Geometric Langlands From Six Dimensions

TL;DR

Geometric Langlands is reframed through a six-dimensional (0,2) theory, where electric–magnetic duality arises from a nonabelian Gerbe-like structure and is transported to four dimensions via compactifications. The work develops a two-pronged description of the space of BPS states and conformal blocks: (i) a 6D-to-5D gauge-theory picture with electric/magnetic gradings tied to the GNO dual group, and (ii) a boundary-WZW/CFT description living on fixed loci in circle fibrations and TN spaces, encoding ramification and duality kernels. By analyzing circle fibrations, singularities, and hyper-Kähler compactifications (notably Taub-NUT spaces), the authors derive a precise correspondence between partition functions, conformal blocks, and affine Kac–Moody representations, yielding a surface-geometric Langlands duality that aligns with the KW program. The framework clarifies how Langlands duality for surfaces emerges from higher-dimensional dualities, highlighting the role of boundary CFTs and the interplay between 2D and 6D degrees of freedom, with concrete constructions via TN_k spaces and WZW models.

Abstract

Geometric Langlands duality is usually formulated as a statement about Riemann surfaces, but it can be naturally understood as a consequence of electric-magnetic duality of four-dimensional gauge theory. This duality in turn is naturally understood as a consequence of the existence of a certain exotic supersymmetric conformal field theory in six dimensions. The same six-dimensional theory also gives a useful framework for understanding some recent mathematical results involving a counterpart of geometric Langlands duality for complex surfaces. (This article is based on a lecture at the Raoul Bott celebration, Montreal, June 2008.)