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

TL;DR

This paper constructs symplectic forms on moduli spaces of flat connections on 2-manifolds using group cohomology, extending Weinstein's techniques, and computes associated moment maps for Hamiltonian group actions.

Contribution

It introduces a new cohomological method to define symplectic structures on twisted and extended moduli spaces of flat connections, generalizing previous constructions.

Findings

Constructed symplectic forms on twisted moduli spaces

Extended the construction to the extended moduli space as a Hamiltonian G-space

Computed the moment map for the G-action on the extended moduli space

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 ${\rm Hom} (\pi_1(\Sigma), G)/G$) 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.