String Bracket and Flat Connections

Abstract

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 $Ψ: \H_{2\ast} (L M) \to ø(\Mc)$, where $\H_{2\ast} (LM)$ 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 $Ω^\ast (M, \adp)$, 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 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.

28 pages. This is the final version

References in corpus (2)

• Notes on string topology
• De Rham model for string topology

Cited by in corpus (4)

• String topology for stacks
• On algebraic structures of the Hochschild complex
• Quantization of the Lie bialgebra of string topology
• On 2-Holonomy