Singularities, Integrability, Symmetries (Scientific Coordinator: Francesco Calogero  Universita' La Sapienza  Roma) 
Local Research Unit: Symmetries, singularities and integrability problems in noncommutative geometry 

Scientific Coordinator
of Research Unit
Ludwik Dabrowski Location of the Research
Unit
Scientific Sectors interested
in the Research
Key Words


Human Resources 

University Personnel
of the Research Unit
 
Scientific Background 

The study of noncommutative geometry fits in a natural manner into the framework of the research
project "Singularities, Integrability and Symmetries"; in fact it yields some methods to
analyze singularities, symmetries and integrability properties and study them in a wider context, both
in the strict geometric sense and from the point of view of applications to quantum field theory,
strings and solvable models in statistical field theory. For example, it is a suitable tool to study the quotient spaces of singular foliations [C]. To such spaces one can associate some noncommutative algebras which are analogues of the algebras of functions on the usual spaces. The most known example is the noncommutative torus [CR] which is associated to the space S^1/tZ, i.e. the quotient of S^1 by the equivalence relation which identifies the points on the orbits of irrational rotations x> x + t mod 1, on S^1. Moreover the study of the quantum completely integrable systems leads in a natural manner to a generalization of usual symmetries known as quantum groups (cf. [FRT] and the references therein). Quantum groups specify also the "braiding" (statistics) of tensor products of two or more identical quantum systems cf. [M.S] and can link spin and statistics [O]. A large class of models in noncommutative geometry originates in the framework of deformation quantization (*products) [K] of the Poisson structures. Classical integrable systems are often characterized by the presence of symmetries related to LiePoisson groups (which can be viewed as the semiclassical limit of quantum groups) and/or the presence of biHamiltonian structures described via pencils of Poisson structures [M.F]. Noncommutative geometry is of great interest also in other areas, among which one can mention models of spacetime at short distances, the unification of fundamental interactions and the regularization of ultraviolet divergences in quantum field theory and string theory. However till now only field theories on very special spaces have been proposed and it is evident that more interesting examples are needed together with field configurations on them. The most studied example is given by YangMills theory on noncommutative tori mentioned above, which has been recently used in string theory [CDS, SW]. On such tori also nonlinear sigma models [DKL] have been considered. Recently some examples corresponding to instantons on spheres, in particular in four dimensions, have been constructed [CL,DLM,BCT,S,BG,DL,HS,V,LM,CDV,AB]. In general, the symmetry aspects of YangMills theories (for instance the orbits of gauge transformations) are classically studied by means of connections on principal fibre bundles. Quantum principal fibre bundles are best described as (Hopf)Galois extensions while (spaces of sections of) vector bundles appear as modules associated to the representations of quantum groups. In this framework also covariant derivatives and principal connections have been studied [BM, HM, DGH]. REFERENCES:[AB] P. Aschieri, F. Bonechi. On the Noncommutative Geometry of Twisted Spheres (math.QA/0108136) Lett. Math. Phys. in stampa[BCT] F. Bonechi, N. Ciccoli, M. Tarlini. Noncommutative Instantons on the 4Sphere from Quantum Groups. (math.QA/0012236) [BG] T. Brzezinski and C. Gonera. Noncommutative 4spheres based on all Podles 2spheres and beyond. (math.QA/0101129) [BM] T. Brzezinski, S. Majid. Quantu m Group Gauge Theory on Quantum Spaces. Commun. Math. Phys. 157 (1993) 591638. [C] A. Connes. Noncommutative geometry. Academic Press 1994. [CDS] A. Connes, M. Douglas and A. Schwarz. Noncommutative geometry and Matrix theory: compactification on tori. JHEP 2 (1998) 3 [CDV] A. Connes, M. DuboisViolette. Noncommutative finitedimensional manifolds. I. Spherical manifolds and related examples. (math.QA/0107070) [CL] A. Connes, G. Landi. Noncommutative manifolds, the instanton algebra and isospectral deformations. (math.QA/0011194) [CR] A. Connes, M. Rieffel. YangMills for Noncommutative TwoTori. Contemp. Math. 62 (1987) 237266 [DGH] L. Dabrowski, H. Grosse, P.M. Hajac. Strong Connections and ChernConnes Pairing in the HopfGalois Theory. Commun. Math. Physics 220 (2001) 301331 [DKL] L. Dabrowski, T. Krajewski, G. Landi. Some Properties of Nonlinear sigma Models in Noncommutative Geometry. Int. J. Mod. Phys. B14 (2000) 23672382 [DL] L. Dabrowski, G. Landi. Instanton algebras and quantum 4spheres. (math.QA/0101177) Differ. Geom. Appl. 16 (2002) 277284 [DLM] L. Dabrowski, G. Landi, T. Masuda. Instantons on the Quantum $4$Spheres $S^4_q$. Commun. Math. Phys. 221 (2001) 161168. [FRT] L. D. Faddeev, N. Y. Reshetikhin, L. A. Takhtajan. Quantization Of Lie Groups And Lie Algebras. Lengingrad Math. J. 1 (1990) 193 [Alg. Anal. 1 (1990) 178]. [HM] P. Hajac, S. Majid. Projective module description of the qmonopole. Commun. Math. Phys. 206 (1999) 247 [HS] J.H. Hong, W. Szymanski. Quantum spheres and projective spaces as graph algebras. University of Newcastle Preprint, January 2001 [K] M. Kontsevich. Deformation quantization of Poisson manifolds, I. (qalg/9709040) [LM] G. Landi, J. Madore. Twisted Configurations over Quantum Euclidean Spheres. (math.QA/0102195) [M.F] F. Magri. A simple model of the integrable Hamiltonian equation. J. Math. Phys. vol. 19, (1978) 1156 [M.S] S. Majid. Foundations of Quantum Group Theory. Cambridge Univ. Press 1995. [O] R. Oeckl. The Quantum Geometry of Supersymmetry and the Generalized Group Extension Problem. (hepth/0106122) [S] A. Sitarz. More noncommutative 4spheres. (mathph/0101001) [SW] N. Seiberg, E. Witten. String Theory and Noncommutative Geometry. JHEP 09 (1999) 32 [V] J.C. Varilly. Quantum symmetry groups of noncommutative spheres. (math.QA/0102065)  
Research Program Description 

The Trieste group will work on classical integrable systems equipped with the biHamiltonian
structure, in the framework of deformation quantization (*products). Families of deformations induced
by the pencils of Poisson structures, the resulting algebras and their representations will be
studied. In particular we plan to investigate the Heisenberg evolution equations, the semiclassical
limit and the existence of Lax pairs with parameter for the integrable system. On this point a
collaboration is expected with other units involved in the project which are active on analogous
topics (e.g. Roma, Napoli). A substantial part of the activity of the local unit will be devoted to the construction of gauge theories on noncommutative spaces previously mentioned, and in particular on so called ConnesLandi spheres, which possess the structure of noncommutative spin manifold. We intend to construct gauge fields (connections) on the projective modules of instantonic type, study them as solutions of suitable field equations, including the selfduality conditions, and calculate the corresponding generalized topological charges (in collaboration with the Napoli unit). On more general kind of quantum spheres the existence and the properties of a suitable Dirac operator, the space of connections, the action of the group of gauge transfromations and the corresponding orbit space will be analysed. Also the theory of nonlinear sigma models introduced by some of us on noncommutative tori and on more general quantum spaces will be developed. A related argument will be the study, from the point of view of spinstatistics relation, of the quantum principal fibre bundle F > SLq(2) > SL(2) (where q is a root of unity) which describes a quantum covering of the Spin group in dimension 3. RECENT PUBLICATIONS (19992001):[1] Bonora, C. Reina, A. Zampa. Enhanced gauge symmetries on elliptic K3. Phys. Lett. B 452 (1999) 244250[2] L. Castellani, G. Landi, F. Lizzi (Editors). Noncommutative Geometry and Hopf Algebras in Field Theory and Particle Physics. World Scientific, Singapore, 2000 [3] A. Connes e G. Landi. Noncommutative Manifolds, the Instanton algebra and isospectral deformations. Commun. Math. Phys. 221 (2001) 141159. [4] L. Dabrowski, P.M. Hajac e P. Siniscalco. Explicit HopfGalois Description of Induced Frobenius Homomorphisms, in: "Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions", D. Kastler et al. (Nova Science Publisher, Commack NY, 1999 [5] L. Dabrowski, T. Krajewski, G. Landi. Some Properties of Nonlinear sigma Models in Noncommutative Geometry. Int. J. Mod. Phys. B14 (2000) 23672382 [6] L. Dabrowski, H. Grosse, P.M. Hajac. Strong Connections and ChernConnes Pairing in the HopfGalois Theory. Commun. Math. Physics 220 (2001) 301331. [7] L. Dabrowski, G. Landi. Instanton Algebras and Quantum 4Spheres. math.QA/0101177; Diff. Geom. Appl. in stampa. [8] L. Dabrowski, G. Landi, T. Masuda. Instantons on the Quantum 4Spheres, Commun. Math. Phys. 221 (2001) 161168. [9] L. Dabrowski, C. Reina, A. Zampa. A[Slq(2)] at roots of unity is a free module over A[Sl(2)]. Lett. Math. Phys. 52 (2000) 339342. [10] G. Dell'Antonio, G. Panati. Zeroenergy resonances and the fluxacrosssurfaces theorem, mathph/0110034. [11] E. Ercolessi, G. Landi, P. TeotonioSobrinho. Ktheory of Noncommutative Lattices. KTheory 18 (1999) 339362 [12] D. Fabbri, P. Fre, L. Gualtieri, C. Reina, A. Tomasiello, A. Zaffaroni, A. Zampa. 3D superconformal theories from Sasakian sevenmanifolds: a new nontrivial evidence for AdS_4/CFT_3. Nucl.Phys. B577 (2000) 547608 [13] G. Falqui, C. Reina, A. Zampa. Super KP equations and Darboux transformations: another perspective on the Jacobian Super KP hirearchy. J. Geom. Phys. 35 (2000) 239272 [14] G. Falqui, C. Reina, A. Zampa. A note on the super Krichever map. J. Geom. Phys. 37 (2001) 169 [15] Eli Hawkins. Geometric Quantization of Vector Bundles. Commun.Math.Phys. 215 (2000) 409432 [16] Eli Hawkins. Quantization of Equivariant Vector Bundles. Commun.Math.Phys. 202 (1999) 517546 [17] G. Landi. An Introduction to Noncommutative Spaces and Their Geometries. Lecture Notes in Physics, m51 SpringerVerlag, BerlinHeidelberg, 1997 [18] G. Landi. Eigenvalues as Dynamical Variables. Quaderno Dipartimento di Scienze Matematiche, Trieste, DSMATS 450, 1999. (grqc/9906044). [19] G. Landi. Noncommutative Geometry (An Introduction to Selected Topics). Quaderno Dipartimento di Scienze Matematiche, Trieste, DSMATS 488, marzo 2001. Special issue of `Acta Applicandae Mathematicae', in stampa. [20] G. Landi. Deconstructing (Super)Monopoles. Seminari di Geometria 19992000, Dipartimento di Matematica, Universita` di Bologna, in stampa. [21] G. Landi. Deconstructing Monopoles and Instantons. Rev. Math. Phys. 12 (2000) 13671390 [22] G. Landi. Projective Modules of Finite Type over the Supersphere S^{2,2} Diff. Geom. Appl. 14 (2001) 95111. [23] G. Landi. Projective Modules of Finite Type and Monopoles over S^2, J. Geom. Phys. 37 (2001) 4762 [24] G. Landi, F. Lizzi. Projective Systems of Noncommutative Lattices as a Pregeometric Substratum, in: "Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions", D. Kastler et al. (Nova Science Publisher, Commack NY, 1999), pp. 85113. [25] G. Landi, F. Lizzi, R. Szabo. String Geometry and the Noncommutative Torus. Commun. Math. Phys. 206 (1999) 603637 [26] G. Landi, F. Lizzi, R.J. Szabo. From Large N Matrices to the Noncommutative Torus, Commun. Math. Phys. 217 (2001) 181201 [27] G. Landi, J. Madore. Twisted Configurations over Quantum Euclidean Spheres. math.QA/0102195. [28] G. Panati, H. Spohn, S. Teufel. SpaceAdiabatic Perturbation Theory. (mathph/0201055) [29] G. Panati, H. Spohn, S. Teufel. Spaceadiabatic Decoupling to All Orders. (quantph/0201123) [30] Panati G., Teta A. The fluxacrosssurfaces theorem for a point interaction hamiltonian. In: Gesztesy, F., Holden H., Jost J., Paycha S., Rockner M., Scarlatti, S. (Eds.) Stochastic Processes, Physics and Geometry: New Interplays. American Mathematical Society Providence, 1999 Rhode Island, 431442 [31] D. Perrot. BRS Cohomology And The Chern Character In NonCommutative Geometry. Lett.Math.Phys. 50 (1999) 135144 [32] D. Perrot. On the Topological Interpretation of Gravitational Anomalies. (mathph/0006003), Commun. Math. Phys., in press [33] D. Perrot. A RiemannRoch Theorem For OneDimensional Complex Groupoids. (mathph/0001040), J. Geom. Phys., in press 