Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model

TL;DR

This work derives the Verlinde formula directly from Chern-Simons theory on Σ×S^1 by an exact gauge-theory reduction to a 2D Abelian BF-type theory and by establishing the equivalence with the G/G gauged WZW model. It shows how the partition function reduces to a Weyl-integral form, with the Weyl determinant reproducing the Weyl denominator and Ray-Singer torsion determining the measure, thereby obtaining the dimension of conformal blocks and fusion data without invoking conformal field theory. A key technical achievement is the k→k+h shift, explained via the Dolbeault index and chiral anomaly in the Abelianized theory. The approach also yields the SU(2) and SU(n) Verlinde formulas, clarifies the role of punctures in fusion rules, and connects CS theory, the G/G model, BF-type theories, and Weyl symmetry in a cohesive topological-field-theory framework. This provides a gauge-theoretic, self-contained derivation of the Verlinde data and illuminates their geometric and topological underpinnings.

Abstract

We give a derivation of the Verlinde formula for the $G_{k}$ WZW model from Chern-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function $Z_{Σ\times S^{1}}$ of $Σ\times S^{1}$ with an arbitrary number of labelled punctures. By a suitable gauge choice, $Z_{Σ\times S^{1}}$ is reduced to the partition function of an Abelian topological field theory on $Σ$ (a deformation of non-Abelian BF and Yang-Mills theory) whose evaluation is straightforward. This relates the Verlinde formula to the Ray-Singer torsion of $Σ\times S^{1}$. We derive the $G_{k}/G_{k}$ model from Chern-Simons theory, proving their equivalence, and give an alternative derivation of the Verlinde formula by calculating the $G_{k}/G_{k}$ path integral via a functional version of the Weyl integral formula. From this point of view the Verlinde formula arises from the corresponding Jacobian, the Weyl determinant. Also, a novel derivation of the shift $k\ra k+h$ is given, based on the index of the twisted Dolbeault complex.