Derivation of the Verlinde formula from Chern-Simons theory and the G/G model

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×S 1 of Σ × S 1 with an arbitrary number of labelled punctures. By what is essentially a suitable gauge choice, Z Σ×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 × 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 → k + h is given, based on the index of the twisted Dolbeault complex.