#### Volume 7, issue 1 (2007)

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String bracket and flat connections

### Hossein Abbaspour and Mahmoud Zeinalian

Algebraic & Geometric Topology 7 (2007) 197–231
 arXiv: math/0602108
##### Abstract

Let $G\to P\to M$ be a flat principal bundle over a compact and oriented manifold $M$ of dimension $m=2d$. We construct a map $\Psi :{H}_{2\ast }^{{S}^{2}}\left(LM\right)\to \mathsc{O}\left(\mathsc{ℳ}\mathsc{C}\right)$ of Lie algebras, where ${H}_{2\ast }^{{S}^{2}}\left(LM\right)$ is the even dimensional part of the equivariant homology of $LM$, the free loop space of $M$, and $\mathsc{ℳ}\mathsc{C}$ is the Maurer–Cartan moduli space of the graded differential Lie algebra ${\Omega }^{\ast }\left(M,adP\right)$, 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 ${H}_{2\ast }^{{S}^{2}}\left(LM\right)$ 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 $G=GL\left(n,ℂ\right)$ and $GL\left(n,ℝ\right)$ together with its natural generalization to other reductive Lie groups.

##### Keywords
free loop space, string bracket, flat connections, Hamiltonian reduction, Chen iterated integrals, generalized holonomy, Wilson loop
##### Mathematical Subject Classification 2000
Primary: 55P35
Secondary: 57R19, 58A10
##### Publication
Received: 15 January 2007
Accepted: 26 January 2007
Published: 29 March 2007
##### Authors
 Hossein Abbaspour Max-Planck Institut für Mathematik Vivatsgasse 7 Bonn 53111 Germany http://guests.mpim-bonn.mpg.de/abbaspou/ Mahmoud Zeinalian Long Island University C W Post College Brookville NY 11548 USA