BF Theories and Group-Level Duality

TL;DR

This paper explores the relationship between 2D topological field theories, Verlinde numbers, and moduli space volumes, revealing a duality that connects large K limits to BF theories with different gauge groups.

Contribution

It completes the computation of moduli space volumes using algebraic conformal field theory results and establishes a duality linking large N limits to BF theories with swapped gauge groups.

Findings

Large K limit of Verlinde numbers relates to moduli space volume

Group-level duality shows a limit yields a BF theory with swapped gauge group

Explicit computation of moduli space volume using algebraic methods

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 $G(K)$.