Gauge equivalence of Dirac structures and symplectic groupoids. (English) Zbl 1026.58019

Bursztyn, Henrique; Radko, Olga

Ann. Inst. Fourier 53, No. 1, 309-337 (2003).

Dirac structures were introduced by T. Courant and A. Weinstein in [Actions hamiltoniennes de groupes. Troisième théorème de Lie, Sémin. Sud-Rhodan. Géom. VIII, Lyon/France 1986, Trav. Cours 27, 39-49 (1988; Zbl 0698.58020)] to provide a geometric framework for the study of constrained mechanical systems. Examples of Dirac structures on a manifold include pre-symplectic forms, Poisson structures and foliations. The authors extend the notion of a symplectic dual pair (defined by A. Weinstein in [J. Differ. Geom. 18, 523-557 (1983; Zbl 0524.58011)] to a more general notion that they call a pre-symplectic pre-dual pair. This generalization consists in the substitution of the symplectic manifold in the dual pair by a pre-symplectic manifold in the pre-dual pair, Poisson manifolds by Dirac manifolds and Poisson manifolds with symplectically orthogonal fibres by Dirac maps verifying a more general condition that includes the kernel of the pre-symplectic form.
Then, the main results of the paper are: 1. Ordinary symplectic dual pairs are obtained as quotient of pre-duals pairs (under natural regularity conditions).
2. Two integrable gauge-equivalent Poisson structures are Morita equivalent (notion defined by P. Xu in Commun. Math. Phys. 142, 493-509 (1991; Zbl 0746.58034), i.e., they form part of a symplectic dual pair verifying some conditions of connectedness on the fibres. An example proves that the converse is not true.
3. A sufficient condition for Morita equivalence of Poisson structures vanishing nearly on a finite number of smooth disjoint curves on a compact connected oriented surface.

Reviewer: E.Outerelo (Madrid)

MSC:

58H05	Pseudogroups and differentiable groupoids
53D17	Poisson manifolds; Poisson groupoids and algebroids

Keywords:

Dirac structures; gauge equivalence; Morita equivalence; symplectic groupoids

Citations:

Zbl 0698.58020; Zbl 0524.58011; Zbl 0746.58034

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