String bracket and flat connections

Abbaspour, Hossein; Zeinalian, Mahmoud

doi:10.2140/agt.2007.7.197

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String bracket and flat connections

Hossein Abbaspour and Mahmoud Zeinalian

Algebraic & Geometric Topology 7 (2007) 197–231

DOI: 10.2140/agt.2007.7.197

arXiv: math/0602108

Abstract

Let $G \to P \to M$ be a flat principal bundle over a compact and oriented manifold $M$ of dimension $m = 2 d$ . 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.

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)