Home page of the Research Program COFIN 2004 
Symmetries, singularities and integrability problems
in
noncommutative geometry 

Trieste Unit

sissa


  • Scientific Coordinator of the research Project
    Francesco Calogero (Universita' La Sapienza - Roma)

  • Scientific Coordinator of the Research Unit
    Ludwik Dabrowski (SISSA)

  • Location of the Research Unit
    SISSA - International School for Advanced Studies - Trieste;

  • Scientific Sectors interested in the Research
    FIS/02, MAT/07 ;

  • Key Words
    Symmetries ; Singularities ; Integrability ; Noncommutative Geometry ; Quantum groups



Human Resources:


University Personnel
Name Institute
Ludwik Dabrowski SISSA
Cesare Reina SISSA
Giovanni Landi Universita' di Trieste
Paolo Furlan Universita' di Trieste
PhD students
Name Institute
Francesco D'Andrea SISSA
Chiara Pagani SISSA
Adriano Parodi SISSA
Lucattilio Tenuta SISSA
Walter Daniel Van Suijlekom SISSA



Scientific Background


Symmetries, singularities, and integrability are natural concepts to study by methods of noncommutative geometry, which permit their analysis from a more general perspective, both in the algebraic and geometric framework as well as their applications to quantum field theory, strings and solvable models in statistical field theory. For example noncommutative geometry offers powerful tools to analyze quotient spaces of singular foliations [C1], starting from the construction of certain noncommutative algebras which are analogues of the algebras of functions on usual spaces. The prototype is the well known noncommutative torus [CR], that can be associated to the quotient space of a circle by the discrete group of irrational rotations. Such a singular quotient space can not be studied by traditional means, whereas it is possible to analyse it in noncommutative geometry, which also incorporates in a natural way its symmetry. In more complicated situations, e.g. the transverse structure of foliations, the symmetry can assume the form of a Hopf algebra describing a diffeomorphism invariant geometry [CM1]. As it is well known, Hopf algebras and quantum groups [M] have developed independently since the eighties as generalized symmetries underlying the quantum completely integrable systems (cf. [FRT] and the references therein) and also 2-dimensional conformal field theory [BLZ]. It is not yet known if analogous structures can be found in conformal field theories in dimensions greater than 2.

Noncommutative geometry is of great interest also in other areas, among which one can mention models of space-time at short distances, the unification of fundamental interactions and the regularization of ultraviolet divergences in quantum field theory and string theory. The most studied example up to now is given by Yang-Mills theory on noncommutative tori mentioned above, recently used in string theory as well [CDS, SW1]. Of particular interest is the construction in [NS] of quantum instantons on a quantum R4. The noncommutativity of the coordinates yields an additional term in the ADHM equations which produces a desingularization of the instantons corresponding to the boundary of the compactified moduli space [N]. Recently other examples of quantum instantons have been constructed (in particular on q-deformed 4-spheres) [CL, DLM, BCT, S, BG, DL, HS, LM, CD-V, AB] that require an analysis analogous to that performed in [NS]. Furthermore, in algebraic geometry novel natural noncommutative compatifications of the modular curve have been constructed, the boundary of which consists of the limiting elliptic curves corresponding to certain noncommutative geometric objects [MM, CM2].

Other interesting examples of spaces are provided by quantum groups (regarded as deformed function algebras). Their study from the point of view of noncommutative differential geometry has started only recently [MNW, CP, C2, DS, SW2, NT, K]. Particularly interesting seems the analysis of their metric and spin structure encoded in terms of a spectral triple (A, H, D) consisting of a C*-algebra A, represented by bounded operators on the Hilbert space H and a suitable operator D generalizing the Dirac operator [C3].

Quantum groups appear also in nonlinear sigma models. As is known, the Poisson brackets of the classical chiral SU(n) WZNW model (see [BDF] for the n=2 case) are governed by certain r-matrix of Poisson-Lie symmetry SL(n).They appear as the classical counterpart of the exchange relations of the quantum chiral SU(n) WZNW model, so the Poisson-Lie symmetry is upgraded upon quantization into a quantum group Uq(sl(n)) symmetry [FHIOPT]. Recently the zero-modes phase space of the chiral SU(n) WZNW model have been studied and the analogous symmetry has been found in [FHT]. It turns out that in this context it is advantageous not to equate the determinant of the zero-modes matrix to unity. In the n=2 case the zero-modes Poisson brackets have been directly determined via an Euler angles parametrization [APH]. In [DKL1, DKL2] nonlinear sigma models on quantum tori with values in two point set have been considered, but it will be extremely interesting to extend this research to other target spaces.

The relevance of the above studies to physics is evident, though extensive work is needed in order to pass to infinite dimensional case (e.g. orbits of gauge connections modulo gauge transformations or metrics modulo diffeomorphisms). This requires a development of the necessary methods and further study of field theories on quantum spaces, e.g. the mentioned Yang-Mills and nonlinear sigma models, together with their symmetries.



References


[AB] P. Aschieri, F. Bonechi. On the Noncommutative Geometry of Twisted Spheres. Lett. Math. Phys. 59 (2002) 133-156

[AFH] L. Atanasova, P. Furlan, L.K. Hadjiivanov. Zero Modes of the SU(2)_k Wess-Zumino-Novikov-Witten Model in Euler Angles Parametrization. (hep-th/0311170)

[BCT] F. Bonechi, N. Ciccoli, M. Tarlini. Noncommutative Instantons on the 4-Sphere from Quantum Groups. (math.QA/0012236)

[BDF] J. Balog, L. Dabrowski, L. Feher. Classical r-matrix and exchange algebra in WZNW and Toda theories. Phys. Lett. B244 (1990) 227-234

[BG] T. Brzezinski and C. Gonera. Non-commutative 4-spheres based on all Podles 2-spheres and beyond. Lett. Math. Phys. 54 (2000) 315-321 [BLZ] V.V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov. Integrable Structure of Conformal Field Theory III. The Yang-Baxter Relation. Commun. Math. Phys. 200 (1999) 297-324

[BM] T. Brzezinski, S. Majid. Quantum Group Gauge Theory on Quantum Spaces. Commun. Math. Phys. 157 (1993) 591--638 [C1] A. Connes. Noncommutative geometry. Academic Press 1994

[C2] A. Connes. Cyclic Cohomology, Quantum group Symmetries and the Local Index Formula for SUq(2). (math.QA/0209142)

[C3] A. Connes. Gravity coupled with matter and foundation of non-commutative geometry. Commun. Math. Phys. 182 (1996) 155-176

[CDS] A. Connes, M. Douglas, A. Schwarz. Noncommutative geometry and Matrix theory: compactification on tori. JHEP 2 (1998) 3

[CD-V] A. Connes, M. Dubois-Violette. Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples. (math.QA/0107070)

[CL] A. Connes, G. Landi. Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys. 221 (2001) 141-159

[CM1] A. Connes, H. Moscovici. Hopf algebras, Cyclic cohomology and the Tranverse Index Theorem. Commun. Math. Phys. 198 (1998) 199--246

[CM2] C. Consani, M. Marcolli. New perspectives in Arakelov geometry. (math.AG/0210357)

[CP] P.S. Chakraborty, A. Pal. Equivariant spectral triples on the quantum SU(2) group. (math.KT/0201004)

[CR] A. Connes, M. Rieffel. Yang-Mills for Non-commutative Two-Tori. Contemp. Math. 62 (1987) 237-266

[DGH] L. Dabrowski, H. Grosse, P.M. Hajac. Strong Connections and Chern-Connes Pairing in the Hopf-Galois Theory. Commun. Math. Physics 220 (2001) 301-331

[DKL1] L. Dabrowski, T. Krajewski, G. Landi. Some Properties of Non-linear sigma Models in Noncommutative Geometry. Int. J. Mod. Phys. B14 (2000)2367-2382

[DKL2] L. Dabrowski, T. Krajewski, G. Landi. Non-linear sigma-models in noncommutative geometry: fields with values in finite spaces. Mod. Phys. Lett. A18 (2003) 2371-2380

[DL] L. Dabrowski, G. Landi. Instanton algebras and quantum 4-spheres. Differ. Geom. Appl. 16 (2002) 277-284

[DLM] L. Dabrowski, G. Landi, T. Masuda. Instantons on the Quantum 4-Spheres S^4_q. Commun. Math. Phys. 221 (2001) 161-168

[DS] L. Dabrowski, A. Sitarz. Dirac Operator on the Standard Podles Quantum Sphere. Banach Center Publications 61, 49-58, 2003

[FHT] P. Furlan, L.K. Hadjiivanov, I.T. Todorov. Chiral Zero Modes of the SU(n) WZNW Model. J. Phys. A 36 (2003) 3855 - 3876

[FHIOPT] P. Furlan, L.K. Hadjiivanov, A.P. Isaev, O.V. Ogievetsky, P.N. Pyatov, I.T. Todorov. Quantum Matrix Algebra for the SU(n) WZNW Model. J. Phys. A 36 (2003) 5497 - 553.

[FP] P. Furlan, V.B. Petkova. Toward solvable 4-dimensional conformal theories. Mod. Phys. Lett. A4 (1989) 227-234

[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 q-monopole. Commun. Math. Phys. 206 (1999) 247.

[HS] J.H. Hong, W. Szymanski. Quantum spheres and projective spaces as graph algebras. Commun. Math. Phys. 232 (2002) 157--188

[K] U. Krahmer. Dirac Operators on Quantum Flag Manifolds. (QA/0305071)

[LM] G. Landi, J. Madore. Twisted Configurations over Quantum Euclidean Spheres. J. Geom. Phys. 45 (2003) 151-163

[M] S. Majid. Foundations of Quantum Group Theory. Cambridge Univ. Press 1995

[MM] Y.I. Manin, M. Marcolli. Continued fractions, modular symbols, and non-commutative geometry. (math.NT/0102006)

[MNW] T. Masuda, Y. Nakagami, J. Watanabe. Noncommutative differential geometry on the quantum SU(2) I: An algebraic viewpoint. K-Theory 4 (1990) 157--180

[N] N.A. Nekrasov. Noncommutative instantons revisited. Commun.Math.Phys. 241 (2003) 143-160

[NS] N. Nekrasov, A. Schwarz. Instantons on noncommutative R4 and (2,0) superconformal six dimensional theory. Commun.Math.Phys. 198 (1998) 689-703

[NST] N.N. Nikolov, Y.S. Stanev, I.T. Todorov. Globally conformal invariant gauge field theory with rational correlation functions. Nucl. Phys. B670 (2003) 373-400

[NT] S. Neshveyev and L. Tuset. A local index formula for the quantum sphere. (math.qa/0309275)

[S] A. Sitarz. More noncommutative 4-spheres. (math-ph/0101001)

[SW1] N. Seiberg, E. Witten. String Theory and Noncommutative Geometry. JHEP 09 (1999) 32

[SW2] K. Schmuedgen, E. Wagner. Dirac operator and a twisted cyclic cocycle on the standard Podles quantum sphere. (math.QA/0305051)



Research Program Description



The Trieste local unit of the project is formed by researchers which have been active for several years in the field of noncommutative geometry, quantum groups and applications in mathematical physics and related areas. It is hosted by the mathematical physics group of SISSA, which is a graduate school with a regular program of doctoral fellowships and an active center of research at an international level. The collaborations with the University of Trieste and the nearby Abdus Salam International Center for Theoretical Physics create a stimulating scientific environment with several joint activities such as seminars, workshops and schools. In particular, in the area of noncommutative geometry an international symposium was organized in March 2001 and another one is planned for October 2004. The infrastructure (library, computer facilities) is of a high level and is adequate for a realization of the proposed research program.

The Trieste group will study symmetries, singularities and integrability using methods of noncommutative geometry. A substantial part of the work will be devoted to the construction of gauge theories on noncommutative spaces mentioned in point 2.4. and in particular on the so called Connes-Landi spheres, that possess the structure of a noncommutative spin manifold. We plan to construct Yang-Mills gauge fields (connections) on the projective modules of instantonic type and study them as solutions of suitable field equations, e.g. the selfduality conditions. Also a systematic analysis of the space of connections, the action of the group of gauge transformations and the corresponding orbit space (moduli space) is planned. Following [NS, N] we also plan to investigate if the ADHM method can be applied and if it provides a desingularization of the instantons on the q-deformed 4-spheres and other quantum spaces on which examples of instantons have been constructed recently. This part of research is planned in collaboration with other local units of the project.

On more general kind of quantum spheres, including the two-parameter family of Podles spheres (e.g. the equatorial one) and the 3-sphere SUq(2), we intend to construct spectral triples with real structure and a Dirac operator asking it to be a first order differential operators (in a suitable sense). Moreover, we plan to analyze the smoothness (regularity) properties. If the spectral triple is regular a pseudodifferential calculus can be introduced, but it seems that on q-deformed spaces a certain modified version of this property can be used instead (cf. [NT]). Another issue to study is the summability of a spectral triple and the question of the metric dimension (or 'dimension spectrum') of a q-deformed space (it is known that there occurs a drop of the cohomological dimesion by 2, cf. [MNW] for the case of quantum 2 and 3 spheres). A related question concerns the orientation axiom: we shall study if it can be appropriately settled on a q-deformed space. This part of research is planned in collaboration with other local units of the project.

An additional argument will be a further study of nonlinear sigma models and their symmetries. In particular on noncommutative tori more complicated target spaces will be considered. It is planned to study also more general quantum spaces as a base space. The future activity will concern also the so-called logarithmic conformal quantum field theories and an investigation of conformal quantum field theories in dimensions larger than two, cf. [NST] which revived the interest in this item. This part of research is planned in collaboration with other local units.