Localization in GWZW and Verlinde formula

TL;DR

This paper shows that Gauged Wess-Zumino-Witten theory for compact groups admits an exact solution by localization, exploiting a fermionic BRST-like symmetry. By connecting GWZW to Chern-Simons theory and analyzing the SU(2) case on arbitrary surfaces, it derives the Verlinde formula as the exact value of the GWZW partition function, thereby counting conformal blocks. The approach also clarifies the role of one-loop corrections and a k-shift $k\to k+c_v$, and provides a covariant treatment for nontrivial bundles. These results deepen the link between two-dimensional gauge theories, topological field theory, and the geometry of moduli spaces, with potential extensions to noncompact groups and integrable systems.

Abstract

Gauged Wess-Zumino-Witten theory for compact groups is considered. It is shown that this theory has fermionic BRST-like symmetry and may be exactly solved using localization approach. As an example we calculate functional integral for the case of SU(2) group on the arbitrary Riemann surface. The answer is the particular case of Verlinde formula for the number of conformal blocks.