A recent attempt to extend the geometric Langlands duality to affine
Kac–Moody groups has led Braverman and Finkelberg 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
AN−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 hyperkähler 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. By adding OM five-planes to
the original analysis, we argue that an analogous relation involving the
non-simply-connected DN 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.