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


Let GPM be a flat principal bundle over a compact and oriented manifold M of dimension m = 2d. We construct a map Ψ : H2S2 (LM)O(C) of Lie algebras, where H2S2 (LM) is the even dimensional part of the equivariant homology of LM, the free loop space of M, and C is the Maurer–Cartan moduli space of the graded differential Lie algebra Ω(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 [Invent Math 85 (1986) 263–302]. We treat different Lie algebra structures on H2S2 (LM) 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, ) and GL(n, ) together with its natural generalization to other reductive Lie groups.

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
Received: 15 January 2007
Accepted: 26 January 2007
Published: 29 March 2007
Hossein Abbaspour
Max-Planck Institut für Mathematik
Vivatsgasse 7
Bonn 53111
Mahmoud Zeinalian
Long Island University
C W Post College
Brookville NY 11548