Gauge equivalence of Dirac structures and symplectic groupoids
Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 309-337.

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.

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.

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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     title = {Gauge equivalence of {Dirac} structures and symplectic groupoids},
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     volume = {53},
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Bursztyn, Henrique; Radko, Olga. Gauge equivalence of Dirac structures and symplectic groupoids. Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 309-337. doi : 10.5802/aif.1945.

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