2022/10/17 by Alekseev, Anton, Naef, Florian, Pulmann, Ján +1 · 1 citation
#FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2210.08944
Let Σ be a compact oriented 2-manifold (possibly with boundary), and let \mathcal GΣ be the linear span of free homotopy classes of closed oriented curves on Σ equipped with the Goldman Lie bracket [⋅, ⋅]Goldman defined in terms of intersections of curves. A theorem of Goldman gives rise to a Lie homomorphism Φeven from (\mathcal GΣ, [⋅, ⋅]Goldman) to functions on the moduli space of flat connections MΣ(G) for G=U(N), GL(N), equipped with the Atiyah-Bott Poisson bracket. The space GΣ also carries the Turaev Lie cobracket δTuraev 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 ∂Σ, we show that the moduli space of flat connections MΣ(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 Φodd\colon \wedge GΣ → Fun(MΣ(Q(N))) which replaces the Goldman map, and which captures the information both on the Goldman bracket and on the Turaev cobracket. The map Φodd is constructed using the "odd trace" function on Q(N).