Moduli Spaces of Flat Connections

Abstract

In this chapter, we consider the moduli space of flat connections on a (trivial) bundle over a surface (defined in § V.1). There is a Hamiltonian group action that allows us to endow this space with a Poisson structure, as we show in § V.2 (this is due to Atiyah and Bott [10] in the case of a closed surface and to Fock and Roslyi [50] in the general case of a surface with boundary). We then look at a special case in § V.3, in which there is an integrable system (due to Goldman [56]) on the moduli space. Following Jeffrey and Weitsman [77], we exhibit a torus action and its momentum mapping, that is, action-angle variables for this system.