String bracket and flat connections (Q2464749)

This paper studies the relation between the Chas-Sullivan string bracket and the Poisson structure on the so-called Maurer-Cartan moduli space. This is a higher dimensional analogue of Goldman's famous result on the Lie algebra of curves on surfaces and the Poisson structure of the space of flat bundles on surfaces. Let us fix a closed manifold of even dimension and a flat \(G\)-principal bundle and let us consider the Maurer-Cartan moduli space \(MC\) which is a natural generalization of the space of solutions of the Maurer-Cartan equation. The authors study the symplectic structure of this space. The space of functions \(O(MC)\) is a Poisson algebra. The \(S^1\)-equivariant homology of the free loop space \(\mathcal LM\) comes equipped with the Chas-Sullivan string bracket or its avatars which is a higher dimensional analogue of Goldman's bracket. Using iterated integrals the authors construct a morphism of Lie algebras \[ H^{S^1}_*(\mathcal LM)\rightarrow O(MC). \]