String Bracket and Flat Connections

Hossein Abbaspour (1) , Mahmoud Zeinalian (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 Long Island University

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 $\Mc$ is the Maurer-Cartan moduli space of the graded differential Lie algebra $\Omega^\ast (M, \adp)$, 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.

Domaines: Topologie algébrique [math.AT]; Topologie géométrique [math.GT]

Hossein Abbaspour, Mahmoud Zeinalian. String Bracket and Flat Connections. Algebraic & Geometric Topology, 2007, 7, pp.197-231. ⟨hal-00442837⟩