Integration of twisted Dirac brackets
arXiv:math/0303180
Abstract
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 over a manifold , we show that multiplicative 2-forms on relatively closed with respect to a closed 3-form on correspond to maps from the Lie algebroid of into the cotangent bundle of , satisfying an algebraic condition and a differential condition with respect to the -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.
42 pages. Minor changes, typos corrected. Revised version to appear in Duke Math. J