Gauge equivalence of Dirac structures and symplectic groupoids

Bursztyn, Henrique; Radko, Olga

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.

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