String bracket and flat connections

Let $G \to P \to M$ be a flat principal bundle over a compact and oriented manifold $M$ of dimension $m = 2 d$ . We construct a map $Ψ : H_{2 *}^{S^{2}} (L M) \to O (ℳ C)$ of Lie algebras, where $H_{2 *}^{S^{2}} (L M)$ is the even dimensional part of the equivariant homology of $L M$ , the free loop space of $M$ , and $ℳ C$ is the Maurer–Cartan moduli space of the graded differential Lie algebra $Ω^{*} (M, ad P)$ , 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 $H_{2 *}^{S^{2}} (L M)$ 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.

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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

References

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

Volume 7, issue 1 (2007)

Hossein Abbaspour and Mahmoud Zeinalian

Abstract