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).
