String bracket and flat connections

Let G --> P --> M be a flat principal bundle over a compact and oriented manifold M of dimension m = 2d. We construct a map Psi: H-2*(S1), (LM) --> O(MC) of Lie algebras, where H-2*(S1) (LM) is the even dimensional part of the equivariant homology of LM, the free loop space of M, and MC is the Maurer-Cartan moduli space of the graded differential Lie algebra Omega* (M, ad P). 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 [17]. We treat different Lie algebra structures on H-2*(S1) (LM) 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 [3] for G = GL(n, C) and GL(n, R) together with its natural generalization to other reductive Lie groups.