Batalin-Vilkovisky structures on moduli spaces of flat connections

Let $\Sigma$ be a compact oriented 2-manifold (possibly with boundary), and let $\mathcal G_{\Sigma}$ be the linear span of free homotopy classes of closed oriented curves on $\Sigma$ equipped with the Goldman Lie bracket $[\cdot, \cdot]_\mathrm{Goldman}$ defined in terms of intersections of curves. A theorem of Goldman gives rise to a Lie homomorphism $\Phi^\mathrm{even}$ from $(\mathcal G_{\Sigma}, [\cdot, \cdot]_\text{Goldman})$ to functions on the moduli space of flat connections $\mathcal{M}_{\Sigma}(G)$ for $G=U(N), GL(N)$, equipped with the Atiyah-Bott Poisson bracket. The space $\mathcal{G}_{\Sigma}$ also carries the Turaev Lie cobracket $\delta_\mathrm{Turaev}$ defined in terms of self-intersections of curves. In this paper, we address the following natural question: which geometric structure on moduli spaces of flat connections corresponds to the Turaev cobracket? We give a constructive answer to this question in the following context: for $G$ a Lie supergroup with an odd invariant scalar product on its Lie superalgebra, and for nonempty $\partial\Sigma$, we show that the moduli space of flat connections $\mathcal{M}_{\Sigma}(G)$ carries a natural Batalin-Vilkovisky (BV) structure, given by an explicit combinatorial Fock-Rosly formula. Furthermore, for the queer Lie supergroup $G=Q(N)$, we define a BV-morphism $\Phi^\mathrm{odd}\colon \wedge \mathcal{G}_{\Sigma} \to \mathrm{Fun}(\mathcal{M}_{\Sigma}(Q(N)))$ which replaces the Goldman map, and which captures the information both on the Goldman bracket and on the Turaev cobracket. The map $\Phi^\mathrm{odd}$ is constructed using the "odd trace" function on $Q(N)$.