2003/03/31 by H. Bursztyn, M. Crainic, A. Weinstein +1 · 2 citations
Mathematics · Physics and Astronomy · #math.DG #math-ph #math.MP #math.SG
published as Integration of twisted Dirac brackets. Duke Math. J. 123 (2004), no. 3, 549--607. · 42 pages. Minor changes, typos corrected. Revised version to appear in Duke Math. J
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 ϕ 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 ϕ-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.