BF Theories and Group-Level Duality

Abstract

It is known that the partition function and correlators of the two-dimensional topological field theory $G_K(N)/ G_K(N)$ on the Riemann surface $\Sigma_{g,s}$ is given by Verlinde numbers, dim($V_{g,s,K}$) and that the large $K$ limit of dim($V_{g,s,K}$) gives Vol(${\cal M}_s$), the volume of the moduli space of flat connections of gauge group $G(N)$ on $\Sigma_{g,s}$, up to a power of $K$. Given this relationship, we complete the computation of Vol(${\cal M}_s$) using only algebraic results from conformal field theory. The group-level duality of $G(N)_K$ is used to show that if $G(N)$ is a classical group, then $\displaystyle \lim_{N\rightarrow \infty} G_K(N) / G_K(N)$ is a BF theory with gauge group $G(K)$. Therefore this limit computes Vol(${\cal M}^\prime_s$), the volume of the moduli space of flat connections of gauge group $G(K)$.