Integrability of Poisson brackets

Details Integrability of Poisson brackets Integrability of Poisson brackets

2002-10-10

arxiv.org/abs/math/0210152v2

Mathematics

Differential Geometry

43 pages, 1 figure

Scientific paper

We show that various notions of integrability for Poisson brackets are all equivalent, and we give the precise obstructions to integrating Poisson manifolds. We describe the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express the obstructions in terms of variations of symplectic areas. As an application of these results, we show that a Poisson manifold admits a complete symplectic realization if, and only if, it is integrable. We discuss also the integration of submanifolds and Morita equivalence of Poisson manifolds.

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