Selfduality and Chern–Simons Theory

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 S 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 threedimensional 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.