On Chern-Simons and WZW Partition Functions

Abstract:Direct analysis of the path integral reduces partition functions in Chern-Simons theory on a three-manifold M with group G to partition functions in a WZW model of maps from a Riemann surface Σ to G. In particular, Chern-Simons theory on S 3 , S 1 × Σ, B 3 and the solid torus correspond, respectively, to the WZW model of maps from S 2 to G, the G/G model for Σ, and Witten's gauged WZW path integral Ansatz for Chern-Simons states using maps from S 2 and from the torus to G. The reduction hinges on the characterization of {\cal A / G}_{n}$ , the space of connections modulo those gauge transformations which are the identity at a point n, as itself a principal fiber bundle with affine-linear fiber.