Torus Actions, Moment Maps, and the Symplectic Geometry of the Moduli Space of Flat Connections on a Two-Manifold

Abstract

We summarize recent work ([W],[JW91a], [JW92]) on the symplectic geometry of the moduli space of flat connections on a two-manifold. This work is based on the existence in these moduli spaces of Hamiltonian torus actions. Using these torus actions and the images of the corresponding moment maps we find a simple description of the moduli spaces, and show how it can be used to compute symplectic volumes and other quantities arising in the geometry and topology of the moduli space.