Integrability of Poisson brackets

TL;DR

This paper establishes the equivalence of various notions of integrability for Poisson brackets, identifies obstructions to integrating Poisson manifolds, and characterizes integration as a symplectic quotient, with implications for symplectic realizations.

Contribution

It provides a unified framework for understanding integrability of Poisson brackets and describes obstructions in terms of symplectic areas, extending the theory of Poisson manifold integration.

Findings

All notions of integrability for Poisson brackets are equivalent.

Obstructions to integrating Poisson manifolds are characterized by symplectic area variations.

A Poisson manifold admits a complete symplectic realization if and only if it is integrable.

Abstract

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.