no. 1
Gauge equivalence of Dirac structures and symplectic groupoids
[Équivalence des structures de Dirac et groupoïdes symplectiques]
Bursztyn, Henrique1 ; Radko, Olga2
1 University of Toronto, Department of Mathematics, Toronto, Ontario M5S 3G3 (Canada)
2 University of California, Department of Mathematics, Berkeley CA 94720 (USA)
Annales de l'Institut Fourier, Tome 53 (2003) no. 1, pp. 309-337.

Nous étudions les transformations de jauge des structures de Dirac et la relation entre les équivalences de jauge et de Morita pour les variétés de Poisson. Nous décrivons comment la structure symplectique d’un groupoïde symplectique est modifiée lors d’une transformation de jauge de la structure de Poisson de la section identité de ce groupoïde et nous prouvons que des structures de Poisson intégrables équivalentes sous une transformation de jauge sont équivalentes au sens de Morita. Comme exemple, nous étudions certains ensembles génériques de structures de Poisson sur les surfaces de Riemann : nous exhibons des invariants complets d’équivalence de jauge pour de telles structures qui, sur la sphère S 2 , donnent un invariant complet d’équivalence de Morita.

We study gauge transformations of Dirac structures and the relationship between gauge and Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a symplectic groupoid is affected by a gauge transformation of the Poisson structure on its identity section, and prove that gauge-equivalent integrable Poisson structures are Morita equivalent. As an example, we study certain generic sets of Poisson structures on Riemann surfaces: we find complete gauge-equivalence invariants for such structures which, on the 2-sphere, yield a complete invariant of Morita equivalence.

DOI : 10.5802/aif.1945
Classification : 57D17, 58H05
Keywords: Dirac structures, gauge equivalence, Morita equivalence, symplectic groupoids
Mot clés : structures de Dirac, équivalence de jauge, équivalence de Morita, groupoïdes symplectiques

Bursztyn, Henrique 1 ; Radko, Olga 2

1 University of Toronto, Department of Mathematics, Toronto, Ontario M5S 3G3 (Canada)
2 University of California, Department of Mathematics, Berkeley CA 94720 (USA) 
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Bursztyn, Henrique; Radko, Olga. Gauge equivalence of Dirac structures and symplectic groupoids. Annales de l'Institut Fourier, Tome 53 (2003) no. 1, pp. 309-337. doi : 10.5802/aif.1945. https://aif.centre-mersenne.org/articles/10.5802/aif.1945/

[1] F. Bayen; M. Flato; C. FrØnsdal; A. Lichnerowicz; D. Sternheimer Deformation Theory and Quantization, Ann. Phys, Volume 111 (1978), pp. 61-151 | DOI | MR | Zbl

[2] A. Blaom A geometric setting for Hamiltonian perturbation theory, Mem. Amer. Math. Soc, Volume 153 (2001) no. 727 | MR | Zbl

[3] J.-L. Brylinski A differential complex for Poisson manifolds, J. Differential Geom, Volume 28 (1988) no. 1, pp. 93-114 | MR | Zbl

[4] H. Bursztyn Semiclassical geometry of quantum line bundles and Morita equivalence of star products, Int. Math. Res. Notices, Volume 16 (2002), pp. 821-846 | DOI | MR | Zbl

[5] H. Bursztyn; S. Waldmann The characteristic classes of Morita equivalent star products on symplectic manifolds, Comm. Math. Physics, Volume 228 (2002) no. 1, pp. 103-121 | DOI | MR | Zbl

[6] A. Cannas da Silva; A. Weinstein Geometric models for noncommutative algebras, American Mathematical Society, Providence, RI, 1999 | MR | Zbl

[7] A. Cattaneo; G. Felder Poisson sigma-models and symplectic groupoids (e-print, math.SG/0003023)

[8] L. S. Charlap Compact flat riemannian manifolds. I, Ann. of Math (2), Volume 81 (1965), pp. 15-30 | DOI | MR | Zbl

[9] A. Coste; P. Dazord; A. Weinstein; i--ii Groupoïdes symplectiques (Nouvelle Série. A), Volume Vol. 2 (1987), pp. 1-62 | Numdam | Zbl

[10] T. Courant Dirac manifolds, Trans. Amer. Math. Soc, Volume 319 (1990) no. 2, pp. 631-661 | DOI | MR | Zbl

[11] T. Courant; A. Weinstein Beyond Poisson structures, Action hamiltoniennes de groupes, Troisième théorème de Lie (Lyon, 1986) (1988), pp. 39-49 | Zbl

[12] M. Crainic Differentiable and algebroid cohomology, van Est isomorphisms and characteristic classes (e-print. To appear in Comm. Math. Helv., math.DG/0008064) | MR | Zbl

[13] M. Crainic; R. Fernandes Integrability of Lie brackets (e-print. To appear in Ann. of Math., math.DG/0105033) | MR | Zbl

[14] P. Dazord; T. Delzant Le problème général des variables actions-angles, J. Differential Geom, Volume 26 (1987) no. 2, pp. 223-251 | MR | Zbl

[15] C. Debord Groupoïdes d'holonomie de feuilletages singuliers, C. R. Acad. Sci. Paris, Sér. I Math, Volume 330 (2000) no. 5, pp. 361-364 | DOI | MR | Zbl

[16] V. L. Ginzburg Grothendieck Groups of Poisson Vector Bundles (e-print. To appear in J. Symplectic Geom., math.DG/0009124) | MR | Zbl

[17] V. L. Ginzburg; J. H. Lu Poisson cohomology of Morita-equivalent Poisson manifolds, Internat. Math. Res. Notices, Volume 10 (1992), pp. 199-205 | DOI | MR | Zbl

[18] M. Gotay; R. Lashof; J. {#x015A;}niatycki; A. Weinstein Closed forms on symplectic fibre bundles, Comment. Math. Helv, Volume 58 (1983) no. 4, pp. 617-621 | DOI | MR | Zbl

[19] P. Hilton; G. Mislin; J. Roitberg Sphere bundles over spheres and non-cancellation phenomena, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, Wash., 1971) (Lecture Notes in Math), Volume Vol. 249 (1971), pp. 34-46 | Zbl

[20] B. Jurco; P. Schupp; J. Wess Noncommutative line bundle and Morita equivalence (e-print, hep-th/0106110) | Zbl

[21] C. Klimcik; T. Strobl Symplectic geometry and Mirror symmetry (Seoul, 2000) (2001), pp. 311-384

[22] M. Kontsevich Deformation Quantization of Poisson Manifolds, I (e-print, q-alg/9709040)

[23] N. Landsman Quantization as a functor (e-print, math-ph/0107023) | MR

[24] J. S. Park Topological open p-branes, J. Geom. Phys, Volume 43 (2002) no. 4, pp. 341-344 | MR

[25] O. Radko A classification of topologically stable Poisson structures on a compact oriented surface (e-print. To appear in J. Symplectic Geom., math.SG/0110304) | MR | Zbl

[26] D. Roytenberg Poisson cohomology of SU(2)-covariant ``necklace'' Poisson structures on S 2 , J. Nonlinear Math. Physics, Volume 9 (2002) no. 3, pp. 347-356 | DOI | MR | Zbl

[27] S. Severa; A. Weinstein Poisson geometry with a 3-form background (2001) (e-print. Proceedings of the International Workshop on Noncommutative Geometry and String Theory, Keio University, math.SG/0107133) | Zbl

[28] A. Weinstein The symplectic "category", Differential geometric methods in mathematical physics (Clausthal, 1980) (1982), pp. 45-51 | Zbl

[29] A. Weinstein The local structure of Poisson manifolds, J. Differential Geom, Volume 18 (1983) no. 3, pp. 523-557 | MR | Zbl

[30] A. Weinstein Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), Volume 16 (1987) no. 1, pp. 101-104 | DOI | MR | Zbl

[31] A. Weinstein Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, Volume 40 (1988) no. 4, pp. 705-727 | DOI | MR | Zbl

[32] A. Weinstein The modular automorphism group of a Poisson manifold, J. Geom. Phys, Volume 23 (1997) no. 3-4, pp. 379-394 | DOI | MR | Zbl

[33] P. Xu Morita equivalence of Poisson manifolds, Comm. Math. Phys, Volume 142 (1991) no. 3, pp. 493-509 | DOI | MR | Zbl

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