Topological Gauge Theory, S-duality and Mirror Symmetry

N=4 d=4 supersymmetric gauge theories with Langlands-dual gauge groups are conjectured to be equivalent on the nonperturbative level; this is known as the S-duality conjecture. Performing a topological reduction of these theories on a Riemann surface, one obtains a pair of topological sigma-models whose target can be identified with the moduli space of Hitchin equations. S-duality implies that these sigma-models are related by mirror symmetry. This leads to a physical interpretation of the Geometric Langlands Program.