Integration of twisted Dirac brackets

TL;DR

This paper extends the correspondence between Poisson structures and symplectic groupoids to twisted Dirac structures, providing new insights into their global objects, with applications to equivariant cohomology, foliation theory, and quasi-hamiltonian spaces.

Contribution

It introduces a framework linking twisted Dirac structures with multiplicative 2-forms on Lie groupoids, generalizing classical correspondences in Poisson geometry.

Findings

Established a correspondence between multiplicative 2-forms and Lie algebroid maps for twisted Dirac structures.

Connected the theory to equivariant cohomology and foliation theory.

Provided a new description of quasi-hamiltonian spaces and group-valued momentum maps.

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 $G$ 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.