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# Advances in Theoretical and Mathematical Physics

## Volume 14 (2010)

### Number 1

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

Pages: 179 – 224

DOI: http://dx.doi.org/10.4310/ATMP.2010.v14.n1.a4

#### Author

#### Abstract

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.