String bracket and flat connections

Abstract

Let $G\to P\to M$ be a flat principal bundle over a compact and oriented manifold $M$ of dimension $m=2d$. We construct a map $\Psi :H_{2\ast }^{S^2}\left(LM\right)\to \mathcal{O}\left(\mathcal{ℳ}\mathcal{C}\right)$ of Lie algebras, where $H_{2\ast }^{S^2}\left(LM\right)$ is the even dimensional part of the equivariant homology of $LM$, the free loop space of $M$, and $\mathcal{ℳ}\mathcal{C}$ is the Maurer–Cartan moduli space of the graded differential Lie algebra ${\Omega }^{\ast }\left(M,adP\right)$, the differential forms with values in the associated adjoint bundle of $P$. For a $2$–dimensional manifold $M$, our Lie algebra map reduces to that constructed by Goldman [Invent Math 85 (1986) 263–302]. We treat different Lie algebra structures on $H_{2\ast }^{S^2}\left(LM\right)$ depending on the choice of the linear reductive Lie group $G$ in our discussion. This paper provides a mathematician-friendly formulation and proof of the main result of Cattaneo, Frohlich and Pedrini [Comm Math Phys 240 (2003) 397–421] for $G=GL\left(n,ℂ\right)$ and $GL\left(n,ℝ\right)$ together with its natural generalization to other reductive Lie groups.