Geodesic
String bracket and flat connections
Abbaspour, Hossein1 ; Zeinalian, Mahmoud2
1Max-Planck Institut für Mathematik, Vivatsgasse 7, Bonn 53111, Germany
2Long Island University, C W Post College, Brookville NY 11548, USA
Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 197-231
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Let G→P→M be a flat principal bundle over a compact and oriented manifold M of dimension m = 2d. We construct a map Ψ : H2∗S2 (LM)→O(ℳC) of Lie algebras, where H2∗S2 (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 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 H2∗S2 (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, ℂ) and GL(n, ℝ) together with its natural generalization to other reductive Lie groups.

DOI : 10.2140/agt.2007.7.197
Keywords: free loop space, string bracket, flat connections, Hamiltonian reduction, Chen iterated integrals, generalized holonomy, Wilson loop

Abbaspour, Hossein  1   ; Zeinalian, Mahmoud  2

1 Max-Planck Institut für Mathematik, Vivatsgasse 7, Bonn 53111, Germany
2 Long Island University, C W Post College, Brookville NY 11548, USA 
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Abbaspour, Hossein; Zeinalian, Mahmoud. String bracket and flat connections. Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 197-231. doi: 10.2140/agt.2007.7.197

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