Symplectic Forms on Moduli Spaces of Flat Connections on 2-manifolds

Abstract

Let $G$ be a compact connected semisimple Lie group. We extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms $\omega$ on \lq twisted' moduli spaces of representations of the fundamental group $\pi$ of a 2-manifold $\Sigma$ (the smooth analogues of $\mathrm{Hom}({\pi }_1(\Sigma ),G)/G$) and on relative character varieties of fundamental groups of 2-manifolds. We extend this construction to exhibit a symplectic form on the extended moduli space [J1] (a Hamiltonian $G$-space from which these moduli spaces may be obtained by symplectic reduction), and compute the moment map for the action of $G$ on the extended moduli space.

12 pages, LaTex version 2.09 This paper will appear in Proceedings of the Georgia International Topology Conference (Athens, GA, August 1993), ed. W. Kazez (International Press). The present version (April 1996) is essentially the version that will appear in print. Several sign errors have been corrected. Some errors have been corrected in the proof of nondegeneracy of the symplectic form (Section 5) on an open dense set in the extended moduli space

Cited by in corpus (8)

• On a self-dual phase space for 3+1 lattice Yang-Mills theory
• Yang-Mills moduli spaces over an orientable closed surface via Fréchet reduction
• Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory
• Extended moduli spaces and the Kan construction
• Finite-dimensional construction of self-duality and related moduli spaces over a closed Riemann surface as stratified holomorphic symplectic spaces
• Representations with Weighted Frames and Framed Parabolic Bundles
• On the variation of the Poisson structures of certain moduli spaces
• Line bundles on moduli and related spaces