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A recent attempt to extend the geometric Langlands duality to affineKac-Moody groups, has led Braverman and Finkelberg [arXiv:0711.2083] toconjecture a mathematical relation between the intersection cohomology of themoduli space of G-bundles on certain singular complex surfaces, and theintegrable representations of the Langlands dual of an associated affineG-algebra, where G is any simply-connected semisimple group. For the A-typegroups, where the conjecture has been mathematically verified to a largeextent, we show that the relation has a natural physical interpretation interms of six-dimensional compactifications of M-theory with coincidentfive-branes wrapping certain hyperkahler four-manifolds; in particular, it canbe understood as an expected invariance in the resulting spacetime BPS spectrumunder string dualities. By replacing the singular complex surface with a smoothmulti-Taub-NUT manifold, we find agreement with a closely related resultdemonstrated earlier via purely field-theoretic considerations by Witten. Byadding OM five-planes to the original analysis, we argue that an analogousrelation involving the non-simply-connected D-type groups, ought to hold aswell. This is the first example of a string-theoretic interpretation of such atwo-dimensional extension to complex surfaces of the geometric Langlandsduality for the A-D groups.

Five-Branes in M-Theory and a Two-Dimensional Geometric Langlands Duality