Toric Structures on the Moduli Space of Flat Connections on a Riemann Surface: Volumes and the Moment Map

Abstract

In earlier papers we constructed a Hamiltonian torus action on an open dense set in the moduli space of flat SU(2) connections on a compact Riemann surface, where the dimension of the torus is half the dimension of the moduli space. This torus action shows that this set can be viewed symplectically as a (noncompact) toric variety. The number of integral points of the moment map for the torus action turns out to be identical to the Verlinde dimension D(g, k). As an application, we furnish a new proof of the relation between the large-k limit of D(g, k) and the volume of the moduli space. From our point of view, this relation follows from the equality between the symplectic volume of a toric variety and the Euclidean volume of the image of the moment map. Similar considerations are shown to give rise to the volumes of moduli spaces of parabolic bundles on a Riemann surface. Knowledge of these volumes has been shown to allow a proof of the Verlinde formula for the dimension of the space of holomorphic sections of line bundles on this space.