Edward Witten

Gauge theory and the Geometric Langlands Program.


N=4 super Yang-Mills theory in four dimensions, compactified to two dimensions on a Riemann surface C, gives, via its S-duality symmetry, a natural framework for understanding the geometric Langlands program for complex Riemann surfaces. The correspondence between the tensor algebra of representations of LG and the Hecke algebra of G comes from S-duality between Wilson loops on one side and 't Hooft loops on the other. The correspondence between LG local systems and D-modules on the moduli space of G-bundles depends on some considerations involving branes, or in other words supersymmetric boundary conditions.