DERIVATION OF THE VERLINDE FORMULA FROM CHERN-SIMONS THEORY AND THE G/G MODEL

We give a derivation of the Verlinde formula for the G(k) WZW model from Chem-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function Z(SIGMA x S1) of SIGMA x S1 with an arbitrary number of labelled punctures. By what is essentially a suitable gauge choice, Z(SIGMA x S1) is reduced to the partition function of an abelian topological field theory on SIGMA (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 SIGMA x S1. 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.