Integration of twisted Dirac brackets

Details Integration of twisted Dirac brackets Integration of twisted Dirac brackets

2003-03-14

arxiv.org/abs/math/0303180v2

Integration of twisted Dirac brackets. Duke Math. J. 123 (2004), no. 3, 549--607.

Mathematics

Differential Geometry

42 pages. Minor changes, typos corrected. Revised version to appear in Duke Math. J

Scientific paper

The correspondence between Poisson structures and symplectic groupoids, analogous to the one of Lie algebras and Lie groups, plays an important role in Poisson geometry; it offers, in particular, a unifying framework for the study of hamiltonian and Poisson actions. In this paper, we extend this correspondence to the context of Dirac structures twisted by a closed 3-form. More generally, given a Lie groupoid $G$ over a manifold $M$, we show that multiplicative 2-forms on $G$ relatively closed with respect to a closed 3-form $\phi$ on $M$ correspond to maps from the Lie algebroid of $G$ into the cotangent bundle $T^*M$ of $M$, satisfying an algebraic condition and a differential condition with respect to the $\phi$-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-hamiltonian spaces and group-valued momentum maps.

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