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**Five-branes in M-theory and a two-dimensional geometric Langlands duality.**
*(English)*
Zbl 1201.81095

Summary: 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 \(A_{N-1}\) 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 hyper-Kähler four-manifolds; in particular, it can be understood as an expected invariance in the resulting space-time 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.

### MSC:

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |

17B20 | Simple, semisimple, reductive (super)algebras |

22E46 | Semisimple Lie groups and their representations |

22E70 | Applications of Lie groups to the sciences; explicit representations |

22E57 | Geometric Langlands program: representation-theoretic aspects |