Editors Steven Duplij and Julius Wess Noncommutative Structures in Mathematics and Physics Proceedings of the NATO Advanced Research Workshop “NONCOMMUTATIVE STRUCTURES IN MATHEMATICS AND PHYSICS” Kiev, Ukraine September 24-28, 2000 kievarwe.tex; 12/03/2001; 3:49; p.1 v Editors Steven Duplij Theory Group Nuclear Physics Laboratory Kharkov National University Kharkov 61077 Ukraine Julius Wess Sektion Physik Ludwig-Maximilians-Universit ¨ at Theresienstr. 37 D-80333 M ¨ unchen Germany Compiling and making-up by Steven Duplij. kievarwe.tex; 12/03/2001; 3:49; p.2 CONTENTS P REFACE viii J. Wess Gauge Theories Beyond Gauge Theory 1 D. Leites, V. Serganova Symmetries Wider Than Supersymmetry 13 K. Stelle Tensions in Supergravity Braneworlds 31 P. Grozman, D. Leites An Unconventional Supergravity 41 E. Bergshoeff, R. Kallosh, A. Van Proeyen Supersymmetry Of RS Bulk And Brane 49 D. Galtsov, V. Dyadichev D-branes And Vacuum Periodicity 61 P. Kosi ´ nski, J. Lukierski, P. Ma ´ slanka Quantum Deformations Of Space-Time SUSY And Noncommutative Superfield Theory 79 D. Leites, I. Shchepochkina The Howe Duality And Lie Superalgebras 93 A. Sergeev Enveloping Algebra Of GL(3) And Orthogonal Polynomials 113 S. Duplij, W. Marcinek Noninvertibility, Semisupermanifolds And Categories Regularization 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 And Quantum Corrections To Superspace Torsion Constraints 153 S. Fedoruk, V. G. Zima Massive Superparticle With Spinorial Central Charges 161 A. Burinskii Rotating Super Black Hole as Spinning Particle 181 F. Toppan Classifying N-extended 1-dimensional Super Systems 195 C. Quesne Para, Pseudo, And Orthosupersymmetric Quantum Mechanics And Their Bosonization 203 A. Frydryszak Supersymmetric Odd Mechanical Systems And Hilbert Q-module Quantization 215 S. Vacaru, I. Chiosa, N. Vicol Locally Anisotropic Supergravity And Gauge Gravity On Noncommutative Spaces 229 T. Kobayashi, J. Kubo, M. Mondrag ´ on, G. Zoupanos Finiteness In Conventional N = 1 GUTs 245 kievarwe.tex; 12/03/2001; 3:49; p.3 CONTENTS vii J. Simon World Volume Realization Of Automorphisms 259 G. Fiore, M. Maceda, J. Madore Some Metrics On The Manin Plane 271 V. Lyubashenko Coherence Isomorphisms For A Hopf Category 283 A. Ganchev Fusion Rings And Tensor Categories 295 V. Mazorchuk On Categories Of Gelfand-Zetlin Modules 299 D. Shklyarov, S. Sinel’shchikov, L. Vaksman Hidden Symmetry Of Some Algebras Of q-differential Operators 309 P. Jorgensen, D. Proskurin, Y. Samoilenko A Family Of ∗-Algebras Allowing Wick Ordering: Fock Representations And Universal Enveloping C ∗ -Algebras 321 A. U. Klimyk Nonstandard Quantization Of The Enevloping Algebra U(so(n)) And Its Applications 331 A. Gavrilik Can the Cabibbo mixing originate from noncommutative extra dimensions? 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 Quasiparticles In Non-commutative Field Theory 369 A. Sergyeyev Time Dependence And (Non)Commutativity Of Symmetries Of Evolution Equations 379 B. Dragovich, I. V. Volovich p-Adic Strings And Noncommutativity 391 G. Djordjevi ´ c, B. Dragovich, L. Ne ˇ si ´ c Adelic Quantum Mechanics: Nonarchimedean And Noncommutative Aspects 401 Y. Kozitsky Gibbs States Of A Lattice System Of Quantum Anharmonic Oscillators 415 D. Vassiliev A Metric-Affine Field Model For The Neutrino 427 M. Visinescu Generalized Taub-NUT Metrics And Killing-Yano Tensors 441 V. Dzhunushaliev An Effective Model Of The Spacetime Foam 453 A. Higuchi Possible Constraints On String Theory In Closed Space With Symmetries 465 A. Alscher, H. Grabert Semiclassical Dynamics 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 The concepts of noncommutative space-time and quantum groups have found 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. Superstrings possess by far the largest set of gauge symmetries ever found in physics, perhaps even large enough to eliminate all divergences in quantum gravity. Not only does superstring’s symmetry include that of Einstein’s theory of general relativity and the Yang-Mills theory, it also includes supergravity and the Grand Unified Theories. One of the exciting new approaches to nonperturbative string theory involves M-theory and duality, which, in fact, force theoretical physicists to reconsider the 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 unification cannot, however, occur at the perturbative level, because it is precisely 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, or more simply ‘branes’, is a natural step to take. The branes are extended objects present in M-theory or in string theories, generally associated to classical solutions of the respective supergravities. 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 groups are interpreted as the endomorphisms of certain noncommutative algebraic varieties defined by quadratic algebras, called quantum linear spaces. L. D. Fad- deev and his collaborators had also interpreted the quantum groups from the point 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 construction of the coaction on the tensor product space involves the flip operator 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 to q-deformations the flip operator is replaced with a braiding giving rise to the quasi-tensor category of k-modules, where a natural 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. Next step in this direction is consideration of noncommutative space-time as 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 mechanical system. The definition of this noncommutative phase space is derived from the noncommutative differential structure on the configuration space. The noncommutative phase space is a q-deformation of the quantum mechanical phase 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 of a noncommutative structure, first derive a calculus entirely based on the algebra and then formulate laws of physics based on this calculus. Bringing together scientists from quantumfieldtheory, string theory and quan- 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, talks and reports. We hope that common projects can be found such that 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 from NATO, East European countries and NIS. We are greatly indebted to the NATO Division of Scientific Affairs for funding of our meeting and to the National Academy of Sciences of Ukraine for help in its local organizing. It is also a great pleasure to thank all the people who contributed to the successful organization of the Workshop, especially members of the Local Organizing Committee Profs. N. Chashchyn and P. Smalko. Finally, we would like to thank all the participants for creating an excellent working atmosphere and for outstanding contributions to this volume. Editors kievarwe.tex; 12/03/2001; 3:49; p.7 GAUGE THEORIES BEYOND GAUGE THEORY JULIUS WESS Sektion Physik der Ludwig-Maximilians-Universit ¨ at Theresienstr. 37, D-80333 M ¨ unchen, Germany and Max-Planck-Institut f ¨ ur Physik (Werner-Heisenberg-Institut) F ¨ ohringer Ring 6, D-80805 M ¨ unchen, Germany 1. Algebraic preliminaries In gauge theories we consider differentiable manifolds as base manifolds and fi- bres that carry a representation of a Lie group. In the following we shall show that it is possible to replace the differentiable manifold by a non-commutative algebra, ref. [1]. For this purpose we first focus our attention on algebraic properties. The coordinates x i x 1 , . . . , x n ∈ R, (1) are considered as elements of an algebra over C subject 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 ideal I R . From the algebraic point of view, we deal with the algebra freely generated by the elements x i and divided by the ideal I R : A x = C  [x 1 , . . . , x n ]  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 expansion at the origin: f(x 1 , . . . , x n ) ∈ A x , (4) f(x 1 , . . . , x n ) = ∞  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 pointwise multiplication of these functions. The monomials of fixed degree form a finite-dimensional subspace of the alge- bra. This algebraic concept can be easily generalized to non-commutative spaces. We consider algebras freely generated by elements ˆx 1 , . . . ˆx n , again calling them coordinates. But now we change the relations to arrive at non-commutative spaces: 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 << ˆ x 1 , . . . , ˆ x n >> I R ˆx,ˆx , (6) ˆ f ∈ A ˆx . In the following we impose one more condition on the algebra: the dimension 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). Only algebras with this property will be considered, among them are the algebras where θ ij is a constant: Canonical structure, ref. [2]: [ˆx i , ˆx j ] = iθ ij , (7) where θ ij is linear in ˆx: Lie structure, ref. [3]: [ˆx i , ˆx j ] = iθ ij k ˆx k , (8) where θ ij is quadratic in ˆx: Quantum space structure, ref.[4]: [ˆx i , ˆx j ] = iθ ij kl ˆx k ˆx l , (9) The constants θ ij k and θ 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 and A ˆx . It is based on the isomorphism of the vector spaces of homogeneous polynomials that have the same degree due to the PBW property. In order to establish the isomorphism we choose a particular basis in the vec- tor space of homogeneous polynomials in the non-commuting variables ˆx and 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 GAUGE THEORIES BEYOND GAUGE THEORY 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 an algebra isomorphism. To establish it we compute the coefficient function of the product of two elements in A ˆx and map it to A x . This defines a product in A § that we denote as diamond product (♦ product). The algebra with this ♦ product we call ♦ A x . There is a natural isomorphism: 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 from the basis of completely symmetrized monomials: (f ∗g)(x) = e i 2 ∂ ∂x i θ ij ∂ ∂y j f(x)g(y)    y⇒x (11) =  d n y δ n (x − y)e i 2 ∂ ∂x i θ ij ∂ ∂y j f(x)g(y). For the Lie structure we can use the Baker-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)    y→x z→x . (13) For the quantum plane we consider the example of the Manin plane ˆxˆy = qˆyˆx, (14) (f ∗g)(x) = q −x  ∂ ∂x  y ∂ ∂y f(x, y)g(x  , y  )    x  →x y  →y . It is natural to use the elements of ♦ A x as objects in physics. Fields of a field theory will be such objects. 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 [...]... that 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... Jurˇ o, P Schupp and J Wess, Noncommutative gauge theory for Poisson manifolds, Nucl c Phys B 584, (2000), 784, hep-th/0005005 B Jurˇ o, P Schupp and J Wess, Nonabelian noncommutative gauge theory and Seibergc Witten map, in preparation N Seiberg and E Witten, String theory and noncommutative geometry, JHEP 9909 (1999) 032, hep-th/9908142 A Dimakis, J Madore, Differential Calculi and Linear Connections,... Heisenberg Algebras, in H Gausterer, H Grosse and L Pittner, eds., Proceedings of the 38 Internationale Universit¨ tswochen f¨ r Kern- und Teilchenphysik, no 543 a u in Lect Notes in Phys., Springer-Verlag, 2000, Schladming, January 1999, math-ph/9910013 F Bayen, M Flato, C Fronsdal, A Lichnerowicz, D Sternheimer, Deformation theory and quantization I Deformations of symplectic structures, Ann Physics 111,... 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, introduced in what follows How noncommutative should F (X) be? To define the space corresponding to an arbitrary algebra is very hard, see Manin’s gloomy remarks in [33], where... language of differential 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 disappear in the theory we propose Specifically, we continue the study started under Berezin’s in uence in [25] (later suppressed under the same in uence in [5], [26]), of algebras just slightly more general than supercommutative... through map: the composition of an embedding h ⊂ g into a minimal ambient and a representation g −→ gl(V )” is too restrictive: the adjoint representation and homomorphisms of Volichenko algebras are ruled out 3) If we abandon the technical hypothesis on epimorphy, do we obtain any simple Volichenko algebras? (Conjecture: we do not.) 4) Describe Volichenko algebras intrinsically, via polynomial identities... superalgebras of interest, e.g., of polynomial growth, cf [16], [17] fr ∂r 2.6 Vectorial Volichenko superalgebras For a vector field D = from vect(m|n) = derC[x, θ], define its inverse order with respect to the nonstandard (if m = 0) grading induced by the grading of C[x, θ] (for which deg xi = 0 and deg θj = 1 for all i and j) and inv.ord(fr ) is the least of the degrees of monomials in the power series... Lie algebras in mathematics and physics, Rev Mod Phys 47, 1975, 573–609 Connes A., Noncommutative geometry Academic Press, Inc., San Diego, CA, 1994 xiv+661 pp 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... Markov M et al (eds) Group–theoretical methods in physics (Zvenigorod, 1982), v 1, Nauka, Moscow, 1983, 274– 278 (Harwood Academic Publ., Chur, 1985, Vol 1–3 , 631–637) Grozman P., Leites D., From supergravity to ballbearings In: Wess J., Ivanov E (eds.) Procedings of the Internatnl seminar in the memory of V Ogievetsky, Dubna 1997, Springer Lect Notes in Physics, 524,1999, 58–67 Grozman P., Leites D.,... 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 therein pertaining to this topic Observe also an obvious connection of Volichenko 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 . and Julius Wess Noncommutative Structures in Mathematics and Physics Proceedings of the NATO Advanced Research Workshop NONCOMMUTATIVE STRUCTURES IN MATHEMATICS. 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.