad cancel, and metric takes the exact Kerr-Newman form. However there are the extra wave fermionic ﬁelds on the bosonic Kerr-Newman background propagating along the Kerr PNC and concentrating near the Kerr singularity. Solution contains also an extra axial singularity which is coupled topologically withsingularringthreading it. kievarwe.tex; 12/03/2001; 3:49; p.191 ROTATINGSUPERBLACKHOLE 185 4. Baglike source ofthe Kerr-Newman solution Theaboveconsiderationofsuper-Kerr-Newmansolutionisbasedonthemassless ﬁelds providing description of the rotating super-black-hole. It could be the end ofstorysince thesourceof arotatingblack holeishidden behind thehorizons. However, the value of angular momentum for spinning particles is very high regarding the mass parameter and the horizons disappear uncovering the Kerr singular ring. To get a regularized solution the massless ﬁelds of the black hole solution have to get a mass in the core region forming a matter source removing theKerrsingularityand twosheetednessof theKerr space. 2 ObtainingaregularKerrsourcerepresentsanoldproblem.Intheﬁrstdisk-like model given by Israel [2] a truncation of the negative sheet was used. As a result there appeared a source distribution on the surface of the disk r= 0. Analyzing theresultingstress-energytensorHamityshowed[13]thatthisdiskhastobeina rigid relativistic rotation and built of an exotic matter having zero energy density and negative pressure. In the development of this model given by L ´opez [5] the truncation is placed at the coordinate surface r=r e = e 2 2m ( whereh= 0), and the region r < r e is replaced by Minkowski space. As a result the source takestheformofthehighlyoblateandinﬁnitelythinellipticshelloftheCompton radiusa= 1 2m and of the thickness of the classical Dirac electron radius r e . For small angular momentum the source takes the form of the Dirac electron model, a charged sphere of the classical size r e . The ﬁelds out of the shell have the exact Kerr-Newman form. Interior of the shell is ﬂat. The shell is charged and rotating, andbuiltofasuperconductingmatter.Incorotatingspaceoneseesthatmatterhas anegativepressure andzeroenergy density. The L´opez source represents a bubble with an inﬁnitely thin domain wall boundary. In the paper [6] an attempt was undertaken to get the source of the Kerr-Newman solution with a smooth matter distribution. Retaining the metric in the Kerr-Schild form (1) and the form and properties of the Kerr PNC, it was assumed that function h(r,θ)takes a more general form h= f(r ) r 2 +a 2 cos 2 θ , where the function f(r)is continuous and takes the usual Kerr-Newman form f KN (r) =mr−e 2 /2intheexternalregion.Inthesametime,inaneighborhood of the Kerr disk r≤r 0 ( the core region ) including the Kerr singularity, the functionf(r)has to satisfy some conditions of regularity to provide ﬁniteness of themetricand the stress-energy tensorofsource. It was shown that this regularity is achieved for the function f(r)∼r n withn≥4. In the case n= 4,f(r) =f 0 (r) =αr 4 , ( in the nonrotational casea= 0) space-time has a constant curvature in the core and generated by a homogeneous matter distribution with energy density ρ= 1 8π 6α. Therefore, assuming that matter in the core has a homogenous distribution one can estimate 2 Thisproblemisactualforblackholephysics,too.Seeforexample[22]andreferencestherein. kievarwe.tex; 12/03/2001; 3:49; p.192 186 A.BURINSKII theboundaryofthecoreregion r 0 asapointofintersectionof f 0 (r)andf KN (r). Regularity of the stress-tensor demands continuity of the function f(r)up to ﬁrst derivative, therefore, the resulting smooth function f(r)must be interpolating betweenfunctions f 0 (r)andf KN (r)neartheboundary ofthe core r≈r 0 . Let us now mention that general metric (1) can be expressed via orthonormal tetradasfollows[6] g µν =m µ m ν +n µ n ν +l µ l ν −u µ u ν ,andthecorresponding stress-energytensorofthesource(followingfromtheEinsteinequations)maybe representedintheform T (af) µν = (8π) −1 [(D+2G)g µν −(D+4G)(l µ l ν −u µ u ν )], whereu µ is the unit time-like four-vector, l µ is the unit vector in radial direction, andn µ ,m µ aretwomore space-like vectors.Here D=−f /prime/prime /(r 2 +a 2 cos 2 θ), (6) G= (f /prime r−f)/(r 2 +a 2 cos 2 θ) 2 , (7) and theBoyer-Lindquistcoordinates t,r,θ,φareused. Like to the results for singular (inﬁnitely thin) shell-like source [13, 5], the stress-energy tensor can be diagonalized in a comoving coordinate system show- ing that the source represents a relativistic rotating disk. However, in this case, the disk is separated into ellipsoidal layers each of which rotates rigidly with its ownangularvelocity ω(r) =a/(a 2 +r 2 ).Inthecomovingcoordinatesystemthe tensorT µν takes theform T µν = 1 8π   2G 0 0 0 0−2G 0 0 0 0 2G+D 0 0 0 0 2 G+D   , (8) that corresponds to energy density ρ= 1 8π 2G, radial pressure p rad =− 1 8π 2G, and tangential pressure p tan = 1 8π (D+ 2G). Settinga= 0for the non-rotating case, we obtain Σ =r 2 , the surfaces r=const.are spheres and we have spherical symmetry for all the above rela- tions. The region described by f(r) =f 0 (r)is the region of constant value of the scalar curvature invariant R= 2D=−2f /prime/prime 0 /r 2 =−24α, and of a constant value of energy density. If we assume that the region of a constant curvature is closelyextendedtotheboundaryofsource r 0 whichisdeterminedasarootofthe equation f 0 (r 0 ) =f KN (r 0 ), (9) then, smoothness of the f(r)in a small neighborhood of r 0 , say|r−r 0 |< δ, implies a smooth interpolation for the derivative of the function f(r)between f /prime 0 (r)| r=r 0 −δ andf /prime KN (r)| r=r 0 +δ .Suchasmoothinterpolationonasmalldistance δshalllead to ashock-likeincreaseof thesecond derivative f /prime/prime (r)byr≈r 0 . kievarwe.tex; 12/03/2001; 3:49; p.193 ROTATINGSUPERBLACKHOLE 187 In charged case for α≤0( AdS internal geometry of core) there exists only onepositiveroot r 0 ,andsecondderivativeofthesmoothfunction f /prime/prime (r)ispositive near this point. Therefore, there appears an extra tangential stress near r 0 caused by the term D=−f /prime/prime (r)/(r 2 +a 2 cos 2 θ)| r=r 0 in the expression (8). It can be interpreted as the appearance of an effective shell ( or a domain wall ) conﬁning the charged ball-like source with a geometry of a constant curvature inside the ball. The case α= 0represents the bubble with a ﬂat interior which has in the limitδ→0aninﬁnitely thinshell.It corresponds tothe L ´opez model. The internal geometry of the ball is de Sitter one for α>0, anti de Sitter one forα<0andﬂat onefor α= 0. Let us consider peculiarities of the rotating Kerr source. In this case the sur- facesr=const.areellipsoidsdescribedbytheequation x 2 +y 2 r 2 +a 2 + z 2 r 2 = 1.Energy density inside the core will be constant only in the equatorial plane cosθ= 0. Therefore, the Kerr singularity is regularized and the curvature is constant in string-like region r < r 0 andθ=π/2near the former Kerr singular ring. The ratio stress| θ =0 stress| θ=π/2 <(r e /a) 4 =e 8 <10 −8 shows a strong increase of the stress near thestring-likeboundary ofthedisk. 5. Fieldmodel: Fromsuperconducting stringsto superconductingbags The known models of the bags and cosmic bubbles with smooth domain wall boundaries are based on the Higgs scalar ﬁeld φwith a Lagrange density of the formL=− 1 2 ∂ µ φ∂ µ φ− λ 2 8 (φ 2 −η 2 ) 2 leading to the kink planar solution ( the wall is placed in xy-plane atz= 0)φ(z) =ηtanh(z/δ),whereδ= 2 λη is the wall thickness. The kink solution describes two topologically distinct vacua <φ> =±ηseparated bythedomain wall. The stress–energy tensor of the domain wall is T ν µ = λ 2 η 4 4 cosh −4 (z/δ)diag(1,1,1,0),indicating a surface stress within the plane of the wall which is equal to the energy density. When applied to the spherical bags or cosmic bubbles [27, 28], the thin wall approximation is usually assumed δ/lessmuchr 0 , and a spherical domain wall separates a false vacuum inside the ball (r<r 0 )<φ> in =−ηfromatrueouter vacuum <φ> out =η. In the gauge string models, the Abelian Higgs ﬁeld provides conﬁnement of themagneticvortexlinesinsuperconductor.Similarly,inthemodelsofsupercon- ducting bags, the gauge Yang-Mills or quark ﬁelds are conﬁned in a bubble ( or cavity)insuperconducting QCD-vacuum. A direct application of the Higgs model for modelling superconducting prop- erties of the Kerr source is impossible since the Kerr source has to contain the external long range Kerr-Newman electromagnetic ﬁeld, while in the models of strings and bags the situation is quite opposite: vacuum is superconducting in external region and electromagnetic ﬁeld acquires a mass there from Higgs ﬁeld kievarwe.tex; 12/03/2001; 3:49; p.194 188 A.BURINSKII turning into a short range ﬁeld. An exclusion represents the U(I)×˜U(I)cosmic string model given by Vilenkin-Shellard and Witten [15, 16] which represents a doubling of the usual Abelian Higgs model. The model contains two sectors, say AandB, with two Higgs ﬁelds φ A andφ B , and two gauge ﬁelds A µ andB µ yieldingtwosortsofsuperconductivity AandB.Itcanbeadaptedtothebag-like source in such a manner that the gauge ﬁeld A µ of theAsector has to describe a long-range electromagnetic ﬁeld in outer region of the bag while the chiral scalar ﬁeld of this sector φ A has to form a superconducting core inside the bag which mustbeunpenetrablefor A µ ﬁeld. The sectorBof the model has to describe the opposite situation. The chiral ﬁeldφ B must lead to a B-superconductivity in outer region conﬁning the gauge ﬁeldB µ insidethebag. ThecorrespondingLagrangianoftheWitten U(I)×˜U(I)ﬁeldmodelisgiven by [16] L=−(D µ φ A )(D µ φ A )−(˜D µ φ B )(˜D µ φ B )− 1 4F µν A F Aµν − 1 4F µν B F Bµν −V, (10) whereF Aµν =∂ µ A ν −∂ ν A µ andF Bµν =∂ µ B ν −∂ ν B µ areﬁeldstresstensors, and thepotentialhas theform V=λ(¯φ B φ B −η 2 ) 2 +f(¯φ B φ B −η 2 )¯φ A φ A +m 2 ¯φ A φ A +µ(¯φ A φ A ) 2 .(11) TwoAbeliangaugeﬁelds A µ andB µ interactseparatelywithtwocomplexscalar ﬁeldsφ B andφ A so that the covariant derivative D µ φ A = (∂ µ +ieA µ )φ A is associated with Asector, and covariant derivative ˜D µ φ B = (∂ µ +igB µ )φ B is associated with Bsector. The model fully retains the properties of the usual bag modelswhicharedescribedby Bsectorprovidingconﬁnementof B µ gaugeﬁeld insidebag,anditacquiresthelongrangeelectromagneticﬁeld A µ intheouter-to- the-bagregiondescribedbysector A.TheAandBsectorsarealmostindependent interacting only through the potential term for scalar ﬁelds. This interaction has to provide synchronized phase transitions from superconducting B-phase inside the bag to superconducting A-phase in the outer region. The synchronization of this transition occurs explicitly in a supersymmetric version of this model given by Morris [23]. 5.1. SUPERSYMMETRIC MORRIS MODEL In Morris model, the main part of Lagrangian of the bosonic sector is similar to the Witten ﬁeld model. However, model has to contain an extra scalar ﬁeld Z providingsynchronizationof thephase transitions in AandBsectors. 3 3 InfacttheMorrismodelcontainsﬁvecomplexchiralﬁelds φ i ={Z,φ − ,φ + ,σ − ,σ + }.How- ever,thefollowingidentiﬁcationoftheﬁeldsisassumed φ=φ + ;¯φ=φ − andσ=σ + ; ¯σ=σ − . In previous notations φ∼φ A andσ∼φ B . kievarwe.tex; 12/03/2001; 3:49; p.195 ROTATINGSUPERBLACKHOLE 189 The effectiveLagrangian ofthe Morris model hasthe form L=−2(D µ φ )(D µ φ)−2(˜D µ σ )(˜D µ σ)−∂ µ Z∂ µ ¯Z − 1 4F µν F µν − 1 4F µν B F Bµν −V(σ,φ,Z ), (12) wherethe potential Visdetermined through the superpotential Was V= 5 /summationdisplay i=1 |W i | 2 = 2|∂W/∂φ| 2 + 2|∂W/∂σ| 2 +|∂W/∂Z| 2 .(13) Thefollowingsuperpotential,yieldingthegaugeinvarianceandrenormalizability ofthemodel,was suggested 4 W=λZ(σ¯σ−η 2 ) + (cZ+m)φ¯φ, (14) wherethe parameters λ,c,m,andηarerealpositive quantities. The resulting scalarpotential Visthen given by V=λ 2 (¯σσ−η 2 ) 2 + 2λc(¯σσ−η 2 )φ¯φ+c 2 (¯φφ) 2 + (15) 2λ 2 ¯ZZ¯σσ+ 2(c¯Z+m)(cZ+m)¯φφ. 5.1.1.Supersymmetricvacua From (13) one sees that the supersymmetric vacuum states, corresponding to the lowestvalueofthe potential, aredetermined bytheconditions F σ =−∂¯W/∂ ¯σ= 0; (16) F φ =−∂¯W/∂ ¯φ= 0; (17) F Z =−∂¯W/∂ ¯Z= 0, (18) and yieldV= 0.These equationslead totwosupersymmetric vacuum states: I)Z= 0;φ= 0;|σ|=η;W= 0; (19) and II)Z=−m/c;σ= 0;|φ|=η /radicalBig λ/c;W=λmη 2 /c.(20) We shall take the state Ifor external region of the bag, and the state IIas a state insidethebag. The treatment of the gauge ﬁeld A µ andB µ inBis similar in many respects because of the symmetry between AandBsectors allowing one to consider the 4 Superpotential is homomorphic function of {Z,φ, ¯φ,σ, ¯σ}. kievarwe.tex; 12/03/2001; 3:49; p.196 190 A.BURINSKII state Σ =ηin outer region as superconducting one in respect to the gauge ﬁeld B µ . FieldB µ acquires the mass m B =gηin outer region, and the ˜U(I)gauge symmetry is broken, which provides conﬁnement of the B µ ﬁeld inside the bag. The bag can also be ﬁlled by quantum excitations of fermionic, or non Abelian ﬁelds.TheinteriorspaceoftheKerrbagisregularizedinthismodelsincetheKerr singularity and twofoldedness are suppressed by function f=f 0 (r). However, a strong increase of the ﬁelds near the former Kerr singularity can be retained leadingtotheappearanceof travelingwavesalong the boundary ofthe disk. 5.2. SUPERSYMMETRIC BUBBLE BASEDON THEMORRIS FIELD MODEL It is shown in [6] that in the planar thin wall approximation, and by neglecting the gauge ﬁelds there is a supersymmetric BPS-saturated domain wall solution interpolatingbetweensupersymmetricvacuaI)andII).Thisdomainwalldisplays the usual structure of stress-energy tensor with a tangential stress. The non-zero componentsofthestress-energytensor taketheform T 00 =−T xx =−T yy = 1 2[δ ij (Φ i , z )(Φ j , z ) +V]; (21) T zz = 1 2[δ ij (Φ i , z )(Φ j , z )−V], (22) where Φ i ={Z,φ − ,φ + ,σ − ,σ + }. One can estimate the mass and energy of a bubble formed by such a domain wall in global supersymmetry setting vacuum I) as external one and vacuum II) as an internal vacuum. Using the Tolman rela- tionM= /integraltext dx 3 √−g(−T 0 0 +T 1 1 +T 2 2 +T 3 3 ), replacing coordinate zon radial coordinater,andintegrating over sphere oneobtains M bubble =−4π /integraldisplay V(r)r 2 dr=−4π /integraldisplay (Φ i , r ) 2 r 2 dr. (23) The resulting effective mass is negative, which is caused by gravitational contri- bution of the tangential stress. The repulsive gravitational ﬁeld was obtained in many singular and smooth models of domain walls [32, 25, 30, 31]. One should note, that similar gravitational contribution to the mass caused by interior of the bag will be M gr.int = /integraltext Dr 2 dr=− 2 3 Λr 3 0 . It depends on the sign of curvature insidethebagand will be negativein deSittercase and positive in AdSone. ThetotalenergyofaunchargedbubbleformingfromthesupersymmetricBPS saturateddomainwallis E 0bubble =E wall = 4π /integraldisplay ∞ 0 ρr 2 dr≈4πr 2 0 /epsilon1 min , (24) wherer 0 is radius of the bubble, and /epsilon1 min =W(0)−W(∞) =λmη 2 /c. Corresponding total mass following from the Tolman relation will be negative kievarwe.tex; 12/03/2001; 3:49; p.197 ROTATINGSUPERBLACKHOLE 191 M 0bubble =−E wall ≈ − 4πr 2 0 /epsilon1 min .It is the known fact showing that the unchargedbubbles areunstableandform thetime-dependent states[30, 31]. Forchargedbubblesthereareextrapositiveterms:contributioncausedbythe energy and mass of the external electromagnetic ﬁeld E e.m. =M e.m. = e 2 2r 0 ,and contribution to mass caused by gravitational ﬁeld of the external electromagnetic ﬁeld ( determined by Tolman relation for the external e.m. ﬁeld) M grav.e.m. = E e.m. = e 2 2r 0 .As aresultthetotalenergyfor charged bubbleis E tot.bubble =E wall +E e.m. = 4πr 2 0 /epsilon1 min +e 2 2r 0 , (25) and thetotalmass willbe M tot.bubble =M 0bubble +M e.m. +M grav.e.m. = (26) −E wall + 2E e.m. =−4πr 2 0 /epsilon1 min +e 2 r 0 . (27) Minimum of the total energy is achieved by r 0 = ( e 2 16π/epsilon1 min ) 1/3 ,which yields the followingexpressions for total mass and energy ofthestationary state M ∗ tot =E ∗ tot =3e 2 4r 0 . (28) One sees that the resulting total mass of charged bubble is positive, however, due tonegativecontributionof M 0bubble itcanbelowerthanBPSenergyboundofthe domainwallformingthisbubble.Thisremarkablepropertyofthebubblemodels( ‘ultra-extreme’statesfortheTypeIdomainwallsin[30])allowsonetoovercome BPS bound [33] and opens the way to get the ratio m 2 /lessmuche 2 which is necessary for particle-like models. 5.3. BAGLIKESOURCE IN SUPERGRAVITY Insupergravitythe scalarpotentialhas amorecomplicate form[21, 30,31, 29] V sg =e k 2 K (K i¯j D i WD j W−3k 2 W¯W), (29) whereKis K¨ahler potential K i¯j = ∂ 2 K ∂Φ i ∂¯Φ j , andk 2 = 8πG N ,G N is the Newton constant.Inthesmall kWlimit,thisexpressionturnsintopotentialofglobalsusy. In this approximation, the above treatment of the charged domain wall bubble will be valid in supergravity. The preserving supersymmetry vacuum state has to satisfy the condition D i W≡W i +k 2 K i W= 0. This condition is satisﬁed for the internal vacuum state II) only in the limit k 2 →0sinceW=λmη 2 /cinside the bag, and D i W≈k 2 K i Wthere. In the order k 2 the vacuum state II) does not kievarwe.tex; 12/03/2001; 3:49; p.198 192 A.BURINSKII preservesupersymmetry.Thereappearsalsoanextracontributiontostress-energy tensorhavingtheleadingterm T µν = 3(k 2 /8π)e k 2 K |W| 2 g µν , (30) and yielding the negative cosmological constant Λ =−3k 4 e k 2 K |W| 2 and to anti-deSitterspace-timeforthebaginterior.Generalexpressionforcosmological constantinside thebaghas theform Λ =k 4 e k 2 K /summationdisplay i {k 2 |K i W| 2 −3|W| 2 }. (31) It yields AdS vacuum if k 2 |K i W| 2 −3|W| 2 <0. InthesametimethevacuumstateI)inexternalregionhas W= 0andΛ = 0, and itpreserves supersymmetryforstrongchiral ﬁelds. 6. Conclusion AregularizedsourceoftheKerr-Newmansolutionisconsideredhavingthestruc- ture of a rotating bag with AdS interior and a smooth domain wall boundary. It is shown that the Witten superconducting string model can be generalized and adapted forming a charged superconducting bag with AdS interior and a long range external gauge ﬁeld which is necessary for description of charged black holes.Since1968asuccessiveaccumulationofevidencesisobservedrelatingthe structure ofKerrgeometrywith physics ofelementary particles. Acknowledgments . We would like to thank organizers of this Workshop for kindinvitationandﬁnancialsupport. References 1. B. Carter, Global structure of the Kerr family of gravitational ﬁelds , Phys. Rev. 174(1968) 1559. 2. W. Israel, Source ofKerr metric , Phys. Rev. D2(1970) 641. 3. A.Burinskii, Microgeons with spin , Sov. Phys. JETP 39(1974)193. 4. D. Ivanenko and A. Burinskii, Gravitational strings in the models of spinning elementary particles, Izv. VUZ Fiz. 5(1975)135. 5. C. A. L ´opez,Extended model of the electron in general relativity , Phys. Rev. D30(1984) 313. 6. A. Burinskii, Supersymmetric superconducting bag as a core of Kerr spinning particle ,e- print hep-th/0008129 . 7. A. Sen, Rotating charged black hole solution in heterotic string theory , Phys.Rev.Lett., 69(1992)1006-1009. 8. A. Sen, Extremal black holes and elementary string states , Modern Phys. Lett. A 10(1995)2081. C.HolzheyandF.Wilczek, Blackholesaselementaryparticles ,Nucl.Phys. B380(1992)447, hep-th/9202014 kievarwe.tex; 12/03/2001; 3:49; p.199 ROTATINGSUPERBLACKHOLE 193 9. A. Burinskii, String - like structures in complex Kerr geometry , in Relativity Today, ( R.P. KerrandZ.Per ´jeseds,), AcademiaiKiado,Budapest1994,pp.149-158, gr-qc/9303003 . ————–, Complex string as source of Kerr geometry , in Espec. Space Explorations, 9 (C2)(1995) 60, Moscow, Belka, hep-th/9503094 . 10. G.C.Debney,R.P.Kerr,A.Schild, SolutionsoftheEinsteinandEinstein-Maxwellequations. , J.Math.Phys. 10(1969)1842. 11. A. Burinskii, R.P. Kerr and Z. Perjes, Nonstationary Kerr Congruences ,e-print gr- qc/9501012 . 12. A. Burinskii, The Kerr geometry, complex world lines and hyperbolic strings , Phys.Lett. A 185(1994)441. 13. V. Hamity, Interior ofKerrmetric , Phys. Let. A56(1976)77. 14. C.A. L ´opez,Material and electromagnetic sources of the Kerr-Newman geometry , Nuovo CimentoB 76(1983)9; 15. A. Vilenkin and E.P.S. Shellard, Cosmic Strings and Other Topological Defects ( Cambrige University Press, 1994) 16. E. Witten, Superconducting strings , Nucl.Phys., B249(1985)557. 17. A. Burinskii, Some properties of the Kerr solution to low energy string theory , Phys.Rev. D 52(1995)5826, hep-th/9504139 . 18. A. Burinskii, Kerr spinning particle, strings and superparticle models , Phys.Rev. D 57 (1998)2392, hep-th/9704102 . ————, Super-Kerr-Newman solution to broken N=2 supergravity , Class. Quantum Grav. 16(1999)3497, hep-th/9903032; ————, Non-trivial supergeneralization of the Kerr-Newman solution , Ann.Phys.(Leipzig), 9(2000) Spec.Issue 34,hep-th/9910045 . 19. S. Deser and B. Zumino, Broken supersymmetry and supergravity , Phys. Rev. Lett. 38(1977) 1433. 20. D.V. Volkov and V.P. Akulov, Possible universal neutrino interaction , JETP Lett. 16(1972)367. 21. J.Wess and J.Bagger, Supersymmetryand supergravity , Princeton, New Jersey 1983. 22. I.Dymnikova, Vacuum nonsingular black hole , Gen.Rel.Grav. 24(1992)235; E.Elizalde and S.R. Hildebrandt, Regular quantum interiors for black holes ,gr- qc/0007030 23. J.R. Morris, Supersymmetry and gauge invariance constraints in a U(I)×U(I) /prime -Higgs superconducting cosmic string model , Phys.Rev. D 53(1996)2078, hep-ph/9511293 . 24. C.A.L ´opez,Internalstructureofaclassicalspinningelectron ,Gen.Rel.Grav. 24(1992)2851. 25. C.A. L ´opez,Dynamics of charged bubbles in general relativity and models of particles Phys.Rev. D 38(1988) 3662. 26. M. G ¨urses and F.G ¨ursey,Lorentz covariant treatment of the Kerr-Schild geometry , Journ. Math.Phys. 16(1975)2385. 27. A. L. Macpherson and B.A. Campbell, Biased discrete supersymmetry breaking and Fermi balls, Phys.Let. B(1995)205. 28. J.R. Morris and D. Bazeia, Supersymmetry breaking and Fermi balls , Phys.Rev. D 54(1996)5217. 29. J.R. Morris, Cosmic stringsinsupergravity , Phys.Rev. D 56(1997)2378. hep-ph/9706302 30. M. Cveti ˇc and H. Soleng, Supergravity domain walls , Phys.Rept. 282(1997)159, hep- th/9604090 31. M. Cveti ˇc, S. Grifﬁes and S.J. Rey, Static domain walls in N=1 supergravity , Nucl.Phys. B 381(1992)301-328, hep-th/9206004 . 32. J.IpserandP.Sikivie, Gravitationallyrepulsivedomainwalls ,Phys.Rev. D30(1983)712-719. 33. G.W. Gibbons and C.M. Hull, A Bogomolny bound for general relativity and solitons in N=2 supergravity , Phys. Lett. B 109(1982)190. kievarwe.tex; 12/03/2001; 3:49; p.200 kievarwe.tex; 12/03/2001; 3:49; p.201 CLASSIFYING N-EXTENDED 1-DIMENSIONAL SUPER SYSTEMS FRANCESCO TOPPAN ∗ CBPF,DCP,RuaDr.Xavier Sigaud150, cep22290-180RiodeJaneiro(RJ), Brazil 1. Introduction In this talk I will report some results obtained in a joint collaboration with A. Pashnev, concerning the classiﬁcation of the irreducible representations of the N-extended Supersymmetry in 1dimension and which ﬁnd applications to the constructionof SupersymmetricQuantumMechanical Systems [1]. This mathematical problem ﬁnds immediate application to the theory of di- mensionally(toonetemporaldimension)supersymmetric 4dtheories,whichgets 4times the number of supersymmetries of the original models (the N= 8su- pergravity being e.g. associated with the a N= 32Supersymmetric Quantum Mechanicaltheory).Duetoalackofsuperﬁeldformalismfor N > 4,onlypartial resultsareknown [2]and [3]. More recently, Supersymmetric and Superconformal Quantum Mechanics havebeenappliedindescribinge.g.thelow-energyeffectivedynamicsofacertain class of black holes, for testing the AdS/CFT correspondence in the case of AdS 2 , ininvestigatingthe light-conedynamicsof supersymmetrictheories. InthisreportoftheworkwithPashnev,twomainresultswillbepresented.At ﬁrst a peculiar property of supersymmetry in one dimension is exhibited, namely that any ﬁnite dimensional multiplet containing dbosons anddfermions in dif- ferent spin states are put into classes of equivalence individuated by irreducible multiplets of just two spin states, where all bosons and all fermions are grouped in the same spin. Later it is shown that all irreducible multiplets of this kind are in one-to-one correspondence with the classiﬁcation of real-valued Clifford ΓmatricesofWeyltype. Thisclassiﬁcationreﬁnes(inthecaseof“non-Euclidean”supersymmetry,see below) the results obtained in [4] and [5]. Another reference where some aspects ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.202 196 F.TOPPAN ofthetheoryoftherepresentationof 1-dimensionalsupersymmetryarediscussed isgiven by[6]. The mathematical problem we are investigating can be stated as follows, ﬁndingtheirreducible representationof thesupersymmetryalgebra {Q i ,Q j }=ω ij H, (1) whereQ i , i= 1,2,···,Nare supercharges and H=−i ∂ ∂t(2) is the Hamiltonian. The constant tensor ω ij can be conveniently diagonalized and normalized in such a way to coincide with a pseudo-Euclidean metric η ij with signature (p,q).Usually the eigenvalues are all assumed being positive (i.e. q= 0), however examples can be given (see [7]), of physical systems whose supersymmetry algebra is characterized by an indeﬁnite tensor. In the following I will discussthe simplestexampleof thiskind. Anygivenﬁnite-dimensionalrepresentationmultipletoftheabovesuperalge- bracanberepresentedin form ofachain of dbosons anddfermions Φ 0 a 0 ,Φ 1 a 1 ,···,Φ M−1 a M−1 ,Φ M a M (3) whose components Φ I a I ,(a I = 1,2,···,d I )are real and alternatively bosonic andfermionic( d=d 0 +d 2 +d 4 +...=d 1 +d 3 +d 5 +...).Forsuchamultiplet theshortnotation{d 0 ,d 1 ,···,d M }willalso be employed. Duetodimensionalityargumentthe i−thsupersymmetrytransformationfor theΦ Ia I componentsisgivenby δ ε Φ I a I =ε i (C I i ) a I a I+1 Φ I+1 a I+1 +ε i (˜C I i ) a I a I−1 d dτΦ I−1 a I−1 , (4) and it simpliﬁes for the end-components (due to the absence of the I=−1and I=M+ 1components). In one dimension it is therefore possible to redeﬁne the last components accordingto Φ M a M = d dτΨ M−2 a M (5) in terms of some functions Ψ M−2 a M . The initial supermultiplet of length M+ 1is now re-expressed as the {d 0 ,d 1 ,···,d M−2 +d M ,d M−1 ,0}supermultiplet of lengthM. By repeating Mtimes the same procedure the shortest supermultiplet {d,d}of length 2can be reached. The above argument outlines the proof of the statement that all supermultiplets are classiﬁed according to the irreducible representations of supermultipletsoflength 2. kievarwe.tex; 12/03/2001; 3:49; p.203 N-EXTENDED 1-DIMSUPER SYSTEMS 197 2. Extended supersymmetriesandreal valued Clifford algebras The main result of the previous Section is that the problem of classifying all N- extendedsupersymmetricquantummechanicalsystemsisreducedtotheproblem of classifying the irreducible representations of length 2. Having this in mind let ussimplifythenotations.Lettheindices a,α= 1,···,dnumberthebosonic(and respectivelyfermionic)elementsintheSUSYmultiplet.Allofthemareassumed todependonthetimecoordinate τ(X a ≡X a (τ),θ α ≡θ α (τ)). In order to be deﬁnite and without loss of generality let us take the bosonic elements to be the ﬁrst ones in the chain {d,d}, which can be conveniently represented alsoasacolumn Ψ = /parenleftbigg X a θ α /parenrightbigg , (6) thesupersymmetrytransformations arereducedto the following setof equations δ ε X a =ε i (C i ) aα θ α ≡i(ε i Q i Ψ) a δ ε θ α =ε i (˜C i ) αb d dτX b ≡i(ε i Q i Ψ) α (7) where,asaconsequence of(1), C i ˜C j +C j ˜C i =iη ij (8) and ˜C i C j +˜C j C i =iη ij (9) Sinceε i ,X a ,θ α arereal,thematrices C i ’s,˜C i ’shavetoberespectivelyimaginary and real. Ifweset(just fornormalization) C i = i √ 2σ i ˜C i = 1 √ 2˜σ i (10) and accommodate σ i ,˜σ i into asingle matrix Γ i = /parenleftbigg 0σ i ˜σ i 0 /parenrightbigg , (11) they form a set of real-valued Clifford Γ-matrices of Weyl type (i.e. block antidiagonal), obeyingthe(pseudo-) Euclideananticommutation relations {Γ i ,Γ j }= 2η ij . (12) kievarwe.tex; 12/03/2001; 3:49; p.204 198 F.TOPPAN Therefore the classiﬁcation of irreducible multiplets of representation of a (p,q) extended supersymmetry is in one-to-one correspondence with the classiﬁcation of the real-valued Clifford algebras C p,q with the further property that the Γ matrices canbe realizedinWeyl(i.e. blockantidiagonal) form. Real-valued Clifford algebras have been classiﬁed in [8] for compact ( q= 0) case,andin[9] forthe non-compactone. Ifollowhere the exposition in [10]. Three cases have to be distinguished for real representations, speciﬁed by the type of most general solution allowed for a real matrix Scommuting with all the Clifford Γ i matrices, i.e. i)thenormalcase,realizedwhen Sisa multipleofthe identity, ii)the almost complex case, for Sbeing given by a linear combination of the identityandofareal J 2 =−1matrix, iii)ﬁnallythequaternioniccase,for Sbeingalinearcombinationofrealmatrices satisfyingthe quaternionic algebra. Real irreduciblerepresentationsofnormaltype existwhenever the condition p−q= 0,1,2mod 8is satisﬁed (their dimensionality being given by 2 [ N 2 ] , whereN=p+q),whilethealmostcomplexandthequaternionictyperepresenta- tionsarerealizedinthe p−q= 3,7mod 8andinthep−q= 4,5,6mod 8 cases respectively. The dimensionality of these representations is given in both casesby 2 [ N 2 ]+1 . We further require the extra-condition that the real representations should ad- mit a block antidiagonal realization for the Clifford Γmatrices. This condition is met forp−q= 0mod 8in the normal case (it corresponds to the standard Majorana-Weyl requirement), p−q= 7mod 8in the almost complex case andp−q= 4,6mod 8in the quaternionic case. In all these cases the real irreducible representation is unique. It is therefore possible to furnish the dimensionality of the irreducible repre- sentations of the of the supersymmetry algebra or, conversely, the allowed (p,q) signatures associated to a given dimensionality of the bosonic and fermionic spaces. The latter result is conveniently expressed by introducing the notion of maximally extended supersymmetry. The C p,q (p−q= 6mod 8) real representation for the quaternionic case can be recovered from the 7mod 8 almost complex C p+1,q representation by deleting one of the Γmatrices; in its turn the latter representation is recovered from the C p+2,q normal Majorana- Weyl representation by deleting another Γmatrix. The dimensionality of the three representations above being the same, the normal Majorana-Weyl repre- sentation realizes the maximal possible extension of supersymmetry compatible with the dimensionality of the representation. In search for the maximal exten- sion of supersymmetry we can therefore limit ourselves to consider the normal Majorana-Weyl representations, as well as the quaternionic ones satisfying the p−q= 4mod 8condition. Let us therefore introduce a parameter /epsilon1, which assumes two values and is kievarwe.tex; 12/03/2001; 3:49; p.205 N-EXTENDED 1-DIMSUPER SYSTEMS 199 used to distinguish the Majorana-Weyl ( /epsilon1= 0) with respect to the quaternionic case(/epsilon1= 1).Aspaceof d= 2 t bosonicand d= 2 t fermionicstatescancarrythe followingset ofmaximally extended supersymmetries (p=t−4z+ 5−3/epsilon1,q=t+ 4z+/epsilon1−3) (13) wherethe integer z=k−lmusttakevaluesin theinterval 1 4(3−t−/epsilon1)≤z≤ 1 4(t+ 5−3/epsilon1) (14) inordertoguarantee the p≥0andq≥0requirements. 3. An applicationand conclusions. One of the most signiﬁcant application of extended supersymmetric quantum mechanics concerns the 1-dimensional σmodels evolving in a target spacetime manifold presenting both bosonic and fermionic coordinates. In general such models present a non-linear kinetic term and the extended supersymmetries put constraints on the metric of the target. In this section let us present here a very simpliﬁed model, which however is illustrative of how invariances under pseudo- Euclidean supersymmetry can arise. Let us in fact consider a model of dbosonic ﬁeldsX a anddspinorsψ α freelymovinginaﬂat d-dimensionaltargetmanifold, not necessarily Minkowskian or Euclidean, endorsed of a pseudo-euclidean η ab . Letusfurthermore introducethefreekineticactionbeing given by S K = /integraldisplay dtL= 1 2 /integraldisplay dt /parenleftBig ˙X a ˙X b η ab +iδ˙ψ α ψ β η αβ /parenrightBig , (15) where the metric η αβ for the spinorial part is assumed to have the same signature as the metric η ab , andδis justasignnormalization( δ=±1). A natural question to be asked is which supersymmetries are invariances of the above free kinetic action. The answer is furnished by accommodating the d bosonicand dfermioniccoordinatesintoa(maximallyextended)irreduciblerep- resentation of the extended supersymmetries, and later counting how many such transformations survive as invariances of the action. The ﬁrst non-trivial example concerns a 2-dimensional target( d= 2), whose two bosonic and two fermionic degrees of freedom carry the {2,2}representation of (2,2)extended supersym- metry. However, only half of these supersymmetries are realized as invariances of the action. The action indeed is invariant under either the (2,0)or the (1,1) extended supersymmetries, whether the target space is respectively Euclidean or Minkowskian. Therefore already in the 2-dimensional Minkowskian case we observe the arising of a pseudo-Euclidean supersymmetry invariance. The next simplestexampleisrealizedbya 4-dimensionaltarget.Thefourbosonicandfour kievarwe.tex; 12/03/2001; 3:49; p.206 200 F.TOPPAN fermioniccoordinatescanbeaccommodatedintothreeirreduciblerepresentations of maximally extended supersymmetry, according to formula (13), namely the (4,0), the (0,4)and the (3,3)extended supersymmetries. The action (15) turns out to be invariant, for Euclidean (4 + 0), Minkowskian (3 + 1)and(2 + 2) signatureforthemetric η, according to thefollowing table (4, 0) (0, 4) (3, 3) (4 + 0) (4, 0) (0, 0) (3, 0)δ= +1 (4 + 0) (0, 0) (0, 4) (0, 3)δ=− 1 (3 + 1) (1, 0) (0, 0) (1, 0)δ= +1 (3 + 1) (0, 0) (0, 1) (0, 1)δ=− 1 (2 + 2) (2, 0) (0, 2) (2, 1)δ= +1 (2 + 2) (2, 0 (0, 2) (1, 2)δ=− 1 which should be understood as follows. The central entries denote how many supersymmetries are realized as invariances of the (15) action for each one of thethreeirreduciblerepresentationsofmaximallyextendedsupersymetry,incor- respondencewiththegivensignatureofspacetimeandsignfor δ.Inthisparticular case invariance under pseudo-Euclidean supersymmetry is guaranteed for the targetofsignature (2 + 2). InthistalkIhavepresentedsomeresultsconcerningtherepresentationtheory for irreducible multiplets of the one-dimensional N= (p,q)extended super- symmetry. A peculiar feature of the one-dimensional supersymmetric algebras consists in the fact that the supermultiplets formed by dbosonic and dfermionic degreesoffreedomaccommodatedinachainwith M+ 1 (M≥2)differentspin states uniquely determines a 2-chain multiplet of the form {d,d}which carries a representationofthe Nextendedsupersymmetry.Furthermore,itisshownthatall such2-chainirreduciblemultipletsofthe (p,q)extendedsupersymmetryarefully classiﬁed;whene.g.thecondition p−q= 0mod8issatisﬁed,theirclassiﬁcation is equivalent to that one of Majorana-Weyl spinors in any given space-time, the numberp+qofextendedsupersymmetriesbeingassociatedtothedimensionality Dofthespacetime,whilethe 2dsupermultipletdimensionalityisthedimension- ality of the corresponding Γmatrices. The more general case for arbitrary values ofpandqhas also beenfully discussed. Thesemathematicalpropertiescanﬁndalotofinterestingapplicationsincon- nection with the construction of Supersymmetric and Superconformal Quantum Mechanical Models. These theories are vastly studied due to their relevance in many different physical domains, to name just a few it can be mentioned the low-energy effective dynamics of black-hole models, the dimensional reduction kievarwe.tex; 12/03/2001; 3:49; p.207 N-EXTENDED 1-DIMSUPER SYSTEMS 201 of higher-dimensional superﬁeld theories, which are a laboratory for the investi- gationof the spontaneous breakingof the supersymmetry, andso on. Acknowledgments. It is a pleasure for me to acknowledge A. Pashnev. The re- sults reported in this talk are fruit of our collaboration. I wish also acknowledge for useful discussions E.A. Ivanov, S. J. Gates Jr., S.O. Krivonos and V. Zima. Finally, let me express my gratitude to the organizers of the ARW conference for theinvitationandthe warmhospitality. References 1. A. Pashnev and F. Toppan, On the Classiﬁcation of N-Extended Supersymmetric Quan- tum Mechanical Systems , CBPF, JINR preprint, CBPF-NF-029/00, JINR E2-2000-193, hep-th/0010135 , Dubna,Rio deJaneiro, 2000. 2. M. De CrombruggheandV. Rittenberg, Ann. of Phys. 151(1983), 99. 3. M.Claudson and M.B. Halpern, Nucl.Phys. B250(1985), 689. 4. B. deWit, A.K. Tollsten and H.Nicolai, Nucl.Phys. B392(1993), 3. 5. S. JamesGates,Jr. and Lubna Rana, Phys. Lett. B352(1995), 50; ibid. B369(1996),262. 6. R.A. Coles and G. Papadopoulos, Class.Quant. Grav. 7(1990),427–438. 7. A. Pashnev, Noncompact Extension of One-Dimensional Supersymmetry and Spinning Parti- cle, JINR preprint, E2-91-536, Dubna,1991. 8. M. Atiyah, R. Bott and A. Shapiro, Topology 3, (Suppl. 1)(1964), 3. 9. I.Porteous, Topological Geometry , van Nostand Rheinhold,London, (1969). 10. S. Okubo, Jou. Math. Phys., 32(1991), 1657; ibid. 1669. kievarwe.tex; 12/03/2001; 3:49; p.208 kievarwe.tex; 12/03/2001; 3:49; p.209 PARA, PSEUDO, AND ORTHOSUPERSYMMETRIC QUANTUMMECHANICSANDTHEIRBOSONIZATION CHRISTIANEQUESNE ∗ PNTPM, Universit ´e Libre de Bruxelles, Campus de la Plaine CP229,BoulevardduTriomphe,B-1050 Brussels, Belgium Abstract. We consider the problem of bosonizing supersymmetric quantum mechanics (SSQM) and some of its variants, i.e., of realizing them in terms of only boson-like operators without fermion-like ones. In the SSQM case, this is realized in terms of the generators of the Calogero- Vasiliev algebra (also termed deformed Heisenberg algebra with reﬂection). In that of the SSQM variants, this is done by considering generalizations of the latter algebra, namely the C λ -extended oscillator algebras, where C λ is thecyclic groupof order λ. 1. Introduction Supersymmetry has established an elegant symmetry between bosons and fermionsandisoneofthecornerstonesofmoderntheoreticalphysics.Itsapplica- tiontoquantummechanicshasprovidedapowerfulmethodofgeneratingsolvable quantum mechanical models. On the other hand, exotic quantum statistics have received considerable attention due to their possible relevance to the fractional quantumHalleffect andanyon superconductivity. By combining both concepts within the framework of quantum mechanics, one gets variants of SSQM: paraSSQM [1–3], pseudoSSQM [4, 5], and or- thoSSQM [6]. They can be realized in terms of bosons and parafermions [7], pseudofermions [4, 5], ororthofermions[8], respectively. By using the Calogero-Vasiliev algebra [9], Plyushchay showed [10] that SSQM can be described in terms of only boson-like operators without fermion- likeones(see also [11]). In the present communication, we shall consider generalizations of the Calogero-Vasiliev algebra, namely the C λ -extended oscillator algebras (where C λ =Z λ is the cyclic group of order λ) [12–14]. We shall show that they hav e ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.210 204 C.QUESNE some interesting applications to variants of SSQM [12, 14], as they provide a bosonizationof the latter analogousto that obtainedby Plyushchay for SSQM. 2. Generalizeddeformedand G-extendedoscillator algebras The generalized deformed oscillator algebras (GDOAs) (see e.g. Refs. [15, 16] and references quoted therein) arose from successive generalizations of the Arik- Coon [17] and Biedenharn-Macfarlane [18, 19] q-oscillators. Such algebras, denoted byA q (G(N)), are generated by the unit, creation, annihilation, and number operators I,a † ,a,N, satisfying the Hermiticity conditions /parenleftBig a † /parenrightBig † =a, N † =N,andthecommutation relations /bracketleftBig N,a † /bracketrightBig =a † , [N,a] =−a, /bracketleftBig a,a † /bracketrightBig q ≡aa † −qa † a=G(N),(1) whereqis some realnumber and G(N)is someHermitian, analyticfunction. On the other hand, G-extended oscillator algebras, where Gis some ﬁnite group,appearedinconnectionwith n-particleintegrablemodels.FortheCalogero model [20],for instance, Gisthe symmetric group S n [21, 22]. For two particles, the S 2 -extended oscillator algebra A (2) κ , whereS 2 = {I,K|K 2 =I}, is generated by the operators I,a † ,a,N,K, subject to theHermiticity conditions /parenleftBig a † /parenrightBig † =a,N † =N,K † =K −1 , andthe relations /bracketleftBig N,a † /bracketrightBig =a † , [N,K ] = 0, K 2 =I, /bracketleftBig a,a † /bracketrightBig =I+κK (κ∈R), a † K=−Ka † , (2) togetherwith theirHermitian conjugates. When theS 2 generatorKis realized in terms of the Klein operator (−1) N , A (2) κ becomes a GDOA characterized by q= 1andG(N) =I+κ(−1) N , and knownasthe Calogero-Vasiliev oscillatoralgebra[9]. Theoperator Kmaybealternativelyconsideredasthegeneratorofthecyclic groupC 2 ofordertwo,sincethelatterisisomorphicto S 2 .Byreplacing C 2 bythe cyclic group of order λ,C λ ={I,T,T 2 ,... ,T λ−1 |T λ =I}, one then gets a new class ofG-extended oscillator algebras [12–14], generalizing that describing thetwo-particleCalogero model. kievarwe.tex; 12/03/2001; 3:49; p.211 PARASUPERSYMMETRICQUANTUM MECHANICS 205 3.C λ -extended oscillatoralgebras Let us consider the algebras generated by the operators I,a † ,a,N,T, satisfying theHermiticity conditions /parenleftBig a † /parenrightBig † =a,N † =N,T † =T −1 , and therelations /bracketleftBig N,a † /bracketrightBig =a † , [N,T] = 0, T λ =I, /bracketleftBig a,a † /bracketrightBig =I+ λ−1 /summationdisplay µ=1 κ µ T µ , a † T=e −i2π/λ Ta † , (3) togetherwiththeirHermitianconjugates[12].Here Tisthegeneratorof(aunitary representation of) the cyclic group C λ (whereλ∈{2,3,4,...}), andκ µ ,µ= 1, 2,...,λ−1,aresomecomplexparametersrestrictedbytheconditions κ ∗ µ =κ λ−µ (so thatthereremainaltogether λ−1independentreal parameters). C λ hasλinequivalent, one-dimensional matrix unitary irreducible represen- tations (unirreps) Γ µ ,µ= 0, 1,...,λ−1, which are such that Γ µ (T ν ) = exp(i2πµν/λ )foranyν= 0,1,...,λ−1.Theprojectionoperatoronthecarrier space of Γ µ may bewrittenas P µ = 1 λ λ−1 /summationdisplay ν=0 e −i2πµν/λ T ν , (4) andconversely T ν ,ν= 0,1,...,λ−1,maybeexpressedintermsofthe P µ ’sas T ν = λ−1 /summationdisplay µ=0 e i2πµν/λ P µ . (5) Thealgebradeﬁningrelations(3)maythereforeberewrittenintermsof I,a † , a,N, andP µ =P † µ ,µ= 0,1,...,λ−1,as /bracketleftBig N,a † /bracketrightBig =a † , [N,P µ ] = 0, λ−1 /summationdisplay µ=0 P µ =I, /bracketleftBig a,a † /bracketrightBig =I+ λ−1 /summationdisplay µ=0 α µ P µ , a † P µ =P µ+1 a † , P µ P ν =δ µ,ν P µ ,(6) where we use the convention P µ /prime =P µ ifµ /prime −µ= 0 modλ(and similarly for otheroperatorsorparametersindexedby µ,µ /prime ).Equation(6)dependsupon λreal parametersα µ = /summationtext λ−1 ν=1 exp(i2πµν/λ )κ ν ,µ= 0, 1,...,λ−1, restricted by the condition /summationtext λ−1 µ=0 α µ = 0. Hence, we may eliminate one of them, for instance α λ−1 ,and denote C λ -extendedoscillatoralgebrasby A (λ) α 0 α 1 ...α λ−2 . kievarwe.tex; 12/03/2001; 3:49; p.212 206 C.QUESNE The cyclic group generator Tand the projection operators P µ can be realized interms ofNas T=e i2πN/λ , P µ = 1 λ λ−1 /summationdisplay ν=0 e i2πν(N−µ)/λ , µ = 0,1,... ,λ−1,(7) respectively. With such a choice, A (λ) α 0 α 1 ...α λ−2 becomes a GDOA, A (λ) (G(N)), characterized by q= 1andG(N) =I+ /summationtext λ−1 µ=0 α µ P µ , whereP µ is given in Eq.(7). For any GDOAA q (G(N)), one may deﬁne a so-called structure func- tionF(N),whichisthesolutionofthedifferenceequation F(N+1)−qF(N) = G(N),suchthatF(0) = 0[15].ForA (λ) (G(N)),weﬁnd F(N) =N+ λ−1 /summationdisplay µ=0 β µ P µ , β 0 ≡0, β µ ≡ µ−1 /summationdisplay ν=0 α ν (µ= 1,2,... ,λ−1). (8) At this point, it is worth noting that for λ= 2, we obtain T=K,P 0 = (I+K)/2,P 1 = (I−K)/2, andκ 1 =κ ∗ 1 =α 0 =−α 1 =κ, so thatA (2) α 0 coincides with the S 2 -extended oscillator algebra A (2) κ andA (2) (G(N))with the Calogero-Vasilievalgebra. InRef.[14],itwasshownthat A (λ) (G(N))(andmoregenerally A (λ) α 0 α 1 ...α λ−2 ) hasonlytwodifferenttypesofunirreps:inﬁnite-dimensionalboundedfrombelow unirreps and ﬁnite-dimensional ones. Among the former, there is the so-called bosonic Fock space representation, wherein a † a=F(N)andaa † =F(N+ 1). Its carrier spaceFis spanned by the eigenvectors |n/angbracketrightof the number operator N, corresponding to the eigenvalues n= 0, 1, 2,..., where|0/angbracketrightis a vacuum state, i.e.,a|0/angbracketright=N|0/angbracketright= 0andP µ |0/angbracketright=δ µ,0 |0/angbracketright.The eigenvectors canbewritten as |n/angbracketright=N −1/2 n /parenleftBig a † /parenrightBig n |0/angbracketright, n = 0,1,2,... , (9) whereN n = /producttext n i=1 F(i). The creation and annihilation operators act upon |n/angbracketrightin theusualway,i.e., a † |n/angbracketright= /radicalBig F(n+ 1)|n+ 1/angbracketright, a|n/angbracketright= /radicalBig F(n)|n−1/angbracketright,(10) whileP µ projects on the µth componentF µ ≡{|kλ+µ/angbracketright|k= 0,1,2,...}of theZ λ -gradedFockspace F= /summationtext λ−1 µ=0 ⊕F µ .ItisobviousthatsuchabosonicFock space representation exists if and only if F(µ)>0forµ= 1, 2,...,λ−1. This givesthe following restrictions onthealgebra parameters α µ , µ−1 /summationdisplay ν=0 α ν >−µ, µ = 1,2,... ,λ−1. (11) kievarwe.tex; 12/03/2001; 3:49; p.213 PARASUPERSYMMETRICQUANTUM MECHANICS 207 In the bosonic Fock space representation, one may consider the bosonic oscillator Hamiltonian,deﬁnedasusual by H 0 ≡ 1 2 /braceleftBig a,a † /bracerightBig . (12) It can be rewrittenas H 0 =a † a+ 1 2   I+ λ−1 /summationdisplay µ=0 α µ P µ   =N+ 1 2I+ λ−1 /summationdisplay µ=0 γ µ P µ ,(13) whereγ 0 ≡ 1 2 α 0 andγ µ ≡ /summationtext µ−1 ν=0 α ν + 1 2 α µ forµ= 1, 2,...,λ−1. The eigenvectors of H 0 are the states|n/angbracketright=|kλ+µ/angbracketright, deﬁned in Eq. (9), and theireigenvaluesare given by E kλ+µ =kλ+µ+γ µ + 1 2 , k = 0,1,2,... , µ = 0,1,... ,λ−1. (14) In eachF µ subspace of the Z λ -graded Fock space F, the spectrum of H 0 is therefore harmonic, but the λinﬁnite sets of equally spaced energy levels, cor- responding to µ= 0, 1,...,λ−1, may be shifted with respect to each other by someamountsdependinguponthealgebraparameters α 0 ,α 1 ,...,α λ−2 ,through theirlinear combinations γ µ ,µ= 0,1,...,λ−1. For the Calogero-Vasiliev oscillator, i.e., for λ= 2, the relation γ 0 =γ 1 = κ/2implies that the spectrum is very simple and coincides with that of a shifted harmonicoscillator.For λ≥3,however,ithasamuchricherstructure.According totheparametervalues,itmaybenondegenerate,ormayexhibitsome( ν+1)-fold degeneracies above some energy eigenvalue, where νmay take any value in the set{1,2,... ,λ−1}. In Ref. [13], the complete classiﬁcation of nondegenerate, twofold and threefold degenerate spectra was obtained for λ= 3in terms ofα 0 andα 1 . In the remaining part of this communication, we will show that the bosonic Fockspacerepresentationof A (λ) (G(N))andthecorrespondingbosonicoscilla- tor Hamiltonian H 0 have some usefulapplications to variantsofSSQM. 4. Applicationtoparasupersymmetric quantummechanicsof order p In SSQM with two supercharges, the supersymmetric Hamiltonian Hand the supercharges Q † ,Q= /parenleftBig Q † /parenrightBig † , satisfy the sqm(2) superalgebra, deﬁned by the relations Q 2 = 0, [H,Q] = 0, /braceleftBig Q,Q † /bracerightBig =H, (15) together with their Hermitian conjugates. Such a superalgebra is most often realized interms ofmutually commutingbosonand fermionoperators. kievarwe.tex; 12/03/2001; 3:49; p.214 208 C.QUESNE Plyushchay[10],however,showedthatitcanalternativelyberealizedinterms ofonlyboson-likeoperators,namelythegeneratorsoftheCalogero-Vasilievalge- braA (2) (G(N))(see also Ref. [11]). The SSQM bosonization can be performed intwodifferentways,bychoosingeither Q=a † P 1 (sothatH=H 0 − 1 2 (K+κ)) orQ=a † P 0 (so thatH=H 0 + 1 2 (K+κ)). The ﬁrst choice corresponds to unbroken SSQM (all the excited states are twofold degenerate while the ground state is nondegenerate and at vanishing energy), and the second choice describes brokenSSQM (allthe statesare twofold degenerateand at positiveenergy). SSQMwasgeneralizedtoparasupersymmetricquantummechanics(PSSQM) of order two by Rubakov and Spiridonov [1], and later on to PSSQM of arbitrary orderpbyKhare[2].Inthelatter case,Eq.(15) isreplaced by Q p+1 = 0 (with Q p /negationslash= 0), [H,Q] = 0, Q p Q † +Q p−1 Q † Q+···+QQ † Q p−1 +Q † Q p = 2pQ p−1 H,(16) and is retrieved in the case where p= 1. The parasupercharges Q,Q † , and the parasupersymmetric Hamiltonian Hare usually realized in terms of mutually commuting bosonandparafermionoperators. A property of PSSQM of order pis that the spectrum of His (p+ 1)-fold degenerateabovethe( p−1)thenergylevel.ThisfactandPlyushchay’sresultsfor p= 1hintatapossibilityofrepresenting Hasalinearcombinationofthebosonic oscillator Hamiltonian H 0 associated withA (p+1) (G(N))and some projection operators. In Ref. [14] (see also Ref. [12]), it was proved that PSSQM of order pcan indeedbebosonizedintermsofthegeneratorsof A (p+1) (G(N))foranyallowed (i.e., satisfying Eq. (11)) values of the algebra parameters α 0 ,α 1 ,...,α p−1 . For suchapurpose,ans ¨atzeof thetype Q= p /summationdisplay ν=0 σ ν a † P ν ,H=H 0 + 1 2p /summationdisplay ν=0 r ν P ν , (17) were chosen. Here σ ν andr ν are some complex and real constants, respectively, to be determined in such a way that Eq. (16) is fulﬁlled. It was found that there arep+ 1families of solutions, which may be distinguished by an index µ∈{0,1,... ,p}and from which one may choose the following representative solutions Q µ = √ 2 p /summationdisplay ν=1 a † P µ+ν , H µ =N+ 1 2 (2γ µ+2 +r µ+2 −2p+ 3)I+ p /summationdisplay ν=1 (p+ 1−ν)P µ+ν ,(18) kievarwe.tex; 12/03/2001; 3:49; p.215 PARASUPERSYMMETRICQUANTUM MECHANICS 209 where r µ+2 = 1 p /bracketleftBigg (p−2)α µ+2 + 2 p /summationdisplay ν=3 (p−ν+ 1)α µ+ν +p(p−2) /bracketrightBigg .(19) The eigenvectors of H µ are the states (9) and the corresponding eigenvalues are easily found. All the energy levels are equally spaced. For µ= 0, PSSQM is unbroken, otherwise it is broken with a ( µ+ 1)-fold degenerate ground state. All theexcitedstatesare( p+ 1)-folddegenerate.For µ= 0,1,...,p−2,theground state energy may be positive, null, or negative depending on the parameters, whereas for µ=p−1orp,it is alwayspositive. Khare [2] showed that in PSSQM of order p,Hhas in fact 2p(and not only two) conserved parasupercharges, as well as pbosonic constants. In other words, there existpindependent operators Q r ,r= 1, 2,...,p, satisfying withHthe set of equations (16), and pother independent operators I t ,t= 2, 3,...,p+ 1, commutingwithH,aswellasamongthemselves.InRef.[14],arealizationofall suchoperatorswasobtained interms of the A (p+1) (G(N))generators. Asaﬁnalpoint,letusnotethatthereexistsanalternativeapproachtoPSSQM of orderp, which was proposed by Beckers and Debergh [3], and wherein the multilinear relationin Eq.(16)isreplaced bythecubic equation /bracketleftBig Q, /bracketleftBig Q † ,Q /bracketrightBig/bracketrightBig = 2QH. (20) In Ref. [12], it was proved that for p= 2, this PSSQM algebra can only be realized by thoseA (3) (G(N))algebras that simultaneously bosonize Rubakov- Spiridonov-Khare PSSQMalgebra. 5. Applicationtopseudosupersymmetricquantum mechanics Pseudosupersymmetric quantum mechanics (pseudoSSQM) was introduced by Beckers, Debergh, and Nikitin [4, 5] in a study of relativistic vector mesons interacting with an external constant magnetic ﬁeld. In the nonrelativistic limit, their theory leads to a pseudosupersymmetric oscillator Hamiltonian, which can be realized in terms of mutually commuting boson and pseudofermion operators, wherethelatterareintermediatebetweenstandardfermionand p= 2parafermion operators. It is then possible to formulate a pseudoSSQM [4, 5], characterized by a pseudosupersymmetric Hamiltonian Hand pseudosupercharge operators Q,Q † , satisfyingthe relations Q 2 = 0, [H,Q] = 0, QQ † Q= 4c 2 QH, (21) and their Hermitian conjugates, where cis some real constant. The ﬁrst two rela- tions in Eq. (21) are the same as those occurring in SSQM, whereas the third one kievarwe.tex; 12/03/2001; 3:49; p.216 210 C.QUESNE is similar to the multilinear relation valid in PSSQM of order two. Actually, for c= 1or 1/2, it iscompatible with Eq.(16)or(20),respectively. InRef.[14],itwasprovedthatpseudoSSQMcanbebosonizedintwodifferent ways in terms of the generators of A (3) (G(N))for any allowed values of the parametersα 0 ,α 1 .Thistime, the ans ¨atze Q= 2 /summationdisplay ν=0 /parenleftBig ξ ν a+η ν a † /parenrightBig P ν ,H=H 0 + 1 22 /summationdisplay ν=0 r ν P ν ,(22) were chosen, and the complex constants ξ ν ,η ν , and the real ones r ν were determinedin sucha waythat Eq.(21)isfulﬁlled. The ﬁrst type of bosonization corresponds to three families of two-parameter solutions,labeledby an index µ∈{0,1,2}, Q µ (η µ+2 ,ϕ) = /parenleftBig η µ+2 a † +e iϕ /radicalBig 4c 2 −η 2 µ+2 a /parenrightBig P µ+2 , H µ (η µ+2 ) =N+ 1 2 (2γ µ+2 +r µ+2 −1)I+ 2P µ+1 +P µ+2 ,(23) where 0<η µ+2 <2|c|,0≤ϕ<2π,and r µ+2 = 1 2c 2 (1 +α µ+2 ) /parenleftBig |η µ+2 | 2 −2c 2 /parenrightBig . (24) Choosing for instance η µ+2 = √ 2|c|, andϕ= 0, hencer µ+2 = 0(producing an overallshift ofthe spectrum), leadsto Q µ =c √ 2 /parenleftBig a † +a /parenrightBig P µ+2 , H µ =N+ 1 2 (2γ µ+2 −1)I+ 2P µ+1 +P µ+2 . (25) A comparison between Eq. (23) or (25) and Eq. (18) shows that the pseudosu- persymmetric and p= 2parasupersymmetric Hamiltonians coincide, but that the corresponding charges are of course different. The conclusions relative to the spectrumand the groundstateenergyare thereforethesame as inSec.4. The second type of bosonization corresponds to three families of one- parametersolutions, againlabeledby anindex µ∈{0,1,2}, Q µ = 2|c|aP µ+2 , H µ (r µ ) =N+ 1 2 (2γ µ+2 −α µ+2 )I+ 1 2 (1−α µ+1 +α µ+2 +r µ )P µ +P µ+1 , (26) wherer µ ∈Rchanges the Hamiltonian spectrum in a signiﬁcant way. The levels are indeed equally spaced if and only if r µ = (α µ+1 −α µ+2 + 3) mod 6 . Ifr µ is small enough, the ground state is nondegenerate, and its energy is negative for µ= 1, or may have any sign for µ= 0or 2. On the contrary, if r µ is large kievarwe.tex; 12/03/2001; 3:49; p.217 PARASUPERSYMMETRICQUANTUM MECHANICS 211 enough, the ground state remains nondegenerate with a vanishing energy in the former case, while it becomes twofold degenerate with a positive energy in the latter. For some intermediate r µ value, one gets a two or threefold degenerate groundstatewithavanishingorpositiveenergy,respectively. 6. Applicationtoorthosupersymmetric quantum mechanics oforder two MishraandRajasekaran[8]introducedorder- porthofermionoperatorsbyreplac- ingthePauliexclusionprinciplebyamorestringentone:anorbitalstateshallnot containmorethanoneparticle,whateverbethespindirection.Thewavefunction is thus antisymmetric in spatial indices alone with the order of the spin indices frozen. Khare, Mishra, and Rajasekaran [6] then developed orthosupersymmetric quantum mechanics (OSSQM) of arbitrary order pby combining boson oper- ators with orthofermion ones, for which the spatial indices are ignored. OS- SQM is formulated in terms of an orthosupersymmetric Hamiltonian H, and 2p orthosuperchargeoperators Q r ,Q † r ,r= 1,2,...,p, satisfying therelations Q r Q s = 0, [H,Q r ] = 0, Q r Q † s +δ r,sp /summationdisplay t=1 Q † t Q t = 2δ r,s H,(27) and their Hermitianconjugates,where randsrunover1,2, ...,p. In Ref. [14], it was proved that OSSQM of order two can be bosonized in terms of the generators of some well-chosen A (3) (G(N))algebras. As ans ¨atze, theexpressions Q 1 = 2 /summationdisplay ν=0 /parenleftBig ξ ν a+η ν a † /parenrightBig P ν , Q 2 = 2 /summationdisplay ν=0 /parenleftBig ζ ν a+ρ ν a † /parenrightBig P ν , H=H 0 + 1 22 /summationdisplay ν=0 r ν P ν , (28) were used, and the complex constants ξ ν ,η ν ,ζ ν ,ρ ν , and the real ones r ν were determined in such a way that Eq. (27) is fulﬁlled. There exist two families of two-parametersolutions,labeled by µ∈{0,1}, Q 1,µ (ξ µ+2 ,ϕ) =ξ µ+2 aP µ+2 +e iϕ /radicalBig 2−ξ 2 µ+2 a † P µ , Q 2,µ (ξ µ+2 ,ϕ) =−e −iϕ /radicalBig 2−ξ 2 µ+2 aP µ+2 +ξ µ+2 a † P µ , H µ =N+ 1 2 (2γ µ+1 −1)I+ 2P µ +P µ+1 , (29) where 0<ξ µ+2 ≤ √ 2and0≤ϕ<2π, provided the algebra parameter α µ+1 is takenasα µ+1 =−1.Asamatteroffact,theabsenceofathirdfamilyofsolutions kievarwe.tex; 12/03/2001; 3:49; p.218 212 C.QUESNE corresponding to µ= 2comes from the incompatibility of this condition (i.e., α 0 =−1)withconditions (11). The orthosupersymmetric Hamiltonian Hin Eq. (29) is independent of the parametersξ µ+2 ,ϕ. All the levels of its spectrum are equally spaced. For µ= 0, OSSQM is broken: the levels are threefold degenerate, and the ground state energy is positive. On the contrary, for µ= 1, OSSQM is unbroken: only the excited states are threefold degenerate, while the nondegenerate ground state has avanishing energy. Suchresults agreewith the general conclusionsof Ref.[6]. Forpvalues greater than two, the OSSQM algebra (27) becomes rather com- plicatedbecausethenumberofequationstobefulﬁlledincreasesconsiderably.A glance at the 18 independent conditions for p= 3led to the conclusion that the A (4) (G(N))algebra is not rich enough to contain operators satisfying Eq. (27). Contrary to what happens for PSSQM, for OSSQM the p= 2case is therefore notrepresentative ofthegeneral one. 7. Conclusion Inthiscommunication,weshowedthatthe S 2 -extendedoscillatoralgebra,which was introduced in connection with the two-particle Calogero model, can be ex- tended to the whole class of C λ -extended oscillator algebras A (λ) α 0 α 1 ...α λ−2 , where λ∈{2,3,...},andα 0 ,α 1 ,...,α λ−2 aresomerealparameters.Inthesameway, the GDOA realization of the former, known as the Calogero-Vasiliev algebra, is generalized to a class of GDOAs A (λ) (G(N)), whereλ∈{2,3,...}, for which one can deﬁne a bosonic oscillator Hamiltonian H 0 , acting in the bosonic Fock space representation. Forλ≥3,thespectrumof H 0 hasaveryrichstructureintermsofthealgebra parametersα 0 ,α 1 ,...,α λ−2 . This can be exploited to provide a bosonization of PSSQM of order p=λ−1, and, forλ= 3, a bosonization of pseudoSSQM and OSSQM ofordertwo. References 1. V.A. Rubakov and V.P. Spiridonov, Parasupersymmetric Quantum Mechanics , Mod. Phys. Lett.A3(1988) 1337. 2. A.Khare, Parasupersymmetry in Quantum Mechanics , J. Math. Phys. 34(1993) 1277. 3. J. Beckers and N. Debergh, Parastatistics and Supersymmetry in Quantum Mechanics , Nucl. Phys.B340(1990)767. 4. J. Beckers, N. Debergh and A.G. Nikitin, On Parasupersymmetries and Relativistic De- scriptions for Spin one Particles: II. The Interacting Context with (Electro)Magnetic Fields , Fortschr. Phys. 43(1995)81. 5. J. Beckers and N. Debergh, From Relativistic Vector Mesons in Constant Magnetic Fields to Nonrelativistic(Pseudo)Supersymmetries , Int.J. Mod.Phys. A10(1995)2783. 6. A.Khare,A.K.Mishra,andG.Rajasekaran, OrthosupersymmetricQuantumMechanics , Int. J.Mod. Phys. A8(1993)1245. kievarwe.tex; 12/03/2001; 3:49; p.219 PARASUPERSYMMETRICQUANTUM MECHANICS 213 7. Y. Ohnuki and S. Kamefuchi, Quantum Field Theory and Parastatistics , Springer-Verlag, Berlin, 1982. 8. A.K. Mishra and G. Rajasekaran, Algebra for Fermions with a New Exclusion Principle , Pramana - J. Phys. 36(1991) 537. 9. M.A. Vasiliev, Higher Spin Algebras and Quantization on the Sphere and Hyperboloid , Int. J.Mod. Phys. A6(1991)1115. 10. M.S.Plyushchay, DeformedHeisenbergAlgebra,FractionalSpinFields,andSupersymmetry without Fermions , Ann. Phys. (N.Y.) 245(1996)339. 11. J. Beckers, N. Debergh, and A.G. Nikitin, Reducibility of Supersymmetric Quantum Mechanics , Int.J. Theor. Phys. 36(1997) 1991. 12. C. Quesne and N. Vansteenkiste, C λ -Extended Harmonic Oscillator and (Para)Supersymmetric Quantum Mechanics , Phys. Lett. A240(1998) 21. 13. C. Quesne and N. Vansteenkiste, Algebraic Realization of Supersymmetric Quantum Mechanics forCyclic Shape Invariant Potentials , Helv.Phys.Acta 72(1999) 71. 14. C. Quesne and N. Vansteenkiste, C λ -Extended Oscillator Algebras and Some of Their Deformations and Applications to Quantum Mechanics , Int. J.Theor. Phys. 39(2000) 1175. 15. C. Quesne and N. Vansteenkiste, Generalized q -Oscillators and Their Hopf Structures , J. Phys.A28(1995) 7019. 16. C. Quesne and N. Vansteenkiste, Representation Theory of Deformed Oscillator Algebras , Helv. Phys. Acta 69(1996)141. 17. M. Arik and D.D. Coon, Hilbert Spaces of Analytic Functions and Generalized Coherent States , J. Math. Phys. 17(1976) 524. 18. L.C. Biedenharn, The Quantum Group SU q (2) and a q -Analogue of the Boson Operators , J. Phys.A22(1989) L873. 19. A.J. Macfarlane, On q -Analogues of the Quantum Harmonic Oscillator and the Quantum GroupSU(2) q , J. Phys. A22(1989) 4581. 20. F. Calogero, Solution of the One-Dimensional N -Body Problems with Quadratic and/or Inversely QuadraticPair Potentials , J. Math.Phys. 12(1971) 419. 21. A.P.Polychronakos, ExchangeOperatorFormalismforIntegrableSystemsofParticles , Phys. Rev. Lett. 69(1992) 703. 22. L. Brink, T.H. Hansson and M.A. Vasiliev, Explicit Solution to the N -Body Calogero Problem , Phys. Lett. B286(1992) 109. kievarwe.tex; 12/03/2001; 3:49; p.220 kievarwe.tex; 12/03/2001; 3:49; p.221 SUPERSYMMETRIC ODD MECHANICAL SYSTEMS AND HILBERT Q-MODULE QUANTIZATION ANDRZEJ FRYDRYSZAK ∗ Instituteof TheoreticalPhysics, University of Wroclaw, pl.Borna 9,50-204Wroclaw,Poland 1. Introduction Supersymmetry can be implemented within a particle model in two ways. The ﬁrst one is commonly exploited and assumes that we use conventional graded Lie algebra approach in the sense that on the classical level we have a Z 2 -graded Lie-Poisson algebra of observables which after quantization is replaced by a Z 2 - graded Lie algebra of operators. Both, graded Poisson bracket and Z 2 -graded commutatorareevenmappings.Thesecondwayofrealizationofsupersymmetry in a particle model is related to the anti-bracket algebras. In this case Lagrangian aswellasHamiltonianofthesupersymmetricsystemisanoddGrassmannalgebra valuedfunctionandtheGrassmannianparityofcanonicalmomentaisoppositeto theparityofrelatedcoordinates.Theanti-bracketisanoddmapping.Realizations of the mentioned type we shall call the even supersymmetric mechanics and the odd supersymmetric mechanics, respectively [1, 2]. The odd mechanics allows particular deformation of geometry of the conﬁguration superspace. The realiza- tion of the supersymmetry algebra after the passage to phase superspace in terms of the Dirac anti-bracket remains conventional [3]. The canonical quantization of both types of models can be done in parallel but in the case of the odd systems onecanintroduceanew Z 2 -gradedalgebrageneralizingcomplexnumbersinsuch a sense that we introduce additional imaginary unit of the odd Grassmannian parity [4, 5]. Such a structure we shall call oddons (referring to the name of quaternions, octonions etc.). The formalism, in both cases, allows to mimic the approach known from the harmonic analysis on the Heisenberg group [6]. In the ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.222 216 A. FRYDRYSZAK even sector it is done for a pair - Heisenberg group and Fermionic Heisenberg group.Intheoddsector it is donefor socalled OddHeisenberg group. In this presentation we shall brieﬂy describe classical aspect of the even and odd models on the example of Z 2 -graded supersymmetric oscillators and moreoverweshalldisplay someissuesof their quantization. 2. Supersymmetric (Superﬁeld)Classical Mechanics. To discuss even and odd mechanical systems on the same footing let us de- ﬁne the following notion of the superﬁeld supersymmetric classical mechanics (SSCM) [7, 3]. Let (N 0 ,N 1 )be a ﬁxed pair of non-negative integers. (N 0 ,N 1 )- dimensionalSSCMisquadruple (Υ;{Q α ,D β ,T}; (M,J,G );S)consistingof: (a)Υ-a”supersymmetrizedtime” (t,θ α ),α= 1,2 (b){Q α ,D β ,T}-super Liealgebra of supertranslations and respectivecovariantderivatives on Υ {Q α ,Q β }= 2iδ αβ T,{D α ,D β }=−2iδ αβ T (1) {Q α ,T}= 0 ={D α ,T},{Q α ,D β }= 0 (2) (c)M-Z 2 -graded conﬁguration space dimM=(N 0 ,N 1 )with gradation map- ping J:M−→M, M =M 0 +M 1 J( s φ) = (−1) ss φ, s φ∈M s ,s= 1,2 (d)G-Z 2 -graded metric in M < s φ, s φ>= N s /summationdisplay i,j=1s G ijs φ is φ j , < s φ, s /prime φ>= 0for s/negationslash=s /primes G ij = (−1) ss G ji (e)S-an action. The action Sisinvariantunder supertranslations Trajectoriesin Mare superﬁelds with thefollowingexpansion 0 φ j (t,ϑ) =x j (t) +iϑ α x α j (t) + 1 2ϑ 2 b j (t) (3) 1 φ j (t,ϑ) =y j (t) +ϑ α y α j (t) + 1 2ϑ 2 f j (t) (4) The actionShas the form S= /integraldisplay L(D α φ,φ)dtdϑ 1 dϑ 2 (5) kievarwe.tex; 12/03/2001; 3:49; p.223 SUPER ODDMECHANICAL SYSTEMS 217 and yieldsthefollowing equationsofmotion δ L δ s φ−(−1) s D α δ L δD αs φ= 0. (6) It is worth mentioning that the mapping Jin the above deﬁnition is the counter- part of the fermionic number operator (−1) F known in supersymmetric quantum mechanics. Such SSCM has two natural realizations in Z 2 -graded conﬁguration space. Namely, Evenrealization:GradedSuperﬁeldOscillator (GSO)[7]. LetN 1 = 2k. S= /integraldisplay 1 4(ε αβ <D α φ,D β φ>−2ω<φ,φ> )dtdϑ 1 dϑ 2 (7) incomponents it gives S= 1 2 /integraldisplay dt 0 G ij [( ˙x i ˙x j −b i b j )−(δ α β x αi ˙x β j )−ω(x i b j +b i x j +i/epsilon1 αβ x α i x β j ) 1 2 /integraldisplay dt 1 G ij [−( ˙y i ˙y j +f i f j ) + (δ α β y αi ˙y β j )−ω(y i f j +f i y j +i/epsilon1 αβ y α i y β j ).(8) We cansummarize the componentcontent ofthe model as follows: −GSO consists of system of bosonic oscillators and rotators and system of fermionicoscillatorsandrotators −full GSOhasadditional symmetry(mixingboth sectors) −momenta have the same grade as conjugate coordinates ⇒Z 2 -graded Pois- son bracket in phase space is an even mapping. The phase space of this model we shall denote (P (0,0) ⊕P (1,1) ;{.,.} 0 ), where (p,q)∈P (0,0) and (Π,Θ)∈P (1,1) Moreover the pair (p,q)describes even coordinates and their conjugated momenta and similarly (Π,Θ)denotes pairs of the odd coordinates andmomenta. Oddrealization:OddGradedSuperﬁeldOscillator (OGSO) [3]. LetN 1 =N 0 . Here we introduce the odd extension of covariant derivatives, in thesensethatinsteadof considering D α ⊗id M wedeﬁne D α ⊗Π, Π 2 =id M ⇒Π α = /parenleftbigg 0q −1 α q α 0 /parenrightbigg , (9) wherec α ∈R. Now S= /integraldisplay ( 1 2<D α φ,Π αβ D β φ>−ω<φ, Πφ>)dtdϑ 1 dϑ 2 (10) kievarwe.tex; 12/03/2001; 3:49; p.224 218 A. FRYDRYSZAK and Π αβ D β0 φ i =/epsilon1 αβ q −1 β D β1 φ i (11) Π αβ D β1 φ i =/epsilon1 αβ q β D β0 φ i . (12) Incomponentsthisactionhasthefollowing form S= /integraldisplay dt /summationdisplay s=1,2s G ij { 1 2[Tr s δ q ( ˙x i ˙y j −b i f j )+ 1 2( s δ qα β x αi ˙y β j − s δ qβ α ˙x α i y βj )] + (−1) s ω(x i f j +i/epsilon1 αβ x α i y β j +b i y j )}.(13) The componentcontentof thismodel canbe characterized asfollows: −OGSO consists of system of bosonic oscillators and rotators and system of fermionicoscillatorsandrotators −momenta have opposite grade with respect to the grade of conjugate coor- dinates⇒”Poisson” bracket in odd phase space i.e. anti-bracket is an odd mapping withshifted gradepropertiesi.e. {A,B} 1 =−(−1) (A+1)(B+1) {B,A} 1 (14) /summationdisplay cycl −(−1) (A+1)(C+1) {A,{B,C} 1 } 1 = 0 (15) The canonical relationsforcomponent ﬁelds,in this case, have the form {F A ,p F B }= (−) |F A |(|F A |+1) δ A B , (16) whereFis generic component ﬁeld and p F is its momentum. Here, in analogy to the previous case we shall denote phase space for this system as (P (0,1) ⊕ P (1,0) ;{·,·} 1 ),where (p,Θ)∈P (0,1) and(Π,q)∈P (1,0) .AsbeforeGreekletters denoteoddentities. 3. Generalization of theHeisenberg group TheHeisenberggroupisconnectedtothestructureofusualphasespace.Weshall need two generalizations of this object. For the even mechanics: a generalization related to the (P (1,1) ,{·,·} 0 )and for the odd mechanics: a generalization related tothe (P (0,1) ,{·,·} 1 ). kievarwe.tex; 12/03/2001; 3:49; p.225 SUPER ODDMECHANICAL SYSTEMS 219 FermionicHeisenbergGroup[8]. Weshallconsiderheretheoddpartofthe phase space i.e. P (1,1) extended by the time dimension. Hence, let P n be the free Q-moduleP n =Q 2n,1 with the ﬁxed basis {e i } 2n i=0 ,|e 0 |= 0,|e i |= 1,i= 1,2,...,2n(Qis the Banach -Grassmann algebra and |·|denotes Grassmannian parity of an element [9]). Moreover, let B(·,·)be the graded symplectic even formdeﬁned on Q 2n,0 with values in Q. Inourbasis v= n /summationdisplay i=1 Π i e i + n /summationdisplay i=1 Θ i e n+i (17) and weﬁxtheformof B(·,·)as follows B(e i ,e j+n ) =δ ij . (18) Hence B(v,v /prime ) =− n /summationdisplay i=1 (Π i Θ i/prime + Θ i Π i/prime ), (19) where|Π i |=|Θ i |= 1.Nowweconsider themodule Pwithcoordinates (v,t)≡(Π,Θ,t) = /parenleftBig Π 1 ,...,Π n ,Θ 1 ,...,Θ n ,t /parenrightBig (20) (wheret∈Q 0 )and withthefollowingmultiplication law (v,t)◦(v /prime ,t /prime ) = /parenleftbigg v+v /prime ,t+t /prime + 1 2B(v,v /prime ) /parenrightbigg . (21) Pequipped with this multiplication forms a group. It is called the Fermionic Heisenberg group and denoted by FH n . As in the conventional case, this group has a matrix realization. For given (Π,Θ,t)∈Q 2n,1 we deﬁne the matrix µ(Π,Θ,t)∈M n+2 (Q)by µ(Π,Θ,t) =   0 Π 1 ...Π n t 0 0...0 Θ 1 ............ 0 0...0 Θ n 0 0...0 0   (22) We have expµ(Π,Θ,t) expµ /parenleftbig Π /prime ,Θ /prime ,t /prime /parenrightbig = expµ /parenleftbigg Π + Π /prime ,Θ + Θ /prime ,t+t /prime − 1 2 /parenleftbig ΠΘ /prime + ΘΠ /prime /parenrightbig/parenrightbigg , (23) kievarwe.tex; 12/03/2001; 3:49; p.226 220 A. FRYDRYSZAK what givesthemultiplication law. The elements µ(Π,Θ,t)form a graded Lie algebra with one even generator Tand2noddgenerators e i =ˆΠ i ande i+n =ˆΘ i andwiththefollowingstructural relations /bracketleftBig ˆΠ i ,ˆΠ j /bracketrightBig + = /bracketleftBig ˆΘ i ,ˆΘ j /bracketrightBig + = 0 /bracketleftBig ˆΠ i ,T /bracketrightBig − = /bracketleftBig ˆΘ i ,T /bracketrightBig − = 0 (24) /bracketleftBig ˆΠ i ,ˆΘ j /bracketrightBig + =δ i j T. Odd Heisenberg Group [5]. To describe the Odd Heisenberg group we shall use a new structure replacing the complex numbers i.e. the algebra of oddons. It providestheoddmultiplicationinthesetofobservables.Thedeﬁnitionofoddons and some of their properties are collected in the Appendix. Let us consider as an extension of the phase space P (0,1) by the time dimension the free Q RO -module T n =Q n|n+1 RO with the basis{E i ,e i ,e 0 } n i=1 , where|e i |=|e 0 |= 0,|E i |= 1 ;i= 1,2,... ,n. LetB(·,·)be the odd symplectic form deﬁned on Q n,n RO with valuesinQ RO .We shall consider vectors ofthe form v= /summationdisplay n i=1 p i E i + /summationdisplay n i=1 Θ i e i (25) and weﬁxB(·,·)as follows B(E i ,e i ) =δ ij (26) ThereforeB(v,v /prime ) = /summationtext n i=1 (p i Θ /primei −Θ i p /primei ).NowletOH n bethesetofvectorsof theform (v,τ) = (p,Θ,τ) = (p 1 ,p 2 ,... ,p n ,Θ 1 ,Θ 2 ,... , Θ n ,τ)(27) whereτ=t·ˆ1,t∈Q R 0 ,Θ i ∈Q R 1 ,p i ∈Q RO 0 . In the setOH n we deﬁne the action inthefollowingform (v,τ)⋆(v /prime ,τ /prime ) = (v+v /prime ,τ+τ /prime + 1 2B(v,v /prime )) (28) The(OH n ,⋆)is a group, we shall call it the Odd Heisenberg group. Its matrix realizationcanbewritteninthefollowingform,forthe (p,Θ,τ)wedeﬁnematrix µ(p i ,Θ i ,τ)∈M n+2 (Q CO ) µ(p,Θ,τ) =   0p 1 ... p n τ 0 0...0 Θ 1 ............ 0 0...0 Θ n 0 0...0 0   (29) kievarwe.tex; 12/03/2001; 3:49; p.227 SUPER ODDMECHANICAL SYSTEMS 221 The odd productofoddexponents givesthefollowing relation exp ⋆ µ(p,Θ,τ)⋆exp ⋆ µ /parenleftbig p /prime ,Θ /prime ,τ /prime /parenrightbig = exp ⋆ µ /parenleftbigg p+p /prime ,Θ + Θ /prime ,τ+τ /prime + 1 2 /parenleftbig pΘ /prime −Θp /prime /parenrightbig/parenrightbigg (30) Elementsµ(p,Θ,τ)formagradedanti-bracketalgebrawithevengenerators ˆe i ,ˆe 0 and oddgenerators ˆE i /bracketleftBig ˆE i ,ˆe j /bracketrightBig 1 =δ ij ˆe 0 (31) 4. HilbertQ-moduleQuantization To describe the quantization of the supersymmetric model we shall use the formalismof theHilbert Q-modules. Q-representations of the Fermionic Heisenberg Group. As in the case of the Heisenberg group, it is possible to consider the Schr ¨odinger representation forFH n . However, due to the nature of the Berezin integral [10] the essentially functional content of it is trivial and the analog of the representation in function spacearisinginthiswayisﬁnitedimensionalandofanalgebraickind.Let S n be the set ofQ C valued functions of n = 2k real Grassmann variables η i ∈Q n R,1 . Let ”∗”denoteconjugationinthe Q C -algebraextendingcomplexconjugation.In S n we introducethe Q C -scalar product /angbracketleftf,g/angbracketright S = /integraldisplay dηf ∗ (η)g(η) (32) dη=dη n ...dη 1 (S n ,/angbracketleft·,·/angbracketright S )is the Hilbert Q-module [6,10]. This is the counterpart of the con- ventionalHilbertspaceofsquareintegrablefunctions.InthisspacetheHermitian conjugate operators tothe differentiationandmultiplication operators are ∂ ∂η i ≡∂ η , ∂ η† =i∂ η (33) ˆη≡η· ˆη † =−iη· (34) This operators are not self-adjoint in S n . However, for the construction of the Q- representation of FH n we need the following Let D=−i∂ η , X i =η i . The operator Π i D i + Θ i X i is self-adjointin S n and expi(t+ ΘX+ ΠD)f(η) = expi /parenleftbigg t+ Θη+ 1 2ΘΠ /parenrightbigg f(η+ Π) (35) kievarwe.tex; 12/03/2001; 3:49; p.228 222 A. FRYDRYSZAK orequivalently e i(ΘX+ΠD) =e i 2 ΘΠ e iΘX e iΠD . (36) The multiplication of e iA /prime e iA yields theFH n group multiplication (6). There existsahomomorphism π 1 :FH n /mapsto→Op(S n ) π 1 (Π,Θ,t) = expi(t+ ΘX+ ΠD) (37) which gives the Q-representation of FH n onS n ,n= 2k.π 1 given by π 1 (Π,Θ,t)f(η) =e i(t+Θη+ 1 2 ΘΠ) f(η+ Π) (38) isaQ-irreducible Q-unitaryrepresentationof FH n .Thematrixcoefﬁcientsof therepresentation π 1 forf, g∈S n aredeﬁned by M(Π,Θ) =/angbracketleftf,π 1 (Π,Θ)g/angbracketright (39) Analogouslytotheconventionaltheorywecanintroducethefunction V(f,g)on theS n by V(f,g)(Π,Θ) =M(Π,Θ) =/angbracketleftf,e i(ΘX+ΠD) g/angbracketright= /integraldisplay dηf ∗ (η− 1 2Π)e iΘη g(η+ 1 2Π) (40) The mapping V:S n ×S n /mapsto→S 2n is the Grassmannian version of the Fourier- Wigner transform (the GFW-transform). In particular, GFW-transform for Grassmannian Gaussian ω 0 ∈S n ,n= 2k, can be written in the following form V(ω 0 ,f)(Π,Θ) = (Pf) − 1 2 e − i 4 z ∗ G −1 z /integraldisplay dηe 1 2 ηGη−ηz− 1 4 zG −1 z f(η),(41) withanewvariable zdeﬁnedas z k =G kj Π j +iΘ k , (42) and ω 0 = (PfG ) − 1 2 e 1 2 ηGη , (43) whereG= (G ij )isan anti-symmetricmatrix and PfGits Pfafﬁanand /angbracketleftω 0 ,ω 0 /angbracketright S = 1. (44) kievarwe.tex; 12/03/2001; 3:49; p.229 SUPER ODDMECHANICAL SYSTEMS 223 ThisallowsustodeﬁnetheGrassmannianBargmanntransform( GB-transform) as (Bf)(z)≡2 − n 4 /integraldisplay dηe 1 2 ηGη−ηz− 1 4 zG −1 z f(η). (45) For furtherconveniencewe shalldenote /bardblz/bardbl 2 = i 2z ∗ G −1 z. (46) One canwrite the FH n group multiplicationfor (z,t)in the form (z,t)◦(z /prime ,t /prime ) = /parenleftbigg z+z /prime ,t+t /prime + 1 2Im(−z ∗ i 2G −1 z) /parenrightbigg ,(47) The transferred representation βcan bedeﬁned as β(z,t)◦B=B◦π 1 (Π,Θ,t) (48) where V(ω 0 ,f)(Π,Θ) =e − 1 2 /bardblz/bardbl 2 (Bf)(z) (49) Letusdeﬁne theGrassmannianBargmann-Fock spaceas F n ={f|fisholomorphicon Q n C,1 and/bardblf/bardbl 2 F = /integraldisplay |dz|e −/bardblz/bardbl 2 f ∗ (z)f(z)} (50) Thebasisinthisspaceisformedbypolynomials {z I k } I k ,k ,whereI k isastrongly ordered multi-index (with increasing entries). The Q- scalar product in F n is deﬁned as /angbracketleftf,g/angbracketright F = /integraldisplay |dz|e −/bardblz/bardbl 2 f ∗ (z)g(z), (51) where |dz|≡− ( i 2) n dzdz ∗ . (F n ,/angbracketleft,/angbracketright F )is a Hilbert Q- module. The operator Hermitian conjugate to the differentiation ∂ z inF n is (∂ z ) † =− i 2G −1 z. (52) We canwrite representation βexplicitly.Let w=Gρ+iσ,forf∈S n we have (β(w)Bf) (z) = (Bπ 1 (ρ,σ)f) (z) =e 1 2 /bardblz/bardbl 2 V(ω 0 ,π 1 (ρ,σ)f)(Π,Θ)(53) kievarwe.tex; 12/03/2001; 3:49; p.230 224 A. FRYDRYSZAK what gives (β(w)Bf) (z) =e − i 2 /bardblw/bardbl 2 e − i 2 zG −1 w ∗ Bf(z). (54) For the Heisenberg group the Bargmann transform relates two distinguished basesintheSchr ¨odingerrepresentationspaceandintheFockspace.Thisproperty also holds for the FH n . The Grassmannian Hermite polynomials are related to thez I k polynomialsformingthebasisoftheFock Q-module.TheGrassmannian Hermite[11] polynomialscan betaken inthe form[12] h i 1 ...i k k =K k e − 1 2 ηGη ∂ α k ...∂ α 1 e ηGη (55) whereK k arenumericalfactors.Thenthe GBtransformof h k ,0≤k≤n,yields (Bh k ) i 1 ...i k (z) = 2 n 4 K k z I k . (56) TheGFWtransform of the Grassmannian Hermite function gives a Grassman- nianLaguerrepolynomial [8] /angbracketleftz I k ,β(w)z I /prime k /angbracketright=e − 1 2 /bardblw/bardbl 2 L (0) I k , L (0) I k = /summationdisplay m,I m ⊂I k N I /prime m I m w ∗ I k−m w I /prime k−m .(57) Q-representations of the Odd Heisenberg Group [5]. The construction of the Schr¨odingerQ-representation known for the Fermionic Heisenberg group can be extended to the Q CO -representation of the Odd Heisenberg group (we consider here only the sector (p,Θ)). Appropriate Grassmannian odd transforms and generalizedGrassmannian oddpolynomials fallto thisscheme as well. LetS n OD be the set of functions on Q n 1 with values in the Q CO , let theQ CO valued scalarproductbegivenintheform /angbracketleftf,g/angbracketright S = /integraldisplay dηf ⋆ (η)g(η), dη =dη n ...dη 1 , η i ∈Q 1 (58) (S n ,/angbracketleft·,·/angbracketright S )formsthe Hilbert Q CO - module. LetD j =−ˆı ∂ ∂η j andX i =η i then the following relations give rise to the deﬁnition ofrepresentation π exp ⋆ ˆı(t+ ΘX+pD)f(η) = (59) exp ⋆ ˆı /parenleftbigg t+ Θη+ 1 2Θ⋆p /parenrightbigg f /parenleftBig η+ˆ1p /parenrightBig =π(p,Θ,t) Letp i =ˆ1Π i ,Π i ∈Q 1 . Therefore algebraic form of relations obtained in the ﬁrst part of the report for the Fermionic Heisenberg group will be here preserved kievarwe.tex; 12/03/2001; 3:49; p.231 SUPER ODDMECHANICAL SYSTEMS 225 moduloodd exponentsand oddunits. The GrassmannianFourier-Wignerodd transformtakes the following form V(f,g)(p,Θ) = /integraldisplay dηf ⋆ (η− 1 2ˆ1p)e ˆıΘη ⋆ g(η+ 1 2ˆ1p) (60) Letˆω O beaGrassmann oddGaussianoftheform ˆω 0 =Ae 1 2 ηˆGη ⋆ , (61) whereAis a normalization factor and ˆG=ˆ1(G ij )withG= (G ij )being an anti-symmetricmatrix in orthogonalform. Deﬁningthe new variable zas z k =G kl p l + ˆıΘ k (62) we canintroduce theGrassmannianBargmannoddtransform as follows (ˆBf)(z)≡2 − n 4 /integraldisplay dηe 1 2 ηˆGη−ηz− 1 4 zˆG −1 z ⋆ f(η). (63) Analogously asfor the FH n thegroup productofthe OH n can be expressed as (z,t)⋆(z /prime ,t /prime ) = /parenleftbigg z+z /prime ,τ+τ /prime + 1 2Im OD (z ⋆ ˆG −1 z) /parenrightbigg ,(64) whereIm OD denotesthe oddonicimaginary part. Modiﬁcation of Grassmannian Hermite polynomials to the odd case is given by theformula ˆh i 1 ...i k k =H k e − 1 2 ηˆGη ⋆ ∂ i k ...∂ i 1 e ηˆGη ⋆ (65) whereincomparisontothefermioniccase,heretheoddexponentsenterthedeﬁ- nition.H k are normalization factors. Grassmannian Bargmann odd transform relates Grassmannian Hermite odd poly- nomials to the z I k basis of the Fock Q CO -module. Grassmannian Laguerre odd polynomialstakevaluesin complexOddonsaswell and havetheform ˆL I k = /summationdisplay m,I m ⊂I k W I /prime m I m z ⋆ I k−m z I /prime k−m , z I k−m , z I /prime k−m ∈Q CO (66) whereW I /prime m I m ∈Q CO are normalization factors. kievarwe.tex; 12/03/2001; 3:49; p.232 226 A. FRYDRYSZAK 5. FinalRemarks We have discussed some issues of the Q-module quantization of the Z 2 -graded mechanical systems, taking as the example, realizations of the same supersym- metry in the even (GSO) and odd (OGSO) superﬁeld model yielding the phase superspace with even superPoisson-bracket and anti-bracket, respectively. The formalismfortheoddsystemcanbedevelopedanalogouslytotheoneknownfor theevensystems,providedthatintheoddcaseweintroduceanoddmultiplication ofobservables. Thishas beendonehere bymeans ofthealgebra of oddons. Appendix Real Oddons. Letˆ1beanelementsuchthat,for homogeneous q s ∈Q R s 1ˆ1 =ˆ1ˆ1 2 = 1q s ˆ1 = (−1) s ˆ1q s (67) The expressionsof theform r=q+ˆ1q /prime , q,q /prime ∈Q R (68) we shall call the real oddons. They form a graded algebra Q RO . This algebra in notgradedcommutative.Despitetheextensionoftheusualproductwecandeﬁne anewoddproduct r⋆r /prime ≡r·ˆ1·r /prime (69) Theˆ1is a unit with respect to the ⋆-multiplication, having the same parity as the multiplication. Complex Oddons. Similarly we can consider the complexiﬁcation of above structure,inthesense that Q CO ≡Q C ⊕ˆıQ C and ˆı 2 =−1,ˆı·1 = ˆı,ˆı·ˆ1 =iˆı·q s = (−1) s q s ˆı, (70) whereq s ∈Q C s .Obviously ˆı·i=−ˆ1.Theproductoftwohomogeneouscomplex oddonstakesthe form z s ·z /prime r =a s a /prime r −(−1) s+1 b s b /prime r + ˆı((−1) s a s b /prime r +b s a /prime r )/negationslash= (−1) rs z /prime r ·z s ,(71) wherez r =a r +ˆıb r ∈Q CO r .Thecomponent aweshallcalltheoddonicrealpart and theb- the oddonic imaginary part. The Q CO can be considered as an algebra withthe odd ⋆product.The even mapping ∗:Q CO −→Q CO (72) suchthat z=a+ ˆıb−→z ∗ =a ∗ −b ∗ ˆı (73) kievarwe.tex; 12/03/2001; 3:49; p.233 SUPER ODDMECHANICAL SYSTEMS 227 we shall consider as oddonic conjugation. Note that we use the same symbol for theextensionof complexconjugation in the Q C . References 1. D.A. Leites, Dokl. Akad. Nauk. SSSR 236(1977),804 2. D. A. Leites, Supplement 3 in F. A. Berezin, M. A. Shubin ”The Schr ¨odinger Equation” (Kluwer Academic Publisher, Dordrecht, 1992) 3. A.Frydryszak, J.Phys. A: Math.Gen. 26(1993), 7227 4. D.V. Volkov, V. A. Soroka, Sov. J. Nucl.Phys 46(1988), 110 5. A.Frydryszak, Lett. Math. Phys. 44(1998), 89 6. M.E.Taylor, ”NoncommutativeHarmonicAnalysis” (AMSProvidence,RhodeIsland,1986) 7. A.Frydryszak, Lett. Math. Phys. 18(1989), 87 8. A.Frydryszak, Lett. Math. Phys. 26(1992), 105 9. A.Frydryszak, L. Jak ´obczyk, Lett. Math. Phys. 16(1988), 101 10. F. A. Berezin, ”The Methodof Second Quantization” (Academic Press, New York, 1966) 11. R. Finkelstein, M.Villasante, Phys. Rev. D 6(1986), 1666 12. A.Frydryszak, Lett. Math. Phys. 20(1990), 159 kievarwe.tex; 12/03/2001; 3:49; p.234 kievarwe.tex; 12/03/2001; 3:49; p.235 LOCALLY ANISOTROPIC SUPERGRAVITY AND GAUGE GRAVITY ON NONCOMMUTATIVE SPACES SERGIU VACARU ∗ Institute of Applied Physics, Academy of Sciences, Academy str. 5, Chisinˇau MD2028,Republic of Moldova IURIECHIOSA,NADEJDA VICOL Faculty of Mathematics and Informatics, gr. 33 MI, State Univer- sity of Moldova, Mateevici str. 60, Chisin ˇau MD2009, Republic of Moldova Abstract. We outline the the geometry of locally anisotropic (la) superspaces and la–supergravity. The approach is backgrounded on the method of anholonomic superframes with associated non- linear connection structure. Following the formalism of enveloping algebras and star product calculus we propose a model of gauge la–gravity on noncommutative spaces. The corresponding Seiberg–Witten maps are established which allow the deﬁnition of dynamics for a ﬁnite number of gravitational gauge ﬁeld components on noncommutativespaces. 1. Introduction Locally anisotropic supergravity was developed as a model of supergravity with anholonomic superframes and associated nonlinear connection (N–connection) structure [13]. This model contain as particular cases supersymmetric Kaluza– Klein and generalized Lagrange and/or Finsler gravities and for nontrivial curva- tures the N–connection describes splittings from higher to lower dimensions of (super)spaces andgenericanholonomiclocal anisotropies. Inordertoavoidtheproblemofformulationofgaugetheoriesonnoncommu- tativespaces[3,10,5,7]withLiealgebravaluedinﬁnitesimaltransformationsand withLiealgebravaluedgaugeﬁeldstheauthorsof[6]suggestedtouseenveloping algebrasoftheLiealgebrasforsettingthistypeofgaugetheoriesandshowedthat inspiteofthefactthatsuchenvelopingalgebrasareinﬁnite–dimensionalonecan restrict them in a way that it would be a dependence on the Lie algebra v alued ∗ [email protected], sergiu − [email protected] kievarwe.tex; 12/03/2001; 3:49; p.236 230 S. VACARU,I.CHIOSA,N.VICOL parametersandtheLiealgebravaluedgaugeﬁeldsandtheirspacetimederivatives only. A still presented drawback of noncommutative geometry and physics is that there is not yet formulated a generally accepted approach to interactions of elementary particles coupled to gravity. There are improved Connes–Lott and Chamsedine–Connes models of nocommutative geometry [2] which yielded ac- tionfunctionalstypingtogetherthegravitationalandYang–Millsinteractionsand gaugebosons the Higgssector (seealsotheapproaches [4] and[8]). In this paper we outline the geometry of locally anisotropoc supergravity and follow the method of restricted enveloping algebras [5, 6] and construct gauge gravitational theories by stating corresponding structures with semisimple or nonsemisimple Lie algebras and their extensions. We consider power series of generators for the afﬁne and non linear realized de Sitter gauge groups and compute the coefﬁcient functions of all the higher powers of the generators of the gauge group which are functions of the coefﬁcients of the ﬁrst power. Such constructions are based on the Seiberg–Witten map [10] and on the formalism of∗–product formulation of the algebra [18] when for functional objects, being functions of commuting variables, there are associated some algebraic noncom- mutativepropertiesencodedinthe ∗–product.Theconceptofgaugegravitytheory onnoncommutativespacesisintroducedinageometricmanner[7]bydeﬁningthe covariant coordinates without speaking about derivatives and this formalism was developed for quantum planes [17]. We prove the existence for noncommutative spacesofgaugemodelsofgravitywhichagreeswithusualgaugegravitytheories [14]beingequivalent,orextending,thegeneralrelativitytheory(seeworks[9,11] forlocallyisotropicspacesandcorrespondingreformulationsandgeneralizations respectively for anholonomic frames [15] and locally anisotropic (super) spaces [16])inthe limit ofcommuting spaces. 2. Locally AnisotropicSupergravity Let us consider a vector superbundle (vs–bundle) /tildewide Eover a supermanifold (s– manifold) /tildewider Mwithsurjectiveprojection π E : /tildewide E→ /tildewider M(forsimplicity,allconstruc- tions are locally trivial). The local supersymmetric coordinates (s–coordinates) on /tildewide Eand /tildewider Mare denoted respectively u= (x,y) ={u α = /parenleftBig x I ,y A /parenrightBig ,where x={x I = /parenleftBig x i ,x /hatwide i /parenrightBig }are(even,odd)coordinateson /tildewider Mandy={y A = /parenleftBig y a ,y /hatwide a /parenrightBig } are(even,odd)coordinatesinﬁbersof π E (indicesrunvaluesdeﬁnedbyevenand odddimensionsofcorrespondingsubmanifolds).Latins–indices I,J,K,L,M,... andA,B,C,D,... willbeusedrespectively forbaseand ﬁbercomponents. A nonlinear connection (N–connection) structure which deﬁnes a global decomposition of T /tildewide Einto horizontal, H /tildewide E,and vertical parts, V /tildewide E, N:T /tildewide E=H /tildewide E⊕V /tildewide E. (1) kievarwe.tex; 12/03/2001; 3:49; p.237 LA-SUPERGRAVITYAND NONCOMMUTATIVE SPACES 231 ThecoefﬁcientsofaN–connection N A I (u)determinethelocallyadapteds–frame (basis, in briefla–frame) δ α =δ/δu α = /parenleftBig δ I =δ/δx I =∂ I −N B I (u)∂ B ,∂ A /parenrightBig , (2) where∂ I =∂/∂x I ,∂ A =∂/∂y A are partials–derivatives,and theduals–frame δ α =δu α = /parenleftBig d I =δx I =dx I ,δ A =δy A =dy A +N A I (u)dx I /parenrightBig .(3) The s–frame(2)isanholonomic [δ J ,δ K }=δ J δ K −(−) |JK| δ K δ J = Ω A JK ∂ A , where|JK|=|J|·|K|is deﬁned by the parity of indices and we write (−) |JK| instead (−1) |JK| ,withanholonomycoefﬁcientscoincidingwiththeN-connection curvature Ω A JK =δ K N A J −(−) |JK| δ J N A K . The geometrical objects on /tildewide Eare given with respect to la–basis (2) and (3) or their tensor products and called ds–tensors, ds–connections (for some addi- tional linear connections), d–spinors and so on. For instance, a metric ds–tensor iswritten /tildewide g=g αβ δ α ⊗δ β =g IJ d I ⊗d J +g AB δ A ⊗δ B . (4) TheLagrangeandFinslerds–metricscanbemodeledonalocallyanisotropic superspace if vs–bundle /tildewide Eover a s–manifold /tildewider Mis substituted by the tangent s–bundleT /tildewider Mandthe coefﬁcientsof ds–metric(4) are taken respectively g IJ (u) = 1 2∂ 2 L(u ) ∂y I ∂y L andg IJ (u) = 1 2∂ 2 F 2 (u ) ∂y I ∂y L where the s–Lagrangian L:T /tildewider M→Λis a s–differentiable function on T /tildewider M,and FisaFinslers–metric functionon T /tildewider M. A linear distinguished connection D,d–connection, in sv–bundle /tildewide Eis a lin- ear connection which preserves by parallelism the horizontal (h) and vertical (v) distribution (1). Ad–connection DΓ ={Γ α βγ = /parenleftBig L, /tildewide L, /tildewide C,C /parenrightBig },isdeterminedbyitsinvariant hh-, hv-,vh-andvv–components,where D (δ K ) δ J =L I JK (u)δ I , D (δ K ) ∂ B =L A BK (u)∂ A , (5) D (∂ C ) δ J =C I JC (u)δ I , D (∂ C ) ∂ B =C A BC (u)∂ A . kievarwe.tex; 12/03/2001; 3:49; p.238 232 S. VACARU,I.CHIOSA,N.VICOL Thereisacanonicald–connection (c) Γdeﬁnedbythecoefﬁcientsofd–metric (4) andofN–connection and satisfyingthe metricitycondition D˜g= 0, (c) L I JK = 1 2g IH (δ K g HJ +δ J g HK −δ H g JK ), (c) L A BK =∂ B N A K + 1 2h AC /parenleftBig δ K H BC −(∂ B N D K )h DC −(∂ C N D K )h DB /parenrightBig , (c) C I JC = 1 2g IK ∂ C g JK , (c) C A BC = 1 2h AD (∂ C h DB +∂ B h DC −∂ D h BC ). ThetorsionT α βγ ofad–connection, T(X,Y ) = [X,DY}−[X,Y},whereX andYareds–vectorsandby [...}wedenotethes–anticommutator,isdecomposed into hv–invariant ds–torsions hT(δ K ,δ J ) =T I JK δ I , vT (δ K ,δ J ) = /tildewide T A JK δ I , hT (∂ A ,δ J ) = /tildewide P I JA δ I , vT(∂ B ,δ J ) =P A JB ∂ A , vT (∂ C ,∂ B ) =S A BC ∂ A , withcoefﬁcients T I JK =L I JK −(−) |JK| L I KJ , /tildewide T A JK =δ K N A J −(−) |KJ| δ J N A K ,(6) /tildewide P I JA =C I JA , P A JB =∂ B N A J −L A BJ , S A BC =C A BC −(−) |BC| C A CB . Theevenandoddcomponentsofds–torsions(6)canbespeciﬁedinexplicitform byusingdecompositionsofindicesintoevenandoddparts, I= (i, /hatwide i),A= (a, /hatwide a) and so on. Thecurvature R α βγτ ofad–connection, R(X,Y )Z=D [X D Y} Z−D [X,Y} Z, whereX,Y,Zare ds–vectors, splits intohv–invariant ds–torsions R(δ K ,δ J )δ H =R I HJK δ I , R(δ K ,δ J )∂ B =R A BJK ∂ A ,(7) R(∂ C ,δ K )δ J = /tildewide P I JKC δ I , R(∂ C ,δ K )∂ B =P A BKC , R(∂ C ,∂ B )δ J = /tildewide S I JBC δ I , R(∂ D ,∂ C )∂ B =S A BCD ∂ A wherethe coefﬁcients are computed R I MJK =δ [K L I |M|J} +L W MJ L I WK −(−) |KJ| L W MK L I WJ +C I KA W A JK /hatwide R A BJK =δ [K L A |B|J} +L C BJ L A CK −(−) |KJ| L C BK L A CJ +C A BC W C JK /tildewide S I JBC =∂ C C I JB −(−) |BC| ∂ B C I JC +C H JB C I HC −(−) |BC| C H JC C I HB S A BCD =∂ D C A BC −(−) |CD| ∂ C C A BD +C E BC C A ED −(−) |CD| C E BD C A EC /tildewide P I JKA =∂ A L I JK −C I JA|K +C I JB P B KA P A BKC =∂ C L A BK −C A BC|K +C A BD P D KC where,forinstance, δ [K L I |M|J} =δ K L I MJ −(−) |KJ| δ J L I MK , C I JA|K =δ K C I JA +L I MK C M IA −L M JK C I MA −L B AK C I JB . kievarwe.tex; 12/03/2001; 3:49; p.239 LA-SUPERGRAVITYAND NONCOMMUTATIVE SPACES 233 Theevenandoddcomponentsofds–curvaturesarecomputedbysplittingindices into evenandoddparts. The torsion and curvature of a d–connection Don a sv–bundle satisfy the identities /summationdisplay SC [(D X T)(Y,Z)−R(X,Y )Z+T(T(X,Y ),Z)] = 0, /summationdisplay SC [(D X R)(U,Y,Z )−R(T(X,Y ),Z)U] = 0, where /summationtext SC means supersymmetric cyclic sums over ds–vectors X,Y,ZandU, fromwhichthegeneralizedBianchiand Ricciidentitiesfollow [1-3]. The Riccids–tensor R βγ =R α βγα has hv–invariantcomponents R IJ =R K IJK , R IA =− (2) P IA =−(−) |KA| /tildewide P K IKA , (8) R AI = (1) P AI =P B AIB , R AB =S C ABC =S AB . Ifads–metric(4)isdeﬁnedon /tildewide E,wecanintroducethesupersymmetricscalar curvature /hatwide R=g αβ R αβ =R+S, whereR=g IJ R IJ andS=h AB S AB . The simplest model of locally anisotropic supergravity (la–supergravity) was constructed by postulating a variant of supersymmetric Einstein--Cartan theory on locally anisotropic superspace /tildewide E,which in invariant hv–components has the fundamental s–ﬁeldequations R IJ − 1 2(R+S−λ)g IJ =k 1 Υ IJ , (1) P AI =k 1 Υ AI , (9) S AB − 1 2(R+S−λ)h AB =k 1 Υ AB , (2) P IA =−k 1 Υ IA , and T α βγ +δ α β T τ γτ −(−) |βγ| δ α γ T τ βτ =k 2 Q α βγ , whereλis the cosmological constant, k 1,2 are respective interaction constants Υ αβ is the energy–momentum ds–tensor and Q α βγ is deﬁned by the supersym- metric spin–density. The bulk of theories of locally isotropic s–gravity are formulated as gauge supersymmetric models based on supervielbein formalism. Similar approaches to la–supergravity on vs–bundles can be developed by considering arbitrary s– framesB α (u) = /parenleftbig B I (u),B C (u) /parenrightbig adapted to the N–connection structure on a vs-bundle /tildewide E= /tildewide E m,l over s–manifold /tildewider M= /tildewider M n,k where (m,l)and(n,k)are re- spective(even,odd)dimensionsofs–manifolds.As–frame B α (u)isrelatedwith kievarwe.tex; 12/03/2001; 3:49; p.240 234 S. VACARU,I.CHIOSA,N.VICOL a standard la–frame (2) via transforms δ α =A α α (u)B α (u),where s–matrices A α α (u) = /parenleftBigg A I I 0 0A C C /parenrightBigg take values into a super Lie group GL m,l n,k (Λ) = GL(n,k,Λ)⊕GL(m,l, Λ)(onsuperspacesthegradedGrassmannalgebrawith Euclidean topology, denoted by Λ,substitutes the real and complex number ﬁelds). We denote by LN /parenleftBig /tildewide E /parenrightBig the set of all adapted to N–connection s–frames in all pointsofvs–bundle /tildewide Eandconsiderthes–bundleoflinearadapteds–frameson /tildewide E deﬁned astheprincipals–bundle LN /parenleftBig /tildewide E /parenrightBig = /parenleftBig LN /parenleftBig /tildewide E /parenrightBig ,π L :LN /parenleftBig /tildewide E /parenrightBig → /tildewide E,GL m,l n,k (Λ) /parenrightBig , for a surjective s–map π L .The canonical basis of standard distinguished s– generatorsI /hatwide α →I α β = /parenleftBigg I I J 0 0I A B /parenrightBigg for the super Lie algebra GL m,l n,k (Λ)of the structural s–group GL m,l n,k (Λ)satisfy s–commutation rules [I /hatwide α ,I /hatwide β }=f /hatwide γ /hatwide α /hatwide β I /hatwide γ . OnLN /parenleftBig /tildewide E /parenrightBig we consider thed–connection 1–form F= Γ α βγ (u)I β α δu γ , where Γ α βγ (u) =A α α A β β Γ α βγ +A α β δ γ A β β , (10) Γ α βγ arethecomponentsofcanonicalvariantofd–connection(5)andthes–matrix A β β is inverse to A α α . The curvatureBofthed–connection (10) B=δF+F∧F =R β αγτ I α β δu γ ∧δu τ (11) has the coefﬁcients R β αγτ =A α α (u)A β β (u)R β αγτ ,whereR β αγτ are deﬁned by ds–curvatures(7). Aside fromLN /parenleftBig /tildewide E /parenrightBig with vs–bundle /tildewide Eis naturally related another s–bundle, thebundleof adaptedtoN–connection afﬁnes–frames AN /parenleftBig /tildewide E /parenrightBig = /parenleftBig AN /parenleftBig /tildewide E /parenrightBig ,π A :AN /parenleftBig /tildewide E /parenrightBig → /tildewide E,AF m,l n,k (Λ) /parenrightBig , withthe afﬁne strucurals–group AF m,l n,k (Λ) =GL m,l n,k (Λ)⊙Λ n,k ⊕Λ m,l . Thed–connectionF(10)inLN /parenleftBig /tildewide E /parenrightBig inducesinalinearCartan d–connection F= (F,χ),inAN /parenleftBig /tildewide E /parenrightBig ,whereχ=e α ⊗A α α (u)δu α ,e α is the standard basis kievarwe.tex; 12/03/2001; 3:49; p.241 LA-SUPERGRAVITYAND NONCOMMUTATIVE SPACES 235 inΛ n,k ⊕Λ m,l ,and,inconsequence,thecurvature B(11)inLN /parenleftBig /tildewide E /parenrightBig inducesthe pair(curvature, torsion)B= (B,T)inAN /parenleftBig /tildewide E /parenrightBig ,where T=δχ+ [F∧γ}=T α βγ e α δu β ∧δu γ , whenT α βγ =A α α T α βγ isdeﬁned bythe coefﬁcientsof d–torsions(6) . By using the ds–metric (4) in /tildewide Eone deﬁnes the (dual for s–forms) Hodge operator∗ /tildewide g .Let the operator∗ −1 /tildewide g be inverse to∗ /tildewide g and /hatwide δ /tildewide g be the adjoint to the absolute derivation /hatwide δ(associated to the scalar product of ds–forms) speciﬁed for (r,s)–forms /hatwide δ /tildewide g = (−1) r+s ∗ −1 /tildewide g ◦ /hatwide δ◦∗ /tildewide g . The supersymmetricvariant oftheKillingform ofthe s–group AF m,l n,k (Λ)is degenerate. In order to generate a metric structure /tildewide g A in the total spaces of the s–bundle AN /parenleftBig /tildewide E /parenrightBig we use and auxiliary nondegenerate bilinear s– form which gives rise to the possibility to deﬁne the Hodge operator ∗ /tildewide g A and /hatwide δ /tildewide gA .Applying the operator of horizontal projection /hatwide Hone deﬁnes the operator /triangle.= /hatwide H◦ /hatwide δ /tildewide gA which does not depend on components of auxiliary biliniar s–form intheﬁber. Following an abstract geometric calculus, by using operators ∗ /tildewide g ,∗ /tildewide g A , /hatwide δ /tildewide g , /hatwide δ /tildewide gA and/triangleone computers /triangleB= (/triangleB,Rt+Ri), (12) wherethe one s–forms Rt= /hatwide δ /tildewide g T+∗ −1 /tildewide g [F,∗ /tildewide g T}, Ri=∗ −1 /tildewide g [χ,∗ /tildewide g B}= (−1) n+k+l+m R αβ g α /hatwide β e /hatwide β δu β areconstructedrespectivelybyusingtheds–torsions(6)andRiccids–tensors(8). Letusintroducethelocallyanisotropicsupersymmetricmatter sourceJcon- structed by using the same formulas from (12) when instead of R αβ is taken k 1 (Υ αβ − 1 2 g αβ Υ)−λ /parenleftBig g αβ − 1 2 g αβ δ τ τ /parenrightBig .Bystraightforwardcalculationswecan proof [3,4]that theYang–Millsequations /triangleB =J (13) for d–connectionF= (F,χ)in s–bundleAN /parenleftBig /tildewide E /parenrightBig ,projected on the base s–manifold, are equivalent to the Einstein equations (9) on /tildewide E.We emphasize that the equations (13) were introduced in a ”pure” geometric manner by using operators∗, /hatwide δand the horizontal projection /hatwide Hbut such gauge s–ﬁeld equations are not variational because of degeneration of the Killing s–form. To construct a kievarwe.tex; 12/03/2001; 3:49; p.242 236 S. VACARU,I.CHIOSA,N.VICOL variationalgaugelikesupersymmetricla–supergravitationalmodelispossible,for instance, by considering a minimal extension of the gauge s–group AF m,l n,k (Λ)to thedeSitters–group S m,l n,k (Λ) =SO m,l n,k (Λ),actingonthes–space Λ m,l n,k ⊕Λand formulating a nonlinearversion ofde Sittergauges–gravity. There are analyzed models of supergravity with generic local anisotropy [13] when instead of s–ﬁeld equations and constraints (9) there are considered an anholonomic generalization of the Wess–Zumino supergravity and some vari- ants induced in low energy limit from superstring theory. The N–connection s–ﬁeld allows us to model generic la–interactions with dynamics and constraints induced by nontrivial (not only via toroidal compactiﬁcations) from higher di- mensionsandthisresultsinageometricaluniﬁcationoftheso–calledgeneralized Finsler–Kaluza–Klein theories. 3. *–Products andEnvelopingAlgebras in Noncommutative Spaces For a noncommutative space the coordinates ˆu i ,(i= 1,...,N )satisfy some noncommutativerelations oftype [ˆu i ,ˆu j ] =    iθ ij , θ ij ∈IC,canonical structure; if ij k ˆu k , f ij k ∈IC,Lie structure; iC ij kl ˆu k ˆu l , C ij kl ∈IC,quantum planestructure(14) whereIC denotesthecomplex number ﬁeld. The noncommutative space is modeled as the associative algebra of IC ;this algebraisfreelygeneratedbythecoordinatesmoduloideal Rgeneratedbythere- lations (one accepts formal power series) A u =IC[[ˆu 1 ,...,ˆu N ]]/R.One restricts attention [6] to algebras having the (so–called, Poincare–Birkhoff–Witt) property that anyelement of A u is deﬁned byitscoefﬁcientfunction and viceversa, /hatwide f= ∞ /summationdisplay L=0 f i 1 ,...,i L : ˆu i 1 ...ˆu i L :when /hatwide f∼{f i }, where : ˆu i 1 ...ˆu i L :denotesthatthebasiselementssatisfysomeprescribedorder (forinstance,thenormalorder i 1 ≤i 2 ≤...≤i L ,or,anotherexample,aretotally symmetric).Thealgebraicpropertiesareallencodedintheso–calleddiamond (♦) product which isdeﬁnedby /hatwide f /hatwide g= /hatwide h∼ {f i }♦{g i }={h i }. In the mentioned approach to every function f(u) =f(u 1 ,... ,u N )of com- muting variables u 1 ,... ,u N one associates an element of algebra /hatwide fwhen the kievarwe.tex; 12/03/2001; 3:49; p.243 LA-SUPERGRAVITYAND NONCOMMUTATIVE SPACES 237 commuting variables aresubstitutedby anticommutingones, f(u) = /summationdisplay f i 1 ...i L u 1 ···u N → /hatwide f= ∞ /summationdisplay L=0 f i 1 ,...,i L : ˆu i 1 ...ˆu i L : when the♦–productleadstoabilinear ∗–product offunctions(see detailsin [7]) {f i }♦{g i }={h i }∼(f∗g) (u) =h(u). The∗–product isdeﬁnedrespectively forthecases (14) f∗g=    exp[ i 2 ∂ ∂u i θ ij ∂ ∂u /primej ]f(u)g(u /prime )| u /prime →u ,canonical str.; exp[ i 2 u k g k (i ∂ ∂u /prime ,i ∂ ∂u /prime/prime )]f(u /prime )g(u /prime/prime )| u /prime →u u /prime/prime →u ,Liestr.; q 1 2 (−u /prime ∂ ∂u/prime v ∂ ∂v +u ∂ ∂u v /prime ∂ ∂v/prime ) f(u,v)g(u /prime ,v /prime )| u /prime →u v /prime →v ,quantum plane , wherethere are consideredvalues oftype e ik n /hatwide u n e ip nl /hatwide u n =e i{k n +p n + 1 2 g n (k,p)} /hatwide u n , (15) g n (k,p) =−k i p j f ij n + 1 6k i p j (p k −k k )f ij m f mk n +..., e A e B =e A+B+ 1 2 [A,B]+ 1 12 ([A,[A,B]]+[B,[B,A]]) +... and for the coordinates on quantum (Manin) planes one holds the relation uv= qvu. A non–abelian gauge theory on a noncommutative space is given by two al- gebraicstructures,thealgebra A u andanon–abelianLiealgebra A I ofthegauge group withgenerators I 1 ,...,I S andtherelations [I s ,I p ] =if s p t I t . (16) In this case both algebras are treated on the same footing and one denotes the generatingelements ofthe bigalgebra by /hatwide u i , /hatwide z i ={ /hatwide u 1 ,..., /hatwide u N ,I 1 ,...,I S }, A z =IC[[ /hatwide u 1 ,..., /hatwide u N+S ]]/R, andthe∗–productformalismistobeappliedforthewholealgebra A z whenthere areconsideredfunctionsofthecommutingvariables u i (i,j,k,... = 1,...,N )and I s (s,p,... = 1,...,S ). For instance, in the case of a canonical structure for the space variables u i we have (F∗G)(u) =e i 2 /parenleftbig θ ij ∂ ∂u/primei ∂ ∂u/prime/primej +t s g s ( i ∂ ∂t/prime ,i ∂ ∂t/prime/prime ) /parenrightbig F /parenleftbig u /prime ,t /prime /parenrightbig G /parenleftbig u /prime/prime ,t /prime/prime /parenrightbig | u /prime →u,u /prime/prime →u t /prime →t,t /prime/prime →t . (17) kievarwe.tex; 12/03/2001; 3:49; p.244 238 S. VACARU,I.CHIOSA,N.VICOL This formalism was developed in [6] for general Lie algebras. In this paper we shall consider those cases when in the commuting limit one obtains the gauge gravityandgeneral relativity theories. 4. Enveloping Algebras for GravitationalGauge Connections To deﬁne gauge gravity theories on noncommutative space we ﬁrst introduce gauge ﬁelds as elements the algebra A u that form representation of the genera- torI–algebra for the de Sitter gauge group. For commutative spaces it is known [9, 11, 16] that an equivalent reexpression of the Einstein theory as a gauge like theory implies, for both locally isotropic and anisotropic spacetimes, the non- semisimplicity of the gauge group, which leads to a nonvariational theory in the total space of the bundle of locally adapted afﬁne frames (to this class one be- long the gauge Poincare theories; on metric–afﬁne and gauge gravity models see original results and reviews in [12]). By using auxililiary biliniear forms, instead of degenerated Killing form for the afﬁne structural group, on ﬁber spaces, the gauge models of gravity can be formulated to be variational. After projection on the base spacetime, for the so–called Cartan connection form, the Yang–Mills equationstransformsequivalentlyintotheEinsteinequationsforgeneralrelativity [9]. A variational gauge gravitational theory can be also formulated by using a minimalextensionoftheafﬁnestructuralgroup Af 3+1 (R)tothedeSittergauge groupS 10 =SO(4 + 1)acting onR 4+1 space. For simplicity, in this paper we restrict our consideration only with the even components of frames, connections and curvaturesofgaugela–supergavity outlined inprevious section. Let now consider a noncommutative space. In this case the gauge ﬁelds are elements of the algebra /hatwide ψ∈A (dS) I that form the nonlinear representation of the de Sitter Lie algebra so (η) (5)when the whole algebra is denoted A (dS) z .Under a nonlinearde Sittertransformationtheelementstransform asfollows δ /hatwide ψ=i /hatwide γ /hatwide ψ, /hatwide ψ∈A u , /hatwide γ∈A (dS) z . So, the action of the generators on /hatwide ψis deﬁned as this element is supposed to form a nonlinear representation of A (dS) I and, in consequence, δ /hatwide ψ∈A u despite /hatwide γ∈A (dS) z .Itshouldbeemphasizedthatindependentofarepresentationtheobject /hatwide γtakes values in enveloping de Sitter algebra and not in a Lie algebra as would be for commuting spaces. The same holds for the connections that we introduce (similarlyto [7]) in order todeﬁne covariant coordinates /hatwide U ν = /hatwide u v + /hatwide Γ ν , /hatwide Γ ν ∈A (dS) z . The values /hatwide U ν /hatwide ψtransforms covariantly, δ /hatwide U ν /hatwide ψ=i /hatwide γ /hatwide U ν /hatwide ψ,if and only if the connection /hatwide Γ ν satisﬁesthetransformationlawoftheenvelopingnonlinearrealized kievarwe.tex; 12/03/2001; 3:49; p.245 LA-SUPERGRAVITYAND NONCOMMUTATIVE SPACES 239 deSitteralgebra, δ /hatwide Γ ν /hatwide ψ=−i[ /hatwide u v , /hatwide γ] +i[ /hatwide γ, /hatwide Γ ν ], whereδ /hatwide Γ ν ∈ A (dS) z .The enveloping algebra–valued connection has inﬁnitely many component ﬁelds. Nevertheless, it was shown that all the component ﬁelds can be induced from a Lie algebra–valued connection by a Seiberg–Witten map ([10,5,6]and[1]for SO(n)andSp(n)).Inthissubsectionweshowthatsimilar constructions could be proposed for nonlinear realizations of de Sitter algebra when the transformationoftheconnectionisconsidered δ /hatwide Γ ν =−i[u ν , ∗ /hatwide γ] +i[ /hatwide γ, ∗ /hatwide Γ ν ]. For simplicity, we treat in more detail the canonical case with the star product (17).Theﬁrstterm inthe variation δ /hatwide Γ ν gives −i[u ν , ∗ /hatwide γ] =θ νµ ∂ ∂u µ γ. Assumingthatthevariationof /hatwide Γ ν =θ νµ Q µ startswithalineartermin θwehave δ /hatwide Γ ν =θ νµ δQ µ , δQ µ = ∂ ∂u µ γ+i[ /hatwide γ, ∗ Q µ ]. We follow the method of calculation from the papers [7, 6] and expand the star product (17)in θbutnoting a andﬁndtoﬁrst orderin θ, γ=γ 1 a I a +γ 1 a b I a I b +...,andQ µ =q 1 µ,a I a +q 2 µ,a b I a I b +...(18) whereγ 1 a andq 1 µ,a are of order zero in θandγ 1 a b andq 2 µ,a b are of second order in θ.The expansion in I b leads to an expansion in g a of the∗–product because the higher order I b –derivatives vanish. For de Sitter case as I b we take the genera- tors, see commutators (16), with the corresponding de Sitter structure constants f b c d /similarequalf α β β (in our further identiﬁcations with spacetime objects like frames and connections we shalluse Greekindices). The result of calculation of variations of (18), by using g a to the order given in(15), is δq 1 µ,a =∂γ 1 a ∂u µ −f b c a γ 1 b q 1 µ,c , δQ τ =θ µν ∂ µ γ 1 a ∂ ν q 1 τ ,b I a I b +..., δq 2 µ,a b =∂ µ γ 2 a b −θ ντ ∂ ν γ 1 a ∂ τ q 1 µ,b −2f b c a {γ 1 b q 2 µ,cd +γ 2 b d q 1 µ,c }. Nextweintroducetheobjects ε,takingthevaluesindeSitterLiealgebraand W µ ,being envelopingde Sitteralgebra valued, ε=γ 1 a I a andW µ =q 2 µ,a b I a I b kievarwe.tex; 12/03/2001; 3:49; p.246 240 S. VACARU,I.CHIOSA,N.VICOL withthe variation δW µ satisfyingtheequation [7, 6] δW µ =∂ µ (γ 2 a b I a I b )− 1 2θ τλ {∂ τ ε,∂ λ q µ }+i[ε,W µ ] +i[(γ 2 a b I a I b ),q ν ].(19) The equation(19)has thesolution(found in [7,10]) γ 2 a b = 1 2θ νµ (∂ ν γ 1 a )q 1 µ,b ,andq 2 µ,a b =− 1 2θ ντ q 1 ν ,a /parenleftBig ∂ τ q 1 µ,b +R 1 τ µ,b /parenrightBig whereR 1 τ µ,b =∂ τ q 1 µ,b −∂ µ q 1 τ ,b +f ec d q 1 τ ,e q 1 µ,e canbeidentiﬁedwiththecoefﬁcients R α βµν ofdeSitternonlineargaugegravitycurvatureifinthecommutativelimit q 1 µ,b /similarequal /parenleftBigg Γ α β l −1 0 χ α l −1 0 χ β 0 /parenrightBigg . The presentedprocedure canbe generalizedto allhigherpowersof θ[6]. 5. NoncommutativeGauge GravityCovariantDynamics The constructions from the previous section are summarized by the conclusion that the de Sitter algebra valued object ε=γ 1 a (u)I a determines all the terms in theenvelopingalgebra γ=γ 1 a I a + 1 4θ νµ ∂ ν γ 1 a q 1 µ,b /parenleftBig I a I b +I b I a /parenrightBig +... and thegaugetransformations are deﬁned by γ 1 a (u)andq 1 µ,b (u),when δ γ 1 ψ=iγ /parenleftBig γ 1 ,q 1 µ /parenrightBig ∗ψ. For de Sitter enveloping algebras one holds the general formula for compositions oftwotransformations δ γ δ ς −δ ς δ γ =δ i(ς∗γ−γ∗ς) which holds alsofor the restrictedtransformationsdeﬁned by γ 1 , δ γ 1 δ ς 1 −δ ς 1 δ γ 1 =δ i(ς 1 ∗γ 1 −γ 1 ∗ς 1 ) . Applyingthe formula(17) wecomputer [γ, ∗ ζ] =iγ 1 a ζ 1 b f a b c I c + i 2θ νµ {∂ v /parenleftBig γ 1 a ζ 1 b f a b c /parenrightBig q µ,c + /parenleftBig γ 1 a ∂ v ζ 1 b −ζ 1 a ∂ v γ 1 b /parenrightBig q µ,b f a b c + 2∂ v γ 1 a ∂ µ ζ 1 b }I d I c . Such commutatorscouldbeused fordeﬁnitionoftensors[7] /hatwide S µν = [ /hatwide U µ , /hatwide U ν ]−i /hatwide θ µν , (20) kievarwe.tex; 12/03/2001; 3:49; p.247 LA-SUPERGRAVITYAND NONCOMMUTATIVE SPACES 241 where /hatwide θ µν is respectively stated for the canonical, Lie and quantum plane structures. Underthe general enveloping algebraone holdsthe transform δ /hatwide S µν =i[ /hatwide γ, /hatwide S µν ]. For instance,thecanonicalcase ischaracterizedby S µν =iθ µτ ∂ τ Γ ν −iθ ντ ∂ τ Γ µ + Γ µ ∗Γ ν −Γ ν ∗Γ µ =θ µτ θ νλ {∂ τ Q λ −∂ λ Q τ +Q τ ∗Q λ −Q λ ∗Q τ }. Byintroducingthegravitationalgaugestrength (curvature) R τλ =∂ τ Q λ −∂ λ Q τ +Q τ ∗Q λ −Q λ ∗Q τ , (21) which could be treated as a noncommutative extension of de Sitter nonlinear gaugegravitationalcurvature onecomputers R τ λ,a =R 1 τ λ,a +θ µν {R 1 τ µ,a R 1 λν ,b − 1 2q 1 µ,a /bracketleftBig (D ν R 1 τ λ,b ) +∂ ν R 1 τ λ,b /bracketrightBig }I b , wherethe gauge gravitation covariantderivative isintroduced, (D ν R 1 τ λ,b ) =∂ ν R 1 τ λ,b +q ν ,c R 1 τ λ,d f cd b . Following thegaugetransformationlawsfor γandq 1 we ﬁnd δ γ 1 R 1 τλ =i /bracketleftBig γ, ∗ R 1 τλ /bracketrightBig withthe restrictedformof γ. Such formulas were proved in references [6, 10] for usual gauge (nongravita- tional)ﬁelds.Herewe reconsideredthemfor gravitational gaugeﬁelds. Following the nonlinear realization of de Sitter algebra and the ∗–formalism we can formulate a dynamics of noncommutative spaces. Derivatives can be in- troducedinsuchawaythatonedoesnotobtainnewrelationsforthecoordinates. Inthiscasea Leibniz rulecan bedeﬁned[6]that /hatwide ∂ µ /hatwide u ν =δ ν µ +d ντ µσ /hatwide u σ /hatwide ∂ τ where the coefﬁcients d ντ µσ =δ ν σ δ τ µ are chosen to have not new relations when /hatwide ∂ µ acts again to the right hand side. In consequence one holds the ∗–derivative formulas ∂ τ ∗f= ∂ ∂u τ f+f∗∂ τ , [∂ l , ∗ (f∗g)] = ([∂ l , ∗ f])∗g+f∗([∂ l , ∗ g]) kievarwe.tex; 12/03/2001; 3:49; p.248 242 S. VACARU,I.CHIOSA,N.VICOL and the Stokes theorem /integraltext [∂ l ,f] = /integraltext d N u[∂ l , ∗ f] = /integraltext d N u ∂ ∂u l f= 0,where, for thecanonicalstructure,theintegralisdeﬁned, /integraldisplay /hatwide f= /integraldisplay d N uf /parenleftBig u 1 ,...,u N /parenrightBig . Anactioncanbeintroducedbyusingsuchintegrals.Forinstance,foratensor oftype(20),when δ /hatwide L=i /bracketleftBig /hatwide γ, /hatwide L /bracketrightBig ,we candeﬁne agauge invariantaction W= /integraldisplay d N uTr /hatwide L, δW = 0, werethetrace hasto be takenforthegroupgenerators. Forthe nonlinear deSitter gaugegravity aproper actionis L= 1 4R τλ R τλ , whereR τλ isdeﬁnedbytheevenpartof(11).Inthiscasethedynamicofnoncom- mutativespaceisentirelyformulatedintheframeworkofquantumﬁeldtheoryof gauge ﬁelds. The method works for matter ﬁelds as well to restrictions to the generalrelativitytheory(seereferences[11,9]). 6. Acknowledgment Theﬁrstauthor(S.V.)isgratefultotheorganizersofNATOARWinKiev,where theresultsof thiswork were communicated,for kind hospitalityand support. References 1. L. Bonora, M. Schnabl, M. M. Sheikh–Jabbari and A. Tomasiello, Noncommutative SO(n) andSp(n) gauge theories, hep–th/0006091. 2. A. Connes, J. Math. Phys. 36 (1995) 6194; A. H. Chamseddine, A. Connes, Phys. Rev. Let- ters 77 (1996) 4868; L. Carminati, B. Iochum, T. Sch ¨ocer,Noncommutative Yang–Mills and noncommutative relativity: a bridgeovertroubledwater, hep–th/9706105. 3. A.Connes, M. R. Douglas and A. Schwarz, JHEP 9802(1998)003, hep–th/0001203. 4. E.Hawkins,Comm.Math.Phys.187(1997)471;gr–qc/9605068;G.Landi,N.A.Viet,K.C. Wali,Phys.Lett.B326(1994)45;hep–th/9402046;A,Sitarz,Class.Quant.Grav.11(1994) 2127; hep–th / 9401145; J. Madoreand J. Mourad, Int. J. Mod. Phys. D3 (1994) 221. 5. B. Jur ˇco andP. Schupp, Eur. Phys. J. C 14, (2000) 367, hep–th/0001032. 6. B.Jur ˇco,S.Schraml,P.ShuppandJ.Wess, Envelopingalgebravaluedgaugetransformations fornon–abelian gauge groups onnon–commutative spaces , hep–th/0006246 7. J.Madore,S.Schraml,P.SchuppandJ.Wess, Gaugetheoryonnoncommutativespaces, Eur. Phys. J.C, inpress, hep–th/0001203. 8. S. Majid, Conceptual Issues for Noncommutative gravity and algebras and ﬁnite sets, math.QA / 0006152; A. H. Chamseddine, Complexiﬁed gravity and noncommutative spaces, hep–th / 0005222; R. Kerner, Noncommutative extensions of classical theories in physics, hep–th / 0004033. kievarwe.tex; 12/03/2001; 3:49; p.249 LA-SUPERGRAVITYAND NONCOMMUTATIVE SPACES 243 9. D. A. Popov, Theor. Math. Phys. 24(1975) 347; D. A. 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Sternheimer, Deformation Quantization: Twenty Years After, math / 9809056. kievarwe.tex; 12/03/2001; 3:49; p.250 kievarwe.tex; 12/03/2001; 3:49; p.251 FINITENESS INCONVENTIONAL N= 1GUTS TATSUOKOBAYASHI ∗ Dept. ofPhys, Kyoto Univ.,Kyoto 606-8502, Japan JISUKEKUBO † Dept. ofPhysics,Kanazawa Univ.,Kanazawa920-1192, Japan MYRIAM MONDRAG ´ON ‡ Inst. de F ´ısica, UNAM, Apdo. Postal 20-364, M ´exico 01000 D.F., M´exico GEORGEZOUPANOS § PhysicsDept., Nat.Technical Univ.,GR-157 80Zografou, Athens,Greece Abstract. Finite Uniﬁed Theories (FUTs) are N= 1supersymmetric GUT’s which have the remarkable feature of being all-loop ﬁnite beyond the uniﬁcation point. They also have impressive predictive power. We present here a review of the recent developments of the softly broken sector ofN= 1FUTs.ThenewcharacteristicpredictionsofFUTsare:1)ThelightestHiggsbosonmass is predicted to be in the window 120-130 GeV, in case the LSP is neutralino, while in case the LSP is the ˜τ(which can be consistently accommodated in presence of bilinear R-parity violating terms) it can be aslight as111 GeV. 2) The s-spectrumstarts aboveseveral hundreds of GeV. 1. Introduction In recent years new frameworks have been developed aiming to provide a uniﬁed description of all interactions including gravity. Theories based on superstrings, non-commutative geometry and quantum groups, although at a different stage of development in each area, have common uniﬁcation targets and share sim- ilar hopes for exhibiting improved renormalization properties in the ultraviolet as compared to ordinary ﬁeld theories. Moreover, recent progress shows that all above theoretical endeavors could be related and thus they might be understood ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] kievarwe.tex; 12/03/2001; 3:49; p.252 246T.KOBAYASHI,J. KUBO, M.MONDRAG ´ON, G. ZOUPANOS in a uniﬁed manner too. However in spite the importance of having frameworks to discuss quantum gravity in a self consistent way, the main goal expected from a uniﬁed description of interactions by the particle physics community is to un- derstand the present day free parameters of the Standard Model (SM) in terms of a few fundamental ones, or in other words to achieve reduction of couplings at a more fundamental level. Unfortunately all the above theoretical frameworks did notofferanythingintheunderstandingofthefreeparametersoftheSM,andinthe bestcasetheyhavemanagedtoaccommodateearliertoolssuchassupersymmetry and ideas like Grand Uniﬁed Theories (GUTs) but without providing any further predictivepower intheseconstructions. Inourrecentstudies[1]-[6],[7]wehavedevelopedacomplementarystrategy in searching for a more fundamental theory possibly at the Planck scale, whose basic ingredients are GUTs and supersymmetry, but its consequences certainly go beyond the known ones. Our method consists of hunting for renormalization group invariant (RGI) relations holding below the Planck scale, which in turn are preserved down to the GUT scale. This programme, called Gauge–Yukawa uniﬁcation scheme, applied in the dimensionless couplings of supersymmetric GUTs, such as gauge and Yukawa couplings, had already noticable successes by predicting correctly, among others, the top quark mass in the ﬁnite and in the minimalN= 1supersymmetric SU(5) GUTs. An impressive aspect of the RGI relationsisthatonecanguaranteetheirvaliditytoall-ordersinperturbationtheory by studying the uniqueness of the resulting relations at one-loop, as was proven in the early days of the programme of reduction of couplings [8]. Even more remarkable is the fact that it is possible to ﬁnd RGI relations among couplings that guaranteeﬁniteness to all-orders inperturbationtheory [9, 10]. Althoughsupersymmetryseemstobeanessentialfeatureforasuccessfulreal- izationoftheaboveprogramme,itsbreakinghastobeunderstoodtoo,sinceithas the ambition to supply the SM with predictions for several of its free parameters. Indeed,thesearchforRGIrelationshasbeenextendedtothesoftsupersymmetry breaking sector (SSB) of these theories [4, 11], which involves parameters of dimension one and two. More recently a very interesting progress has been made [12]-[17]concerningtherenormalizationpropertiesoftheSSBparametersbased conceptually and technically on the work of ref. [18]. In ref. [18] the powerful supergraph method [19] for studying supersymmetric theories has been applied to the softly broken ones by using the “spurion” external space-time independent superﬁelds[20].Inthelattermethodasoftlybrokensupersymmetricgaugetheory is considered as a supersymmetric one in which the various parameters such as couplings and masses have been promoted to external superﬁelds that acquire “vacuum expectation values”. Based on this method the relations among the soft term renormalization and that of an unbroken supersymmetric theory have been derived.Inparticularthe β-functionsoftheparametersofthesoftlybrokentheory areexpressedintermsofpartialdifferentialoperatorsinvolvingthedimensionless kievarwe.tex; 12/03/2001; 3:49; p.253 FINITENESSINCONVENTIONAL N=1GUTS 247 parametersoftheunbrokentheory.Thekeypointinthestrategyofrefs.[15]-[17] in solving the set of coupled differential equations so as to be able to express all parameters in a RGI way, was to transform the partial differential operators involved to total derivative operators. This is indeed possible to be done on the RGIsurface whichisdeﬁned by thesolution ofthereduction equations. On the phenomenological side there exist some serious developments too. Previouslyanappealing“universal”setofsoftscalarmasseswasasummedinthe SSB sector of supersymmetric theories, given that apart from economy and sim- plicity (1) they are part of the constraints that preserve ﬁniteness up to two-loops [21,22],(2)theyareRGIuptotwo-loopsinmoregeneralsupersymmetricgauge theories, subject to the condition known as P= 1/3Q[11] and (3) they appear intheattractivedilatondominatedsupersymmetrybreakingsuperstringscenarios [23]. However, further studies have exhibited a number of problems all due to the restrictive nature of the “universality” assumption for the soft scalar masses. For instance (a) in ﬁnite uniﬁed theories the universality predicts that the lightest supersymmetric particle is a charged particle, namely the superpartner of the τ lepton ˜τ(b) the MSSM with universal soft scalar masses is inconsistent with the attractiveradiativeelectroweaksymmetrybreaking[24]and(c)whichistheworst of all, the universal soft scalar masses lead to charge and/or colour breaking min- ima deeper than the standard vacuum [25]. Therefore, there have been attempts to relax this constraint without loosing its attractive features. First an interesting observation was made that in N= 1Gauge–Yukawa uniﬁed theories there exists a RGI sum rule for the soft scalar masses at lower orders; at one-loop for the non-ﬁnite case [5] and at two-loops for the ﬁnite case [6]. The sum rule manages to overcome the above unpleasant phenomenological consequences. Moreover it was proven [17] that the sum rule for the soft scalar masses is RGI to all-orders for both the general as well as for the ﬁnite case. Finally the exact β-function for the soft scalar masses in the Novikov-Shifman-Vainstein-Zakharov (NSVZ) scheme [26] for the softly broken supersymmetric QCD has been obtained [17]. Armedwiththeabovetoolsandresultsweareinapositiontostudythespectrum of the full ﬁnite and minimal supersymmetric SU(5) models in terms of few free parameters with emphasis on the predictions for the masses of the lightest Higgs and LSPand onthe constraintsimposed by havinga large tanβ. 2. Reduction ofCouplingsand Finiteness in N= 1SUSY Gauge Theories A RGI relation among couplings, Φ(g 1 ,···,g N ) = 0, has to satisfy the partial differential equation (PDE) µdΦ/dµ = /summationtext N i=1 β i ∂Φ/∂g i = 0, whereβ i is theβ-function of g i . There exist ( N−1) independent Φ’s, and ﬁnding the completesetofthesesolutionsisequivalenttosolvetheso-calledreductionequa- tions (REs), β g (dg i /dg) =β i , i= 1,···,N, wheregandβ g are the primary coupling and its β-function. Using all the (N−1) Φ’s to impose RGI relations, kievarwe.tex; 12/03/2001; 3:49; p.254 248T.KOBAYASHI,J. KUBO, M.MONDRAG ´ON, G. ZOUPANOS one can in principle express all the couplings in terms of a single coupling g. Thecompletereduction,whichformallypreservesperturbativerenormalizability, can be achieved by demanding a power series solution, whose uniqueness can be investigated at the one-loop level. The completely reduced theory contains only one independent coupling with the corresponding β-function. This possibility of couplinguniﬁcationisattractive,butitcanbetoorestrictiveandhenceunrealistic. Inpracticeonemay usefewer Φ’sas RGI constraints. It is clear by examining speciﬁc examples, that the various couplings in supersymmetric theories have easily the same asymptotic behaviour. Therefore searchingforapowerseriessolutiontotheREsisjustiﬁed.Thisisnotthecasein non-supersymmetric theories. Let us then consider a chiral, anomaly free, N= 1globally supersym- metric gauge theory based on a group G with gauge coupling constant g. The superpotential ofthe theoryisgiven by W= 1 2m ij Φ i Φ j + 1 6C ijk Φ i Φ j Φ k , (1) wherem ij andC ijk aregaugeinvarianttensorsandthematterﬁeld Φ i transforms accordingto theirreduciblerepresentation R i ofthe gaugegroup G. The one-loop β-functionofthegauge coupling gis given by β (1) g = dg dt=g 3 16π 2 [ /summationdisplay i l(R i )−3C 2 (G) ], (2) wherel(R i )is the Dynkin index of R i andC 2 (G)is the quadratic Casimir of the adjointrepresentationofthegaugegroup G.Theβ-functionsof C ijk ,byvirtueof the non-renormalization theorem, are related to the anomalous dimension matrix γ j i of the matterﬁelds Φ i as β ijk C = d dtC ijk =C ijp /summationdisplay n=1 1 (16π 2 ) n γ k(n) p + (k↔i) + (k↔j).(3) At one-looplevel the γ j i are given by γ j(1) i = 1 2C ipq C jpq −2g 2 C 2 (R i )δ j i , (4) whereC 2 (R i )isthequadraticCasimiroftherepresentation R i ,andC ijk =C ∗ ijk . AsonecanseefromEqs.(2)and(4)alltheone-loop β-functionsofthetheory vanishifβ (1) g andγ j(1) i vanish, i.e. /summationdisplay i /lscript(R i ) = 3C 2 (G), 1 2C ipq C jpq = 2δ j i g 2 C 2 (R i ). (5) kievarwe.tex; 12/03/2001; 3:49; p.255 FINITENESSINCONVENTIONAL N=1GUTS 249 A very interesting result is that the conditions (5) are necessary and sufﬁcient for ﬁniteness atthe two-looplevel. The one- and two-loop ﬁniteness conditions (5) restrict considerably the pos- siblechoicesoftheirreps. R i foragivengroup GaswellastheYukawacouplings in the superpotential (1). Note in particular that the ﬁniteness conditions cannot be applied to the supersymmetric standard model (SSM), since the presence of a U(1)gauge group is incompatible with the condition (5), due to C 2 [U(1)] = 0. This naturally leads to the expectation that ﬁniteness should be attained at the grand uniﬁed level only, the SSM being just the corresponding, low-energy, effective theory. A natural question to ask is what happens at higher loop orders. There exists a very interesting theorem [9] which guarantees the vanishing of the β-functions toallordersinperturbationtheory,ifwedemandreductionofcouplings,andthat all the one-loop anomalous dimensions of the matter ﬁeld in the completely and uniquelyreduced theoryvanish identically. 3. Soft Supersymmetry Breaking-SumRuleof softscalarmasses The above described method of reducing the dimensionless couplings has been extended [4] to the soft supersymmetry breaking (SSB) dimensionful parameters ofN= 1supersymmetric theories. In addition it was found [5] that RGI SSB scalarmassesinGauge-Yukawauniﬁedmodelssatisfyauniversalsumrule.Here we will describe ﬁrst how the use of the available two-loop RG functions and the requirement of ﬁniteness of the SSB parameters up to this order leads to the soft scalar-mass sumrule[6]. Consider the superpotential given by (1) along with the Lagrangian for SSB terms −L SB = 1 6h ijk φ i φ j φ k + 1 2b ij φ i φ j + 1 2(m 2 ) j i φ ∗i φ j + 1 2Mλλ +h.c.,(6) wheretheφ i arethescalarpartsofthechiralsuperﬁelds Φ i ,λarethegauginosand Mtheir uniﬁed mass. Since we would like to consider only ﬁnite theories here, weassumethatthegaugegroupisasimplegroupandtheone-loop β-functionof thegaugecoupling gvanishes.Wealsoassumethatthereductionequationsadmit power series solutionsoftheform C ijk =g /summationdisplay n=0 ρ ijk (n) g 2n . (7) Accordingtotheﬁnitenesstheoremofref.[9],thetheoryisthenﬁnitetoallorders inperturbationtheory,if,amongothers,theone-loopanomalousdimensions γ j(1) i vanish.The one-and two-loop ﬁniteness for h ijk can be achieved by h ijk =−MC ijk +···=−Mρ ijk (0) g+O(g 5 ). (8) kievarwe.tex; 12/03/2001; 3:49; p.256 250T.KOBAYASHI,J. KUBO, M.MONDRAG ´ON, G. ZOUPANOS With the above assumptions (and a couple of minor ones [6]) we ﬁnd the followingsoft scalar-masssumrule (m 2 i +m 2 j +m 2 k )/MM † = 1 +g 2 16π 2 ∆ (1) +O(g 4 ) (9) for i, j,kwith ρ ijk (0) /negationslash= 0, where ∆ (1) isthe two-loop correction ∆ (1) =−2 /summationdisplay l [(m 2 l /MM † )−(1/3)]T(R l ), (10) which vanishes for the universal choice in accordance with the previous ﬁndings ofref.[22]. If we know higher loop β-functions explicitly, we can follow the same pro- cedure and ﬁnd higher loop RGI relations among SSB terms. However, the β-functionsofthesoftscalarmassesareexplicitlyknownonlyuptotwoloops.In order to obtain higher loop results, we need something else instead of knowledge ofexplicitβ-functions, e.g.somerelationsamong β-functions. The recent progress made using the spurion technique [19, 20] leads to the followingall-looprelationsamong SSB β-functions[12]-[16], β M = 2O /parenleftbigg β g g /parenrightbigg , (11) β ijk h =γ il h ljk +γ jl h ilk +γ kl h ijl −2γ i 1l C ljk −2γ j 1l C ilk −2γ k 1l C ijl , (12) (β m 2 ) ij = /bracketleftbigg ∆ +X ∂ ∂g /bracketrightbigg γ ij , (13) O= /parenleftbigg Mg 2 ∂ ∂g 2 −h lmn ∂ ∂C lmn /parenrightbigg , (14) ∆ = 2OO ∗ + 2|M| 2 g 2 ∂ ∂g 2 +˜C lmn ∂ ∂C lmn +˜C lmn ∂ ∂C lmn ,(15) where (γ 1 ) ij =Oγ ij ,C lmn = (C lmn ) ∗ ,and ˜C ijk = (m 2 ) il C ljk + (m 2 ) jl C ilk + (m 2 ) kl C ijl . (16) It wasalso found[16] thattherelation h ijk =−M(C ijk ) /prime ≡−MdC ijk (g ) dlng, (17) among couplings is all-loop RGI. Furthermore, using the all-loop gauge β- functionof Novikov etal.[26]givenby β NSVZ g =g 3 16π 2 /bracketleftbigg/summationtext l T(R l )(1−γ l /2)−3C(G ) 1−g 2 C(G)/8π 2 /bracketrightbigg , (18) kievarwe.tex; 12/03/2001; 3:49; p.257 FINITENESSINCONVENTIONAL N=1GUTS 251 it wasfound the all-loopRGI sumrule[17], m 2 i +m 2 j +m 2 k =|M| 2 { 1 1−g 2 C(G)/(8π 2 )dlnC ij k dlng+ 1 2d 2 lnC ij k d(lng) 2 } + /summationdisplay l m 2 l T(R l ) C(G)−8π 2 /g 2 dlnC ij k dlng. (19) In addition the exact β-function for m 2 in the NSVZ scheme has been obtained [17] fortheﬁrst time and isgiven by β NSVZ m 2 i = /bracketleftBigg |M| 2 { 1 1−g 2 C(G)/(8π 2 ) d dlng+ 1 2d 2 d(lng) 2 } + /summationdisplay l m 2 l T(R l ) C(G)−8π 2 /g 2 d dlng /bracketrightBigg γ NSVZ i . (20) 4. FiniteUniﬁedTheories Inthissectionweexaminetwoconcrete SU(5)ﬁnitemodels,wherethereduction of couplings in the dimensionless and dimensionful sector has been achieved. A predictive Gauge-Yukawa uniﬁed SU(5)model which is ﬁnite to all orders, in addition to the requirements mentioned already, should also have the following properties: 1. One-loopanomalous dimensions arediagonal,i.e., γ (1)j i ∝δ j i . 2. Three fermion generations, 5 i (i= 1,2,3), obviously should not couple to 24. Thiscanbe achieved forinstance byimposing B−Lconservation. 3. ThetwoHiggsdoubletsoftheMSSMshouldmostlybemadeoutofapairof Higgsquintet and anti-quintet,whichcoupletothe thirdgeneration. Inthefollowingwe discusstwoversions ofthe all-order ﬁnite model. A:Themodelofref. [1]. B:Aslightvariationofthemodel A,whosedifferencesfrom Awillbecomeclear inthefollowing. The superpotential whichdescribes thetwo modelstakes the form[1, 6] W= 3 /summationdisplay i=1 [ 1 2g u i 10 i 10 i H i +g d i 10 i 5 i H i ] +g u 23 10 2 10 3 H 4 (21) +g d 23 10 2 5 3 H 4 +g d 32 10 3 5 2 H 4 + 4 /summationdisplay a=1 g f a H a 24H a +g λ 3(24) 3 , whereH a andH a (a= 1,..., 4)stand forthe Higgs quintetsandanti-quintets. kievarwe.tex; 12/03/2001; 3:49; p.258 252T.KOBAYASHI,J. KUBO, M.MONDRAG ´ON, G. ZOUPANOS Thenon-degenerateandisolatedsolutionsto γ (1) i = 0forthemodels{A,B} are: (g u 1 ) 2 ={ 8 5, 8 5}g 2 ,(g d 1 ) 2 ={ 6 5, 6 5}g 2 ,(g u 2 ) 2 = (g u 3 ) 2 ={ 8 5, 4 5}g 2 ,(22) (g d 2 ) 2 = (g d 3 ) 2 ={ 6 5, 3 5}g 2 ,(g u 23 ) 2 ={0, 4 5}g 2 ,(g d 23 ) 2 = (g d 32 ) 2 ={0, 3 5}g 2 , (g λ ) 2 = 15 7g 2 ,(g f 2 ) 2 = (g f 3 ) 2 ={0, 1 2}g 2 ,(g f 1 ) 2 = 0,(g f 4 ) 2 ={1,0}g 2 . Accordingtothetheoremofref.[9]thesemodelsareﬁnitetoallorders.Afterthe reductionofcouplings thesymmetryof Wisenhanced [1, 6]. The main difference of the models AandBis that three pairs of Higgs quintets and anti-quintets couple to the 24forBso that it is not necessary to mixthemwith H 4 andH 4 inordertoachievethetriplet-doubletsplittingafterthe symmetrybreakingof SU(5). In the dimensionful sector, the sum rule gives us the following boundary conditions at the GUTscale[6]: m 2 H u + 2m 2 10 =m 2 H d +m 2 5 +m 2 10 =M 2 forA, (23) m 2 H u + 2m 2 10 =M 2 , m 2 H d −2m 2 10 =−M 2 3, m 2 5 + 3m 2 10 =4M 2 3forB, (24) where we use as free parameters m 5 ≡ m 5 3 andm 10 ≡m 10 3 for the model A, andm 10 forB, in additionto M. 5. Predictions ofLowEnergyParameters Since the gauge symmetry is spontaneously broken below M GUT , the ﬁnite- nessandGauge-Yukawauniﬁcationconditionsdonotrestricttherenormalization property at low energies, and all it remains are boundary conditions on the gauge and Yukawa couplings (22), the h=−MCrelation (8) and the soft scalar-mass sum rule (9) at M GUT , as applied in the various models. So we examine the evolution of these parameters according to their renormalization group equations at two-loop for dimensionless parameters and at one-loop for dimensionful ones withtherelevantboundaryconditions.Below M GUT theirevolutionisassumedto begovernedbytheMSSM.Wefurtherassumeauniquesupersymmetrybreaking scaleM s so thatbelow M s the SMisthecorrecteffective theory. The predictions for the top quark mass M t are∼183and∼174GeV in models AandBrespectively. Comparing these predictions with the most recent experimental value M t = (173.8±5.2)GeV, and recalling that the theoretical values forM t may suffer from a correction of less than ∼4%[7], we see that kievarwe.tex; 12/03/2001; 3:49; p.259 FINITENESSINCONVENTIONAL N=1GUTS 253 0.4 0.8 1.2 1.6 m 10 [TeV] 0.1160.1170.118 m h [TeV] Figure 8.m h as function of m 10 forM= 0.8(dashed) 1.0(solid) TeVfor theﬁnite model B. they are consistent with the experimental data. In addition the value of tanβis obtained as tanβ= 54and48 formodels AandBrespectively. In the SSB sector, besides the constraints imposed by reduction of couplings and ﬁniteness, we also look for solutions which are compatible with radiative electroweaksymmetrybreaking. Concerning the SSB sector of the ﬁnite theories AandB, besides the gaug- ino mass we have two and one more free parameters respectively, as previously mentioned. Thus, we look for the parameter space in which the lighter ˜τmass squaredm 2 ˜τ is larger than the lightest neutralino mass squared m 2 χ (which is the LSP). In the case where all the soft scalar masses are universal at the uniﬁcation scale, there is no region of M s =MbelowO(few) TeV in which m 2 ˜τ > m 2 χ is satisﬁed.Butoncetheuniversalityconditionisrelaxedthisproblemcanbesolved naturally (provided the sum rule). More speciﬁcally, using the sum rule (9) and imposing the conditions a) successful radiative electroweak symmetry breaking b)m ˜τ 2 >0and c)m ˜τ 2 > m χ 2 , we ﬁnd a comfortable parameter space for both models(although model Brequireslarge M∼1TeV). InTables1and2wepresentrepresentativeexamplesofthevaluesobtainedfor thesparticlespectraineachofthemodels.ThevalueofthelightestHiggsphysical massm h hasalreadytheone-loopradiativecorrectionsincluded,evaluatedatthe appropriatescale[27]. Finally, we calculate BR(b→sγ)[28], whose experimental value is 1× 10 −4 0.For kievarwe.tex; 12/03/2001; 3:49; p.260 254T.KOBAYASHI,J. KUBO, M.MONDRAG ´ON, G. ZOUPANOS TABLE II. A representative example of the predictions for the s-spectrum for the ﬁnite model AwithM= 1.0TeV, m 5 = 0.8TeV andm 10 = 0.6TeV . m χ =m χ 1 (T eV) 0.45m ˜b 2 (T eV) 1.76 m χ 2 (T eV) 0.84m ˜τ =m ˜τ 1 (T eV) 0.63 m χ 3 (T eV) 1.49m ˜τ 2 (T eV) 0.85 m χ 4 (T eV) 1.49m ˜ν 1 (T eV) 0.88 m χ ± 1 (T eV) 0.84m A (T eV) 0.64 m χ ± 2 (T eV) 1.49m H ± (T eV) 0.65 m ˜t 1 (T eV) 1.57m H (T eV) 0.65 m ˜t 2 (T eV) 1.77m h (T eV) 0.122 m ˜b 1 (T eV) 1.54 TABLE III. A representative example of the predictions of the s-spectrum for the ﬁnite model BwithM= 1TeV and m 10 = 0.65TeV . m χ =m χ 1 (T eV) 0.45m ˜b 2 (T eV) 1.70 m χ 2 (T eV) 0.84m ˜τ =m ˜τ 1 (T eV) 0.47 m χ 3 (T eV) 1.30m ˜τ 2 (T eV) 0.67 m χ 4 (T eV) 1.31m ˜ν 1 (T eV) 0.88 m χ ± 1 (T eV) 0.84m A (T eV) 0.73 m χ ± 2 (T eV) 1.31m H ± (T eV) 0.73 m ˜t 1 (T eV) 1.51m H (T eV) 0.73 m ˜t 2 (T eV) 1.73m h (T eV) 0.118 m ˜b 1 (T eV) 1.56 kievarwe.tex; 12/03/2001; 3:49; p.261 FINITENESSINCONVENTIONAL N=1GUTS 255 example,theminimalmodelwith M= 1TeVleadsto BR(b→sγ) = 3.8×10 −4 forµ<0. 6. Conclusions The programme of searching for exact RGI relations among dimensionless cou- plings in supersymmetric GUTs, started few years ago, has now supplemented with the derivation of similar relations involving dimensionful parameters in the SSB sector of these theories. In the earlier attempts it was possible to derive RGI relations among gauge and Yukawa couplings of supersymmetric GUTs, which couldleadeventoall-loopﬁnitenessundercertainconditions.Thesetheoretically attractive theories have been shown not only to be realistic but also to lead to a successful prediction of the top quark mass. The new theoretical developments include the existence of a RGI sum rule for the soft scalar masses in the SSB sector ofN= 1supersymmetric gauge theories exhibiting gauge-Yukawa uni- ﬁcation. The all-loop sum rule substitutes now the universal soft scalar masses and overcomes its phenomenological problems. Of particular theoretical interest is the fact that the ﬁnite uniﬁed theories, which could be made all-loop ﬁnite in the supersymmetric sector can now be made completely ﬁnite. In addition it is interesting to note that the sum rule coincides with that of a certain class of stringmodelsinwhichthemassivestringmodesareorganizedinto N= 4super- multiplets. Last but not least in ref. [17], the exact β-function for the soft scalar massesintheNSVZschemewasobtainedfortheﬁrsttime.Ontheotherhandthe above theories have a remarkable predictive power leading to testable predictions oftheirspectrumintermsofveryfewparameters.Inadditiontothepredictionof the top quark mass, which holds unchanged, the characteristic features that will judge the viability of these models in the future are 1) the lightest Higgs mass is found to be around 120 GeV and the s-spectrum starts beyond several hundreds of GeV. Therefore the next important test of Gauge-Yukawa and Finite Uniﬁed theories will be given with the measurement of the Higgs mass, for which these models show an appreciable stability, which is alarmingly close to the IR quasi ﬁxed point prediction of the MSSM for large tan β[29]. Our preliminary search in the available parameter space of the above models shows that in case we relax the requirement that the mass of the s-tau should be smaller than the neutrali- nos masses, we obtain a wider window in the prediction of the lightest Higgs mass starting from 111 GeV. This possibility has no obvious problem in case we introduce bilinear R-parity violating terms that preserve ﬁniteness. Actually, the introduction of such terms might be unavoidable given that it is a necessary ingredient of the only known mechanism to introduce neutrino masses in these models[30]. kievarwe.tex; 12/03/2001; 3:49; p.262 256T.KOBAYASHI,J. KUBO, M.MONDRAG ´ON, G. ZOUPANOS Acknowledgements It is a pleasure to thank the Organizing Committee for the very warm hospital- ity offered to one of us (G.Z.). Supported by the projects PAPIIT-125298 and ERBFMRXCT960090. References 1. KapetanakisD.,Mondrag ´on,M.andZoupanos,G.,(1993) Zeit.f.Phys. C60181;Mondrag ´on, M. andZoupanos, G. (1995) Nucl. 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B4983; Carena, M.et. al., (1995) Phys. Lett. B355209. 28. Bertolini, S., Borzumati, F.,Masiero, A. and Ridolﬁ, G. (1991) Nucl. Phys. B353591. 29. Jurˇciˇsin, M and Kazakov, D.I. (1999) Mod.Phys. Lett. A14671. 30. Hirsch,M. et.al.,(2000) Phys. Rev. D62113008. kievarwe.tex; 12/03/2001; 3:49; p.264 kievarwe.tex; 12/03/2001; 3:49; p.265 WORLDVOLUMEREALIZATIONOFAUTOMORPHISMS JOAN SIMON ∗ DepartamentofParticlePhysics,TheWeizmannInstituteofScience, 2 Herzl Street,76100 Rehovot,Israel Abstract. The relation among spacetime supersymmetry algebras and the world volume approach to string theory is reviewed. The realization of some of the automorphism transformations of these superalgebras on the world volume theory is discussed. We distinguish among linear realizations and non-local ones. The consistency ofthe latter with dualityin M/stringtheory is checked. 1. Introduction Our contribution to the NATO Advanced Research Workshop on ’NonCommu- tative Structures in Mathematics and Physics’ is devoted to the relation among supersymmetry algebras and reparametrization invariant ﬁeld theories describ- ing the low energy dynamics of branes. In particular, we shall concentrate on branes propagating in SuperPoincar ´e, and consequently, on maximally extended SuperPoincar ´ealgebras. The study of M/String theory spectrums can be done along purely alge- braic methods or ﬁeld theory ones. The algebraic approach is based on the assumption that the N= 1supersymmetry in eleven dimensions (or the cor- respondingN= 2supersymmetries in ten dimensions) is valid at any energy, so that the M-theory (string theory) spectrum must be organized into represen- tations of the SuperPoincar ´e algebra. This approach entirely characterizes BPS states, those preserving some amount of supersymmetry, thus ﬁlling in short irreducible representations of the forementioned algebra. Given a maximally extended supersymmetry algebra[1], [2] {Q α ,Q β }=−MI αβ + Γ(Z) αβ , (1) where Γ(Z) αβ stands for the traceless part of the supersymmetry anticom- mutation relations, and given any state |α >, the positivity of the matrix < ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.266 260 J. SIMON α|{Q α ,Q β }|α> 1 implies a bound on the rest mass M. When the latter is satu- rated, there is a linear combination of the supersymmetry generators annihilating the state. This means that the symmetric matrix {Q α ,Q β }has at least one zero eigenvalue (det{Q α ,Q β }= 0).Thus,generically,thesearchforsuchBPSstates isequivalent tothe resolutionof theeigenvalueproblem[4] Γ(Z)|α>=M|α> . (2) Any solution to equation (2) describes a Clifford valued BPS state |α >by its massMandtheamountofsupersymmetrypreserved (ν),whichwillgenerically be determined by some set of mutually commuting constant operators {P i }such thatP i |α >=|α >∀i. Both depend on the charges Zcarried by|α >. A partial analysis of equation (2) was done in [5], where a whole family of BPS states, called factorizable states, were classiﬁed. We refer the reader to [5] for a discussion onequation(2) andsome oftheirsolutions. Theworldvolumeapproach isbasedonbraneeffectiveactions,whicharesup- posed to describe the low energy dynamics of string theory when the string scale vanishes (α /prime →0)andgravitydecouples.Thedynamicsofbranespropagatingin SuperPoincar ´earedescribedbyreparametrizationsusy-kappainvariantﬁeldtheo- riesprovidinguswithaﬁeldtheoryrealizationoftheprevioussuperalgebras.The algebraic saturation of the BPS bound has its ﬁeld theory counterpart in the satu- ration of the Bogomolny’ type bound derived from the energy density computed onthebrane[6].Onlycertainﬁeldtheoryconﬁgurationsdosaturatesuchbounds, thesearethesocalled BPSconﬁgurations .Onewayofsystematicallylookingfor suchconﬁgurationsistheresolutionofthe kappasymmetrypreservingcondition . This method is based on the search for the subset of supersymmetry transfor- mations that leave bosonic conﬁgurations (θ= 0)invariant. Since fermions do transform inhomogeneouslyin brane effective actions, δθ=/epsilon1+ (1 + Γ κ )κ+O(θ) (3) where/epsilon1is the global supersymmetry parameter (the Killing spinor of the back- ground geometry) and κis the local kappa symmetry one, the above invariance requirementis satisﬁedwhenever [7] Γ κ /epsilon1=/epsilon1. (4) Γ κ is a spinor valued matrix being ﬁeld and background dependent. It satisﬁes Γ 2 κ =Iand tr Γ κ = 0, conditions that allow kappa symmetry to remove half of the fermionic degrees of freedom on the brane, a necessary condition to get a supersymmetric ﬁeldtheoryon thebrane,but notasufﬁcient one. 1 It is assumed that Q α satisﬁes the necessary requirements for this positivity to hold. In M- theory, the Majorana charges do certainly satisfy them. See [3], for a discussion on this point in arbitrary spacetimesignatures. kievarwe.tex; 12/03/2001; 3:49; p.267 WORLDVOLUME AUTOMORPHISMS 261 In the case of SuperPoincar ´e backgrounds, /epsilon1is a 32constant spinor. In less symmetric background superspaces, it will generically depend on the point. Solving equation(4)givesriseto 1. someconstraints ontheconﬁgurationspace f i [φ j ] = 0 2. somesupersymmetrypreserving conditions P i /epsilon1=/epsilon1∀i wheref i [φ j ]standsforsomefunctionalrelationinvolvingthedynamicalﬁeldson the brane{φ i }and their derivatives {∂φ i ,∂∂φ i ,...}. On the other hand, P i is a constant spinor valued matrix satisfying P 2 i = 1and trP i = 0. IfP i = Γ [a 1 ...a i ] equals the antisymmetrized product of gamma matrices, we shall call it single projector. Constraints 1. become BPS equations . This can be checked by computing the energydensityfunctionaloftheﬁeldtheorywhich canalwaysbe written as 2 E 2 = (E 0 +Z) 2 + /summationdisplay i /parenleftBig t i f i [φ j ] /parenrightBig 2 (5) if weare describing aBPS stateat threshold(intersectionofbranes) or as E 2 =E 2 0 +Z 2 + /summationdisplay i /parenleftBig t i f i [φ j ] /parenrightBig 2 (6) foranon-thresholdBPSstate.BothexpressionsshowtheBPSequationcharacter oftheconstraints f i [φ j ] = 0. Conditions 2. determine the amount of supersymmetry preserved (ν)and the kind of branes involved in the state due to the one to two correspondence among singlebranesandsingle projectors 3 . Thus, all in all, one gets a ﬁeld theory realization of the previous algebraic BPS states (|α >). They are indeed the same because they are characterized by the same supersymmetry projection conditions (P i )and they do have the same energy (M=E). Once the connection among brane effective actions and supersymmetry alge- brashasbeenestablished,itisnaturaltoaskabouttheextentofsuchaconnection regardingthemaximalautomorphismgroupsofSuperPoincar ´ealgebras.Inpartic- ular, theN= 1D= 11superalgebra admits a GL(32,R)automorphism group [5][8][9][10][11]. One of the ﬁrst consequences of such automorphism structure is the existence of SO(32)transformations relating ν= 1 2 non-threshold bound stateswithν= 1 2 boundstatesatthreshold,havingthesamemass 4 .Without loss 2 We have assumed the existence of a single Zcharge in the above derivation, but the exten- sion to more general conﬁgurations is straightforward. E 0 stands for the vacuum energy of the conﬁguration. 3 This is because given any single projector P i , there always exists ˜P i such thatP i ˜P i =I. So, ifP i /epsilon1=/epsilon1⇒˜P i /epsilon1=/epsilon1. 4 there exist similar phenomena for less supersymmetric BPS states, see [5]. kievarwe.tex; 12/03/2001; 3:49; p.268 262 J. SIMON ofgenerality,consider a non-threshold boundstate described by (cosβΓ 1 + sinβΓ 2 )|α> =|α> ,{Γ 1 ,Γ 2 }= 0 (7) M= /radicalBig Z 2 1 +Z 2 2 (8) where Γ i i= 1,2satisﬁesanalogouspropertiestothoseof P j andβisaconstant parameter.Therealwaysexists U β =e βΓ 2 Γ 1 /2 ∈SO(32), suchthat(7) becomes U β Γ 1 U t β |α>=|α>⇔ Γ 1 |α /prime >=|α /prime >, (9) which allows us to reinterpret it in terms of an SO(32) related BPS state |α /prime >at threshold havingthesame mass(8). Motivatedbythepreviousdiscussion,itseemsrathernaturaltolookforworld volume realizations of such automorphisms. Since the Lorentz group in eleven dimensions can be seen as a subgroup of GL(32,R), it is obvious that such subgroup will be linearly realized on the brane (before any gauge ﬁxing). This is because any brane effective action propagating in SuperPoincar ´e is manifestly (quasi-)invariant under the superisometries of the background [12]. In section 2, we will discuss a particular example of such linear realizations and the way they act on BPS conﬁgurations, showing explicitly the connection among non- thresholdandthresholdboundstatesillustratedinthealgebraicapproach.Besides this linear realizations, the analysis done in [5] shows that central charges Z’s aregenerically’rotated’amongthemselvesunderautomorphismtransformations. Since for bosonic conﬁgurations, such topological charges are given by world space integrals involving derivatives of the brane dynamical ﬁelds, one should alsoexpect,ifany,theexistenceofnon-localtransformationsleavingcertainbrane theoriesinvariant.Wereviewtheresultsof[13]concerningthatpointinsection3. Startingfromthenon-localtransformationsleavingtheD3-braneactioninvariant [14], which are the world volume realization of the S-duality automorphism for theN= 2D= 10typeIIBSuperPoincar ´ealgebra,weperformaT-dualityalong a world volume direction to get some new non-local transformations of the D2- brane in type IIA. The latter have a natural M-theory interpretation as rotations involving the world volume scalar (y)which becomes a one form (V (1) )after the world volume dualization relating both effective theories in three dimensions [15,16].Thisdualizationexplainstheoriginofsuchnon-localtransformationsin typeIIA theory. These results illustrate that part of the automorphism group is realized on the world volume ﬁeld theory, either as linear realizations or as non-local ones. It would be interesting to clarify which is the symmetry structure that is being real- ized on brane effective actions. Along the same lines, it would also be interesting to understand the existing relation among the automorphism group and U-duality groups.Asitwaspointedoutin[13],the N= 2D= 10typeIIBSuperPoincar ´e algebra admits SL(2,R)in its maximal automorphism group, the latter being kievarwe.tex; 12/03/2001; 3:49; p.269 WORLDVOLUME AUTOMORPHISMS 263 the U-duality group for type IIB superstring theory. When compactifying several dimensions and using T-duality adequately, one may suspect of deriving some relation among the corresponding U-duality group and the automorphism group ofthedimensionally reduced superalgebra. 2. Linear realizations Given any brane effective action S[φ i ], the set of dynamical ﬁelds can always be splitted into{φ i }={x m ,θ,V (p) },x m andθbeing superspace coordinates and V (p) some p-form degrees of freedom on the brane. These actions are invariant (δS[φ i ] = 0)under some set of global and local transformations. We shall con- centrate on the global ones. These include the superisometries of the background geometry, so since we are considering SuperPoincar ´e backgrounds, it certainly includestheSO (1,D−1)Lorentztransformations δθ= 1 4ω mn Γ mn θ , δx m =ω mn η np x p , δV (p) = 0.(10) Let us concentrate on M2-brane effective actions in M-theory. We are thus considering three dimensional ﬁeld theories probing eleven dimensional Super- Poincar´espace[17].Toillustratepreviousideas,weshalllookforaworldvolume soliton onan M2-branecorresponding tothenon-threshold boundstate M2 : 1 2 M2 : 2 3 M2 : 1 3 . By setting the static gauge (x µ =σ µ µ= 0,1,2)and exciting one transverse scalar (x 3 =x), one can check that the kappa symmetry preserving condition (4) issolvedby x= tanα /parenleftBig cosβσ 1 + sinβσ 2 /parenrightBig , (11) whereαandβare arbitrary constants,whenever /epsilon1satisﬁes {cosαΓ 012 + sinα(cosβΓ 023 + sinβΓ 013 )}/epsilon1=/epsilon1, (12) whichindeedcorrespondstotheforementioned ν= 1 2 non-thresholdboundstate. According to our discussion in the introduction, there must exist an SO(32) transformationrelatingsuchaconﬁgurationwitha ν= 1 2 boundstateatthreshold, corresponding in this particular case, to a single membrane lying in the 12-plane. We will explicitly check that this is indeed the case by considering the following SO(32)group element U=U α U β =e −αΓ 13 /2 e −βΓ 12 /2 . (13) kievarwe.tex; 12/03/2001; 3:49; p.270 264 J. SIMON Bycomputing its ﬁnitetransformationon thescalar coordinates,wederive ˜x 2 = cosβσ 2 + sinβσ 1 ,˜x 1 = cos β cosα σ 1 − sin β cosα σ 2 ˜x= 0 , (14) whichshowsthereisnotransversescalarexcitedintherotatedconﬁguration (˜x= 0). This is understood as having no more membranes in the conﬁguration than just the deﬁning one. This interpretation is further conﬁrmed by rewriting the supersymmetryprojection conditionin termsofthe transformed Killing spinor Γ 012 /epsilon1 /prime =/epsilon1 /prime , /epsilon1 /prime =U t /epsilon1. (15) Equation(15) describesasinglemembrane inthe 12-plane, asexpected. 3. Non-local realizations Inthissectionweshallreviewtheresultsreportedin[13].Weshallstartouranaly- sisbystudyingD3-braneeffectiveactions.Theseprovideaﬁeldtheoryrealization ofsome truncationof N= 2D= 10typeIIBSuperPoincar ´e algebra[3] {Q i ,Q j }=P + Γ M Y ij M +P + 1 3!Γ MNP /epsilon1 ij Y MNP +P + 1 5!Γ M 1 ...M 5 Y +ij M 1 ...M 5 , (16) where the central charges are given by Y ij M =δ ij Y (0) M +τ ij 1 Y (1) M +τ ij 3 Y (3) M and Y +ij M 1 ...M 5 =δ ij Y +(0) M 1 ...M 5 +τ ij 1 Y +(1) M 1 ...M 5 +τ ij 3 Y +(3) M 1 ...M 5 . If we consider an SL(2,R)transformation ˜Q i = (UQ) i ,U λ =e λiτ 2 /2 ∈ SL(2,R), the latter belongs to the type IIB automorphism group if the charges transform as ˜Z ij = /parenleftBig UZU t /parenrightBig ij . (17) Notice thatU λ ∈SO(2)subgroup of SL(2,R)which rotates /parenleftBig Y (1) M ,Y (3) M /parenrightBig and /parenleftBig Y +(1) M 1 ...M 5 ,Y +(3) M 1 ...M 5 /parenrightBig as doublets, whereas Y mnp andY +(0) m 1 ...m 5 remain invariant. This is consistent with the S-duality interpretation of U π/2 , which interchanges D-strings and fundamental strings, D5-branes and NS5-branes, while leaving D3 and KK5Bmonopolesself-dual. ThisSO(2)transformation is reminiscent of the electro-magnetic duality in four dimensions, and it was indeed proved in [14] that the off-shell transforma- tionsgiving riseto sucha rotationare given by δx m = 0, δθ = λ 2 iτ 2 θ (18) δF µν =λK µν , δK µν =−λF µν (19) kievarwe.tex; 12/03/2001; 3:49; p.271 WORLDVOLUME AUTOMORPHISMS 265 whereK µν =− 1 2 ε µνρσ ˜K ρσ5 and ˜K ρσ = 1 √ − det G∂L D 3 ∂F ρσ ,L D3 being the La- grangian density for an abelian D3-brane propagating in SuperPoincar ´e [18–21]. Itisremarkablethattheinﬁnitesimaltransformationforthefermionicﬁeldagrees with the inﬁnitesimal transformation of the supersymmetry generator. Notice that it isF=dVthe one entering in previous linear transformations (19). So, when rewritten in terms of the gauge potential V, they become non-local transformations[22]. To get a more physical understanding of these transformations, we shall eval- uate them on-shell; in particular, on Bion conﬁgurations [23, 24]. These are ν= 1/4solitons representing fundamental strings ending on the brane. As all BPS conﬁgurations, they are characterized by some BPS equations F 0a =∂ a y a= 1,2,3andsomesupersymmetryconditions Γ 0123 iτ 2 /epsilon1=/epsilon1 (20) Γ 0y τ 3 /epsilon1=/epsilon1 (21) corresponding to thearray D3 : 1 2 3 F1 : 4 . Ifwecompute K µν whenweareon-shell,weget K 0a = 0,K ab =/epsilon1 abc F 0c ,which give rise to δE a = 0andδB a =λE a , whose ﬁnite form generates an SO(2) rotation ˜E a = cosλE a ,˜B a = sinλE a , whereE a andB a correspond to the electric and magnetic ﬁelds, respectively. Thus the rotated conﬁguration is both electrically and magnetically charged: it is a dyon. This interpretation is further conﬁrmed by rewriting the supersymmetry condition (21) in terms of the transformedKilling spinor, ˜/epsilon1=U t /epsilon1 Γ 0y (cosατ 3 + sinατ 1 ) ˜/epsilon1= ˜/epsilon1, (22) which indeed describes a non-threshold bound state of fundamental strings ( τ 3 factor)andD-strings( τ 1 factor). We could have also analyzed the energy of such conﬁgurations. The starting BIonveriﬁes E BIon =E D3 +Y (3) 4 ,whereY (3) 4 = /integraltext D3 /vectorE·/vector∇yisthechargecarried by the fundamental string along the y(x 4 )direction, whereas E D3 stands for the energy of an inﬁnite planar D3-brane. After the SO(2) transformation, E dyon = E BIon =E D3 + /radicalbigg/parenleftBig ˜Y (3) 4 /parenrightBig 2 + /parenleftBig ˜Y (1) 4 /parenrightBig 2 , where ˜Y (3) 4 = /integraltext D3 cosλ/vectorE·/vector∇yand ˜Y (1) 4 = /integraltext D3 sinλ/vectorB·/vector∇y. In this way, we check that the ﬁeld theory SO(2) 5 ε µνρσ denotes the covariantly constant antisymmetric tensor with indices raised and lowered byG µν . kievarwe.tex; 12/03/2001; 3:49; p.272 266 J. SIMON transformations(18-19)indeedrotatethechargesofthespacetimesupersymmetry algebra. In the following we shall check the consistency of the previous set of trans- formations with the known web of dualities in M/string theory. The ﬁrst step will be to perform a longitudinal T-duality transformation, that is, along one of the D3-brane world volume directions, to study the corresponding symmetry struc- ture in type IIA. Finally, the M-theory origin for such type IIA symmetry will be explained. As before, these checks can be studied either from an algebraic perspectiveor fromaﬁeldtheory one. The realization of T-duality at the level of superalgebras is known to be a mappingrelatingthesupersymmetrychargesasfollows Q + =Q 2 , Q − = Γ s Q 1 , (23) whereQ ± are the type IIA supercharges and sstands for the spacelike direction along we perform the transformation. Such a mapping, does change the chirality of one of the generators and induces some transformation on the charges Z’s [3] which agrees with the known T-duality rules among BPS single branes. In this way, the previous U λ automorphism can be rewritten as U s =e λ/2 Γ s Γ 11 , which indeed belongs to SO (32), the subgroup of type IIA automorphisms preserving energy. The latter statement can be straightforwardly derived from the M-algebra analysis done in [5]. Notice that Γ 11 is the ten dimensional chirality operator, so thatU s can not be interpreted as an spacetime rotation. This transformation will “rotate” several doublets of charges appearing in type IIA, while keeping some others invariant. In particular, charges Z sm andZ m corresponding to D2-branes and fundamentalstringswillforman SO (2)doubletunder U s transformations. Moving back to the world volume approach, the analysis done in [25, 26] will be used to derive the symmetry structure inherited on the D2-brane after performing the longitudinal T-duality. Since δx m = 0in (18), there will be no compensating diffeomorphism transformation coming from the partial gauge ﬁx- ing locally identifying (x s =ρ)one world volume direction (ρ)with one target space direction (x s ). It is then straightforward to derive a set of non-local trans- formations leaving the D2-brane invariant, just by double dimensional reduction of(18-19) δθ= λ 2 Γ m Γ 11 θ, (24) δK m ˆµˆν =−λ m F ˆµˆν , δF ˆµˆν =λ m K m ˆµˆν (25) δK ˆµρ =−λ m ∂ ˆµ ˜x m , δ∂ ˆµ ˜x m =λ m K ˆµρ (26) whereK m ˆµˆν andK ˆµρ wherecomputed explicitly in[13]. Notice that whereas in type IIB there was a single transformation (λ), in type IIA we have a set of them (λ m ). This enhancement of symmetry is typical of T- duality on symmetric backgrounds. The performance of T-duality is manifestly kievarwe.tex; 12/03/2001; 3:49; p.273 WORLDVOLUME AUTOMORPHISMS 267 non-covariant, but in the limit R→∞, the isometries of the background allow us to recover target space covariance. A much more algebraic way to reach the sameconclusionistocomputethecommutatorofarotation (ω)withourprevious non-localtransformation (λ s ) [δ ω ,δ λ s ] =δ λ ˜s ˜s/negationslash=s, (27) which generates all the forementioned transformations. Another difference be- tween this set of transformations and type IIB ones, is that bosonic matter ﬁelds do transform (δx m /negationslash= 0), its origin being the component of the original gauge ﬁeld(V ρ )along whichwe performthe T-duality. Just as for the D3-brane case, we shall analyze the behaviour of some partic- ular BPS conﬁguration under these new transformations. We shall consider the T-dualconﬁguration ofa typeIIBdyon.Thisisgiven by the array D2 : 1 2 F1 : 4 D2 : 3 4 . This supersymmetric conﬁguration is describedby theBPS equations E ˆa = cosα∂ ˆa y (28) /epsilon1 ˆaˆb ∂ ˆb ˜x 3 = sinαδ ˆaˆb ∂ ˆb yˆa,ˆb= 1,2 (29) and supersymmetryprojection conditions Γ 012 /epsilon1=/epsilon1 (30) (cosαΓ 0y Γ 11 + sinαΓ 03y )/epsilon1=/epsilon1. (31) The further condition F 12 = 0states that there are no D0-branes being described by our conﬁguration as can be seen from inspection of equations (30-31). Notice that whenα= 0, we recover the usual BIon describing a fundamental string ending on the D2-brane, whereas for α= π 2 , we recover the Cauchy-Riemann equations describing the intersection of two D2-branes at a point, D2⊥D2(0). Both conﬁgurations are related to each other by application of transformations (25) and (26).Computing themwhen (28)-(29) aresatisﬁedwe get δ/vectorE=−λ⋆∇˜x 3 , δ /parenleftBig ⋆∇˜x 3 /parenrightBig =λ/vectorE, (32) where we are using the standard two dimensional calculus notation, that is, /vector∇= (∂ 1 ,∂ 2 )and⋆/vector∇= (∂ 2 ,−∂ 1 ).Itsﬁnitetransformation is /vectorE /prime = cos (α+λ)/vector∇y , ⋆/vector∇˜x /prime3 = sin (α+λ)/vector∇y (33) kievarwe.tex; 12/03/2001; 3:49; p.274 268 J. SIMON Thus, as expected, by ﬁne tuning the global parameter λ, we interpolate between BIon conﬁgurationsand D2⊥D2(0)intersections. The SO(2) rotation described by (32) ﬁts with the supersymmetry algebra picture. In this case, the charge carried by the fundamental string is given by the worldspace integral Z y = /integraltext D2 /vectorE·/vector∇y, whereas the charge carried by the second D2-brane admits the ﬁeld theory realization Z 3y = /integraltext D2 ⋆/vector∇x 3 ·/vector∇y. Thus we see thatZ y ,Z 3y are indeed rotated under (32) transformations, as the pure algebraic digressionwassuggesting to us. We shall conclude with the M-theory interpretation of the latter set of trans- formations. Since the eleven dimensional supersymmetry generator decomposes asQ=Q + +Q − , it is pretty clear that the previous type IIA automorphism transformations become rotations in eleven dimensions, and as such, they should belinearlyrealizedonthemembraneeffectiveactionasin(10).Itisactuallyquite simpletounderstandtherelationamongtheselineartransformationsandthenon- localonesfoundintheD2-braneaction.Asitisknown[15,16],theworldvolume dualization of a scalar in three dimensions gives rise to a one form. When doing suchadualizationonthemembraneaction,therelationamongtheirﬁeldstrengths is given by∂ ˆµ y=K ˆµρ . Thus, linear transformations among eleven dimensional scalar ﬁelds generate linear transformations among K ˆµρ and∂ ˆµ x m . The above relationexplainstheoriginofthenon-localsymmetriesintypeIIA.Furthermore, it matcheswiththeenhancementofsymmetry derived previously fromT-duality. We shall conclude by analyzing the uplifted conﬁguration corresponding to thetypeIIAone discussedabove.Thisisdescribed by the array M2 : 1 2 M2 : 4 5 M2 : 3 4 . Setting the static gauge x µ =σ µ µ= 0,1,2and exciting three transverse scalars x i i= 3,4,5one can check that a solution to the kappa symmetry preserving conditionisfound whenthe followingBPSequations are satisﬁed cosα/vector∇x 4 =⋆/vector∇x 5 ,sinα/vector∇x 4 =⋆/vector∇x 3 , (34) whenever/epsilon1satisﬁes Γ 012 /epsilon1=/epsilon1 (35) (cosαΓ 045 + sinαΓ 034 )/epsilon1=/epsilon1. (36) Notice that (34) interpolate among M2⊥M2(0)conﬁgurations in deﬁnite directionsfor α= 0, π 2 . It is straightforward to check that the rotation in the 35-plane generated by U=e αΓ 35 /2 relates the previous conﬁguration with one in which ˜x 3 has a con- stant value, and is no longer excited. Such a conﬁguration corresponds to two kievarwe.tex; 12/03/2001; 3:49; p.275 WORLDVOLUME AUTOMORPHISMS 269 membranes intersecting at a point. This interpretation can also be checked by rewriting equation(36) interms of thetransformedKillingspinor /epsilon1 /prime =U t /epsilon1 Γ 045 /epsilon1 /prime =/epsilon1 /prime , (37) which indeed corresponds to a membrane along 45-plane, while equation (35) is not modiﬁed (Γ 012 /epsilon1 /prime =/epsilon1 /prime ). Furthermore, all previous results on the D2-brane can be easily recovered from M-theory, this being the last check of consistency betweenthepresentednon-localtransformationsanddualitiesinM/stringtheory. Acknowledgements JSwassupportedbyafellowshipfromComissionatperaUniversitatsiRecercade laGeneralitatdeCatalunyaandispresentlybeingsupportedbyafellowshipfrom the Feinberg Graduate School. This work was supported in part by AEN98-0431 (CICYT), GC1998SGR (CIRIT). J.S.thanksNATO for partial support. References 1. P. K. Townsend, M-theory from its superalgebra , in ’Strings, branes and dualities’, Carg `ese 1997, ed. L. Baulieu et al.,Kluwer Academic Publ. 1999, p.141, hep-th/9712004. 2. P. K. Townsend, M(embrane) theory on T 9 , Nucl. Phys. Proc. Suppl. 67(1998) 88-92, hep-th/9708034. 3. E. Bergshoeff and A. van Proeyen, The many faces of OSp(1—32) Class. Quant. Grav. 17 (2000)3277-3304, hep-th/0003261. 4. N. A. Obers and B. Pioline, U-duality and M-theory Phys. Rep. 318 (1999) 113-225, hep- th/9809039. 5. J. Molins and J. Sim ´on, BPS states and Automorphisms , to appear in Phys. Rev. D. hep- th/0007253. 6. J. Gauntlett, J. Gomis and P. K. 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B585(2000) 219-252,hep-th/0003211. kievarwe.tex; 12/03/2001; 3:49; p.277 SOMEMETRICSON THEMANINPLANE GAETANO FIORE ∗ Dip. diMatematicae Applicazioni,Fac.di Ingegneria Universit `adi Napoli,V.Claudio 21, 80125 Napoli MARCO MACEDA † Laboratoirede Physique Th ´eorique etHautes Energies Universit ´e de Paris-Sud,B ˆatiment211, F-91405 Orsay JOHN MADORE ‡ Laboratoirede Physique Th ´eorique etHautes Energies Universit ´e de Paris-Sud,B ˆatiment211, F-91405 Orsay 1. Introduction and notation LetAbe a∗-algebra with differential calculus Ω 1 (A)[1] and suppose that it has a frame [2], a set of 1-forms θ i dual to a set of inner derivations e i =adλ i and which thereforecommuteswith theelementsof thealgebra: θ i f=fθ i . (1) The differential calculus will be real [4] if the λ i are anti-hermitian. Using the framewecanset df=e i fθ i (2) fromwhichitfollowsthat the module structureof Ω 1 (A)is given by fdg= (fe i g)θ i , dgf = (e i g)fθ i . ∗ [email protected] † [email protected] ‡ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.278 272 G. FIORE,M.MACEDA,J.MADORE If a frame exists the module Ω 1 (A)is free of rank nas a left or right module. It canthereforebe identiﬁed with thedirectsum Ω 1 (A) = n /circleplusdisplay 1 A (3) ofncopies ofA. In this representation θ i is given by the element of the direct sumwiththeunitinthe i-thpositionandzeroelsewhere.Weshallrefertothein- tegernasthedimensionofthegeometry.Usingtheframeformalismweconsider some possible metrics on the Manin plane. We require that the metric be real and symmetric. In practice this means that we use the freedom of noncommutative geometry to impose a different ‘ σ-symmetry’, which is chosen so that a complex metric is hermitian and an un-symmetric metric is σ-symmetric. The notion of reality and symmetry are changed so that the deﬁnition of hermitian does not change.Werefertoalongerarticle[3]formoredetailsaswellasforacomparison withotherdeﬁnitionsof metrics. Letπbe theproduct in Ω ∗ (A)andset π(θ i ⊗θ j ) =P ijkl θ k ⊗θ l , P ijkl ∈Z(A). Sinceπis aprojectionwehave P ijmn P mnkl =P ijkl (4) and theproduct θ i θ j satisﬁes θ i θ j =P ijkl θ k θ l . (5) If theθ i anti-commutethen P ijkl = 1 2(δ i k δ j l −δ j k δ i l ). (6) Sincethe exterior derivative of θ i is a2-formitcannecessarilybe written as dθ i =− 1 2C ijk θ j θ k . where, because of (5), the structure elements can be chosen to satisfy the constraints C ijk P jklm =C ilm . Fromthegenerators θ i wecanconstructa1-form θ=−λ i θ i (7) inΩ 1 (A)whichplays therole[1] ofaDirac operator: df=−[θ,f]. kievarwe.tex; 12/03/2001; 3:49; p.279 METRICSON THEMANIN PLANE 273 Fromtheidentity d 2 = 0one ﬁndsthat d(θf−fθ) = [dθ,f] + [θ,[θ,f]] = [dθ+θ 2 ,f] = 0. It followsthat ifwe write dθ+θ 2 =− 1 2K ij θ i θ j (8) thecoefﬁcients K ij mustlieinZ(A).Againfrom(5)theycanbechosentosatisfy theconstraints K jk P jklm =K lm . It will alsobeconvenient tointroducethe quantities C ijkl =δ i k δ j l −2P ijkl . (9) Then from(4) weﬁndthat C ijkl C klmn =δ i m δ j n . (10) Fromthecondition d 2 = 0itcanbe shown that 2P ijkl λ K λ l −F ikl λ i −K ij = 0 for somearrayofnumbers F ijk . Weintroduceaﬂip σ: Ω 1 (A)⊗ A Ω 1 (A) σ −→Ω 1 (A)⊗ A Ω 1 (A). (11) Intermsoftheframe it isgiven by S ijkl ∈Z(A)deﬁned by σ(θ i ⊗θ j ) =S ijkl θ k ⊗θ l and which must satisfy theconstraint (S jikl ) ∗ S lkmn =δ i m δ j n . (12) We useσto imposethereality condition S ijkl g kl = (g ji ) ∗ (13) on the metric. This is a combination of a ‘twisted’ symmetry condition and the ordinary condition of hermiticity on a complex matrix. A covariant derivative on the module Ω 1 (A)must satisfy both a left and a right Leibniz rule. We use the ordinaryleftLeibnizrule anddeﬁne therightLeibniz rule as D(ξf) =σ(ξ⊗df) + (Dξ)f (14) kievarwe.tex; 12/03/2001; 3:49; p.280 274 G. FIORE,M.MACEDA,J.MADORE for arbitrary f∈Aandξ∈Ω 1 (A). Usingσone can also impose [5] a reality conditionon thecurvature. Foreverydifferentialcalculusandﬂiponecanconstructthelinearconnection ω ijk =λ l (S iljk −δ l j δ i k ). (15) The connection1-formisgiven by ω ik =λ l S iljk θ j +δ i k θ. (16) WhenF ijk = 0thecurvatureofthecovariantderivative Ddeﬁnedin(15)canbe readilycalculated. Oneﬁndsthe expression 1 2R i jkl =S imrn S npsj P rskl λ m λ p − 1 2δ i j K kl . This can also bewritten intheform 1 2R i jkl =−S imrn S npsj S rsuv P uvkl λ m λ p − 1 2δ i j K kl . The relation(18) suggeststhat wedeﬁne a Riccimapby the action Ric(θ i ) = 1 2R ik θ k , R ik =R ijkl g lj on theframe. In complete analogy with the commutative case a metric gcan be deﬁned as anA-bilinear,nondegeneratemap[6] Ω 1 (A)⊗ A Ω 1 (A) g −→A (17) and as such it can [7] be used to deﬁne a ‘distance’ between ‘points’. It is impor- tant to notice here that the bilinearity is an alternative way of expressing locality. Inordinarydifferentialgeometryif ξandηare1-formsthenthevalueof g(ξ⊗η) at a given point depends only on the values of ξandηat that point. Bilinearity is anexactexpressionofthisfact.Ingeneralthealgebraintroducesacertainamount of non-locality via the commutation relations and it is important to assure that all geometric quantities be just that nonlocal and not more. Without the bilinearity condition it is not possible to distinguish for example in ordinary space-time a metric which assigns a function to a vector ﬁeld in such a way that the value at a given point depends only on the vector at that point from one which is some sort ofconvolution overthe entire manifold. Wedeﬁne framecomponentsof themetricby g ij =g(θ i ⊗θ j ). kievarwe.tex; 12/03/2001; 3:49; p.281 METRICSON THEMANIN PLANE 275 Theylienecessarilyinthecenter Z(A)ofthealgebra.Theconditionthat(15)be metric-compatiblecan bewrittenas S imln g np S jkmp =g ij δ k l . (18) One can understand this odd condition by introducing a ‘covariant derivative’ D i X j ofa constant‘vector’by theformula D i X j =ω jik X k . The covariant derivative D i (X j Y k )of the product of two such ‘vectors’ must be deﬁned as D i (X j Y k ) =D i X j Y k +S jlim X m D l Y k since there is a ‘ﬂip’ as the index on the derivation crosses the index on the ﬁrst ‘vector’.Thecondition (18) becomesthen simply D i g jk = 0. We shallrequire thatthemetric be symmetric inthe sense g◦π= 0 (19) that itannihilates the 2-forms.Weshall imposealsothe condition π◦(σ+ 1) = 0 (20) that the antisymmetric part of a symmetric tensor vanish. This can be considered as a condition on the product or on the ﬂip. In ordinary geometry it is the deﬁ- nition ofπ; a 2-form can be considered as an antisymmetric tensor. Because of this condition the torsion is a bilinear map [6]. The most general solution can be writtenintheform 1 +σ= (1−π)◦τ (21) whereτis arbitrary. Supposethat τis invertible.Then becauseof the identity 1 =π+ (1 +σ)◦τ −1 one can identify the second term on the right-hand side as the projection onto the symmetric part of the tensor product. The choice τ= 2yields the value σ= 1−2π.Ifτisnotinvertiblethentherearisesthepossibilitythatpartofthetensor product isneither symmetric norantisymmetric. Itissometimes convenient towrite themetricas asum g ij =g ij S +g ij A kievarwe.tex; 12/03/2001; 3:49; p.282 276 G. FIORE,M.MACEDA,J.MADORE of a symmetric and an antisymmetric part (in the usual sense of the word) The inverse matrix wewriteasasum g ij =η ij +B ij of a symmetric and an antisymmetric term. We shall choose as normalization whenpossibletheconditionthat η ij bethestandardMinkowskioreuclideanform. 2. TheWess-Zumino calculus The extended quantum plane is the ∗-algebraAgenerated by the hermitian elementsuandvwith theirinverses u −1 andv −1 and therelation uv=qvu, q =e iα (22) as well astheusualrelations between inverses. Wedeﬁne, for q 4 /negationslash= 1, λ 1 =q 4 q 4 −1u −2 v 2 , λ 2 =−q 2 q 4 −1u −2 . The important fact is that the λ a are singular in the limit q→1and that they are anti-hermitianif qisofunit modulus.Weﬁnd for q 2 /negationslash=−1 e 1 u=−q 2 (q 2 + 1)u −1 v 2 , e 1 v=−q 4 q 2 + 1u −2 v 3 , e 2 u= 0, e 2 v=q 2 q 2 + 1u −2 v.(23) Thesederivationsareagainextendedtoarbitrarypolynomialsinthegeneratorsby theLeibnizrule. Using themweﬁnd du=−q 2 (q 2 + 1)u −1 v 2 θ 1 , dv =−q 2 q 2 + 1u −2 v(q 2 v 2 θ 1 −θ 2 )(24) and solvingfor the θ i weobtain θ 1 =−q 2 (q 2 + 1)uv −2 du, θ 2 =−(q 2 + 1)u(uv −1 dv−du). The module structure which follows from the condition (1) that the θ i commute withthe elementsofthe algebraisgiven by[8] udu=q 2 duu, udv =qdvu + (q 2 −1)duv, vdu=qduv, vdv =q 2 dvv.(25) One can show that they are invariant under the coaction of the quantum group SL q (2,C).This invariance was encodedinthechoice of λ a . kievarwe.tex; 12/03/2001; 3:49; p.283 METRICSON THEMANIN PLANE 277 Considerthechange of generators deﬁnedby u= ˜u −2 , v = ˜q 2 ˜u −2 ˜v 2 . Ifonesetsalso q= ˜q −4 thenoneﬁndsthattheWess-Zuminorelations(25)written usingthegenerators ˜uand˜vbecome udu=qduu, udv =qdvu, vdu=q −1 duv, vdv =q −1 dvv.(26) What we have done in fact is use the λ a as generators of the algebra and the differential calculus; otherwise nothing has been changed. Properly renormalized thenwehave λ 1 =q 1/ 2 q−1v, λ 2 =−q 1/ 2 q−1u. and solvingfor the θ i one obtains θ 1 =−q −1/2 (u −1 v) −1 d(u −1 ), θ 2 =q 1/2 (u −1 v)d(v −1 ). It followsthat the volumeelementisan exactform: θ 1 θ 2 =−d(u −1 )d(v −1 ). Thisformulahasbeenobtainedbyastraight-forwardchangeofgeneratorsand,in- dependentoftheperhapsnot-too-convincingargumentsofthefollowingsections, suggests that u −1 andv −1 are light-cone coordinates in the commutative limit. Theframeissingularalongthelightconethroughtheorigin.Ifinarepresentation one forces the original ˜uand˜vto be hermitian then the uandvmust be positive operators. One concludes then that |t|>|x|andxmust therefore be a bounded operator. The structure ofthe differentialalgebra isgivenby therelations (θ 1 ) 2 = 0, (θ 2 ) 2 = 0, θ 1 θ 2 +qθ 2 θ 1 = 0. This can be written in the form (5) with C 1221 =qandC 2112 =q −1 . The reality ofthedifferentialimplies thatthestructureelementsmustsatisfy theconditions ((C ijk ) ∗ +C ijk )P jklm = 0 fromwhichfollows that (C i21 ) ∗ =−C i12 =q −1 C i21 , (C i12 ) ∗ =−C i21 =qC i12 . aregivenby C 112 = (q −1 −1)λ 2 , C 212 = (q −1 −1)λ 1 . kievarwe.tex; 12/03/2001; 3:49; p.284 278 G. FIORE,M.MACEDA,J.MADORE Withthe changeof generators t= 1 √ 2(u −1 −v −1 ), x = 1 √ 2(u −1 +v −1 ). (27) thecommutationrelation canbewrittenas [t,x] =−itan(α/2)(t 2 −x 2 ). 3. Themetricsandtheir connections With our index conventions the metric is written as g ij = (g 1 ,g 2 ,g 3 ,g 4 )and so thecondition(18) canbe writtenin thematrixform   S 11 S 12 S 13 S 14 S 21 S 22 S 23 S 24 S 31 S 32 S 33 S 34 S 41 S 42 S 43 S 44  /parenleftBig S (g) /parenrightBig =   g 1 0g 3 0 0g 1 0g 3 g 2 0g 4 0 0g 2 0g 4   (28) wherewehaveintroducedthematrix S (g) deﬁned by S (g) =   S 11 g 1 +S 12 g 3 ··· ···S 33 g 1 +S 34 g 3 S 11 g 2 +S 12 g 4 ··· ···S 33 g 2 +S 34 g 4 S 21 g 1 +S 22 g 3 ··· ···S 43 g 1 +S 44 g 3 S 21 g 2 +S 22 g 4 ··· ···S 43 g 2 +S 44 g 4   .(29) If weintroduce thematrix P= 1 2   0 0 0 0 0 1−q0 0−q −1 1 0 0 0 0 0   (30) offrame componentsfor πthen thecondition(19) isequivalent to the relation g 2 =qg 3 . (31) The consistencycondition(20) isequivalent totheconditions S 13 =qS 12 , S 23 =q(S 22 + 1), S 33 =qS 32 −1, S 43 =qS 42 . (32) The equations to be solved then are Equations (28), (31) and (32). We are especially interested in real solutions, which satisfy therefore also (13). We have foundthatthereareseveraltypesofsolutions[3],fourofwhichweshalldescribe inthefollowingsubsections.Onecanshowthattherearenosolutionswith τ= 2. kievarwe.tex; 12/03/2001; 3:49; p.285 METRICSON THEMANIN PLANE 279 Acompleteclassiﬁcationhasbeengiven[9]ofthesolutionstothebraidequation aswell[10,11]asofthosewhichsatisfyaweakermodiﬁedequation.Inanycase to within four arbitrary constants we can write the coefﬁcients of the metric with respecttothebasis d˜uandd˜v.Ifweintroducethecomponents ˜g ij =g(d˜u i ⊗d˜u j ) thenweﬁndfrom(24) thatinthe limit q→1 ˜g ij = 1 4˜u −4 ˜v 4 /parenleftBigg g 1 ˜u 2 ˜u(g 2 ˜v+g 3 ˜v −1 ) ˜u(g 2 ˜v+g 3 ˜v −1 )g 2 ˜v 2 −2g 3 +g 4 ˜v −2 /parenrightBigg . The line element is determined by the inverse of this matrix. A metric g /prime deﬁned by setting ˜g /primeij = /parenleftBigg 1 0 0 1 /parenrightBigg necessarily then cannotbe bilinear. 3.1. SOLUTION I AfamilyofsolutionscanbefoundwithaMinkowski-signaturemetric.Theseare themostinterestingsolutions.Withtheconvenientnormalizationofthemetricso thatg 3 =q −1/2 the ﬂipis givenby the matrix S=   q−q −1/2 (q−1)g 1 −q 1/2 (q−1)g 1 q −1 (q+ 1) −1 (q−1)(q 2 + 1) 0 0 q−q −1/2 (q−1)g 1 0q −1 0 q −3/2 (q−1)g 1 0 0 0 q −1   . It tends to the ordinary ﬂip as q→1and forg 1 = 0is a solution to the braid equation.The correspondingmetricgivenby g ij = /parenleftBigg g 1 q 1/2 q −1/2 0 /parenrightBigg . (33) From (31) one sees that it is σ-symmetric for all g 1 and hermitian if g 1 = 0. In thiscaseσisgivenby S=   q0 0 0 0 0q0 0q −1 0 0 0 0 0q −1   . (34) Theσandπarerelatedas in(21)with(usingthesame conventions) T=   1 +q0 0 0 0 2 0 0 0 0 2 0 0 0 0 1 + q −1   . (35) kievarwe.tex; 12/03/2001; 3:49; p.286 280 G. FIORE,M.MACEDA,J.MADORE The fact that Tis not proportional to the identity is due to the fact that the map (1 +σ)/2is not a projector and that we would like it to act as such and be the complementary to π. The metric is of indeﬁnite signature and in ‘light-cone’ coordinates. Ifwe use theexpression q=e iα weﬁnd that g ij S = cos( α 2) /parenleftbigg 0 1 1 0 /parenrightbigg , g ij A =isin( α 2) /parenleftbigg 0 1 −1 0 /parenrightbigg .(36) The inversemetric componentsare deﬁnedbythe equation g ij g jk =δ k i . This matrix also can be split. If we rescale so that the symmetric part is of the standard form we ﬁnd η ij = /parenleftbigg 0 1 1 0 /parenrightbigg , B ij =itan( α 2) /parenleftbigg 0 1 −1 0 /parenrightbigg . The metric connection has vanishing curvature. The linear connection (15) is givenby ω ij = (1−q) /parenleftBigg 1 0 0−q −1 /parenrightBigg θ. Because oftheidentities dθ= 0, θ 2 = 0 the curvature vanishes; with the choice (34) of ﬂip the quantum plane is ﬂat. In thecommutative limit thelineelementisgivenby ds 2 =g ij θ i ⊗ S θ b = 2θ 1 ⊗ S θ 2 =d(u −1 )⊗ S d(v −1 ) =dt 2 −dx 2 . The subscript Sindicatesa symmetrized tensorproduct. 3.2. SOLUTION II Afamilyofsolutionsdeﬁnedbyﬂipswhicharenotsolutionstothebraidequation isgiven by S=   −q 2 0 0 0 0 0 q 0 0−q −2 −1−q −1 0 0 0 0 q −1   (37) The metricisgivenagain by (33).The curvature Curvis deﬁned by Ω ij =−(q 2 −1)q −3 (1 +q+q 2 ) /parenleftbigg 0 0 1 0 /parenrightbigg (λ 1 ) 2 θ 1 θ 2 . kievarwe.tex; 12/03/2001; 3:49; p.287 METRICSON THEMANIN PLANE 281 It diverges as (q−1) −1 whenq→1. This is then the case of a regular metric which hasasingular metric connection. 3.3. SOLUTION III Athird familysatisﬁes no reality conditions S= 1 q 2 + 1   2q 0 0 1−q 2 0 1−q 2 2q 0 0 2q q 2 −1 0 q 2 −1 0 0 2 q   . (38) Aσ-symmetricmetricisgiven by S 1221 =S 2112 =2 q q 2 + 1. Inthelimitq→1thisbecomes Ω ij = /parenleftBigg 0−1 1 0 /parenrightBigg (u 2 +v 2 )θ 1 θ 2 . 3.4. NON-SOLUTIONS There are a certain number of partial solutions which are unsatisfactory for some reason or other. As an example, to underline the possibility of exotic metrics which are neither symmetric nor anti-symmetric according to our deﬁnitions, we considerσdeﬁned by thematrix S=   0 0 0 γ 0−1 0 0 0 0−1 0 γ −1 0 0 0   whereγ∈Risaparameter.Thisvalueof Sisasolutiontothebraidequation. Theσandπarerelatedas in(21)with(usingthesame conventions) T=   1 0 0γ 0 0 0 0 0 0 0 0 γ −1 0 0 1   . (39) Thismeansthat τisnotinvertibleandthecaseisdegenerate.Theproblemhereis that(1 +σ)/2cannotevenbetwistedto a projector.The metricis given by g ij =i /parenleftbigg 1 0 0−γ −1 /parenrightbigg . (40) kievarwe.tex; 12/03/2001; 3:49; p.288 282 G. FIORE,M.MACEDA,J.MADORE One hasτ= 1 +σand the ﬂip is degenerate. Instead of interchanging g 2 andg 3 asdoestheordinaryﬂip,itinterchanges g 1 andg 4 .Italsochangesthesign,which accounts for the iin the metric components. Also g◦(1 +σ) = 0so in a certain sensethemetrichasvanishingsymmetricaswellasantisymmetricparts.Werefer toσnonetheless asa‘ﬂip’ becauseitsatisﬁes(20).The curvature is given by Ω ij =q −1 (q 2 −1)δ i j λ 1 λ 2 θ 1 θ 2 It is singularin thecommutative limit. Finally we notice that here is no solution using the ˆR-matrix to construct σ. AsimilarproblemwasfoundbyCotta-Ramusino&Rinaldiintryingtoconstruct holonomygroups[12]. Acknowledgment The authors would like to thank A. Chakrabarti for enlightening conversations. Oneofthem(JM)wouldliketothankDieterL ¨ustforhishospitalityattheInstitut f¨ur Physik,Berlin,were partof this researchwascarried out. References 1. A.Connes, Noncommutative Geometry , Academic Press, 1994. 2. A. Dimakis and J. Madore, Differential calculi and linear connections , J. Math. Phys. 37 (1996), 4647–4661. 3. G.Fiore, M. Maceda and J. Madore, Metrics on the Manin Plane , Preprint (to appear). 4. A.Connes, Noncommutative geometryand reality , J.Math. Phys. 36(1995), 6194. 5. G.FioreandJ.Madore, Leibnizrulesandrealityconditions , Euro.Phys.Jour.C math/980607 (to appear). 6. M. Dubois-Violette, J. Madore, T. Masson, and J. Mourad, On curvature in noncommutative geometry, J.Math. Phys. 37(1996), 4089–4102. 7. B. L. Cerchiai, R. Hinterding, J. Madore, and J. Wess, The geometry of a q-deformed phase space, Euro. Phys. Jour.C 8(1999), 533–546. 8. J. Wess and B. Zumino, Covariant differential calculus on the quantum hyperplane , Nucl. Phys. (Proc. Suppl.) 18B(1990), 302. 9. J. Hietarinta, Solving the two-dimensional constant quantum Yang-Baxter equation , J. Math. Phys.34(1993), 1725. 10. M.GerstenhaberandA.Giaquinto, BoundarysolutionsofthequantumYang-Baxterequation andsolutions in three dimentions , q-alg/9710033 (toappear). 11. B. Aneva, D. Arnaudon, A. Chakrabarti, V. Dobrev, and S. Mihov, On combined standard- nonstandard or hybrid (q,h)-deformations , math.QA/0006206 (to appear). 12. P.Cotta-RamusinoandM.Rinaldi, Link-diagrams,Yang-Baxterequationandquantumholon- omy, inQuantumGroupswithapplicationstoPhysic,(M.GerstenhaberandJ.Stasheff,eds.), Vol. 134Amer. Math. Soc., Providence,RhodeIsland, 1992,pp. 19–44. kievarwe.tex; 12/03/2001; 3:49; p.289 COHERENCEISOMORPHISMSFOR AHOPF CATEGORY VOLODYMYRLYUBASHENKO ∗ Instituteof Mathematics,Kyiv,Ukraine Crane and Frenkel proposed a notion of a Hopf category in [1]. It was moti- vated by Lusztig’s approach to quantum groups – his theory of canonical bases. In particular, Lusztig obtains braided deformations U q n + of universal envelop- ing algebras U n + for some nilpotent Lie algebras n + together with canonical bases of these braided Hopf algebras [2–4]. The elements of the canonical basis are identiﬁed with certain objects of equivariant derived categories, contained in semisimple abelian subcategories of semisimple complexes. Conjectural proper- tiesofthesecategorieswerecollectedintoasystemofaxiomsofaHopfcategory, equipped with functors of multiplication and comultiplication, isomorphisms of associativity, coassociativity and coherence which satisfy four equations [1]. Crane and Frenkel gave an example of a Hopf category resembling the semisim- ple category encountered in Lusztig’s theory corresponding to one-dimensional Lie algebra n + – nilpotent subalgebra of sl(2). The mathematical framework and some furtherexamplesofHopfcategories wereprovided by Neuchl [5]. We discuss an example of a related notion – triangulated Hopf category – the whole equivariant derived category equipped with operations-functors and structureisomorphisms.Theadditiverelationsbetweenoperationsproposedin[1] arereplacedwithdistinguishedtriangles.Thepreliminarystudyofthesubjectcan be found in [6, 7]. In the present paper we construct the coherence isomorphisms in full required generality. The essential ingredient – the equation for coherence isomorphisms isstillnotproven. 1. Operationsin a gradedHopf algebra LetQ + be a commutative monoid additively generated by elements of a ﬁnite setI. DenoteR=Z[q,q −1 ]. LetHbe aQ + -graded braided Hopf R-algebra, for instance,thealgebra U q n + ofLusztig[4].Thecomultiplicationin H=⊕ v∈Q + H v ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.290 284 V. LYUBASHENKO canbe written as ∆ = /summationdisplay u,v∈Q + ∆ u,v , ∆ u,v :H u+v →H u ⊗ R H v . Similarlyforiterated comultiplication ∆ (b) = (∆ (b−1) ⊗1)◦∆ :H→H ⊗b , ∆ (b) = /summationdisplay v j ∈Q + ∆ (b) v 1 ,...,v b , ∆ (b) v 1 ,...,v b :H v 1 +···+v b →H v 1 ⊗ R ···⊗ R H v b . The associativity, thecoassociativityandthe bialgebra axiom imply the equation ∆ (b) (x 1 )·...·∆ (b) (x a ) = ∆ (b) (x 1 ·...·x a ) (1) for arbitrary elements x i ∈H. Note that the multiplication in the left hand side uses the braiding. Apply equation (1) to homogeneous elements x i of degreev i and write downits homogeneouscomponent of multidegree (v 1 ,...,v b )∈Q b + : /summationdisplay /summationtext j v i j =v i /summationtext i v i j =v j ∆ (b) v 1 1 ,...,v 1 b (x 1 )·...·∆ (b) v a 1 ,...,v a b (x a ) = ∆ (b) v 1 ,...,v b (x 1 ·...·x a ).(2) Each summand in the left hand side can be viewed as an operation with ainputs andboutputs. These operations are not distinguished in algebra setup. However, ingraded Hopfcategoriestheirexplicit use seemsadvantageous. 2. Themain ingredients Categories will be equivariant derived categories X G :=D b,c G (X), whereXis a complexalgebraicvariety,equippedwiththeactionofacomplexalgebraicgroup G, asdeﬁned by BernsteinandLunts[8]. The functors will be compositions of functors of the three types (see [8]). Let φ:G→Hbeagrouphomomorphism,let XbeaG-space,letYbeanH-space, and letf:X→Ybe aφ-equivariantmap. Thenthereare — theinverseimage functor f φ∗ : Y H → X G , — ifφ:G→His surjective, K= Ker(φ),XisK-free, andY=K\X, the directimagefunctor (inthiscaseit is anequivalence) f φ∗ : X G → Y H . —ifφ= 1 :G=Histheidentity,thedirectimagefunctorwithpropersupports f 1! : X G → Y G . Quiver. Let(H,I)beaﬁniteorientedgraphwiththesetofvertices I,thesetof edgesH,thestructuremap H→I×I,h/mapsto→(h /prime ,h /prime/prime ),whereh /prime ∈Iisthesource ofh∈H, andh /prime/prime ∈Iisthetargetof h∈H,such thath /prime /negationslash=h /prime/prime . kievarwe.tex; 12/03/2001; 3:49; p.291 COHERENCE ISOMORPHISMSFOR HOPFCATEGORY 285 LetVbe a ﬁnite dimensional I-gradedC-vector space, (a function V:I→ ObC-vect,i/mapsto→V(i)). Itsautomorphismgroup is G V = Aut I-grad -vect V= /productdisplay i∈I GL(V(i)). Deﬁnealinear space E V =⊕ h∈H Hom C (V(h /prime ),V(h /prime/prime )). The union of all E V is the class of representations of the quiver. The group G V acts onE V by(g.x) h =g h /prime/prime x h g −1 h /prime . The union of all G V \E V is the set of iso- morphism classes of representations of the quiver. We consider the collection of equivariantderivedcategories E V G V as ourHopfcategory. Filtrations. To introduceoperationsweneedto considerdecompositionsof V V:V 1 ⊕V 2 ⊕···⊕V k =V intoI-graded subspaces.Associate withit aﬁltration of V 0 =V (0) ⊂V (1) ⊂···⊂V (k) =V, V (m) =V 1 ⊕···⊕V m . Associate with ittheparabolic group P V P V ={g∈G V |∀m g(V (m) )⊂V (m) }. The unipotentradical of P V is denotedU V .The group L V ={g∈G V |∀m g(V m )⊂V m }= k /productdisplay m=1 G V m /similarequalP V /U V isaLevisubgroup of P V . LetF V be thelinear subspaceof E V respecting the ﬁltration: F V ={x∈E V |∀m,h x h (V (m) (h /prime ))⊂V (m) (h /prime/prime )}. The groupP V actsinF V . Operations. Let twodecompositionsof Vintoadirect sumbe given: V:V 1 ⊕V 2 ⊕···⊕V k∼ →V, W:W 1 ⊕W 2 ⊕···⊕W l∼ →V. kievarwe.tex; 12/03/2001; 3:49; p.292 286 V. LYUBASHENKO LetO⊂G V be a leftP W -invariant and right P V -invariant subset. We associate withitan operation X I O;V W = V 1 V 2 V k O W 1 W 2 W l = V 1 V 2 V k O W 1 W 2 W l =⑂ V W ◦Ψ O;V W . The components of it are the generalized multiplication and comultiplication functors. Multiplicationhalf. The multiplication half operationis V 1 V 2 V k O = Ψ O;V W = /parenleftbigg/producttext k i=1 E V i L Vφ ∗ →O×F V P W ×P Vπ ∗ →O× P V F V P Wα ! →E V P W /parenrightbigg . The schemeofmultiplicationissimilar tothatofLusztig [2–4]: k /productdisplay i=1 E V i ← φ O×F Vπ →O× P V F Vα →E V , whereφ(o,f) =κ(f)is the forgetful map, κ:F V → /producttext k i=1 E V i is the natural projection,πis the canonical projection, α(o,f) =o.ι(f)is induced from the action map, and ι:F V →E V is thenaturalembedding. Comultiplication half. The comultiplicationhalf operation functor is W 1 W 2 W l =⑂ V W = /parenleftbigg E V P Wι ∗ →F W L Wκ ! → /producttext l j=1 E W j L Wτ → /producttext l j=1 E W j L W /parenrightbigg , whereτis theshift τL=L[2 /summationdisplay r>s;h dimW r (h /prime )·dimW s (h /prime/prime )]. Theschemeofcomultiplicationismadeofthenaturalembedding ιandthenatural projectionκ(as inLusztig [2–4]): E V ← ι F Wκ → l /productdisplay j=1 E W j . kievarwe.tex; 12/03/2001; 3:49; p.293 COHERENCE ISOMORPHISMSFOR HOPFCATEGORY 287 Braiding. Foramodule MoverG A 1 ×···×G A k andamodule NoverG B 1 × ···×G B l whereA m ,B n aresomeI-gradedvectorspaces,wedeﬁnethebraiding as the functor M× N /producttext G A m × /producttext G B n τ − →M× N /producttext G A m × /producttext G B n σ ∗ −→N× M /producttext G B n × /producttext G A m , whereσisthepermutationisomorphismofgroupsandmodulesandthefunctor τ istheshift τ(L) =L /bracketleftBig −2 /summationdisplay m,n i∈I dimA m (i) dimB n (i) + 2 /summationdisplay m,n h∈H dimA m (h /prime ) dimB n (h /prime/prime ) /bracketrightBig . Distinguishedtriangles. Toclarifythemeaningofoperations X I O ,noticethatthe orbits of the action of P W ×P V inG V are in natural bijection with the orbits of the action of G V in the space of pairs of ﬁltrations P W \G V ×G V /P V . By [9] these orbits are in bijection with a×b-matrices (v i j )with elements in Q + =Z I + , such that /summationtext j v i j =v i is the dimension of V i and /summationtext i v i j =v j is the dimension of W j . Thus, the orbits are in bijection with the summands in the left hand side of equation (2). The P W ×P V -invariant subsets are unions of orbits, thereby, they arerepresentedby sumsofseveral summandsin(2). The additive relation (2) in algebra is replaced for our Hopf category by a systemoffunctorial distinguishedtriangles X I O U →X I O X →X I O F → givenforanybi-invariantsubset O X ⊂G V andabi-invariantclosedsubset O F ⊂ O X withO U =O X −O F .Thefollowingdiagrammadewithgivendistinguished trianglesis anoctahedron X I O R →X I O W d = X I O Z ← 1 ← = d X I O Q 1 ↑ ←1 → X I O F ↓ → X I O R →X I O W = d X I O U → → d = X I O Q 1 ↑ ←1 ← X I O F ↓1 ← for any pair of closed embeddings O F ⊂O Z ⊂O W , whereO U =O W −O F , O Q =O Z −O F ,O R =O W −O Z . This means commutativity of two squares formedby diagonal maps andofthefour triangles marked “=”. Coherence isomorphism. Both associativity isomorphism and coassociativity isomorphism of [7] are particular cases of the general coherence isomorphism. kievarwe.tex; 12/03/2001; 3:49; p.294 288 V. LYUBASHENKO For any collection of indices and for any collection of bi-invariant subsets (O /prime 1 ,...,O /prime a ,O /prime/prime 1 ,...,O /prime/prime b ), which may occur in the following diagram, there existsabi-invariant subset Oandacoherenceisomorphism O /prime 1 O /prime a b b b σ a,b O /prime/prime 1 a a O /prime/prime b a ≡ O /prime 1 O /prime a O /prime/prime 1 O /prime/prime b coher → O . Hereσ a,b = (s a,b ) ∼ + is the braid, corresponding to the permutation s a,b of the set {1,2,...,ab}, s a,b (1 +r+kb) = 1 +k+rafor0≤r<b, 0≤k<a, underthestandardsplitting S ab →B ab ,whichmapstheelementarytranspositions tothegeneratorsofthebraidgroup.The subset Oiscomputed asfollo ws O=U V · /productdisplay m O /prime m = /productdisplay m O /prime m ·U V ⊂P V , O=U W · /productdisplay r O /prime/prime r = /productdisplay r O /prime/prime r ·U W ⊂P W , O= OP U × P U O= O× P W ∩P U O= O ·O⊂G V . The general coherence isomorphismisbuilt asthe composition Y V O /prime 1 O /prime a U X W O /prime/prime 1 O /prime/prime b Z = O /prime 1 O /prime a P V 1 P Va O /prime/prime 1 O /prime/prime b P Z 1 P Zb coher → O /prime 1 P V1 O /prime a P Va OP U P Z P Z1 P Zb assoc coass → O P Z = O . kievarwe.tex; 12/03/2001; 3:49; p.295 COHERENCE ISOMORPHISMSFOR HOPFCATEGORY 289 The threecomponents ofthe coherence isomorphismare deﬁned next. /producttext E Ym s /producttext L Y m φ ∗ → /producttext O /prime m ×F Y m /producttext P Vm ×P Y m π ∗ → /producttext O /prime m × PYm F Y m /producttext P V m α ! → /producttext E V m /producttext P Vm O×F Y P Z ×P Y φ ∗ ↓ π ∗ → OP U ×O×F Y P Z ×P U ×P Y φ ∗ ↓ π ∗ → OP U ×O× PY F Y P Z ×P U φ ∗ ↓ 1×α ! → OP U ×F V P Z ×P U φ ∗ ↓ O×F Y P Z ×P Y π ∗ ↓ π ∗ →Id = = = = = = = = /arrowrighttophalf/arrowrightbothalf O× PY F Y P Z π ∗ ↓ β ! → OP U × PU F V P Z π ∗ ↓ assoc E V P Z α ! ↓ α ! → E V P Z ι ∗ → F W /producttext r P Z r κ ! → /producttext r E W r /producttext r P Zr F Z L Z ι ∗ ↓κ ! →ι ∗ → /producttext r F Z r /producttext r L Zr ι ∗ ↓ coass /producttext n,r E Zn r L Z κ ! ↓κ ! → The isomorphism coherispresented inFigure9, where thenumbers A= /summationdisplay m<n ;r>s /summationdisplay i∈I dimV m r (i)·dimV n s (i), B= /summationdisplay m>n ;r>s /summationdisplay h∈H dimV m r (h /prime )·dimV n s (h /prime/prime ) are,actually, dimensionsofthespaces A=U W /(U W ∩P V ) =⊕ m<n ;r>s Hom(V m r ,V n s ), B=⊕ h∈H;m>n ;r>s Hom C (V m r (h /prime ),V n s (h /prime/prime ))⊂F W ∩F V , and weuse the notation F=F W ∩F V /B. The whole coherenceisomorphismis presentedin Figure10. 3. Elementaryisomorphisms. The coherence isomorphisms are pastings of isomorphisms and their inverses of thefollowing10 types: kievarwe.tex; 12/03/2001; 3:49; p.296 290 V. LYUBASHENKO Figure 9. The isomorphism coher. kievarwe.tex; 12/03/2001; 3:49; p.297 COHERENCE ISOMORPHISMSFOR HOPFCATEGORY 291 Figure 10. The whole coherence isomorphism. kievarwe.tex; 12/03/2001; 3:49; p.298 292 V. LYUBASHENKO a) g ψ∗ f φ∗∼ − → /parenleftbig f φ g ψ /parenrightbig ∗ ; b) f φ∗ g ψ∗∼ − → /parenleftbig f φ g ψ /parenrightbig ∗ ; c) f 1! g 1!∼ − → /parenleftbig f 1 g 1 /parenrightbig ! ; d)basechangeisomorphism,where W=X× Y Z,andh,jaretheprojections X G f 1! → Y G W H h φ∗ ↓ j 1! → ⇐= = = = = = = = = Z H g φ∗ ↓ e) the isomorphism of X G f φ ∗ → Y H g ψ ∗ → Z B with X G h ξ ∗ → W K j χ ∗ → Z B , where W=X× Y Z,K=G φ × Hψ Bandh,j,ξ,χare the projections; it is given by thepasting X G f φ∗ → Y H X G/epsilon1 ⇐=/harpoondownleft/harpoondownright /arrowvertexdbl/arrowvertexdbl/arrowvertexdbl f φ∗ ← Z B g ψ∗ ↓ W K h ξ∗ ↓ j χ∗ → ← /arrowdbltp/arrowvertexdbl /arrowdbltp/arrowvertexdbl j χ∗ ← Z Bη ⇐=/harpoondownleft/harpoondownright /arrowvertexdbl/arrowvertexdbl/arrowvertexdbl f) the isomorphism of X G f 1 ! → Y G g φ ∗ → Z H with X G j φ ∗ → W H h 1 ! → Z H , where K= Ker(φ: G→H),W=K\X,h=K\f:W=K\ X→K\Y= Z,andjisthequotient map; itisgiven bythe pasting X G = = = = = = =/arrowrighttophalf/arrowrightbothalf X G f 1! → Y G /epsilon1 −1 /arrowdblbt/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl W H j φ∗ ↓ h 1! → j φ∗ → Z H /arrowdblbt/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl = = = = = = =/arrowrighttophalf/arrowrightbothalf g φ∗ → η −1 /arrowdblbt/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl Z H g φ∗ ↓ And4moretypes ofelementaryisomorphisms: kievarwe.tex; 12/03/2001; 3:49; p.299 COHERENCE ISOMORPHISMSFOR HOPFCATEGORY 293 i) whenever j|φis an induction map and π,q=π◦(j|φ)are quotient maps, there is anisomorphism H× G X H j φ∗ → X G H× G X Hη −1 ⇐= /harpoondownleft/harpoondownright /arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl /arrowvertexdbl π ∗ → j φ∗ ← Y B ⇐=q ∗ ↓ q)whenever π: X G → Y H isaquotient map,thereisanisomorphism X G ←π ∗ Y H Y Hη ⇐=/harpoondownleft/harpoondownright /arrowvertexdbl/arrowvertexdbl/arrowvertexdbl π ∗ → s) whenever ˜Pis a split extension of ˜L,U= Ker( P p ⊿ L)is contractible and˜Eisa˜P-space,onwhich Uactstrivially,then 1 p∗ : ˜ E ˜L → ˜ E ˜P isanequivalence and there isanisomorphism ˜ E ˜L ˜ E ˜P = = = = = = = = = = = = = = = = = = = = /arrowrighttophalf/arrowrightbothalf 1 i∗ → /arrowdblbt/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl/arrowvertexdbl ˜ E ˜P 1 p∗ → v)whenever G-maph:E→Bisavectorbundle, there is an isomorphism B G E G ⇐= = = = = = = = =h ∗ ← B G T −2 dim C h ↓h ! → Theorem17. The2-categoryformed by — objects: equivariantderived categories; — 1-morphisms: compositions of functors of 3 types: inverse image functors, direct image functors for quotient maps, direct image functors with proper supports; kievarwe.tex; 12/03/2001; 3:49; p.300 294 V. LYUBASHENKO — 2-morphisms: compositions of isomorphisms of 6 types a)–f) or their inverses is a 2-groupoid, that is, for any 1-morphisms FandGwith the common source and target the set Hom(F,G)either is empty or has exactly one element (and all 2-morphismsareinvertible). If the above theorem would hold for all 10 types of isomorphisms, it would mean that all equations between coherence isomorphisms, which can be written, holdtrue. Sucha generalization isnot proven yet. References 1. L.CraneandI.B.Frenkel, Fourdimensionaltopologicalquantumﬁeldtheory,Hopfcategories, and the canonicalbases , J. Math.Phys. 35(1994), 5136–5154. 2. G. Lusztig, Canonical bases arising from quantized enveloping algebras , J. American Math. Soc.3(1990), 447–498. 3. ———, Quivers, perverse sheaves, and quantized enveloping algebras , J. American Math. Soc.4(1991), 365–421. 4. ———, Introduction toQuantum groups , Birkh¨auser, Boston, 1993. 5. M. Neuchl, Representation Theory of Hopf Categories , PhD thesis. To appear in Adv. in Math. under the title Higher-dimensional algebra VI: Hopf categories. Available at http://www.mathematik.uni-muenchen.de/ ∼neuchl. 6. V.V.Lyubashenko, ExampleofatriangulatedHopfcategory , V¯ısnikKi¨ıv.Un¯ıv.Ser.F¯ız.-Mat. Nauki2(1999), 50–58 (in Ukrainian). 7. ———, Operations and isomorphisms in a triangulated Hopf category , Methods of Func. Analysis and Topology 5(1999),37–53. 8. J. Bernstein and V. Lunts, Equivariant Sheaves and Functors , Vol. 1578 of Lecture Notes in Math., Springer, Berlin, Heidelberg,1994. 9. A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformations of gl n , DukeMath. J. 61(1990), 655–677. kievarwe.tex; 12/03/2001; 3:49; p.301 FUSIONRINGSANDTENSORCATEGORIES ALEXANDER GANCHEV ∗ INRNE, Tsarigradsko chausse72, BG1784 Soﬁa Thedeﬁnitionofafusionring F[1],[2],[3]isanabstractionoftheproperties of the Grothendieck ring K 0 (C)of a rigid braided semisimple monoidal category C. For certain issues it is convenient to pass to an algebra (over the complex numbers) thus a fusion algebra Fis a unital associative and commutative algebra with a chosen basis Isuch that the fusion rules N c ab ,a,b,c∈I, i.e., the structure constants in this basis, a·b= /summationtext c N c ab c, are in Z + and their is an involutive automorphism a→¯asuch thatN 1 ab =δ ¯a,b . The setIcorresponds to the sectors, i.e., the equivalence classes of simple objects or irreps, the monoidal structure in Cis responsible for the structure of unital associative ring, the braiding for the commutativity,whiletherigidity translatesin the involutative automorphism. Fusion rings/algebras appear in many occasions (we consider only ﬁnite di- mensionalones):thecategory CinK 0 (C)couldbeRep(ﬁnite(quantum)group); Rep(U q (g))/Zwithq p = 1,ga simple Lie algebra, and Zthe ideal of zero quantumdimensionalmodules;Also CcouldbetheMoore-Sibergcategoryof2- dimensionalrationalconformalﬁeldtheory(2D-RCFT)ortheDoplicher-Roberts categoryoflocalizableautomorphismsofthealgebraofobservablesofa2D-QFT (quantum ﬁeld theory) with Ilabeling the superselection sectors (the generalized charges). The last three are typically non Tannakian categories and in particular thestatisticaldimensions(=ranks)ofthesectorsareingeneralonlyalgebraicinte- gers. Most generally Cis the rep category of a quasitriangular weak Hopf algebra (or quantum gropoid). On many occasions (2D-RCFT, 2D-QFT) one has more structurewith Cbeingribbon(=tortile)andinfactaTuraevmodularcategorywith Icomprising a representation of the modular group SL 2 (Z)with modular Sand Tmatrices. The Splays the role of characters and diagonalizes the fusion rules (Verlinde’sfamousformula)while Tisdiagonalwiththebalancingphasesonthe diagonal. NowIbrieﬂymentionseveralinmyviewimportantproblems:structuretheory of fusion rings and tensor categories, classiﬁcation of particular cases of fusion ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.302 296 A. GANCHEV rings and tensor categories, categoriﬁcation, i.e., reconstructing a tensor category fromitsfusionringand ﬁnallyexplicitformulas forcertainfusion rings. Fusion rules with a generator of dimension <2are classiﬁed in [1]. For modular fusion algebras (i.e., reps of SL 2 (Z)with Verlinde giving fusion rules) there areinitial stepstowardsaclassiﬁcation[4]. Most anythingelse isopen. Fusion algebras are particular cases of table algebras [5]. For a table algebra the requirements that the structure constants N c ab are positive integers and N 1 ab = δ ¯a,b are relaxed to N c ab ∈R + andN 1 ab /negationslash= 0iff¯a=b. Table algebras have been extensively studied by Arad, Blau and coworkers. Particular cases of table algebras with generators of dimension 2 or 3 have been classiﬁed. Though they are not directly relevant to fusion rule algebra classiﬁcation one again encounters for the fusion graphs a 1-dimensional structure (afﬁne Dynkin diagrams) for the caseofadimension2generatoranda2-dimensionalstructure(thefusiongraphof the fundamental irrep of sl(3), a tringular tesselation of the corresponding Weyl chamber, orfoldings of it)[6]. Forﬁnitegroupsitisclearthatsimplegroupshavefusionrulesalgebraswhich have no notrivial subfuison rule algebras, hence such fusion rule algebras is nat- ural to call simple. More generally if a group Ghas a normal subgroup Hthen K 0 (G/H )is a subfusion rule algebra of K 0 (G). This extends to Hopf algebras [7] and [8]. For table algebras there is a more developed structure theory [9] – in particular one has composition series for table algebras. What is the theory of extensionsforfusionrulealgebrasisanopensubject.Since K 0 isonlyhalfexact one will probably have to use the higher Kfunctors and the long exact sequence inKtheorytorelateinformationaboutthestructureoftensorcategoriesandtheir fusionrulealgebras. Categoriﬁcation,i.e.,reversingthe K 0 functor,isaverychallengingproblem. Some very initial “experimental” work of solving the pentagon equations to ob- tain categories from given fusion rules was done in [10]. For the fusion rules of truncatedsl(n)with the relevant Hecke algebra the corresponding braided tensor categorieswerereconstructedin[11].Thepentagonisa(ingeneralanonabelian) 3-cocycle condition – a preliminary sketch of how to attack the relevant non- abeliancohomologyproblemisgivenin[12].Forthecaseofabelianfusionrules (Fis the group algebra of an abelian group) it is an ordinary group cohomology problem solved in [1]. The categoriﬁcation of the fusion rules of the quaternionic or the rank 8 dihedral group and their generalizations (where all but one of the sectors are abelian) was done in [13]. In general, for a tensor category with a nontrivial abelian subfusion algebra one can characterize all 6jsymbols involv- ing an abelian lable in terms of abelian group cohomology and moreover there is also an action on all 6jsymbols by the abelian group (work in preparation). One would like to characterize the image of K 0 in the category of all fusion rule algebras and ﬁnd “moduli” distinguishing categories with the same fusion rules. In the case of modular categories one is tempted to conjecture that the balancing kievarwe.tex; 12/03/2001; 3:49; p.303 FUSION RINGS 297 phases(theTmatrix)separatescategorieswiththesamefusionrules(=“character table”=modular Smatrix) and that K 0 is a bijection from (equivalent classes of) modular categories to modular fusion algebras [14]. Since it seems to be the case that two different simple ﬁnite groups cannot have the same fusion rules one can trytoexploreaconjecturethatifasimplefusionrulealgebrahasacategoriﬁcation thanitisunique. For certain classes of fusion rules, e.g. fusion rules of WZW models based on afﬁne Kac-Moody algebras ˆg k at integer levels k(same as truncated U q (g)for q k+h ∨ = 1) one has nice formulas for N c ab generalizing a classical formula of Weyl[15],[16],[17].Forthemuchharderandlessstudiedcaseoffractionalevel WZW one knows the fusion rules only for g=sl(2)andsl(3)([18] and [19] respectively). Very little is known for the fusion rules of more general models of 2D-RCFT. In particular one would like to know the fusion rules of fractional level afﬁne sl(n)and on the other hand to relate them to the fusion rules of W-algebras obtained from these models by quantum hamiltonian reduction or cosetting. Even for the case of the Polyakov-Bershadski W (2) 3 which is obtained as the nonprincipal reduction of sl(3)at fractional levels the fusion rules are not known in general. A better understanding of the structure theory of fusion rules hopefully could help in such problems. On the other hand the fusion rules of fractionalsl(3)do not look like anything coming from a known algebraic object (ﬁnite group, Lie (super) algebra) hence it is interesting to try to categorify these fusionrules. References 1. J. Fr¨olich and T. Kerler, Quantum groups, quantum categories and quantum ﬁeld theory , LectureNotes in Math. 1524, Springer-Verlag, Berlin , 1993. 2. J.Fuchs, Fusion rules inconformal ﬁeld theory , Fortschr. Phys. 42(1994), 1. 3. P. Di Francesco, P. Mathieu, and D. S ´en´echal,Conformal Field Theory , Springer-Verlag, Berlin, 1997. 4. W. Eholzer, On the classiﬁcation of modular fusion algebras , Commun. Math. Phys. 172 (1995), 623. 5. Z. Arad and H. Blau, Table algebras and applications to ﬁnite group theory , J. Algebra, 138 (1991), 137-185. 6. H. Blau, B. Xu, Z. Arad, E. Fisman, V. Miloslavsky, and M. Muzychuk, Homogeneous Integral Table Algebras of Degree Three: A Trilogy Memoairs of the AMS, vol. 144, no. 684(2000). 7. W. Nichols, M. B. Richmond, The Grothendieck Group of a Hopf Algebra , J. Pure and Appl. Algebra106(1996), 297-306. 8. D. Nikshych, K 0 -Rings and Twisting of Finite Dimensional Semisimple Hopf Algebras Commun. Algebra 26(1998), 321-342. 9. H.Blau, Quotient Structures in C-Algebras , J.Algebra 175(1995), 24-64. 10. J. Fuchs, A. Ganchev, and P. Vecserny ´es,Rational Hopf algebras: polynomial equations, gauge ﬁxing, and low dimensional examples , Int.J. Mod. Phys. A10(1995), 3431; kievarwe.tex; 12/03/2001; 3:49; p.304 298 A. GANCHEV 11. D. Kazhdan and H. Wenzl Reconstructing Monoidal Categories Adv. Sov. Math. 16(1993), 111-136. 12. A. Davydov, On some Hochschild cohomology classes of fusion algebras ,q- alg/9711025 . 13. D.TambaraandS.Yamagami, TensorCategorieswithFusionRulesofSelf-DualityforFinite Abelian Groups , J. Algebra, 209(1998),692-707. 14. A.Ganchev, Fusionrules,modularcategoriesandconformalmodels , in:NewtrendsinQFT, (A. Ganchev, R.Kerler, and I. Todorov, eds.), Heron Press, Soﬁa, 1996, pp.142-145. 15. M. Walton, Fusion rules for WZW models , Nucl. Phys. 340(1990), 777. 16. P. Furlan, A. Ganchev, and V. Petkova, Quantum groups and fusion rule multiplicities , Nucl. Phys.343(1990), 205. 17. V. Kac, Inﬁnite dimensional Lie algebras (3rd edition) , Cambridge Univ. Press, Cambridge, 1990. 18. H. Awata and Y. Yamada, Fusion rules for fractional level sl(2)algebra, Mod. Phys. Lett. A7(1992), 1185. 19. P. Furlan, A. Ganchev, and V. Petkova, An extension of the character ring of /hatwide sl(3)and its quantization , Commun. Math. Phys. 202(1999),701. kievarwe.tex; 12/03/2001; 3:49; p.305 ON CATEGORIESOF GELFAND-ZETLINMODULES VOLODYMYRMAZORCHUK ∗ G¨oteborgUniversity,Sweden 1. Theorigins Although the theory of Gelfand-Zetlin modules can be developed for all serial complexsimpleﬁnite-dimensionalLiealgebrasandtheir(non-standard)quantum analogs, in this paper we will discuss the most classical case of the Lie algebra g= gl(n,C)andwillgiveashortoverviewofknownresultsinothercasesinthe end of the paper. We will denote by e i,j ,1≤i,j≤n, the matrix units and will always abbreviateGelfand-Zetlinby GZ. Thistheorystartsfromthefamousoriginalpaper[9]byGelfandandZetlin,in which, using a step by step reduction to the smaller subalgebras, the authors con- structedaveryspecialandnicebasisineachsimpleﬁnite-dimensional g-module. It is well-known that simple ﬁnite-dimensional g-modules are parametrized by the vectors m= (m 1 ,m 2 ,...,m n )with complex coefﬁcients, satisfying m i − m i+1 ∈N.Thesevectorsrepresentthe(shifted)highestweightofthecorrespond- ingsimplemodulewithrespecttothestandardCartansubalgebra hof gconsisting of diagonal matrices. We will denote the simple module, which corresponds to m, byV(m). To formulate the result of Gelfand and Zetlin we have to introduce the notion of tableau. By a tableau,[l], we will mean a doubly-indexed complex vector (l i,j ), where 1≤i≤nand1≤j≤i. Theorem 18. V(m)possesses a basis, indexed by all tableaux [l], satisfying the followingconditions: l n,j =m j ,1≤j≤n,andl i,j ≥l i−1,j >l i,j+1 ,1<i≤n, 1≤j <i.Moreover,theactionofthegeneratorsof ginthisbasisisgivenby the ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.306 300 V.MAZORCHUK following Gelfand-Zetlin formulae: e i,i+1 [l] =− i /summationdisplay j=1i+1 /productdisplay k=1 (l i,j −l i+1,k ) /productdisplay k/negationslash=i (l i,j −l i,k )[l+δ i,j ], e i+1,i [l] = i /summationdisplay j=1i−1 /productdisplay k=1 (l i,j −l i−1,k ) /productdisplay k/negationslash=i (l i,j −l i,k )[l−δ i,j ], e i,i [l] =   i /summationdisplay j=1 l i,j − i−1 /summationdisplay j=1 l i,j   [l]. 2. GenericGelfand-Zetlinmodules The idea to use Theorem 18 to construct new g-modules goes back to Drozd, Ovsienko and Futorny ([4, 5]). This was based on the observation that GZ- formulae contain only rational functions in parameters, so, if one takes a set of tableaux, closed under the shifts, coming from the action of generators, such that allfunctionsinGZ-formulaewillbewell-deﬁned,theresultingspaceshouldbea g-module. Thiscanbeformallypresentedin thefollowing statement. Theorem 19. Let[t]be a tableau satisfying t i,j −t i,k /negationslash∈Zfor all 1≤i<nand 1≤j/negationslash=k≤i. Denote by P([t])the set of all tableaux [l]satisfyingl n,j =t n,j , 1≤j≤nandl i,j −t i,j ∈Zforallpossible i,j.LetV([t])denoteavectorspace, whereP([t])is a basis. Then GZ-formulae deﬁne on V([t])the structure of a g-moduleofﬁnite length. Ideaofthe proofoftheﬁrststatement. To prove the ﬁrst part of the theorem (that V([t]) is a g-module) it is sufﬁcient to check that any relation in U( g)is satisﬁed onV([l]). In our ﬁxed basis P([t])this relation can be rewritten as a collec- tion of rational functions in entries of tableaux, which have to be shown to be zero. The last is easy cause ﬁnite-dimensional modules give sufﬁciently many points,inwhichthesefunctionstakezerovalues.Thelastargumentusescrucially Theorem 18. ToprovethesecondpartweneedtorecallonemorepropertyoftheGZ-basis ofV(m),which willlead usto thenotionof Gelfand-Zetlin subalgebra. kievarwe.tex; 12/03/2001; 3:49; p.307 CATEGORIESOFGELFAND-ZETLINMODULES 301 3. Gelfand-Zetlinsubalgebra As we have already mentioned, Theorem 18 was obtained using step by step reduction to the smaller subalgebras. Now we make this statement more precise. We consider achainofsubalgebras gl(1,C)⊂ gl(2,C)⊂···⊂ gl(n,C) embedded with respect to the left upper corner. This chain induces the chain of thecorrespondinguniversalenveloping algebras U( gl(1,C))⊂U( gl(2,C))⊂···⊂U( gl(n,C)). Denote byZ k thecenterZ( gl(k,C))ofthe algebra U( gl(k,C)),1≤k≤n. The idea to get the GZ-basis of V(m)was the following: we take V(m)and consider it as gl(n−1,C)-module. The last is completely reducible and we can consider all components as gl(n−2,C)-module, decompose them and proceed till gl(1,C). Now we recall that simple ﬁnite-dimensional gl(k,C)-modules are completely determined by their central character. It is also important that, if we decomposeasimpleﬁnite-dimensional gl(k,C)-moduleintoadirectsumofsim- ple gl(k−1,C)submodules, all latter will occur with multiplicity 1. Altogether this mean that the resulting GZ basis will be an eigenbasis for all algebras Z k , or, inotherwords,forthecommutativesubalgebra Γ⊂U=U( gl(n,C)),generated by allZ k . Moreover, the remark about the multiplicities implies that Γin fact separatesthe elements oftheGZ-basisof V(m). Drozd, Ovsienko and Futorny called Γthe Gelfand-Zetlin subalgebra of U. It is well-known that Γis a polynomial algebra in n(n+ 1)/2variables. It was observed by Zhelobenko ([22]), that there is a set of generators, γ i,j ,1≤i≤n, 1≤j≤i, ofΓsuch that the eigenvalue of the action of γ i,j on a tableaux, [l], occurring in V(m), should be computed as the j-th symmetric polynomial in variables (l i,1 ,l i,2 ,...,l i,i ). Using the arguments analogous to that, presented in Section2,onegets thatthesameistruein all V([t]). Ideaofthe proofofthesecond statementofTheorem 19. As we saw, the basis P([t])ofV([t])is an eigenbasis for Γ. Moreover, it is easy to get that Γin fact separates the elements of P([t]). Hence, any submodule of V([t])has a basis, which is a subset of P([t]). Now if one draws a graph with elements of P([t]) as vertices and joins thous pairs, who mutually appear with non-zero coefﬁcients in GZ-formulae, one gets a graph with a ﬁnite number of connected components (this numbercanbeeasily computed).Thisﬁnishesthe proof. Remarkthata complete proofof Theorem19can befound in[16]. kievarwe.tex; 12/03/2001; 3:49; p.308 302 V.MAZORCHUK 4. Categoryof Gelfand-Zetlinmodules The introduction of GZ-subalgebra caused a natural deﬁnition of an abstract no- tion of Gelfand-Zetlin modules, analogous to the notion of the weight module. This was also done by Drozd, Ovsienko and Futorny. They proposed to call a Gelfand-Zetlin module any g-module,V, which decomposes into a direct sum of ﬁnite-dimensionalmodules,whenviewedas Γ-module.Thenbythecategory, GZ, of Gelfand-Zetlin modules it is natural to understand the full subcategory of the categoryofall g-modules,consistingofallGZ-modules.AsexamplesofGelfand- Zetlin modules one can take ﬁnite-dimensional modules, h-weight modules with ﬁnite-dimensional weight spaces (in particular, all highest weight modules) or genericGelfand-Zetlinmodules. Now we recall that tableaux naturally parameterize (not bijectively!) sim- ple ﬁnite-dimensional Γ-modules, moreover, non-isomorphic Γ-simples do not have non-trivial extensions. Hence, any GZ-module, V, comes together with itsGelfand-Zetlin support ,gzsupp(V), i.e. the set of all tableaux parameter- izing all simple Γ-modules, occurring in V. We have to note that the product G=S 1 ×S 2 ×···×S n of symmetric groups naturally acts on the space of all tableaux permuting the components in the rows. Any fundamental domain of this action bijectively parameterizes Γ-simples and, by deﬁnition, gzsupp(V)is invariantunderthisaction.Hencetheorbitsof Gactingon gzsupp(V)bijectively parameterize Γ-simples appearingin V. Calltwotableaus, [l]and[t],equivalentprovided l n,j =t n,j andl i,j −t i,j ∈Z for alli,j. Let Ddenote the set of equivalence classes of tableaux. First basic result about the category of Gelfand-Zetlin modules was the following statement, due toDrozd,OvsienkoandFutorny ([6]). Theorem20. ThecategoryGZdecomposesinto a directsum, GZ=⊕ P∈ D GZ P , of full subcategories, where the category GZ P consists of all Gelfand-Zetlin modulesVsuchthat gzsupp(V)⊂G◦P. Proof.Is not difﬁcult if one reminds that GZ-formulae preserve the equivalence classesof tableaux. In fact, Drozd, Ovsienko and Futorny embedded this special case of U−Γ relative situation in a wide framework of Harish-Chandra subalgebras , which is very convenient (and very general) for study of the whole category of Gelfand- Zetlin modules. It is not our aim to discuss this approach and we refer the reader totheoriginalpaper [5]. kievarwe.tex; 12/03/2001; 3:49; p.309 CATEGORIESOFGELFAND-ZETLINMODULES 303 5. Afew theorems ofOvsienko As soon as one has formulated the notion of a GZ-module, there is a natural and basic question arising: Is it true that each character of Γcan be continued to a g-module.Equivalently:isittruethateach GZ P isnotempty.Itiseasytoanswer “yes” forn= 1,2. Forn= 3the same was prooved in [4]. The general case was recentlycompleted by Ovsienko ([21]), but thepaper has not appearedyet. Theorem21. EachGZ P is notempty. Ideaofthe proof. The proof is hard and technical. In fact, the result appears as a biproduct to a special geometrical statement. One should look at the image of Γin gr(U). This image of{γ i,j }deﬁnes a certain algebraic variety, which is the variety of the so-called strongly nilpotent matrices (i.e. matrices, all main minors of which are nilpotent). The statement will follow from abstract nonsense if one proves that the sequence {γ i,j }is regular. The last can be derived if one proves that the variety of strongly nilpotent matrices is a complete intersection, i.e. that alltheirreduciblecomponentsofithavethesamedimension.Thelastisthemost difﬁcult and technical part of the proof and is the main result of the mentioned paper ofOvsienk o. From Theorem 21 it follows that for any tableau [l]there exists a simple GZ- module,V,suchthat [l]∈gzsupp(V).UsingtheconvenienttechniqueofHarish- Chandra subalgebras, mentioned above, Ovsienko managed to give much more usefulinformation about simpleGZ-modules. Theorem22. 1. Foreach [l]thereexistsonlyﬁnitelymany(uptoisomorphism) simple GZ-modules Vwith[l]∈gzsupp(V). 2. LetVbe a simple ﬁnite-dimensional g-module and Fbe a simple ﬁnite- dimensional Γ-module. Then the multiplicity of FinV(the last is viewed asΓ-module)isﬁnite. I have also to note that [21] contains a complete proof of the statement that Γis a maximal commutative subalgebra of U( g). This statement can be found (without proof!) in all classical monographs (e.g. [22]). The proof in [21] is the ﬁrstcompleteI have seen. 6. GeneralizedVerma modulesand Gelfand-Zetlin modules It seems that the ﬁrst time, when it was understood that generic Gelfand-Zetlin modules are very convenient for computations was the paper [18], where the authors investigated the question about the structure of the so-called generalized Verma modules. Consider the inclusion gl(k,C)⊂ gl(n,C) = gwith respect to the left upper corner. Let Pdenote the parabolic subalgebra of g, generated by kievarwe.tex; 12/03/2001; 3:49; p.310 304 V.MAZORCHUK gl(k,C)and the standard Borel subalgebra op upper-triangular matrices. Take a simple gl(k,C)-module,V, set that the rest of the Cartan subalgebra acts on it viasomecharacter,say λ,andtherestoftheBorelsubalgebraannihilatesit.Thus Vbecomes a P-module. The induced module M(V,λ) =U⊗ U( P) Vis called ageneralized Verma module . It turned out that taking Vto be a simple generic GZ-module, V([t]), the structure of M(V([t]),λ)can be described in terms of the Weyl group acting on the space of parameters, as it was done for the classical Verma modulesbyBernstein,I.Gelfandand S.Gelfand ([2]). It is trivial that M(V([t]),λ)is a GZ-module over g. One can also see that it is generated by the elements (annihilated by the nilpotent radical of P), cor- responding to the tableaux [l], satisfying the following condition: l i,j =l i−1,j , k < i≤n. The Weyl group S n acts naturally on the set of such tableaux, permuting the elements of the upper row (which also causes the corresponding changes in all rows with i > k). For a transposition, (i,j)∈S n ,i < j, write (i,j)[l]≤[l]providedl n,i −l n,j ∈Z + and close the relation ≤transitively. The next statementisthe mainresult of [18]. Theorem 23. Let[l](resp. [l /prime ]) be the tableau of a canonical generator of M(V([t]),λ)(resp.M(V([t /prime ]),λ /prime )). Assume that l i,j =l /prime i,j for alli < kand allj.Thenthefollowing statements areequivalent: 1.M(V([t]),λ)⊂M(V([t /prime ]),λ /prime ). 2. Theuniqueirreduciblequotientof M(V([t]),λ)isacompositionsubquotient ofM(V([t /prime ]),λ /prime ). 3.[l]≤[l /prime ]. The proof of this theorem, presented in [18] goes the general line of the original proof in [2], but uses some calculations with generic GZ-modules. In particular,oneofthemainsthingsoneneedshereisamoreorlessprecisedescrip- tion ofM(V([t]),λ)as a gl(k,C)-module. This question easily reduces to the calculationof F⊗V([t]),whereFisasimpleﬁnite-dimensional gl(k,C)-module. If one recalls that simple generic GZ-modules correspond to certain characters of Γand the last one is generated by a sequence of centers, one can use the famous Theorem of Kostant ([12]), which tells how one can compute the action of the center onF⊗V([t]). In this way one easily derives all potential subquotients of F⊗V([t]). This (and existence of some of them, which is easy) was enough for thegoalsofTheorem 23. 7. Categories of gl(n,C)-modules generated by a simple generic Gelfand- Zetlinmodule The necessity to study F⊗V([t])deeper was understood in [8], where some categoriesofLiealgebramoduleswhereconstructed,whicharebasedonthecate- goriesofmodulesbehavingwellundertensoringwithﬁnitedimensionalmodules. kievarwe.tex; 12/03/2001; 3:49; p.311 CATEGORIESOFGELFAND-ZETLINMODULES 305 As the main example of the latter, a category, generated by a simple generic GZ- module, was presented. Let V([t])be a simple generic GZ-module. Denote by C([t])the full subcategory, consisting of all subquotients of modules F⊗V([t]), whereFissimpleﬁnite-dimensional.Itturnedoutthatthiscategoryhasrelatively easy structure. Theorem 24.C([t])decomposes into a direct sum of full subcategories, each of whichisequivalenttothemodulecategoryofaﬁnite-dimensionalassociativeand local algebra.In particular, C([t])hasenough projective objects. Ideaofthe proof. One of the main ingredients of the proof is the following lemma: Lemma 25. The module F⊗V([t])has length dim(F), all simple subquotients of it are simple generic GZ-modules and the multiplicity of V([s])inF⊗V([t]), wheres i,j =t i,j ,i<n,equals /summationtext dim(F µ ),wherethesumistakenoverall µsuch that the vector (t n,j ) j=1,...,n +µcoincides witha permutation of (s n,j ) j=1,...,n . Lemma 25 is proved by a direct calculation, using GZ-formulae and the Littelwood-Richardson rule. It also represents a “generic behaviour” of simple genericGZ-modulesin contrast withﬁnite-dimensional modules. AfterLemma25onecanﬁrstdescribeallsimplemodulesin C([t]).Thesewill beV([s]), withs i,j −t i,j ∈Z. Then it is easy to ﬁnd among them a projective module and prove the existence of projectives using the exactness os F⊗ − . Decomposition withrespect to centralcharacterscompletes the proof. In two subsequent papers ([13, 14]) it was noticed that the category C([t]) closely connected to various categories of g-modules, independently appeared in differentcontexts.Theresultsofthesetwopaperscanbecollectedinthefollowing statement. Theorem 26. Assume that t n,j ∈Zfor allj. Then the following categories of g-modulesare equivalent: 1. ThecategoryC([t]). 2. The category of complete (in the sense of Enright, [7]) weight extensions of highestweight modules with integral support. 3. A certain category of algebraic Harish-Chandra bimodules in the sense of Bernstein andS.Gelfand([1]). Ideaofthe proof. The equivalence of the ﬁrst and the second categories is the content of [13]. It is based on a precise construction of the equivalence functor, which is a generalization of the Mathieu’s twist functor ([15]). The equivalence of the second and the third categories is proved in [14], using an intermediate equivalence of the second category with a category of injectively copresented modulesintheBernstein-Gelfand-Gelfandcategory O ([3]). kievarwe.tex; 12/03/2001; 3:49; p.312 306 V.MAZORCHUK 8. Case ofclassical and quantumalgebrasandopen problems An analogue of Theorem 18 for orthogonal algebras (simple ﬁnite dimensional complex Lie algebras of type B n andD n ) was obtained also by Gelfand and Zetlin in [10]. The corresponding generic modules were constructed in [17]. For symplectic Lie algebras (type C n ) an analogue of Theorem 18 is a recent result of Molev, [20]. For U q ( gl n )the classical result was obtained by Jimbo ([11]) and generic modules were constructed by Turowska and the author ([19]). For non- standard quantum deformations of orthogonal algebras the classical construction of Gelfand-Zetlin basis in ﬁnite-dimensional modules can be found in a series of recent papers by Klimyk and Jorgov, available at “xxx.lanl.gov”, where one can also ﬁndinformation about correspondingresults for root of unity case. Finally,wewanttogivealistofsomequestionsandopenproblemsrelatedto Gelfand-Zetlinmodules: 1. Classifyandgivea precise construction ofall simple GZ-modules. 2. Find a criterion, when a given character of Γhas only one extension to a simple g-module. 3. LetFbeasimpleﬁnitedimensional gl(n,C)-module.ConsidertwoGelfand- Zetlin basis of it, with respect to the inclusions of subalgebras into left upper andinto rightlower corners.Whatwill bethetransformationmatrix? 4. LetVbeasimpleGelfand-Zetlinmoduleand Fbeaﬁnite-dimensionalmod- ule. DoesV⊗Fhave a ﬁnite length? Is it possible to compute composition subquotients and multiplicities of V⊗F? 5. Are there any analogues of Gelfand-Zetlin construction for exceptional Lie algebras? 6. Extend all already known for gl(n,C)results to the case of orthogonal and symplecticalgebras.Alsoﬁndinthosecasessolutionstotheaboveproblems. References 1. I.Bernstein and S.Gelfand, Tensor product of ﬁnite and inﬁnite-dimensional representations of semisimple Lie algebras , Compositio Math., 41(1980), 245–285. 2. I.Bernstein, I.Gelfand and S.Gelfand, Structure of representations that are generated by vectors of highest weight , Funktsional. Anal.i Prilozhen., 5(1971),1–9. 3. I.Bernstein, I.Gelfand and S.Gelfand, A certain category of g-modules, Funktsional. Anal. i Prilozhen., 10(1976),1–8. 4. Yu.A.Drozd, S.A.Ovsienko and V.M.Futorny, Irreducible weighted sl(3)-modules, Funkt- sional. Anal. i Prilozhen., 23(1989), 57–58. 5. ———, Harish-Chandra subalgebras and Gelfand-Zetlin modules , Math. and Phys. Sci., 424(1994), 72–89. 6. ———, On Gelfand-Zetlin modules , Rend. Circ. Mat. Palermo (2) Suppl., 26(1991), 143– 147. 7. T.Enright, On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions andmultiplicity formulae , Annals Math., 110(1979),1–82. kievarwe.tex; 12/03/2001; 3:49; p.313 CATEGORIESOFGELFAND-ZETLINMODULES 307 8. V.Futorny, S.K ¨onig and V.Mazorchuk, Categories of induced modules and projectively stratiﬁed algebras , Univ. Bielefeld preprint, 99-024, Bielefeld, 1999, to appear in Algebr. Represent. Theory. 9. I.M.Gelfand, M.L.Zetlin, Finite-dimensional representations of the group of unimodular matrices, DokladyAkad. Nauk SSSR(N.S.), 71(1950), 825–828. 10. ———, Finite-dimensional representations of the group of orthogonal matrices , Doklady Akad. Nauk SSSR(N.S.), 71(1950), 1017–1020. 11. M.Jimbo, QuantumR-matrix for the generalized Toda system: an algebraic approach , in Field Theory, quantum gravity and strings, Lecture notes in Physics, 246, Springer-Verlag, Berlin-New York, 1986, pp. 335–361. 12. B.Kostant, On the tensor product of a ﬁnite and inﬁnite dimensional representations , Journal of Func.analisis, 20(1975),257–285. 13. S.K ¨onig and V.Mazorchuk, An equivalence of two categories of sl(n,C)-modules, Univ. Bielefeld preprint, 99-114, Bielefeld, 1999, toappear inAlgebr. Represent. Theory. 14. ———, Enright’scompletionsandinjectivelycopresentedmodules , Univ.Bielefeld preprint, 99-130,Bielefeld, 1999. 15. O.Mathieu, Classiﬁcation of simple weight modules , Annales Inst. Fourier, 50(2000), 537– 592. 16. V.Mazorchuk, Generalized Verma modules , Univ. Bielefeld preprint, E-99-006, Bielefeld, 1999, to be publishedas a monograph by Lviv Scientiﬁc Publisher. 17. ———, On Gelfand-Zetlin modules over orthogonal Lie algebras , Univ. Bielefeld preprint, 98-106,Bielefeld, 1998, to appear inAlgebra Colloq. 18. V.Mazorchuk, S.Ovsienko, Submodule structure of generalized Verma modules induced from generic Gelfand-Zetlin modules , Algebr. Represent. Theory, 1(1998),3–26. 19. V.Mazorchuk, L.Turowska, On Gelfand-Zetlin modules over U q (gl(n)), Czech. J. Phys., 50 (2000), 139–144. 20. A.Molev, A basis for representations of symplectic Lie algebras ,preprint, math.QA/9804127 . 21. S.Ovsienklo, Someﬁniteness statements for Gelfand-Zetlinmodules , to appear. 22. D.P.Zhelobenko, Compact Lie groups and their representation , Translation of Mathematical Monographs, Vol. 40, American MAthematical Society, Providence, R.I., 1973. kievarwe.tex; 12/03/2001; 3:49; p.314 kievarwe.tex; 12/03/2001; 3:49; p.315 HIDDENSYMMETRY OFSOME ALGEBRAS OF q-DIFFERENTIAL OPERATORS DMITRY SHKLYAROV, SERGEY SINEL’SHCHIKOV ∗ and LEONID VAKSMAN † InstituteforLowTemperaturePhysics&Engineering,47Leninave, 61164Kharkov,Ukraine 1. Introduction Let us explain the meaning of the words ”q-differential operators” and ”hidden symmetry”.Let C[z] q bethealgebraofpolynomialsin zovertheﬁeldofrational functionsC(q 1/2 )(we assume this ﬁeld to be the ground ﬁeld throughout the paper).Wedenoteby Λ 1 (C) q theC[z] q -bimodulewiththegenerator dzsuchthat z·dz=q −2 dz·z. Letdbethelinear map C[z] q →Λ 1 (C) q givenby the twoconditions: d:z/mapsto→dz, d(f 1 (z)f 2 (z)) =d(f 1 (z))f 2 (z) +f 1 (z)d(f 2 (z)). (ThelaterconditionisjusttheLeibnizrule).Thebimodule Λ 1 (C) q (togetherwith themapd)isawellknownﬁrstorderdifferentialcalculusoverthealgebra C[z] q . Thedifferential dallowsonetointroduceanoperatorof”partialderivative” d dz in C[z] q : d(f(z)) =dz·d f dz(z). Let us introduce also the notation /hatwide zfor the operator in C[z] q of multiplication by z: /hatwide z:f(z)/mapsto→zf(z) . ∗ [email protected] † [email protected] kievarwe.tex; 12/03/2001; 3:49; p.316 310 D. SHKLYAROV, S.SINEL’SHCHIKOV, L.VAKSMAN LetD(C) q be the subalgebra in the algebra End C(q 1/2 ) (C[z] q )(of all endomor- phisms of the linear space C[z] q ) containing 1and generated by d dz , /hatwide z. It is easy tocheckthat d dz· /hatwide z=q −2 /hatwide z· d dz+ 1. Thusthealgebra D(C) q isananalogue ofthe Weyl algebra A 1 (C). Letλ∈C(q 1/2 ).Onechecks that themap /hatwide z/mapsto→λ· /hatwide z, d dz/mapsto→λ −1 · d dz is extendable up to an automorphism of the algebra D(C) q . Such automorphisms are ”evident” symmetries of D(C) q . It turn out that they belong to a wider set of symmetries of D(C) q . This set does not consists of automorphisms only. Let us turn toprecise formulations. Tostartwith,werecallonethedeﬁnitionofthequantumuniversalenveloping algebraU q sl 2 [5].It is i)thealgebra given by thegenerators E,F,K,K −1 ,and the relations KK −1 =K −1 K= 1, KE =q 2 EK, KF =q −2 FK, EF−FE=K−K − 1 q−q −1 ; ii) the Hopf algebra : the comultiplication ∆, the antipode S, and the counit ε aredeterminedby ∆(E) =E⊗1 +K⊗E, ∆(F) =F⊗K −1 + 1⊗F, ∆(K) =K⊗K, S(E) =−K −1 E, S (F) =−FK, S (K) =K −1 , ε(E) =ε(F) = 0, ε (K) = 1. There is a well known structure of U q sl 2 -module in the space C[z] q . Let us describe itexplicitly: E:f(z)/mapsto→−q 1/2 z 2 f(z)−f(q 2 z ) z−q 2 z, F:f(z)/mapsto→q 1/2 f(z)−f(q −2 z ) z−q −2 z, K ±1 :f(z)/mapsto→f(q ±2 z). It can be checked that C[z] q is aU q sl 2 -module algebra, i.e. for any ξ∈U q sl 2 , f 1 ,f 2 ∈C[z] q ξ(1) =ε(ξ)·1, (1) kievarwe.tex; 12/03/2001; 3:49; p.317 HIDDEN SYMMETRY OFq-OPERATORS 311 ξ(f 1 f 2 ) = /summationdisplay j ξ /prime j (f 1 )ξ /prime/prime j (f 2 ), (2) with∆(ξ) = /summationtext j ξ /prime j ⊗ξ /prime/prime j . R EMARK . This observation is an analogue of the following one. The group SL 2 (C)acts onCP 1 via the fractional-linear transformations. Thus the universal envelopingalgebra U sl 2 actsviadifferentialoperatorsinthespaceofholomorphic functions on theopen cell C⊂CP 1 . LetVbe aU q sl 2 -module. Then the algebra End(V)admits a ”canonical” structure ofU q sl 2 -module:for ξ∈U q sl 2 ,T∈End(V) ξ(T) = /summationdisplay j ξ /prime j ·T·S(ξ /prime/prime j ), (3) where ∆(ξ) = /summationtext j ξ /prime j ⊗ξ /prime/prime j ,Sis the antipode, and the elements in the right-hand sidearemultipliedwithinthealgebra End(V).Itiswellknownthatthisactionof U q sl 2 inEnd(V)makes End(V)intoaU q sl 2 -modulealgebra(i.e.for ξ∈U q sl 2 , T 1 ,T 2 ∈End(V)(1),(2)holdwith f 1 ,f 2 beingreplacedby T 1 ,T 2 ,respectively). Theobjectsconsideredabovearethesimplestamongoneswedealwithinthe present paper. In this simplest case our main result can be formulated as follows: the algebra D(C) q is aU q sl 2 -module subalgebra in the U q sl 2 -module algebra End C(q 1/2 ) (C[z] q )(where theU q sl 2 -action is given by (3)). This U q sl 2 -module structure in thealgebra D(C) q iswhatwecall”hiddensymmetry” of D(C) q . R EMARK . InthesettingofthepreviousRemarktheanalogousfactisevident: forξ∈ sl 2 the action (3) is just the commutator of the differential operators ξ andTin the space of holomorphic functions on C. The commutator is again a differentialoperator. Wecan describe the U q sl 2 -actioninD(C) q explicitly: E( /hatwide z) =−q 1/2 /hatwide z 2 , F ( /hatwide z) =q 1/2 , K ±1 ( /hatwide z) =q ±2 /hatwide z, E( d dz) =q −3/2 (q −2 + 1) /hatwide z d dz, F ( d dz) = 0, K ±1 ( d dz) =q ∓2 d dz. (The action of U q sl 2 on an arbitrary element of D(C) q can be produced via the rule(2).) The paper is organizedasfollows. In Section 2 we recall one deﬁnitions of the quantum universal enveloping algebraU q sl N , aU q sl N -module algebra C[Mat m,n ] q of holomorphic polynomi- als on a quantum matrix space Mat m,n , and a well known ﬁrst order differential calculus Λ 1 (Mat m,n ) q overC[Mat m,n ] q (inthisIntroductionthecase m=n= 1 was considered). Then we introduce an algebra D(Mat m,n ) q of q-differential operators inC[Mat m,n ] q and formulate a main theorem concerning a hidden symmetryofthisalgebra. Section 3containsasketchof theproofof the main theorem. kievarwe.tex; 12/03/2001; 3:49; p.318 312 D. SHKLYAROV, S.SINEL’SHCHIKOV, L.VAKSMAN InSection4wediscussbrieﬂypossiblegeneralizationsofourresults.Specif- ically, the space Mat m,n is an example of a prehomogeneous vector space of commutative parabolic type [6]. In [9] q-analogs of all such vector spaces were introduced.Ourresultsadmitageneralizationonthecaseofanarbitraryquantum prehomogeneous vector spaceofcommutativeparabolictype. We take this opportunity to thank Prof. H. P. Jakobsen and Prof. T. Tanisaki who attracted our attention to other approaches to the notion of quantum differentialoperators. This research was partially supported by Award No.UM1-2091 of the U.S. Civilian Research andDevelopmentFoundation. 2. Themain theorem In this Section we deal with a well known q-analogue of the polynomial algebra onthespace Mat m,n ofm×nmatrices(intheIntroductionweconsideredthecase m=n= 1). Let the ground ﬁeld be the ﬁeld of rational functions C(q 1/2 ). The algebraC[Mat m,n ] q is the unital algebra given by its generators z α a ,a= 1,...n, α= 1,...m,andthefollowingrelations z α a z β b = =    qz β b z α a , a=b&α<β oraβ z β b z α a + (q−q −1 )z β a z α b , a<b &α<β,(4) The Hopf algebra U q sl N is determined by the generators E i ,F i ,K i ,K −1 i , i= 1,... ,N−1,and the relations K i K j =K j K i , K i K −1 i =K −1 i K i = 1, K i E j =q a ij E j K i , K i F j =q −a ij F j K i , E i F j −F j E i =δ ij (K i −K −1 i )/(q−q −1 ) E 2 i E j −(q+q −1 )E i E j E i +E j E 2 i = 0,|i−j|= 1 (5) F 2 i F j −(q+q −1 )F i F j F i +F j F 2 i = 0,|i−j|= 1 [E i ,E j ] = [F i ,F j ] = 0,|i−j|/negationslash= 1. The comultiplication ∆,theantipode S,and thecounit εare determined by ∆(E i ) =E i ⊗1 +K i ⊗E i ,∆(F i ) =F i ⊗K −1 i + 1⊗F i ,∆(K i ) =K i ⊗K i , (6) S(E i ) =−K −1 i E i , S (F i ) =−F i K i , S (K i ) =K −1 i ,(7) kievarwe.tex; 12/03/2001; 3:49; p.319 HIDDEN SYMMETRY OFq-OPERATORS 313 ε(E i ) =ε(F i ) = 0, ε (K i ) = 1. The algebraC[Mat m,n ] q possesses a structure of U q sl N -module algebra with N=m+n. Explicit formulae for the action of U q sl N inC[Mat m,n ] q are as follows(see[7]): K n z α a =    q 2 z α a , a=n&α=m qz α a , a=n&α/negationslash=mora/negationslash=n&α=m z α a ,otherwise,(8) F n z α a =q 1/2 · /braceleftbigg 1, a=n&α=m 0,otherwise, (9) E n z α a =−q 1/2 ·    q −1 z m a z α n , a/negationslash=n&α/negationslash=m (z m n ) 2 , a=n&α=m z m n z α a ,otherwise, (10) and withk/negationslash=n K k z α a = =    qz α a , k<n &a=kork>n &α=N−k q −1 z α a , k<n &a=k+ 1 ork>n &α=N−k+ 1 z α a ,otherwise,(11) F k z α a =q 1/2 ·    z α a+1 , k<n &a=k z α+1 a , k>n &α=N−k 0,otherwise, (12) E k z α a =q −1/2 ·    z α a−1 , k<n &a=k+ 1 z α−1 a , k>n &α=N−k+ 1 0,otherwise. (13) R EMARKS .i) In the classical case the corresponding action of U sl N in the space of holomorphic functions on Mat m,n can be produced via an embedding Mat m,n into the Grassmanian Gr m,N as an open cell (we describe a q-analogue oftheembeddingin [7]). ii) Using the structure of U q sl N -module inC[Mat m,n ] q we can deﬁne the structure of U q sl N -module algebra in End C(q 1/2 ) (C[Mat m,n ] q )via (3) with ξ∈ U q sl N ,T∈End C(q 1/2 ) (C[Mat m,n ] q ). Now let us recall a deﬁnition of a well known ﬁrst order differential calculus overC[Mat m,n ] q . Let Λ 1 (Mat m,n ) q be theC[Mat m,n ] q -bimodule given by its generatorsdz α a ,a= 1,...n,α= 1,...m, andthe relations kievarwe.tex; 12/03/2001; 3:49; p.320 314 D. SHKLYAROV, S.SINEL’SHCHIKOV, L.VAKSMAN z β b dz α a = m /summationdisplay α /prime ,β /prime =1n /summationdisplay a /prime ,b /prime =1 R β /prime α /prime βα R b /prime a /prime ba dz α /prime a /prime ·z β /prime b /prime , (14) with R b /prime a /prime ba =    q −1 , a=b=a /prime =b /prime 1, a/negationslash=b&a=a /prime &b=b /prime q −1 −q , a<b &a=b /prime &b=a /prime 0,otherwise.(15) The mapd:z α a /mapsto→dz α a can be extended up to a linear operator d: C[Mat m,n ] q →Λ 1 (Mat m,n ) q satisfyingtheLeibnizrule.Itwasnotedfortheﬁrst timein[8],thatthereexistsauniquestructureofa U q sl N -modulein Λ 1 (Mat m,n ) q suchthatthemap disamorphismof U q sl N -modules.Thepair /parenleftbig Λ 1 (Mat m,n ) q ,d /parenrightbig istheﬁrstorder differential calculusover C[Mat m,n ] q . Let us introduce an algebra D(Mat m,n ) q of q-differential operators on Mat m,n . For this purpose, we deﬁne the linear operators ∂ ∂z αa inC[Mat m,n ] q via thedifferential d: df= n /summationdisplay a=1m /summationdisplay α=1 dz α a ·∂ f ∂z αa , f∈C[Mat m,n ] q , and theoperators /hatwider z αa by /hatwider z αa f=z α a ·f, f∈C[Mat m,n ] q . ThenD(Mat m,n ) q istheunitalsubalgebrain End C(q 1/2 ) (C[Mat m,n ] q )generated by theoperators ∂ ∂z αa , /hatwider z αa ,a= 1,...n,α = 1,...m. Tostartwith, we describe D(Mat m,n ) q interms ofgenerators and relations. Proposition 2.1. The complete list of relations between the generators /hatwider z αa , ∂ ∂z αa , a= 1,...n,α= 1,...m,ofD(Mat m,n ) q isas follows /hatwide z α a /hatwide z β b = =    q /hatwide z β b /hatwide z α a , a=b&α<β oraβ /hatwide z β b /hatwide z α a + (q−q −1 ) /hatwide z β a /hatwide z α b , aβ ∂ ∂z αa ∂ ∂z β b + (q−q −1 ) ∂ ∂z β a ∂ ∂z α b , a<b &α<β, (17) ∂ ∂z αa /hatwide z β b = n /summationdisplay a /prime ,b /prime =1m /summationdisplay α /prime ,β /prime =1 R b /prime a ba /prime R β /prime α βα /prime /hatwide z β /prime b /prime ∂ ∂z α /prime a /prime +δ ab δ αβ , (18) withδ ab ,δ αβ beingtheKroneckersymbols, and R b /prime a ba /prime given by (15). Now wepresent themain resultofthepaper Theorem 2.2. i) The algebra D(Mat m,n ) q is aU q sl N -module subalgebra in the U q sl N -module algebra End C(q 1/2 ) (C[Mat m,n ] q ). ii) TheU q sl N -module structure in D(Mat m,n ) q is described explicitly as follows: U q sl N acts on the generators /hatwide z α a via formulae (8)-(13) (where z α a should be replacedby /hatwide z α a );forthegenerators ∂ ∂z αa theformulaeare K n ∂ ∂z αa =    q −2 ∂ ∂z αa , a=n&α=m q −1 ∂ ∂z αa , a=n&α/negationslash=mora/negationslash=n&α=m ∂ ∂z αa ,otherwise,(19) F n ∂ ∂z αa = 0a= 1,...n, α = 1,...m, (20) E n ∂ ∂z αa =q −3/2 · ·    n /summationtext b=1 /hatwide z m b ∂ ∂z m b + m /summationtext β=1 /hatwide z β n ∂ ∂z β n + (q −2 −1) n /summationtext b=1m /summationtext β=1 /hatwide z β b ∂ ∂z β b , a=n&α=m /summationtext m β=1 /hatwide z β n ∂ ∂z β a , a/negationslash=n&α=m /summationtext n b=1 /hatwide z m b ∂ ∂z α b , a=n&α/negationslash=m 0 ,otherwise, (21) kievarwe.tex; 12/03/2001; 3:49; p.322 316 D. SHKLYAROV, S.SINEL’SHCHIKOV, L.VAKSMAN andwithk/negationslash=n K k ∂ ∂z αa = =    q −1 ∂ ∂z αa , k<n &a=kork>n &α=N−k q ∂ ∂z αa , k<n &a=k+ 1 ork>n &α=N−k+ 1 ∂ ∂z αa ,otherwise,(22) F k ∂ ∂z αa =−q 3/2 ·    ∂ ∂z α a−1 , k<n &a=k+ 1 ∂ ∂z α−1 a , k>n &α=N−k+ 1 0,otherwise,(23) E k ∂ ∂z αa =−q −3/2 ·    ∂ ∂z α a+1 , k<n &a=k ∂ ∂z α+1 a , k>n &α=N−k 0,otherwise.(24) 3. Sketchofthe proof Let us outline an idea of the proof of the main theorem. To prove the statement i) ofthetheoremwehavetoexplainwhyforarbitrary ξ∈U q sl N ,T∈D(Mat m,n ) q ξ(T)∈D(Mat m,n ) q . (25) The mapz α a /mapsto→ /hatwide z α a , a= 1,...n,α= 1,...m, is extendable up to an embedding of algebras J:C[Mat m,n ] q /arrowhookleft→End C(q 1/2 ) (C[Mat m,n ] q ). Evidently, Jintertwines the actions of U q sl N inC[Mat m,n ] q andEnd C(q 1/2 ) (C[Mat m,n ] q ) (this is a corollary of the fact that C[Mat m,n ] q is aU q sl N -module algebra). This observation proves (25) for Tof the formJ(f),f∈C[Mat m,n ] q ,as well as the ﬁrst part of the statement ii) of the theorem. What remains is to prove (25) for T= ∂ ∂z αa ,a= 1,...n,α= 1,...m. The space End C(q 1/2 ) (C[Mat m,n ] q )can be made into a left C[Mat m,n ] q - moduleasfollows: z α a (T) = /hatwide z α a ·T, witha= 1,...n,α= 1,...m,T∈End C(q 1/2 ) (C[Mat m,n ] q ). This structure is compatible withthe action of U q sl N .Deﬁne the U q sl N -module Λ 1 (Mat m,n ) q /circlemultiplydisplay C[Mat m,n ] q End C(q 1/2 ) (C[Mat m,n ] q ). kievarwe.tex; 12/03/2001; 3:49; p.323 HIDDEN SYMMETRY OFq-OPERATORS 317 The differential d:C[Mat m,n ] q →Λ 1 (Mat m,n ) q is a morphism of the U q sl N - modules.This implies U q sl N -invariance oftheelement n /summationdisplay a=1m /summationdisplay α=1 dz α a ⊗ ∂ ∂z αa ∈Λ 1 (Mat m,n ) q /circlemultiplydisplay C[Mat m,n ] q End C(q 1/2 ) (C[Mat m,n ] q ), i.e. forallξ∈U q sl N n /summationdisplay a=1m /summationdisplay α=1 /summationdisplay j ξ /prime j dz α a ⊗ξ /prime/prime j ∂ ∂z αa =ε(ξ) n /summationdisplay a=1m /summationdisplay α=1 dz α a ⊗ ∂ ∂z αa (26) withεbeing the counit of U q sl N ,∆(ξ) = /summationtext j ξ /prime j ⊗ξ /prime/prime j (∆is the coproduct in U q sl N ).Aswasprovedin[7], Λ 1 (Mat m,n ) q isthefreeright C[Mat m,n ] q -module with the generators dz α a ,a= 1,...n,α= 1,...m. Thus, forξ∈U q sl N there existsauniqueset f b,α β,a (ξ)∈C[Mat m,n ] q ,a= 1,...n,α= 1,...m,b= 1,...n, β= 1,...m,such that ξdz α a = n /summationdisplay b=1m /summationdisplay β=1 dz β b f b,α β,a (ξ). Usingthe later equality, we canrewrite (26) asfollows: n /summationdisplay a,b=1m /summationdisplay α,β=1 /summationdisplay j dz β b ⊗f b,α β,a (ξ /prime j )ξ /prime/prime j ∂ ∂z αa =ε(ξ) n /summationdisplay a=1m /summationdisplay α=1 dz α a ⊗ ∂ ∂z αa .(27) Now one can obtain formulae (19) - (24) (and thus prove (25) for T= ∂ ∂z αa , a= 1,...n,α= 1,...m)viaapplying(27)tothegenerators E i ,F i ,K i ,K −1 i of U q sl N . 4. Concludingnotes The space Mat m,n ofm×nmatrices considered in the present paper is the sim- plest example of a prehomogeneous vector space of commutative parabolic type [6]. Such vector spaces are closely related to non-compact Hermitian symmetric spaces.Speciﬁcally,anynon-compactHermitiansymmetricspacecanberealized (viatheso-calledHarish-Chandraembedding)asaboundedsymmetricdomainin some prehomogeneousvector spaceof commutativeparabolictype. In[9]aq-analogueofanarbitraryprehomogeneousvectorspaceofcommuta- tiveparabolictypewasconstructed.Moreprecisely,let Ubeaboundedsymmetric domain, g −1 the corresponding prehomogeneous vector space, and gthe Lie al- gebra of the automorphism group of U. In the paper [9] a U q g-module algebra C[ g −1 ] q andacovariantﬁrstorderdifferentialcalculus /parenleftbig Λ 1 ( g −1 ),d /parenrightbig overC[ g −1 ] q kievarwe.tex; 12/03/2001; 3:49; p.324 318 D. SHKLYAROV, S.SINEL’SHCHIKOV, L.VAKSMAN were introduced. Using the ﬁrst order differential calculus, one can produce a deﬁnitionofq-differentialoperatorsin C[ g −1 ] q justasitwasdoneinSection2in thecase g −1 = Mat m,n . LetD( g −1 ) q be the algebra of q-differential operators in C[ g −1 ] q . In this general setting it can also be proved that D( g −1 ) q is aU q g-module subalgebra in theU q g-modulealgebra End(C[ g −1 ] q ).Indeed,iteasytoseethattheproofofour maintheorem(Section3)doesnotuseaspeciﬁcnatureofthecase g −1 = Mat m,n . 5. Appendix:q-Differential operatorsin holomorphicq-bundles. InthisAppendix’ C[Mat m,n ] q -module’meansright C[Mat m,n ] q -module. LetΓbe a ﬁnitely generated free C[Mat m,n ] q -module, i.e. there exists an isomorphism ofthe C[Mat m,n ] q -modules π: Γ→V /circlemultiplydisplay C[Mat m,n ] q , withVbeingaﬁnitedimensionalvectorspace.Theisomorphism πwillbecalled a trivialization of Γ. Elements of Γare q-analogs of sections of a holomorphic bundle on Mat m,n . Let us consider two such C[Mat m,n ] q -modules Γ 1 ,Γ 2 to- gether with their trivializations π 1 : Γ 1 →V 1 /circlemultiplytext C[Mat m,n ] q ,π 2 : Γ 2 → V 2 /circlemultiplytext C[Mat m,n ] q .Set D(Γ 1 ,Γ 2 ) q = {D∈Hom(Γ 1 ,Γ 2 )|π 2 ·D·π 1−1 ∈Hom(V 1 ,V 2 ) /circlemultiplydisplay D(Mat m,n ) q }. Elements of D(Γ 1 ,Γ 2 ) q can be treated as q-analogues of differential operators in sections ofholomorphic bundles. To see that D(Γ 1 ,Γ 2 ) q is well deﬁned, we need to verify its independence of the choice of trivializations. Let π /prime 1 : Γ 1 →V /prime 1 /circlemultiplytext C[Mat m,n ] q ,π /prime 2 : Γ 2 → V /prime 2 /circlemultiplytext C[Mat m,n ] q be other trivializations of Γ 1 andΓ 2 , respectively. Evidently, it is sufﬁcient to prove, that for an arbitrary D /prime ∈Hom(V 1 ,V 2 ) /circlemultiplytext D(Mat m,n ) q themapπ /prime 2 ·π −1 2 ·D /prime ·π 1 ·(π /prime 1 ) −1 belongsto Hom(V /prime 1 ,V /prime 2 ) /circlemultiplytext D(Mat m,n ) q .But this follows from the fact that π 1 ,π 2 ,π /prime 1 ,π /prime 2 are morphisms of the C[Mat m,n ] q - modules, and, thus, π 1 ·(π /prime 1 ) −1 ∈Hom(V /prime 1 ,V 1 ) /circlemultiplytext J(C[Mat m,n ] q )andπ /prime 2 · π 2−1 ∈Hom(V 2 ,V /prime 2 ) /circlemultiplytext J(C[Mat m,n ] q )(withJ(C[Mat m,n ] q )being the unital subalgebra in D(Mat m,n ) q generatedby /hatwide z α a ,a= 1,...n,α= 1,...m). In applications ﬁnitely generated free C[Mat m,n ] q -modules with some addi- tional properties arise. We will discuss two special types of such C[Mat m,n ] q - modules. Theﬁrsttypeconsistsofthoseﬁnitelygeneratedfree C[Mat m,n ] q -modules Γ which,inaddition,are U q sl N -module.Itmeansthat ΓisaU q sl N -moduleandthe multiplication map Γ /circlemultiplytext C[Mat m,n ] q →Γis amorphism of the U q sl N -modules. ForC[Mat m,n ] q -modules of this type a result analogous to the main theorem (Section2) canbe obtained. Letus turnto precise formulations. kievarwe.tex; 12/03/2001; 3:49; p.325 HIDDEN SYMMETRY OFq-OPERATORS 319 IfV 1 ,V 2 are modules over a Hopf algebra Athen the space Hom(V 1 ,V 2 ) admits the following ”canonical” structure of an A-module: for ξ∈A,T∈ Hom(V 1 ,V 2 ) ξ(T) = /summationdisplay j ξ /prime j ·T·S(ξ /prime/prime j ), (28) where ∆(ξ) = /summationtext j ξ /prime j ⊗ξ /prime/prime j (∆isthecoproduct), Sistheantipode,andtheproduct intheright-handsidemeansthecompositionofthemaps S(ξ /prime/prime j )∈End(V 1 ),T∈ Hom(V 1 ,V 2 ),ξ /prime j ∈End(V 2 ).Itiswellknownthatthisactionmakes Hom(V 1 ,V 2 ) intoanA-moduleleft End(V 2 )-moduleandan A-moduleright End(V 1 )-module, i.e. the compositionmap End(V 2 ) /circlemultiplydisplay Hom(V 1 ,V 2 ) /circlemultiplydisplay End(V 1 )→Hom(V 1 ,V 2 ) isamorphismof the A-modules. We can use the above construction to equip Hom(Γ 1 ,Γ 2 )(where Γ 1 ,Γ 2 are U q sl N -module ﬁnitely generated free C[Mat m,n ] q -modules) with the structure of aU q sl N -module.Usingour main theorem,one canprovethat the subspace D(Γ 1 ,Γ 2 ) q ⊂Hom(Γ 1 ,Γ 2 ) is U q sl N -invariant; thus, the compositionmap D(Γ 2 ) q /circlemultiplydisplay D(Γ 1 ,Γ 2 ) q /circlemultiplydisplay D(Γ 1 ) q →D(Γ 1 ,Γ 2 ) q (here D(Γ) q denotes D(Γ,Γ) q ) makes D(Γ 1 ,Γ 2 ) q into a U q sl N -module left D(Γ 2 ) q -module and a U q sl N -moduleright D(Γ 1 ) q -module. The second type of C[Mat m,n ] q -modules consists of those U q sl N -module C[Mat m,n ] q -modules which admit good trivializations. Let U q ( f+ p − )be the Hopf subalgebra in U q sl N generated by F i ,K ±1 i ,i= 1,...N−1, andE j , j= 1,...n−1,n+ 1,...N−1. Suppose that a ﬁnitely generated free C[Mat m,n ] q -module ΓisU q sl N -module(inparticular, ΓisaU q ( f+ p − )-module C[Mat m,n ] q -module). A trivialization π: Γ→V /circlemultiplytext C[Mat m,n ] q is called good trivialization if it satisﬁes the following conditions: i) Vis a ﬁnite dimensional U q ( f+ p − )-modulewiththeproperty F n v= 0foranyv∈V;ii)πisamorphism of theU q ( f+ p − )-modules (here V /circlemultiplytext C[Mat m,n ] q is endowed with U q ( f+ p − )- module structure via the coproduct ∆ :U q ( f+ p − )→U q ( f+ p − ) /circlemultiplytext U q ( f+ p − )). Itturnoutthatthesetofgoodtrivializationsofa C[Mat m,n ] q -module Γisnot too wide: if π 1 : Γ→V 1 /circlemultiplytext C[Mat m,n ] q ,π 2 : Γ→V 2 /circlemultiplytext C[Mat m,n ] q are two good trivializations,then π 2 ·π −1 1 =T⊗1 (29) withT∈Hom U q ( f+ p − ) (V 1 ,V 2 ). kievarwe.tex; 12/03/2001; 3:49; p.326 320 D. SHKLYAROV, S.SINEL’SHCHIKOV, L.VAKSMAN We distinguish this type of C[Mat m,n ] q -modules because for them the notion ofaq-differentialoperatorwithconstantcoefﬁcientsiswell-deﬁned.Speciﬁcally, letD(Mat m,n ) 0 q be the unital subalgebra in D(Mat m,n ) q generated by ∂ ∂z αa ,a= 1,...n,α= 1,...m. Suppose that Γ 1 ,Γ 2 areU q ( f+ p − )-moduleC[Mat m,n ] q - modules with good trivializations π 1 : Γ 1 →V 1 /circlemultiplytext C[Mat m,n ] q ,π 2 : Γ 2 → V 2 /circlemultiplytext C[Mat m,n ] q .We set D(Γ 1 ,Γ 2 ) 0 q ={D∈D(Γ 1 ,Γ 2 ) q |π 2 ·D·π 1−1 ∈Hom(V 1 ,V 2 ) /circlemultiplydisplay D(Mat m,n ) 0 q }. Elements of D(Γ 1 ,Γ 2 ) 0 q can be treated as q-analogues of the differential opera- tors with constant coefﬁcients in sections of holomorphic bundles. Independence D(Γ 1 ,Γ 2 ) 0 q of trivializations directly follows from the relationship (29) between twoarbitrary goodtrivializationsof a C[Mat m,n ] q -module. References 1. E. E. Demidov, Modules over quantum Weyl algebras , Vestnik MGU, Mathematics and Mechanics, 1(1993), 53–56. 2. H. P. Jakobsen, Quantized Hermitian Symmetric Spaces , In ”Lie theory and its applications in physics” (Clausthal, 1995), 105 – 116. 3. ———-, Q-Differential Operators , E-print: math.QA/9907009, 1999. 4. A. Kamita, Y. Morita, and T. Tanisaki, Quantum deformations of certain prehomogeneous spaces I, Hiroshima Math. J., 28(1998),527 – 540. 5. M. Rosso, Representations des groups quantiques , Seminaire BOURBAKI, 744(1991), 443 – 483. 6. H.Rubenthaler, Lespairesdualesdanslesalg `ebresdeLier ´eductives, Asterisque, 219(1994). 7. D. Shklyarov, S. Sinel’shchikov, and L. Vaksman, Quantum matrix ball: differential and integral calculi , E-print: math.QA/9905035,1999. 8. S. Sinel’shchikov and L. Vaksman, Hidden symmetry of the differential calculus on the quantum matrix space , J. Phys. A. 30(1997), 23 – 26. 9. ———-, On q-analogues of bounded symmetric domains and Dolbeault complexes , Math. Phys., Anal., and Geom., 1(1998), 75 – 100; E-print: q-alg/9703005, 1997. kievarwe.tex; 12/03/2001; 3:49; p.327 A FAMILY OF∗-ALGEBRAS ALLOWING WICK ORDERING: FOCK REPRESENTATIONSANDUNIVERSALENVELOPING C ∗ -ALGEBRAS PALLEJORGENSEN ∗ Department of Mathematics, The University of Iowa, Iowa City, Iowa52242-1419U.S.A. DANIIL PROSKURIN † Kyiv Taras Shevchenko University, Cybernetics Department, Volodymyrska,64, Kyiv,01033,Ukraine YURIISAMOILENKO ‡ Institute of Mathematics, National Academy of Sciences, Tereschenkivska, 3,Kyiv,01601,Ukraine Abstract. WeconsideranabstractWickorderingasafamilyofrelationsonelements a i anddeﬁne ∗-algebrasbytheserelations.Therelationsaregivenbyaﬁxedoperator T: h⊗ h→ h⊗ h,where his one-particle space, and they naturally deﬁne both a ∗-algebra and an inner-product space H T , /angbracketleft·,·/angbracketright T . Ifa ∗ i denotes the adjoint, i.e., /angbracketlefta i ϕ,ψ/angbracketright T =/angbracketleftϕ,a ∗ i ψ/angbracketright T , then we identify when /angbracketleft·,·/angbracketright T is positive semideﬁnite (the positivity question!). In the case of deformations of the CCR-relations (theq ij -CCRandthetwistedCCR’s),weworkouttheuniversal C ∗ -algebras A,andweprovethat, in these cases, the Fock representations ofthe A’s are faithful. 1. Introduction In recent papers [1–6], the applications of Lie superalgebras, quantum groups, q- algebrasinmathematicalphysicshavestimulatedinterestinthe ∗-algebrasdeﬁned by generators and relations and their representations by Hilbert space operators. For example, the representations of various deformations of canonical commu- tation relations (CCR), in particular Fock representaion, were used to construct non-classical models of theoretical physics and probability, such as the free quon gas(see[7]), q-Gaussian processes(see[8]) etc. ∗ [email protected] † [email protected] ‡ yurii [email protected] kievarwe.tex; 12/03/2001; 3:49; p.328 322 P.JORGENSEN,D.PROSKURIN,Y. SAMOILENKO The constructions are interesting from both physical and mathematical points of view. They give a canonical realisation of a given deformed relation like the Fock representation, or a realisation by differential operators. When the rela- tions can be realised by bounded operators, it is useful to study the universal envelopingC ∗ -algebrasforthemandthestabilityofisomorphismclassesofthese C ∗ -algebras on parameters (see for example [9, 10]). The stability question [10] referstohowthe C ∗ -isomorphismclassesdependonvariationsinthedeformation variables; in some cases there are open regions in parameter space where the C ∗ -isomorphism classisconstant. In the present paper we give a review of some results concerning a wide class ofdeformed relationsofthe following form a ∗ i a j =δ ij 1 + d /summationdisplay k,l=1 T kl ij a l a ∗ k , i,j = 1,... ,d, (1) whereT kl ij ∈C, such thatT kl ij =¯T lk ji . These relations generate a ∗-algebra al- lowing Wick ordering or Wick algebra (see [4, 11–13]). The ∗-algebra A T has a naturally deﬁned Fock vacuum “state” or functional and there is a corresponding inner-productspace H T ,/angbracketleft·,·/angbracketright T ,suchthat,intheassociatedGNS-representation, theidentity/angbracketlefta i ϕ,ψ/angbracketright T =/angbracketleftϕ,a ∗ i ψ/angbracketright T holds.Butthevacuumfunctionalisgenerally notpositive,andtheoperatorsintherepresentationnotbounded,andthereforethe Hermitian inner product /angbracketleft·,·/angbracketright T is then generally not positive semideﬁnite. The positivityquestion,andthefaithfulnessoftheFockrepresentation,arethefociof thispaper. Note that (1) generalizes some well-known types of deformed commutation relations, quantum groups, etc. (see [1, 3, 5, 6, 8, 12, 14, 15]). The basic exam- ples for us will be the q ij -CCR introduced and studied by M. Bo ˙ zejko and R. Speicher (see [8, 12]), and the twisted canonical commutation relations (TCCR) constructedbyW.PuszandS.L.Woronowicz(see[6]).Theywerefurtherstudied in[16]wherethetraditionalCuntzalgebraof[17]wasconsideredasabase-point, corresponding to q ij = 0, and the variation of the C ∗ -isomorphism class was considered asafunction of q ij . E XAMPLE 1.q ij -CCR, 2dgenerators: C/angbracketlefta i , a ∗ i |a ∗ i a j =δ ij 1 +q ij a j a ∗ i , i,j = 1,... ,d, q ji =q ij ∈C,|q ij |≤1/angbracketright E XAMPLE 2. TheWick algebrafor TCCR: a ∗ i a i = 1 +µ 2 a i a ∗ i −(1−µ 2 ) /summationdisplay k<i a k a ∗ k , i= 1,... ,d a ∗ i a j =µa j a ∗ i , i/negationslash=j,0<µ< 1 kievarwe.tex; 12/03/2001; 3:49; p.329 ∗-ALGEBRAS ALLOWING WICKORDERING 323 We present some sufﬁcient conditions on the coefﬁcients {T kl ij }for the exis- tenceoftheFockrepresentation,andwedescribethestructureoftheFockspace. We also give conditions for the faithfulness of Fock representation and describe itskernel inthedegenerated case (see Sec. 3). Furtherweconsidertheuniversal C ∗ -algebrasfortheexamplesabove.Specif- ically we show that the universal C ∗ -algebras for q ij -CCR (TCCR) can be generatedbyisometries(partialisometries)satisfyingacertainalgebraicrelation. The description of the C ∗ -isomorphism classes for different values of parameters ispresented. We also show that the Fock representations of q ij -CCR for some values of parameters, and TCCR for any value of parameter, are faithful on the C ∗ -level, i.e., the Fock representations of the corresponding C ∗ -algebras are faithful (see Sec. 4). The complete proofs of all results presented here can be found in [4, 10, 11, 18, 19]. For detailed information about ∗-representations of ﬁnitely generated ∗- algebras see [20]. 2. Basic deﬁnitions Firstly let us construct a canonical realization of Wick algebra, i.e., the ∗-algebra on the relations (1), with coefﬁcients {T kl ij }: we denote it by W(T). To do it consider a ﬁnite-dimensional Hilbert space H=/angbracketlefte 1 ,... ,e d /angbracketright. Construct the full tensoralgebraover H,H ∗ , denotedbyT(H,H ∗ ).Then W(T)∼=T(H,H ∗ )//angbracketlefte ∗ i ⊗e j −δ ij 1− /summationdisplay T kl ij e i ⊗e ∗ j /angbracketright, (2) dividingoutbythetwo-sidedidealontherelations(1).Notethatinthisrealization thesubalgebraof W(T)generatedby{a i }isidentiﬁed withthe T(H). The following operators were presented in [11] as a useful tool for computa- tion with Wickalgebrasand their Fockrepresentations. T:H⊗H/mapsto→H⊗H , Te k ⊗e l = /summationdisplay i,j T lj ik e i ⊗e j , T=T ∗ T i :H ⊗n /mapsto→H ⊗n , T i = 1⊗···⊗ 1 /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright i−1 ⊗T⊗1⊗···⊗ 1 /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright n−i−1 , R n :H ⊗n /mapsto→H ⊗n , R n = 1 +T 1 +T 1 T 2 +···+T 1 T 2 ···T n−1 , P n :H ⊗n /mapsto→H ⊗n , P 2 =R 2 , P n+1 = (1⊗P n )R n+1 . (3) The sequences of operators P 0 = 1 vac ,R 1 = 1 +T,P 1 = (1⊗1)(1 +T)∼= 1 +T,R 2 ,...,P n aredeﬁnedrecursively.Itisthesequence P n whichentersinto the positivity question. The other one is only intermediate. The Hermitian inner product/angbracketleft·,·/angbracketright T onT n (H)isthen /angbracketleftφ,ψ/angbracketright T n (H) :=/angbracketleftϕ,P n ψ/angbracketright tensor (4) kievarwe.tex; 12/03/2001; 3:49; p.330 324 P.JORGENSEN,D.PROSKURIN,Y. SAMOILENKO where/angbracketleft·,·/angbracketright tensor is just the usual inner product on T n (H)induced by/angbracketleft·,·/angbracketrighton H.Hence,weneedconditionson T:H⊗H→H⊗H whichmaketheoperators P n positiveforall n.Forexample,intermsoftheseoperatorswecandescribethe procedure of Wick ordering, i.e., the commutation formula for ﬁxed generator a ∗ i and anyhomogeneous polynomialin a k , k= 1,... ,d(see [21]). Proposition 27. LetX∈H ⊗n .Then e ∗ i ⊗X=µ(e ∗ i )R n X+µ(e ∗ i ) d /summationdisplay k=1 T 1 T 2 ···T n (X⊗e k )e ∗ k ,(5) whereµ(e ∗ i ):T(H)/mapsto→T(H)is deﬁnedasfollows µ(e ∗ i )1 = 0, µ(e ∗ i )e i 1 ⊗···⊗e i n =δ ii 1 e i 2 ⊗···⊗e i n . Forourexamples theoperator Thave thefollowingform: E XAMPLE 3. Te i ⊗e j =q ij e j ⊗e i ,i,j= 1,... ,d. E XAMPLE 4. Te i ⊗e i =µ 2 e i ⊗e i Te i ⊗e j =µe j ⊗e i , i<j Te i ⊗e j =−(1−µ 2 )e i ⊗e j +µe j ⊗e i , i>j. Note that for both examples, the operator Tsatisﬁes a braid condition , i.e., on theH ⊗3 we have T 1 T 2 T 1 =T 2 T 1 T 2 . (6) The operators presented above appear naturally in construction of Fock represen- tation ofW(T). This notion is induced in the obvious way from the classical one for CCR,however,in general,theFock spaceis notalways symmetric(see [11]). Deﬁnition28. Therepresentation λ 0 actingonthe space T(H)by formulas λ 0 (a i )e i 1 ⊗···⊗e i n =e i ⊗e i 1 ⊗···⊗e i n , n∈N∪{0} λ 0 (a ∗ i )1 vac = 0 where the action of λ 0 (a ∗ i )on the monomials of degree n≥1is determined inductivelyusing thebasicrelations, iscalledtheFock representation. It is easy to see that λ 0 (a i )are the classical creation operators and λ 0 (a ∗ i )are twistedannihilationones.Evidentlyinthiswaywehaveconstructedarepresenta- tionofW(T),butnotyeta∗-representation.Todoitonehastosupplythe T(H) by theappropriateinnerproduct (see[11]). This iswhere formula(4) comes in. kievarwe.tex; 12/03/2001; 3:49; p.331 ∗-ALGEBRAS ALLOWING WICKORDERING 325 Deﬁnition 29. The Fock inner product (see [11]) is the unique semilinear Hermitian form/angbracketleft,/angbracketright T onT(H)such that /angbracketleftλ 0 (a i )X,Y/angbracketright T =/angbracketleftX,λ 0 (a ∗ i )Y/angbracketright T , X,Y∈T(H). Similarly to the deﬁnition of Fock representation, the Fock inner product on T(H)canbecomputedinductively.Itiseasytoseethatfor X∈H ⊗m ,Y∈H ⊗n , n/negationslash=m,wehave/angbracketleftX,Y/angbracketright T = 0.Onthecomponentsofpowers 0,1,theFockinner product concideswith thestandard one.Forany X,Y∈H ⊗n ,n≥2, wehave /angbracketleftX,Y/angbracketright T =/angbracketleftX,P n Y/angbracketright, which agreeswith(4) above. Theoperator P n =P n (T)aregiven in (3). Evidently, if we want to extend the Fock representation of W(T)to the∗- representationbyHilbert-spaceoperators,weshouldrequirethatalltheoperators P n ,n= 2,..., bepositive semideﬁnite, andthatthesubspace I= /circleplusdisplay n≥2 KerP n determines the kernel of the Fock inner product. Consequently the Hilbert-space structure ofthe Fock spaceemerges. 3. ThestructureoftheFock representation In this section we present some sufﬁcient conditions posed on the operator Tfor the positive-deﬁnite property of the Fock inner product, and we show that the kernel of the Fock representation is generated as a ∗-idealby the kernel of the Fock inner product. In particular, when the Fock inner product is strictly positive deﬁnite (i.e., when it has zero kernel), the Fock representation π F is faithful, i.e., Ker(π F ) = 0. There are several sufﬁcient conditions on the operator Tfor the Fock inner producttobepositive.Itwasshownin[10]thatforsufﬁcientlysmallcoefﬁcients wehavestrictpositivityoftheFockinnerproduct.Thisresultisacorollaryofthe stability of the universal enveloping C ∗ -algebra for the Wick algebra around the zero basepoint(see Sec.4). Theorem 30. If the operator Tsatisﬁes the norm bound /bardblT/bardbl< √ 2−1, then P n >0,n≥2,where>refersto strictpositivity. Anotherkind ofsufﬁcientconditionispositivity ofoperator T(see[11]). Theorem31. IfT≥0thenP n >0,n≥2. In the present paper we will suppose that the operator Tsatisﬁes the braid condition (6). It was shown by M. Bo ˙ zejko and R. Speicher (see [12]) that, in kievarwe.tex; 12/03/2001; 3:49; p.332 326 P.JORGENSEN,D.PROSKURIN,Y. SAMOILENKO this case, the operators P n ,n≥2, have a natural description in terms of quasi- multiplicative operator-valued mappings on the Coxeter group S n . The following is a corollary of a much more general result proved in [12] for mappings on the generalCoxetergroup. Theorem 32. LetTsatisfy the braid condition (6)and suppose−1≤T≤1. ThenP n ≥0. Moreover, if/bardblT/bardbl≤1, thenP n >0, and the operators of the Fock representationarebounded,i.e.,theFockrepresentationisbyboundedoperators. (Recall,theFockrepresentationof theundeformed CCR-algebra is unbounded. ) We present a more precise version of this theorem. Namely, we give the de- scription of kernel of P n in the degenerate case. As an immediate corollary of this result we have the strict positivity of P n ,n≥2, for braided Tsatisfying the inequality−1<T≤1(see [4]). Theorem 33. LetW(T)be a Wick algebra with braided operator Tsatisfying thenormbound/bardblT/bardbl≤1. Thenfor any n≥1, KerP n+1 = /summationdisplay k+l=n−1 H ⊗ k ⊗Ker(1 +T)⊗H ⊗ l = n /summationdisplay k=1 Ker(1 +T k ). Letusillustrate thisresulton theexamples. E XAMPLE 5. Forq ij -CCRwe havethealternatives: −|q ij |<1foranyi,j= 1,... ,d. In thiscase−1<T < 1and theFockinnerproduct is strictlypositive. −|q ij |= 1,i/negationslash=j. Forthese valuesof parameterswe have −1≤T≤1and Ker(1 +T) =/angbracketlefta j a i −q ij a i a j , i<j/angbracketright. E XAMPLE 6. Forthe TCCRWickalgebra, wehave −1≤T≤1,and Ker(1 +T) =/angbracketlefta j a i −µa i a j , i<j/angbracketright. The following proposition shows that, for algebras with braided operator T, thekerneloftheFockrepresentationisgeneratedasa ∗-idealbythekernelofthe Fockinner product, i.e., I= /circleplusdisplay n≥2 KerP n . Proposition34. LetW(T)beaWickalgebrawithbraidedoperator Tandletthe Fockrepresentation λ 0 bepositive (i.e.,theFockinnerproductispositivedeﬁnite ). Then Kerλ 0 =I⊗T (H ∗ ) +T(H)⊗I ∗ . kievarwe.tex; 12/03/2001; 3:49; p.333 ∗-ALGEBRAS ALLOWING WICKORDERING 327 Combiningthis proposition withTheorem 33,weget: Theorem 35. LetW(T)be a Wick algebra with the braided operator T,−1≤ T≤1. Then the kernel of the Fock representation is generated as a ∗-ideal by Ker(1 +T). This theorem implies that, for q ij -CCR,|q ij |<1, the Fock representation is faithful. For the TCCR Wick algebra, and for q ij -CCR, the kernels of the Fock representations are generated by the families a j a i −µa i a j ,i < j, anda j a i − q ij a i a j ,i < j, respectively; and hence the Fock representations of quotients of these algebrasby the ∗-ideals generatedby thesefamiliesare faithful. 4. Universal bounded representation Inthissectionwediscussuniversalenveloping C ∗ -algebrasfor q ij -CCRandWick TCCR. Let us recall that the universal C ∗ -algebra for a certain ∗-algebraAis also called the universal bounded representation. It is the C ∗ -algebra Awith natural homomorphism ψ:A→Asuchthat,foranyhomomorphism ϕ:A→B,where Bis aC ∗ -algebra, there exists a unique homomorphism θ:A→Bsatisfying θ◦ψ=ϕ. It can be obtained by the completion of A/Jwith the following C ∗ -seminormonA: /bardbla/bardbl= sup π /bardblπ(a)/bardbl, where supis taken over all bounded representations of A, andJis the kernel of this seminorm. Obviously this process requires that sup π /bardblπ(a)/bardbl<∞for any a∈A. Note thatforourexamples thisconditionissatisﬁed. Theuniversalboundedrepresentationfor q ij -CCRwasstudiedin[9,10].The followingproposition follows fromthemain resultofpaper [10]. Proposition 36. LetA {q ij } be the universal enveloping C ∗ -algebra for q ij -CCR, |q ij |< √ 2−1. Thenthereexiststhe naturalisomorphism A {q ij } ∼=A 0 , where A 0 isaC ∗ -algebrageneratedbytheisometries s i ,i= 1,... ,d,satisfying s ∗ i s j = 0, i/negationslash=j i.e.,isomorphismwiththe Cuntz-Toeplitz algebra. This impliesthatthe Fock representationof A {q ij } is faithful. Let us consider the A {q ij } ,|q ij |= 1, for anyi/negationslash=jandq ii :=q i ,|q i |<1 (i.e., unimodular off-diagonal terms). In this case, we do not have stability on the whole setof parameters(see [18]). kievarwe.tex; 12/03/2001; 3:49; p.334 328 P.JORGENSEN,D.PROSKURIN,Y. SAMOILENKO Proposition 37. If for anyi/negationslash=jwe have|q ij |= 1, then A {q ij } is isomorphic to theC ∗ -algebra A 0,{q i } generatedbyisometries {s i , i= 1,... ,d}satisfying s ∗ i s j =q ij s j s ∗ i , s j s i =q ij s i s j , i/negationslash=j, andtheFockrepresentationof A {q ij } is faithful. Finally for the universal C ∗ -algebra A µ for the Wick TCCR, we have the isomorphism A µ ∼=A 0 for any−1< µ < 1, where the C ∗ -algebra A 0 is generatedbythepartialisometries {s i , i= 1,... ,d}satisfying the relations s ∗ i s j =δ ij   1− /summationdisplay k<i s k s ∗ k   , i,j = 1,... ,d. The Fockrepresentationof A µ is faithfulalso(see [19]). A CKNOWLEDGEMENTS . P. J. was partially supported by the NSF under grants DMS-9700130andINT-9722779. References 1. L.C. Biedenharn, The quantum group SU q (2)and aq-analogue of the boson operators , J. Phys. A22(1989), L873–L878. 2. I.M. Burban and A.U. Klimyk, On spectral properties of q-oscillator operators , Lett. Math. Phys.29(1993), 13–18. 3. D.I. Fivel, Interpolation between Fermi and Bose statistics using generalized commutators , Phys. Rev. Lett. 65(1990), 3361–3364. 4. P.E.T. Jørgensen, D.P. Proskurin and Yu. S. Samo ˘ılenko,The kernel of Fock representations of Wick algebras with braided operator of coefﬁcients , accepted for publication in Paciﬁc J. Math., math-ph/0001011. 5. A.J.Macfarlane, Onq-analoguesofthequantumharmonicoscillatorandthequantumgroup SU(2) q , J. Phys. A 22(1989), 4581–4588. 6. W. Pusz and S.L. Woronowicz, Twisted second quantization , Rep. Math. Phys. 27(1989), 251–263. 7. R.F. Werner, The free quon gas suffers Gibbs’ paradox , Phys. Rev. D (3) 48(1993), 2929– 2934. 8. M.Bo ˙ zejkoandR.Speicher, AnexampleofageneralizedBrownianmotion ,Commun.Math. Phys.137(1991), 519–531. 9. K.DykemaandA.Nica, OntheFockrepresentationofthe q-commutationrelations ,J.Reine Angew. Math. 440(1993), 201–212. 10. P.E.T. Jørgensen, L.M. Schmitt and R.F. Werner, q-canonical commutation relations and stability of the Cuntzalgebra , Paciﬁc J. Math. 165(1994), 131–151. 11. , PositiverepresentationsofgeneralcommutationrelationsallowingWickordering ,J. Funct. Anal. 134(1995), 33–99. 12. M. Bo ˙ zejko and R. Speicher, Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces , Math. Ann. 300(1994), 97–120. 13. W. Marcinek and R. Ralowski, On Wick algebras with braid relations , J. Math. Phys. 36 (1995), 2803–2820. kievarwe.tex; 12/03/2001; 3:49; p.335 ∗-ALGEBRAS ALLOWING WICKORDERING 329 14. O.W. Greenberg, Particles with small violations of Fermi or Bose statistics , Phys. Rev. D (3) 43(1991), 4111–4129. 15. W. Marcinek, On commutation relations for quons , Rep.Math. Phys. 41(1998),155–172. 16. P.E.T. Jørgensen and R.F. Werner, Coherent states of the q-canonical commutation relations , Commun. Math. Phys. 164(1994), 455–471. 17. J.Cuntz, SimpleC ∗ -algebrasgeneratedbyisometries ,Commun.Math.Phys. 57(1977),173– 185. 18. D. Proskurin, Stability of a special class of q ij -CCR and extensions of higher-dimensional noncommutative tori , to appear in Lett.Math. Phys. 19. D. Proskurin and Yu. Samoilenko, Stability of a C ∗ algebra associated with the TCCR , submitted to Algebras andRepresentation Theory. 20. V. Ostrovsky ˘ı and Yu. Samo ˘ılenko,Introduction to the Theory of Representations of Finitely Presented∗-Algebras, I: Representations by bounded operators , The Gordon and Breach Publishing Group, London, 1999. 21. D.P.Proskurin, HomogeneousidealsinWick ∗-algebras ,Proc.Amer.Math.Soc. 126(1998), 3371–3376. kievarwe.tex; 12/03/2001; 3:49; p.336 kievarwe.tex; 12/03/2001; 3:49; p.337 NONSTANDARD QUANTIZATION OF THE ENEVLOPING ALGEBRA U(so(n)) ANDITSAPPLICATIONS A. U.KLIMYK ∗ Institutefor Theoretical Physics, Kiev,Ukraine 1. Introduction Quantum orthogonal groups, quantum Lorentz groups and their corresponding quantum algebras are of special interest for modern mathematical physics (see, forexample,[1]and[2]).M.Jimbo[3]andV.Drinfeld[4]deﬁned q-deformations (quantumalgebras) U q (g)forallsimplecomplexLiealgebras gbymeansofCar- tan subalgebras and root subspaces (see also [5] and [6]). Reshetikhin, Takhtajan and Faddeev [7] deﬁned quantum algebras U q (g)in terms of the quantum R- matrixsatisfyingthequantumYang–Baxterequation.However,theseapproaches donotgiveasatisfactorypresentationofthequantumalgebra U q (so(n,C))froma viewpointofsomeproblemsinquantumphysicsandrepresentationtheory.When considering representations of the quantum groups SO q (n+ 1)andSO q (n,1) we are interested in reducing them onto the quantum subgroup SO q (n). This reductionwouldgiveananalogueoftheGel’fand–Tsetlinbasisfortheserepresen- tations. However, deﬁnitions of quantum algebras mentioned above do not allow the inclusions U q (so(n+ 1,C))⊃U q (so(n,C))andU q (so n,1 )⊃U q (so n ). To be able to exploit such reductions we have to consider q-deformations of the Lie algebra so(n+1,C)deﬁnedintermsofthegenerators I k,k−1 =E k,k−1 −E k−1,k (whereE is is the matrix with elements (E is ) rt =δ ir δ st )rather than by means ofCartansubalgebrasandrootelements.Toconstructsuchdeformationswehave to deform trilinear relations for elements I k,k−1 instead of Serre’s relations (used in the case of Jimbo’s quantum algebras). As a result, we obtain the associative algebrawhichwillbedenotedas U /prime q (so(n,C)).Thisq-deformationwasﬁrstcon- structedin[8].Itpermitsonetoconstructthereductionsof U /prime q (so(n+ 1,C)onto U /prime q (so(n,C) . ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.338 332 A.U.KLIMYK In the classical case, the imbedding SO(n)⊂SU(n)(and its inﬁnitesimal analogue)isofgreatimportancefornuclearphysicsandinthetheoryofRieman- niansymmetricspaces.ItiswellknownthatintheframeworkofDrinfeld–Jimbo quantumgroupsandalgebrasonecannotconstructthecorrespondingembedding. The algebra U /prime q (so(n,C))allows to deﬁne such an embedding [9], that is, it is possible to deﬁne the embedding U /prime q (so(n,C))⊂U q (sl n ), whereU q (sl n )is a Drinfeld-Jimboquantum algebra. As a disadvantage of the algebra U /prime q (so(n,C))we have to mention the difﬁ- culties with Hopf algebra structure. Nevertheless, U /prime q (so(n,C))turns out to be a coidealinU q (sl n )(see[9])andthisfactallowsustoconsidertensorproductsofﬁ- nitedimensionalirreduciblerepresentationsof U /prime q (so(n,C))formanyinteresting cases(see[10]for the case U /prime q (so(3,C))). For convenience, below we denote the Lie algebra so(n,C)byso n and the q-deformedalgebra U /prime q (so(n,C))byU /prime q (so n ). Finite dimensional irreducible representations of the algebra U /prime q (so n )were constructed in [8]. The formulas of action of the generators of U /prime q (so n )upon the basis (which is a q-analogue of the Gel’fand–Tsetlin basis) are given there. A proof of these formulas and some their corrections were given in [11]. However, ﬁnite dimensional irreducible representations described in [8] and [11] are rep- resentations of the classical type. They are q-deformations of the corresponding irreducible representations of the Lie algebra so n , that is, atq→1they turn into representations of so n . The algebra U /prime q (so n )has other classes of ﬁnite dimensional irreducible rep- resentations which have no classical analogue. These representations are singular at the limitq→1. They are described in [12]. Note that the description of these representations for thealgebra U /prime q (so 3 )isgiven in[10]. AsinthecaseofDrinfeld–Jimboquantumalgebras,when qisarootofunity, then the representation theory of U /prime q (so n )is much more rich. In this case all ir- reducible representations of U /prime q (so n )are ﬁnite dimensional. The corresponding theorem is proved by means of an analogue of the Poincar ´e–Birkhoff–Witt theo- rem forU /prime q (so n )(this analogue was announced in [13]) and use central elements ofthisalgebra for qa root ofunity (they arederived in[14]). 2. Theq-deformed algebra U /prime q (so n ) The universal enveloping algebra U(so n )of the Lie algebra so n has two different structures.TheﬁrstoneisdeterminedbyrootsandrootelementsoftheLiealgebra so n . A deformation of U(so n )equipped with this structure leads to the Drinfeld– Jimbo quantum algebra U q (so n ). The second structure of U(so n )is related to the basis of the Lie algebra so n consisting of skew-symmetric matrices. A de- formation of U(so n )equipped with this structure leads to the algebra U /prime q (so n ) considered inthispaper. kievarwe.tex; 12/03/2001; 3:49; p.339 NONSTANDARDQUANTIZATION OF U(so(n)) 333 Inordertoobtain U /prime q (so n )wehavetotakedeterminingrelationsforthegener- ating elements I 21 ,I 32 ,···,I n,n−1 ofU(so n )and to deform these relations. The elementsI 21 ,I 32 ,···,I n,n−1 belong to the basis I ij ,i > j, of the Lie algebra so n . The matrices I ij ,i > j, are deﬁned as I ij =E ij −E ji , whereE ij is the matrix with entries (E ij ) rs =δ ir δ js . The universal enveloping algebra U(so n )is generated by a part of the basis elements I ij ,i>j, namely, by the elements I 21 , I 32 ,···,I n,n−1 .Theseelements satisfythe relations I 2 i,i−1 I i+1,i −2I i,i−1 I i+1,i I i,i−1 +I i+1,i I 2 i,i−1 =−I i+1,i , I i,i−1 I 2 i+1,i −2I i+1,i I i,i−1 I i+1,i +I 2 i+1,i I i,i−1 =−I i,i−1 , I i,i−1 I j,j−1 −I j,j−1 I i,i−1 = 0 for|i−j|>1. The following theoremistrue[15]for theenveloping algebra U(so n ). Theorem 1. The universal enveloping algebra U(so n )is isomorphic to the com- plex associative algebra (with a unit element) generated by the elements I 21 , I 32 ,···,I n,n−1 satisfying theaboverelations. Wemakethe q-deformationoftheserelationsby 2→[2] := (q 2 −q −2 )/(q− q −1 ) =q+q −1 .Asaresult,weobtainthecomplexassociativealgebragenerated by elements I 21 ,I 32 ,···,I n,n−1 satisfyingtherelations I 2 i,i−1 I i+1,i −(q+q −1 )I i,i−1 I i+1,i I i,i−1 +I i+1,i I 2 i,i−1 =−I i+1,i ,(1) I i,i−1 I 2 i+1,i −(q+q −1 )I i+1,i I i,i−1 I i+1,i +I 2 i+1,i I i,i−1 =−I i,i−1 ,(2) I i,i−1 I j,j−1 −I j,j−1 I i,i−1 = 0 for|i−j|>1. (3) This algebra was introduced by us in [8] and is denoted by U /prime q (so n ). Hereqtakes anycomplexvalue suchthat q/negationslash= 0,±1. Let us formulate for the algebra U /prime q (so n )an analogue of the Poincar ´e– Birkhoff–Witt theorem. For this we determine (see [16] and [17]) in U /prime q (so n ) elements analogous to the matrices I ij ,i > j, of the Lie algebra so n . In order to give them we use the notation I k,k−1 ≡I + k,k−1 ≡I − k,k−1 . Then fork > l + 1 we deﬁnerecursively I + kl := [I l+1,l ,I k,l+1 ] q ≡q 1/2 I l+1,l I k,l+1 −q −1/2 I k,l+1 I l+1,l , (4) I − kl := [I l+1,l ,I k,l+1 ] q −1 ≡q −1/2 I l+1,l I k,l+1 −q 1/2 I k,l+1 I l+1,l . The elements I + kl ,k>l,satisfythecommutationrelations [I + ln ,I + kl ] q =I + kn ,[I + kl ,I + kn ] q =I + ln ,[I + kn ,I + ln ] q =I + kl fork>l>n, (5) [I + kl ,I + nr ] = 0 for k>l>n>r andk>n>r>l, (6) kievarwe.tex; 12/03/2001; 3:49; p.340 334 A.U.KLIMYK [I + kl ,I + nr ] q = (q−q −1 )(I + lr I + kn −I + kr I + nl ) fork>n>l>r. (7) ForI − kl ,k > l, the commutation relations are obtained from these relations by replacingI + kl byI − kl andqbyq −1 . The algebra U /prime q (so n )can be considered as an associative algebra (with unit element) generated by I + kl ,1≤l < k≤n, satisfying the relations (5)–(7). Similarly,U /prime q (so n )is an associative algebra generated by I − kl ,1≤l < k≤n, satisfying the corresponding relations. Now the Poincar ´e–Birkhoff–Witt theorem for thealgebra U /prime q (so n )canbeformulated asfollows. Theorem2. Theelements I + 21m 21 I + 31m 31 ···I + n1m n1 I + 32m 32 I + 42m 42 ···I + n2m n2 ···I + n,n−1m n,n−1 , m ij ∈N, form a basis of the algebra U /prime q (so n ). This assertion is true if I + ij ,i < j, are replacedby thecorrespondingelements I − ij . The proofof thistheoremis given in [18]. 3. Theembedding U /prime q (so n )→U q (sl n ) The algebra U /prime q (so n )can be embedded into the Drinfeld–Jimbo quantum algebra U q (sl n )(see [9]). This quantum algebra is generated by the elements E i ,F i , K ±1 i =q ±H i ,i= 1,2,···,n−1,satisfyingtherelations K i K j =K j K i , K i K −1 i =K −1 i K i = 1, K i E j K −1 i =q a ij E j , K i F j K −1 i =q −a ij F j ,[E i ,F j ] =δ ij K i −K −1 i q−q −1 , E 2 i E i±1 −(q+q −1 )E i E i±1 E i +E i±1 E 2 i = 0, F 2 i F i±1 −(q+q −1 )F i F i±1 F i +F i±1 F 2 i = 0, [E i ,E j ] = 0,[F i ,F j ] = 0 for|i−j|>1, wherea ij are elementsof theCartanmatrixoftheLie algebra sl n . Letusintroduce the elements ˜I j,j−1 =F j−1 −qq −H j−1 E j−1 , j = 2,3,···,n, ofU q (sl n ). It is proved in [9] that there exists the algebra homomorphism ϕ: U /prime q (so n )→U q (sl n )uniquely determined by the relations ϕ(I i+1,i ) = ˜I i+1,i , i= 1,2,···,n−1. The following theorem states that this homomorphism is an isomorphism. Theorem 3. The homomorphism ϕ:U /prime q (so n )→U q (sl n )determined by the relationsϕ(I i+1,i ) =˜I i+1,i ,i= 1,2,···,n−1,isanisomorphismof U /prime q (so n )to U q (sl n ). kievarwe.tex; 12/03/2001; 3:49; p.341 NONSTANDARDQUANTIZATION OF U(so(n)) 335 In [16] the authors of that paper state that this homomorphism is an isomor- phism and say that it can be proved by means of the Diamond Lemma. However, wecouldnotrestoretheirproofandfoundanotheronein[18].Theorem3hasthe followingimportant corollary,provedin[18]: Corollary. Finite dimensional irreducible representations of U /prime q (so n )separate elements of this algebra, that is, for any a∈U /prime q (so n )there exists a ﬁnite dimensionalirreduciblerepresentation TofU /prime q (so n )suchthatT(a)/negationslash= 0. This corollary istrue for qnota rootof unityaswellas for qa root ofunity. Problems: Wethinkthatthealgebra U /prime q (so n )isconnectedwithsomeextensionof the Drinfeld–Jimbo quantum algebra U q (so n ). This conjecture is proved in [10] for the case n= 3. It is shown there that there is an isomorphism ϕ:U /prime q (so 3 )→ ˆU q (sl 2 ),where ˆU q (sl 2 )isan extensionof the quantumalgebra U q (sl 2 ). 4. Centralelementsof U /prime q (so n ) Letusform theelements J ± k 1 ,k 2 ,...,k 2r =q ∓ r(r− 1) 2 /summationdisplay /prime s∈S 2r ε q ±1 (s)I ± k s(2) ,k s(1) I ± k s(4) ,k s(3) ···I ± k s(2r) ,k s(2r−1) ,(8) of the algebra U /prime q (so n )(see [13]), where 1≤k 1 < k 2 <···< k 2r ≤nand summationrunsoverall permutations sof indicesk 1 ,k 2 ,···,k 2r such that k s(2) >k s(1) , k s(4) >k s(3) , ... ,k s(2r) >k s(2r−1) , k s(2) <k s(4) <...<k s(2r) . The symbol ε q ±1 (s)≡(−q ±1 ) /lscript(s) stands for the q-analogue of Levi–Chivita an- tisymmetric tensor, /lscript(s)means the length of permutation s. Note that in the limit q→1both sets in (8) reduce to the set of components of rank 2rantisymmetric tensoroperatorof Liealgebra so n . Theorem4. Theelements C (2r) n = /summationdisplay 1≤k 1 <k 2 <...<k 2r ≤n q k 1 +k 2 +...+k 2r −r(n+1) J + k 1 ,k 2 ,...,k 2r J − k 1 ,k 2 ,...,k 2r ,(9) wherer= 1,2,···,{n/2}({a}means the integral part of a), are Casimir ele- ments ofU /prime q (so n ), that is, they belong to the center of this algebra. If nis even, then the elements C (n)+ n ≡J + 1,2,···,n andC (n)− n ≡J − 1,2,···,n also belong to the centerofU /prime q (so n ). Central elements of this theorem are found in [13]. It was conjectured in [13] thatforqnotarootofunitythesetofcentralelements C (2r) n ,r= 1,2,···,{(n− kievarwe.tex; 12/03/2001; 3:49; p.342 336 A.U.KLIMYK 1)/2}, and the element C (n)+ n (ifnis even) generates the center of the algebra U /prime q (so n ). Letusgiveexplicitlysomecentralelements.For U /prime q (so 3 )andU /prime q (so 4 )wehave C (2) 3 =q −1 I 2 21 +I + 31 I − 31 +qI 2 32 =qI 2 21 +I − 31 I + 31 +q −1 I 2 32 , C (2) 4 =q −2 I 2 21 +I 2 32 +q 2 I 2 43 +q −1 I + 31 I − 31 +qI + 42 I − 42 +I + 41 I − 41 , C (4)+ 4 =C (4)− 4 =q −1 I 21 I 43 −I + 31 I + 42 +qI 32 I + 41 =qI 21 I 43 −I − 31 I − 42 +q −1 I 32 I − 41 . The quadraticcentralelement of U /prime q (so n )isof the form C (2) n = /summationdisplay 1≤i<j≤n q i+j−n−1 I + ji I − ji . Ifqis a root of unity, then (as in the case of Drinfeld–Jimbo quantum alge- bras) there exist additional central elements of U /prime q (so n )which are given by the followingtheorem, proved in [14]: Theorem5. Letq k = 1fork∈Nandq j /negationslash= 1for0<j <k.Thentheelements C (k) (I + rl ) = {(k−1)/2} /summationdisplay j=0 /parenleftbigg k−j j /parenrightbigg 1 k−j /parenleftBig i q−q −1 /parenrightBig 2j I + rlk−2j , r>l, (10) where{(k−1)/2}is the integral part of the number (k−1)/2, belong to the centerofU /prime q (so n ). It is well-known that a Drinfeld–Jimbo algebra U q (g)forqa root of unity (q k = 1) is a ﬁnite dimensional vector space over the center of U q (g). The same assertionistrueforthealgebra U /prime q (so n ).ByTheorem5,anyelement (I + ij ) s ,s≥k, can be reduced to a linear combination of (I + ij ) r ,r < k, with coefﬁcients from the centerCofU /prime q (so n ). Now our assertion follows from this sentence and from Poincar´e–Birkhoff–Witt theorem for U /prime q (so n ). Using this assertion, it is proved thefollowingtheorem[18]: Theorem 6. Ifqis a root of unity, then any irreducible representation of U /prime q (so n ) isﬁnitedimensional. It can be proved more strong assertion: there exists a ﬁxed positive integer rsuch that dimension of any irreducible representation of U /prime q (so n )atqa root of unity does not exceed r. Of course, the number rdepends onk(recall thatkis deﬁned byq k = 1). 5. Irreducible representationsof U /prime q (so n ) We ﬁrst assume that qis not a root of unity. Then the algebra U /prime q (so n )has two types of irreducible ﬁnitedimensional representations: kievarwe.tex; 12/03/2001; 3:49; p.343 NONSTANDARDQUANTIZATION OF U(so(n)) 337 (a)representationsoftheclassicaltype(at q→1theygivethecorresponding ﬁnitedimensional irreducible representations oftheLie algebra so n ); (b)representationsofthenonclassicaltype(theydonotadmitthelimit q→1 since inthispoint therepresentationoperators aresingular). Let us describe the classical type representations of the algebras U /prime q (so n ), n≥3, which are q-deformations of the ﬁnite dimensional irreducible repre- sentations of the Lie algebra so n . As in the case of irreducible representations of the Lie algebra so n , they are given by sets m n consisting of{n/2}numbers m 1,n ,m 2,n ,...,m {n/2},n (here{n/2}denotes integral part of n/2) which are all integralor allhalf-integraland satisfythedominance conditions m 1,2p+1 ≥m 2,2p+1 ≥...≥m p,2p+1 ≥0, (11) m 1,2p ≥m 2,2p ≥...≥m p−1,2p ≥|m p,2p | (12) forn= 2p+ 1andn= 2p, respectively. These representations are denoted byT m n . For a basis in a representation space we can take the q-analogue of the Gel’fand–Tsetlinbasiswhichisobtainedbysuccessivereductionoftherepresen- tationT m n to the subalgebras U /prime q (so n−1 ),U /prime q (so n−2 ),···,U /prime q (so 3 ),U /prime q (so 2 ) := U(so 2 ). As in the classical case, its elements are labeled by Gel’fand–Tsetlin tableaux {ξ n }≡   m n m n−1 ... m 2   , (13) wherethe componentsof m r andm r−1 satisfythe betweennessconditions m 1,2p+1 ≥m 1,2p ≥m 2,2p+1 ≥m 2,2p ≥...≥m p,2p+1 ≥m p,2p ≥−m p,2p+1 , (14) m 1,2p ≥m 1,2p−1 ≥m 2,2p ≥m 2,2p−1 ≥...≥m p−1,2p−1 ≥|m p,2p |.(15) The explicit formulas for the operators T m n (I j,j−1 ),j= 2,3,···,n, of the representation T m n ofU /prime q (so n )andtheirproofsare given in[11]. The representations, described above, are called representations of the classi- caltype,sinceunderthelimit q→1theoperators T m n (I j,j−1 )turnintothecorre- spondingoperators T m n (I j,j−1 )forirreducibleﬁnitedimensionalrepresentations withhighest weights m n oftheLie algebra so n . Let us give the explicit expressions for Casimir operators (corresponding to the central elements, described in Theorem 4) in the classical type irreducible representationsof U /prime q (so n ).Forthiswedeﬁnethegeneralizedfactorialelementary symmetric polynomials. Fixing an arbitrary sequence of complex numbers a= (a 1 ,a 2 ,···), for eachr= 0,1,2,···,N, we introduce these polynomials in N variablesz 1 ,z 2 ,···,z N bytheformula e r (z 1 ,z 2 ,... ,z N |a) = kievarwe.tex; 12/03/2001; 3:49; p.344 338 A.U.KLIMYK = /summationdisplay 1≤p 1 <p 2 <···<p r ≤N (z p 1 −a p 1 )(z p 2 −a p 2 −1 )...(z p r −a p r −r+1 ). BySchurLemma,Casimiroperatorsintheirreducibleﬁnitedimensionalrep- resentations,characterizedbythenumbers (m 1,n ,m 2,n ,... ,m N,n ),N={n/2}, aremultipleto theidentityoperator: T m n (C (2r) n ) =χ (2r) m n 1. Theorem7 [13].The eigenvalueoftheoperator T m n (C (2r) n )is χ (2r) m n = (−1) r e r ([l 1,n ] 2 ,[l 2,n ] 2 ,... , [l N,n ] 2 |a), where a= ([/epsilon1] 2 ,[/epsilon1+ 1] 2 ,[/epsilon1+ 2] 2 ,...),l k,n =m k,n +N−k+/epsilon1. (Here/epsilon1= 0 forn= 2Nand/epsilon1= 1 2 forn= 2N+ 1.)Ifn= 2Niseven, then T m n (C (n)+ n ) =T m n (C (n)− n ) = /parenleftbig √ −1 /parenrightbig N [l 1,n ][l 2,n ]...[l N,n ]1. The algebra U /prime q (so n )has also irreducible ﬁnite dimensional representations Tof nonclassical type, that is, such that the operators T(I j,j−1 )have no clas- sical limitq→1. They are given by sets /epsilon1:= (/epsilon1 2 ,/epsilon1 3 ,···,/epsilon1 n ),/epsilon1 i =±1, and by sets m n consisting of{n/2}half-integral (but not integral) numbers m 1,n ,m 2,n ,···,m {n/2},n (here{n/2}denotes the integral part of n/2) that satisfythedominance conditions m 1,2p+1 ≥m 2,2p+1 ≥...≥m p,2p+1 ≥1/2, m 1,2p ≥m 2,2p ≥...≥m p−1,2p ≥m p,2p ≥1/2 forn= 2p+ 1andn= 2p, respectively. These representations are denoted by T /epsilon1,m n . For a basis in the representation space we can use an analogue of the basis (13).Itselements arelabeledby thetableaux {ξ n }≡   m n m n−1 ... m 2   , wherethe componentsof m 2p+1 andm 2p satisfy thebetweenness conditions m 1,2p+1 ≥m 1,2p ≥m 2,2p+1 ≥m 2,2p ≥...≥m p,2p+1 ≥m p,2p ≥1/2, m 1,2p ≥m 1,2p−1 ≥m 2,2p ≥m 2,2p−1 ≥...≥m p−1,2p−1 ≥m p,2p . Explicit formulas for the operator T /epsilon1,m n (I j,j−1 ),j= 2,3,···,n, of the representation T /epsilon1,m n ofU q (so n )are given in[12]. Theorem 8. The representations T /epsilon1,m n are irreducible. The representations T /epsilon1,m n andT /epsilon1 /prime ,m /primen are pairwise nonequivalent for (/epsilon1,m n )/negationslash= (/epsilon1 /prime ,m /prime n ). For any kievarwe.tex; 12/03/2001; 3:49; p.345 NONSTANDARDQUANTIZATION OF U(so(n)) 339 admissible (/epsilon1,m n )andm /prime n the representations T /epsilon1,m n andT m /primen are pairwise nonequivalent. The algebra U /prime q (so n )has non-trivial one-dimensional representations. They are special cases of the representations of the nonclassical type. They are de- scribed as follows. Let /epsilon1:= (/epsilon1 2 ,/epsilon1 3 ,···,/epsilon1 n ),/epsilon1 i =±1, and let m n = (m 1,n ,m 2,n ,···,m {n/2},n ) = ( 1 2 , 1 2 ,···, 1 2 ). Then the corresponding represen- tationsT /epsilon1,m n areone-dimensional andare given by the formulas T /epsilon1,m n (I k+1,k )|ξ n /angbracketright=/epsilon1 k +1 q 1/2 −q −1/2 |ξ n /angbracketright. Thus, to every /epsilon1:= (/epsilon1 2 ,/epsilon1 3 ,···,/epsilon1 n ),/epsilon1 i =±1, there corresponds a one- dimensionalrepresentationof U /prime q (so n ). Conjecture. Ifqis not a root of unity, then every irreducible ﬁnite dimensional representation of U /prime q (so n )is equivalent to one of the representations T m n of the classicaltype or tooneof the representations T /epsilon1,m n ofthenonclassical type. This conjecture isprovedfor thealgebra U /prime q (so 3 )(see [19]). Irreducible representations of the algebra U /prime q (so n )forqa root of unity are describedin[18].Forconstructionoftheseirreduciblerepresentationsof U /prime q (so n ), it is used the method of D. Arnaudon and A. Chakrabarti [20] for construction of irreducible representations of the quantum algebra U q (sl n )whenqis a root of unity. Ifq p = 1andpis an odd integer, then there exists the series of irreducible representationsof U /prime q (so n )whichacton p N -dimensionalvectorspace(where Nis thenumberofpositiverootsoftheLiealgebra so n )andaregivenby r= dim so n complex parameters. These representations are irreducible for generic values of these parameters. These representations constitute the main class of irreducible representations of U /prime q (so n ). For some special values of the representation param- etersinC r therepresentationsarereducible.Thesereduciblerepresentationsgive many other classes of (degenerate) irreducible representations which are given by less number of parameters or by parameters, values of which cover subsets of C r of Lebesgue measure 0. As in the case of irreducible representations of the quantum algebra U q (sl n ), it is difﬁcult to enumerate all irreducible representa- tions of these classes. However, the most important classes of these degenerate representations can be constructed. In particular, in [18] we give 2 n−1 classes of these representations, which are an analogue of the nonclassical type irreducible representations of U /prime q (so n )forqnotaroot ofunity. kievarwe.tex; 12/03/2001; 3:49; p.346 340 A.U.KLIMYK 6. Restriction of representationsof U q (sl n )toU /prime q (so n ) In this section we assume that qis not a root of unity. The algebra U /prime q (so n )is a subalgebra of the quantum algebra U q (sl n ). Therefore, we may restrict irre- ducibleﬁnitedimensionalrepresentationsofthealgebra U q (sl n )tothesubalgebra U /prime q (so n ). Generally speaking, such a restriction leads to reducible representations of the subalgebra. It was proved in [16] that each irreducible ﬁnite dimensional representation of U q (sl n )under restriction to U /prime q (so n )decomposes into a direct sum of irreducible representations of this subalgebra. N. Iorgov has proved (will bepublished)thatsuchadecompositioncontainsonlyirreduciblerepresentations of the classical type. However, explicit formula for the decomposition is known onlyfortherestriction U q (sl 3 )→U /prime q (so 3 ). Irreducible ﬁnite dimensional representations of U q (sl 3 )are given by three integers/lscript= (l 1 ,l 2 ,l 3 )such thatl 1 ≥l 2 ≥l 3 . We denote such the representation byR /lscript .Irreducibleﬁnitedimensionalclassicaltyperepresentationsof U /prime q (so 3 )are denotedbyT k , wherekis anonnegativeintegralor half-integralnumber. In order to ﬁnd which irreducible representations of U /prime q (so 3 )are contained in the decomposition of R /lscript ↓ U /primeq (so 3 ) we split in [21] the spectrum SpecR /lscript (I 21 )of therepresentationoperator R /lscript (I 21 )intospectraofoperators T k (I 21 )ofirreducible representations T k ofU /prime q (so 3 ). (It is proved in [21] that such splitting is unique.) As a result,we havethat R /lscript ↓ U /primeq (so 3 ) = /summationdisplay s/primes+l 2 −l 3 /summationdisplay k=s T k ifl 1 −l 2 is oddand R /lscript ↓ U /primeq (so 3 ) = /summationdisplay s/primes+l 2 −l 3 /summationdisplay k=s T k ⊕ /summationdisplay r/prime T r ifl 1 −l 2 iseven,where /summationtext /prime s meansthesummationoverthevalues l 1 −l 2 ,l 1 −l 2 − 2,l 1 −l 2 −4,···,1 (or 2)and the last sum /summationtext /prime r is over the values l 2 −l 3 ,l 2 − l 3 −2,l 2 −l 3 −4,···,0 (or 1).Notethatthesedecompositionscoincidewiththe corresponding decompositionsfor the reduction SU(3)→SO(3). 7. Applications There are the following main applications of the algebra U /prime q (so n )and its irre- duciblerepresentations: 1.Thetheoryoforthogonalpolynomialsandspecialfunctions(especially,the theory ofq-orthogonal polynomials and basic hypergeometric functions). This kievarwe.tex; 12/03/2001; 3:49; p.347 NONSTANDARDQUANTIZATION OF U(so(n)) 341 directionisnotgoodworkedout.Someideasofsuchapplicationscanbefoundin [22]. 2.Thealgebra U /prime q (so n )(espesiallyitsparticularcase U /prime q (so 3 ))isrelatedtothe algebra of observables in 2+1 quantum gravity on the Riemmanian surfaces (see thepapers[23]–[25]). 3. Aq-analogue of the Riemannian symmetric space SU(n)/SO(n)is con- structed by means of the algebra U /prime q (so n ). This construction is fulﬁlled in the paper [9]. 4. Aq-analogue of the theory of harmonic polynomials ( q-harmonic polyno- mials on quantum vector space R n q ) is constructed by using the algebra U /prime q (so n ). In particular, a q-analogue of different separations of variables for the q-Laplace operator is given by means of this algebra and its subalgebras. This theory is containedin the papers[16] and[26]. 5. The algebra U /prime q (so n )also appear in the theory of links in the algebraic topology(see[27]). 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Przytycki, Multiplicative structure of Kauffman bracket skein module quantization ,e-print: math.QA/9902117. kievarwe.tex; 12/03/2001; 3:49; p.349 CAN THE CABIBBOMIXING ORIGINATEFROM NONCOMMUTATIVEEXTRADIMENSIONS? ALEXANDRE GAVRILIK ∗ BogolyubovInstitutefor TheoreticalPhysics,Kiev, Ukraine Abstract. Treatinghadronicﬂavorsymmetrieswithquantumalgebras U q (su n )leadstointeresting consequences such as: new mass sum rules for hadrons 1 − , 1 2+ , 3 2+ of improved accuracy; possi- bilitytolabeldifferentﬂavorstopologically-bytoruswindingnumber;properlyﬁxeddeformation parameterqin case of baryons is linked in a simplest way to the Cabibbo angle θ C , that suggests forθ C the exact value π 14 . In this connection, we discuss the possibility that this angle and the Cabibbo mixing as a whole take its origin in noncommutativity of some additional, with regard to 3+1, space-time dimensions. 1. Introduction The problem of fermion ﬂavors, mixings and masses (see e.g., [1]) belongs to most puzzling ones in particle physics. The Cabibbo mixing ﬁrst introduced for three lightest ﬂavors in the context of weak decays [2] involves the angle θ C . Importanceofthisconceptwasfurtherconﬁrmedafteritsgeneralizationtomixing of 3 families [3]. Due to Wolfenstein parametrization [4] of CKM matrix, the Cabibbo angle now plays a prominent role: not only CKM matrix elements V ij , but also the quark (and even lepton) mass ratios are often expressed as powers of small parameter λ= sinθ C ≈0.22. No doubt, it is necessary to know the value ofλas precise as possible. In this respect, the main bonus of our approach to ﬂavor symmetries, based on quantum algebras, is that it suggests theoretically motivated exact value for θ C , namely,θ C = π 14 . As further implication, it leads us to a conjecture of possible noncommutative-geometric origin of the Cabibbo mixing, and our aim here is to argue this may indeed be the case. Below, when treatingbaryonmasses,werestrictourselveswith4ﬂavorsincludingu-,d-,s-,and c-quarks.Basictooloftheapproachusedistherepresentationtheoryofquantum algebras [5] U q (su n )adopted, instead of conventional SU(n), to describe ﬂavor symmetries classifyinghadronsinto multiplets. ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.350 344 A.GAVRILIK 2. Vectormeson masses: q-deformationreplaces (singlet) mixing We use 1 Gelfand-Tsetlin basis vectors for meson states from (n 2 −1)-plet of ’n-ﬂavor’U q (u n )embedded into{(n+1) 2 −1}-plet of ’dynamical’ U q (u n+1 ); construct mass operator ˆM n invariant under the ’isospin+hypercharge’ q-algebra U q (u 2 )from generators of ’dynamical’ algebra U q (u n+1 )(e.g., ˆM 3 =M 0 1+ γ 3 A 34 A 43 +δ 3 A 43 A 34 ); calculate the expressions for masses m V i ≡/angbracketleftV i |ˆM 3 |V i /angbracketright - these involve symmetry breaking parameters γ 3 , δ 3 and theq-parameter. In particular,for n= 3weobtain m ρ =M o , m K ∗ =M o −γ 3 , m ω 8 =M o −2[2] q [3] q γ 3 ,(1) where [x] q ≡ q x −q − x q−q −1 istheq-numberthat’deforms’anumber xand,tohaveequal masses for particles and their anti’s, δ 3 =γ 3 was set.q-Dependence appears only inthemassof ω 8 (isosingletin U q (su 3 )-octet).Excluding M 0 ,γ 3 ,theq-analogof Gell–Mann -Okubo(GMO) relation is[8]: m ω 8 + /parenleftbigg 2[2] q [3] q −1 /parenrightbigg m ρ = 2[2] q [3] q m K ∗ . (2) In the limitq= 1(i.e., at [2] [3] = 2 3 ), this reduces to usual GMO formula 3m ω 8 + m ρ = 4m K ∗ which needssinglet mixing[9].However, italso yields m ω 8 +m ρ = 2m K ∗ ifq=e iπ/5 (then,[2] q = [3] q ).(3) Withm ω 8 ≡m φ , and no mixing, eq.(3) coincides with nonet mass formula of Okubo[10]agreeing ideallywithdata [11]. For3≤n≤6mass operator is constructed analogously. Again, calcula- tions show: only isosinglets ω 15 ,ω 24 ,ω 35 of(n 2 −1)-plets ofU q (u n )contain q-dependence.As result,we get the q-deformedmass relations [8,6,7]: [n] (q) m ω n2−1 + (b n;q + 2n−4)m ρ = 2m D ∗n + (c n;q + 2) n−1 /summationdisplay r=3 m D ∗r ,(4) b n;q ≡nc n;q −6 [n] 2 (q) + /parenleftbigg 24 [2] q −1 /parenrightbigg [n] (q) , c n;q ≡2 [n] 2 (q) − 8 [2] q [n] (q) , where [n] (q) ≡[n] q /[n−1] q . Then, natural ﬁxation by setting [n] q = [n−1] q , n= 4,5,6,leadstothehigheranalogs ofOkubo’ssum rule: m ω 15 + (5−8/[2] q 4 )m ρ = 2m D ∗ + (4−8/[2] q 4 )m K ∗ (5) 1 For more detailsconcerning this approach see refs.[6, 7, 13]. kievarwe.tex; 12/03/2001; 3:49; p.351 CABIBBO MIXINGFROM EXTRA DIMENSIONS? 345 m ω 24 + (9−16/[2] q 5 )m ρ = 2m D b∗ + (4−8/[2] q 5 )(m D ∗ +m K ∗ ) (6) m ω 35 + (13−24/[2] q 6 )m ρ = 2m D t∗ + (4−8/[2] q 6 )(m D b∗ +m D ∗ +m K ∗ ). (7) Hereq n =e iπ/(2n−1) are the values that solve eqns. [n] q −[n−1] q = 0. Like in the case with m ω 8 ≡m φ ,it is meant in (5)-(7) that J/ψis put in place of ω 15 ,Υ inplaceofω 24 ,toponium in placeof ω 35 (i.e.,no mixing!). Theq-polynomials [n] q −[n−1] q havea topologicalmeaning. 3. Torusknotsandtopological labellingof ﬂavors Polynomials [n] q −[n−1] q ≡P n (q),by their roots, reduce q-analogs (2), (4) to realistic mass sum rules (MSR) (3), (5)-(7). And, due to property (i)P n (q) = P n (q −1 ),(ii)P n (1) = 1,they coincide [8, 7] with such knot invariants as Alexanderpolynomials ∆(q){(2n−1) 1 }of(2n−1) 1 -torus knots.E.g., [3] q −[2] q =q 2 +q −2 −q−q −1 + 1≡∆(q){5 1 }, [4] q −[3] q =q 3 +q −3 −q 2 −q −2 +q+q −1 −1≡∆(q){7 1 } correspond to the 5 1 - and 7 1 -knots. Since the q-deuce in (4) can be linked to the trefoil (or 3 1 -) knot: [2] q −1 =q+q −1 −1≡∆(q){3 1 },all theq-dependence inmassesof ω n 2 −1 andincoefﬁcientsin(2),(4)isexpressiblethroughAlexander polynomials.Namely, [3] q [2] q = 1 + ∆{5 1 } [2] q = 1 + ∆{5 1 } ∆{3 1 }+1 , [n] q [n−1] q = 1 +∆{(2n−1) 1 } [n−1] q = 1 +∆{(2n−1) 1 } 1 + /summationtext n−1 r=2 ∆{(2r−1) 1 }, n= 4,5,6. (8) Thevaluesq n arethusrootsofrespectiveAlexanderpolynomials.Foreach n,the ’senior’ (numerator) polynomial in [3] q [2] q and (8) is speciﬁed: by its root, it ’singles out’ thecorrespondingMSR from q-deformedanalog. Thus,theq-parameterforeach nisﬁxedinarigidwayasaroot q n of∆{(2n− 1) 1 },contrarytothechoiceof qbyﬁttinginotherphenomenologicalapplications [12].Moreover,usingﬂavor q-algebrasalongwith’dynamical’ q-algebrasaccord- ingtoU q (u n )⊂U q (u n+1 ),wegain:thetorusknots 5 1 ,7 1 ,9 1 ,11 1 areputinto correspondence [6, 7] with vector quarkonia s¯s,c¯c,b¯b, andt¯trespectively. In a sense,thepolynomial P n (q)≡[n] q −[n−1] q byitsrootq(n) determinesthevalue ofq(deformation strength) for each nand thus serves as deﬁning polynomial kievarwe.tex; 12/03/2001; 3:49; p.352 346 A.GAVRILIK for the MSR/quarkonium/ﬂavor corresponding to n. Hence, the applying of q- algebras suggests a possibility of topological labeling of ﬂavors : ﬁxed number n corresponds to 2n−1overcrossings of 2-strand braids whose closure gives these (2n−1) 1 -torus knots. With the form (2n−1,2)of same torus knots this means the correspondence n↔w≡2n−1, wbeing the winding number around tube oftorus(winding number aroundhole is2). 4. Deﬁning q-polynomials foroctet baryonmasssum rules Analogous scheme was applied to baryons 1 2+ too. Excluding undetermined con- stantsM 0 ,α,βfrom ﬁnal obtained expressions for M N ,M Ξ ,M Λ ,M Σ leads to theq-deformedmass relations(MRs)oftheform [6, 7, 13] [2]M N + [2] [2]−1M Ξ = [3]M Λ + /parenleftbigg [2] 2 [2]−1−[3] /parenrightbigg M Σ +A q B q /parenleftBig M Ξ + [2]M N −[2]M Σ −M Λ /parenrightBig (9) whereA q andB q are certain polynomials of [2] q with non-overlapping sets of zeros. It is important that different dynamical representations produce differing pairsA q ,B q . AnyA q possesses the factor ([2] q −2)and thus the ’classical’ zero q= 1. In the limit q= 1eachq-deformed mass relation reduces to the standard GMO sum rule M N +M Ξ = 1 2 M Σ + 3 2 M Λ for octet baryons (its accuracy is 0.58%). At some values of qwhich are zeros of particular A q other thanq= 1, we obtain MSRs which hold with better accuracy than the GMO one. The two newMSRs q=e iπ/6 ⇒M N +1+ √ 3 2M Ξ = 2 √ 3M Λ +9− √ 3 6M Σ (0.22%)(10) q=e iπ/7 ⇒M N + 1 [2] q 7 −1M Ξ = 1 [2] q 7 −1M Λ +M Σ (0.07%)(11) result[6,7,13]fromtwodifferentdynamicalrepresentations D (1) andD (2) whose respective polynomials A (1) q andA (2) q possess zeros q=e iπ/6 andq=e iπ/7 . The choice with q=e iπ/7 turnsoutto bethe bestpossibleone. 2 The sum rule (10) was ﬁrst derived [6] from a speciﬁc dynamical representa- tion(irrep)D (1) ofU q (u 4,1 ).However,the’compact’dynamical U q (u 5 )isequally wellsuited.Amongtheadmissibledynamicalirrepsthereexistanentireseries of 2 In sec. 8 we arguethat this value of qis linkedto theCabibboangle: θ 8 = π 7 =2θ C . kievarwe.tex; 12/03/2001; 3:49; p.353 CABIBBO MIXINGFROM EXTRA DIMENSIONS? 347 irreps (numbered by integer m,6≤m <∞) which produce the corresponding inﬁnitesetofMSRs: M N + 1 [2] q m −1M Ξ =[3] q m [2] q m M Λ + /parenleftBig [2] q m [2] q m −1−[3] q m [2] q m /parenrightBig M Σ (12) withq m =e iπ/m .Each of these shows better agreement with data than the clas- sical GMO one. Few of them, including the MSRs (10), (11) and the ’classical’ GMO whichcorresponds to q ∞ = 1,areshownin the table. θ= π m (RHS−LHS),M eV |RHS−LHS | RHS , % π/ ∞ 26.2 0.58 π/ 30 25.42 0.56 π/ 12 20.2 0.44 π/ 8 10.39 0.23 π/ 7 3.26 0.07 π/ 6 -10.47 0.22 Comparing (12) with (9) shows that the vanishing of A q B q is crucial for obtaining this discrete set of MSRs and for providing a kind of ’discrete ﬁtting’. Hence, A q serves as deﬁningpolynomial for the correspondingMSR. Since [2] q 7 =q 7 + 1 q 7 = 2 cos π 7 ,the MSR(11)takes the equivalentform M Ξ −M N +M Σ −M Λ = (2 cos π 7)(M Σ −M N ) (13) which exhibits somesimilarity with decuplet massformula given below. 5. Decuplet baryons: universal q-deformedmass relation In the case of SU(3)-decuplet baryons 3 2+ , the convensional 1st order symmetry breakingyields[9]equalspacingrule(ESR)forisopletmembersin 10-plet.Em- piricaldatashowfor M Σ ∗ −M ∆ ,M Ξ ∗ −M Σ ∗ andM Ω −M Ξ ∗ noticeabledeviation from ESR: 152.6MeV↔148.8MeV↔139.0MeV. Use of the q-algebras U q (su n )instead ofSU(n)provides natural improvement. From evaluations of decuplet masses in two particular irreps of the dynamical algebra U q (u 4,1 ), the q-deformedmassrelation (1/[2] q )(M Σ ∗ −M ∆ +M Ω −M Ξ ∗ ) =M Ξ ∗ −M Σ ∗ , [2] q ≡q+q −1 , (14) was derived [14]. As proven there, this mass relation is univ ersal - it results from each admissible irrep (which contains U q (su 3 )-decuplet embedded in 20-plet of kievarwe.tex; 12/03/2001; 3:49; p.354 348 A.GAVRILIK U q (su 4 )) of the dynamical U q (u 4,1 ). With empirical masses [11], the formula (14) is successful if [2] q /similarequal1.96.Pure phase q=e iθ (or[2] q = 2 cosθ) with θ=θ 10 /similarequal π 14 provides excellent agreement with data (below, we argue that θ 10 =θ C ).Notice a similarityofeq.(14)with the MR (1/2)(M Σ ∗ −M ∆ +M Ω −M Ξ ∗ ) =M Ξ ∗ −M Σ ∗ (15) obtained earlier in diverse contexts [15]: by tensor method, in additive quark model with general pair interaction, in a diquark–quark model, in modern chi- ral perturbation theory. Such model-independence of (15) stems because each of these approachesaccounts 1st and2nd order ofSU(3)-breaking. Theq-deformed MSR (14) is universal even in a wider sense: it results from admissible irreps (containing U q (su 4 )20-plet) of both U q (su 4,1 )and the ’compact’ dynamical U q (su 5 ). Say, within a dynamical irrep {4000}ofU q (su 5 ) calculation yields: M ∆ =M 10 +β, M Σ ∗ =M 10 + [2]β+α, M Ξ ∗ = M 10 + [3]β+ [2]α, M Ω =M 10 + [4]β+ [3]α,from which (14) stems. On theother hand,thesefourmassescanbe comprisedby single formula M D i =M /parenleftbig Y(D i ) /parenrightbig =M 10 +α[1−Y(D i )] +β[2−Y(D i )](16) with explicit dependence on Y(hypercharge). If q= 1, this reduces to M D i = ˜M 10 +aY(D i ),i.e.,lineardependenceonhypercharge Y(orstrangeness)where a=−α−β,˜M 10 =M 10 +α+ 2β. 6. Nonpolynomial SU(3)-breakingeffectsin baryon masses Formula (16) involves highly nonlinear dependence of mass on hypercharge (it is Ythat causesSU(3)-breaking for decuplet). Since for q-number [N]we have [N] =q N−1 +q N−3 +...+q −N+3 +q −N+1 (Nterms) this shows expo- nentialY-dependence of masses. Such high nonlinearity makes (14) and (16) radically different from the abovementioned result (15) of traditional treatment that accounts foreffectslinear andquadraticin Y. For octet baryon masses, high nonlinearity ( nonpolynomiality ) inSU(3)- breaking effectively accounted by the model was demonstrated in [13]. For this, the expressions for (isoplet members of) octet masses with explicit dependence on hypercharge Yand isospinI, throughI(I+ 1), are used. The typical matrix element(µ 1 ,µ 2 are functions ofirrep labels m 15 ,m 55 ): /angbracketleftB i |A 34 A 45 A 54 A 43 |B i /angbracketright= [2] −1 [3] −1 /parenleftBig [Y/2][Y/2+1]−[I][I+1] /parenrightBig µ 1 (m 15 ,m 55 ) −[2] −1 [5] −1 /parenleftBig [Y/2−1][Y/2−2]−[I][I+ 1] /parenrightBig µ 2 (m 15 ,m 55 ), kievarwe.tex; 12/03/2001; 3:49; p.355 CABIBBO MIXINGFROM EXTRA DIMENSIONS? 349 contributingtooctetbaryonmasses,illustratesthedependence.Fromdeﬁnitionof q-bracket [n] = sin(nh ) sin(h) ,q=exp(ih), it is clearly seen that baryon masses depend on hypercharge Yand isospin I(hence, onSU(3)-breaking effects) in highly nonlinear- nonpolynomial -fashion. Theabilitytotakeintoaccounthighlynontrivialsymmetrybreakingeffectsby applyingq-analogsU q (su n )of ﬂavor symmetries is much alike the fact demon- stratedin[16]that,byexploitingappropriate freeq-deformedstructureoneisable to efﬁciently study the properties of (undeformed) quantum-mechanical systems withcomplicatedinteractions. 7. Touseornot tousetheHopf-algebrastructure An alternative, as regards (9), version of q-deformed analog can be derived [13] using for the symmetry breaking part of mass operator a component of q-tensor operator - this clearly implies [17] the Hopf algebra structure (comultiplication, antipode) of the U q (su n )quantum algebras. Let us brieﬂy discuss such version. We useq-tensor operators (V 1 ,V 2 ,V 3 )resp. (V ¯1 ,V ¯2 ,V ¯3 )formed from elements ofU q (su 4 )and transforming as 3resp.3 ∗ under the adjoint action of U q (su 3 ). WithH 1 ,H 2 as Cartan elements and with notation [X,Y ] q ≡XY−qYX, the components (V 1 ,V 2 ,V 3 )read V 1 = [E + 1 ,[E + 2 ,E + 3 ] q ] q q −H 1 /3−H 2 /6 , V 2 = [E + 2 ,E + 3 ] q q H 1 /6−H 2 /6 , V 3 =E + 3 q H 1 /6+H 2 /3 , (17) and similarlyfor (V ¯1 ,V ¯2 ,V ¯3 )(see [13]),of whichwehereonly give V ¯3 =q H 1 /6+H 2 /3 E − 3 . (18) Clearly,U q (su 3 )is broken to U q (su 2 ). Like in the nondeformed case of su(3) brokento its isospinsubalgebra su(2),theformofmass operator is ˆM=ˆM 0 +ˆM 8 (19) where ˆM 0 isU q (su 3 )-invariant and ˆM 8 transforms as I= 0,Y= 0component of tensor operator of 8-irrep ofU q (su 3 ).If|B i /angbracketrightis a basis vector of carrier space of8which corresponds to some baryon B i , the mass of B i is given byM B i = /angbracketleftB i |ˆM|B i /angbracketright.Theirrep 8occurstwiceinthedecompositionof 8⊗8.This,andthe Wigner-Eckarttheoremfor U q (su n )[18]appliedto q-tensoroperatorsunderirrep 8ofU q (su 3 ),leadtothemassoperatoroftheform ˆM=M 0 1+αV (1) 8 +βV (2) 8 and thusto M B i =/angbracketleftB i |(M 0 1+αV (1) 8 +βV (2) 8 )|B i /angbracketright (20) kievarwe.tex; 12/03/2001; 3:49; p.356 350 A.GAVRILIK whereV (1) 8 andV (2) 8 are two dictinct tensor operators which both transform as I=0,Y=0componentofirrep 8ofU q (su 3 );M 0 ,α,β-undeterminedconstants depending on details of dynamics. From 3⊗3 ∗ =1⊕8,3 ∗ ⊗3=1⊕8it is seen that the operators V 3 V ¯3 andV ¯3 V 3 from (17),(18) are just the isosinglets needed in eq.(20). As result, mass operator in (20) with redeﬁned M 0 ,α,βis ˆM=M 0 1+αV 3 V ¯3 +βV ¯3 V 3 ,or ˆM=M 0 1+αE + 3 E − 3 q Y +βE − 3 E + 3 q Y (21) whereY= (H 1 + 2H 2 )/3is hypercharge. Matrix elements (20) with ˆMfrom (21)areevaluatedbyembedding 8inaparticularrepresentationof U q (su 4 ).Say, if one takes the adjoint 15ofU q (su 4 ), the evaluation of baryon masses yields: M N =M 0 +βq, M Σ =M 0 , M Λ =M 0 + [2] [3] (α+β), M Ξ =M 0 +αq −1 . ExcludingM 0 ,αandβ, weﬁnallyobtain [3]M Λ +M Σ = [2](q −1 M N +qM Ξ ). (22) This alternative q-analog of octet mass relation looks much simpler than the for- merq-analog(9).Thissame q-relation(22)resultsfromembedding 8inanyother admissible dynamical representation. What concerns empirical validity [11] of (22), there is no other way to ﬁx the q-parameter as by usual ﬁtting (for each of the values q 1,2 =±1.035,q 3,4 =±0.903 √−1, theq-MR (22) indeed holds within experimental uncertainty). This is in sharp contrast with the q-analogs (9) for which there exists an appealing possibility to ﬁx qin a rigid way by zeros of relevantpolynomial A q . Summarizingweshouldstressthat,althoughtheuseofHopf-algebrastructure leads to simple and mathematically appealing result eq.(22), from the physical (phenomenological) viewpoint the version (9) of q-analog obtained by apply- ing only the tools of representation theory of quantum algebras and not strictly q-covariant symmetry breaking part in mass operator, provides much more inter- estingresults.Amongtheseisthedegeneracyliftingandthepossibilitytochoose among a variety of dynamical representations, deﬁning polynomials and, thus, within discrete set of viable mass sum rules. That led us to the best MSR (11) (or (13))foroctetbaryons. 8. On the connection:deformation parameter ↔Cabibbo angle In 3-ﬂavor case of vector mesons, the deformation angle π 5 that determines φ- meson in (3) coincides remarkably with ω-φmixing angle (known [11] to be θ ωφ = 36 ◦ ) of traditional SU(3)-based scheme. In other words, the concept of q-deformedﬂavor symmetries isclosely related with the issueof singlet mixing. Forpseudoscalar(PS) mesons,the generalization [19]ofGMO-formula f 2 π m 2 π + 3f 2 η m 2 η = 4f 2 K m 2 K with 1/f 2 π + 3/f 2 η = 4/f 2 K ,(23) kievarwe.tex; 12/03/2001; 3:49; p.357 CABIBBO MIXINGFROM EXTRA DIMENSIONS? 351 involvesdecay constantsascoefﬁcients. Presentedin the equivalent form 3 m 2 π +9f 2 K /f 2 π 4−f 2 K /f 2π m 2 η = 4f 2 K f 2π m 2 K , (24) itistobecomparedwithour q-analog(2)ofGMOrewrittenforPSmesons(with massessquared), namely m 2 π + [3] 2[2]−[3]m 2 η 8 = 2[2] 2[2]−[3]m 2 K . (25) Without singlet mixing, it is satisﬁed for (the mass of) physicalη-meson put insteadofη 8 at properly ﬁxed q=q PS ,andjustthisis meant below. Thetwogeneralizations(24)resp.(25)yieldthestandardGMOmassformula in the corresponding limit of single parameter, f K f π →1resp.q→1. Moreover, thefollowingidentiﬁcationisvalid: f 2 K f 2π ←→ 1 2 [2] 2[2]−[3],3f 2 K /f 2 π 4−f 2 K /f 2π ←→ 1 3 [3] 2[2]−[3],(26) fromwhich,using [3] q = [2] 2 q −1, weget [2] ± = 1−ξ π,K ± /radicalBig/parenleftbig 1−ξ π,K /parenrightbig 2 + 1, ξ π,K ≡(4f 2 K /f 2 π ) −1 .(27) The ratiof K /f π is related to the Cabibbo angle. This is evident either from the formula(see [20]): tan 2 θ C = m 2 π m 2 K /bracketleftBig f K f π − m 2 π m 2 K /bracketrightBig −1 ,or from the formula Γ K→ µν Γ π→µν = (tanθ C ) 2 f 2 K f 2π M K M π /parenleftBigg 1−(M µ /M K ) 2 1−(M µ /M π ) 2 /parenrightBigg 2 for the ratio of weak decay rates usually applied to determine [21, 11] f K /f π in termsoftheCabibboangle,withknownempiricaldataondecayratesandmasses. Thus, the value of f K /f π is expressible through θ C . Together with (26), (27) this implies: within our scheme, the (realistic value q PS of)deformation parameter is directlyconnected with the Cabibbo angle . Similarconclusioncanbearrivedatinanother,moregeneralcontext.In[22], theq-deformed lagrangian for gauge ﬁelds of the Weinberg - Salam (WS) model invariantunderthequantum-groupvaluedgaugetransformationswasconstructed. The obtainedformula[22] F 0 µν = Tr q (F µν ) [2(q 2 +q −2 )] −1/2 =B µν cosθ+F 3 µν sinθ, (28) 3 Note that having used the additional constraint in (23) we are led to the single dimensionless quantity f K f π involvedinthe multipliers of masses. kievarwe.tex; 12/03/2001; 3:49; p.358 352 A.GAVRILIK F 3 µν =∂ µ A 3 ν −∂ ν A 3 µ + ie ab3 (A a µ A b ν −A a ν A b µ ) + [A 3 µ ,B ν ]−[A 3 ν ,B µ ], B µν =∂ µ B ν −∂ ν B µ + [B µ ,B ν ] + [A a µ ,A a ν ] where tanθ= (1−q 2 )/(1 +q 2 ), (29) exhibits a mixing of the U(1)-component B µ with nonabelian components A a µ (the third one). Introducing the new potentials ˜A µ =B µ cosθ+A 3 µ sinθ, Z µ =−B µ sinθ+A 3 µ cosθyields nothing but deﬁnition of physical photon ˜A µ andZ-boson of WS model, where θcoincides with the Weinberg angle, θ=θ W . Since atθ= 0the potentials B µ andA 3 µ get completely unmixed whereas nonzero θ(i.e., nontrivial q-deformation) provides proper mixing as a characteristic feature of the WS model, it is thus seen that the weak mixing is adequately modelled by the q-deformation . Moreover, formula analogous to (29), i.e.,tanθ W =q /radicalbig [4]/([2][3]) [1/2] [3/2],was obtained [23] within somewhat different approachto q-deforming the standardmodel. Hence,theq-deformationrealizespropermixinginthesectorofgaugeﬁelds, thus providing explicit connection between the weak angle and the deformation parameterq. Ontheother hand,therelation found in[24],namely θ W = 2(θ 12 +θ 23 +θ 13 ), (30) connectsθ W withtheCabibboangle θ 12 ≡θ C (andtwootherKobayashi-Maskawa anglesθ 13 ,θ 23 ;aswedealwithtwolightestfamilies,wehavetodiscard θ 13 ,θ 23 ). The importance of (30) consists in that it links two apparently different mixings: oneinvolvedin bosonic(interaction)sector,theotherin fermionic (matter)sector oftheelectroweakstandard model. Combining (29) and (30) ( θ 23 ,θ 13 omitted) we conclude: the Cabibbo angle should be connected with theq-parameter of a quantum-group (or quantum- algebra) basedstructureappliedin thefermion sector. It remains to recall that all our treatment in secs.4-7 using the q-algebras U q (su n )concerned just the fermion sector although at the level of baryons as 3- quarkboundstatesoffundamentalfermions.Hence,itisnaturaltoassertthatthere exists direct connection of the q-parameter involved in (13), (14) with fermion mixing angle. Setting θ 10 =g(θ C )andθ 8 =h(θ C )we ﬁnd for the functions g(θ C )andh(θ C )remarkably simpleexplicit form: θ 10 =θ C , θ 8 = 2θ C . (31) Withθ 8 = π 7 (see(11)) this suggests forCabibbo angle the exactvalue π 14 . kievarwe.tex; 12/03/2001; 3:49; p.359 CABIBBO MIXINGFROM EXTRA DIMENSIONS? 353 9. Discussion Quantum groups and their Hopf dual counterpart - quantum universal envelop- ing algebras (QUEA) incorporate transformation/covariance properties of related quantum vector spaces [25]. In the context of quantum homogeneous spaces (see e.g., [26]) the corresponding quantum groups act (say, on their noncommuting ’coordinates’)inanonlinearway,asitwasexempliﬁed[27]withquantum CP qn . Both quantum groups and their dual QUEA provide necessary tools in construct- ing[28,17]covariantdifferentialcalculiandparticularnoncommutativegeometry on quantum spaces. In the case at hand the internalsymmetries, underlying our treatment of baryon mass sum rules in secs. 4-7 and based on the broken U q (su n ) (n≥3)as wellasunbrokenisospin U q (su 2 )q-algebras,arecloselyrelatedtocertaininternal orextra(asregardstheMinkowskispace M 3,1 )spacetimedimensions.Fromthis we infer the following. The above justiﬁed direct link (31) between the Cabibbo angleθ C = π 14 and theq-parameter, which measures strength of q-deformation for theq-algebrasU q (su n )of ﬂavor symmetry, can be viewed as an indication ofnoncommutative-geometricoriginoffermionmixing.Inthiscontext,thevalue θ C = π 14 of the Cabibbo angle would serve as the noncommutativity measure of relevantquantumspace(responsibleforthemixingandexplicitlyasyetunknown) in extra dimensions. Concerning the latter, one can assert that their number is not lessthan2. Acknowledgements. Iwouldliketothanktheorganizersforcreatingstimulating and warm atmosphere at this NATO workshop. 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Suppl. 18 B(1991), 302–312. kievarwe.tex; 12/03/2001; 3:49; p.362 kievarwe.tex; 12/03/2001; 3:49; p.363 NONCLASSICAL TYPE REPRESENTATIONS OF NONSTANDARD QUANTIZATION OF ENVELOPING ALGEBRAS U(so(n)), U(so(n,1)) AND U(iso(n)) NIKOLAI IORGOV ∗ BogoliubovInstitute forTheoretical Physics, Kiev, Ukraine 1. Introduction Quantum orthogonal groups, quantum Lorentz group and their corresponding quantumalgebrasareofspecialinterestformodernphysics[1].M.Jimbo[2]and V. Drinfeld [3] deﬁned q-deformations (quantum algebras) U q (g)for all simple complex Lie algebras gby means of Cartan subalgebras and root subspaces (see also [4]). However, this approach does not give a satisfactory presentation of the quantum algebra U q (so(n,C))from a viewpoint of some problems in quantum physicsandrepresentationtheory.Whenconsideringrepresentationsofthequan- tum algebras U q (so n+1 )andU q (so n,1 )we are interested in reducing them onto the quantum subalgebra U q (so n ). This reduction would give the analogue of the Gel’fand-Tsetlin basis for these representations. However, deﬁnitions of quan- tum algebras mentioned above do not allow the inclusions U q (so(n+ 1,C))⊃ U q (so(n,C))andU q (so n,1 )⊃U q (so n ). To be able to exploit such reductions we have to consider q-deformations of the Lie algebra so(n+ 1,C)deﬁned in terms of the generators I k,k−1 =E k,k−1 −E k−1,k (whereE is is the matrix with elements (E is ) rt =δ ir δ st )rather than by means of Cartan subalgebras and root elements. To construct such deformations we have to deform trilinear relations forelements I k,k−1 insteadofSerre’srelations(asinthecaseofJimbo’squantum algebras). As a result, we obtain the associative algebra which will be denoted as U /prime q (so(n,C)). Theseq-deformations were ﬁrst constructed in [5]. They permit one to con- struct the reductions of U /prime q (so n+1 )andU /prime q (so n,1 )ontoU /prime q (so n ). Theq -deformed ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.364 358 N. IORGOV algebraU /prime q (so(n,C))leads forn= 3to theq-deformed algebra U /prime q (so(3,C)) deﬁned by A.Odesskii[6]and D.Fairlie [7]. In the classical case, the embedding SO(n)⊂SU(n)(and its inﬁnitesimal analogue)isofgreatimportancefornuclearphysicsandinthetheoryofRieman- niansymmetricspaces.ItiswellknownthatintheframeworkofDrinfeld–Jimbo quantumgroupsandalgebrasonecannotconstructthecorrespondingembedding. The algebra U /prime q (so(n,C))allows to deﬁne such an embedding [8,9], that is, it is possible to deﬁne the embedding U /prime q (so(n,C))⊂U q (sl n ), whereU q (sl n )is the Drinfeld–Jimboquantumalgebra. As a disadvantage of the algebra U /prime q (so(n,C))we have to mention the difﬁ- culties with Hopf algebra structure. Nevertheless, U /prime q (so(n,C))turns out [8,9] to bea coideal in U q (sl n ). Finite dimensional irreducible representations of algebra U /prime q (so(n,C))were constructed in [5]. The formulas of action of the generators of the algebra upon theq-analogue of the Gel’fand–Tsetlin basis are given there. A proof of these formulas and some their corrections were given in [10]. However, ﬁnite dimen- sional irreducible representations described in [5] and [10] are representations of the classical type. They are q-deformations of the corresponding irreducible representations of the Lie algebra so(n,C), that is, at q→1they turn into representations of so(n,C). The algebra U /prime q (so(n,C))has other classes of ﬁnite dimensional irreducible representations which have no classical analogue. These representations are sin- gularatthelimit q→1.Theyweredescribedin[11].Notethatthedescriptionof theserepresentationsforthealgebra U /prime q (so(3,C))isgivenin[12].Aclassiﬁcation ofirreducible∗-representationsofrealformsofthealgebra U /prime q (so(3,C))isgiven in[13]. Thereexistsanalgebra,closelyrelatedtothealgebra U /prime q (so(n,C)),whichisa q-deformationoftheuniversalenvelopingalgebra U(iso n )oftheLiealgebra iso n of the Euclidean group ISO(n)(see [14]). It is denoted as U q (iso n ). Irreducible representations of the classical type of the algebra U q (iso n )were described in [14]. A proof of the corresponding formulas was given in [15]. However, the algebraU q (iso n ),q∈R, has irreducible representations of the nonclassical type. A description of these representations is the aim of this paper. Note that the description of these representations for U q (iso 2 )is given in [16]. The second aim of this paper is to describe irreducible representations of nonclassical type of the algebra U /prime q (so n,1 )which is a real form of the algebra U /prime q (so(n+ 1,C)). Representations of the classicaltype ofthisalgebra are describedin [5] and[17]. We assume throughout the paper that qis a ﬁxed positive number. Thus, we giveformulasforrepresentationsforthesevaluesof q.However,theserepresenta- tionscanbeconsideredforanyvaluesof qnotcoincidingwitharootofunity.For this we have to treat appropriately square roots in formulas for representations or to rescale basis vector in such a way that formulas for representations would not kievarwe.tex; 12/03/2001; 3:49; p.365 NONCLASSICAL REPS OF NONSTANDARDQUANTIZATION 359 containsquare roots. For convenience, we denote the Lie algebra so(n,C)byso n and the algebra U /prime q (so(n,C))byU /prime q (so n ). 2. Theq-deformed algebras U /prime q (so n )andU q (iso n ) In our approach [5] to the q-deformation of the algebras U(so n )we deﬁne the q-deformed algebra U /prime q (so(n,C))as the associate algebra (with a unit) generated by theelements I i,i−1 ,i= 2,3,...,nsatisfyingthe deﬁning relations I i,i−1 I 2 i−1,i−2 −(q+q −1 )I i−1,i−2 I i,i−1 I i−1,i−2 +I 2 i−1,i−2 I i,i−1 =−I i,i−1 ,(1) I 2 i,i−1 I i−1,i−2 −(q+q −1 )I i,i−1 I i−1,i−2 I i,i−1 +I i−1,i−2 I 2 i,i−1 =−I i−1,i−2 ,(2) I i,i−1 I j,j−1 =I j,j−1 I i,i−1 ,|i−j|>1. (3) In the limit q→1formulas (1)–(3) give the relations deﬁning the universal enveloping algebra U(so n ). Note also that relations (1) and (2) principally differ fromtheq-deformedSerrerelationsintheapproachofJimbo[2]andDrinfeld[3] to quantum orthogonal algebras by a presence of nonzero right hand side and by possibility ofthereduction U /prime q (so n )⊃U /prime q (so n−1 )⊃···⊃U /prime q (so 3 ). Recall that in the standard Jimbo–Drinfeld approach to the deﬁnition of quantum algebras, the algebras U q (so 2m )and the algebras U q (so 2m+1 )are distinct series ofquantumalgebraswhichare constructed independently ofeach other. Various real forms of the algebras U /prime q (so n )are obtained by imposing cor- responding∗-structures. The compact real form U /prime q (so(n))is deﬁned by the ∗-structure I ∗ i,i−1 =−I i,i−1 , i= 2,3,...,n. The noncompact q-deformed algebras U /prime q (so p,r )wherer=n−pare singled out respectivelyby means ofthe ∗-structures I ∗ i,i−1 =−I i,i−1 , i/negationslash=p+ 1, i≤n, I ∗ p+1,p =I p+1,p . Among the noncompact real q-algebrasU /prime q (so p,r ),the algebras U /prime q (so n−1,1 )(a q-analogueoftheLorentz algebras)are ofspecial interest. Wealsodeﬁnethealgebra U q (iso n )whichisanonstandarddeformationofthe universal enveloping algebra of the Lie algebra iso n of the Euclidean Lie group ISO(n). It is the associative algebra (with a unit) generated by the elements I 21 ,I 32 ,···,I n,n−1 ,T n such that the elements I 21 ,I 32 ,···,I n,n−1 satisfy the deﬁning relations of the subalgebra U /prime q (so n )and the additional deﬁning relations are I 2 n,n−1 T n −(q+q −1 )I n,n−1 T n I n,n−1 +T n I 2 n,n−1 =−T n , kievarwe.tex; 12/03/2001; 3:49; p.366 360 N. IORGOV T 2 n I n,n−1 −(q+q −1 )T n I n,n−1 T n +I n,n−1 T 2 n = 0, I k,k−1 T n =T n I k,k−1 ifk<n (see [14]). If q= 1, then these relations deﬁne the classical algebra U(iso n ). Let us note that the deﬁning relations for U q (iso n )can be expressed by bilinear relations[15]. 3. Finitedimensional classicaltyperepresentationsof U /prime q (so n ) Inthissectionwedescribe(intheframeworkofa q-analogueofGel’fand–Tsetlin formalism) irreducible ﬁnite dimensional representations of the algebra U /prime q (so n ), n≥3, which areq-deformations of the ﬁnite dimensional irreducible representa- tions of the Lie algebra so n . They are given by the sets m n consisting of⌊n/2⌋ numbersm 1,n ,m 2,n ,...,m ⌊n/2},n (here⌊n/2⌋denotesintegralpartof n/2)which areallintegralor allhalf-integral andsatisfythe dominance conditions m 1,2p+1 ≥m 2,2p+1 ≥...≥m p,2p+1 ≥0, m 1,2p ≥m 2,2p ≥...≥m p−1,2p ≥|m p,2p | forn= 2p+ 1andn= 2p, respectively. These representations are denoted by T m n . For a basis in a representation space we take the q-analogue of Gel’fand– Tsetlinbasiswhichisobtainedbysuccessivereductionoftherepresentation T m n to the subalgebras U /prime q (so n−1 ),U /prime q (so n−2 ),···,U /prime q (so 3 ),U /prime q (so 2 ) :=U(so 2 ). As intheclassical case,itselementsare labelledby Gel’fand–Tsetlin tableaux {ξ n }≡    m n m n−1 ... m 2    ≡{m n ,ξ n−1 }≡{m n ,m n−1 ,ξ n−2 }, (4) wherethe componentsof m k andm k−1 satisfythe “betweenness” conditions m 1,2p+1 ≥m 1,2p ≥m 2,2p+1 ≥m 2,2p ≥...≥m p,2p+1 ≥m p,2p ≥−m p,2p+1 , m 1,2p ≥m 1,2p−1 ≥m 2,2p ≥m 2,2p−1 ≥...≥m p−1,2p−1 ≥|m p,2p |. The basiselementdeﬁnedbytableau {ξ n }isdenotedas|ξ n /angbracketright. Itisconvenientto introduce theso-called l-coordinates l j,2p+1 =m j,2p+1 +p−j+ 1, l j,2p =m j,2p +p−j, (5) forthenumbers m i,k .Inparticular, l 1,3 =m 1,3 + 1andl 1,2 =m 1,2 .Theoperator T m n (I 2p+1,2p )of the representation T m n ofU /prime q (so n )acts upon Gel’fand–Tsetlin kievarwe.tex; 12/03/2001; 3:49; p.367 NONCLASSICAL REPS OF NONSTANDARDQUANTIZATION 361 basiselements,labeledby(4),by theformula T m n (I 2p+1,2p )|ξ n /angbracketright= p /summationdisplay j=1 A j 2p (ξ n ) q l j,2p +q −l j,2p |(ξ n ) +j 2p /angbracketright− p /summationdisplay j=1 A j 2p ((ξ n ) −j 2p ) q l j,2p +q −l j,2p |(ξ n ) −j 2p /angbracketright (6) and theoperator T m n (I 2p,2p−1 )of therepresentation T m n actsas T m n (I 2p,2p−1 )|ξ n /angbracketright= p−1 /summationdisplay j=1 B j 2p−1 (ξ n ) [2l j,2p−1 −1][l j,2p−1 ]|(ξ n ) +j 2p−1 /angbracketright − p−1 /summationdisplay j=1 B j 2p−1 ((ξ n ) −j 2p−1 ) [2l j,2p−1 −1][l j,2p−1 −1]|(ξ n ) −j 2p−1 /angbracketright+ iC 2p−1 (ξ n )|ξ n /angbracketright. (7) In these formulas, (ξ n ) ±j k means the tableau (4) in which j-th component m j,k in m k isreplacedby m j,k ±1.Thecoefﬁcients A j 2p ,B j 2p−1 ,C 2p−1 in(6)and(7)are givenbytheexpressions A j 2p (ξ n ) = /parenleftBigg/producttext p i=1 [l i,2p+1 +l j,2p ][l i,2p+1 −l j,2p − 1] /producttext p i/negationslash=j [l i,2p +l j,2p ][l i,2p −l j,2p ] × /producttext p−1 i=1 [l i,2p−1 +l j,2p ][l i,2p−1 −l j,2p − 1] /producttext p i/negationslash=j [l i,2p +l j,2p + 1][l i,2p −l j,2p −1] /parenrightBigg 1/2 , (8) and B j 2p−1 (ξ n ) =  /producttext p i=1 [l i,2p +l j,2p−1 ][l i,2p −l j,2p−1 ] /producttext p−1 i/negationslash=j [l i,2p−1 +l j,2p−1 ][l i,2p−1 −l j,2p−1 ] × /producttext p−1 i=1 [l i,2p−2 +l j,2p−1 ][l i,2p−2 −l j,2p−1 ] /producttext p−1 i/negationslash=j [l i,2p−1 +l j,2p−1 −1][l i,2p−1 −l j,2p−1 −1]   1/2 , (9) C 2p−1 (ξ n ) = /producttext p i=1 [l i,2p ] /producttext p−1 i=1 [l i,2p−2 ] /producttext p−1 i=1 [l i,2p−1 ][l i,2p−1 −1], (10) wherenumbersin squarebracketsmean q-numbers deﬁned by [a] :=q a −q − a q−q −1 . It is seen from (5) that C 2p−1 in (10) identically vanishes if m p,2p ≡l p,2p = 0. A proof of the fact that formulas (6)-(10) indeed determine a representation of U /prime q (so n )isgivenin [10]. kievarwe.tex; 12/03/2001; 3:49; p.368 362 N. IORGOV 4. Finitedimensional nonclassical type representationsof U /prime q (so n ) The representations of the previous section are called representations of the classical type, because at q→1the operators T m n (I j,j−1 )turn into the corre- spondingoperators T m n (I j,j−1 )forirreducibleﬁnitedimensionalrepresentations withhighest weights m n oftheLie algebra so n . Thealgebra U /prime q (so n )alsohasirreducibleﬁnitedimensionalrepresentations T of nonclassical type, that is, such that the operators T(I j,j−1 )have no classical limitq→1. They are given by sets /epsilon1:= (/epsilon1 2 ,/epsilon1 3 ,···,/epsilon1 n ),/epsilon1 i =±1, and by sets m n consisting of⌊n/2⌋half-integral numbersm 1,n ,m 2,n ,... ,m ⌊n/2⌋,n (here ⌊n/2⌋denotesintegral partof n/2) thatsatisfythedominanceconditions m 1,2p+1 ≥m 2,2p+1 ≥...≥m p,2p+1 ≥1/2, m 1,2p ≥m 2,2p ≥...≥m p−1,2p ≥m p,2p ≥1/2 forn= 2p+ 1andn= 2p, respectively. These representations are denoted by T /epsilon1,m n . For a basis in the representation space we use the analogue of the basis of the previoussection.Its elementsare labeledby tableaux {ξ n }≡    m n m n−1 ... m 2    ≡{m n ,ξ n−1 }≡{m n ,m n−1 ,ξ n−2 }, (11) wherethe componentsof m k andm k−1 satisfythe “betweenness” conditions m 1,2p+1 ≥m 1,2p ≥m 2,2p+1 ≥m 2,2p ≥...≥m p,2p+1 ≥m p,2p ≥1/2, m 1,2p ≥m 1,2p−1 ≥m 2,2p ≥m 2,2p−1 ≥...≥m p−1,2p−1 ≥m p,2p . The basiselementdeﬁnedbytableau {ξ n }isdenotedas|ξ n /angbracketright. It is convenient to introduce the l-coordinates as in (5) The operator T /epsilon1,m n (I 2p+1,2p )of the representation T /epsilon1,m n ofU q (so n )acts upon our basis elements, labeledby (11), by theformulas T /epsilon1,m n (I 2p+1,2p )|ξ n /angbracketright=δ m p,2p ,1/2 /epsilon1 2p +1 q 1/2 −q −1/2 D 2p (ξ n )|ξ n /angbracketright + p /summationdisplay j=1 A j 2p (ξ n ) q l j,2p −q −l j,2p |(ξ n ) +j 2p /angbracketright− p /summationdisplay j=1 A j 2p ((ξ n ) −j 2p ) q l j,2p −q −l j,2p |(ξ n ) −j 2p /angbracketright, wherethesummationinthelastsummustbefrom1to p−1ifm p,2p = 1/2,and theoperator T m n (I 2p,2p−1 )oftherepresentation T m n acts as T /epsilon1,m n (I 2p,2p−1 )|ξ n /angbracketright= p−1 /summationdisplay j=1 B j 2p−1 (ξ n ) [2l j,2p−1 −1][l j,2p−1 ] + |(ξ n ) +j 2p−1 /angbracketright kievarwe.tex; 12/03/2001; 3:49; p.369 NONCLASSICAL REPS OF NONSTANDARDQUANTIZATION 363 − p−1 /summationdisplay j=1 B j 2p−1 ((ξ n ) −j 2p−1 ) [2l j,2p−1 −1][l j,2p−1 −1] + |(ξ n ) −j 2p−1 /angbracketright+/epsilon1 2p ˆC 2p−1 (ξ n )|ξ n /angbracketright, where [a] + =q a +q − a q−q −1 . In these formulas, (ξ n ) ±j k means the tableau (11) in which j-th component m j,k inm k is replaced by m j,k ±1.Matrix elements A j 2p andB j 2p−1 are given by the same formulasasin (6) and(7)(that is,by theformulas (8) and(9))and ˆC 2p−1 (ξ n ) = /producttext p s=1 [l s,2p ] + /producttext p−1 s=1 [l s,2p−2 ] + /producttext p−1 s=1 [l s,2p−1 ] + [l s,2p−1 −1] + . D 2p (ξ n ) = /producttext p i=1 [l i,2p+1 − 1 2 ] /producttext p−1 i=1 [l i,2p−1 − 1 2 ] /producttext p−1 i=1 [l i,2p + 1 2 ][l i,2p − 1 2 ]. Theorem 1. The representations T /epsilon1,m n are irreducible. The representations T /epsilon1,m n andT /epsilon1 /prime ,m /primen are pairwise nonequivalent for (/epsilon1,m n )/negationslash= (/epsilon1 /prime ,m /prime n ). For any admissible (/epsilon1,m n )andm /prime n the representations T /epsilon1,m n andT m /primen are pairwise nonequivalent. The algebra U /prime q (so n )has non-trivial one-dimensional representations. They are special cases of the representations of the nonclassical type. They are describedas follows. Let/epsilon1= (/epsilon1 2 ,/epsilon1 3 ,···,/epsilon1 n ),/epsilon1 i =±1, and let m n =(m 1,n , m 2,n ,···, m ⌊n/2⌋,n ) = ( 1 2 , 1 2 ,···, 1 2 ). Then the corresponding representations T /epsilon1,m n are one-dimensionaland are given by theformulas T /epsilon1,m n (I k+1,k )|ξ n /angbracketright=/epsilon1 k +1 q 1/2 −q −1/2 |ξ n /angbracketright. Thus, to every /epsilon1:= (/epsilon1 2 ,/epsilon1 3 ,···,/epsilon1 n ),/epsilon1 i =±1, there corresponds a one- dimensionalrepresentationof U /prime q (so n ). 5. Deﬁnition ofrepresentationsof U /prime q (so n,1 )andU q (iso n ) Let us recall that we assume that qis a positive number. We give the following deﬁnition of inﬁnite dimensional representations of the algebras U /prime q (so n,1 )and U q (iso n )(wedenotethesealgebrasby A).Itisahomomorphism R:A→L (H) ofAto the spaceL(H)of linear operators (bounded or unbounded) on a Hilbert spaceHsuchthat (a) operators R(a),a∈ A, are deﬁned on an invariant everywhere dense subspaceD⊂H; kievarwe.tex; 12/03/2001; 3:49; p.370 364 N. IORGOV (b)R↓U /prime q (so n )decomposes into a direct sum of irreducible ﬁnite dimen- sional representations of U /prime q (so n )(with ﬁnite multiplicities if Ris irreducible); (c) subspacesof irreduciblerepresentations of U /prime q (so n )belongtoD. Twoinﬁnitedimensionalirreduciblerepresentations RandR /prime ofAonspaces HandH /prime , respectively, are called (algebraically) equivalent if there exists an everywhere dense invariant subspaces V⊂DandV /prime ⊂D /prime and a one-to-one linear operator A:V→V /prime such thatAR(a)v=R /prime (a)Avfor alla∈Aand v∈V. Remarkthatourdeﬁnitionofinﬁnitedimensionalrepresentationsof U /prime q (so n,1 ) andU q (iso n )corresponds to the deﬁnition of Harish-Chandra modules for the pairs (so n,1 ,so n )and(iso n ,so n ),respectively.Thus,modulesdeterminedbyrep- resentations of the above deﬁnition can be called q-Harish-Chandra modules of thepairs (U /prime q (so n,1 ),U /prime q (so n ))and(U q (iso n ),U /prime q (so n )), respectively. 6. Representationsof U q (iso n ) Therearethe followingclassesofirreducible representations of U q (iso n ): (a) Finite dimensional irreducible representations RofU /prime q (so n ). They are irreducible representationsof U q (iso n )withR(T n ) = 0. (b)Inﬁnite dimensionalirreducible representations of theclassicaltype. (c) Inﬁnitedimensionalirreducible representations ofthe nonclassical type. Representations R λ,m of class (b) are given in [14,15]. Let us describe rep- resentations of class (c), that is, representations Rfor which there exists no limitq→1for the operators R(T n )andR(I i,i−1 ). These representations are given by/epsilon1:= (/epsilon1 2 ,/epsilon1 3 ,···,/epsilon1 n+1 ), non-zero complex parameter λand by num- bersm= (m 2,n+1 ,m 3,n+2 ,···,m ⌊(n+1)/2⌋,n+1 ),m 2,n+1 ≥m 3,n+2 ≥···≥ m ⌊(n+1)/2⌋,n+1 ≥1/2, describing irreducible representations of the nonclassical type of the subalgebra U /prime q (so n−1 )(see section 4). We denote the corresponding representations of U q (iso n )byR /epsilon1,λ,m . Inorderto describethespaceof therepresentation R /epsilon1,λ,m wenote that R /epsilon1,λ,m ↓U /prime q (so n ) = /circleplusdisplay m n T /epsilon1 /prime ,m n ,m n = (m 1,n ,···,m ⌊n/2⌋,n ), (12) where/epsilon1 /prime = (/epsilon1 2 ,···,/epsilon1 n )is the part of the set /epsilon1, the summation is over all irreducible nonclassical type representations T /epsilon1 /prime ,m n ofU /prime q (so n )for which the componentsof m n satisfythe “betweenness”conditions m 1,2k ≥m 2,2k+1 ≥m 2,2k ≥...≥m k,2k+1 ≥m k,2k ≥1/2 ifn= 2k, m 1,2k−1 ≥m 2,2k ≥m 2,2k−1 ≥...≥m k−1,2k−1 ≥m k,2k ifn= 2k−1. kievarwe.tex; 12/03/2001; 3:49; p.371 NONCLASSICAL REPS OF NONSTANDARDQUANTIZATION 365 The carrier space ˆH /epsilon1,m of the representation R /epsilon1,λ,m decomposes as ˆH /epsilon1,m = /circleplustext m n H /epsilon1 /prime ,m n , where the summation is such as in (12) and H /epsilon1 /prime ,m n are the sub- spaces, where the representations T /epsilon1 /prime ,m n ofU /prime q (so n )are realized. We choose a basis in every subspace H /epsilon1 /prime ,m n as in section 4. The set of all these bases gives a basis of the space ˆH /epsilon1,m . We denote the basis elements by |m n ,M/angbracketright, whereM are the corresponding tableaux. The numbers m ij from|m n ,M/angbracketrightdetermine the numbersl ij as insection 3.The numbers m i,n+1 determinethe numbers l i,2k+1 =m i,2k+1 +k−i+ 1, n= 2k, l i,2k =m i,2k +k−i, n = 2k−1. The operators R /epsilon1,λ,m (I i,i−1 )are given by formulas of the nonclassical type rep- resentationsofthealgebra U /prime q (so n )fromsection4.Fortheoperators R /epsilon1,λ,m (T 2k ) andR /epsilon1,λ,m (T 2k−1 )wehave theexpressions R /epsilon1,λ,m (T 2k−1 )|m 2k−1 ,M/angbracketright=λ k−1 /summationdisplay j=1 ˜B j 2k−1 (m 2k−1 ,M ) [2l j,2k−1 −1][l j,2k−1 ] + |m +j 2k−1 ,M/angbracketright +λ k−1 /summationdisplay j=1 ˜B j 2k−1 (m −j 2k−1 ,M ) [2l j,2k−1 −1][l j,2k−1 −1] + |m −j 2k−1 ,M/angbracketright +i/epsilon1 2k λˆC 2k−1 (m 2k−1 ,M)|m 2k−1 ,M/angbracketright, R /epsilon1,λ,m (T 2k )|m 2k ,M/angbracketright= iλδ m p,2p ,1/2 /epsilon1 2k +1 q 1/2 −q −1/2 D 2k |m 2k ,M/angbracketright +λ k /summationdisplay j=1 ˜A j 2k (m 2k ,M ) q l j,2k −q −l j,2k |m +j 2k ,M/angbracketright+λ k /summationdisplay j=1 ˜A j 2k (m −j 2k ,M ) q l j,2k −q −l j,2k |m −j 2k ,M/angbracketright, wherethesummationinthelastsummustbefrom1to k−1ifm k,2k = 1/2,and ˜A j 2k (m 2k ,M) = /parenleftBigg/producttext k i=2 [l i,2k+1 +l j,2k ][l i,2k+1 −l j,2k − 1] /producttext i/negationslash=j [l i,2k +l j,2k ][l i,2k −l j,2k ] × /producttext k−1 i=1 [l i,2k−1 +l j,2k ][l i,2k−1 −l j,2k − 1] /producttext i/negationslash=j [l i,2k +l j,2k + 1][l i,2k −l j,2k −1] /parenrightBigg 1/2 , (13) ˜B j 2k−1 (m 2k−1 ,M) = /parenleftBigg/producttext k i=2 [l i,2k +l j,2k−1 ][l i,2k −l j,2k−1 ] /producttext i/negationslash=j [l i,2k−1 +l j,2k−1 ][l i,2k−1 −l j,2k−1 ] × /producttext k−1 i=1 [l i,2k−2 +l j,2k−1 ][l i,2k−2 −l j,2k−1 ] /producttext i/negationslash=j [l i,2k−1 +l j,2k−1 −1][l i,2k−1 −l j,2k−1 −1] /parenrightBigg 1/2 ,(14) ˆC 2k−1 (M) = /producttext k s=2 [l s,2k ] + /producttext k−1 s=1 [l s,2k−2 ] + /producttext k−1 s=1 [l s,2k−1 ] + [l s,2k−1 −1] + , (15) kievarwe.tex; 12/03/2001; 3:49; p.372 366 N. IORGOV D 2k = /producttext k i=2 [l i,2k+1 − 1 2 ] /producttext k−1 i=1 [l i,2k−1 − 1 2 ] /producttext k−1 i=1 [l i,2k + 1 2 ][l i,2k − 1 2 ]. (16) Theorem 2. The representations R /epsilon1,λ,m are irreducible. The representations R /epsilon1,λ,m andR /epsilon1 /prime ,λ /prime ,m /prime are equivalent if and only if /epsilon1=/epsilon1 /prime ,m=m /prime andλ=±λ /prime . The operators R /epsilon1,λ,m (T n )are bounded. The representation R /epsilon1,λ,m is equivalent tonoof therepresentations R λ /prime ,m /prime ofclassical type. 7. Representationsof U /prime q (so n,1 ) Irreducible representations of classical type of algebra U /prime q (so n,1 )are given in [5,17]. Here we describe irreducible representations of nonclassical type (that is, representations Rfor which there exists no limit q→1for the operators R(I i,i−1 )).Theserepresentationsaregivenbytheset /epsilon1:= (/epsilon1 2 ,/epsilon1 3 ,···,/epsilon1 n+1 ),bya complexparameter candbytheset m= (m 2,n+1 ,m 3,n+1 ,···,m ⌊(n+1)/2⌋,n+1 ), m 2,n+1 ≥m 3,n+2 ≥···≥m ⌊(n+1)/2⌋,n+1 ≥1/2, describing irreducible rep- resentations of the nonclassical type of the subalgebra U /prime q (so n−1 )(see section 4). We denote thecorrespondingrepresentationsof U q (so n,1 )byR /epsilon1,c,m . Inorderto describethespaceof therepresentation R /epsilon1,c,m we note that R /epsilon1,λ,m ↓U /prime q (so n ) = /circleplusdisplay m n T /epsilon1 /prime ,m n ,m n = (m 1,n ,···,m ⌊n/2⌋,n ), (17) where/epsilon1 /prime = (/epsilon1 2 ,···,/epsilon1 n ), the summation is over all irreducible nonclassical type representations T /epsilon1 /prime ,m n of the subalgebra U /prime q (so n )for which the components of m n satisfythe“betweenness” conditions m 1,2k ≥m 2,2k+1 ≥m 2,2k ≥...≥m k,2k+1 ≥m k,2k ≥1/2 ifn= 2k m 1,2k−1 ≥m 2,2k ≥m 2,2k−1 ≥...≥m k−1,2k−1 ≥m k,2k ifn= 2k−1. The carrier space ˆH /epsilon1,m of the representation R /epsilon1,c,m decomposes as ˆH /epsilon1,m = /circleplustext m n H /epsilon1,m n , where the summation is such as in (17) and H /epsilon1 /prime ,m n are the sub- spaces, where the representations T /epsilon1 /prime ,m n ofU /prime q (so n )are realized. We choose the basis in every subspace H /epsilon1,m n as in section 4. The set of all these bases gives a basis of the space ˆH /epsilon1,m . We denote the basis elements by |m n ,M/angbracketright, whereM are the corresponding tableaux. The numbers m ij from|m n ,M/angbracketrightdetermine the numbersl ij as in section 3. The numbers m i,n+1 determine the numbers l i,n+1 as in section 6. The operators R /epsilon1,c,m (I i,i−1 ),i≤n, are given by formulas of the nonclassical type representations of the algebra U /prime q (so n )as in section 4. For the operatorsR /epsilon1,c,m (I 2k+1,2k )ifn= 2kandR /epsilon1,c,m (T 2k,2k−1 )ifn= 2k−1wehave theexpressions R /epsilon1,c,m (I 2k,2k−1 )|m 2k−1 ,M/angbracketright kievarwe.tex; 12/03/2001; 3:49; p.373 NONCLASSICAL REPS OF NONSTANDARDQUANTIZATION 367 = k−1 /summationdisplay j=1 ([c+l j,2k−1 ][c−l j,2k ]) 1/2 ˜B j 2k−1 (m 2k−1 ,M ) [2l j,2k−1 −1][l j,2k−1 ] + |m +j 2k−1 ,M/angbracketright − k−1 /summationdisplay j=1 ([c+l j,2k−1 + 1][c−l j,2k + 1]) 1/2 ˜B j 2k−1 (m −j 2k−1 ,M ) [2l j,2k−1 −1][l j,2k−1 −1] + |m −j 2k−1 ,M/angbracketright +/epsilon1 2k [c] + ˆC 2k−1 (m 2k−1 ,M)|m 2k−1 ,M/angbracketright, R /epsilon1,c,m (I 2k+1,2k )|m 2k ,M/angbracketright =δ m k,2k ,1/2 [c−1/2]/epsilon1 2k +1 q 1/2 −q −1/2 D 2k (m 2k ,M)|m 2k ,M/angbracketright + k /summationdisplay j=1 ([c+l j,2k ][c−l j,2k −1]) 1/2 ˜A j 2k (m 2k ,M ) q l j,2k −q −l j,2k |m +j 2k ,M/angbracketright− − k /summationdisplay j=1 ([c+l j,2k −1][c−l j,2k ]) 1/2 ˜A j 2k (m −j 2k ,M ) q l j,2k −q −l j,2k |m −j 2k ,M/angbracketright, wherethesummationinthelastsummustbefrom1to k−1ifm k,2k = 1/2,and ˜A j 2k ,˜B j 2k−1 ,ˆC 2k−1 ,D 2k aresuch asin(13)–(16). Theorem3. Therepresentation R /epsilon1,c,m ofU /prime q (so 2k,1 )isirreducibleifandonlyif c isnothalf-integeroroneofthenumbers c,1−ccoincideswithoneofthenumbers l j,2k+1 ,j= 2,3,···,k. The representation R /epsilon1,c,m ofU /prime q (so 2k−1,1 )is irreducible if and only if cis not half-integer or |c|coincides with one of the numbers l j,2k , j= 2,3,···,k,or|c|<l k,2k . Acknowledgements The author is thankful to Prof. A. U. Klimyk for the fruitful discussions. The researchcontainedinthispaperwassupportedinpartbyAwardNo.UP1-2115of the Civilian Research and Development Foundation for the Independent States of theFormerSovietUnion (CRDF). References 1. W. B. Schmidke, J. Wess, and B. Zumino, Aq-deformed Lorentz algebra , Z. Phys. C 52 (1991), 471–476. 2. M.Jimbo, Aq-differenceanalogueof U(g)andtheYang–BaxterEquation , Lett.Math.Phys. 10(1985), 63–69. 3. V. G. Drinfeld, Hopf algebra and Yang–Baxter equation , Sov. Math. 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A.M.GavrilikandN.Z.Iorgov, Representationsofthenonstandardalgebras U q (so(n))and U q (so(n−1,1))in Gel’fand–Tsetlin basis , Ukr. J. Phys. 43(1998), 791–797. kievarwe.tex; 12/03/2001; 3:49; p.375 QUASIPARTICLES IN NON-COMMUTATIVEFIELD THEORY KARL LANDSTEINER ∗ Theory DivisionCERN, 1211Geneva23, Switzerland Abstract. After a short introduction to the UV/IR mixing in non-commutative ﬁeld theories we reviewthepropertiesofscalarquasiparticlesinnon-commutativesupersymmetricgaugetheoriesat ﬁnite temperature. In particular we discuss theappearance of superluminous wave propagation. 1. Introduction Given the experience of quantum mechanics it seems a rather natural idea that spacetime at very small distance-scales might be described by non-commuting coordinates [1, 2]. Keeping the example of quantum mechanics in mind one is leadtowrite downa commutationrelationfor the coordinates such as [x m ,x n ] =iθ mn . (1) In order to study quantum ﬁeld theory on such non-commuting spaces it is useful to make some further simplifying assumptions, in particular we will take θ mn to bean elementof thecenter ofthe algebra deﬁnedby(1). A convenient way of thinking about non-commutativity is by deformation of the product on the space of ordinary function. Using θ mn as deformation parameterwedeﬁne theso-calledMoyalproduct (or star-product)by f(x)∗g(x) := lim y→x e i 2 θ mn ∂ x m ∂ y n f(x)g(y). (2) Inmomentumspace it takestheform f(x)∗g(x) = /integraldisplay d n k (2π) n /integraldisplay d n q (2π) n ˜f(k)˜g(q)e −i(k+q)x e − i 2 k m θ mn q n .(3) Animmediateconsequenceisthatwecanalwaysdeleteonestarundertheintegral because the additional terms by which the Moyal product differs from the usual ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.376 370 K.LANDSTEINER k 1 k 2 k 3 k 4 =exp(g Π 2 a<b kθ k b ) ai 2-- Figure11. Feynman rulefor non-commutative Φ 4 vertex. k 2 k 1 k 3 k 4 (a) k 1 k 2 k 3 k 4 (b) Figure 12. Corrections to the two-point function can be either planar as in (a) or non-planar as in (b) product are total derivatives thankstotheantisymmetryof θ mn /integraldisplay f(x)∗g(x)d n x= /integraldisplay/parenleftbigg f(x).g(x) + i 2θ mn ∂ m f(x)∂ n g(x) +··· /parenrightbigg d n x.(4) This furthermoreimpliescyclic symmetry underintegral /integraldisplay f∗g∗h= /integraldisplay f.g∗h= /integraldisplay g∗h.f= /integraldisplay g∗h∗f. (5) We have now all the ingredients do start discussing ﬁeld theory. Before doing so we will introduce one further simpliﬁcation, namely we will assume from that time is an ordinary commuting coordinate, i.e. θ m0 = 0. This has the advantage that we are still dealing with a system with a ﬁnite number of time derivatives. Although a canonical formalism for theories with an inﬁnite number of time derivatives can be developed [3] it turns out that quantum ﬁeld theory on spaces withtime-spacenon-commutativityarenotunitary atthe one-looplevel [4] 1 . Non-commutative ﬁeld theories can be viewed as non-local deformations of localﬁeldtheories.Forﬁeldsofspinzeroorone-halfwecantakeaLagrangianof an ordinary ﬁeld theory and deform the product of ﬁelds according to the Moyal product (4), i.e. we replace the ordinary product by the star product. For spin one-ﬁeldswealsohavetoconsiderthatthegaugesymmetryisdeformed, δA m = ∂ m λ+i{A m ,λ} ∗ ,where{.,.} ∗ denotestheMoyalbracket {f,g} ∗ =f∗g−g∗f. The non-commutative ﬁeld strength of a gauge ﬁeld is deﬁned accordingly as F mn =∂ m A n −∂ n A m +i{A m ,A n } ∗ and the covariant derivative as D m .= ∂ m .+i{A m ,.} ∗ [5]. 1 This applies to the time-like case, i.e. in all coordinate systems with θ mn =constthe commutator (1) involvesthe time coordinate. kievarwe.tex; 12/03/2001; 3:49; p.377 QUASIPARTICLES IN NON-COMMUTATIVE FIELD THEORY 371 Letusconsidernowascalar Φ 4 theoryoninfourdimensions.Withoutfurther loss of generality we assume θ 23 =−θ 32 =θ. Because one can drop one star- productintheLagrangianthefreetheoryisunchangedwithrespecttotheoneon ordinaryIR n .The treelevel propagator isthen theusual one /angbracketleftΦ(p)Φ(−p)/angbracketright= i p 2 −m 2 . (6) The one-loop corrections to the two point function that arise from the Φ 4 vertex are shown in ﬁgure 2(a) and 2(b). Because of the cyclic symmetry of the vertex wehavetwodistinctclassesofgraphs[6].Ifweconnectneighbouringlinesofthe vertex in ﬁgure (1) the dependence of the exponential on the internal momentum k=k 1 =−k 2 cancels.Thusthediagram2(a)givesrisetoaquadraticdivergence inthesamewayasit happens in ordinary Φ 4 theory. Ifwecontracthowevernon-neighbouringlinesthedependenceontheinternal momentum of the exponent does not cancel. The distinct classes of Feynman diagrams in non-commutative ﬁeld theories are called planar if they are of type 2(a) andnon-planar iftheyare of type2(b). Thedivergenceisregulatedbytherapidoscillationoftheexponentialfunction atlargeinternalmomentumandwe ﬁnd 4g 2 /integraldisplay d 4 k (2π) 4 e i˜p k k 2 −m 2 =ig 2 4π 2 ˜p 2 +···, (7) Where we introduced the notation ˜p n =p m θ mn and the dots indicate terms that are less singular for ˜p→0. Resummation gives rise to a corrected two-point functiononthe one looplevelofthe form Γ 2 (p) =p 2 −m 2 R +g 2 π 2 ˜p 2 . (8) Thequadraticdivergenceintheplanargraphgivesrisetoarenormalizationofthe mass.Thenon-planargraphresultsinadramaticchangeoftheinfraredbehaviour of the theory. On a technical level the origin of this infrared divergence is easily understood. The non-planar diagram is regulated by the phase factor stemming from the star product. This phase is absent if the external momentum ﬂowing intothediagramvanishes.Thustheultravioletdivergencehasbeenconvertedinto an infrared divergence. This phenomenon UV/IR mixing has ﬁrst been discussed in [7] and has been further investigated in [9]- [33]. Notice also that the IR- singularity is present even in the massive theory. Since it is induced by modes inthefarUV circlingin the loopit isinsensitive tothe presenceofa massterm. It should be emphasized that there are usually also subleading logarithmic infrared divergencies. In the infrared these become important at momenta of the order of p=O(e − 1 g2 ). Down to these non-perturbatively small momenta kievarwe.tex; 12/03/2001; 3:49; p.378 372 K.LANDSTEINER the infrared behaviour is dominated by the effects stemming from the quadratic divergencies. In the following we will always concentrate on the leading order IR-behaviour and thusneglect thecontributionsfrom the logarithms. In supersymmetric theories quadratic divergencies in four dimensions are ab- sent. However at ﬁnite temperature supersymmetry is broken and the one-loop dispersion relation will again show effects from UV/IR mixing in non-planar graphs.BecausetemperatureactsasacutoffnoIR-singularitiesaretobeexpected. The next section reviews these effects in the example of N= 4supersymmetric Yang-Mills theory. 2. Quasiparticles in non-commutative N=4SYM We limit ourselves to the study of a non-commutative U(1)N= 4gauge theory. The spectrum of the theory consists of six scalars, four Majorana Fermions and a vector ﬁeld. The Lagrangiantakestheform L= 1 g 2 /integraltext/parenleftBig − 1 4 F mn F mn + 1 2 D m Φ ab D m Φ ab + 1 4 {Φ ab ,Φ cd } ∗ {Φ ab ,Φ cd } ∗ + +iλ a σ m D m ¯λ a +i{λ a ,λ b } ∗ Φ ab +i{¯λ a ,¯λ b } ∗ Φ ab /parenrightBig . (9) The theory has a global SU(4)symmetry under which the fermions transform under the 4,¯4. The 6 scalars transform in the antisymmetric. This symmetry is indicatedby indices a,b. Wewillstudythedispersionrelationofthe N=4scalarsatﬁnitetemperature andonelooplevel.FinitetemperatureisimplementedintheMatsubaraformalism by considering the theory on S 1 ×IR×IR 2 nc . The last factor indicates the two- dimensional non-commutative plane. The fermions are taken to have antiperiodic boundary conditions on the S 1 factor. Non-commutative ﬁeld theories at ﬁnite temperaturehavebeeninvestigated in[34]-[37] The scalarself-energyis given by Σ T = 32g 2 /integraldisplay d 3 k (2π) 3 sin 2˜p· k 2 k(n B (k) +n F (k)) + 4g 2 P 2 ¯Σ,(10) n B (k)andn F (k)denote Bose-Einstein and Fermi-Dirac distributions. Four mo- mentum is denoted by P 2 =p 2 0 −p 2 , lowercase denotes three-momentum. Momenta alongthe non-commutativedirectionsas will be called transverse. The ﬁrst term in (10) vanishes at T= 0because of supersymmetry. The second term contributes to the ﬁnite temperature wave-function renormalization of the scalar ﬁeld. It affects the position of the pole only to O(g 4 )and we will dropitinthesequel. kievarwe.tex; 12/03/2001; 3:49; p.379 QUASIPARTICLES IN NON-COMMUTATIVE FIELD THEORY 373 0.5 1 1.5 2 2.5 Pnc 0.5 1 1.5 2 2.5 3 Energy Figure 13. Dispersion relation for scalars in N= 4Yang-Mills for different temperatures. The momentumpis taken to lie entirely in the non-commutative directions. The dashed line shows the light coneω=p. The dotted line shows the momentum p c below which the group velocity ∂ ω ∂p is bigger than one. Using the relation sin 2˜p k 2 = 1 2 (1−cos ˜pk)we can separate the planar and non-planarcontributionsto the self-energy.The dispersion relation becomes ω 2 =p 2 + 2g 2 T 2 −4g 2 T π|˜p|tanhπ|˜p| T 2. (11) Aplotofthedispersionrelationisshowninﬁgure(3).Thehyperbolictangent arises solelyfrom the non-planarcontributionto thedispersion relation. Forlargetransverseexternalmomentathenon-planarcontributionissublead- ingwith respect tothe planar one, ω 2 ≈p 2 + 2g 2 T 2 −4g 2 T π|˜p|, T ˜p/greatermuch1. (12) The second term comes from the planar diagrams and gives a mass to the scalar excitations. The subdominant term linear in Tarises solely from soft bosons in non-planardiagrams.Thesearemodeswithcharacteristicmomentum k/lessmuchTand large occupation number n B ≈T/k/greatermuch1, Σ np ∼ /integraldisplay d 3 k 1 kcos ˜pk T k∼ T ˜p. (13) In usual space-time the approximation n B ≈T/k/greatermuch1results in the well known ultraviolet catastrophe of classical ﬁeld theory. In the non-planar sector of non-commutative space this does not happen as long as ˜pis different from zero. This is yet another manifestation of the UV/IR mixing of non-commutative ﬁeld theories: to leading order at high temperature, the non-planar contribution is effectivelypurely classical[36]. At low transverse external momenta, the non-planar contribution tends to cancel the planar one. For zero external transverse momentum the interaction switchesoff.Thetheorybecomesafree,gapless U(1)gaugetheorywith ω 2 ≈p 2 3 . kievarwe.tex; 12/03/2001; 3:49; p.380 374 K.LANDSTEINER Let us consider now the case where the momentum lies along the non- commutative directions. Since ω(0) = 0and for large p,ω(p)≈ /radicalbig p 2 + 2g 2 T 2 , which lies above the lightcone, there is a region in between with ∂ω(p ) ∂p >1. Thus thegroup velocitymustexceed thespeedof lightfor smalltransverse momenta! ω 2 ≈ /parenleftBigg 1 +g 2 π 2 T 4 θ 2 6 /parenrightBigg p 2 . (14) The low momentum excitations are massless, but propagate with an index of re- fractionn=p/ωsmaller than one. Because the interactions switch off at low momenta, we expect these modes to be long-lived. In ﬁgure (3) the momentum p c below which the group velocity exceeds one is depicted by a dotted line. The dashedlineshowsthelightcone ω=p. Let us emphasize that these qualitative features should be quite general and not an artifact of our one loop approximation, as they simply arise from the fact that the theory is non-interacting at zero transverse momentum and develops a massgapotherwise 2 . We now investigate the consequences of the dispersion relation (14) for wave propagation. Imagine that some disturbance of the scalar ﬁeld is created in the thermal bath at time t= 0. To simplify matters we will consider only a one dimensional problem with momentum pointing in a non-commutative direction. Thefastestmovingmodesaretheoneswithlongestwavelength.Thesearealsothe modes which are long lived in the thermal bath. For these it is possible to obtain theexactasymptoticbehaviourbynotingthatthedispersionrelationaround k= 0 is ω(k) =c 0 k−γk 3 +O(k 5 ), (15) withc 0 = /radicalBig 1 + g 2 π 2 T 4 θ 2 6 andγ= g 2 π 4 θ 4 T 6 120c 0 .Thisisthedispersionrelationofthe linearisedKorteweg-deVriesequationwhosesolutionisexpressedintermsofthe Airy function Ai(z). We can express the solution for the head of a wavetrain by [38] Φ = A 2(3γt) 1 3 Ai /parenleftBigg x−c 0 t (3γt) 1 3 /parenrightBigg . (16) The Airy function has oscillatory behaviour for negative argument and decays exponentially for positive argument. Thus the wavetrain decays exponentially ahead ofx=c 0 t. Behind the wave becomes oscillatory. In this region one can 2 One might also be worried if these effects are gauge dependent. A model without gauge sym- metry can be obtained if one sets the gauge ﬁeld and one fermion (the ﬁeld content of an N= 1 vector multiplet)to zero. This would result in a N= 1Wess-Zumino model with Moyal bracket interactions. It wouldonly change the overall factor in (10). kievarwe.tex; 12/03/2001; 3:49; p.381 QUASIPARTICLES IN NON-COMMUTATIVE FIELD THEORY 375 match the Airy function with the asymptotics obtained from a stationary wave approximation. In between the oscillatory region and the exponential decay there is a transition region of width proportional to (γt) 1 3 aroundx=c 0 t. In this region the wavetrain has its ﬁrst crest which therefore is moving with a velocity approximatelygiven by c 0 . Group velocities faster than the speed of light do also appear in conventional physics, e.g. it is well-known that this happens for light propagation in media close to an absorption line. Since the dispersive effects are however large, the group velocity loses its meaning as the velocity of signal transportation. In our case, it is interesting to notice that as the temperature increases, not only c 0 but alsoγgrows. This implies that at high temperatures the soft transverse momenta become very dispersive. In such situations it is useful to introduce the concept of a front velocity which is the velocity of the head of the wavetrain. For the propagation of light in a medium it can be shown that this front velocity never exceeds the speed of light even if the group velocity can be faster than the speed of light [39]. In our case the front velocity can be deﬁned as the velocity of the ﬁrst crest of the wavetrain. According to (14) and (15) this is always bigger that the speed of light. The advance of the ﬁrst crest with respect to an imagined light front is (c 0 −1)t. Since its spread grows as (γt) 1 3 , the ﬁrst crest is well deﬁned outsidethelightconeforlarge enoughtime, t>t 0 wheret 0 = /radicalBig γ (c 0 −1) 3 . The question arises if this superluminosity implies a violation of causality. This is not necessarily the case. Violation of causality needs both ingredients: superluminosityandtherelativityprinciple.ImagineanobserverAemittingsome signal with superluminous velocity c 0 . If the relativity principle is valid another observer B in a boosted frame relative to A could then catch the signal. B could send an answer also with superluminous speed c 0 . The answer would reach ob- serverAbeforehesentouttheoriginalsignal.Thecrucialpointisofcoursethatin thenon-commutativespace-timeweareconsideringboostsarenotanymoresym- metries.InparticularonlyintheframeofobserverAtimeisordinary,commuting time. Any other frame involving a boost in a non-commutative direction implies that also time is non-commutative. To obtain an answer if causality is violated onewouldhavetocalculatethedispersionrelationalsoinsuchaframeandstudy wave propagation then. Finite temperature ﬁeld theory with non-commutative timeishoweverdifﬁculttoformulateduetotheinﬁnitenumberoftimederivatives appearing in the star product. This is an open question though progress could possiblybeachievedalong thelines in[3]. 3. Discussionand Outlook We have concentrated on reviewing the properties of scalar quasiparticles at ﬁ- nite temperature in non-commutative N= 4gauge theory. Another system that has been studied in [37] is the non-commutative Wess-Zumino model with star- kievarwe.tex; 12/03/2001; 3:49; p.382 376 K.LANDSTEINER productinteractionsinsteadofMoyal-brackets.Theone-loopself-energyisgiven by a similar expression as (10) except that sin ˜ pk 2 is substituted by cos ˜ pk 2 . It turns out that this has the effect that for temperatures T >T 0 ≈ 1 √ gθ the minimum of thedispersionrelationisdisplacedfrom p= 0!Ithasbeenarguedthatthismakes Bose-condensation of scalar modes impossible for temperatures higher than T 0 [37] 3 . Anothersystemthathasbeenstudiedin[37]was N=2gaugetheoryatﬁnite density. The results are qualitatively analogous to the case with temperature. The roleofthetemperatureisthen played bythechemical potential. Non-commutative ﬁeld theories in the setup discussed here appear also in stringtheory.In[41]itwasshownthatthephysicsofD-branesina B-ﬁeldback- groundinaparticularscalinglimitwith α /prime →0isdescribedbynon-commutative supersymmetric gauge theories. It has been suggested that the effects of UV/IR mixing could be understood from a string perspective [7]. The UV/IR mixing in this stringy context has been considered in [43]-[49]. 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Introduction Nearlyallknowntodayintegrablesystemsarehomogeneouswithrespecttosome scaling. For such systems no generality is lost in assuming the homogeneity of symmetries, master symmetries, recursion operators, etc., and this considerably simpliﬁes theirﬁndingandstudy,see e.g.[1]–[10]. In the present paper we combine this well-known idea with our new results on the structure of time-dependent (cf. e.g. [6, 11–14] for the time-independent case)formalsymmetriesforanaturalgeneralizationofthesystems,consideredin [11,12,15],namely,for(1+1)-dimensionalnondegenerateweaklydiagonalizable (NWD) evolution systems with constraints. This enables us to ﬁnd simple sufﬁcient conditions for the commutativity and time-independence of higher order symmetries and for the existence of inﬁnite number of such symmetries for homogeneous NWD systems with constraints. Note that the majority of known [8, 10, 12] and recently found, see e.g. [7, 16, 17], integrable evolution systems in (1+1) dimensions ﬁt into this class. Moreover, our results, unlike the majority ofalreadyknownones,arevalidforthesystemswithtime-dependentcoefﬁcients as well,cf. e.g. [18],andarenotrestrictedto scalarequations. Let us stress that the proofs and the application of our results involve just an easy veriﬁcation of some weight-related conditions and do notrely on the existence of a master symmetry or e.g. (hereditary) recursion operator. Hence, the results of present paper (except for those on existence of inﬁnitely many symmetries) can be applied to non-integrable systems as well. On the other hand, the simplicity of use makes our results particularly helpful in the study of ne w ∗ [email protected], [email protected] kievarwe.tex; 12/03/2001; 3:49; p.386 380 A.SERGYEYEV integrablesystemsforwhichonlyafewhigherordersymmetriesand(sometimes) a‘candidate’forthemastersymmetryareknown,butnorecursionoperatorisyet found.Indeed,weshowthatthecheckofasmallnumberofconditionsforthelow ordersymmetriescanreplacetediouschecks,cf.[19],thattime-independentsym- metriesofsufﬁcientlyhighordercommute,thata‘candidate’formastersymmetry is a nontrivial master symmetry and that its action yields the symmetries being well-deﬁned (cf. [10] for recursion operators and localsymmetries) functions of local variables x,t,u,u 1 ,...andof nonlocalvariables ω γ deﬁned below. Note that, unlike [4, 5, 19], in order to prove the existence of inﬁnitely many symmetries we do not make a prioriextra assumptions, say, about the existence of “negative” master symmetries τ j ,j < 0[19]: all we need is a suitable ‘candidate’τfor the master symmetry and a higher order time-independent symmetry. We also show that in order to verify the commutativity of allhigher order time-independent homogeneous symmetries at once, it sufﬁces to check only a small number of conditions for the time-independent symmetries of order lower than two. Moreover, checks of this kind are almost entirely algorithmic, so computeralgebra softwarecan bereadily applied to perform them. The paper is organized as follows. In Section 2 we give some deﬁnitions and facts, being the straightforward extension of those from [11, 12, 15] to the case of explicitly time-dependent evolution systems with constraints. In Section 3 we present the sufﬁcient conditions of well-deﬁniteness of the symmetries generated by means of master symmetry for the general evolution systems with constraints. In Section 4 we deﬁne nondegenerate weakly diagonalizable (NWD) systems with constraints and present some results on structure of their formal symmetries. In Section 5 we ﬁnd the sufﬁcient conditions for commutativity and time-independence of higher order symmetries and for the existence of inﬁnite hierarchiesoftime-independenthigherordersymmetriesforhomogeneousNWD systemswithconstraints. 2. Basic deﬁnitions andstructures Letusconsider asystem ofevolution equationswith constraints(cf. [15]) ∂u/∂t=F(x,t,u,...,u n /prime ,/vector ω) (1) for the vector function u= (u 1 ,...,u s ) T . Here u j =∂ j u/∂x j ,u 0 ≡uand F= (F 1 ,...,F s ) T ;/vector ω= (ω 1 ,...,ω c ) T ; T denotesthematrixtransposition.The nonlocal variables ω α are deﬁnedhere bymeans ofthe relations [15,20] ∂ω α /∂x=X α (x,t,u,u 1 ,...,u h ,/vector ω), (2) ∂ω α /∂t=T α (x,t,u,u 1 ,...,u h ,/vector ω). (3) We shalldenote by Ωthesetof nonlocalvariables ω γ ,γ= 1,...,c. kievarwe.tex; 12/03/2001; 3:49; p.387 SYMMETRIESOF EVOLUTIONEQUATIONS 381 LetA j,k (Ω)be the algebra of all locally analytic scalar functions of x,t,u,u 1 ,...,u j ,ω 1 ,...,ω k with respect to the standard multiplication, A≡A (Ω) = /uniontext c k=1 /uniontext ∞ j=0 A j,k (Ω), and letA loc ={f∈A|∂f/∂/vector ω = 0}be the subalgebra of localfunctions inA. Note that we do not exclude the case c=∞. The operatorsoftotal x- andt-derivativesonAhavetheform D≡D x = ∂ ∂x+ ∞ /summationtext i=0 u i+1 ∂ ∂u i + c /summationtext α=1 X α ∂ ∂ω α , D t = ∂ ∂t+ ∞ /summationtext i=0 D i (F) ∂ ∂u i + c /summationtext α=1 T α ∂ ∂ω α . Following [15, 20], we require that [D x ,D t ] = 0 or, equivalently, D t (X α ) =D x (T α )forα= 1,...,c. We shall denote by ImDthe image ofAunderD. Throughout this paper except for Section 3 we make a blanket assumption that the kernel of DinA consistssolely offunctionsof t. Consider the set Mat p (A)[ [D −1 ] ]offormal series in powers of Dof the form H= /summationtext q j=−∞ h j D j , whereh j arep×pmatrices with entries from A, cf. e.g. [11, 12].We shallwritefor short A[ [D −1 ] ]insteadof Mat 1 (A)[ [D −1 ] ]. The greatest m∈Zsuch thath m /negationslash= 0is called the degreeof H∈ Mat p (A)[ [D −1 ] ]and is denoted as m= deg H. We assume that deg 0 =−∞, cf. e.g. [1]. The formal series Hof degreemis called nondegenerate [12], if deth m /negationslash= 0. For H= /summationtext m j=−∞ h j D j ∈A[ [D −1 ] ],h m /negationslash= 0, itsresidueandloga- rithmic residue are deﬁnedas res H=h −1 andres ln H=h m−1 /h m [11, 12]. The set Mat p (A)[ [D −1 ] ]is an algebra under the multiplication law, given by the“generalized Leibnizrule”,cf.[1], aD i ◦bD j =a ∞ /summationdisplay q=0 i(i−1)···(i−q+ 1) q!D q (b)D i+j−q for monomials aD i ,bD j ,a,b∈Mat p (A), and extended by linearity to the whole Mat p (A)[ [D −1 ] ]. The commutator [ A, B] = A◦ B− B◦ Amakes Mat p (A)[ [D −1 ] ]intoa Liealgebra.Belowwe omit ◦ifthis is notconfusing. G∈A s iscalled, seee.g. [1–3], a symmetry for(1)–(3), if ∂G/∂t+ [F,G] = 0, (4) where [·,·]istheLiebracket [K,H] =H /prime [K]−K /prime [H].Thedirectionalderivative of any (smooth) function f∈A q along H∈A s is deﬁned here as f /prime [H] = (df(x,t,u+/epsilon1H,u 1 +/epsilon1D x (H),...)/d/epsilon1) /epsilon1=0 . Extending the technique of [15] to thecaseoftime-dependentsystems(1)–(3),wecaneasilyshowthatforany f∈A we havef /prime ∈A[ [D −1 ] ]. For anyf∈A q we shall deﬁne its formal order asfordf= degf /prime . This naturally generalizes thenotionoforderforlocal functions,cf. e.g.[1, 12]. kievarwe.tex; 12/03/2001; 3:49; p.388 382 A.SERGYEYEV LetS F (A)bethesetofallsymmetries G∈A s for(1)–(3),S (k) F (A) ={G∈ S F (A)|fordG≤k},Ann F (A) ={G∈S F (A)|∂G/∂t= 0}. In general, forA /negationslash=A loc neitherA s norS F (A)are closed under the Lie bracket, but if [P,Q]∈A s forsome P,Q∈S F (A),then we have [P,Q]∈S F (A). A formal series R= /summationtext r j=−∞ η j D j ∈Mat s (A)[ [D −1 ] ]is called [1, 11, 15] theformalsymmetry ofrankmfor(1) (or,rather,for(1)–(3)), provided deg(D t ( R)−[F /prime , R])≤degF /prime + deg R−m. (5) The derivative D t ( R)is deﬁnedhere as D t ( R) = /summationtext r j=−∞ D t (η j )D j . ThesetFS (q) F (A)ofallformalsymmetriesofranknotlowerthan qofsystem (1)–(3) is a Lie algebra, because for the formal symmetries Pand Qof ranksp andqwe have [ P, Q]∈FS (r) F (A)forr= min(p,q),cf.[12]. Eq.(4) is well known to be nothing but the compatibility condition for (1) and∂u/∂τ =G. Provided G∈ A s , we have∂(∂u/∂τ)∂t=D t (G)and ∂(∂u/∂t)∂τ=F /prime [G]. HenceEq.(4)maybe rewritten as D t (G) =F /prime [G]. LetF /prime ≡ n /summationtext i=−∞ φ i D i andn 0 = /braceleftbigg 1−j,ifφ i =φ i (x,t),i=n−j,...,n, 2otherwise. SinceD t (G) = F /prime [G]impliesD t (G /prime )−[F /prime ,G /prime ]−F /prime/prime [G] = 0, and degF /prime/prime [G]≤degF /prime +n 0 −2,wereadily seethat G /prime ∈FS (fordG−n 0 +2) F (A). 3. Actionof mastersymmetries ontime-independent symmetries As we have already mentioned above, for P,Q∈A s in general [P,Q]/negationslash∈A s . In particular, when we repeatedly commute a master symmetry τ∈ A s with some time-independent symmetry Q∈Ann F (A), it is by no means obvious thatQ i = ad i τ (Q) = [τ,Q i−1 ]belong toA s , except for the case A=A loc . In some cases we can make the conditions [τ,Q i ]∈A s or[P,Q]∈A s hold by introducing new nonlocal variables ˜ω κ and thus replacing Aby a larger algebra ˜A. But in order that [P,Q]∈A s forP,Q∈A s it obviously sufﬁces to require thatω /prime µ [P]∈Afor thoseω µ on which Qactually depends and ω /prime ν [Q]∈Afor thoseω ν onwhich Pactually depends, cf.Ch. 6in[20]. Moreover,we have Proposition 1. Letτ,Q∈A s ,ω /prime γ [Q]∈Aandω /prime γ [τ]∈Aforγ= 1,...,c. ThenQ l = ad l τ (Q)∈A s foralll= 1,2,.... Proof.Let us use induction. To prove that [τ,Q l ]∈ A s , if Q l = [τ,Q l−1 ]∈A s , it sufﬁces to prove that ω /prime ν ([τ,Q l−1 ])∈Afor allω ν whichτdepends on and that ω /prime δ [τ]∈Afor allω δ which [τ,Q l−1 ]depends on. Asω /prime ν ([τ,Q l−1 ]) = (ω /prime ν [Q l−1 ]) /prime [τ]−(ω /prime ν [τ]) /prime [Q l−1 ], it sufﬁces that ω /prime γ [τ]∈A for allω γ , which [τ,Q l−1 ]andω /prime ν [Q l−1 ]depend on, and ω /prime κ [Q l−1 ]∈Afor all ω κ whichτandω /prime ν [τ]depend on, inorderthat [τ,Q l ]∈A s ./square kievarwe.tex; 12/03/2001; 3:49; p.389 SYMMETRIESOF EVOLUTIONEQUATIONS 383 It appears that nearly all known master symmetries of integrable systems (1)–(3) satisfy the conditions of Proposition 1 for a suitably chosen set Ωof nonlocal variables ω γ , so their action indeed yields the symmetries from A s . For instance, if∂F/∂/vector ω = 0andA=A(Ω UAC,F ), then by virtue of the results of [20] Proposition 1 holds true for any τ,Q∈S F (A). Here Ω UAC,F is the set of all nonlocal variables ω γ associated with the universal abelian covering (see [20] for its deﬁnition) over (1). Let us stress that Proposition 1 is valid for any τand Qmeeting the relevant requirements, no matter whether τis a master symmetry andQisasymmetryfor(1)–(3). Note that Proposition 1 is obviously valid for more general systems of PDEs with constraints than (1)–(3), if we suitably redeﬁne for them the Lie bracket, the directionalderivativeand thealgebra A. 4. Thestructureofformal symmetriesforNWD systems Consider a particular class of evolution systems with constraints (1)–(3) such thatn≡fordF≥2and the leading coefﬁcient Φof the formal series F /prime (i.e.,F /prime ≡ΦD n +...) hassdistinct eigenvalues λ i and can be diagonalized by means of a matrix Γ = Γ(x,t,u,...,u n /prime ,/vector ω), i.e., the matrix Λ = ΓΦΓ −1 is diagonal, cf. [11, 12]. For these systems there exists a unique formal series T= Γ + Γ /summationtext ∞ j=1 Γ j D −j ∈Mat s (A)[ [D −1 ] ]such that all coefﬁcients of the formalseries V= TF /prime T −1 +(D t ( T)) T −1 arediagonalmatricesandthediagonal entriesofmatrices Γ j ,j= 1,2,...,areequaltozero.Theproofisessentiallythe same as for Proposition 3.1 from [11]. We shall call the systems with constraints (1)–(3) having the above properties and such that det Φ/negationslash= 0nondegenerate weaklydiagonalizable(NWD) .Notethatwhen uisscalar,i.e., s= 1,anysystem (1)–(3) with n≡fordF≥2obviously is an NWD system with constraints, having T= 1and V=F /prime . Belowin thissection(1)–(3)will be anNWDsystem with constraints. Eq.(5) yields [11, 12] deg(D t (˜ R)−[ V,˜ R])≤deg V+ deg ˜ R−m, where ˜ R= TRT −1 , whence we ﬁnd (cf. [6, 12, 13]) that any R∈FS (n+1) F (A)can be represented inthe form R= T −1 /parenleftBigg r /summationtext j=r−n+1 c j (t) V j/n /parenrightBigg T+ 1 nT −1 /parenleftBig D −1 /parenleftBig ˙c r (t)Λ −1/n −rc r (t)D t (Λ −1/n ) /parenrightBig/parenrightBig V r−n +1 n T+ N,deg N<r−n+ 1.(6) Likewise, for R∈FS (m) F (A)withm= 2,...,nwe have R= T −1 /parenleftBigg r /summationtext j=r−m+2 c j (t) V j/n /parenrightBigg T+ N,deg N<r−m+ 2.(7) kievarwe.tex; 12/03/2001; 3:49; p.390 384 A.SERGYEYEV Herer= deg R, N=bD ν +··· ∈ Mat s (A)[ [D −1 ] ]is some formal series,ν=r−nin (6) andν=r−m+ 1in (7);c j (t)andΓbΓ −1 are diagonals×smatrices; for V≡diag( V 1 ,..., V s ), V i ∈ A[ [D −1 ] ], we set V j/n = diag( V j/n 1 ,..., V j/n s )[11]; dotstands forthe partialderivative w.r.t. t. Inthissectionweassumethatanyfunction ˜h+a(t),wherea(t)isanarbitrary functionoft, canbetaken for D −1 (h), ifh=D(˜h)andh,˜h∈A. Form= 2,...,n + 1Eqs. (6), (7) represent a general solution of (5) for any NWD system with constraints (1)–(3). Hence, if at least one entry of the matrix (˙c r (t)Λ −1/n −rc r (t)D t (Λ −1/n ))does not belong to ImD, then (1)–(3) has no formalsymmetriesfrom FS (n+1) F (A)withagiven c r (t). Forany P≡ T −1 c p (t) V p/n T+···and Q≡ T −1 d q (t) V q/n T+···wehave [ P, Q] = T −1 (1/n)(pc p (t)˙d q (t)−qd q (t)˙c p (t)) V p+q− n n T+ K(8) by virtue of (6), provided P, Q∈FS (n+1) F (A). Here K∈Mat s (A)[ [D −1 ] ]is some formal series, deg K<p+q−n. LetP,Q∈A s ,R≡[P,Q]. Then R /prime =Q /prime/prime [P]−P /prime/prime [Q]−[P /prime ,Q /prime ]. If P,Q∈S F (A), then (5) and (6) for R=P /prime and R=Q /prime imply degP /prime/prime [Q]≤ p+n 0 −2<p+q−nanddegQ /prime/prime [P]≤q+n 0 −2<p+q−nforp,q>n +n 0 −2, p≡fordP,q≡fordQ. This resultand(8)for P=P /prime , Q=Q /prime yield [P,Q] /prime =− T −1 (1/n)(pc p (t)˙d q (t)−qd q (t)˙c p (t)) V p+q− n n T+˜ K,(9) where ˜ K∈Mat s (A)[ [D −1 ] ]issomeformal series, deg˜ K<p+q−n. So,ifP,Q∈S F (A),p,q>n +n 0 −2,then fordR≤p+q−n.IfR∈A s , thenR∈S (p+q−n) F (A),andR∈S (p+q−n−1) F (A),ifpc p (t)˙d q (t) =qd q (t)˙c p (t). Let (1)–(3) have a nondegenerate formal symmetry R∈Mat s (A)[ [D −1 ] ], r≡deg R/negationslash= 0, of rankq > n. ThenD t (ρ a j )∈ImD, i.e.,ρ a j are conserved densities, for a= 1,...,sandj=−1,0,...,q−n−2, where ρ a 0 = res ln(( TRT −1 ) 1/r ) aa andρ a j = res(( TRT −1 ) j/r ) aa forj/negationslash= 0, cf. [11]. Forn 0 <2we haveρ a j ∈ImDforalla= 1,...,sandj=−1,0,...,−n 0 . Proposition2. LetanNWDsystemwithconstraints(1)–(3)haveanondegenerate formal symmetry R∈Mat s (A)[ [D −1 ] ],r≡deg R/negationslash= 0,q≡rank R>n; let for a= 1,...,sthere existm a ∈{− 1,1,2,...,min(n−2,q−n−2)}such that m a /negationslash= 0,ρ a m a /negationslash∈ImDandρ a j ∈ImDforj=−1,1...,m a −1,j/negationslash= 0.Thenfor each P∈FS (m+n+2) F (A),m= max a m a , there exists a constant s×sdiagonal matrixcsuch that P= T −1 c R p/r T+···,p≡deg P. kievarwe.tex; 12/03/2001; 3:49; p.391 SYMMETRIESOF EVOLUTIONEQUATIONS 385 Proof.Since R∈FS (n+1) F (A), by (6) we have R= T −1 h(t) V r/n T+···. For any P∈FS (n+1) F (A)we can (cf.[6, 14] and(6))represent ˜ P≡ TPT −1 as ˜ P= p /summationtext j=p−n+1 c j (t)˜ R j/r + 1 n /parenleftBig D −1 /parenleftBig ˙c p (t)(h(t)) n/r ρ −1 /parenrightBig/parenrightBig ˜ R p−n +1 r +˜ N.(10) Here ˜ N≡ /summationtext p−n j=−∞ ˜b j D j ∈Mat s (A)[ [D −1 ] ],c j (t),h(t),˜b p−n are diagonal s×s matrices,ρ −1 ≡diag(ρ 1 −1 ,...,ρ s −1 ),˜ R= TRT −1 ; the fractional powers ˜ R j/r aredeﬁned sothat theirﬁrst rcoefﬁcientsarediagonal, cf. [11, 12]. For P∈FS (d) F (A)we have deg(D t (˜ P)−[ V,˜ P])≤n+p−d, and thus deg(D t (˜ P i )−[ V,˜ P i ])≤n+p+i−min(q,d)for˜ P i ≡˜ P˜ R i/r . Hence, for −p−2<i< min(q,d)−n−p−1wehave res(D t (˜ P i )−[ V,˜ P i ]) = 0. Letusplug(10)intothisequalityfor −p−2<i< min(q,d,2n)−n−p−1 and break it into sscalar equations. Since res([ V,˜ P i ]) aa ∈ImDby Adler’s formula, see e.g. [12], and D t (ρ a j+i )∈ImDby assumption, we easily ﬁnd that for any P∈FS (m+n+2) F (A)we have (˙c p (t)) aa ρ a m a = 0modulo the terms from ImDfor alla= 1,...,s. So, ˙c p (t) = 0, andtheresult follows. /square Corollary 1. Under the assumptions of Proposition 2, for any G∈S F (A),k≡ fordG≥m+n+n 0 , we have G /prime = T −1 c R k/r T+···, wherecis a constant s×sdiagonal matrix. 5. Symmetries ofhomogeneousNWD systems Let (1)–(3) possess a scaling symmetry D=αtF+xu 1 +βu, whereβ= diag(β 1 ,...,β s )is a diagonal matrix, α,β j = const, and let the determining equations (2), (3) for ω γ ,γ= 1,...,c, be homogeneous with respect to D. Then weshallcalltheevolutionsystemwithconstraints(1)–(3) homogeneous w.r.t.D, cf.e.g.[7,8,10,20].Ifaformalvectorﬁeld G∂/∂uishomogeneousofweight κ w.r.t.D, then we shall say for short that G∈A s itself is homogeneous of weight κandwrite wt(G) =κ. Forhomogeneoussystems(1)–(3)thereusuallyexistsabasisin S F (A)made ofhomogeneoussymmetries,andhencetherequirementofhomogeneityof P,Q andτbelowisbynomeansrestrictive.So,thephraselike“forall(homogeneous) H∈Mthecondition Pistrue”belowmeansthatthereexistsabasisin Msuch that all its elements are homogeneous w.r.t. D, and for all of them the condition Pholdstrue.Wehaveanobvious Lemma 1. Let (1)–(3) be a homogeneous system with constraints, and homogeneous P,Q∈S F (A)be such that [P,Q]∈ M, whereMis a subspace ofA s . Suppose that wt(G)/negationslash= wt([ P,Q]) = wt( P) + wt( Q)for all (homogeneous) G∈S (p+q) F (A)∩M,p≡fordP,q≡fordQ.Then [P,Q] = 0. kievarwe.tex; 12/03/2001; 3:49; p.392 386 A.SERGYEYEV This result, as well as other results below, allows to prove the commutativity for large familiesof symmetries at once. Examples below show that we can usually choose the subspaces like Mlarge enough so that the condition [P,Q]∈Mcan be veriﬁed for all symmetries in the family without actually computing [P,Q]. On the other hand, by proper choice of these subspaces we can considerably reduce the number of weight-related conditions to be veriﬁed, and thusmake theapplicationof ourresults trulyefﬁcient. Below in this section we assume that (1)–(3) is a homogeneous NWD system with constraints and P,Q∈S F (A)are itshomogeneous symmetries, p≡fordP,q≡fordQ. Note that if p,q > n +n 0 −2, then by (9) we should verify the conditions of Lemma 1 only for G∈S (p+q−n) F (A)∩M(for G∈S (p+q−n−1) F (A)∩M,ifinaddition pc p (t)˙d q (t)−qd q (t)˙c p (t) = 0). 5.1. COMMUTATIVITYAND TIME DEPENDENCE OFSYMMETRIES Corollary 2. Letα/negationslash= 0,∂Φ/∂t= 0and∂X γ /∂t=∂T γ /∂t= 0,γ= 1,...,c. Let homogeneous P,Q∈Ann F (A)be such that [P,Q]∈ L, whereLis a subspace ofA s . Letp,q≥b F ≡min(max(n 0 ,0),n+n 0 −1), where p≡fordP,q≡fordQ. Suppose that wt(G)/negationslash= (p+q)α/nfor all (homogeneous) G∈S (n 0 −1) F (A)∩Ann F (A)∩L.Then [P,Q] = 0. Proof.IfP,Q∈Ann F (A),[P,Q]∈A s ,p,q≥b F , then, using (6), (7) and (9), weﬁndthat [P,Q]∈N≡S (p+q−1) F (A)∩Ann F (A).Eqs.(6)or(7)for R=G /prime imply wt(G) =kα/n/negationslash= wt([ P,Q]) = (p+q)α/nforallhomogeneous G∈N withk≡fordG≥n 0 . Hence, under our assumptions wt(G)/negationslash= (p+q)α/nfor all homogeneous G∈N∩L≡M , andthusby Lemma1 [P,Q] = 0./square For instance, for the integrable [21] equation u t =D 2 (u −1/2 1 ) +u 3/2 1 ≡K withn 0 = 2andα= 3/2the spaceS (1) K (A loc )∩Ann K (A loc )is spanned by 1 andu 1 , and wt(1),wt(u 1 )≤1< α(p+q)/n= (p+q)/2forp,q≥b K = 2. Hence, by Corollary 2 all (homogeneous) time-independent local generalized symmetries offormal order p>1forthisequation commute. Likewise, using Corollary 2, we can easily show that for any λ-homogeneous integrable evolution equation with λ≥0from [8] all its x,t-independent homogeneouslocal generalizedsymmetriesofformalorder k>0commute. Ifn 0 ≤0and, in addition to the conditions of Corollary 2 for PandQ, the commutator [P,Q]∈S F (A loc ),[P,Q]isx,t-independent and wt([P,Q])/negationslash= 0, then[P,Q] = 0. The weight-related conditions are automatically satisﬁed, as the onlyx,t-independent symmetries in S (n 0 −1) F (A loc )are constant ones, and their weight is zero. In particular, for anyhomogeneous (with α/negationslash= 0) NWD system oftheform u t = Φ(x)u n + Ψ(x,t)u n−1 +f(x,t,u,...,u n−2 ),where Φ,Ψare s×smatrices, allhomogeneous x,t-independent local generalized symmetries offormalorder k>0commute. kievarwe.tex; 12/03/2001; 3:49; p.393 SYMMETRIESOF EVOLUTIONEQUATIONS 387 Let R∈FS (2) F (A)be a nondegenerate formal symmetry for (1)–(3), r≡deg R/negationslash= 0. Then by (7) R= Γ −1 h(t)Λ r/n ΓD r +···, where h(t)≡diag(h 1 (t),...,h s (t))is as×sdiagonal matrix. Assume that h(t)is homogeneous w.r.t. Dandζ R ≡(α/n + wt(h(t))/r)/negationslash= 0. Let Z F, R (A) ={G∈S F (A)|k≡fordG≥n 0 ;thereexists adiagonal matrixc(t),wt(c(t)) = 0,such that G /prime = Γ −1 c(t)(h(t)) k/r Λ k/n ΓD k +···}. We set here (h(t)) k/r ≡ diag((h 1 (t)) k/r ,..., (h s (t)) k/r ). Let also St F, R (A) ={G∈Z F, R (A)|c(t)is aconstant matrix }, and N (j) F, R (A)be the set of symmetries G∈S F (A)such thatk≡fordG≥n 0 ,k≤j, and G /prime = Γ −1 c(t)(h(t)) k/r Λ k/n ΓD k +···, wherec(t)is ans×sdiagonal matrix, different for different Gandk, and the entries of c(t)are linear combinations of functions of t, say,ψ b (t), such that for all bwe have wt(ψ b (t))< ζ R (j−k) forζ R >0andwt(ψ b (t))> ζ R (j−k)forζ R <0. For any homogeneous G∈N (j) F, R (A)we have wt(G)<jζ R forζ R >0andwt(G)>jζ R forζ R <0, sowt(H)/negationslash= wt( P)foranyhomogeneous P∈Z F, R (A)andH∈N (fordP) F, R (A). LetP,Q∈S F (A)be homogeneous, and [P,Q]∈L 1 ∪L 2 , whereL 1 is a subspace of N (j) F, R (A)for somejand R, andL 2 is a subspace of S (d) F (A)for somed. Assume that Rsatisﬁes the above conditions, wt([P,Q])≥jζ R for ζ R >0andwt([P,Q])≤jζ R forζ R <0, and wt(H)/negationslash= wt([ P,Q])for all (homogeneous) H∈L 2 /(L 2 ∩N (j) F, R (A)).Thenby Lemma 1 [P,Q] = 0. Suppose that, in addition to the above conditions for [P,Q], we haved <0, wt([P,Q])>0forζ R >0andwt([P,Q])<0forζ R <0, and [P,Q]belongs toS F (A loc )and can be represented (as function of tandx) as a polynomial in variablesχ(t)andξ(x)such that wt(χ(t))<0andwt(ξ(x))<0forζ R >0, andwt(χ(t))>0andwt(ξ(x))>0forζ R <0. Then [P,Q] = 0, and there is nofurtherweight-relatedconditionstoverify.Indeed, S (d) F (A loc )foranyd<0is spanned by the symmetries of the form G=G(x,t), and for any homogeneous symmetry H=H(x,t)being a polynomial in χ(t)andξ(x)we obviously have wt(H)/negationslash= wt([ P,Q]). Note that under the assumptions of Proposition 2 all G∈S F (A)with fordG≥m+n+n 0 belongto St F, R (A)byCorollary1.Supposethat Rsatisﬁes the conditions, given above. Let d= min(m+n+n 0 −1,p+q). Then for any P,Q∈S F (A)such that [P,Q]∈A s we have [P,Q]∈N (p+q) F, R (A)∪S (d) F (A). Then [P,Q] = 0for homogeneous P,Q∈Z F, R (A), once wt(H)/negationslash= wt([ P,Q]) forall(homogeneous) H∈S (d) F (A)/(S (d) F (A)∩N (p+q) F, R (A)).Ifp,q>n +n 0 −2, then by (9) we can take d= min(m+n+n 0 −1,p+q−n)(or d= min(m+n+n 0 −1,p+q−n−1),ifpc p (t)˙d q (t)−qd q (t)˙c p (t) = 0). If∂F/∂t=∂X γ /∂t=∂T γ /∂t= 0,γ= 1,...,c, then F∈S F (A), and ∂P/∂t= [P,F]∈S (p) F (A)forP∈S F (A).So,taking Q=Fandimposingthe kievarwe.tex; 12/03/2001; 3:49; p.394 388 A.SERGYEYEV extra condition d≤pinthree previousparagraphsyields valid results. Wealsohavethe following Proposition 3. Letα/negationslash= 0and∂F/∂t = 0,∂X γ /∂t =∂T γ /∂t = 0, γ= 1,...,c; let homogeneous P∈S F (A)be such that p≡fordP≥n 0 , ford∂P/∂t < p and[P,F]∈L, whereLis a subspace ofA s . Suppose that wt(G)/negationslash= (p+n)α/nfor all (homogeneous) G∈S (p−1) F (A)∩Lsuch that G/negationslash∈N (p+n) F,F /prime (A). Then [P,F] = 0,andthus∂P/∂t= 0andP∈Ann F (A). Proof.Asford∂P/∂t<p,wehave∂P/∂t= [P,F]∈S (p−1) F (A)∩L≡M . The conditions ford∂P/∂t 0,∂F/∂t = 0,∂X γ /∂t =∂T γ /∂t = 0,γ= 1,...,c, and homogeneous P,Q∈St F,F /prime (A),p,q≥n 0 , be polynomials in t. If we take the space of symmetries from S F (A)polynomial in time t, for ˜L, and setL 1 = N (p+q) F,F /prime (A)∩˜L,L 2 =S (n 0 −1) F (A)∩˜L,d=n 0 −1, then [P,Q]∈L 1 ∪L 2 ≡M, and thus the weight-related conditions of Lemma 1, Corollary 2, Proposition 3, etc., are to be checked only for (homogeneous) G∈L 2 . Furthermore, if n 0 ≤0, thenS (n 0 −1) F (A loc )contains only the symme- triesG=G(x,t), and soanyhomogeneous local generalized symmetry Kof formal order k >0being polynomial in tandxand such that ∂ 2 K/∂u k ∂t= 0 is in fact time-independent, and any two such symmetries commute. This result applies e.g. to anyhomogeneous NWD system with α > 0having the form u t = Φ(x)u n + Ψ(x)u n−1 +f(x,u,...,u n−2 ),where Φ,Ψares×smatrices. 5.2. MASTER SYMMETRIES OF HOMOGENEOUS NWD SYSTEMS Corollary 3. Letα/negationslash= 0,∂Φ/∂t= 0and∂X γ /∂t=∂T γ /∂t= 0,γ= 1,...,c. Suppose that there exist a homogeneous Q∈Ann F (A)and a homogeneous τ∈A s such that∂τ/∂t= 0,∂[τ,F]/∂t= 0,K=τ+t[τ,F]∈S F (A), q≡fordQ> n +n 0 −2,b≡ford[τ,F]>max(fordτ,n), the formal series ([τ,F]) /prime is nondegenerate, [[τ,F],Q]∈ L, whereLis a subspace of A s ,[τ,Q]∈ A s . Let wt(H)/negationslash= (b+q)α/nfor all (homogeneous) H∈ L∩S (n 0 −1) F (A)∩Ann F (A). Then Q 1 = [τ,Q]∈Ann F (A),and fordQ 1 >q. Proof.From (4) with G=Kit clear that [τ,F]∈Ann F (A), so by Corollary 2 [[τ,F],Q] = 0, whence, using [F,Q] = 0and the Jacobi identity, we ﬁnd that [F,[τ,Q]] = 0, so[τ,Q]∈Ann F (A). By (9) the nondegeneracy of ([τ,F]) /prime readilyimplies ford[τ,Q] = ford[ K,Q] =b+q−n>q./square Theorem 1. Let the conditions of Corollary 3 be satisﬁed, ad j [τ,F] (Q)∈ L j , whereL j are some subspaces of A s ,Q j ≡ad j τ (Q)∈ A s , and wt(H)/negationslash= kievarwe.tex; 12/03/2001; 3:49; p.395 SYMMETRIESOF EVOLUTIONEQUATIONS 389 ((b−n)j+q+n)α/nforall(homogeneous) H∈L j ∩S (n 0 −1) F (A)∩Ann F (A), j= 2,...,i.Then Q j ∈Ann F (A)andfordQ j >fordQ j−1 ,j= 1,...,i. The proof of the theorem consists in replacing in Corollary 3 the symmetry QbyQ j = ad j τ (Q)andrepeateduseofthiscorollaryfor j= 2,...,i.Notethat we caneasilyverify that ad j τ (Q)∈A s ,using Proposition 1. Thus, Proposition 1, Corollary 3 and Theorem 1 enable us to ensure that τ indeed is a nontrivial master symmetry, producing a sequence of symmetries of inﬁnitely growing formal orders, without assuming a priorithe existence of hereditary recursion operator [5] or e.g. of “negative” master symmetries τ j , j < 0[19]. So, our results provide a useful complement to the known general resultsonmastersymmetries, cf. e.g.[3–5,19]. It is important to stress that in general the symmetries Q i are not obliged to commute pairwise. The check of their commutativity and picking out the commutative subset in the sequence of Q i can be performed using either the resultsof presentpaper orothermethods, seee.g. [1, 4,5, 19]. We often can take [τ,F]orFforQ, and then in order to use Theorem 1 it sufﬁces to knowonly asuitable‘candidate’ τforthemaster symmetry. Forinstance,integrableHarryDymequation u t =u 3 u 3 ≡H,seee.g.[1,14], satisﬁestheconditionsofProposition1andofTheorem1forall i= 2,3,...with α= 3,b= 5,A=A(Ω UAC,H ),τ=u 3 D 3 (uω 1 )≡τ 0 +u 3 u 3 ω 1 ,τ 0 ∈A loc , Q= [τ,u 3 u 3 ] = 3u 5 u 5 +···∈ Ann H (A).Inparticular,thenonlocalvariable ω 1 inτisdeﬁnedbymeansoftherelations ∂ω 1 /∂t=−uu 2 −u 2 1 /2,∂ω 1 /∂x=u −1 (informally, ω 1 =D −1 (u −1 )). Thus, by Theorem 1 Q j = ad j τ (Q)∈Ann H (A), j= 1,2,..., together with Q −1 ≡u 3 u 3 ∈Ann H (A loc )andQ 0 ≡Qform the inﬁnite hierarchy of time-independent symmetries for the Harry Dym equation. The commutativity of Q j ,j=−1,0,1,..., readily follows from Corollary 2. Notethatitispossibletoshowthat Q j ,j= 0,1,...,areinfact localgeneralized symmetries of Harry Dym equation and coincide with the members of hierarchy generated by means of the recursion operator R=u 3 D 3 ◦u◦D −1 ◦u −2 from theseedsymmetry u 3 u 3 , uptothe constantmultiples. Acknowledgements I am sincerely grateful to the organizers of NATO ARW Noncommutative Structures in Mathematics and Physics for inviting me to participate and to give a talk and for their hospitality. It is also my great pleasure to thank Profs. M. Błaszak, B. Fuchssteiner, B. Kupershmidt, P. Olver and V. V. Sokolov for stimulating discussions and comments. Last but not least, I acknowledge with deep gratitude Dr. M. Marvan’s reading the drafts of this paper and making a lot ofprecise remarks, which considerablyimprovedit. This research was supported by the Ministry of Education, Youth and Sports ofCzech Republic, Grants CEZ:J10/98:192400002andVS 96003. kievarwe.tex; 12/03/2001; 3:49; p.396 390 A.SERGYEYEV References 1. P. Olver, Applications of Lie Groups to Differential Equations , Springer-Verlag, New York, 1993. 2. A.S. Fokas, Symmetries andintegrability , Stud. Appl. Math. 77: 3 (1987), 253–299. 3. M.Błaszak, Multi-HamiltonianTheoryofDynamicalSystems , Springer-Verlag,Heildelberg, 1998. 4. B. Fuchssteiner, Mastersymmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolutionequations , Progr. Theor. Phys. 70(1983), 1508–1522. 5. W.Oevel, Ageometricalapproachtointegrablesystemsadmittingtimedependentinvariants , in Topics in Soliton Theory and Exactly Solvable Nonlinear Equations (Oberwolfach, 1986) (M. Ablowitz, B. Fuchssteiner, M. 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Sokolov, The symmetry approach to classiﬁcation of integrable equations , in What is Integrability? (V.E. Zakharov, ed.), Springer-Verlag, New York,1991, pp. 115–184. 13. A.V. Mikhailov, R.I. Yamilov, Towards classiﬁcation of (2+1)-dimensional integrable equations. Integrability conditions. I , J. Phys. A: Math. Gen. 31(1998), 6707–6715. 14. N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics , D. Reidel Publishing Co., Boston, 1985. 15. F.Kh. Mukminov, V.V. Sokolov, Integrable evolution equations with constraints , Mat. Sb.(N.S.) 133(175):3 (1987), 392–414. 16. V.V.Sokolov,T.Wolf, Asymmetrytestforquasilinearcoupledsystems , InverseProblems 15 (1999), L5–L11. 17. M.V. Foursov, Classiﬁcation of certain type integrable coupled potential KdV and modiﬁed KdV-type equations , J.Math. Phys. 41(2000), 6173–6185. 18. B. Fuchssteiner, Integrable nonlinear evolution equations with time-dependent coefﬁcients , J.Math. Phys. 34(1993), 5140–5158. 19. I.Dorfman, DiracStructuresandIntegrabilityofNonlinearEvolutionEquations , JohnWiley & Sons, Chichester, 1993. 20.Symmetries and Conservation Laws for Differential Equations of Mathematical Physics (I.S. Krasil’shchik and A.M. Vinogradov, eds.), American Mathematical Society, Providence, 1999. 21. J.A. Cavalcante and K. Tenenblat, Conservation laws for nonlinear evolution equations , J. Math. Phys. 29(1988), 1044–1049. kievarwe.tex; 12/03/2001; 3:49; p.397 p-ADIC STRINGS AND NONCOMMUTATIVITY BRANKO DRAGOVICH 1,2∗ andIGORV.VOLOVICH 1 1 Steklov Mathematical Institute, Gubkin St. 8, 117966 Moscow, Russia 2 InstituteofPhysics, P.O.Box 57,11001 Belgrade,Yugoslavia Abstract. Some possible connections between p-adic string theory and noncommutativity are considered.TheirrelationtotheuncertaintyinspacemeasurementsatthePlanckscaleisdiscussed. Existenceofnew p-adicstringamplitudesispointedout.Somesimilaritiesbetween p-adicsolitonic branes and noncommutative scalar solitons are emphasized. More explicit and deeper connections between string ﬁeld theory and p-adic string theory couldemerge in the near future. 1. Introduction Itiswell-known(forarecentreview,see[1])thattheinterplaybetweenquantum- mechanical and general relativity principles gives an uncertainty ∆xon the measurementsof distances xin theform ∆x≥/lscript 0 = /radicalBigg /planckover2pi1 G c 3 ∼10 −33 cm, (1) where/lscript 0 is the Planck length. This fact requires reconsideration of many our basic concepts about the spacetime structure at the Planck scale. It leads to the investigation of some new and more fundamental mathematical notions. To this end, we will consider here two very natural approaches. From the one point of view, the uncertainty (1) means at least restriction on the dominance of real num- bers and archimedean geometry in their applications at the Planck scale. Namely, this formula has been derived with the implicit use of the real numbers and any archimedean geometry. In this way we see that the usual physical theory predicts its breakdown at the Planck scale. A graceful exit from this situation should be in the use of adeles and adelic topology, which contain archimedean as well as nonarchimedeangeometries.Fromtheotherpointofview,theuncertainty(1) has ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.398 392 B. DRAGOVICH, I.V. VOLOVICH to be a consequence of some noncommutativity between space coordinates. This conclusion follows from the analogous situation in ordinary quantum mechanics: the uncertainty ∆x∆k≥ /planckover2pi1 2 is a direct consequence of the noncommutativity in the form of the Heisenberg algebra [ˆx,ˆk] =i/planckover2pi1between coordinates xandk of the phase space. Thus, we see that the uncertainty (1) leads to consider also noncommutative geometry at the Planck scale. M-theory is the best candidate to describe physics at this scale. It contains strings and branes. By now, it seems that an employment of nonarchimedean geometry based on p-adic numbers and noncommutativegeometry givenby thecommutation relation [ˆx i ,ˆx j ] =i/planckover2pi1θ ij (2) is unavoidable in a further progress of the ”theory of everything”. In the sequel wewillmainlyconsidersomeaspectsof p-adicstringsandtheirpossibleconnec- tion with noncommutative geometry. A notion of p-adic string was introduced in [2], where the hypothesis on the existence of nonarchimedean geometry at the Planck scale was made, and string theory with p-adic numbers was initiated. In particular, generalization of the usual Veneziano and Virasoro-Shapiro ampli- tudes with complex valued multiplicative characters over various number ﬁelds was proposed and p-adic valued Veneziano amplitude was constructed by means ofp-adic interpolation. Very successful p-adic analogues of the Veneziano and Virasoro-Shapiroamplitudeswereproposedin[3]asthecorrespondingGel’fand- Graev[4]betafunctions.Usingthisapproach,FreundandWittenobtained[5]an attractive adelic formula, which states that the product of the crossing symmetric Veneziano (or Virasoro-Shapiro) amplitude and its all p-adic counterparts equals unit (or a deﬁnite constant). This gives possibility to consider an ordinary four- point function, which is rather complicate, as an inﬁnite product of its inverse p-adic analogues, which have simple forms. These ﬁrst papers induced an inter- est in various aspects of p-adic string theory (for a review, see [6, 7]). A recent interest inp-adic string theory has been mainly related to the generalized adelic formulas for four-point string amplitudes [8], the tachyon condensation [9], and thenewpromisingadelicapproach[10].Inadditiontotheexpression(1),onecan motivate the application of p-adic numbers in physics by the fact that the ﬁeld of rational numbers Qis dense not only in Rbut also in the ﬁeld of p-adic numbers Q p (pdenotes any prime number). Another motivation may be a conjecture that fundamental physical laws should be invariant under change R←→Q p [11]. One of the very interesting and fruitful recent developments in string theory (for a reviev, see [12, 13]) has been noncommutative geometry and the corresponding noncommutative ﬁeld theory. This subject started to be very actual after Connes, Douglas and Schwarz shown [14] that gauge theory on noncommutative torus describes compactiﬁcations of M-theory to tori with constant background three- form ﬁeld. Noncommutative ﬁeld theory (see, e.g. [15]) may be regarded as a deformation of the ordinary one in which ﬁeld multiplication is replaced by the kievarwe.tex; 12/03/2001; 3:49; p.399 p-ADIC STRINGSANDNONCOMMUTATIVITY 393 Moyal(star)product (f ⋆g)(x) = exp /bracketleftbigg i /planckover2pi1 2θ ij ∂ ∂y i ∂ ∂z j /bracketrightbigg f(y)g(z)| y=z=x , (3) wherex 1 ,x 2 ,···,x d denote coordinates of noncommutative space, and θ ij = −θ ji are noncommutativity parameters. There are many properties of D-brane dynamics which may be studied by noncommutative ﬁeld theory. In particular, it enables to investigate a mixing of the UV and IR effects, and the tachyon condensation. Replacing the ordinary product between coordinates by the Moyal product (3) wehave x i ⋆x j −x j ⋆x i =i/planckover2pi1θ ij , (4) which resembles the usual Heisenberg algebra. In the next Section we provide readerwithsomeverybasicfactson p-adicanalysis.Section3isdevotedtothe p- adic string amplitudes. After that we consider an effective ﬁeld theory of bosonic p-adicstringsanditsconnectionwithnoncommutativescalarsolitons.Attheend we discusstheobtainedresultsand possible prospects. 2.p-Adic numbers andtheir functions When we wish to introduce p-adic numbers it is instructive to start from Q, since Qis the simplest ﬁeld of numbers of characteristic 0and it contains results of all physicalmeasurements.Anynon-zerorationalnumbercanbepresentedasinﬁnite expansionsinto thetwoquite differentforms.The usual oneis tothe base 10, i.e. −∞ /summationdisplay k=n a k 10 k , a k = 0,···,9, (5) and theother oneistothe base p(pis a primenumber)and reads +∞ /summationdisplay k=m b k p k , b k = 0,···,p−1, (6) wherenandmaresomeintegers.Theserepresentationshavetheusualrepetition of digits, but, in a sense, expansions are in the mutually opposite directions. The series (5) and (6) are convergent with respect to the usual absolute value |·| ∞ andp-adic absolute value |·| p , respectively. Allowing arbitrary combinations for digits, we obtain standard representation of real numbers (5) and p-adic numbers (6).RandQ p exhaust all number ﬁelds which contain Qas a dense subﬁeld. They have many distinct geometric and algebraic properties. Geometry of p-adic numbers is the nonarchimedean one. For much more on p-adic numbers and p- adic analysis one can see, e.g. [4, 7, 16]. There are mainly two kinds of analysis kievarwe.tex; 12/03/2001; 3:49; p.400 394 B. DRAGOVICH, I.V. VOLOVICH onQ p based on two different mappings: Q p →Q p andQ p →C. We use both of them, in classical and quantum p-adic models, respectively. Elementary p-adic functions are given by the same series as in the real case, but their regions of convergence are usually different. For instance, expx= /summationtext ∞ n=0x n n! andlnx= /summationtext ∞ n=1 (−1) n+1(x−1) n n converge if|x| p <|2| p and|x−1| p <1, respectively. Derivativesof p-adicvaluedfunctionsarealsodeﬁnedasintherealcase,butusing p-adic norm instead of the absolute value. As a deﬁnite p-adic valued integral we take difference of the corresponding antiderivative in end points. Usual complex- valuedp-adicfunctionsare: (i)anadditivecharacter χ p (x) = exp 2πi{x} p ,where {x} p is the fractional part of x∈Q p ,(ii)a multiplicative character π s (x) =|x| s p , wheres∈C, and(iii)locally constant functions with compact support, like, e.g. Ω(|x| p ) = 1if|x| p ≤1andΩ(|x| p ) = 0otherwise. There is well deﬁned Haar measure andintegration. Forexample, /integraldisplay Q p χ p (αx 2 +βx)dx=λ p (α)|2α| − 1 2 p χ p /parenleftBigg −β 2 4α /parenrightBigg , α/negationslash= 0,(7) whereλ p (α)is anarithmetic function[7].Anadele x[4] isan inﬁnitesequence x= (x ∞ ,x 2 ,···,x p ,···), wherex ∞ ∈Randx p ∈Q p with the restriction that for all but a ﬁnite set Sof primespwe havex p ∈Z p . Componentwise addition and multiplication can be appliedtoadeles.Itisusefultopresenttheringofadeles Ainthefollowingform: A=∪ S A(S),A(S) =R× /productdisplay p∈S Q p × /productdisplay p/negationslash∈S Z p , whereZ p ={x∈Q p :|x| p ≤1}is the ring of p-adic integers.Ais also locallycompacttopologicalspace.Therearetwokindsofanalysisover A,which generalize thecorresponding analysis over RandQ p . 3.p-Adicstringamplitudes Like in the ordinary string theory, the starting point in an investigation of p-adic strings is a construction of the corresponding scattering amplitudes. Recall that the ordinary crossing symmetric Veneziano amplitude can be presented in the followingforms: A ∞ (k 1 ,···,k 4 )≡ A ∞ (a,b) =g 2 /integraldisplay R |x| a−1 ∞ |1−x| b−1 ∞ dx (8) =g 2 /bracketleftbigg Γ(a)Γ(b ) Γ(a+b)+Γ(b)Γ(c ) Γ(b+c)+Γ(c)Γ(a ) Γ(c+a) /bracketrightbigg (9) kievarwe.tex; 12/03/2001; 3:49; p.401 p-ADIC STRINGSANDNONCOMMUTATIVITY 395 =g 2 ζ(1−a ) ζ(a)ζ(1−b ) ζ(b)ζ(1−c ) ζ(c)(10) =g 2 /integraldisplay DXexp /parenleftbigg − i 2π /integraldisplay d 2 σ∂ α X µ ∂ α X µ /parenrightbigg 4 /productdisplay j=1 /integraldisplay d 2 σ j exp /parenleftBig ik (j) µ X µ /parenrightBig ,(11) where/planckover2pi1= 1, T= 1/π,anda=−α(s) =−1− s 2 , b=−α(t), c=−α(u)with thecondition s+t+u=−8,i.e.a+b+c= 1.Tointroducethecorresponding p-adic Veneziano amplitude there is a sense to consider p-adic analogs of all the abovefourexpressions. p-Adicgeneralizationoftheﬁrstexpressionwasproposed in[3]and itreads A p (a,b) =g 2 p /integraldisplay Q p |x| a−1 p |1−x| b−1 p dx, (12) where|·| p denotesp-adic absolute value. In this case only string world-sheet parameterxis treated as p-adic variable, and all other quantities maintain their usual (real)valuation.An attractiveadelic formula ofthe form A ∞ (a,b) /productdisplay p A p (a,b) = 1 (13) wasfound[5],where A ∞ (a,b)denotestheusualVenezianoamplitude(8).Asim- ilarproductformulaholdsalsofortheVirasoro-Shapiroamplitude.Theseinﬁnite products are divergent, but they can be successfully regularized. Unfortunately, there is a problem to extend this formula to the higher-point functions. p-Adic analogs of (9) and (10) were also proposed in [2] and [17], respectively. In these cases, world-sheet, string momenta and amplitudes are manifestly p-adic. Since string amplitudes are p-adic valued functions, it is not so far enough clear their physicalinterpretation.Expression(11)isbasedonFeynman’sfunctionalintegral method, which is generic for all quantum systems and has successful p-adic gen- eralization[18].Its p-adiccounterpart,proposedin[10],hasbeenelaborated[19] and deserves further study. Note that in this approach, p-adic string amplitude is complex valued, while not only the world-sheet parameters but also target space coordinates and string momenta are p-adic variables. Such p-adic generalization is a natural extension of the formalism of p-adic [20] and adelic [21] quantum mechanicstostringtheory.Intheframeworkofthisnewapproachwewillpresent here some results concerning the p-adic Veneziano amplitude. Instead of the start with the very expression (11) we will take in the real case as a starting point the followingformula A ∞ (k 1 ,···,k 4 ) =g 2 ∞4 /productdisplay j=1 /integraldisplay dx j exp   2 hT /summationdisplay i<j k i k j ln|x i −x j | ∞   ,(14) kievarwe.tex; 12/03/2001; 3:49; p.402 396 B. DRAGOVICH, I.V. VOLOVICH which canbederived from(11), andafter somestandardevaluation [22] onehas A ∞ (k 1 ,···,k 4 ) =g 2 ∞ /integraldisplay Q ∞ dx|x| 2k1k 2 hT ∞ |1−x| 2k2k 3 hT ∞ . (15) Intheconstructionof p-adicamplitude wetake p-adic analogueof (14), whichis A p (k 1 ,···,k 4 ) =g 2 p /integraldisplay Q p dxχ p   1 hT /summationdisplay i<j k i k j ln(x i −x j )   .(16) Notethatfrom(16)onecannotobtain(12)sincelogarithmicfunction lnisp-adic valuedandadditivecharacter χ p iscomplexvaluedfunction.Thus,wehaveherea newtypeofp-adicstringamplitudes.When k i k j hT ∈Q p \Z p additivecharacterwill be different from 1and we have non-trivial p-adic amplitude. The corresponding adelicstringamplitude is A(k (1) ,···,k (4) ) =A ∞ (k (1) ∞ ,···,k (4) ∞ ) /productdisplay p∈S A p (k (1) p ,···,k (4) p ) /productdisplay p/negationslash∈S A p (k (1) p ,···,k (4) p ),(17) wherek (i) isan adele,i.e. k (i) = (k (i) ∞ ,k (i) 2 ,···,k (i) p ,···) (18) with the restriction that k (i) p ∈Z p for all but a ﬁnite set Sof primesp. The topological ring of adeles Aprovides a framework for simultaneous and uniﬁed considerationofrealand p-adicnumbers.Rationalnumbersarealsoembeddedin the space of adeles. If k (i) p k (j) p hT ∈Z p for all primes pthenA p (k (1) p ,···,k (4) p ) = g 2 p /producttext 4 j=1 /integraltext dx j ,sinceχ p (a) = 1whena∈Z p . In this case, p-adic effects con- tribute only to the effective coupling constant, and adelic amplitude is equal to the ordinary one. When k (i) p k (j) p hT ∈Q p \Z p then additive character may give non- trivial contributions to adelic amplitude, what also depends on adelic state of the world-sheet. 4.p-Adic solitonicbranes andnoncommutative scalar solitons There is an effective tachyon ﬁeld theory in terms of real numbers with an exact action which describes p-adic strings with amplitude (12). The corresponding Lagrangian [23,24]in d-dimensional Minkowskispace ( /planckover2pi1= 1) is L= 1 g 2 p 2 p−1 /bracketleftbigg − 1 2ϕp − 1 2 /square ϕ+ 1 p+ 1ϕ p+1 /bracketrightbigg , (19) kievarwe.tex; 12/03/2001; 3:49; p.403 p-ADIC STRINGSANDNONCOMMUTATIVITY 397 where/squaredenotes the Laplacian, ϕis the tachyon ﬁeld and pis an arbitrary prime number. Note that this Lagrangian has been recently considered in the context of tachyon condensation and brane descent relations [9]. The above Lagrangian yieldstheequationofmotion p − 1 2 /square ϕ=ϕ p . (20) Inadditiontosolutions ϕ= 0andϕ= 1thereisalso solutionof theform ϕ(x) =p n 2(p−1) exp /parenleftBigg −p− 1 2plnp n /summationdisplay i=1 x 2 i /parenrightBigg , (21) wheren≤d−1. This conﬁguration can be called the p-adic solitonic q-brane solution,where q=d−n−1.In particularcase, n= 2andp= 2, onehas ϕ(x 1 ,x 2 ) = 2 exp /parenleftbigg − 1 4 ln 2(x 2 1 +x 2 2 ) /parenrightbigg . (22) On theotherhand thereisa noncommutativescalarsoliton [25] φ(x 1 ,x 2 ) = 2 exp /parenleftbigg − 1 θ(x 2 1 +x 2 2 ) /parenrightbigg (23) which is the simplest nontrivial (trivial solutions are φ= 0andφ= 1) solution oftheequation (φ⋆φ )(x) =φ(x), (24) where⋆denotes the Moyal product (3) with θ ij =θε ij . The solution (23) of the equation (24) extremises energy in noncommutative scalar ﬁeld theory [25] with thepotential V(φ) = 1 2m 2 φ⋆φ− 1 3φ⋆φ⋆φ, (25) wherem= 1and the kinetic term is neglected in the limit θ−→∞. It is evident that the above solitonic solutions (22) and (23) are equal if θ= 4 ln 2 . This noncommutativescalarﬁeldmodelcanbeextendedtothemoregeneralcasewith V(φ) = 1 2m 2 φ 2 −c k +1 k+ 1φ k+1 , (26) whereﬁeldsaremultipliedbythestarproduct,and φ≡φ(x 1 ,···,x n )witheven nspatial directions. Thecorrespondingequation c k+1 φ k (x) =m 2 φ(x) (27) kievarwe.tex; 12/03/2001; 3:49; p.404 398 B. DRAGOVICH, I.V. VOLOVICH hasthe solution φ(x) = 2 n 2 /parenleftBigg m 2 c k+1 /parenrightBigg 1 k exp /parenleftBigg − 1 θ n /summationdisplay i=1 x 2 i /parenrightBigg . (28) The solutions (21) and (28) may be identiﬁed taking the corresponding values for massmand noncommutativity parameter θ. Thus, we see that there is an intriguing similarity between p-adic solitonic branes and noncommutative scalar solitons. Discussionand concluding remarks In the previous Section we considered two nonlocal scalar ﬁeld theories. Their potentials involve inﬁnitely many derivatives. The corresponding differential equations are of the inﬁnite order, and they extremize the action and the energy, respectively. It seems that there is a sense to expect something noncommutative in the effective p-adic Lagrangian (19), as well as something p-adic (nonar- chimedean) in noncommutative scalar ﬁeld theory with potential (26). Moreover, some more explicit relations between string ﬁeld theory and p-adic string theory could be found in the coming years (see also comments in [9]). We believe that thereisanunderlyingprinciple,whichconnectsthefollowingthreespaceproper- ties: noncommutativity, nonarchimedean geometry and the uncertainty relation (1). Let us also mention that various aspects of possible connection between quantum groups, nonarchimedean geometry and p-adic strings are discussed in [26, 27]. On q-deformation of the Veneziano amplitude one can see [28] and referencestherein.Itisworthnotingthatonecanintroduce[29]theMoyalproduct inp-adicquantummechanics anditreads( h= 1) (ˆf∗ˆg)(x) = /integraldisplay Q dp /integraldisplay Q dp dkdk /prime χ p (−(x i k i +x j k /prime j ) + 1 2k i k /prime j θ ij )˜f(k)˜g(k /prime ),(29) whereddenotes spatialdimensionality. Acknowledgements . B.D. wishes to thank Profs. J. Wess and S. Duplij for theirinvitationtoparticipateandgiveatalkattheARW”NoncommutativeStruc- tures in Mathematics and Physics”. The work on this paper was supported in part by RFFIgrant 990100866. References 1. L.J. Garay, Int. J. Mod. Phys. A10(1995) 145. 2. I.V. Volovich, p-Adic String , Class. Quantum Grav. 4(1987) L83. 3. P.G.O. Freundand M. Olson, Phys. Lett. B199(1987) 186. kievarwe.tex; 12/03/2001; 3:49; p.405 p-ADIC STRINGSANDNONCOMMUTATIVITY 399 4. I.M.Gel’fand,M.I.GraevandI.I.Pyatetski-Shapiro, RepresentationTheoryandAutomorphic Functions , Saunders, London, 1966. 5. P.G.O. Freundand E.Witten, Adelic String Amplitudes , Phys. Lett. B 199(1987) 191. 6. L. Brekke and P.G.O. Freund, p-Adic numbers in physics , Phys. Rep. 233(1993) 1. 7. V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, p-Adic Analysis and Mathematical Physics , World Scientiﬁc,Singapore, 1994. 8. V.S. Vladimirov, Adelic Formulas for Gamma and Beta Functions of One-Class Quadratic Fields: Applications to 4-Particle Scattering string amplitudes , in Proc. Steklov Math. Institute228(2000) 67, math-ph/0004017. 9. D.GhoshalandA.Sen, TachyonCondensationandBraneDescentRelationsin p-AdicString Theory, hep-th/0003278. 10. B. Dragovich, On AdelicStrings , hep-th/0005200. 11. I.V. Volovich, Number Theory as the Ultimate Physical Theory , Preprint CERN-TH. 4781 (1987). 12. J.H. Schwarz, Recent Progressin Superstring Theory , hep-th/0007130. 13. A.Sen, Recent Developments in Superstring Theory , hep-lat/0011073. 14. A. Connes, M.R. Douglas and A. Schwarz, Noncommutative Geometry and Matrix Theory: Compactiﬁcationon Tori , JHEP9802(1998)003, hep-th/9711162. 15. I.Ya. Aref’eva and I.V. Volovich, Noncommutative Gauge Fields on Poisson Manifolds , hep- th/9907114. 16. W.H. Schikhof, Ultrametric Calculus , Cambridge U.P., Cambridge, 1984. 17. I.Ya. Aref’eva, B. Dragovich and I.V. Volovich, On the adelic string amplitudes , Phys. Lett. 209B(1988)445. 18. G.S. Djordjevi ´c and B. Dragovich, p-Adic Path Integral for Quadratic Actions , Mod. Phys. Lett.A 12(1997) 1455. 19. B. Dragovich, P. Rodi ´c and I.V. Volovich, New Amplitudes for pAdic and Adelic Bosonic Strings,in preparation. 20. V.S. Vladimirov and I.V. Volovich, p-Adic Quantum Mechanics , Commun. Math. Phys. 123 (1989)659. 21. B. Dragovich, Adelic Model of Harmonic Oscillator , Theor. Math. Phys. 101(1994) 1404; Adelic Harmonic Oscillator , Int. J. Mod.Phys. A 10(1995)2349. 22. M.B.Green,J.H.SchwarzandE.Witten, SuperstringTheory,I ,CambridgeU.P.,Cambridge, 1987. 23. L. Brekke, P.G.O.Freund,M. Olson and E. Witten,Nucl. Phys. B302(1988)365. 24. P.H. Framptonand Y. Okada, Phys. Rev. Lett. 60(1988)484. 25. R. Gopakumar, S. Minwalla and A. Strominger, NoncommutativeSolitons , hep-th/0003160. 26. I.Ya. Aref’eva and I.V. Volovich, Quantum group particles and non-archimedean geometry , Phys. Lett. B 268(1991)179. 27. P.G.O. Freund, On the Quantum Group- p-Adics Connection , Preprint EFI 90-90, December, 1990. 28. L.J.Romans, DeformingtheVenezianoModel(”q-strings”) ,intheICTPSeriesinTheoretical Physics - Volume 6: 1989 Summer School in High Energy Physics and Cosmology, pp. 278- 287, World Scientiﬁc,Singapore. 29. G.Djordjevi ´c,B.DragovichandLj.Ne ˇs´c,AdelicQuantumMechanics:Nonarchimedeanand NoncommutativeAspects , in the Proceedings of this Workshop. kievarwe.tex; 12/03/2001; 3:49; p.406 kievarwe.tex; 12/03/2001; 3:49; p.407 ADELIC QUANTUMMECHANICS:NONARCHIMEDEAN AND NONCOMMUTATIVEASPECTS GORAN DJORDJEVI ´C 1,2∗ , BRANKO DRAGOVICH 3,4 and LJUBIˇSANEˇSI´C 1 1 DepartmentofPhysics,FacultyofSciences,UniversityofNi ˇs,P.O. Box91,18001Ni ˇs,Yugoslavia 2 Sektion Physik, Universit ¨at M¨unchen, Theresienstr. 37, D-80333 M¨unchen,Germany 3 Instituteof Physics,P.O.Box 57,11001 Belgrade, Yugoslavia 4 Steklov Mathematical Institute, Gubkin St. 8, 117966, Moscow, Russia Abstract. We present a short review of adelic quantum mechanics pointing out its non- Archimedean and noncommutative aspects. In particular, p-adic path integral and adelic quantum cosmologyareconsidered.Somesimilaritiesbetween p-adicanalysisandq-analysisarenoted.The p-adicMoyal productis introduced. 1. Introduction There is now a common belief that the usual picture of spacetime as a smooth pseudo-Riemannian manifold should breakdown somehow at the Planck length l p ∼10 −33 cm, due to the quantum gravity effects. We consider here two possibilities, which come from modern mathematics and mathematical physics: non-Archimedean geometry related to p-adic numbers, and noncommutative geometrywithspace coordinatesgivenby noncommuting operators [ˆx i ,ˆx j ] =i/planckover2pi1θ ij (1) ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.408 402 G.DJORDJEVI ´C,B.DRAGOVICH, L. NE ˇSI´C or by q-deformation x i x j =qx j x i . Some noncommutativity of conﬁguration space should not be a surprise in physics since quantum phase space with the canonical commutation relation (9) is the well-known example of noncommu- tative geometry. We will mostly review our recent results concerning adelic quantummechanics.Weillustratesomefeaturesofadelicquantummechanicsby itsapplicationinquantumcosmology.Afewremarkablesimilaritiesbetweennon- Archimedeanandnoncommutativestructuresarenoted.TheusualMoyalproduct isextendedto p-adicandadelicquantummechanics.Since1987,therehavebeen manyinterestingapplicationsof p-adicnumbersandnon-Archimedeangeometry in various parts of modern theoretical and mathematical physics (for a review, see [1–3]). However we restrict ourselves here to p-adic and adelic quantum mechanics as well as to some related topics. In particular, we review Feynman’s p-adicpathintegralmethod.Afundamentalroleofintegralapproachto p-adicand adelic quantum mechanics (and adelic quantum cosmology) is emphasized. The obtainedp-adicprobabilityamplitudeforone-dimensionalsystemswithquadratic Lagrangians has the form as that one in ordinary quantum mechanics. It is well known that measurements give rational numbers Q, whereas theoretical models traditionally use real Rand complexCnumber ﬁelds. A completion of Qwith respect to the p-adic norms gives the ﬁelds of p-adic numbers Q p (pis a prime number) in the same way as completion with absolute value yields R. The paper of Volovich [4] initiated a series of articles on p-adic string theory and many other branches of theoretical and mathematical physics. The metric introduced byp-adic norm is the non-Archimedean (ultrametric) one. Possible existence of such space around the Planck length is the main motivation to study p-adic quantum models. However, p-adic analysis also plays a role in some areas of ”macroscopicphysics”as,forexample:spinglasses,quasicrystalsandsomeother complex systems. In order to investigate possible p-adic quantum phenomena it is necessary to have the corresponding theoretical formalism. An important step in this direction is a formulation of p-adic quantum mechanics [5, 6]. Because of total disconnectedness of p-adic spaces and different valuations of variables and wave functions, the quantization is performed by the Weyl procedure. A unitary representation of the evolution operator U p (t)on the Hilbert space L 2 (Q p )of complex-valued functions of a p-adic argument is an appropriate way to describe quantum dynamics of p-adic systems. Recently formulated adelic quantum me- chanics [7] successfully uniﬁes ordinary and all p-adic quantum mechanics. The appearance of space-time discreteness in adelic formalism (see, e.g. [8]) is an encouragement for the further investigations. This paper is organized as follows. We start with a short introduction to p-adic numbers, adeles and their functions. After that,p-adic and adelic quantum mechanics based on the Weyl quantization and Feynman’s path integral are presented. In Section 4 we review our previuos results concerning one-dimensional p-adic propagator. In Section 5 we will see how adelic quantum mechanics can be useful in investigation of the very early kievarwe.tex; 12/03/2001; 3:49; p.409 ADELIC QUANTUMMECHANICS 403 universe, where in a natural way space-time discreteness emerges in minisuper- space models of adelic quantum cosmology. In the last Section we give some of interesting relations between non-Archimedean and noncommutative analysis. We alsodeﬁneanddiscuss thecorresponding p-adic Moyal product. 2.p-Adic numbers andadeles Anyx∈Q p can be presentedinthe form[9] x=p ν (x 0 +x 1 p+x 2 p 2 +···), ν∈Z, (2) wherex i = 0,1,···,p−1are digits.p-Adic norm of any term x i p ν+i in the canonicalexpansion(2)is |x i p ν+i | p =p −(ν+i) andthestrongtriangleinequality holds,i.e.|a+b| p ≤max{|a| p ,|b| p }. It follows that|x| p =p −ν ifx 0 /negationslash= 0. There is no natural ordering on Q p . However one can introduce a linear order on Q p by the following deﬁnition: x < yif|x| p <|y| p or when|x| p =|y| p there exists such index m≥0that digits satisfy x 0 =y 0 ,x 1 =y 1 ,···,x m−1 = y m−1 ,x m < y m . Derivatives of p-adic valued functions ϕ:Q p →Q p are deﬁned as in the real case, but with respect to the p-adic norm. There is no in- tegral /integraltext ϕ(x)dxin a sense of the Lebesgue measure [2], but one can introduce /integraltext b a ϕ(x)dx= Φ(b)−Φ(a)asafunctionalofanalyticfunctions ϕ(x),where Φ(x) is an antiderivative of ϕ(x). In the case of map f:Q p →Cthere is well-deﬁned Haarmeasure. We usehere theGauss integral /integraldisplay Q υ χ υ (ax 2 +bx)dx=λ υ (a)|2a| − 1 2 υ χ v /parenleftbig −b 2 4a /parenrightbig , a/negationslash= 0,(3) whereindex υdenotesreal( υ=∞)andp-adiccases, i.e.υ=∞,2,3,5,···.χ υ is an additive character: χ ∞ (x) = exp(−2πix),χ p (x) = exp(2πi{x} p ), where {x} p is the fractional part of x∈Q p .λ υ (a)is the complex-valued arithmetic function [2]. An adele [10] is an inﬁnite sequence a= (a ∞ ,a 2 ,...,a p ,...), where a ∞ ∈R≡Q ∞ ,a p ∈Q p witharestrictionthat a p ∈Z p forallbutaﬁniteset Sof primesp.Thesetofalladeles Amayberegardedasasubsetofdirecttopological productQ ∞ × /producttext p Q p whoseelements satisfy theabove restriction, i.e. A=∪ S A(S),A(S) =R× /productdisplay p∈S Q p × /productdisplay p/∈S Z p . (4) Ais a topological space, and can be considered as a ring with respect to the com- ponentwise addition and multiplication. An elementary function on adelic ring A is ϕ(x) =ϕ ∞ (x ∞ ) /productdisplay p ϕ p (x p ) = /productdisplay v ϕ v (x v ) (5) kievarwe.tex; 12/03/2001; 3:49; p.410 404 G.DJORDJEVI ´C,B.DRAGOVICH, L. NE ˇSI´C with the main restriction that ϕ(x)must satisfy ϕ p (x p ) = Ω(|x p | p )for all but a ﬁnitenumber of p,where Ω(|x| p ) = /braceleftbigg 1,0≤|x| p ≤1, 0,|x| p >1,(6) isacharacteristicfunctiononthesetof p-adicintegersZ p ={x∈Q p :|x| p ≤1}. ItshouldbenotedthattheFouriertransformofthecharacteristicfunction(vacuum state) Ω(|x p |)isΩ(|k p |). All ﬁnite linear combinations of elementary functions (5) make the setD(A)of the Schwartz-Bruhat functions. The Fourier transform ofϕ(x)∈D(A)(that mapsD(A)ontoD(A))is ˜ϕ(y) = /integraldisplay A ϕ(x)χ(xy)dx= /integraldisplay R ϕ ∞ (x)χ ∞ (xy)dx /productdisplay p /integraldisplay Q p ϕ p (x)χ p (xy)dx,(7) wheredx=dx ∞ dx 2 ...dx p ...is the Haar measure on A. The Hilbert space L 2 (A)is a space of complex-valued functions ψ 1 (x),ψ 2 (x),..., with the scalar product and norm (ψ 1 ,ψ 2 ) = /integraldisplay A ¯ψ 1 (x)ψ 2 (x)dx,||ψ||= (ψ,ψ) 1/2 <∞. (8) Abasisoftheabovespacemaybegivenbytheorthonormaleigenfunctionsofan evolutionoperator [7]. 3. Adelic quantum mechanics In foundations of standard quantum mechanics (over R) one usually starts with a representationofthe canonical commutation relation [ˆq,ˆk] =i/planckover2pi1, (9) whereqis a coordinate and kis the corresponding momentum. It is well known thattheprocedureofquantizationisnotunique.Informulationof p-adicquantum mechanics [5, 6] the multiplication ˆqψ→xψhas no meaning for x∈Q p and ψ(x)∈C. Also, there is no possibility to deﬁne p-adic ”momentum” or ”Hamil- tonian” operator. In the real case they are inﬁnitesimal generators of space and time translations, but, since Q p is disconnected ﬁeld, these inﬁnitesimal transfor- mationsbecomemeaningless.However,ﬁnitetransformationsremainmeaningful and the corresponding Weyl and evolution operators are p-adically well deﬁned. For the onedimensionalsystems whichclassicalevolutioncan be describedby z t =T t z, z t = /parenleftbigg q(t) k(t) /parenrightbigg , z= /parenleftbigg q(0) k(0) /parenrightbigg , (10) kievarwe.tex; 12/03/2001; 3:49; p.411 ADELIC QUANTUMMECHANICS 405 whereq(0)andk(0), are initial position and momentum, respectively, and T t is a matrix. Canonical commutation relation in p-adic case can be represented by the Weyloperators ( h= 1) ˆQ p (α)ψ p (x) =χ p (αx)ψ p (x) (11) ˆK p (β)ψ(x) =ψ p (x+β). (12) Now,to therelation(9)in the realcase,corresponds ˆQ p (α)ˆK p (β) =χ p (αβ)ˆK p (β)ˆQ p (α) (13) inthep-adic one.It ispossibletointroduce the family of unitary operators ˆW p (z) =χ p (− 1 2qk)ˆK p (β)ˆQ p (α), z∈Q p ×Q p , (14) that is a unitary representation of the Heisenberg-Weyl group. Recall that this group consistsoftheelements (z,α)with thegroup product (z,α)·(z /prime ,α /prime ) = (z+z /prime ,α+α /prime + 1 2B(z,z /prime )), (15) whereB(z,z /prime ) =−kq /prime +qk /prime is a skew-symmetric bilinear form on the phase space. Dynamics of a p-adic quantum model is described by a unitary operator of evolution U(t)without using the Hamiltonian. Instead of that, the evolution operatorhas beenformulated in terms ofits kernel K t (x,y) U p (t)ψ(x) = /integraldisplay Q p K t (x,y)ψ(y)dy. (16) Thenextsectionwillbedevotedtothepathintegralformulationandcalculationof thequantumpropagator K t (x,y)onp-adicspaces.Inthisway[5] p-adicquantum mechanics isgiven byatriple (L 2 (Q p ),W p (z p ),U p (t p )). (17) Keeping in mind that standard quantum mechanics can be also given as the cor- responding triple, ordinary and p-adic quantum mechanics can be uniﬁed in the formof adelicquantummechanics [7] (L 2 (A),W(z),U(t)). (18) L 2 (A)is the Hilbert space on A,W(z)is a unitary representation of the Heisenberg-Weyl group on L 2 (A)andU(t)is a unitary representation of the evolutionoperator on L 2 (A).Theevolutionoperator U(t)is deﬁned by U(t)ψ(x) = /integraldisplay A K t (x,y)ψ(y)dy= /productdisplay v /integraldisplay Q v K (v) t (x v ,y v )ψ (v) (y v )dy v .(19) kievarwe.tex; 12/03/2001; 3:49; p.412 406 G.DJORDJEVI ´C,B.DRAGOVICH, L. NE ˇSI´C The eigenvalueproblemfor U(t)reads U(t)ψ αβ (x) =χ(E α t)ψ αβ (x), (20) whereψ αβ areadeliceigenfunctions, E α = (E ∞ ,E 2 ,...,E p ,...)iscorresponding energy,indices αandβdenoteenergylevelsandtheirdegeneration.Notethatany adeliceigenfunctionhas theform Ψ(x) = Ψ ∞ (x ∞ ) /productdisplay p∈S Ψ p (x p ) /productdisplay p/negationslash∈S Ω(|x p | p ), x∈A, (21) where Ψ ∞ ∈L 2 (R),Ψ p ∈L 2 (Q p ). Adelic quantum mechanics takes into ac- count alsop-adic quantum effects and may be regarded as a starting point for constructionofamorecompletesuperstringandM-theory.Inthelow-energylimit adelicquantum mechanics becomes ordinary one. 4.p-Adic pathintegrals A suitable way to calculate propagator in p-adic quantum mechanics is by p-adic generalizationofFeynman’spathintegral.Fortheclassicalaction ¯S(x /prime/prime ,t /prime/prime ;x /prime ,t /prime ) which is a polynomial quadratic in x /prime/prime andx /prime it is well known that in ordinary quantummechanics theFeynman path integralis K(x /prime/prime ,t /prime/prime ;x /prime ,t /prime ) = /parenleftbigg i h∂ 2 ¯ S ∂x /prime/prime ∂x /prime /parenrightbigg 1/2 exp /parenleftbigg 2π i h¯S(x /prime/prime ,t /prime/prime ;x /prime ,t /prime ) /parenrightbigg .(22) p-Adic generalization of the Feynman path integral was suggested in [5] and can bewrittenona p-adiclineas K p (x /prime/prime ,t /prime/prime ;x /prime ,t /prime ) = /integraldisplay χ p /parenleftbigg −S[q ] h /parenrightbigg Dq= /integraldisplay χ p /parenleftbigg − 1 h /integraldisplay t /prime/prime t /prime L(q,˙q,t)dt /parenrightbigg/productdisplay t dq(t). (23) In (23) we take h∈Qandq,t∈Q p . This path integral is elaborated, for the ﬁrst time,for theharmonicoscillator [11].It was shownthat there exists the limit K p (x /prime/prime ,t /prime/prime ;x /prime ,t /prime ) = lim n→∞ K (n) p (x /prime/prime ,t /prime/prime ;x /prime ,t /prime ) = lim n→∞ N (n) p (t /prime/prime ,t /prime ) × /integraldisplay Q p ··· /integraldisplay Q p χ p /parenleftbigg − 1 h n /summationdisplay i=1 ¯S(q i ,t i ;q i−1 ,t i−1 ) /parenrightbigg dq 1 ···dq n−1 ,(24) whereN (n) p (t /prime/prime ,t /prime )isthecorrespondingnormalizationfactorfortheharmonicos- cillator. The subdivision of p-adic time segment t 0 < t 1 <···< t n−1 < t n is made according to linear order on Q p and|t i −t i−1 | v →0for every kievarwe.tex; 12/03/2001; 3:49; p.413 ADELIC QUANTUMMECHANICS 407 i= 1,2,···,n, whenn→ ∞. In the similar way we have calculated path integrals for: a particle in a constant external ﬁeld [12], some minisuperspace cosmological models and a relativistic free particle [8], as well as for a harmonic oscillator with a time-dependent frequency [12]. p-Adic classical mechanics has thesameanalyticformasintherealcase.If q(t) = ¯q(t) +y(t)denotesapossible quantum path, with conditions y(t /prime ) =y(t /prime/prime ) = 0, where ¯q(t)is ap-adic classical pathwithδS[¯q] = 0,we havethe following actionfor quadratic Lagrangians: S[q] =S[¯q] + 1 2!δ 2 S[¯q] =S[¯q] + 1 2 /integraldisplay t /prime/prime t /prime /parenleftbigg y ∂ ∂q+ ˙y ∂ ∂˙q /parenrightbigg (2) L(q,˙q,t)dt.(25) Putting(25) into (23),and usingcondition /integraldisplay Q p K ∗ p (x /prime/prime ,t /prime/prime ;x /prime ,t /prime )K p (z,t /prime/prime ;x /prime ,t /prime )dx /prime =δ p (x /prime/prime −z), (26) withquadraticexpansionofactionaswellasthegeneralformofthenormalization factor N p (t /prime/prime ,t /prime ) =|N p (t /prime/prime ,t /prime )| ∞ A p (t /prime/prime ,t /prime ), we obtaingeneralexpression forthepropagator (forsome details, see [13]) K p (x /prime/prime ,t /prime/prime ;x /prime ,t /prime ) =λ p /parenleftbigg − 1 2h∂ 2 ¯ S ∂x /prime/prime ∂x /prime /parenrightbigg 1 h∂ 2 ¯ S ∂x /prime/prime ∂x /prime  1 2 p χ p /parenleftbigg − 1 h¯S(x /prime/prime ,t /prime/prime ;x /prime ,t /prime ) /parenrightbigg . (27) This result exhibits some very important properties. For instance, replacing an indexpwithvin (27) we can write quantum-mechanical amplitude Kin or- dinary and all p-adic cases in the same compact form. It points out a generic behaviourofquantumpropagationinArchimedeanandnon-Archimedeanspaces and emphasizes the fundamental role of the Feynman path integral method in quantum theory. Also, considering the most general quadratic p-adic Lagrangian L(x,˙x,t) =a(t) ˙x 2 +2b(t) ˙xx+c(t)x 2 +2d(t) ˙x+2e(t)x+f(t)withanalyticco- efﬁcients,wefoundaconnection[14]betweenthesecoefﬁcientsandthesimplest p-adic quantum state Ω(|x| p ), that is necessary for existence of adelic quantum dynamics. For space-time discreteness in adelic models, see [8]. It is worth men- tioning that this approach can be extended to systems with the two, three and more dimensions, and results will be presented elsewhere. The above results are also a starting point for a further elaboration of adelic quantum mechanics and for a semiclassical computation of the p-adic path integrals with non-quadratic Lagrangians. 5. Adelic quantum cosmology Adelicquantumcosmology[15]isanapplicationofadelicquantummechanicsto the universe as a whole. It uniﬁes ordinary and p-adic quantum cosmology. Here, kievarwe.tex; 12/03/2001; 3:49; p.414 408 G.DJORDJEVI ´C,B.DRAGOVICH, L. NE ˇSI´C path integral formalism occurs to be quite appropriate tool to take integration over both Archimedean and non-Archimedean geometries on the equal footing. In this approach we introduce υ-adic complex-valued cosmological amplitudes by a functionalintegral /angbracketlefth /prime/prime ij ,φ /prime/prime ,Σ /prime/prime |h /prime ij ,φ /prime ,Σ /prime /angbracketright υ = /integraldisplay D(g µν ) υ D(Φ) υ χ υ (−S υ [g µν ,Φ]).(28) In practice, it is not possible to deal with full superspace (the space of all 3- metrics and matter ﬁeld conﬁgurations). Instead, one exploits minisuperspace (a ﬁnitenumberofcoordinates (h ij ,φ)).Afterthissimpliﬁcation, υ-factorsofadelic minisuperspacepropagatorare given bytherelation /angbracketleftq α/prime/prime |q α/prime /angbracketright υ = /integraldisplay dNK υ (q α/prime/prime ,N|q α/prime ,0), (29) whereK υ is an ordinary quantum-mechanical propagator with ﬁxed minisuper- space coordinates q α and the lapse function N. We illustrate adelic quantum cosmologyby BianchiImodel (k= 0).Using Lorentz metric [16] ds 2 =σ 2 /bracketleftBigg −N 2 (t ) a 2 (t)dt 2 +a 2 (t)dx 2 +b 2 (t)dy 2 +c 2 (t)dz 2 /bracketrightBigg (30) and replacements: x=bc+a 2 2, y=bc−a 2 2,˙z 2 =a 2 ˙b˙c, (31) we obtainthecorresponding action S p [x,y,z ] = 1 2 /integraldisplay 1 0 dt /bracketleftBigg − 1 N /parenleftBigg ˙x 2 −˙y 2 2+ ˙z 2 /parenrightBigg −λN(x+y) /bracketrightBigg ,(32) and equationsofmotion ¨x+λN 2 = 0,¨y−λN 2 = 0,¨z= 0. (33) Taking into account conditions x(0) =x /prime , y(0) =y /prime , z(0) =z /prime ,x(1) = x /prime/prime , y(1) =y /prime/prime , z(1) =z /prime/prime , the quantum transition amplitude can be written as K p (x /prime/prime ,y /prime/prime ,z /prime/prime ,N|x /prime ,y /prime ,z /prime ,0) =λ p (−2N ) /vextendsingle/vextendsingle/vextendsingle 4 1 3 N /vextendsingle/vextendsingle/vextendsingle 3 2 p χ p /parenleftbig −¯S(x /prime/prime ,y /prime/prime ,z /prime/prime ,N|x /prime ,y /prime ,z /prime ,0) /parenrightbig . (34) kievarwe.tex; 12/03/2001; 3:49; p.415 ADELIC QUANTUMMECHANICS 409 Conditions for the existence of the vacuum state Ω(|x| p )Ω(|y| p )Ω(|z| p )can be calculated from the equality /integraldisplay |x /prime | p ≤1 /integraldisplay |y /prime | p ≤1 /integraldisplay |z /prime | p ≤1 K p (x /prime/prime ,y /prime/prime ,z /prime/prime ,N|x /prime ,y /prime ,z /prime ,0)dx /prime dy /prime dz /prime = Ω(|x /prime/prime | p )Ω(|y /prime/prime | p )Ω(|z /prime/prime | p ), and thesimplest vacuum stateis Ψ p (x,y,z,N ) = /braceleftbigg Ω(|x| p )Ω(|y| p )Ω(|z| p ),|N| p ≤1,|λ| p ≤1, p/negationslash= 2, Ω(|x| 2 )Ω(|y| 2 )Ω(|z| 2 ),|N| 2 ≤ 1 2 ,|λ| 2 ≤2, p= 2. (35) According to (21) adelic wave function Ψ(x,t)offers more information on a physical system than only its standard part Ψ ∞ (x,t). In quantum-mechanical experiments, as well as in all measurements, numerical results belong to the ﬁeld of rational numbers Q. For the Bianchi I model, as well as for any adelic quan- tum model, according to the usual interpretation of the wave function we have to consider|Ψ(x,t)| 2 ∞ at rationalspace-time points.In the above adeliccase we get |Ψ(x,y,z,N )| 2 ∞ =|Ψ ∞ (x,y,z,N )| 2 ∞ /productdisplay p Ω(|x| p )Ω(|y| p Ω(|z| p ) = /braceleftbigg |Ψ ∞ (x,y,z,N )| 2 ∞ , x,y,z∈Z, 0, x,y,z ∈Q\Z.(36) Here we used the following properties of the Ω-function: Ω 2 (|x| p ) = Ω(|x| p ), /producttext p Ω(|x| p ) = 1ifx∈Z, and /producttext p Ω(|x| p ) = 0ifx∈Q\Z. Thus, it means that positions x,y,zmay have only discrete values: x= 0,±1,±2,.... Since theΩ-function is invariant under the Fourier transformation, there is also discrete momentumspace.Whensystemisinsomeexcitedstate,thesharpnessofthedis- crete structure disappears and space demonstrates usual continuous properties. It isworthmentioningthataspace-timediscretenessisalsonotedintheframework ofq-deformed quantum mechanics[17]. 6.p-Adic analysisand q-analysis.TheMoyal product Some connections between p-adic analysis and quantum deformations has been noticed [18] in a variety of cases during the last ten years or so. It was shown [19] that the two parameter Sklyanin quantum algebra and its generalizations provideapromisingconnectionbetweenthe p-adicsandquantumdeformation.A similar connection has been indicated by Macdonald’s paper [20] on orthogonal polynomials associated with the root systems. In [19] it was also pointed out that kievarwe.tex; 12/03/2001; 3:49; p.416 410 G.DJORDJEVI ´C,B.DRAGOVICH, L. NE ˇSI´C elliptic quantum group and its generalizations unify the p-adic and real versions of a Lie group (e.g. SL(2)). This result is connected with adelic approach and the possibility of establishing q-deformed Euler products. In some other contexts it has been observed that the Haar measure on SU q (2)coincides with the Haar measureontheﬁeldof p-adicnumbers Q p ifq= 1 p [21].Namely,Tomea-Jackson integralin q-analysis /integraldisplay 1 0 f(x)d q x= (1−q) ∞ /summationdisplay n=0 f(q n )q n , (37) and theintegral in p-adicanalysis /integraldisplay |x| p ≤1 f(|x| p )dx= (1− 1 p) ∞ /summationdisplay n=0 f(p −n )p −n , (38) areequalifq= 1 p ,i.e. /integraldisplay 1 0 f(x)d 1/p x= /integraldisplay |x| p ≤1 f(|x| p )dx. (39) Inq-analysisthereisthefollowingdifferentialoperator(relatedtotheq-deformed momentum in the coordinaterepresentation[21]) ∂ q f(x) =f(x)−f(qx ) (1−q)x. (40) Inp-adicanalysis,whenoneconsidersacomplex-valuedfunction f(x)depending onap-adicvariable xwearenotabletousestandarddeﬁnitionofdifferentiation. Insteadofthat itis possibleto use Vladimirov’soperator D α ψ(x) =p− 1 1−p −1−α /integraldisplay f(x)−f(y ) |x−y| α+1p dy (41) which in a sense resembles (40). Moreover, there is a potential such that the spectrumof the p-adic Schr ¨odinger- like(diffusion) equation [22] Dψ(x) +V(|x| p )ψ(x) =Eψ(x) (42) is the same one as in the case of q-deformed oscillator found by Biedenharn [23] and Macfarlane [24] for q= 1/p. For more details, see [21]. Recently [25], it has been proposed a new pseudodifferential operator with rational part of p- adic numbers{x} p . In such case, energy levels for p-adic free particle exhibit discrete dependence on the corresponding momentum: {E} p ={k} 2 p . Note also a proposal for q-deformation of Vladimirov’s operator [26]. We see that there are some interesting relations between p-adic and q-analysis, and in a sense be- tween adelic quantum mechanics and noncommutative one. It would be fruitful kievarwe.tex; 12/03/2001; 3:49; p.417 ADELIC QUANTUMMECHANICS 411 to ﬁnd some deeper reasons for these connections, between theories which pre- tend to give us more insights on the space-time structure at the Planck scale. By now it is not enough understood. It seems to be reasonable to formulate a non- commutative adelic quantum mechanics that may connect non-Archimedean and noncommutative effects and structures. As the ﬁrst step in this direction one has to consider a p-adic and adelic generalization of the Moyal product. Let us con- sider D-dimensional classical space with coordinates x 1 ,x 2 ,···,x D . Letf(x) be a classical function f(x) =f(x 1 ,x 2 ,···,x D ). Then, with the respect to the Fouriertransformations, wehave ˜f(k) = /integraldisplay Q Dυ dxχ v (kx)f(x), (43) f(x) = /integraldisplay Q Dυ dkχ v (−kx)˜f(k). (44) According tothe usualWeyl quantization ˆf(x) = /integraldisplay Q D∞ dkχ ∞ (−kˆx)˜f(k)≡f(ˆx). (45) Letusnowhave twoclassical functions f(x)andg(x)with ˆf(x) = /integraldisplay Q D∞ dkχ ∞ (−kˆx)˜f(k), (46) ˆg(x) = /integraldisplay Q D∞ dkχ ∞ (−kˆx)˜g(k). (47) In the coordinate representation we can write the same above expressions replac- ingˆxbyxand extend it to all p-adic cases. Now we are interested in product ˆf(x)ˆg(x).In therealcasethis operatorproduct is oftheform (ˆf·ˆg)(x) = /integraldisplay /integraldisplay dkdk /prime χ ∞ (−kˆx)χ ∞ (−k /prime ˆx)˜f(k)˜g(k /prime ).(48) Using the Baker-Campbell-Hausdorff formula, the relation (1) and then the coordinaterepresentationone ﬁndstheMoyalproductin theform (f∗g)(x) = /integraldisplay /integraldisplay dkdk /prime χ υ /parenleftbigg −(k+k /prime )x+ 1 2k i k /prime j θ ij /parenrightbigg ˜f(k)˜g(k /prime ),(49) where we already used our generalization from Q ∞ toQ υ . Note that in the real case weuse k i →−(i/2π)(∂/∂x i )andobtainthe wellknown form (f∗g)(x) =χ ∞ /parenleftBigg −θ ij 2(2π) 2 ∂ ∂y i ∂ ∂z j /parenrightBigg f(y)g(z)| y=z=x .(50) kievarwe.tex; 12/03/2001; 3:49; p.418 412 G.DJORDJEVI ´C,B.DRAGOVICH, L. NE ˇSI´C Thus, asthe p-adicMoyalproduct wetake (ˆf∗ˆg)(x) = /integraldisplay Q Dp /integraldisplay Q Dp dkdk /prime χ p (−(x i k i +x j k /prime j ) + 1 2k i k /prime j θ ij )˜f(k)˜g(k /prime ).(51) AstheﬁrststepinadelizationonecanconsidertheMoyalproducton R× /producttext p∈S Q p × /producttext p/negationslash∈S Z p space. Various adelic aspects of the Moyal product will be presented elsewhere. Acknowledgments .AuthorsG.Dj.andB.D.wishtothanktheco-Directorsof ARW ”Noncommutative Structures in Mathematics and Physics” Profs. J. Wess and S. Duplij for their invitation to participate and give a talk. G.Dj. is partially supported by DFG Project “Noncommutative space-time structure - Cooperation with Balkan Countries”. The work of B.D. was supported in part by RFFI grant 990100866. References 1. L. Brekke and P.G.O. Freund, p-Adic numbers in physics , Phys. Rep. 233,1 (1993). 2. V.S. Vladimirov, I.V. Volovich and E.I. Zelenov, p-Adic Analysis and Mathematical Physics , (World Scientiﬁc, Singapore, 1994). 3. A. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Kluwer Acad. Publ., Dordrecht, 1994). 4. I.V. Volovich, p-Adic String , Class. Quantum Grav. 4, L83 (1987). 5. V.S.VladimirovandI.V.Volovich, p-AdicQuantumMechanics ,Comm.Math.Phys. 123,659 (1989). 6. Ph. Ruelle, E. Thiran, D. Verstegen and J. Weyers, Quantum Mechanics on p-Adic Fields , J. Math. Phys. 30, 2854 (1989). 7. B. Dragovich, Adelic Harmonic Oscillator , Int. J. Mod. Phys. A10, 2349 (1995). 8. B.Dragovich, AdelicWaveFunctionoftheUniverse ,in:Proc.oftheThirdA.FriedmannInt. SeminaronGrav.andCosmology,FriedmannLab.Publishing,St.Petersburg,1995,pp.311– 321; G.S. Djordjevi ´c, B. Dragovich, Lj. Ne ˇsi´c,p-Adic and Adelic Free Relativistic Particle , Mod. Phys.Lett. A 14, 317 (1999). 9. W.H. Schikhof, Ultrametric Calculus , (Cambridge U.P., Cambridge,1984). 10. I.M. Gel’fand, M.I. Graev and I.I. Pyatetskii-Shapiro, Representation Theory and Automor- phicFunctions , (Saunders, London, 1966). 11. E.I. Zelenov, p-Adic path integrals , J. Math.Phys. 32,147 (1991). 12. G.S.Djordjevi ´candB.Dragovich, Onp-AdicFunctionalIntegration ,inProc.oftheIIMath. Conf. in Pri ˇstina, Priˇstina(Yugoslavia), 1996. 13. G.S. Djordjevi ´c and B. Dragovich, p-Adic Path Integral for Quadratic Actions , Mod. Phys. Lett.A12, 1455 (1997). 14. G.S. Djordjevi ´c and B. Dragovich, Adelic connection between classical and quantum dynamics , in Proc. of the XIII Conf.on Appl. Mathematics,Igalo’98(Yugoslavia) 23(2000). 15. B.DragovichandLj.Ne ˇsi´c,p-AdicandAdelicGeneralizationofQuantumCosmology ,Grav. Cosm.5, 222 (1999). 16. A. Ishikawa and H. Ueda, The Wave Function of the Universe by the New Euclidean Path- integral Approach inQuantum Cosmology, Int. J.Mod. Phys. D2, 249(1993). 17. J.Wess, q-Deformed Heisenberg Algebras , mat-ph/9910013. kievarwe.tex; 12/03/2001; 3:49; p.419 ADELIC QUANTUMMECHANICS 413 18. P.G.O. Freund, On the Quantum Group - p-Adics Connection , in Quarks, Symmetries and Strings, (M. Kaku, A. Jevicki and K. Kikkawa, eds,), World Scientiﬁc, Singapore, 1991, pp. 267-275. 19. P.G.O. Freund and A.V. Zabrodin, Macdonald Polynomials from Sklyanin Algebras: A Con- ceptual Basis for the p-Adics-Quantum Group Connection , Commun. Math. Phys. 147, 277 (1992). 20. I.G. Macdonald, in Orthogonal Polynomials: Theory and Practice, (P. Nevai ed.), Kluwer Acad. Publ., Dordrecht, 1990, p.311. 21. I.Ya.Aref’evaandI.V.Volovich, QuantumGroupParticlesandNon-ArchimedeanGeometry , Phys. Lett. B268, 179(1991). 22. V.S.VladimirovandI.V.Volovich, p-AdicSchr ¨odingerTypeEquation ,Lett.Math.Phys. 18, 43(1989). 23. L.C. Biedenharn, The quantum group SU q (2)and a q-analogue of the boson operators , J. Phys. A: Math. Gen. 22, L873 (1989). 24. A.J.Macfarlane, Onq-analoguesofthequantumharmonicoscillatorandthequantumgroup SU(2) q , J. Phys. A: Math. Gen. 22, 4581(1989). 25. D. Dimitrijevi ´c, G.S. Djordjevi ´c and B. Dragovich On Schr¨odinger-type equation on p-adic spaces,to bepublished in Bal. Phys. Lett. 26. S.V.Kozyrev, TheElementsofnoncommutativeanalysison RandQ p ,Ph.D.Thesis,Moscow, 1995(in Russian). kievarwe.tex; 12/03/2001; 3:49; p.420 kievarwe.tex; 12/03/2001; 3:49; p.421 GIBBSSTATESOF A LATTICESYSTEMOF QUANTUM ANHARMONIC OSCILLATORS YURIKOZITSKY ∗† Instituteof Mathematics,MarieCurie-Sklodowska University, Lublin 20-031,Poland 1. Introduction Gibbs states of interacting quantum lattice systems are constructed as positive functionals on von Neumann algebras whose elements (observables) represent physical quantities [8], [13]. For the systems, the algebra of observables of every subsysteminaﬁnitesubsetofthelatticemayberepresentedasthe C ∗ -algebraof bounded operators on a Hilbert space, the theory of Gibbs states is quite well elaborated [8]. But if one needs to include into consideration also unbounded operators,thesituationbecomesmuchmorecomplicated.In1975anapproachto theconstructionofGibbsstates,whichusestheintegrationtheoryinpathspaces, has been initiated [1] (see also [5], [6], [7], [11], [13], [15]). Here the state at a temperature T=β −1 is deﬁned by means of a probability measure µ β on a certain inﬁnite-dimensional space, analogously to the Euclidean quantum ﬁeld theory.Thatisthereasonwhy µ β is known as theEuclidean Gibbs state . In this paper we consider the following model. To each point of the lattice L=Z d ,d∈Nthere is attached a quantum particle (oscillator) with the reduced mass m= m ph //planckover2pi1 2 ( m ph isthephysicalmass),whichhasanunstableequilibrium position at this point. Such particles perform D-dimensional oscillations around their equilibrium positions and interact via attractive potential. Similar objects havebeenstudiedformanyyearsasquiterealisticmodelsofcrystallinesubstance undergoingstructural phasetransitions (seee.g.[16]). In Section 2, following [2], [3], [4], we summarize main aspects of the con- struction of the Euclidean Gibbs state for the model considered. In Section 3, we ∗ [email protected] † SupportedinpartbythePolishScientiﬁcResearchCommitteeundertheGrant2P03A02915 kievarwe.tex; 12/03/2001; 3:49; p.422 416 Y.KOZITSKY provide a number of assertions describing such states. In particular, we show that strongzero-pointoscillationssuppresscriticalpointanomalies.Thelatterresultis astrengthening ofsimilar onesgiven in[2],[14]. 2. EuclideanFormalismforQuantum GibbsStates The oscillations of the particle having its equilibrium position at l∈Lare de- scribed by the momentum and displacement operators {p l ,q l }, densely deﬁned on the complex Hilbert space H l =L 2 (R D ). The whole system is described by theformal Hamiltonian H= 1 2 /summationdisplay l,l /prime d ll /prime (q l ,q l /prime ) + /summationdisplay l H l , (1) H l = 1 2 m(p l ,p l ) + 1 2(q l ,q l ) +V(q l ), (2) where (., .)stands for scalar product in R D andd ll /prime form a dynamical matrix. The potential Vis chosenasfollows V(x) =v((x,x)), (3) wherevis a polynomial, convex on R +def = [0,+∞). Some of our results were obtained underassumptionthat v(ξ) = 1 2aξ+ r /summationdisplay s=2 b s ξ s , r≥2, a∈R, b s ≥0, b r >0. (4) Forp∈Z,let S p = /braceleftBigg {x l ,l∈L}| /summationdisplay l (1 +|l|) 2p x 2 l <∞ /bracerightBigg , (5) where|l|isthe Euclidean normon L=Z d ⊂R d .Let also S def = /intersectiondisplay S p ,S /primedef = /uniondisplay S −p , p∈N 0def =N∪{0}. (6) The dynamical matrix is supposed to be invariant under translations on L, and attractive(d ll /prime ≤0).Wealsosupposethatforevery l∈L,thesequence{d ll /prime ,l /prime ∈ L}belongs toS. Set Λ ={l= (l 1 , ..., l d )|l 0 j ≤l j ≤l 1 j , l 0 j <l 1 j , l 0 j ,l 1 j ∈Z, j= 1, ...,d}. Givenabox Λ,letL(Λ)denotethepartitionof Lbytheboxeswhichareobtained as translations of Λ. Let also Gbe the group of all translations of L, and G(Λ) = kievarwe.tex; 12/03/2001; 3:49; p.423 GIBBS STATES OFQUANTUMSYSTEMS 417 {t∈ G|t(Λ)∈L(Λ)}, wheret(Λ) ={t(l), l∈Λ}. Then the dynamical matrix (d Λ ll /prime ) l,l /prime ∈Λ obeying periodic conditions on the boundaries of Λand the periodic local Hamiltonian H Λ are d Λ ll /prime = min{d lt(l /prime ) :t∈ G(Λ)}, (7) H Λ = 1 2 /summationdisplay l,l /prime ∈Λ d Λ ll /prime (q l ,q l /prime ) + /summationdisplay l∈Λ H l . (8) The latter is an essentially self-adjoint lower bounded operator acting in H Λ = L 2 /parenleftBig R D|Λ| /parenrightBig (|·|stands forcardinality). For a box Λand an inverse temperature β=T −1 , a periodic Gibbs state γ β,Λ isthefollowingfunctional γ β,Λ (A) =trace(Ae −βH Λ ) trace(e −βH Λ ), (9) deﬁned on the C ∗ -algebra A Λ of linear bounded operators on H Λ . Given Λand t∈R,wedeﬁneanautomorphism of A Λ a Λ t (A) = exp (itH Λ )Aexp (−itH Λ ). (10) AsigniﬁcantroleintheconstructionoftheGibbsstatesofourmodelisplayedby multiplication operators. Bounded multiplication operators form a commutative subalgebra of A Λ . The components of the displacement operator q (k) l ,l∈Λare multiplication operators, but they do not belong to A Λ since they are unbounded. In[12]there was proved thefollowingassertion(see also[1], [11]). Proposition 38. Lett 1 ,...,t n ∈RandA 1 ,...A n be bounded continuous func- tionsA j :R D|Λ| →C. Then A Λ is the smallest strongly closed linear space containingall operatorsoftheform a Λ t 1 (A 1 ) a Λ t 2 (A 2 )... a Λ t n (A n ). ForA 1 ,...,A n ∈ A Λ andt 1 ,...t n ∈R, a temporal Green function corresponding to theperiodicboundaryconditions is G β,Λ A 1 ,...,A n (t 1 ,...,t n ) =γ β,Λ /parenleftBig a Λ t 1 (A 1 )... a Λ t n (A n ) /parenrightBig . (11) Foranopensubset O⊂C n ,letHol(O)standforthesetofallholomorphicin O complexvaluedfunctions.Let also D β ndef ={(t 1 ,...,t n )∈C n |0</Ifractur(t 1 )</Ifractur(t 2 )···</Ifractur(t n )<β}.(12) By means of the arguments which were used in a similar situation in [1], Sect. 3 and [12],Sect. 2,one canprovethefollowingstatement. kievarwe.tex; 12/03/2001; 3:49; p.424 418 Y.KOZITSKY Lemma39. ForeveryA 1 ,...,A n ∈ A Λ , (a)G β,Λ A 1 ,...,A n may be extended toaholomorphic functionon D β n ; (b)thisextension(whichwill alsobe written as G β,Λ A 1 ,...,A n ) iscontinuous on the closur eD β n ofD β n ,moreover, forall (t 1 ,...,t n ) ∈D β n , /vextendsingle/vextendsingle/vextendsingle G β,Λ A 1 ,...,A n (t 1 ,...,t n ) /vextendsingle/vextendsingle/vextendsingle ≤/bardblA 1 /bardbl·····/bardblA n /bardbl, (13) where/bardbl·/bardblstands for operatornorm; (c)for everyξ 1 ,...,ξ n ∈R,theset D β n (ξ 1 ,...,ξ n ) def ={(t 1 ,...,t n )∈D β n |/Rfractur(t j ) =ξ j , j= 1,...,n}, issuchthat forarbitrary F,G∈Hol(D β n ),their equalityon D β n (ξ 1 ,...,ξ n )impliesthat FandGareequal on theD β n . The restrictionofthe function(11)to D β n (0,..., 0),i.e. Γ β,Λ A 1 ,...,A n (τ 1 ,...τ n ) =G β,Λ A 1 ,...,A n (iτ 1 ,...iτ n ), (14) isatemperature(Matsubara)Greenfunction, whichhas suchaproperty Γ β,Λ A 1 ,...,A n (τ 1 +θ,...τ n +θ) = Γ β,Λ A 1 ,...,A n (τ 1 ,...τ n ), (15) for everyθ∈I βdef = [0,β],whereadditionismodulo β. In view of Proposition 38, the Green functions, deﬁned by (11) with bounded multiplication operators, fully determine the state γ β,Λ . Claim (c) of the latter as- sertionyieldsinturnthatthisstateisdeterminedbytheMatsubarafunctions(14). IntheEuclideanapproachthesefunctionsareobtainedasmomentsofprobability measures. We begin their construction with introducing corresponding measure spaces.Given β >0andΛ, weset Ω β,Λ ={ω Λ = (ω l ) l∈Λ |ω l ∈C(I β →R D ), ω Λ (0) =ω Λ (β)}.(16) Inthesequel,C β willstandfor Ω β,Λ withaone-point Λ.LetalsoX β standforthe real Hilbertspace L 2 (I β →R D )equipped with scalarproduct andnorm /angbracketleftω,ω /prime /angbracketright β = /integraldisplay I β (ω(τ),ω /prime (τ))dτ,/bardblω/bardbl β = /radicalBig /angbracketleftω,ω/angbracketright β . (17) Further X β,Λ ={ω Λ = (ω l ) l∈Λ |ω l ∈X β }. (18) kievarwe.tex; 12/03/2001; 3:49; p.425 GIBBS STATES OFQUANTUMSYSTEMS 419 Since Λisﬁnite, Ω β,Λ andX β,Λ maybeequippedwiththeusualBanachspaceand Hilbert space structures respectively. Let B(Ω β,Λ )stand for the Borel σ–algebra ofthesubsetsof Ω β,Λ .Considerthefollowingstrictlypositivetraceclassoperator onX β S β = (− m∆ β + 1) −1 1, (19) where ∆ β is the Laplace operator in L 2 (I β )and1is the identity operator in R D . It determines onX β aO(D)–invariantGaussianmeasure χ β ,for which /integraldisplay X β exp{/angbracketleftϕ,ω/angbracketright β }χ β (dω) = exp /braceleftbigg 1 2/angbracketleftS β ϕ,ϕ/angbracketright β /bracerightbigg . (20) This measure is concentrated on C β ⊂X β [1], [11]. It describes a D-dimensional quantumharmonicoscillatorwiththemass m.Onecanshow(seee.g.[1])thatfor anyτ∈I β , /integraldisplay X β exp [α(ω(τ),ω(τ))]χ β (dω)<∞,∀α<α ∗ , (21) where α ∗ = 2 √m·exp(β/ √m)− 1 exp(β/ √m) + 1. (22) Givenabox Λ,we write χ β,Λ (dω Λ ) = /circlemultiplydisplay l∈Λ χ β (dω l ), (23) E V β,Λ (ω Λ ) = 1 2 /summationdisplay l,l /prime ∈Λ d Λ ll /prime /angbracketleftω l ,ω l /prime /angbracketright β + /summationdisplay l∈Λ /integraldisplay I β V(ω l (τ))dτ. (24) Under the assumptions regarding Vandd ll /prime ,E V β,Λ is a continuous function from Ω β,Λ toR.Aperiodic localEuclidean Gibbsmeasure is µ β,Λ (dω Λ ) = 1 Z β,Λ exp /braceleftBig −E V β,Λ (ω Λ ) /bracerightBig γ β,Λ (dω Λ ). (25) It is a probability measure on the Hilbert space X β,Λ , supported on Ω β,Λ .Z β,Λ is the normalizing constant. Therefore, the Green functions (14) constructed with multiplication operators A 1 ,...A n ∈ A Λ may bewritten follows [1],[11] Γ β,Λ A 1 ,...,A n (τ 1 ,...,τ n ) (26) = /integraldisplay X β,Λ A 1 (ω Λ (τ 1 ))...A n (ω Λ (τ n ))µ β,Λ (dω Λ ). kievarwe.tex; 12/03/2001; 3:49; p.426 420 Y.KOZITSKY TheGibbsstatesofthewholesystemwhichcorrespondtotheperiodicbound- ary conditions are constructed as limits of the above states γ β,Λ when Λ/arrownortheastL. More precisely, let Lbe a sequence of boxes ordered by inclusion and such that ∪ Λ∈L Λ =L.For Λ 1 ⊂Λ 2 ,onemayintroduceanaturalnorm-preservingembed- ding A Λ 1 ⊂ A Λ 2 ,whichdeﬁnesanincreasingsequenceofalgebras { A Λ ,Λ∈L}. In a standard way [8], this sequence deﬁnes a quasi-local algebra of observables. Two sequencesL,L /prime are called equivalent if the corresponding quasi-local alge- bras coincide. A standard sequence Lis the sequence of boxes {Λ L ,L∈N}, Λ L = (−L,L] d ∩Z d . In the sequel, all (thermodynamic) limits Λ/arrownortheastLare taken over a sequence L, which is equivalent to the standard one. The existence ofperiodic Gibbsstates forsimilarmodels wasshownin [7]. The great advantage of the Euclidean approach lies in the fact that due to the above relationship between the Green functions and local Gibbs measures one may apply to the quantum case the machinery of conditional distributions, which formthebaseofmodernclassicalequilibriumstatisticalphysics(seee.g.[9],[10] and the references therein). To this end we will employ the spaces Ω β,Λ , deﬁned by (16), (18), also for inﬁnite subsets Λ. In particular, Ω β will stand for Ω β,Λ with Λ =L. These spaces are equipped with the product topology and with the σ-algebras B(Ω β,Λ )generated by cylinder subsets. For ∆⊂Λ⊂L, we write ω ∆ ×ζ Λ\∆ for the conﬁguration (ξ l ) l∈Λ such thatξ l =ω l forl∈∆, andξ l =ζ l forl∈Λ\∆. Given a sequence of boxes L, in order to have the collections {Ω β,Λ ,Λ∈L}ordered by inclusion, we introduce the following mappings. For ∆⊂Λ, we putω ∆ /mapsto→ω ∆ ×0 Λ\∆ ∈Ω β,Λ , where 0 Λ is the zero conﬁguration in Ω β,Λ . Hence we consider every conﬁguration ω ∆ as an element of all Ω β,Λ with ∆⊂Λ.Besides, wedeﬁne Ω β,Λ /ownerω Λ /mapsto→(ω Λ ) Λ /prime ∈Ω β,Λ /prime , as aconﬁguration such that ω l = 0forl∈Λ /prime \Λ.Let Ω t βdef ={ζ∈Ω β |{/bardblζ l /bardbl β ,l∈L}∈S /prime }. (27) Forζ∈Ω β and a box Λ, we deﬁne the local Gibbs measure, subject to ζ, as the followingconditionalprobabilitymeasure. Weput µ β,Λ (B|ζ) = 0, ζ∈Ω β \Ω t β , B∈ B(Ω β,Λ ), (28) and forevery ζ∈Ω t β , µ β,Λ (dω Λ |ζ) = 1 Z β,Λ (ζ)exp /braceleftBig −E V β,Λ (ω Λ |ζ) /bracerightBig χ β,Λ (dω Λ ).(29) Here Z β,Λ (ζ) def = /integraldisplay Ω β,Λ exp /braceleftBig −E V β,Λ (ω Λ |ζ) /bracerightBig χ β,Λ (dω Λ ), kievarwe.tex; 12/03/2001; 3:49; p.427 GIBBS STATES OFQUANTUMSYSTEMS 421 isthelocalpartition functionsubjecttotheexternalboundary condition ζ Λ c ,and E β,Λ (ω Λ |ζ) = 1 2 /summationdisplay l,l /prime ∈Λ d ll /prime /angbracketleftω l ,ω l /prime /angbracketright β + /summationdisplay l∈Λ,l /prime ∈Λ c d ll /prime /angbracketleftω l ,ζ l /prime /angbracketright β ,(30) E V β,Λ (ω Λ |ζ) =E β,Λ (ω Λ |ζ) + /summationdisplay l∈Λ /integraldisplay I β V(ω l (τ))dτ, (31) whereVis given by (3). Under the assumptions regarding Vandd ll /prime , both E β,Λ (·|ζ),E V β,Λ (·|ζ)are continuous functions from Ω β,Λ toRfor allζ∈Ω t β . The function E β,Λ (·|ζ)describes the interaction of the particles in Λbetween themselvesandwith theﬁxedconﬁguration ζ Λ c ,Λ c =L\Λ. Thus, along with (26), one may introduce the temperature Green function which corresponds totheexternal boundarycondition ζ Λ c Γ ζ,β,Λ A 1 ,...,A n (τ 1 ,...,τ n ) (32) = /integraldisplay X β,Λ A 1 (ω Λ (τ 1 ))...A n (ω Λ (τ n ))µ β,Λ (dω Λ |ζ). HereA 1 ,...,A n aremultiplicationoperatorssuch thatfor every τ 1 ,...,τ n ∈I β ,thefunction Ω β,Λ /ownerω Λ /mapsto→A 1 (ω Λ (τ 1 ))...A n (ω Λ (τ n )), isµ β,Λ (·|ζ)integrable for every ζ∈Ω β , that holds for A 1 ,...,A n ∈ A Λ . Note that the above temperature Green function is deﬁned only for multiplication op- erators, there are no a prioriinformation regarding its analytic and continuity properties (except for ζ= 0), even inthecase ofbounded operators. ForB∈ B(Ω β )andω∈Ω β , letδ B (ω)take values 1, resp. 0, if ωbelongs, resp.doesnotbelong,to B.Thenonecanintroduceafamilyofprobabilitykernels {π β,Λ |Λ⊂L,|Λ|<∞}, on(Ω β , B(Ω β )) π β,Λ (B|ζ) def = /integraldisplay Ω β,Λ δ B (ω Λ ×ζ Λ c )µ β,Λ (dω Λ |ζ). (33) Theysatisfytheconsistencyconditions(formore detailssee e.g. [10]) π β,Λ π β,∆ (B|ζ) def = /integraldisplay Ω β π β,Λ (dω|ζ)π β,∆ (B|ω) =π β,Λ (B|ζ),(34) which holds for arbitrary pairs of ﬁnite subsets ∆⊂Λ⊂Land anyB∈B(Ω β ), ζ∈Ω t β . kievarwe.tex; 12/03/2001; 3:49; p.428 422 Y.KOZITSKY Deﬁnition 40. A probability measure µon the space (Ω β ,B(Ω β ))is said to be a Euclidean Gibbs state at the inverse temperature βif it satisﬁes the Dobrushin- Lanford-Ruelle (DLR)equilibrium equation /integraldisplay Ω β µ(dω)π β,Λ (B|ω) =µ(B), (35) for allﬁnite Λ⊂LandB∈ B(Ω β ). 3. TheResults Bymeansoftherepresentation(26)weextendtheGreenfunctionstounbounded multiplication operators. Theorem 41. Let the functions A 1 ,...,A n :R D|Λ| →Cbe such that for everyβ > 0and everyτ∈ I β , the functions Ω β,Λ /ownerω Λ /mapsto→A j (ω Λ (τ)), j= 1,...n, areµ β,Λ –integrable. Then, for the corresponding multiplication operatorsA 1 ,...,A n , the Green function (26) may be analytically continued on thedomainD β n deﬁned by (12). Incontrasttothecaseofboundedoperators(c.f.claim(b)ofLemma39),one cannot expect that such extended Green functions are uniformly bounded onD β n and continuousonitsboundaries. Deﬁnition42. Acontinuousfunction A:R D|Λ| →Cbelongstothefamily F (D) Λ if forarbitrary α>0,thefunction R D|Λ| /ownerx Λ /mapsto→|A(x Λ )|exp    −α /summationdisplay l∈Λ |x l | 2    , (36) isboundedon R D|Λ| . Inthecase of one-pointboxes,i.e. for |Λ|= 1, wewrite F (D) . Corollary 43. For arbitrary A 1 ,...,A n ∈ F (D) Λ , the temperature Green function (26) maybecontinuedanalyticallyinaccordance with Theorem 41. Indeed, by (21), functions from F (D) Λ are integrable. As it has been already mentioned, the above analyticity does not imply continuity of the temperature Green functions. To prove it we have used the tightness of the local Gibbs measures. Theorem44. Givenabox Λ,letA 1 ,...,A n belongto F (D) Λ .Thenforall ζ∈Ω β , theGreenfunctions(26), (32) arecontinuous on I n β /owner(τ 1 ,...,τ n ). kievarwe.tex; 12/03/2001; 3:49; p.429 GIBBS STATES OFQUANTUMSYSTEMS 423 Theorem 45. [FKG Inequality] Given Λandζ∈Ω β , letµstand for any of the local Gibbs measures (25), (29) with D= 1. Then for any functions F,G∈ F (1) Λ , whichgrowwhenevery chosen ω l (τ)increases,the following inequality holds <FG> µ ≥<F > µ <G> µ , (37) where<·> µ standsfor expectationwith respecttothe measure µ. Theorem 46. [GKS Inequalities] Given Λ, let the local Gibbs measure be de- ﬁnedby(25)with D= 1.Letalsotherealvaluedfunctions A 1 ,...,A n+m ∈ F (1) Λ , n,m∈Nhave thefollowing properties: (a)everyA j dependsonly onthevaluesof x l j withcertain l j ∈Λ; (b)everyA j iseitheranoddmonotone growingfunctionof x l j oran even positivefunction,monotone growingon [0,+∞). Then for the Green functions (26), (32), the following inequalities hold for arbitraryτ 1 ,...,τ n+m ∈I β : Γ β,Λ A 1 ,...,A n (τ 1 ,...,τ n )≥0,Γ 0,β,Λ A 1 ,...,A n (τ 1 ,...,τ n )≥0, (38) Γ β,Λ A 1 ,...,A n+m (τ 1 ,...,τ n+m )≥ Γ β,Λ A 1 ,...,A n (τ 1 ,...,τ n )× (39) Γ β,Λ A n+1 ,...,A n+m (τ n+1 ,...,τ n+m ) Γ 0,β,Λ A 1 ,...,A n+m (τ 1 ,...,τ n+m )≥ Γ 0,β,Λ A 1 ,...,A n (τ 1 ,...,τ n )× Γ 0,β,Λ A n+1 ,...,A n+m (τ n+1 ,...,τ n+m ). Now the model (1) - (3) with D∈Nwill be compared with the scalar model described by the same local Hamiltonian with D= 1. In order to distinguish vector and scalar objects we will supply the latter ones with tilde, writing /tildewide H Λ , /tildewide γ β,Λ , /tildewide Γ β,Λ . Inthesequel, thepolynomial vissupposedto be ofthe form (4). Theorem 47. [Scalar Domination] GivenA 1 ,...,A n ∈ F (D) Λ , let there exist k= 1,...,Dandthefunctions /tildewide A 1 ,... /tildewide A n ∈ F (1) Λ ,satisfyingtheconditionsofthe above theorem, such that A j (x Λ ) = /tildewide A j (x (k) Λ ),j= 1,...,n. Then for arbitrary τ 1 ,...,τ n ∈I β 0≤Γ β,Λ A 1 ,...,A n (τ 1 ,...,τ n )≤ /tildewide Γ β,Λ /tildewide A 1 ,..., /tildewide A n (τ 1 ,...,τ n ). (40) kievarwe.tex; 12/03/2001; 3:49; p.430 424 Y.KOZITSKY R EMARK 3. Note that all A j depend onx (k) Λ with one and the same k. The ﬁrst aboveinequalityisa D-dimensionalversionof(38).Thesecondinequalityin(40) describesscalardomination. In the model considered, the structural phase transition, breaking O(D)- symmetry,isassociatedwiththeappearanceoflargeﬂuctuationsofdisplacements ofparticles. To describethemweintroduce ﬂuctuation operators Q Λ = 1 /radicalbig |Λ| /summationdisplay l∈Λ q l , (41) correspondingto normalﬂuctuations.IftheGreenfunctions(14)constructedwith A=Q (k) Λ , remain bounded when Λ/arrownortheastL, the ﬂuctuations are regarded as nor- mal. At the critical point the ﬂuctuations become so large that to preserve the boundedness of the Green functions one should use an abnormal normalization, i.e. Q λ,Λ =λ(Λ)Q Λ =λ (Λ) /radicalbig |Λ| /summationdisplay l∈Λ q l , where{λ(Λ)∈R,Λ∈L}isaconvergingtozerosequence.Given F 1 ,...,F n ∈ F (D) , letA λ j stand forF j (Q λ,Λ ),j= 1,...,n. Deﬁnition48. Givenβ >0, lettheconvergence Γ β,Λ A λ 1 ,...,A λn (τ 1 ,...,τ n )−→F 1 (0)...F n (0),Λ/arrownortheastL, (42) hold for all n∈N, allτ 1 ,...,τ n ∈ I β , allF 1 ,...,F n ∈ F (D) , arbitraryL, and any converging to zero sequence {λ(Λ),Λ∈L}. Then the ﬂuctuations of displacementsofparticles aresaidtobe normal. Set J=− /summationdisplay l /prime d ll /prime , T = /tildewide H l +J /parenleftBig q (1) l /parenrightBig 2 , (43) where the sum is taken over the whole lattice L. The operator Thas a purely discretenon-degenerate spectrum.Denote Tψ n =/epsilon1 n ψ n ,∆ = min{/epsilon1 n+1 −/epsilon1 n , n∈N}. Theorem 49. Let the mass m, the spectral parameter ∆, and the interaction parameterJobey thecondition m∆ 2 >2J. (44) Then for any D∈N, the ﬂuctuations of displacements of particles in the D- dimensionalmodel remain normalatall temperatures. kievarwe.tex; 12/03/2001; 3:49; p.431 GIBBS STATES OFQUANTUMSYSTEMS 425 References 1. S. Albeverio, R. 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Phys., 51(2000), 71–81. 15. B. Simon, Functional Integration and Quantum Physics, Academic Press, New York San Francisco London, 1979. 16. S. Stamenkovi ´c,Uniﬁed Model Description of Order-Disorder and Displacive Structural Phase Transitions, Condensed Matter Physics, 1(14)(1998), 257-309. kievarwe.tex; 12/03/2001; 3:49; p.432 kievarwe.tex; 12/03/2001; 3:49; p.433 AMETRIC-AFFINEFIELDMODELFOR THE NEUTRINO DMITRIVASSILIEV ∗ Department ofMathematical Sciences, University of Bath, BathBA2 7AY,UK 1. Main result We deﬁne space-time as a real oriented 4-manifold Mequipped with a non- degenerate metric g(not necessarily symmetric) and an afﬁne connection Γ. We write space-time as a triple {M,g, Γ}. The 16 components of the metric tensor g µν andthe64connectioncoefﬁcients Γ λµν aretheunknownsinourmodel,asis themanifold Mitself. This approach is known as the Einstein–Schr ¨odinger metric-afﬁne ﬁeld the- ory;see,forexample,AppendixIIin[1],or[2].Duringtheperiodfromthe1920s tothe1950smanymathematiciansandphysicistscontributedtothissubject,with the list of authors containing names such as M.Born, A.S.Eddington, L.Infeld, T.Levi-Civita and H.Weyl. In modern theoretical physics metric-afﬁne ﬁeld theo- riesarenotamainstreamsubject;reviewsofsomeofthemorerecentworkinthis area canbefound in[3], [4], [5], [6]. The immediate motivation for our paper comes from [7] where it was shown that it is possible to give a sensible tensor interpretation of the Dirac equation in ﬂat Minkowski 3-space by treating the electromagnetic ﬁeld as an afﬁne con- nection in the embedding Minkowski 4-space. The “electromagnetic” connection suggestedin[7] is the metric compatibleconnectioncorresponding to torsion T=e∗A whereeis the electron charge, Ais the (given) real-valued vector potential of the electromagnetic ﬁeld, and ∗is the Hodge star; here we use a system of units in whichboththespeedoflight candPlanck’sconstant /planckover2pi1havevalue1.Inparticular, suchaninterpretationofelectromagnetismresolvestheproblemof distinguishing ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.434 428 D.VASSILIEV the electron from the positron without resorting to “negative frequencies”. Re- garding the afﬁne connection itself as an unknown quantity is the next obvious step. Weconstructourmathematicalmodelforthe neutrinoasfollows. Firstly, weconsider theYang–Millsequationforthe afﬁneconnection: δ YM R= 0 (1) whereRis the Riemann curvature tensor (10) and δ YM is the divergence on curvatures (13). Secondly, weconsider theEinstein equation: Ric= 0 (2) whereRicis the Ricci curvature tensor. Equation (2) describes the absence of sources of gravitation. The objective of this paper is the study of the combined system (1), (2) which is a system of 80 real non-linear partial differential equations with 80 real unknownsg µν ,Γ λµν . In other words, we are combining the basic equation of relativistic quantum mechanics (Yang–Mills equation) with the basic equation of generalrelativity(Einsteinequation). R EMARK 4. If the metric is symmetric and the connection is that of Levi-Civita then(2)implies (1).Inthegeneralcase(1)and(2) are independent. We deﬁne Minkowski space M 4 as a real 4-manifold which admits a global coordinatesystem (x 0 ,x 1 ,x 2 ,x 3 )andisequippedwith the metric g µν = diag(+1,−1,−1,−1). (3) Our deﬁnition of M 4 speciﬁes two elements of the triple {M,g, Γ}, namely, the manifoldMand themetric g, but doesnotspecifythe connection Γ. Our mainresultis Theorem 50. Letube a complex-valued vector function which is a plane wave solution ofthe polarisedMaxwellequation ∗du=±idu (4) inM 4 .LetΓbethe metriccompatibleconnection corresponding totorsion T λµν = Re(u λ (du) µν ). (5) Then the space-time {M 4 ,Γ}is asolution of(1),(2). Note that the vector equation (4) forms the basis of the mathematical model in [7]. It is shown in [7] that under certain circumstances equation (4) produces effects normally attributed to spinors. kievarwe.tex; 12/03/2001; 3:49; p.435 FIELDMODELFOR NEUTRINO 429 Letusrewrite (4) as ∗du=iαdu, (6) α=±1. The non-trivial ( du/negationslash≡0) plane wave solutions of (6) can, of course, be writtendownexplicitly:up toaproper Lorentztransformation theyare u(x) =we −ik·x (7) where w µ =C(0,1,−iα,0), k µ =β(1,0,0,1), (8) β=±1,andC∈R + isan arbitraryconstant (amplitude). Substitution of(7) into(5)produces T λµν = Re(−iw λ (k∧w) µν e −2ik·x ). (9) Thus, the space-time in Theorem 50 is a wave of torsion which, up to a proper Lorentz transformation,isgivenbytheexplicitformulae (9), (8). The paper hasthefollowingstructure. InSection2 wespecify ournotation. Section 3 is a brief description of Yang–Mills theory in our particular setting (afﬁneconnectionovervectors). In Section 4 we prove Theorem 50. The crucial element of the proof is the linearisationansatz (17), (16). InSection5weestablishgeneralinvariantpropertiesofoursolutions(3)–(5). ItturnsoutthatourRiemanncurvaturetensorspossess allthesymmetryproperties ofthe“usual”curvaturetensorsgeneratedbyLevi-Civitaconnections.Thismeans that in observing such connections we might be led to believe (mistakenly) that we liveinaLevi-Civitauniverse. In Section 6 we show that the Riemann curvature tensors corresponding to our solutions (3)–(5) have an algebraic structure which makes them equivalent to bispinors. It turns out that these bispinors satisfy the Weyl equation (Dirac equation for massless particle), which justiﬁes our interpretation of space-times (3)–(5) as the neutrino and antineutrino. We show that our model explains the well known fact that neutrinos are always left-handed whereas antineutrinos are always right-handed. In Section 7 we compare our results with those of Einstein who suggested [8] a double duality equation as a possible model for elementary particles. We show thatourspace-times(3)–(5)satisfythisequation.Herethecrucialpointisthatwe getthesignpredicted by Einstein. In Section 8 we vary the Yang–Mills Lagrangian (12) with respect to the metricand show that our solutions (3)–(5) provide stationary points. This fact kievarwe.tex; 12/03/2001; 3:49; p.436 430 D.VASSILIEV is highly unusual and does not follow from abstract Yang–Mills theory which guarantees onlyconformal invariance. 2. Basic notation Wedenote∂ µ =∂/∂x µ anddeﬁnethecovariantderivativeofavectorfunctionas ∇ µ v λ :=∂ µ v λ + Γ λµν v ν .Wedeﬁnethetorsiontensoras T λµν := Γ λµν −Γ λνµ , theRiemanncurvaturetensoras R κλµν :=∂ µ Γ κνλ −∂ ν Γ κµλ + Γ κµη Γ ηνλ −Γ κνη Γ ηµλ ,(10) and theRiccicurvature tensoras Ric λν :=R κλκν . Wedeﬁnethecontravariantmetrictensorasthesolutionofthelinearalgebraic systemg µν g νκ =δ µκ .Wehavetotakegreatcarewhenraisingorloweringtensor indices because in our statement of the problem the metric is not assumed to be symmetric and the connection is not assumed to be metric compatible. Only when it is clear that we are in a situation when the metric is symmetric and the connection is metric compatible we gain the full freedom of writing any tensor with either upper or lower indices (in any combinations), the raising or lower- ing being achieved via contraction with the contravariant or covariant symmetric metric tensor. Givenascalar function fwe write forbrevity /integraldisplay f:= /integraldisplay M f /radicalBig |detg|dx 0 dx 1 dx 2 dx 3 ,detg:= det(g µν )/negationslash= 0. We deﬁne the Hodge star as (∗Q) µ q+1 ...µ 4 := (q!) −1 /radicalbig |detg|Q µ 1 ...µ q ε µ 1 ...µ 4 whereεis the totally antisymmetric quantity. We put ε 0123 :=±1, where +or− istakendependingonwhethertheorientationofthecoordinatesystemispositive ornegative, respectively. When dealing with a connection which is compatible with a given symmetric metricitisconvenienttointroducethe contortion tensorK λµν := Γ λµν − /braceleftBig λ µν /bracerightBig , where /braceleftBig λ µν /bracerightBig := 1 2 g λκ (∂ µ g νκ +∂ ν g µκ −∂ κ g µν )is the Christoffel symbol. Con- tortion has the antisymmetry property K λµν =−K νµλ .A symmetric metric andcontortionuniquelydeterminethemetriccompatibleconnection.Torsionand contortion are related as(see [9],formula (7.35)) T λµν =K λµν −K λνµ , K λµν = /parenleftbig T λµν +T µλν +T νλµ /parenrightbig /2.(11) A bispinor in M 4 is a column of four complex numbers /parenleftbig ξ 1 ξ 2 η ˙1 η ˙2 /parenrightbig T whichchangeunderLorentztransformationsinaparticularway,seeSections18, 19and26in[10]fordetails;amorecompactexpositionisgiveninthebeginning ofSection3in [11].The Pauliand Diracmatricesare I= /parenleftbigg 1 0 0 1 /parenrightbigg , σ 1 = /parenleftbigg 0 1 1 0 /parenrightbigg , σ 2 = /parenleftbigg 0−i i0 /parenrightbigg , σ 3 = /parenleftbigg 1 0 0−1 /parenrightbigg , kievarwe.tex; 12/03/2001; 3:49; p.437 FIELDMODELFOR NEUTRINO 431 γ 0 = /parenleftbigg 0I I0 /parenrightbigg , γ j = /parenleftbigg 0−σ j σ j 0 /parenrightbigg , γ 5 =−iγ 0 γ 1 γ 2 γ 3 = /parenleftbigg −I0 0I /parenrightbigg . 3. TheYang–Millsequation PutR κλρν :=g ρµ R κλµν whereR κλµν is the Riemann curvature tensor (10). The Yang–MillsLagrangian forthe afﬁneconnectionis L YM :=− 1 2 /integraldisplay R κλρν R λκνρ . (12) The Yang–Mills equation (1) is the Euler–Lagrange equation obtained from (12) by varying the connection coefﬁcients Γ λµν (but not the metric). The explicit formulaforthedifferentialoperator δ YM appearingin (1) is (δ YM R) ρ = 1 2 /radicalbig |detg|(∂ σ + [Γ σ ,·]) /parenleftbigg /radicalBig |detg|(g ρµ g νσ +g µρ g σν )R µν /parenrightbigg . (13) Inwriting(13)weusedmatrixnotationtohidetwoindices: R µν =R κλµν ,Γ σ = Γ κσλ , withκenumerating the rows and λthe columns. By [·,·]we denote the commutator,i.e., [L,N] τ λ :=L τκ N κλ −N τκ L κλ . Note that the operator (13) is invariant under the transposition of the metric, g µν →˜g µν :=g νµ . For more details concerning transposition invariance and its possible physicalsigniﬁcance see [1] p.142–143. From now on, until Section 8, we work only in Minkowski space and only with metric compatible connections. This leads to a number of simpliﬁcations. Connection coefﬁcients now coincide with contortion, for which we continue usingmatrixnotation K σ =K κσλ .Formula(10) becomes R µν =∂ µ K ν −∂ ν K µ + [K µ ,K ν ], (14) and theYang–Millsequation (1),(13) becomes (∂ ν + [K ν ,·])R µν = 0. (15) The Yang–Mills equation (15) appears to be overdetermined as it is a system of64equationswithonly24unknowns(24isthenumberofindependentcompo- nents of the contortion tensor). However 40 of the 64 equations are automatically fulﬁlled. This is a consequence of the fact that the Lie algebra of real antisym- metric rank 2 tensors is a subalgebra of the general Lie algebra of real rank 2 tensors. kievarwe.tex; 12/03/2001; 3:49; p.438 432 D.VASSILIEV 4. Proof ofTheorem50 The fundamental difﬁculty with the Yang–Mills equation (15) as well as with the Einstein equation (2) is that these equations are non-linear with respect to the un- knowncontortion K.Thefollowinglemmaplaysacrucialroleinourconstruction by allowingus toget rid ofthe non-linearities. Lemma51. LetLbeacomplex rank 2antisymmetrictensor satisfying ∗L=±iL. (16) Then [ReL,ImL] = 0. Proof.Theresult follows fromthegeneral formula [∗L,N] =∗[L,N]. Lemma 51 can be rephrased in the following way: the Lie algebra of real antisymmetricrank2tensorshas2-dimensionalabeliansubalgebraswhichcanbe explicitly describedinterms of theeigenvectorsof theHodge star. Lemma51immediatelyimpliesthe followinglinearisation ansatz. Corollary52. Supposecontortionis of the form K κνλ (x) = Re(L κλ v ν (x) ) (17) whereLis a constant complex antisymmetric tensor satisfying (16) and vis a complex-valued vector function. Then the non-linear terms in the formula for Riemann curvature(14)andin theYang–Millsequation (15) vanish. Substituting (17)into (14), (15)wereduce equations(1), (2) to δdv= 0, (18) L κλ (dv) κν = 0. (19) Heredis the exterior derivative and δis its adjoint, so that (18) is the Maxwell equation. Letuslookforplanewavesolutions, i.e., v(x) =−iwe −2ik·x (20) wherew/negationslash= 0is a constant complex vector and k/negationslash= 0is a constant real vector. Here we put the extra factor −iatwas well as the extra factor 2in the exponent forthesakeofconvenience;thereasonfordoingthisistoachieveagreementwith (9).Substituting(20) into(18), (19)weget k ν (k∧w) µν = 0, (21) kievarwe.tex; 12/03/2001; 3:49; p.439 FIELDMODELFOR NEUTRINO 433 L κλ (k∧w) κν = 0. (22) We have reduced our original system of partial differential equations (1), (2) to the purely algebraic problem (16), (21), (22). Straightforward analysis shows thatthespace-timesdescribedinTheorem50aresolutionsof(16),(21),(22),and, moreover,the onlynon-trivial( R/negationslash≡0) solutions. 5. Invariantproperties ofour solutions Itisknown[4],[5],[6]thatthe24-dimensionalspaceofrealtorsionsdecomposes intothefollowing3irreduciblesubspaces:tensortorsions,tracetorsions,andaxial torsions. Thedimensions are 16, 4, and4, respectively. Lemma53. The torsions inTheorem 50are purely tensor. Proof.Thetracecomponentofatorsiontensor T λµν iszeroiffT λλν = 0,and the axial component is zero iff T λµν ε λµνκ = 0. These identities are established by direct examinationof theexplicitformulae(9), (8). Letusmentionthe followingusefulgeneral result. Lemma54. Iftheaxialcomponentofatorsioniszerothenthistorsioncoincides, uptoanaturalreorderingofindices,withthecorresponding(see(11))contortion: T λµν =K µλν . Lemma54explainswhythetorsionofourspace-timeshasthesimplestructure (5), (4). Our linearisation ansatz (17), (16) required us to work with contortion rather than torsion, and in the end in order to calculate torsion we had to use the ﬁrst formula (11). We did not get a cumbersome expression for torsion only because its axialcomponentiszero. Lemma 55. The Riemann curvatures of space-times from Theorem 50 have all thesymmetry properties ofcurvatures in theLevi-Civitasetting,that is, R κλµν =−R λκµν =−R κλνµ =R µνκλ , (23) R κλµν ε κλµν = 0. (24) Proof.Letus deﬁnethecomplex Riemanncurvaturetensor CR κλµν :=F κλ F µν (25) where F:=du (26) kievarwe.tex; 12/03/2001; 3:49; p.440 434 D.VASSILIEV anduisfrom(6). Lemmas 53,54 andCorollary52imply R κλµν = Re(CR κλµν ). (27) Directexaminationof formulae(25)–(27),(7), (8)establishes(23), (24). 6. Weyl’sequation The torsions (and, therefore, space-times) from Theorem 50 are described, up to aproperLorentztransformationandascalingfactor C∈R + ,byapairofindices α,β=±1;see(9),(8).Itmayseemthatthisgivesus4essentiallydifferentspace- times.However,formula(9)containstheoperationoftakingtherealpartand,asa result,thetransformation {α,β}→{−α,−β}doesnotchangeourtorsion.Thus, Theorem50providesuswithonlytwoessentiallydifferentspace-timeslabeledby the indexτ:=αβ=±1. The purpose of this section is to show that it is natural tointerpretthesetwospace-times astheneutrinoand antineutrino. We base our interpretation on the analysis of the Riemann curvature tensor. Wechosetoanalysecurvatureratherthantorsionbecausecurvatureisanaccepted physical obervable. In our analysis of the Riemann curvature tensor we will work with the com- plex curvature (25) rather than the real curvature (27) because the complex one has a simpler structure. Indeed, according to formula (25) the complex Riemann curvaturetensor CRfactorizesasthesquareofarank2tensor Fandis,therefore, completely determined byit. Workingwiththerank2tensor Fismucheasierthanwiththeoriginalrank4 tensorCR,butonewouldliketosimplifytheanalysisevenfurtherbyfactorizing Fitself. It is impossible to factorize Fas the square of a vector but it is possible tofactorizeFas the squareofa bispinor. Lemma56. A complexrank 2antisymmetric tensor Fsatisfying F µν F µν = 0, (∗F) µν F µν = 0 (28) isequivalent to abispinor ψ, therelationship between the twobeing F µν =− i 4ψ T γ 0 γ 2 γ µ γ ν ψ. (29) Proof.Formula (29) is a special case of the general equivalence relation between rank 2 antisymmetric tensors and rank 2 symmetric bispinors, see end of Section 19 in [10]. Conditions (28) are necessary and sufﬁcient for the factorization of thesymmetricrank2spinors assquares ofrank 1spinors. kievarwe.tex; 12/03/2001; 3:49; p.441 FIELDMODELFOR NEUTRINO 435 R EMARK 5. The corresponding text in the end of Section 19 in [10] contains mistakes.These canbe corrected byreplacing everywhere iby−i. R EMARK 6. For a given tensor Fformula (29) deﬁnes the individual spinors ξ= /parenleftbig ξ 1 ξ 2 /parenrightbig T andη= /parenleftbig η ˙1 η ˙2 /parenrightbig T uniquely up to choice of sign. This is in agreementwiththegeneralfactthataspinordoesnothaveaspeciﬁcsign,seethe beginningofSection19in [10]. R EMARK 7. Conditions (28)are equivalent to detF= 0,det∗F= 0. R EMARK 8. Formula (29) is invariant under proper Lorentz transformations and space inversion, butnot undertimeinversion. Our particular tensor Fdeﬁned in accordance with formula (26) satisﬁes the conditions(28).Indeed, F µν F µν = 0isthestatementthatthecomplexscalarcur- vature is zero (consequence of the complex Ricci curvature being zero), whereas (∗F) µν F µν = 0is the statement that the complex Riemann curvature tensor CR satisﬁes thecyclic sumidentity, cf.(24). Thus, the complex Riemann curvature tensor (25) has an algebraic struc- ture which makes it equivalent to a bispinor. Direct calculations show that the corresponding bispinorfunction ψ(x)satisﬁes theWeyl equation γ µ ∂ µ ψ= 0 (30) as well astheadditionalcondition γ 5 ψ=−αψ (31) whereα=±1is from (6). Conversely, any plane wave solution of (30), (31) generatesa complexRiemann curvaturetensor ofthe type (25). A non-trivial ( ψ(x)/negationslash≡const) plane wave solution of (30), (31) can, up to a proper Lorentz transformation, be written as ψ(x) =ϕe − i 2 k·x whereϕis a constant bispinor and kis given by (8). Recall that the formula for kcontains the parameter β=±1which determines whether the wave vector klies on the forward(β= +1)orbackward ( β=−1) lightcone. Non-trivial plane wave solutions of (30), (31) with β= +1are called neutri- nos whereas those with β=−1are called antineutrinos. A neutrino is said to be left-handed if α=−1and right-handed if α= +1. An antineutrino is said to be left-handedif α= +1andright-handed if α=−1. R EMARK 9. Theabovedeﬁnitionsagreewiththeoperationofchargeconjugation (see formula (26.6) in [10]) in that the left-handed neutrino and left-handed an- tineutrino are charge conjugates of one another, as are the right-handed neutrino and right-handedantineutrino. As explained in the beginning of this section, the transformation {α,β}→ {−α,−β}doesnotchangetheresultingspace-time.Thismeansthatinourmodel kievarwe.tex; 12/03/2001; 3:49; p.442 436 D.VASSILIEV theleft-handedneutrinoisidenticaltotheleft-handedantineutrino,andtheright- handedneutrinois identical totheright-handedantineutrino. 7. Einstein’sdouble dualityequation Theonly apriorisymmetrypropertiesoftheRiemanncurvaturetensorgenerated by a connection compatiblewithasymmetricmetric are R κλµν =−R λκµν =−R κλνµ . (32) LetRbethe36-dimensionallinearspaceofrealrank4tensorssatisfying(32). We consider thefollowing twoendomorphisms in R: R→R T , (R T ) κλµν :=R µνκλ , (33) R→ ∗ R ∗ , ( ∗ R ∗ ) κλµν := (|detg|/4)ε κ /prime λ /prime κλ R κ /prime λ /prime µ /prime ν /prime ε µ /prime ν /prime µν .(34) R EMARK 10. It is easy to see that the endomorphisms (33), (34) are well de- ﬁned even if the manifold is not orientable. In the case of (34) this observation is a consequence of a much deeper fact established in [12]: the rank 8 tensor (detg)ε κ /prime λ /prime κλ ε µ /prime ν /prime µν is a purely metrical quantity in that it is expressed via the metric tensor.Thisis aspecial featureofdimension 4. The endomorphisms (33), (34) have the following properties: (i) they com- mute, (ii) their eigenvalues are ±1, (iii) they have no associated eigenvectors. Therefore,Rdecomposes intoa directsum of4 invariant subspaces R=⊕ a,b=± R ab ,R ab :={R∈R|R T =aR, ∗ R ∗ =bR}.(35) The decomposition (35) was suggested in [13] and developed in [8], [12]. Actually,thepapers[13],[8],[12]dealonlywiththecaseofaLevi-Civitaconnec- tion,butthegeneralizationtothecaseofanarbitraryafﬁneconnectioncompatible with a symmetric metric is straightforward. Lanczos called tensors R∈Rself- dual (respectively, antidual) if ∗ R ∗ =−R(respectively, ∗ R ∗ =R). Such a choice of terminology is due to the fact that Einstein and Lanczos deﬁned their double dualityendomorphismas R→(sgn detg) ∗ R ∗ . (36) Theadvantageof(36)isthatthislinearoperatorisexpressedviathemetrictensor as a rational function. The endomorphism (36) is, in a sense, even more invariant than(34)as itdoes not “feel”thesignatureofthe metric. Lemma57. (Rainich[13]) ThesubspacesR ++ andR +− havedimensions9and 12, respectively. kievarwe.tex; 12/03/2001; 3:49; p.443 FIELDMODELFOR NEUTRINO 437 R EMARK 11. In Rainich’s paper the dimensions are actually given as 9 and 11. The reason behind this is that Rainich imposed on curvatures the cyclic sum con- dition (24). This excludes from R +− curvatures of the type R κλµν =ε κλµν and, therefore,reduces thedimension by1. Lemma 58. (Einstein [8]) LetR∈R ++ . Then the corresponding Ricci tensor is symmetric and trace free. Moreover, Ris uniquely determined by its Ricci tensor andthemetric tensoraccording to theformula R κλµν = (g κµ Ric λν +g λν Ric κµ −g κν Ric λµ −g λµ Ric κν )/2.(37) Einstein’s goal in [8] was to construct a mathematical model for the electron; note that this paper was published a year before Dirac discovered his equation. Einstein argued that the Riemann curvature tensor of the electron should lie in an eigenspace of the endomorphism (34). As in this particular paper Einstein restricted his analysis to the case of a Levi-Civita connection he had to make the choice between the invariant subspaces R ++ andR +− . The difference between thesetwoinvariantsubspacesisfundamental:ithasnothingtodowiththechoice offorwardandbackwardlightconesorthechoiceoforientationofthecoordinate system, and, as a consequence, it has nothing to do with the notions of “particle” and “antiparticle”orthe notionsof“left-handedness”and “right-handedness”. Lemmas 57 and 58 led Einstein to the conclusion that curvatures from R ++ aretootrivialandthedimensionofthesubspacetoolow(9insteadoftheexpected 10 which is the number of independent components of the energy–momentum tensor)toassociateitwiththeelectron.Einstein’sconjecturewasthattheRiemann curvature tensor of the electron should lie in the invariant subspace R +− , that is, it shouldsatisfy the equation ∗ R ∗ =−R. (38) Formulae(25)–(27), (4)imply that ourspace-times (3)–(5) satisfy(38). Our paper falls short of constructing an afﬁne ﬁeld model for the electron. Nevertheless, we ﬁnd it encouraging that our afﬁne ﬁeld model for the neutrino agreeswith Einstein’s doubleduality equation(38). 8. Variationof the metric Variation of the Yang–Mills Lagrangian (12) with respect to the metric produces thefollowingEuler–Lagrangeequation: H−(trH/4)g= 0 (39) whereH µσ :=R κλµν g νρ R λκρσ ,trH:=H µσ g σµ . In deriving (39) we did not makeanyassumptions onthe symmetryofthemetric. Note the fundamental difference between our original equations (1), (2) and equation (39):(1), (2)are linear incurvature, whereas (39) isquadratic. kievarwe.tex; 12/03/2001; 3:49; p.444 438 D.VASSILIEV Lemma 59. Let the metric be symmetric and Lorentzian, and let Rbe of the form (27) where CRis a complex rank 4 tensor which factorises as the product of antisymmetric rank 2 tensors, CR κλµν =F κλ G µν , such that∗F=iαF, ∗G=iα /prime G,α,α /prime =±1. ThenRsatisﬁestheequation(39). Proof.TheLemma is proved bya straightforward Maple TM calculation. Lemma59and formulae (25)–(27),(4) immediatelyimply Corollary 60. Our space-times (3)–(5) provide stationary points of the Yang– Mills Lagrangian (12)with respecttothevariation of the metric. InordertoillustratehowunusualCorollary60isletusexaminewhathappens in the case of the Maxwell equation, which is the simplest example of a Yang– Millsequation.StraightforwardcalculationsshowthattheMaxwellequationona Lorentzian manifold does not have nontrivial solutions which provide stationary points oftheMaxwellLagrangianwith respecttothe variationof the metric. We see that afﬁne connections are very special in that they produce effects which arenot manifestin theabstract Yang–Mills theory. Acknowledgements The author is indebted to D. V. Alekseevsky, F. E. Burstall and A. D. King for stimulating discussions. The author’s research was supported by a Leverhulme Fellowship. References 1. A.Einstein, The meaning ofrelativity , 6th edition, Methuen &Co, London, 1960. 2. E. Schr ¨odinger,Space-time structure , Cambridge University Press, 1985. 3. E. W. Mielke, Geometrodynamics ofgauge ﬁelds , Akademie-Verlag, Berlin, 1987. 4. F. W. Hehl, J. D.McCrea, E. W. Mielke, and Y. Ne’eman, Metric-afﬁne gauge theory of gravity:ﬁeldequations,Noetheridentities,worldspinorsandbreakingofdilationinvariance , PhysicsReports 258(1995), 1–171. 5. F. Gronwald, Metric-afﬁne gauge theory of gravity I. Fundamental structure and ﬁeld equations , International Journal of Modern PhysicsD 6(1997), 263–303. 6. F. W. Hehl and A. Macias, Metric-afﬁne gauge theory of gravity II. Exact solutions , International Journal ofModernPhysics D 8(1999),399–416. 7. D. Vassiliev, A tensor interpretation of the 2D Dirac equation , preprint , http://xxx.lanl.gov/abs/math-ph/0006019 , 2000. 8. A. Einstein, ¨Uber die formale Beziehung des Riemannschen Kr ¨ummungstensors zu den Feldgleichungen der Gravitation , MathematischeAnnalen 97(1927), 99–103. 9. M. Nakahara, Geometry, Topology and Physics , Institute of Physics,Bristol, 1998. 10. V. B. Berestetskii, E. M. Lifshitz and L. P. Pitaevskii, Quantum Electrodynamics , Course of Theoretical Physics vol. 4, 2ndEdition, Pergamon Press, Oxford, 1982. kievarwe.tex; 12/03/2001; 3:49; p.445 FIELDMODELFOR NEUTRINO 439 11. D. Elton and D. Vassiliev, The Dirac equation without spinors , in Rostock Conference on Functional Analysis, Partial Differential Equations and Applications (J.Rossmann, P.Tak ´ac, and G.Wildenhain eds.), series Operator Theory: Advances and Applications vol. 110, Birkh¨auser Verlag, Basel,1999,133–152. 12. C. Lanczos, The splitting of the Riemann tensor , Reviews of Modern Physics 34(1962), 379–389. 13. G.Y. Rainich, Electricity in curved space-time , Nature115(1925),498. kievarwe.tex; 12/03/2001; 3:49; p.446 kievarwe.tex; 12/03/2001; 3:49; p.447 GENERALIZEDTAUB-NUTMETRICSANDKILLING-YANOTENSORS MIHAIVISINESCU ∗ Department ofTheoretical Physics, National Institute for Physicsand Nuclear Engineering, P.O.Box M.G.-6,Magurele,Bucharest,Romania Abstract. The relation between ”hidden” symmetries encapsulated in the St ¨ackel-Killing tensors and the Killing-Yano tensors is investigated. A necessary condition that a St ¨ackel-Killing tensor of valence2bethecontractedproductofaKilling-Yanotensorofvalence2withitselfisre-derivedfor a Riemannian manifold. This condition is applied to the generalized Euclidean Taub-NUT metrics whichadmitaKeplertypesymmetry.ItisshownthatingeneraltheSt ¨ackel-Killingtensorsinvolved in the Runge-Lenz vector cannot be expressed as a product of Killing-Yano tensors. The only exceptionis the original Taub-NUT metric. 1. Introduction It is known that spacetime isometries give rise to constants of motion along geodesics. However not all conserved quantities along geodesics arise from isometries of the manifold and associated Killing vector ﬁelds. Such integrals of motion are related to ”hidden” symmetries of the manifold encapsulated in the St¨ackel-Killing tensors. A St¨ackel-Killing tensor of valence ris a tensorK µ 1 ...µ r which is completely symmetric and whichsatisﬁesa generalizedKilling equation K (µ 1 ...µ r ;λ) = 0. (1) On manifolds like the four-dimensional Kerr-Newman and Taub-NUT man- ifolds, the geodesic equations are integrable because of the existence of a St¨ackel-Killingtensor K µν ofvalence 2[1]allowingtheconstructionofaconstant ofmotionquadraticinparticle’sfour-momentum p µ : k= 1 2K µν (x)p µ p ν = 1 2K µν (x) ˙x µ ˙x ν (2) wherethe overdotdenotes ordinary proper-timedifferentiation d dτ . ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.448 442 M.VISINESCU The Killing condition (1) is actually equivalent with the conservation of K, i.e.Kcommuteswiththeworld-line Hamiltonian H= 1 2g µν p µ p ν (3) inthesense ofPoissonbrackets. Related to this, the Klein-Gordon, Schr ¨odinger and Dirac equations are separablein Kerr-Newman[2,3]and Taub-NUT spaces[4,5]. Moreover Carter and McLenagham [6] showed the existence of a Dirac-type linear differential operator which commutes with the standard Dirac operator in the Kerr-Newman space. The construction of this operator depends upon the re- markablefactthattheSt ¨ackel-KillingtensoroftheKerr-Newmangeometryhasa certainroot K µν =f µλ f λ ν (4) wheref µν is a Killing-Yano tensor. A tensor f µ 1 ...µ r is called a Killing-Yano tensorof valence r[7]if it istotally antisymmetric andit satisﬁesthe equation f µ 1 ...(µ r ;λ) = 0. (5) The role of the Killing-Yano tensors can also be noticed for spinning man- ifolds [8, 9]. The conﬁguration space of spinning particles (spinning space) is an extension of an ordinary Riemannian manifold, parametrized by local coordi- nates{x µ }, to a graded manifold parametrized by local coordinates {x µ ,ψ µ }, with the ﬁrst set of variables being Grassmann-even (commuting) and the second set Grassmann-odd (anticommuting). The equations of motion of the pseudo-classical Dirac particlecan be derived fromthe action S= /integraldisplay b a dτ /parenleftbigg 1 2g µν (x) ˙x µ ˙x ν + i 2g µν (x)ψ µ Dψ ν Dτ /parenrightbigg . (6) where the covariant derivative of the Grassmann-valued spin variable ψ µ is deﬁned by Dψ µ Dτ=˙ψ µ + ˙x λ Γ µ λν ψ ν . (7) The action(6) is invariantunder the supersymmetry δx µ =−i/epsilon1ψ µ , δψ µ = ˙x µ /epsilon1 (8) wherethe inﬁnitesimalparameter /epsilon1ofthe transformation is Grassmann-odd. This supersymmetry transformation are obtained from the conserved super- charge Q= ˙x µ ψ µ ,˙Q= 0 (9) kievarwe.tex; 12/03/2001; 3:49; p.449 GENERALIZEDTAUB-NUTMETRICS 443 by takingthe bracket δF=i/epsilon1{Q,F}. (10) ThatQisconservedandtheabovesupertransformationrepresentasymmetry followsfromthebracket relations {Q,Q}−− 2iH ,{Q,H}= 0. (11) Additional conserved supercharges exist if the background geometry admits a Killing-Yano tensor f µ 1 ...µ r . In such a geometry there exist an additional superinvariant constant of motion Q f deﬁnedby [10] Q f =f µ 1 ...µ r Π µ 1 ψ µ 2 ...ψ µ r + i r+ 1(−1) r+1 f [µ 1 ...µ r ;µ r+1 ] ·ψ µ 1 ...ψ µ r+1 . (12) which issuperinvariant {Q f ,Q}= 0. (13) The existence of a new supersymmetry of this kind implies automatically the existence of a new Grassmann-even constant of motion Zdeﬁned by the bracket ofQ f withitself {Q f ,Q f }=−2iZ. (14) The explicitformof Zisgivenin[9]forKilling-Yanotensors ofvalence 2. This paper is devoted to the relations between the St ¨ackel-Killing and the Killing-Yano tensors for a 4-dimensional Riemannian manifold. The general re- sultsareappliedtothecaseofthegeneralizedEuclideanTaub-NUTmetricswhich admitaKepler-typesymmetry[11]. The Euclidean Taub-NUT metric is involved in many modern studies in phy- sics.Hawking[12]hassuggestedthattheEuclideanTaub-NUTmetricmightgive risetothegravitationalanalogoftheYang-Millsinstanton.InthiscaseEinstein’s equations are satisﬁed with zero cosmological constant and the manifold is R 4 withaboundarywhichisatwistedthree-sphere S 3 possessingadistortedmetric. The Kaluza-Klein monopole was obtained by embedding the Taub-NUT gravita- tional instanton into ﬁve-dimensional Kaluza-Klein theory. On the other hand, in thelong-distancelimit,neglectingradiation,therelativemotionoftwomonopoles isdescribed bythe geodesicsof thisspace[13]. From the symmetry viewpoint, the geodesic motion in Taub-NUT space ad- mits a “hidden” symmetry of the Kepler type if a cyclic variable is gotten rid of [14]. Moreover in the Taub-NUT geometry there are four Killing-Yano tensors [7]. Three of these are complex structure realizing the quaternionic algebra and kievarwe.tex; 12/03/2001; 3:49; p.450 444 M.VISINESCU the Taub-NUT manifold is hyper-K ¨ahler [14]. In addition to these three vector- like Killing-Yano tensors, there is a scalar one which has a non-vanishing ﬁeld strength andit exists byvirtue ofthemetricbeing type D. For the geodesic motions in the Taub-NUT space, the conserved vector anal- ogous to the Runge-Lenz vector of the Kepler type problem is quadratic in 4-velocities,itscomponentsareSt ¨ackel-Killingtensorsandtheycanbeexpressed as symmetrized productsofKilling-Yano tensors[14–16,10, 17]. In the last time, Iwai and Katayama [18–20] extended the Taub-NUT metric so that it still admits a Kepler-type symmetry. This class of metrics, of course, includestheoriginalTaub-NUT metric. In what follows we investigate if the St ¨ackel-Killing tensors involved in the conserved Runge-Lenz vector of the extended Taub-NUT metrics can also be expressedinterms ofKilling-Yanotensors. The relationship between Killing tensors and Killing-Yano tensors has been studied to the purpose of the Lorentzian geometry used in general relativity [21, 22]. In the next section we re-examine the conditions that a Killing tensor of valence 2be the contracted product of a Killing-Yano tensor of valence 2with itself. The procedure is quite simple and devoted to the Riemannian geometry appropriateto EuclideanTaub-NUT metrics. InSection3weshowthatingeneraltheKillingtensorsinvolvedintheRunge- Lenz vector cannot be expressed as a product of Killing-Yano tensors. The only exceptionisthe original Taub-NUTmetric. Our commentsandconcluding remarksare presentedin Section4. 2. Therelationshipbetween Killingtensors and Killing-Yano tensors We consider a 4−dimensional Riemannian manifold Mand a metric g µν (x)on Min local coordinates x µ . We write the metric in terms of the local orthonormal vierbeinframe e a µ ds 2 =g µν (x)dx µ dx ν = /summationdisplay a=0,1,2,3 (e a ) 2 (15) wheree a =e a µ dx µ . Greek indices µ,ν,...are raised and lowered with g µν or its inverseg µν , while Latin indices a,b,...are raised and lowered by the ﬂat metric δ ab ,a,b = 0,1,2,3.Vierbeinsandinversevierbeinsinter-convertLatinandGreek indiceswhennecessary. LetΛ 2 bethespaceoftwo-forms Λ 2 := Λ 2 T ∗ (R 4 −{0}).Wedeﬁneself-dual and anti-selfdualbases for Λ 2 usingthevierbein one-forms e a : basisof Λ 2 ± =    λ 1 ± =e 0 ∧e 1 ±e 2 ∧e 3 λ 2 ± =e 0 ∧e 2 ±e 3 ∧e 1 ,∗λ i ± =±λ i ± λ 3 ± =e 0 ∧e 3 ±e 1 ∧e 2 (16) kievarwe.tex; 12/03/2001; 3:49; p.451 GENERALIZEDTAUB-NUTMETRICS 445 Letfbe a Killing-Yano tensor of valence 2 and ∗fits dual. The symmetric combination of fand∗fisaself-dualtwo-form f+∗f= /summationdisplay i=1,2,3 y i λ i + (17) whiletheir difference isananti-self-dual two-form f−∗f= /summationdisplay i=1,2,3 z i λ i − . (18) Anexplicit evaluationshowsthat (f+∗f) 2 =− /summationdisplay i=1,2,3 (y i ) 2 · 1 1, (19) (f−∗f) 2 =− /summationdisplay i=1,2,3 (z i ) 2 · 1 1 (20) where 1 1 is4×4identity matrix. Let us suppose that a St ¨ackel-Killing tensor K µν can be written as the contractedproduct of aKilling-Yanotensor f µν with itself: K µν =f µλ ·f λ ν = (f 2 ) µν , µ,ν = 0,1,2,3. (21) We infer from thelastequationsthat: K+ 1 16 /bracketleftBigg/summationdisplay i (y 2 i −z 2 i ) /bracketrightBigg 2 K −1 + 1 2 /summationdisplay i (y 2 i +z 2 i )· 1 1= 0.(22) OntheotherhandtheKillingtensor Kissymmetricanditcanbediagonalized with the aid of an orthogonal matrix. Its eigenvalues satisfy an equation of the seconddegree: λ 2 α + 1 2 /summationdisplay i (y 2 i +z 2 i )λ α + 1 16 /bracketleftBigg/summationdisplay i (y 2 i −z 2 i ) /bracketrightBigg 2 = 0 (23) withatmosttwo distinctroots. In conclusion a St ¨ackel-Killing tensor Kwhich can be written as the square ofaKilling-Yanotensorhas at themosttwo distincteigenvalues. kievarwe.tex; 12/03/2001; 3:49; p.452 446 M.VISINESCU 3. GeneralizedTaub-NUT metrics For a special choice of coordinates the generalized Euclidean Taub-NUT metric considered byIwai andKatayama[18–20]takesthe form: ds 2 G =f(r)[dr 2 +r 2 dθ 2 +r 2 sin 2 θdϕ 2 ] +g(r)[dχ+ cosθdϕ] 2 (24) wherer > 0is the radial coordinate of R 4 −{0}, the angle variables (θ,ϕ,χ ),(0≤θ < π, 0≤ϕ < 2π,0≤χ < 4π)parameterize the unit sphere S 3 ,andf(r)andg(r)arearbitraryfunctionsof r. Wedecomposethemetric(24)intotheorthogonal vierbeinbasis: e 0 =g(r) 1 2 (dχ+ cosθdϕ), e 1 =rf(r) 1 2 (sinχdθ−sinθcosχdϕ), e 2 =rf(r) 1 2 (−cosχdθ−sinθsinχdϕ), e 3 =f(r) 1 2 dr. (25) Spaceswithametricoftheformabovehaveanisometrygroup SU(2)×U(1). TherearefourKillingvectors D A =R µ A ∂ µ ,A= 0,1,2,3, (26) corresponding to the invariance of the metric (24) under spatial rotations ( A= 1,2,3)andχtranslations( A= 0). LetusconsidergeodesicﬂowsofthegeneralizedTaub-NUTmetricwhichhas theLagrangian Lonthe tangentbundle T(R 4 −{0}) L= 1 2f(r)[ ˙r 2 +r 2 (˙θ 2 + sin 2 θ˙ϕ 2 )] + 1 2g(r)( ˙χ+ cosθ˙ϕ) 2 (27) where ( ˙r,˙θ,˙ϕ,˙χ,r,θ,ϕ,χ )standforcoordinatesinthetangentbundle.Since χis acyclicvariable q=g(r)(˙θ+ cosθ˙ϕ) (28) is a conserved quantity. This is known in the literature as the “relative electric charge”. Takingintoaccountthiscyclicvariable,thedynamicalsystemforthegeodesic ﬂow onT(R 4 −{0})can be reduced to a system on T(R 3 −{0}). The reduced system admits manifest rotational invariance, and hence has a conserved angular momentum: → J= → r× → p+q → r r(29) kievarwe.tex; 12/03/2001; 3:49; p.453 GENERALIZEDTAUB-NUTMETRICS 447 where → rdenotes the three-vector → r= (r,θ,ϕ )and → p=f(r)˙ → ris the mechanical momentum. Iff(r)andg(r)are takentobe f(r) =4m+ r r, g(r) =16m 2 r 4m+r(30) the metricds 2 G becomes the original Euclidean Taub-NUT metric. The parameter mcan be positive or negative, depending on the application; for m> 0the four- dimensional Taub-NUT metric represents a non-singular solution of the self-dual EuclideanEinsteinequationandassuchisinterpretedasagravitationalinstanton. As observed in [14], the Taub-NUT geometry also possesses four Killing- Yano tensors of valence 2. The ﬁrst three are rather special: they are covariantly constant(withvanishing ﬁeld strength) f i = 8m(dχ+ cosθdϕ)∧dx i −/epsilon1 ijk (1 +4 m r)dx j ∧dx k , D µ f ν iλ = 0, i,j,k = 1,2,3. (31) Theyare mutually anticommuting andsquare the minus unity: f i f j +f j f i =−2δ ij . (32) Thustheyarecomplexstructuresrealizingthequaternionalgebra.Indeed,the Taub-NUTmanifolddeﬁnedby (24)and(30) ishyper-K ¨ahler. In addition to the above vector-like Killing-Yano tensors there also is a scalar one f Y = 8m(dχ+ cosθdϕ)∧dr+ 4r(r+ 2m)(1 + r 4m) sinθdθ∧dϕ(33) which hasanon-vanishing component oftheﬁeldstrength f Yrθ;ϕ = 2(1 + r 4m)rsinθ. (34) In the original Taub-NUT case there is a conserved vector analogous to the Runge-Lenzvector ofthe Kepler-type problem: → K= 1 2 → K µν ˙x µ ˙x ν = → p× → j+ /parenleftBigg q 2 4m−4mE /parenrightBigg → r r(35) wherethe conservedenergy E,fromeq. (3),is E= → p 2 2f(r)+q 2 2g(r). (36) kievarwe.tex; 12/03/2001; 3:49; p.454 448 M.VISINESCU The components K iµν involved with the Runge-Lenz type vector (35) are Killing tensors and they can be expressed as symmetrized products of the Killing-Yanotensors f i (31)andf Y (33)[16, 10]: K iµν − 1 8m(R 0µ R iν +R 0ν R iµ ) =m /parenleftBig f Yµλ f iλ ν +f Yνλ f iλ µ /parenrightBig .(37) Returning to the generalized Taub-NUT metric, on the analogy of eq. (35), Iwai and Katayama [18–20] assumed that in addition to the angular momentum vector thereexist aconserved vector → Sof thefollowing form: → S= → p× → J+κ → r r(38) withanunknownconstant κ. It was found that the metric (24) still admits a Kepler type symmetry (38) if thefunctions f(r)andg(r)take,respectively, theform f(r) =a+ br r, g(r) =ar+br 2 1 +cr+dr 2 (39) wherea,b,c,dare constants. The constant κinvolved in the Runge-Lenz vector (38) is κ=−aE+ 1 2cq 2 . (40) Ifab > 0andc 2 −4d < 0orc > 0,d > 0, no singularity of the metric appears inR 4 −{0}. On the other hand, if ab< 0a manifest singularity appears atr=−a/b[19]. Itisstraightforwardtoverifythatthecomponentsofthevector → SareSt¨ackel- Killing tensors in the extended Taub-NUT space (24) with the function f(r)and g(r)given by (39). Moreover the Poisson brackets between the components of → J and → Sare[18]: {J i ,J j }=/epsilon1 ijk J k , {J i ,S j }=/epsilon1 ijk S k , {S i ,S j }= (dq 2 −2bE)/epsilon1 ijk J k (41) asitisexpectedfromthesamerelationsknownfortheoriginalTaub-NUTmetric. OurtaskistoinvestigateifthecomponentsoftheRunge-Lenzvector(38)can be the contracted product of Killing-Yano tensors of valence 2. On the model of eq.(37) from the original Taub-NUT case it is not required that a component S i of the Runge-Lenz vector (38) to be directly expressed as a symmetrized product of Killing-Yano tensors. Taking into account that → Stransforms as a vector under kievarwe.tex; 12/03/2001; 3:49; p.455 GENERALIZEDTAUB-NUTMETRICS 449 rotationsgeneratedby → J,eq.(41),thecomponents S iµν canbecombinedwithtriv- ialSt¨ackel-Killingtensorsoftheform (R 0µ R iν +R 0ν R iµ )togettheappropriate tensorwhich hastobedecomposedinaproduct ofKilling-Yanotensors. In order to use the results from the previous section, we shall write the sym- metrized product of two different Killing-Yano tensors f /prime andf /prime/prime as a contracted product off /prime +f /prime/prime with itself, extracting adequately the contribution of f /prime2 and f /prime/prime2 . Since the generalized Taub-NUT space (24) does not admit any other non- trivial St¨ackel-Killing tensor besides the metric g µν and the components S iµν of (38),f /prime2 andf /prime/prime2 should be connected with the scalar conserved quantities E, → J 2 ,q 2 throughthetensors g µν , /summationtext A=1,2,3 R Aµ R Aν andR 0µ R 0ν . In conclusion we shall consider a general linear combination between a com- ponentS i oftheRunge-Lenzvector(38)andsymmetrizedpairsofKillingvectors oftheform S iab +α 13 /summationdisplay A=1 R Aa R Ab +α 2 R 0a R 0b +α 3 (R 0a R ib +R ia R 0b )(42) whereα i are constants. We are looking for the conditions the above tensor be the contracted product of a Killing-Yano tensor with itself. For this purpose we evaluate the eigenvalues of the matrix (42) and we get that it has at the most two distincteigenvaluesifand only if α 1 +α 2 = 0, α 3 =− c 4, d=c 2 4. (43) Hence the constants involved in the functions f,gare constrained, restricting accordingly their expressions. It is worth to mention that if relation (43) between the constants canddis satisﬁed, the metric is conformally self-dual or anti-self- dualdepending uponthesign ofthe quantity 2 +cr[19]. Finally the condition stated for a St ¨ackel-Killing tensor to be written as the square of a skew symmetric tensor in the form (21) must be supplemented with eq.(5) which deﬁnes a Killing-Yano tensor. To verify this last condition we shall usetheNewman-PenroseformalismforEuclideansignature[23].Weintroducea tetradwhichwillbegivenasanisotropiccomplexdyaddeﬁnedbythevectors l,m togetherwith theircomplexconjugatessubject tothe normalization conditions l µ ¯l µ = 1, m µ ¯m µ = 1 (44) withall othersvanishing andthemetricisexpressedin theform ds 2 =l⊗¯l+¯l⊗l+m⊗¯m+ ¯m⊗m. (45) kievarwe.tex; 12/03/2001; 3:49; p.456 450 M.VISINESCU For a St¨ackel-Killing tensor Kwith two distinct eigenvalues one can choose thetetrad insuchthat K µν = 2λ 2 1 l (µ ¯l ν) + 2λ 2 2 m (µ ¯m ν) . (46) The skew symmetrictensor f µν which enterdecomposition (21) has theform f µν = 2λ 1 l [µ ¯l ν] + 2λ 2 m [µ ¯m ν] . (47) Again, a standard evaluation shows that the above quantity is a Killing-Yano tensoronlyif c=2 b a. (48) With this constraint, together with (43), the extended metric (24) coincides, up toaconstantfactor, withtheoriginal Taub-NUTmetric on setting a/b= 4m. 4. Concludingremarks The aim of this paper is to show that the extensions of the Taub-NUT geometry do notadmit aKilling-Yano tensor,even ifthey possessSt ¨ackel-Killing tensors. This result is not unexpected. The conserved quantities K iµν which enter eq.(37) are the components of the Runge-Lenz vector → Kgiven in (35). In the original Taub-NUT case these components K iµν are related to the symmetrized products between the Killing-Yano tensors f i (31) andf Y (33). Adequately the threeKilling-Yanotensors f i transformasvectorsunderrotationsgeneratedby → J liketheRunge-Lenz vector(41),while f Y isa scalar. The extended Taub-NUT metrics are not Ricci ﬂat and, consequently, not hyper-K¨ahler. On the other hand the existence of the Killing-Yano tensors f i is correlatedwith the hyper-K ¨ahler,self-dualstructure of themetric. The non-existence of the Killing-Yano tensors makes the study of ”hidden” symmetries more laborious in models of relativistic particles with spin involving anticommuting vectorial degrees of freedom. In general the conserved quantities from the scalar case receive a spin contribution involving an even number of Grassmann variables ψ µ . For example, starting with a Killing vector K µ , the conserved quantity inthe spinningcaseis J(x,˙x,ψ) =K µ ˙x µ + i 2K [µ;ν] ψ µ ψ ν . (49) The ﬁrst term in the r.h.s. is the conserved quantity in the scalar case, while the lasttermrepresents thecontributionof the spin. The generalized Killing equations on spinning spaces in the presence of a St¨ackel-Killing tensor are more involved. Unfortunately it is not possible to write kievarwe.tex; 12/03/2001; 3:49; p.457 GENERALIZEDTAUB-NUTMETRICS 451 closed, analytic expressions of the solutions of these equations using directly the components of the St ¨ackel-Killing tensors. However, assuming that the St ¨ackel- Killing tensors can be written as symmetrized products of pairs of Killing-Yano tensors,theevaluation ofthespincorrections isfeasible [9,16, 10,17]. IftheKilling-Yanotensorsaremissing,totakeupthequestionoftheexistence of extra supersymmetries and the relation with the constants of motion we are forced to enlarge the approach to Killing equations (5), (1). In fact, in ref.[9], supersymmetries are shown to depend on the existence of a tensor ﬁeld f µν sat- isfying eq.(5) which will be referred to as the f-symbol. The general conditions for constants of motion were derived, and it was shown that one can have new supercharges which do not commute with the original supercharge Q(9) if one allows thef-symbols to have a symmetric part. It was shown that in this case theantisymmetricpartdoesnotsatisfytheKilling-Yanocondition(5).Weshould like to remark that the general conditions of ref.[9] allow more possibilities than Killing-Yanotensorsforthe constructionofsupercharges. Summing up, we believe that the relation between the f-symbols and the Killing-Yano tensors could be fruitful and that it should deserve further studies. An analysis of the f-symbols in the generalized Taub-NUT geometry is under way. Acknowledgements IshouldliketoacknowledgethegenerosityofNATOinitssupportforthiswork- shop. 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Nutku, Gravitational instantons admit hyper-K ¨ahler structure , Class.Quant.Grav. 16(1999)189-210. kievarwe.tex; 12/03/2001; 3:49; p.459 ANEFFECTIVE MODEL OFTHESPACETIME FOAM VLADIMIR DZHUNUSHALIEV ∗ Kyrgyz-RussianSlavicUniversity,Bishkek, Kyrgyzstan 1. Introduction The notion of a spacetime foam was introduced by Wheeler [1, 2] for the de- scription of the possible complex structure of the spacetime on the Planck scale (L Pl ≈10 −33 cm). This hypothesized spacetime foam is a set of quantum worm- holes (WH) (handles) appearing in the spacetime on the Planck scale level (see Fig.1). For the macroscopic observer these quantum ﬂuctuations are smoothed and we have an ordinary smooth manifold with the metric submitting to Einstein equations.Theexactmathematicaldescriptionofthisphenomenonisverydifﬁcult and even though there is a doubt: does the Feynman path integral in the gravity contain a topology change of the spacetime ? This question spring up because ∗ [email protected] +- +- +- 5D quantum handles mouthes +-schematic designation for quantum handles Figure14. Attheleftsideoftheﬁgureispresentedahypothesizedspacetimefoam.Ifweneglect of the cross section of handle then (at the right) hand we have a schematic designation for the spacetime foam. kievarwe.tex; 12/03/2001; 3:49; p.460 454 V. DZHUNUSHALIEV 2 3 3 1 4 Figure 15. Here whole spacetime is 5D but in the external spacetime (3) G 55 is nonvariable and we have Kaluza-Klein theory in its initial interpretation as 4D gravity + electromagnetism. In the throat (2)G 55 component of the 5D metric is equivalent to 4D gravity+electromagnetism+scalar ﬁeld. Near the event horizon (4) the metric is the Reissner-Nordstrom metric and the throat is a solution of the 5D Kaluza-Klein theory. We should join these metrics on the event horizons. (1) is the forceline of theelectric ﬁeld. (according to the Morse theory) the singular points must arise by the topology change. In such points the time arrow is undeﬁned that leads in difﬁculties at deﬁnition of the Lorentzian metric, curvature tensor and so on. The main goal of thispaperis tosubmit an effectivemodel ofthespacetime foam. 2. Model ofa singlequantumwormhole At ﬁrst we present a model of a single handle in the spacetime foam, see Fig(2). The 5Dmetric[3–5]for thethroatis ds 2 =η AB ω A ω B = −r 2 0 ∆(r)(dχ−ω(r)dt) 2 + ∆(r)dt 2 −dr 2 − a(r) /parenleftBig dθ 2 + sinθ 2 dϕ 2 /parenrightBig , (1) a=r 2 0 +r 2 ,∆ =±2r 0 qr 2 +r 2 0 r 2 −r 2 0 , ω=±4r 2 0 q r r 2 −r 2 0 . (2) whereχis the 5 th extra coordinate; η AB = (±,−,−,−,∓),A,B = 0,1,2,3,5; r,θ,ϕare the 3Dpolar coordinates; r 0 >0andqare some constants. We can see that there are two closed ds 2 (5) (±r 0 ) = 0hypersurfaces at the r=±r 0 . In kievarwe.tex; 12/03/2001; 3:49; p.461 MODELOFTHESPACETIMEFOAM 455 some sense these hypersurfaces are like to the event horizon and in Ref.[6] such hypersurfacesarenamedasa D-holes.Onthesehypersurfacesweshouldjoin[7]: −the ﬂux of the 4D electric ﬁeld (deﬁned by the Maxwell equations) with the ﬂuxof the5D electricﬁelddeﬁned by R 5t = 0Kaluza-Klein equation. −the area of the Reissner-Nordstr ¨om event horizon with the area of the ds 2 (5) (±r 0 ) = 0hypersurface. Itisnecessarytonotethatbothsolutions(Reissner-Nordstr ¨omblackholeand5D throat) have only two integration constants 1 and on the event horizon takes place an algebraic relation between these 4D and 5D integration constants. Another explanationofthefactthatweuseonlytwojoiningconditionisthefollowing(see Ref.[8] for the more detailed explanations): in some sense on the event horizon holdsa“holographyprinciple”.Thismeansthatinthepresenceoftheeventhori- zon the 4D and 5D Einstein equations lead to a reduction of the amount of initial data. For example the Einstein - Maxwell equations for the Reissner-Nordstr ¨om metric ds 2 = ∆dt 2 −dr 2 ∆−r 2 /parenleftBig dθ 2 + sin 2 dϕ 2 /parenrightBig , (3) A µ = (ω,0,0,0)) (4) (whereA µ istheelectromagneticpotential, κisthegravitationalconstant)canbe writtenas −∆ /prime r+1− ∆ r 2 = κ 2ω /prime2 , (5) ω /prime = q r 2 . (6) For the Reissner - Nordstr ¨om black hole the event horizon is deﬁned by the con- dition ∆(r g ) = 0, wherer g is the radius of the event horizon. Hence in this case we seethatontheeventhorizon ∆ /prime g = 1 r g − κ 2r g ω /prime g2 , (7) here (g) means that the corresponding value is taken on the event horizon. Thus, Eq.(5),whichistheEinsteinequation,isaﬁrst-orderdifferentialequationsinthe whole spacetime (r≥r g ). The condition (7) tells us that the derivative of the metric on the event horizon is expressed through the metric value on the event horizon. This is the same what we said above: the reduction of the amount of initial data takes place by such a way that we have only two integration constants (massmand chargeefor the Reissner-Nordstr ¨om solution and qandr 0 for the 5D throat). 1 in fact, for the Reissner-Nordstr ¨om black hole thisleads to the“no hair”theorem. kievarwe.tex; 12/03/2001; 3:49; p.462 456 V. DZHUNUSHALIEV Figure 16. The left mouth of the quantum WH entraps the force lines of the electric ﬁeld and looks as (-) electric charge. The force lines outcome from the right mouth of WH which one looks as (+) charge. The 5D throat has an interesting property [9]. We see that the signs of the η 55 andη 00 arenotdeﬁned.Weremarkthatthis5Dmetricislocatedbehindtheevent horizonthereforethe4Dobserverisnotabletodeterminethesignsofthe η 55 and η 00 .Moreoverthis5Dmetriccanﬂuctuatebetweenthesetwopossibilities.Hence the external 4D observer is forced to describe such composite WH by means of something like spinor. Another interesting characteristic property of this solution is that we have the ﬂuxofelectricﬁeldthroughthethroat, i.e.eachmouthcanentraptheelectricforce lines and this leads that this mouth is like to electric charge for the external 4D observer,seeFig.16.Wecanneglectthecrosssectionofthethroatandinthiscase eachmouthispoint-likeandwecantrytodescribethesemouthswithhelpofsome effectiveﬁeld.Takingintoaccountthespinor-likepropertiesofquantumhandles, weassumethat spacetimefoamcanbedescribedwithhelpofaneffectivespinor ﬁeld. 3. Approximatemodelof thespacetimefoam The physical meaning of the spinor ﬁeld depends on the method of attaching the quantumhandlestothe external space,see Fig.(17). 3.1. QUANTUM WORMHOLES WITH SEPARATEDMOUTHS In this case|ψ| 2 is a density of the mouths in the external space and e|ψ| 2 is a densityoftheelectriccharge [10]. kievarwe.tex; 12/03/2001; 3:49; p.463 MODELOFTHESPACETIMEFOAM 457 +- - + +- - + 1 2 Figure 17. At the left hand of the ﬁgure quantum handles connect two spaces. At the right hand the mouths of quantum handles areseparated indistanceof the order l Pl . Following this way we write differential equations for the gravitational + electromagnetic ﬁeldsin thepresenceof thespacetime foam (ψ)asfollows R µν − 1 2g µν R=T µν , (8) /parenleftbigg iγ µ ∂ µ +eA µ − i 4ω ¯a¯bµ γ µ γ [¯a γ ¯b] −m /parenrightbigg ψ= 0, (9) D ν F µν = 4πe /parenleftbig ¯ψγ µ ψ /parenrightbig , (10) For ourmodel we usethefollowing ansatz: thesphericallysymmetric metric ds 2 =e 2ν(r) ∆(r)dt 2 −dr 2 ∆(r)−r 2 /parenleftBig dθ 2 + sin 2 dϕ 2 /parenrightBig , (11) theelectromagnetic potential A µ = (−φ,0,0,0), (12) and thespinor ﬁeld ˜ψ=e −iωt e −ν/ 2 r∆ 1/4 /parenleftBig f,0,igcosθ,igsinθe iϕ /parenrightBig . (13) The following is very important for us: the ansatz (13) for the spinor ﬁeld ψhas theT tϕ component of the energy-momentum tensor and the J ϕ = 4πe(¯ψγ ϕ ψ) componentofthecurrent.Letweremindthat ψdeterminesthestochasticalgasof the virtual WH’s which can not have a preferred direction in the spacetime. This meansthataftersubstitutionexpression(11)-(13)intoﬁeldequationstheyshould be averaged by the spin direction of the ansatz (13). After this averaging we have kievarwe.tex; 12/03/2001; 3:49; p.464 458 V.DZHUNUSHALIEV T tϕ = 0andJ ϕ = 0and we have the following equations system describing our spherically symmetric spacetime f /prime √ ∆ = f r−g /parenleftBigg (ω−eφ)e − ν √ ∆+m /parenrightBigg , (14) g /prime √ ∆ =f /parenleftBigg (ω−eφ)e − ν √ ∆−m /parenrightBigg − g r, (15) r∆ /prime = 1−∆−κe −2 ν ∆(ω−eφ) /parenleftBig f 2 +g 2 /parenrightBig −r 2 e −2ν φ /prime2 ,(16) r∆ν /prime =κe −2 ν ∆(ω−eφ) /parenleftBig f 2 +g 2 /parenrightBig −κe − ν r √ ∆fg− κ 2me − ν √ ∆ /parenleftBig f 2 −g 2 /parenrightBig , (17) r 2 ∆φ /prime/prime =−8πe /parenleftBig f 2 +g 2 /parenrightBig − /parenleftBig 2r∆−r 2 ∆ν /prime /parenrightBig φ /prime , (18) whereκis some constant. This equations system was investigated in [11] and result is the following. A particle-like solution exists which has the following expansionsnear r= 0 f(r) =f 1 r+O(r 2 ), g(r) =O(r 2 ), (19) ∆(r) = 1 +O(r 2 ), ν(r) =O(r 2 ), φ(r) =O(r 2 )(20) and thefollowingasymptoticalbehaviour ∆(r)≈1−2m ∞ r+(2e ∞ ) 2 r 2 , ν(r)≈const, (21) φ(r)≈2e ∞ r, (22) f≈f 0 e −αr , g≈g 0 e −αr , f 0 g 0 = /radicalBigg m ∞ + ω m ∞ −ω, α 2 =m 2 ∞ −ω 2 , (23) wherem ∞ is the mass for the observer at inﬁnity and 2e ∞ is the charge of this solution. The solution exists for both cases (|e ∞ |/m ∞ )>1and(|e ∞ |/m ∞ )<1 but for us is essential the ﬁrst case with (|e ∞ |/m ∞ )>1. In this case the clas- sical Einstein-Maxwell theory leads to the “naked” singularity. The presence of the spacetime foam drastically changes this result: the appearance of the vir- tual wormholes can prevent the formation of the “naked” singularuty in the Reissner-Nordstr ¨omsolution with|e|/m> 1. Our interpretation ofthis solution ispresented onthe Fig.(18). kievarwe.tex; 12/03/2001; 3:49; p.465 MODELOFTHESPACETIMEFOAM 459 e/m > 1 e/m > 1 e/m < 1 e/m < 1 1 22 Figure18. 1arethequantum(virtual)WHs, 2aretwosolutionswith |e ∞ |/m ∞ >1.Suchobject can be named as the wormhole with quantum throat . +- +- +- +- Figure 19. The distance between mouths of the quantum handle isof order l Pl . 3.2. QUANTUM WORMHOLES WITH NON-SEPARATED MOUTHS The secondpossibility [12] ispresentedonthe Fig.(19). We will consider the 5D Kaluza-Klein theory + torsion + spinor ﬁeld. The Lagrangian inthiscase is L= √ −G /braceleftbigg − 1 2k /parenleftBig R (5) −S ABC S ABC /parenrightBig + /planckover2pi1 c 2 /bracketleftbigg i¯ψ /parenleftbigg γ C ∇ C − mc i/planckover2pi1 /parenrightbigg ψ+h.c. /bracketrightbigg/bracerightbigg (24) where∇ C =∂ C − 1 4 (ω ¯A¯BC +S ¯A¯BC )γ [¯A γ ¯B] is the covariant derivative, Gis the determinant of the 5D metric, R (5) is the 5D scalar curvature, S ABC is the antisymmetricaltorsiontensor, A,B,Carethe5Dworldindexes, ¯A,¯B,¯Carethe 5-beinindexes, γ B =h B¯A γ ¯A ,h B¯A isthe5-bein, γ ¯A arethe5Dγmatriceswithusual kievarwe.tex; 12/03/2001; 3:49; p.466 460 V.DZHUNUSHALIEV deﬁnitionsγ ¯A γ ¯B +γ ¯B γ ¯A = 2η ¯A¯B ,η ¯A¯B = (+,−,−,−,−)is the signature of the5Dmetric, ψisthespinorﬁeldwhicheffectivelyandapproximatelydescribes the spacetime foam, []means the antisymmetrization, /planckover2pi1,candmare the usual constants.Afterdimensional reductionwehave L= √−g /braceleftbigg − 1 2k /parenleftbigg R+ 1 4F αβ F αβ /parenrightbigg + /planckover2pi1 c 2 /bracketleftbigg i¯ψ /parenleftbigg γ µ ˜∇ µ − 1 8F ¯α¯β γ ¯5 γ [¯α γ ¯β] − 1 4l 2 Pl /parenleftBig γ [¯A γ ¯B γ ¯C] /parenrightBig/parenleftBig i¯ψγ [¯A γ ¯B γ ¯C] ψ /parenrightBig − mc i/planckover2pi1 /parenrightbigg ψ+h.c. /bracketrightbigg/bracerightbigg (25) S ¯A¯B¯C = 2l 2 Pl /parenleftBig i¯ψγ [¯A γ ¯B γ ¯C] ψ /parenrightBig (26) wheregis the determinant of the 4D metric, ˜∇ µ =∂ µ − 1 4 ω ¯a¯bµ γ [¯a γ ¯b] is the 4D covariant derivative of the spinor ﬁeld without torsion, Ris the 4D scalar curvature,F αβ =∂ α A β −∂ β A α is the Maxwell tensor, A µ =h ¯5 µ is the electro- magnetic potential, α,β,µare the 4D world indexes, ¯α,¯β,¯µare the 4D vier-bein indexes,h ¯µ ν is the vier-bein, γ ¯µ are the 4D γmatrices with usual deﬁnitions γ ¯µ γ ¯ν +γ ¯ν γ ¯µ = 2η ¯µ¯ν ,η ¯µ¯ν = (+,−,−,−)is the signature of the 4D metric. Varyingwith respectto g µν ,¯ψandA µ leadsto thefollowing equations R µν − 1 2g µν R= 1 2 /parenleftbigg −F µα F α ν + 1 4g µν F αβ F αβ /parenrightbigg + 4l 2 Pl /bracketleftBig/parenleftBig i¯ψγ µ ˜∇ ν ψ+i¯ψγ ν ˜∇ µ ψ /parenrightBig +h.c. /bracketrightBig − 2l 2 Pl /bracketleftBig F µα /parenleftBig i¯ψγ ¯5 γ [ν γ α] ψ /parenrightBig +F να /parenleftBig i¯ψγ ¯5 γ [µ γ α] ψ /parenrightBig/bracketrightBig − 2g µν l 4 Pl /parenleftBig i¯ψγ [¯A γ ¯B γ ¯C] ψ /parenrightBig/parenleftBig i¯ψγ [¯A γ ¯B γ ¯C] ψ /parenrightBig , (27) D ν H µν = 0, H µν =F µν +˜F µν , ˜F µν = 4l 2 Pl /parenleftBig i¯ψγ ¯5 γ [µ γ ν] ψ /parenrightBig = 4l 2 Pl E µναβ /parenleftBig i¯ψγ [α γ β] ψ /parenrightBig , (28) iγ µ ˜∇ µ ψ− 1 8F ¯α¯β /parenleftBig iγ ¯5 γ [¯α γ ¯β] ψ /parenrightBig − 1 2l 2 Pl /parenleftBig iγ [¯A γ ¯B γ ¯C] ψ /parenrightBig/parenleftBig i¯ψγ [¯A γ ¯B γ ¯C] ψ /parenrightBig = 0,(29) whereω ¯a¯bµ is the 4D Ricci coefﬁcients without torsion, E µναβ is the 4D abso- lutely antisymmetric tensor. The most interesting for us is the Maxwell equation (28) which permits us to discuss the physical meaning of the spinor ﬁeld. We would like to show that this equation in the given form is similar to the electro- dynamic in the continuous media. Let we remind that for the electrodynamic in thecontinuousmediatwotensors ¯F µν and¯H µν areintroduced[13]forwhichwe kievarwe.tex; 12/03/2001; 3:49; p.467 MODELOFTHESPACETIMEFOAM 461 havethe followingequations system(in the Minkowski spacetime) ¯F αβ,γ +¯F γα,β +¯F βγ,α = 0, (30) ¯H αβ ,β = 0 (31) and thefollowingrelationsbetween thesetensors ¯H αβ u β =ε¯F αβ u β , (32) ¯F αβ u γ +¯F γα u β +¯F βγ u α =µ /parenleftbig ¯H αβ u γ +¯H γα u β +¯H βγ u α /parenrightbig (33) whereεandµarethedielectricandmagneticpermeabilityrespectively, u α isthe 4-vectorofthe matter.Fortherestmediaand inthe 3Ddesignation we have ε¯E i =¯E i + 4π¯P i =¯D i ,where ¯E i =¯F 0i ,¯D 0i =¯H 0i , (34) µ¯H i =¯H i + 4π¯M i =¯B i ,where ¯B i =/epsilon1 ijk ¯F jk ,¯H i =/epsilon1 ijk ¯H jk ,(35) whereP i is the dielectric polarization and M i is the magnetization vectors, /epsilon1 ijk is the 3D absolutely antisymmetric tensor. Comparing with the (28) Maxwell equation forthespacetime foam inthe 3Dform E i +˜E i =D i whereE i =F 0i ,˜E i =˜F 0i , D i =H 0i (36) B i +˜B i =H i whereB i =/epsilon1 ijk F jk ,˜B i =/epsilon1 ijk ˜F jk , H i =/epsilon1 ijk H jk (37) we seethatthefollowingnotationscan beintroduced. ˜E i = 4l 2 Pl /epsilon1 ijk /parenleftBig i¯ψγ [j γ k] ψ /parenrightBig (38) isthepolarization vector ofthespacetime foamand ˜B i =−4l 2 Pl /epsilon1 ijk /parenleftBig i¯ψγ ¯5 γ [j γ k] ψ /parenrightBig (39) isthemagnetization vector ofthespacetime foam. Thephysicalreasonforthisisevidently:eachquantumWHisliketoamoving dipole (see Fig.(20)which producesmicroscopicalelectric and magneticﬁelds. 4. Supergravityasa possiblemodelofthespacetime foam From the above-mentioned arguments we see that the most important for such kind models of the spacetime foam is the presence of the nonminimal interac- tion term (in Lagrangian) between spinor and electromagnetic ﬁelds. Let we note that the N=2 supergravity [14] which contains the vier-bein e a µ , Majorana Rarita- Schwinger ﬁeld ψ µ , photonA µ and a second Majorana spin- 3 2 ﬁeldϕ µ has the followingtermin Lagrangian L se = κ √ 2¯ψ µ /parenleftbigg eF µν + 1 2γ 5 ˜F µν /parenrightbigg ϕ ν +···, ˜F µν =e µναβ F αβ (40) kievarwe.tex; 12/03/2001; 3:49; p.468 462 V. DZHUNUSHALIEV + - + - +- + - + - +- + - +- + -- --- ++++ dipoles mouthes5D quantum wormholes Figure 20. For the 4D observer each mouth looks as a moving electric charge. This allows us in some approximation imagine the spacetime foam as acontinuous media with apolarization. The term like this usually occur in supergravities which have some gauge multi- plet of supergravity and some matter multiplet. Taking into account the previous reasonings we can suppose that supergravity theories can be considered as approximatemodels ofthe spacetimefoam. 5. Conclusions Thus, here we have proposed the approximate model for the description of the spacetime foam. This model is based on the assumption that the whole spacetime is5dimensionalbut G 55 isthedynamicalvariableonlyinthequantumtopological handles(wormholes).Inthiscase5Dgravityhasthesolutionwhichwehaveused as a model of the single quantum wormhole. The properties of this solution is such that we can assume that the quantum topological handles (wormholes) can beapproximatelydescribed bysome effectivespinor ﬁeld. The topological handles of the spacetime foam either can be attached to one space or connect two different spaces. In the ﬁrst case we have something like to strings between two D-branes (or wormhole with the quantum throat) and such objectcandemonstrateamodelofpreventingtheformationthenakedsingularity with relation e > m. In the second case the spacetime foam looks as a dielectric withquantumhandlesasdipoles. Such model leads to the very interesting experimental consequences. We see that the spacetime foam has 5D structure and it connected with the electric ﬁeld. This observation allows us to presuppose that the very strong electric ﬁeld can open a door into 5 dimension! The question is: as is great should be this ﬁeld ? The electric ﬁeld E i in the CGSE units and e i in the “geometrized” units can be kievarwe.tex; 12/03/2001; 3:49; p.469 MODELOFTHESPACETIMEFOAM 463 connected by formula e i =G 1/ 2 c 2 E i = /parenleftBig 2.874×10 −25 cm −1 /gauss /parenrightBig E i , (41) [e i ] =cm −1 ,[E i ] =V/cm (42) Asweseethevalueof e i isdeﬁnedbysomecharacteristiclength l 0 .Itispossible thatl 0 is a length of the 5 th dimension. If l 0 =l Pl thenE i ≈10 57 V/cmand this ﬁeldstrengthisinthePlanckregion,andiswillbeyondexperimentalcapabilities tocreate.Butif l 0 hasadifferentvalueitcanleadtomuchmorerealisticscenario for theexperimentalcapability toopen doorinto 5 th dimension. Another interesting conclusion of this paper is that supergravity theories hav- ing nonminimal interaction between spinor and electromagnetic ﬁleds can be considered asapproximateand effective modelsof the spacetimefoam. 6. Acknowledgment I would like to acknowledge the generosity of NATO in its support for this workshopand ICTP (grantKR-154). References 1. C. Misner and J. Wheeler, Ann. of Phys., 2, 525 (1957); J. Wheeler, Ann. of Phys., 2, 604(1957). 2. J.Wheeler, Neutrinos, Gravitation and Geometry (Princeton Univ. Press, 1960). 3. A.Chodos andS. Detweiler, Gen. Rel. Grav. 14(1982)879-890. 4. G. Cl ´ement,Gen. Rel. Grav. 16(1984) 477-489; G. Cl ´ement,Gen. Rel. Grav. 16(1984) 131- 138. 5. V. Dzhunushaliev, Grav. Cosmol., 3, 240(1997). 6. Bronnikov K.,Int.J.Mod.Phys. D4, 491(1995), Grav. Cosmol., 1, 67(1995). 7. V. Dzhunushaliev, Mod.Phys.Lett. A 13, 2179 (1998). 8. V. Dzhunushaliev, “Matching condition on the event horizon and the holography principle”, gr-qc/9907086, tobepublished in Int. J. Mod.Phys.D. 9. V. Dzhunushaliev, H.-J.Schmidt, Grav. Cosmol. 5, 187 (1999). 10. V.Dzhunushaliev,“WormholewithQuantumThroat”,gr-qc/0005008,tobepublishedinGrav. Cosmol. 11. F. Finster, J. Smoller, S.-T. Yau, Phys. Lett. A259, 431 (1999). 12. V. Dzhunushaliev, “An Approximate Model of the SpacetimeFoam”, gr-qc/0006016. 13. L.D. Landau and E.M. Lifshitz, “Electrodynamics of Continuous Media”, (Pergamon Press, Oxford - London - New Jork - Paris, 1960). 14. S. Ferrara and P. V. Nieuwenhuizen, Phys. Rev. Lett. 37, 1669 (1976). kievarwe.tex; 12/03/2001; 3:49; p.470 kievarwe.tex; 12/03/2001; 3:49; p.471 POSSIBLE CONSTRAINTS ON STRING THEORY IN CLOSED SPACE WITH SYMMETRIES ATSUSHIHIGUCHI ∗ Department ofMathematics, University of York,YORK, YO10 5DD,UnitedKingdom Abstract. Itiswellknownthatcertainquadraticconstraintshavetobeimposedonlinearizedgrav- ityinclosedspacewithsymmetries.Wereviewthisphenomenonanddiscussoneoftheconstraints which arise in linearized gravity on static ﬂat torus in detail. Then we point out that the mode with negative kinetic energy, which is necessary for satisfying this constraint, appears to be missing in the freebosonic string spectrum. 1. Introduction (Super)stringtheoryistheleadingcandidateforauniﬁedtheoryincludinggravity. Inparticular,itcontainsandgeneralizesEinstein’sgeneralrelativity[1–3].There- fore,itisnaturaltoexpectthatthetheoryincorporatesdiffeomorphisminvariance. However, this invariance is not manifest in the perturbative deﬁnition of string theory starting from non-interacting string. Now, it is well known that a solution of linearized Einstein equations (with or without matter ﬁelds) in compact back- ground space with Killing symmetries cannot be extended to an exact solution unless the linearized solution satisﬁes certain quadratic constraints [4, 5]. This phenomenon, called linearization instability, is a consequence of diffeomorphism invariance of the full theory. (This fact can be seen most clearly in the quantum context.) Therefore, one may obtain some insight into how diffeomorphism in- variance is incorporated in string theory by investigating the way linearization instabilitiesmanifestthemselves. Inthisarticlewereviewthephenomenonoflinearizationinstabilityingeneral relativity with emphasis on the case with static ﬂat torus space. In particular, we point out that in this space a mode with negative kinetic term is essential in satisfying one of the constraints and that this mode seems to be missing in the spectrum of free bosonic string theory. The rest of the article is org anized ∗ [email protected] kievarwe.tex; 12/03/2001; 3:49; p.472 466 A.HIGUCHI as follows. In Section 2 the phenomenon of linearization instability in classical and quantum general relativity is reviewed. In Section 3 one of the constraints occurring in ﬂat torus space is discussed in detail and the importance of a mode with negative kinetic term is emphasized. In Section 4 it is pointed out that this mode is absent in a seemingly natural treatment of the zero-momentum sector of closedbosonicstringinthisspace.InSection5asummaryofthisarticleisgiven. The metricsignatureis (−+ +···+)throughoutthis article. 2. Linearization instabilitiesingeneralrelativity Consider classical general relativity with any bosonic matter ﬁelds. Suppose we wanttoﬁndasolutioninthistheoryorderbyorderinperturbationtheorystarting from a (globally-hyperbolic) background spacetime satisfying the vacuum Ein- steinequations R ab = 0.Tododosowewritethemetric g ab andthematterﬁelds φ i as g ab =g (0) ab +h (1) ab +h (2) ab +···, φ i =φ (1) i +φ (2) i +···, whereg (0) ab is the background metric and where h (k) ab andφ (k) i are the ﬁelds ob- tained as the k-th order approximation. (The ﬁelds φ i are assumed to vanish at zero-th order for simplicity.) The ﬁrst-order approximation (h (1) ab ,φ (1) i )corre- sponds to non-interacting waves in the background spacetime. The second-order perturbationofthemetric, h (2) ab ,canberegardedasthegravitationalﬁeldgenerated by thefree ﬁelds h (1) ab andφ (1) i . Let the stress-energy tensor of the ﬁelds h (1) ab andφ (1) i in the background spacetime with metric g (0) ab beT (1) ab . We note ﬁrst that the linear contribution to theEinstein tensor E ab =R ab − 1 2g ab R withg ab =g (0) ab +h ab is E (L) ab (h) = 1 2(∇ c ∇ b h ca +∇ c ∇ a h cb −∇ c ∇ c h ab −∇ a ∇ b h cc ) − 1 2g (0) ab (∇ c ∇ d h cd −∇ c ∇ c h dd ). Here the covariant derivatives are compatible with the metric g (0) ab and indices are raisedand loweredby this metric. Theﬁeld h (2) ab mustsatisfy E (L) ab (h (2) ) =κT (1) ab , (1) kievarwe.tex; 12/03/2001; 3:49; p.473 STRING THEORYIN CLOSEDSPACE 467 whereκis a constant. The stress-energy tensor T (1) ab is divergence-free, i.e., ∇ a T (1) ab = 0, if the linear equations of motion are satisﬁed. On the other hand theequation ∇ a E (L) ab (h) = 0 (2) holds for anyh ab . This is a consequence of the Bianchi identity ˜∇ a E ab = 0, where ˜∇ a is the covariant derivative compatible with the full metric g ab . For this reasonEq. (2)is calledthebackgroundBianchi identity. Now, supposethatthere isaKillingvectorﬁeld X a satisfying ∇ a X b +∇ b X a = 0. Then, it is easy to verify that the current j a X ≡T (1)ab X b is conserved. The corresponding conservedNoether charge is givenby Q X ≡ /integraldisplay Σ dΣn a j a X , where the integration is over any Cauchy surface Σandn a is the unit normal to the Cauchy surface. (Since Q X comes from a stress-energy tensor of the free ﬁeldsh (1) ab andφ (1) i , it is quadratic in these ﬁelds.) If the vector X a is a time- translation Killing vector, then the charge Q X is nothing but the energy. If it is a space-translation Killing vector, then Q X is a component of the momentum. We notethat E (L)ab (h)X b = 1 2∇ b K ab (h), whereK ab (h)isananti-symmetric tensorgiven by K ab (h) =X a ∇ b h cc −X b ∇ a h cc +X c ∇ a h bc −X c ∇ b h ac +X c ∇ a h bc −X c ∇ b h ac +h ca ∇ b X c −h cb ∇ a X c . Hence, the integral of E (L)ab (h)X b over the Cauchy surface can be expressed as asurfaceintegral as /integraldisplay Σ dΣn a E (L)ab (h)X b = 1 2 /integraldisplay ∂Σ dSn a r b K ab (h), where∂Σis the “boundary” of the Cauchy surface at inﬁnity and r a is the unit vectornormaltotheboundaryalongtheCauchysurface.Byusingthisexpression and Eq.(1)onecan write theNoethercharge Q X asasurface integral: Q X = 1 2κ /integraldisplay ∂Σ dSn a r b K ab (h (2) ). (3) kievarwe.tex; 12/03/2001; 3:49; p.474 468 A.HIGUCHI In asymptotically-ﬂat spacetime this equation allows us to express energy and momentum ofanisolated systemas surfaceintegrals atspacelike inﬁnity[6]. Now, suppose that the Cauchy surface is compact, i.e., that the space is “closed”. Then, the right-hand side of Eq. (3) must vanish for any h ab because there is nosurfaceterm.Hence, Q X = 0. (4) Thus, the conserved charge Q X is constrained to vanish. Note that this constraint cannotbederivedfromthelinearizedtheoryalone.Itarisesinthefulltheorywhen we try to ﬁnd the correction to the linear theory. Solutions of the linearized ﬁeld equations are not extendible to exact solutions unless they satisfy this constraint. (The background spacetime here is said to be linearization unstable because of the existence of spurious solutions to the linearized equations. The constraint (4) issometimes called a linearizationstability condition.) Althoughwewillconcentrateonclassicaltheory,itisinterestingtonotewhat theconstraint(4)impliesinquantumtheory.IntheDiracquantization,constraints areimposedon thephysicalstates. Thus,thequantum versionof (4) reads Q X |phys/angbracketright= 0, (5) where|phys/angbracketrightis any physical state and Q X is the quantum operator correspond- ing to the conserved Noether charge Q X . Since the operator Q X generates the spacetime symmetry associated with the Killing vector ﬁeld X a , the constraint (5)impliesthatallphysicalstatesmustbeinvariantunderthisspacetimesymme- try [7]. This requirement might seem absurdly strong at ﬁrst sight. For example, in linearized gravity in de Sitter spacetime allphysical states are required to be de Sitter invariant. 1 However, in the (formal) Dirac quantization of full gen- eral relativity, the states are (roughly speaking) required to be diffeomorphism invariant. The constraint (5) can be interpreted to be inforcing the part of the diffeomorphism invariance of the physical states that has not been broken by the backgroundmetric. 3. TheHamiltonianconstraintof linearizedgravity onﬂattorus In this section we discuss linearized gravity in static ﬂat (D−1)-dimensional torus space with all directions compactiﬁed. This spacetime has space- and time- translationinvariance.Therefore,theenergyandmomentumoflinearizedgravity are conserved and are both constrained to vanish. Below we concentrate on the linearization stability condition which requires that energy be zero since it will 1 The vacuum state is the only de Sitter invariant state if one insists on using the original Fock spaceoflinearizedgravity,butonecanconstructinﬁnitelymanyinvariantstatesbyusingadifferent Hilbert space[8]. kievarwe.tex; 12/03/2001; 3:49; p.475 STRING THEORYIN CLOSEDSPACE 469 be important later in the discussion of string theory. We ﬁnd that there is a mode with negative kinetic term and that there would be no excitation as a result of the linearizationstabilityconditionifitwerenotforthismode.Weconsideronlypure gravityfor simplicity. Letusimposethe standard(“Lorenz”orHilbert) gauge condition ∂ a h ab = 1 2∂ b h, (6) whereh=h cc .ThentheHamiltoniandensityreads H= 1 4 /bracketleftBig ∂ t ˜h ab ∂ t ˜h ab +∂ i ˜h ab ∂ i ˜h ab /bracketrightBig −D− 2 4D /bracketleftBig (∂ t h) 2 +∂ i h∂ i h /bracketrightBig , where ˜h ab =h ab − 1 D g ab his the traceless part of h ab . The indexiruns from 1to D−1,i.e.,it isa spacelikeindex.Theﬁeldequations aresimply /square˜h ab = 0,/squareh= 0. The modes with nonzero momentum kare proportional to e −ik 0 t+ik·x , where (k 0 ) 2 −k 2 = 0.Ontheother hand, themodeswith k= 0take the form ˜h ab , h∝At+B, whereAandBareconstants. 2 The Hamiltonian can be writtenas H= /integraldisplay d D−1 xH=H 0 +H /prime , whereH 0 is the energy in the modes with k= 0and whereH /prime is the energy in themodeswith k/negationslash= 0.Forthemodeswith k/negationslash= 0thetracehcanbegaugedaway and thephysical modeshave theform ˜h ab ∝H ab e −ik 0 t+ik·x , whereH ab is a constant symmetric tensor satisfying H tb = 0,H ii = 0and k i H ij = 0. Then we can easily see that H /prime ≥0. The situation is rather different for the modes with k= 0. Since these modes are constant in space, they satisfy ∂ i ˜h ab =∂ i h= 0.Hence,theconditions comingfrom(6)are ∂ t ˜h ti = 0and ∂ t ˜h tt =−D− 2 2D∂ t h . 2 Note that the energy corresponding to these modes would be inﬁnite for A/negationslash= 0if the space were not compactiﬁed. This iswhy these modes would notbe present in uncompactiﬁed space. kievarwe.tex; 12/03/2001; 3:49; p.476 470 A.HIGUCHI Letuswrite ˜h ab =˜h (0) ab +˜h /prime ab , h=h (0) +h /prime , where ˜h (0) ab andh (0) are the zero-momentum parts of ˜h ab andh. Then the zero- momentum Hamiltonian H 0 is givenby H 0 = /integraldisplay d D−1 x /bracketleftBigg 1 4∂ t ˜h (0) ij ∂ t ˜h (0)ij −D 2 − 4 8D(∂ t h (0) ) 2 /bracketrightBigg . Noticethatthe tracemode h (0) has anegative kineticterm. Since the Hamiltonian is the Noether charge corresponding to the time- translationsymmetryofthebackgroundspacetime,thediscussionintheprevious sectionshowsthat H=H 0 +H /prime = 0. The solutions of the linearized equations which do not satisfy this condition cannotbeextendedto exactsolutions.Thisequation can bere-expressedas −D 2 − 4 8D /integraldisplay d D−1 x(∂ t h (0) ) 2 +H /prime/prime = 0, (7) where H /prime/prime =H /prime + 1 4 /integraldisplay d D−1 x∂ t ˜h (0) ij ∂ t ˜h (0)ij ≥0. Now, the quantity 1 2 h (0) V, whereVis the volume of the background space, is the change in the volume of the space. Hence, Eq. (7) relates the expan- sion/contraction rate of space to the energy due to the excitation of the system. In fact this equation is the linearized version of a familiar equation in cosmology. Notice that the trace mode h (0) plays a vital role in satisfying Eq. (7). If this modewereabsent,Eq.(7)wouldimplythattherewerenoexcitationsonﬂattorus compactiﬁedin all directions. 4. Masslesssectorofbosonicstring inthepositionrepresentation Massless excitations of closed string include gravitons, i.e., linearized gravity is presentamongthemodesoffreeclosedbosonicstringinMinkowskispacetime. 3 Thisfactisoneofthemostimportantfeaturesofstringtheoryasauniﬁedtheory. It is natural to expect that this feature persists in string theory in static ﬂat torus compactiﬁedinalldirections.Therefore,thetotalenergyandmomentuminstring (ﬁeld) theory are expected to vanish in this spacetime. We also expect that there 3 This fact goes beyond the linearized level as iswell known[1–3]. kievarwe.tex; 12/03/2001; 3:49; p.477 STRING THEORYIN CLOSEDSPACE 471 is a mode with negative kinetic term among the closed-string modes so that the linearizationstabilitycondition(7)canbesatisﬁedbynon-vacuumstates(instring ﬁeld theory). However, we will ﬁnd in the “old covariant approach” that there is nomasslessstringexcitationwhichcorrespondstothezero-momentummode h (0) withnegativekineticenergyifwetreatthezero-momentummodesinawaywhich seems mostnatural. Letusstartwithadiscussionofopenstringinﬂat (D−1)-dimensionaltorus. The masslessstatesinthe oldcovariantapproach aredenoted by α a −1 |0;p/angbracketright, where the state|0;p/angbracketrightwith momentum p a has no string excitation (see, e.g., Ref. [9]). The creation operator α a −1 creates the lowest harmonic-oscillator mode on the string in the a-direction and the annihilation operator α a 1 annihilates it. As is well known, the physical state conditions lead to p 2 = 0andp·α 1 |phys/angbracketright= 0, where [α a 1 ,α b −1 ] =g ab andp·α 1 ≡p a α a 1 . [Here,g ab = diag(−1,1,1,... , 1).] Letusconsider awave-packetstate |ψ/angbracketright= /integraldisplay d D p (2π) D ˆA a (p)α a −1 |0;p/angbracketright, where ˆA a (p)is a function of p a . The physical state conditions then read p 2 ˆA a (p) = 0andp a ˆA a (p) = 0. Now, deﬁne the (spacetime) position repre- sentation ofthiswave packetas A a (x) = /integraldisplay d D p (2π) D ˆA a (p)e −ip·x . Then the physical state conditions become /squareA a = 0and∂ a A a = 0. Thus, we recovertheequationssatisﬁedbyanon-interacting U(1)gaugeﬁeldintheLorentz gauge.The zero-momentummodes inﬂat (D−1)-dimensionaltorussatisfy ∂ t A t = 0, ∂ 2 t A i = 0. Theseimplythat A t = constandA i =E i t+A (0) i .Theconstant A t canbegauged away, but the constants E i (the electric ﬁeld) and A (0) i represent physical degrees offreedom. Next, we will apply the above procedure to a closed string on static ﬂat torus and examine whether or not there is a mode with negative kinetic term. The massless excitations ofa closedbosonic stringare α a −1 ˜α b −1 |0;p/angbracketright. The operator α a −1 (˜α a −1 ) creates the lowest left-moving (right-moving) mode on the string in the a-direction, and the operator α a 1 and˜α a 1 annihilate them. The kievarwe.tex; 12/03/2001; 3:49; p.478 472 A.HIGUCHI physical state conditions lead to p 2 = 0andp·α 1 |phys/angbracketright=p·˜α 1 |phys/angbracketright= 0, where [α a 1 ,α b −1 ] = [˜α a 1 ,˜α b −1 ] =g ab .We againconsidera wave-packet state |Ψ/angbracketright= /integraldisplay d D p (2π) D ˆH ab (p)α a −1 ˜α b −1 |0;p/angbracketright. (Noteherethatthetensor ˆH ab (p)isnotnecessarilysymmetric.)Thephysicalstate conditionsread p 2 ˆH ab (p) = 0andp a ˆH ab =p b ˆH ab = 0.Inthespacetimeposition representation, H ab (x) = /integraldisplay d D p (2π) D ˆH ab (p)e −ip·x , thephysical stateconditions are /squareH ab = 0and ∂ a H ab =∂ b H ab = 0. (8) The equation/squareH ab = 0naturally come from the following Lagrangian density: L=− 1 4∂ a H bc ∂ a H bc . (9) Theconstraints(8)canbeimposedbyhand.Oneﬁndsthemodescorrespondingto gravitons, anti-symmetric tensor particles and dilatons in the nonzero momentum sector of this theory as in Minkowski spacetime. The constraints (8) for the zero- momentum sectorread ∂ t H ta =∂ t H at = 0 for alla.The energyinthe zero-momentum sectoris E 0 = 1 4 /integraldisplay d D−1 x∂ t H ij ∂ t H ij , wherei,j= 1,2,···D−1. There is no mode with negative kinetic term in this expression, and E 0 is positive deﬁnite. Thus, the negative-energy mode, which is necessary for non-vacuum states to satisfy the constraint (7), does not appear in a seeminglynaturalpositionrepresentationofthemasslesssectorofclosedbosonic string. 5. Summary In this article, we reviewed the fact that quadratic constraints arise in linearized gravity if the background spacetime allows Killing symmetries and has compact Cauchy surfaces. This implies that the total energy and momentum in free string (ﬁeld)theoryshouldbeconstrainedtovanishinﬂattorusspacewithalldirections compactiﬁed. We examined one of these constraints in linearized gravity in this kievarwe.tex; 12/03/2001; 3:49; p.479 STRING THEORYIN CLOSEDSPACE 473 space, emphasizing that a mode with negative kinetic energy is essential in satis- fying this constraint. Then we analyzed free closed bosonic string theory in this space and found that this mode does not appear in a seemingly natural treatment ofthemassless sector. It is possible that the Lagrangian density (9) is wrong, and a more careful analysis may lead to a Lagrangian density describing the usual linearized gravity, anti-symmetrictensorgaugeﬁeldanddilatonscalarﬁeldafterall.Itwillbeinter- esting to see how this can be achieved. The situation is rather puzzling, however, becausestringtheoryisformulatedintermsofaphysicalobject,i.e.,astring,and doesnot seemtoallowany negative-energy mode. References 1. T.Yoneya, Quantumgravityandthezero-slopelimitofthegeneralizedVirasoromodel ,Nuovo Cim. Lett. 8(1973), pp. 951–955. 2. T. Yoneya, Connection of dual models to electrodynamics and gravidynamics , Prog. Theor. Phys.51(1974), pp. 1907–1920. 3. J.ScherkandJ.Schwarz, Dualmodelsfornon-hadrons ,Nucl.Phys. B81(1974),pp.118–144. 4. D.BrillandS.Deser, Instabilityofclosedspacesingeneralrelativity , Commun.Math.Phys. 32(1973), pp. 291–304. 5. A. Fischer and J. Marsden, Linearization stability of Einstein equations , Bull. Am. Math. Soc.79(1973), pp. 997–1003. 6. R. Arnowitt, S. Deser and C. W. Misner, The dynamics of general relativity , in “Gravitation: an introduction to current research”, ed.L. Witten, Wiley, New York, 1962, pp. 226–265. 7. V. Moncrief, Invariant states and quantized gravitational perturbations , Phys. Rev. D 18 (1978), pp. 983–989. 8. A. Higuchi, Quantum linearization instabilities of de Sitter spacetime: II , Class. Quantum Grav.8(1991), pp. 1983–2004. 9. M.B.Green,J.H.SchwarzandE.Witten, Superstingtheory:vol.1.Introduction , Cambridge University Press, Cambridge, 1987, pp. 113–116. kievarwe.tex; 12/03/2001; 3:49; p.480 kievarwe.tex; 12/03/2001; 3:49; p.481 SEMICLASSICALDYNAMICSOF SU(2)MODELS ADRIAN ALSCHER ∗ and HERMANNGRABERT † Fakult¨atf¨urPhysik,Albert-Ludwigs-Universit ¨at Freiburg, Hermann-Herder-Str.3, D-79106 Freiburg, Germany Withinthescopeofsimplequantummechanicswepresentasemiclassicaltheory whichisexact.Whilethesemiclassicaltheoryofcanonicalphasespacepathinte- grals is now well established [1, 2] we examine here the case where the classical phase space is the two-sphere. After summarizing some relevant features of a classicalspin,webrieﬂydiscussthelocalizationofclassicalphasespaceintegrals andthenpresentanextensionforaquantumspin.Thesemiclassicalpropagatoris employed tosolvethe Jaynes-Cummings model. 1. Classical spin Aclassicalspinisdescribedby aclassical Blochvector onthe two-sphere /vectorS∈S 2 (s) = /braceleftBig (S x ,S y ,S z )∈R 3 |S=s /bracerightBig . We makeuse ofsphericalcoordinates U={Ω = (ϑ,ϕ)|0<ϑ<π, 0<ϕ< 2π}. This coordinate system cannot be extended over the whole S 2 (s). However, as S 2 (s)isembeddedinR 3 ,anappropriatemetric gandvolumeform ωareinduced g=s 2 (dϑ⊗dϑ+ sin 2 (ϑ)dϕ⊗dϕ), ω=ssin(ϑ)dϑ∧dϕ. The symplectic volume form is closed and non degenerate. Hence, the pair (S 2 (s),ω)generates a symplectic differential manifold. Now, Hamiltonian dy- namicsisdetermined bytheHamiltonianvectorﬁeld X H ω(X H ,·) =dH , ∗ [email protected] † [email protected] kievarwe.tex; 12/03/2001; 3:49; p.482 476 A.ALSCHER, H.GRABERT leadingtothedynamical system ssin(ϑ)˙ϑ=∂ H ∂ϕ, ssin(ϑ) ˙ϕ=−∂ H ∂ϑ. These classical equations of motion can also be derived by introducing the classical action S[Ω(t)] = /integraldisplay T 0 dt /bracketleftBig θ ϑ ˙ϑ+θ ϕ ˙ϕ−H /bracketrightBig , withthe symplectic potential θ=s[cos(ϑ)dϕ+dG]. For classical spin dynamics the localization of oscillating phase space integrals was observed [3]. To see this we examine the symplectic form αof the external algebraofthe cotangentbundle α=e −iT(H−ω) , which is equivariantly closed. The integral over the whole sphere can be written as Z= /integraldisplay S 2 (s) α= /integraldisplay S 2 (s) αe −νD H β , (1) withtheequivariantexactform D H β=dg(X H ,·) +g(X H ,X H ).Now,theright handsideofEq.(1)does notdepend on ν,allowingfor the localization of Z[4] Z= lim ν→∞ /integraldisplay S 2 (s) αe −νD H β . The stationaryphaseapproximation resultsin theBerlinge-Vergueformula Z=−2π /summationdisplay Ω∈U fp α (0) (Ω) /radicalbig detdX H (Ω), and only the sum over the ﬁx points U fp ={Ω∈U|X H (Ω) = 0}has to be considered. Therefore, the question arises whether there exists a similar saddle point approximationofpathintegralsforquantum mechanicalspins. 2. Quantum spin Niemi and Pasanen [5] have proposed a supersymmetric formulation of a path integral which leads to a semiclassical localization formula. However, it only kievarwe.tex; 12/03/2001; 3:49; p.483 SEMICLASSICALDYNAMICSOF SU(2)MODELS 477 describes correct quantum mechanics if the action is supersymmetrically exact, leadingtothenecessarycondition θ(X H ) =H.Anotherapproach[6]isbasedon geometric quantization. Here we make use of a path integral in the spin coherent staterepresentationofthequantum mechanicalspinHilbert space [7, 8] |ψ g /angbracketright=D s (g)|↑/angbracketright, where the (2s+ 1)-dimensional irreducible representation of g∈SU(2)acts on |↑/angbracketright=|s,m =s/angbracketright. The spin coherent states |ψ g /angbracketrightand|ψ g /prime /angbracketrightdescribe the same physical stateif g∼g /prime ⇔g /prime ∈gU(1), which gives rise to the ﬁber bundle representation of SU(2)overS 2 (s)≡ SU(2)/U(1). Distinct spin coherent states are canonically isomorphic to the left cosets which becomes obvious if we parameterize any g∈SU(2)with Euler angles (ϑ,ϕ,χ ): |Ω/angbracketright=|ψ g /angbracketright=e −isχ e −iϕS z e −iϑS y |↑/angbracketright. Wemakeuseofasectionofthe SU(2)bundleandchooseonespecialmemberin every leftcoset.In particularweﬁx χ= 0for every|Ω/angbracketright. Thescalar product /angbracketleftΩ /prime/prime |Ω /prime /angbracketright= /bracketleftBig cos(ϑ /prime/prime /2) cos(ϑ /prime /2)e i 2 (ϕ /prime/prime −ϕ /prime ) + sin(ϑ /prime/prime /2) sin(ϑ /prime /2)e − i 2 (ϕ /prime/prime −ϕ /prime ) /bracketrightBig 2s gives rise to a gauge invariant metric and volume form which are identical to the geometric structures of S 2 (s)[9]. Hence, a representation of quantum states is found which is useful in order to understand quantum systems with discrete degreesof freedom interms ofclassicalmechanics. Weconsiderthemostgeneral SU(2)model described bythe Hamiltonian H(t) =B x (t)S x +B y (t)S y +B z (t)S z . (2) Following the lines of [10] the propagator can be represented as the limit of a Wienerregularizedphasespacepathintegral /angbracketleftΩ /prime/prime |U(T)|Ω /prime /angbracketright= lim ν→∞ /integraldisplay dµ w exp{iS[Ω(t)]} (3) withthe sphericalWiener measure dµ w =N T /productdisplay t=0 dcos[ϑ(t)]dϕ(t) exp /braceleftBigg − 1 2sν /integraldisplay T 0 dt /bracketleftBig g ϑϑ ˙ϑ 2 +g ϕϕ ˙ϕ 2 /bracketrightBig/bracerightBigg . This enforces that only continuous Brownian motion paths contribute to the path integral. Now, the dominant path approximation of the right hand side of Eq. (3) canbe shown tocoincide withtheexactquantumresult [10] exp{iS cl [Ω(t)]}=/angbracketleftΩ /prime/prime |U(T)|Ω /prime /angbracketright. (4) kievarwe.tex; 12/03/2001; 3:49; p.484 478 A.ALSCHER, H.GRABERT ForSU(2)models (2) no contributions of ﬂuctuations around the dominant path havetobetakeninto account. Apart from an extension of the localization of classical phase space integrals to the case of quantum propagators, the formula (4) is also useful to study spins coupled with other degrees of freedom. Here, we apply it to an exactly solvable model. 3. Jaynes-Cummingsmodel The Jaynes-Cummingsmodel ischaracterizedby theHamiltonian [11,12] H=a † a+ (1 + ∆)S z +λ(aS + +a † S − ), whereais the canonical annihilation operator of a bosonic ﬁeld mode and S ± = S x ±iS y ,S z areoperatorsofaspin- 1 2 .ItiswellknownthattheJaynes-Cummings model allowsapartfrom Hforanothertimeindependent operator[14] N=a † a+S z . Hence,thetimeevolutionoperator U(T) =e −iHT =e −iNT e −iCT , whereC=H−N.Representingthespinoperatorsintheeigenbasisof S z formed by theeigenvectors |↑/angbracketrightand|↓/angbracketright e −iNT =e −ia † aT /parenleftBig e − i 2 T |↑/angbracketright/angbracketleft↑| +e + i 2 T |↓/angbracketright/angbracketleft↓| /parenrightBig . Introducingfurthertheeigenketsof a † a,invariantsubspacesaredistinguished.In particular the kets|↑n−1/angbracketright≡|↑/angbracketright|n−1/angbracketrightand|↓n/angbracketright≡|↓/angbracketright|n/angbracketrightspan the subspace withN= (n− 1 2 ). In this subspace the time independent operator Cgenerates SU(2)dynamics. In termsoftheoperators J x = 1 2 /parenleftBig |↑n−1/angbracketright/angbracketleft↓n|+|↓n/angbracketright/angbracketleft↑n−1| /parenrightBig , J y = i 2 /parenleftBig −|↑n−1/angbracketright/angbracketleft↓n|+|↓n/angbracketright/angbracketleft↑n−1| /parenrightBig , J z = 1 2 /parenleftBig |↑n−1/angbracketright/angbracketleft↑n−1|−|↓n/angbracketright/angbracketleft↓n| /parenrightBig , we have C= 2λ √nJ x + ∆J z . Accordingly, /angbracketleftΩ /prime/prime |e −iCT |Ω/angbracketright= lim ν→∞ /integraldisplay dµ w exp{iS[ϑ(t),ϕ(t)]}, kievarwe.tex; 12/03/2001; 3:49; p.485 SEMICLASSICALDYNAMICSOF SU(2)MODELS 479 withthe action S[ϑ(t),ϕ(t)] = /integraldisplay T 0 dt /bracketleftbigg 1 2cos(ϑ) ˙ϕ−C(ϑ,ϕ) /bracketrightbigg , where C(ϑ,ϕ) =/angbracketleftϑϕ|C|ϑϕ/angbracketright =λ √nsin(ϑ) cos(ϕ) + ∆ 2cos(ϑ). Now thedominantpathapproximation(4)gives exp{iS cl [Ω(t)]}= exp /braceleftbigg −i /integraldisplay T 0 dtC(¯ϑ /prime/prime (t),¯ϕ /prime/prime (t)) /bracerightbigg /angbracketleftΩ /prime/prime |Ω /prime /angbracketright,(5) Introducing thecomplex variables ζ= tan /parenleftBigg ¯ ϑ 2 /parenrightBigg e i¯ϕ , η= tan /parenleftBigg ¯ ϑ 2 /parenrightBigg e −i¯ϕ , (6) thedominantpath isdeterminedby ˙ζ=−iλ √n(1−ζ 2 ) +i∆ζ, ˙η=iλ √n(1−η 2 )−i∆η, with boundary conditions ζ(0) =ζ /prime andη(T) =η /prime/prime . Hence, the endpoint of the classical trajectoryobeys ζ(T) =2Ω n ζ /prime cos(Ω n T) +i[∆ζ /prime −λ √n] sin(Ω n T ) 2Ω n ζ /prime cos(Ω n T)−i[λ √nζ /prime + ∆] sin(Ω n T), η(T) =η /prime/prime , withthe Rabifrequency Ω n = /radicalBigg λ 2 n+∆ 2 4. Intermsofthecomplex variables(6) weget C(ζ(T),η /prime/prime ) =i d dTlog /braceleftbigg (1 +ζ /prime η /prime/prime ) cos(Ω n T) − i Ω n /bracketleftBig λ √n(ζ /prime +η /prime/prime ) + ∆ 2(1−ζ /prime η /prime/prime ) /bracketrightBig sin(Ω n T) /bracerightbigg . kievarwe.tex; 12/03/2001; 3:49; p.486 480 A.ALSCHER, H.GRABERT Now,the integralinEq.(5) isreadilysolved andthepropagator takesthe form e iS cl =a(T) cos /parenleftbigg ϑ /prime/prime 2 /parenrightbigg cos /parenleftbigg ϑ /prime 2 /parenrightbigg e i 2 (ϕ /prime/prime −ϕ /prime ) +a ∗ (T) sin /parenleftbigg ϑ /prime/prime 2 /parenrightbigg sin /parenleftbigg ϑ /prime 2 /parenrightbigg e − i 2 (ϕ /prime/prime −ϕ /prime ) +b(T) cos /parenleftbigg ϑ /prime/prime 2 /parenrightbigg sin /parenleftbigg ϑ /prime 2 /parenrightbigg e i 2 (ϕ /prime/prime +ϕ /prime ) −b ∗ (T) sin /parenleftbigg ϑ /prime/prime 2 /parenrightbigg cos /parenleftbigg ϑ /prime 2 /parenrightbigg e − i 2 (ϕ /prime/prime +ϕ /prime ) , where a(T) = cos(Ω n T)−i ∆ 2Ω n sin(Ω n T), b(T) =−iλ √ n Ω n sin(Ω n T). This givesindeedtheexact propagator[13] ofthe model. This work was supported by a grant from the Deutsche Forschungsgemein- schaft(DFG). References 1. I.Daubechies, J.R. Klauder, J.Math.Phys. 26(1985),2239. 2. J.R. Klauder, Phys.Rev.Lett. 56, 897 (1986). 3. E. Keski-Vakkuri, A.J. Niemi, G. Semenoff and O. Tirkkonen, Phys.Rev. D44(1991), 3899. 4. N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac-Operators Springer, Berlin, 1991. 5. A.J. Niemi and P. Pasanen, Phys.Lett. B253(1991), 349. 6. E.A. Kochetov, J.Phys. A31(1998), 4473. 7. J.M. Radcliffe, J.Phys. A4(1971), 313. 8. A.M.Perelomov, GeneralizedCoherentStatesandTheirApplications Springer,Berlin,1986. 9. J.P. Provost and G.Vallee, Comm.Math.Phys. 76(1980), 289. 10. A.Alscher and H. Grabert, J.Phys. A32(1999), 4907. 11. E.T. Jaynes and F.W. Cummings, Proc.IEEE 51(1963), 89. 12. S. Stenholm, Phys.Rep. C6(1973), 1. 13. B.W. Shore and P.L. Knight, J.Mod.Opt. 40(1993),1195. 14. J.R. Ackerhalt and K. Rzazewski, Phys.Rev. A12(1975),2549. kievarwe.tex; 12/03/2001; 3:49; p.487 LIST OF SPEAKERSAND E-PRINTS 481 L IST OF SPEAKERS AND THEIR E- PRINTS ( ARW contributionsare inbold ) 1.Adrian Alscher [email protected] quant-ph/0006072, quant-ph/0004046, quant-ph/9904102, nucl-th/9606011 2.Andrzej Borowiec [email protected] gr-qc/0011103, math.QA/0007151, math-ph/0007031, math.QA/9910018, gr- qc/9906043, math-ph/9906012, gr-qc/9806116, hep-th/9801126, q-alg/9710006, gr-qc/9705025, dg-ga/9612009, gr-qc/9611067, hep-th/9312023 3.Friedemann Brandt [email protected] hep-th/0010155 hep-th/0009133, hep-th/0006152, hep-th/0005086, hep-th/0002245, hep-th/9910177 4.AlexanderBurinskii [email protected] hep-th/0011188 hep-th/0008129, gr-qc/0008055, hep-th/9910045, hep-th/9908198, gr-qc/9904012, hep-th/9903032,hep-th/9802110,hep-th/9801177,hep-th/9704102,hep-th/9504139, hep-th/9503094, gr-qc/9501012, gr-qc/9303003 5.Goran Djordjevic [email protected] hep-th/0005216, quant-ph/0005027, math-ph/0005026, math-ph/0005025 6.Branko Dragovich [email protected] math-ph/0010023, hep-th/0005216, hep-th/0005200, gr-qc/0005103, quant-ph/0005027, math-ph/0005026, math-ph/0005025, math-ph/0005020 7.Steven Duplij [email protected] math-ph/0012039 physics/0008231, physics/0006062, math.FA/0006001, math-ph/0005033, math- ph/9910045, hep-th/9809089, q-alg/9609022, funct-an/9609002, alg-geom/9510013, alg-geom/9506004, hep-th/9505179 8.Vladimir Dzhunushaliev [email protected] gr-qc/0010029 hep-th/0010185, gr-qc/0006016, gr-qc/0005123, gr-qc/0005008, cond-mat/0001257, hep-th/9912194, gr-qc/9912018, gr-qc/9911120, gr-qc/9911080, gr-qc/9910092, gr- qc/9908076, gr-qc/9908074, gr-qc/9908049, gr-qc/9907086, gr-qc/9905104, gr- qc/9903075, hep-th/9902076, hep-th/9810094, gr-qc/9810050, hep-ph/9807239, gr- qc/9807086, gr-qc/9807080, hep-th/9806073, gr-qc/9806046, gr-qc/9805104, gr- qc/9712068, gr-qc/9711033, hep-th/9707039, cond-mat/9704062, gr-qc/9612047, hep-th/9611096, gr-qc/9607007, hep-th/9606125, hep-th/9606124, hep-th/9603120, gr-qc/9603007, gr-qc/9512014, hep-th/9510056, supr-con/9510001 9.Andrzej Frydryszak [email protected] math-ph/9807036,hep-th/9601020 10.DmitriGaltsov [email protected] hep-th/0012059 gr-qc/0008076, hep-th/0007228, hep-th/0006242, gr-qc/0006087, hep-th/0005099, hep-th/9912127,hep-th/9910171,hep-th/9908133,hep-th/9908132,hep-th/9901130, hep-th/9810070, gr-qc/9808002, hep-th/9801160, gr-qc/9712024, gr-qc/9712003, hep-th/9709181, gr-qc/9706067, gr-qc/9706063, hep-th/9702039, gr-qc/9612067, gr-qc/9612007, gr-qc/9608023, gr-qc/9608021, hep-th/9607043, hep-th/9606042, hep-th/9606041, gr-qc/9606014, hep-th/9507164, hep-th/9507005, hep-th/9504155, hep-th/9503092,hep-th/9410217,hep-th/9409041,hep-th/9407155,hep-th/9308068, hep-th/9305112, hep-th/9212153, gr-qc/9209008 11.AlexanderGanchev [email protected] hep-th/9906139, math.QA/9807106, physics/9803038, hep-th/9709103, hep- kievarwe.tex; 12/03/2001; 3:49; p.488 482 LIST OF SPEAKERSAND E-PRINTS th/9608018, hep-th/9407013, hep-th/9403075, hep-th/9402153, hep-th/9308038, hep-th/9308037, hep-th/9207032, hep-th/9201080, dg-ga/9606011 12.Alexandre Gavrilik [email protected] hep-ph/0011057 hep-ph/0010019, hep-ph/9912222, math.QA/9911201, hep-th/9911120, gr- qc/9911094, nucl-th/9906034, hep-ph/9807559, hep-ph/9712411, q-alg/9709036, hep-ph/9504233, q-alg/9511017 13.AtsushiHiguchi [email protected] gr-qc/0011070, gr-qc/0011062, quant-ph/0006125, quant-ph/0005013, gr- qc/0004079, gr-qc/9901006, quant-ph/9812036, gr-qc/9806093, gr-qc/9804066, gr-qc/9609025, gr-qc/9605030, gr-qc/9603045, gr-qc/9508051, gr-qc/9505035, gr-qc/9505009, gr-qc/9412048, gr-qc/9407038, gr-qc/9406009 14.NikolayIorgov [email protected] hep-ph/0010019, math.QA/0007105, hep-ph/9912222, math.QA/9911201, math.QA/9911129, nucl-th/9906034, math.QA/9905059, hep-ph/9807559, math.QA/9805032, q-alg/9709036,q-alg/9709036, q-alg/9511017 15.Anatolij Klimyk [email protected] math.QA/0007105, math.QA/9911130, math.QA/9911129, math.QA/9911114, math.QA/9905059, math.QA/9901080, math.QA/9805048, math.QA/9805032, q-alg/9709035 16.YuriKozitsky [email protected] math.DS/9909182, math-ph/9812017 17.Karl Landsteiner [email protected] hep-th/0011003 hep-th/0006210,hep-th/0004115,hep-th/9911124,hep-th/9909166,hep-th/9908010, hep-th/9901143,hep-th/9806137,hep-th/9805158,hep-th/9801002,hep-th/9708118, hep-th/9705199,hep-th/9609059,hep-th/9606146,hep-th/9507008,hep-th/9502147, hep-th/9412198, hep-th/9408033, hep-th/9309111 18.Dimitry Leites [email protected] hep-th/9710045, hep-th/9702120, hep-th/9702073 19.Jerzy Lukierski [email protected] hep-th/0011053 hep-th/0012056, hep-th/0011214, hep-th/0009120, hep-th/0007102, math.QA/0007065, math.QA/0005145, hep-th/0005112, hep-th/9912264, hep- th/9912051, hep-th/9907113, hep-th/9904109, gr-qc/9903066, hep-th/9902037, hep-th/9812074, hep-th/9812063, hep-th/9811022, math-ph/9807036, hep- th/9706031, hep-th/9612017, hep-th/9610230, hep-th/9606170, hep-th/9504110, hep-th/9412114,hep-th/9411115,hep-th/9405076,hep-th/9312153,hep-th/9312068, hep-th/9310117, hep-th/9204086, hep-th/9108018 20.Volodymyr Lyubashenko [email protected] q-alg/9510004, hep-th/9405168, hep-th/9405167, hep-th/9403189,hep-th/9311095 21.John Madore [email protected] hep-th/0009230, hep-th/0005273, math.QA/0004011, math.QA/0002215, math.QA/0002007, hep-th/0001203, math.QA/9907023, gr-qc/9906059, math.QA/9904027, hep-th/9903239, math.QA/9812141, math.QA/9809160, math.QA/9807123, math.QA/9806071, q-alg/9709007, gr-qc/9709002, gr- qc/9708053, gr-qc/9706047, gr-qc/9705083, q-alg/9702030, gr-qc/9611026, gr-qc/9607065, gr-qc/9607060, hep-th/9601169, hep-th/9601120, hep-th/9506183, kievarwe.tex; 12/03/2001; 3:49; p.489 LIST OF SPEAKERSAND E-PRINTS 483 hep-th/9506041, hep-th/9502017, hep-th/9411127, hep-th/9410199, gr-qc/9307030, hep-ph/9209226 22.Vladimir Mazorchuk [email protected] 23.Jan Naudts [email protected] hep-th/0012209, math-ph/0012051, cond-mat/0011225, math-ph/0009031, math-ph/9908025, math-ph/9907008, quant-ph/9904110, cond-mat/9904070, math-ph/9903002,quant-ph/9809061 24.Irina Shchepochkina (Paramonova) [email protected] physics/9703022, hep-th/9702122, hep-th/9702121, hep-th/9702120 25.Christiane Quesne [email protected] math-ph/0012033 math-ph/0008034, math-ph/0008020, math-ph/0007016, math-ph/0004027, quant-ph/0003085, math-ph/0003025, math-ph/9911004, math-ph/9908022, math-ph/9908021, math.QA/9903151, math-ph/9901016, math.QA/9811064, math.QA/9810161, solv-int/9808017, quant-ph/9802066, physics/9708004, hep- th/9706067, q-alg/9706002, quant-ph/9703037, q-alg/9701031, q-alg/9701030, q-alg/9701029, hep-th/9612173, hep-th/9607035, q-alg/9605041, hep-th/9604132, q-alg/9512032, hep-th/9510006, hep-th/9507078, hep-th/9505071,hep-th/9505011 26.YuriiSamoilenko yurii − [email protected] math.QA/0010308, math-ph/0001011,math-ph/9910018 27.AlexanderSergeev [email protected] math.RT/9904079, math.RT/9810148, math.RT/9810113, math.RT/9810111, math.RT/9810110, math.RT/9810109 28.ArturSergyeyev [email protected] solv-int/9902002 29.JoanSimon [email protected] hep-th/0010242,hep-th/0007253,hep-th/0003211,hep-th/9910177,hep-th/9909005, hep-th/9907022,hep-th/9812095,hep-th/9807113,hep-th/9803196,hep-th/9803040, hep-th/9712125, hep-th/9707063 30.KelloggStelle [email protected] hep-th/0011167,hep-th/0007120,hep-th/9911156,hep-th/9907202,hep-th/9903057, hep-th/9812086,hep-th/9810159,hep-th/9807051,hep-th/9806051,hep-th/9803259, hep-th/9803235,hep-th/9803116,hep-th/9710244,hep-th/9708109,hep-th/9707207, hep-th/9706207,hep-th/9701088,hep-th/9608173,hep-th/9605082,hep-th/9602140, hep-th/9511203,hep-th/9508042,hep-th/9502108,hep-th/9412168,hep-th/9404170, hep-th/9401007,hep-th/9212037,hep-th/9212017,hep-th/9209111,hep-th/9206108, hep-th/9201020, hep-th/9110015 31.Francesco Toppan [email protected] hep-th/0010135, hep-th/0005035, hep-th/0005034, solv-int/9912003, hep- th/9907148, hep-th/9904134, hep-th/9810145, hep-th/9809003, hep-th/9805147, solv-int/9710001, hep-th/9705109, hep-th/9703224, hep-th/9612245, hep- th/9610038, hep-th/9608036, hep-th/9603187, hep-th/9506133, hep-th/9504138, hep-th/9503122,hep-th/9411046,hep-th/9409126,hep-th/9409125,hep-th/9405095, hep-th/9312045,hep-th/9310062,hep-th/9307106,hep-th/9303073,hep-th/9210020, hep-th/9208048 32.Sergiu Vacaru [email protected] hep-th/0011221 hep-th/0009163, gr-qc/0009039, gr-qc/0009038, gr-qc/0005025, gr-qc/0001060, gr- kievarwe.tex; 12/03/2001; 3:49; p.490 484 LIST OF SPEAKERSAND E-PRINTS qc/0001057, gr-qc/0001020, gr-qc/9905053, gr-qc/9811048, hep-th/9810229, hep- th/9807214, gr-qc/9806080, physics/9801016, physics/9706038, physics/9705030, physics/9704024,hep-th/9611091,hep-th/9611034,dg-ga/9609004,hep-th/9607196, hep-th/9607195,hep-th/9607194,gr-qc/9604017,gr-qc/9604016,gr-qc/9604015,gr- qc/9604014, gr-qc/9604013, gr-qc/9602010 33.Leonid Vaksman [email protected] math.QA/9904173, math.QA/9809018, math.QA/9808015, math.QA/9803074, math.QA/9909036, math.QA/9905035, math.QA/9904173, math.QA/9809038, math.QA/9809018, math.QA/9809002, math.QA/9808047, math.QA/9808037, math.QA/9808015, math.QA/9803110, math.QA/9803074, q-alg/9703005, q-alg/9603012, q-alg/9511007 34.A. VanProeyen [email protected] hep-th/0012110 hep-th/0010195,hep-th/0010194,hep-th/0007044,hep-th/0006179,hep-th/0003261, hep-th/0003023, math.DG/0002122, hep-th/9912049, hep-th/9910030, hep- th/9907124, hep-th/9904085, hep-th/9904066, hep-th/9902100, hep-th/9901060, hep-th/9812066,hep-th/9804177,hep-th/9804099,hep-th/9803228,hep-th/9801206, hep-th/9801140,hep-th/9801112,hep-th/9801102,hep-th/9712092,hep-th/9711161, hep-th/9710166,hep-th/9703082,hep-th/9703081,hep-th/9611112,hep-th/9606073, hep-th/9512139,hep-th/9510195,hep-th/9510186,hep-th/9509035,hep-th/9506075, hep-th/9505123,hep-th/9505097,hep-th/9503022,hep-th/9502072,hep-th/9412200, hep-th/9410162,hep-th/9407061,hep-th/9310067,hep-th/9307126,hep-th/9306147, hep-th/9210068, hep-th/9207091, hep-th/9206097, hep-th/9205027, hep-th/9112027 35.Pierre VanHove [email protected] hep-th/0010182,hep-th/0010167,hep-th/9910056,hep-th/9910055,hep-th/9903050, hep-th/9809130, hep-th/9712079, hep-th/9707126, hep-th/9706175, hep-th/9704145 36.DmitriVassiliev [email protected] gr-qc/0012046, math-ph/0006019 37.Mihai Visinescu [email protected] hep-th/0008181,hep-th/9911126,hep-th/9911014,hep-th/9805116,hep-th/9707175, hep-th/9610097, hep-th/9602015, hep-th/9407130, hep-th/9401036, hep-th/9304022 38.Julius Wess [email protected] hep-th/0009230, hep-th/0006246, math.QA/0006179, hep-th/0005005, math.QA/0004011, hep-th/0001203, math-ph/9910013, math.QA/9809160, math.QA/9808024, math.QA/9807123, math.QA/9801104, hep-th/9605161, hep-th/9511106, hep-ph/9505291, q-alg/9502007 39.Vladimir Zima oleﬁ[email protected] hep-th/0009166, hep-th/9807192, hep-th/9802032, hep-th/9409117 40.GeorgeZoupanos [email protected] hep-ph/0010141, hep-ph/0010069, hep-ph/0006262, hep-ph/9910277, hep-ph/9812221, hep-th/9808178, hep-th/9804074, hep-ph/9803217, hep- th/9803095, hep-ph/9802280, hep-ph/9802267, hep-th/9711157, hep-ph/9708225, hep-ph/9707425, hep-ph/9704218, hep-ph/9703289, hep-ph/9702391, hep- ph/9609218, hep-ph/9606434, hep-ph/9604216, hep-ph/9512435, hep-ph/9512400, hep-ph/9512258, hep-ph/9511304, hep-ph/9510279, hep-ph/9509434, hep- th/9506092, hep-th/9502017, hep-ph/9411222, hep-th/9409106, hep-th/9409032, hep-th/9409003, hep-ph/9210218 kievarwe.tex; 12/03/2001; 3:49; p.491