String Theory Journal Club

Let \Sigma be a closed, oriented surface equipped with a Riemannian metric and let G be a compact, connected Lie group. For this data, we introduce the Migdal model, by defining a discrete partition function. It happens to be independent of the choice of the decomposition of the surface into polygons, thus being a good model for the infinite dimensional Yang-Mills integral. I will introduce also the moduli space of flat connections \mathcal{M}, which is the space of flat G-connection over \Sigma modulo the gauge transfomations. It can be equipped with a canonical symplectic form and its symplectic volume can be computed as a certain limit of the Yang-Mills partition function. Moreover there is a bijection between \mathcal{M} and the G-representations of the fundamental group of \Sigma.