Homotopy Poisson and BV${}_{\infty }$ structures on derived moduli spaces of flat connections

Abstract

We show that quasi-Poisson and quasi-BV structures on (super)manifolds $M$ admitting an action of a Lie (super)group $G$ can be identified with a special class of homotopy Poisson and respectively $B{V}_{\infty }$ structures on the associated differential (super)manifolds of the form $\mathcal{M}:=M\times \Pi g$, $g$ being the Lie (super)algebra of $G$. The construction is based on the proven existence of an essentially unique family of higher Jacobiators for any Lie (super)algebra $g$ with an invariant (odd or even) scalar product. We apply these arguments to the moduli spaces $M$ of flat $G$-connections on a surface with boundaries using Fock-Rosly construction.

35p