String Bracket and Flat Connections

Let $G\to P\to M$ be a flat principal bundle over a closed and oriented manifold $M$ of dimension $m=2d$. We construct a map of Lie algebras $\Psi: \H_{2\ast} (L M) \to {\o}(\Mc)$,where$\H_{2\ast} (LM)$ is the even dimensional part of the equivariant homology of $LM$, the free loop space of $M$, and $\text{Mc}$ is the Maurer-Cartan moduli space of the graded differential Lie algebra ${\mathrm{\Omega }}^{\ast }(M,\text{adp})$, 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 in \cite{G2}. We treat different Lie algebra structures on \H_{2\ast}(LM) depending on the choice of the linear reductive Lie group $G$ in our discussion.