Grégory Ginot:A (generalized) Goldman bracket for smooth stacks and equivariant (co)homology with respect to a group stack action.

This is joint work with Behrang Noohi. The Goldman bracket defines a Lie algebra structure on the free abelian group spaned by free homotopy classes of loops on a surface. It was introduced as a universal Lie algebra structure mapping to the Poisson algebras of functions on the character variety of a reductive group. This bracket was one of the main inspiration behind string topology which provides a generalization of it for oriented manifolds  of any dimension, that is  a graded Lie algebra structure on the SO(2)-equivariant homology of the free loop space of the manifold.

In this talk we explain how to extend these brackets to the case of stacks, for instance orbifolds.  To do so, one needs to define a nice theory of stacky group action and its (co)homological invariant. In the first part of the talk we will mainly discuss the traditional Goldman bracket, its string topology presentation and states our main results. We will provide some details on the quotient of a stack by a group stack and the corresponding (co)homology in the second part and explain how to deduce the Goldman bracket out of it.