Batalin-Vilkovisky structures on moduli spaces of flat connections

Abstract

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

58 pages, many figures. Accepted version

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