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

TL;DR

This paper explores a string-theoretic interpretation of a two-dimensional extension of the geometric Langlands duality, linking M-theory five-branes, hyperkahler manifolds, and affine Kac-Moody groups, with new insights into dualities and moduli spaces.

Contribution

It provides the first string-theoretic interpretation of the geometric Langlands duality extension for A-D groups using M-theory five-branes and hyperkahler manifolds.

Findings

Relation between intersection cohomology and affine G-algebra representations confirmed for A-type groups.

Agreement with Witten's field-theoretic results when replacing singular surfaces with multi-Taub-NUT manifolds.

Proposed extension to D-type groups involving OM five-planes, suggesting broader applicability.

Abstract

A recent attempt to extend the geometric Langlands duality to affine Kac-Moody groups, has led Braverman and Finkelberg [arXiv:0711.2083] 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-type 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. By adding OM five-planes to the original analysis, we argue that an analogous relation involving the non-simply-connected D-type 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.