## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 14, Number 1 (2010), 179-224.

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

#### 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.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 14, Number 1 (2010), 179-224.

**Dates**

First available in Project Euclid: 31 August 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1283281760

**Mathematical Reviews number (MathSciNet)**

MR2684980

**Zentralblatt MATH identifier**

1201.81095

#### Citation

Tan, Meng-Chwan. Five-Branes in M-Theory and a Two-Dimensional Geometric Langlands Duality. Adv. Theor. Math. Phys. 14 (2010), no. 1, 179--224. https://projecteuclid.org/euclid.atmp/1283281760