Integration of twisted Dirac brackets
Henrique Bursztyn, Marius Crainic, Alan Weinstein, Chenchang Zhu
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 $T^*M$ 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 $\phi$-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.