Gauge equivalence of Dirac structures and symplectic groupoids
[Équivalence des structures de Dirac et groupoïdes symplectiques]
Bursztyn, Henrique ; Radko, Olga
Annales de l'Institut Fourier, Tome 53 (2003), p. 309-337 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 2003-01-01
DOI : https://doi.org/10.5802/aif.1945
Classification:  57D17,  58H05
Mots clés: structures de Dirac, équivalence de jauge, équivalence de Morita, groupoïdes symplectiques 
@article{AIF_2003__53_1_309_0,
     author = {Bursztyn, Henrique and Radko, Olga},
     title = {Gauge equivalence of Dirac structures and symplectic groupoids},
     journal = {Annales de l'Institut Fourier},
     volume = {53},
     year = {2003},
     pages = {309-337},
     doi = {10.5802/aif.1945},
     mrnumber = {1973074},
     zbl = {1026.58019},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2003__53_1_309_0}
}

Bursztyn, Henrique; Radko, Olga. Gauge equivalence of Dirac structures and symplectic groupoids. Annales de l'Institut Fourier, Tome 53 (2003) pp. 309-337. doi : 10.5802/aif.1945. http://gdmltest.u-ga.fr/item/AIF_2003__53_1_309_0/

[1] F. Bayen; M. Flato; C. Frønsdal; A. Lichnerowicz; D. Sternheimer Deformation Theory and Quantization, Ann. Phys, Tome 111 (1978), pp. 61-151 | Article | MR 496157 | Zbl 0377.53024

[2] A. Blaom A geometric setting for Hamiltonian perturbation theory, Mem. Amer. Math. Soc, Tome 153 (2001) no. 727 | MR 1848237 | Zbl 1003.70002

[3] J.-L. Brylinski A differential complex for Poisson manifolds, J. Differential Geom, Tome 28 (1988) no. 1, pp. 93-114 | MR 950556 | Zbl 0634.58029

[4] H. Bursztyn Semiclassical geometry of quantum line bundles and Morita equivalence of star products, Int. Math. Res. Notices, Tome 16 (2002), pp. 821-846 | Article | MR 1891209 | Zbl 1031.53120

[5] H. Bursztyn; S. Waldmann The characteristic classes of Morita equivalent star products on symplectic manifolds, Comm. Math. Physics, Tome 228 (2002) no. 1, pp. 103-121 | Article | MR 1911250 | Zbl 1036.53068

[6] A. Cannas Da Silva; A. Weinstein Geometric models for noncommutative algebras, American Mathematical Society, Providence, RI (1999) | MR 1747916 | Zbl 01515267

[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), Tome 81 (1965), pp. 15-30 | Article | MR 170305 | Zbl 0132.16506

[9] A. Coste; P. Dazord; A. Weinstein; I--Ii Groupoïdes symplectiques, Publications du Département de Mathématiques, Univ. Claude-BernardLyon (Nouvelle Série. A) Tome Vol. 2 (1987), pp. 1-62 | Numdam | Zbl 0668.58017

[10] T. Courant Dirac manifolds, Trans. Amer. Math. Soc, Tome 319 (1990) no. 2, pp. 631-661 | Article | MR 998124 | Zbl 0850.70212

[11] T. Courant; A. Weinstein Beyond Poisson structures, Action hamiltoniennes de groupes, Troisième théorème de Lie (Lyon, 1986), Hermann, Paris (1988), pp. 39-49 | Zbl 0698.58020

[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 2016690 | Zbl 1041.58007

[13] M. Crainic; R. Fernandes Integrability of Lie brackets (e-print. To appear in Ann. of Math., math.DG/0105033) | MR 1973056 | Zbl 1037.22003

[14] P. Dazord; T. Delzant Le problème général des variables actions-angles, J. Differential Geom, Tome 26 (1987) no. 2, pp. 223-251 | MR 906389 | Zbl 0634.58003

[15] C. Debord Groupoïdes d'holonomie de feuilletages singuliers, C. R. Acad. Sci. Paris, Sér. I Math, Tome 330 (2000) no. 5, pp. 361-364 | Article | MR 1751671 | Zbl 0948.57022

[16] V. L. Ginzburg Grothendieck Groups of Poisson Vector Bundles (e-print. To appear in J. Symplectic Geom., math.DG/0009124) | MR 1959580 | Zbl 1032.53072

[17] V. L. Ginzburg; J. H. Lu Poisson cohomology of Morita-equivalent Poisson manifolds, Internat. Math. Res. Notices, Tome 10 (1992), pp. 199-205 | Article | MR 1191570 | Zbl 0783.58026

[18] M. Gotay; R. Lashof; J. {#X015A;}Niatycki; A. Weinstein Closed forms on symplectic fibre bundles, Comment. Math. Helv, Tome 58 (1983) no. 4, pp. 617-621 | Article | MR 728456 | Zbl 0536.53040

[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), Springer, Berlin (Lecture Notes in Math) Tome Vol. 249 (1971), pp. 34-46 | Zbl 0227.55016

[20] B. Jurco; P. Schupp; J. Wess Noncommutative line bundle and Morita equivalence (e-print, hep-th/0106110) | Zbl 1036.53070

[21] C. Klimcik; T. Strobl Symplectic geometry and Mirror symmetry (Seoul, 2000), World Sci. Publishing, NJ (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 1958827

[24] J. S. Park Topological open p-branes, J. Geom. Phys, Tome 43 (2002) no. 4, pp. 341-344 | MR 1882334

[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 1959058 | Zbl 1093.53087

[26] D. Roytenberg Poisson cohomology of SU(2)-covariant ``necklace'' Poisson structures on S 2 , J. Nonlinear Math. Physics, Tome 9 (2002) no. 3, pp. 347-356 | Article | MR 1916390 | Zbl 1032.53073

[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 1029.53090

[28] A. Weinstein The symplectic "category", Differential geometric methods in mathematical physics (Clausthal, 1980), Springer, Berlin (1982), pp. 45-51 | Zbl 0486.58017

[29] A. Weinstein The local structure of Poisson manifolds, J. Differential Geom, Tome 18 (1983) no. 3, pp. 523-557 | MR 723816 | Zbl 0524.58011

[30] A. Weinstein Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), Tome 16 (1987) no. 1, pp. 101-104 | Article | MR 866024 | Zbl 0618.58020

[31] A. Weinstein Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, Tome 40 (1988) no. 4, pp. 705-727 | Article | MR 959095 | Zbl 0642.58025

[32] A. Weinstein The modular automorphism group of a Poisson manifold, J. Geom. Phys, Tome 23 (1997) no. 3-4, pp. 379-394 | Article | MR 1484598 | Zbl 0902.58013

[33] P. Xu Morita equivalence of Poisson manifolds, Comm. Math. Phys, Tome 142 (1991) no. 3, pp. 493-509 | Article | MR 1138048 | Zbl 0746.58034