Five-branes in M-theory and a two-dimensional geometric Langlands duality

A recent attempt to extend the geometric Langlands duality to affine Kac-Moody groups has led Braverman and Finkelberg [1] to conjecture a mathematical relation between the intersection cohomology of the moduli space of G-bundles on certain singular complex surfaces, and the integrable representations of the Langlands dual of an associated affine G-algebra, where G is any simply-connected semisimple group. For the A(N-1) groups, where the conjecture has been mathematically verified to a. large extent, we show that the relation has a natural physical interpretation in terms of six-dimensional compactifications of M-theory with coincident five-branes wrapping certain hyperkahler four-manifolds; in particular, it can be understood as an expected invariance in the resulting spacetime BPS spectrum under string dualities. By replacing the singular complex surface with a smooth multi-Taub-NUT manifold, we find agreement with a closely related result demonstrated earlier via purely field-theoretic considerations by Witten [2]. By adding OM five-planes to the original analysis, we argue that an analogous relation involving the non-simply-connected D-N groups ought to hold as well. This is the first example of a string-theoretic interpretation of such a two-dimensional extension to complex surfaces of the geometric Langlands duality for the A-D groups.