arXiv2026
Let $Σ$ be a compact oriented 2-manifold (possibly with boundary), and let $\mathcal G_Σ$ be the linear span of free homotopy classes of closed oriented curves on $Σ$ 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 $Φ^\mathrm{even}$ from $(\mathcal G_Σ, [\cdot, \cdot]_\text{Goldman})$ to functions on the moduli space of flat connections $\mathcal{M}_Σ(G)$ for $G=U(N), GL(N)$, equipped with the Atiyah-Bott Poisson bracket. The space $\mathcal{G}_Σ$ also carries the Turaev Lie cobracket $δ_\mathrm{Turaev}$ 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 $\partialΣ$, we show that the moduli space of flat connections $\mathcal{M}_Σ(G)$ 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 $Φ^\mathrm{odd}\colon \wedge \mathcal{G}_Σ \to \mathrm{Fun}(\mathcal{M}_Σ(Q(N)))$ which replaces the Goldman map, and which captures the information both on the Goldman bracket and on the Turaev cobracket. The map $Φ^\mathrm{odd}$ is constructed using the "odd trace" function on $Q(N)$.