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 S3, S1×Σ, B3 and the solid torus correspond, respectively, to the WZW model of maps from S2 to G, the G/G model for Σ, and Witten's gauged WZW path integral Ansatz for Chern–Simons states using maps from S2 and from the torus to G. The reduction hinges on the characterization of \(\), 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.