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

We give a derivation of the Verlinde formula for the Gk WZW model from Chern-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function ZΣ×S1 of Σ × S 1 with an arbitrary number of labelled punctures. By a suitable gauge choice, ZΣ×S1 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× S1. We derive the Gk/Gk model from Chern-Simons theory, proving their equivalence, and give an alternative derivation of the Verlinde formula by calculating the Gk/Gk 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. e-mail: [email protected] e-mail: [email protected]