String bracket and flat connections

Abstract

Let G P M be a flat principal bundle over a compact and oriented manifold M of dimension m=2d. We construct a map Ψ:HS22* (LM) O(MC) of Lie algebras, where HS22* (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 Ω*(M,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 [Invent Math 85 (1986) 263-302]. We treat different Lie algebra structures on HS22*(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 [Comm Math Phys 240 (2003) 397 -421] for G=GL(n,C) and GL(n,R) together with its natural generalization to other reductive Lie groups. © 2007 Mathematical Sciences Publishers.