Selfduality and Chern-Simons Theory

Abstract: We propose a relation between the operator of S-duality (of N=4 super Yang-Mills theory in 3+1D) and a topological theory in one dimension lower. We construct the topological theory by compactifying N=4 super Yang-Mills on a circle with an S-duality and R-symmetry twist. The S-duality twist requires a selfdual coupling constant. We argue that for a sufficiently low rank of the gauge group the three-dimensional low-energy description is a topological theory, which we conjecture to be a pure Chern-Simons theory. This conjecture implies a connection between the action of mirror symmetry on the sigma-model with Hitchin's moduli space as target space and geometric quantization of the moduli space of flat connections on a Riemann surface.