A physics perspective on geometric Langlands duality

TL;DR

This paper reviews the gauge theory approach to geometric Langlands duality, emphasizing the role of S-duality in N=4 supersymmetric theories and exploring six-dimensional frameworks to unify and extend the theory.

Contribution

It discusses the development of geometric Langlands duality via gauge theory, highlighting the potential of six-dimensional formulations to make the duality a manifest symmetry and extend results to algebraic surfaces.

Findings

Connection between S-duality and geometric Langlands duality

Proposal of six-dimensional framework for manifest symmetry

Extension of duality concepts to algebraic surfaces

Abstract

We review the approach to the geometric Langlands program for algebraic curves via S-duality of an N=4 supersymmetric four dimensional gauge theory, initiated by Kapustin and Witten in 2006. We sketch some of the central further developments. Placing this four dimensional gauge theory into a six dimensional framework, as advocated by Witten, holds the promise to lead to a formulation which makes geometric Langlands duality a manifest symmetry (like coavariance in differential geometry). Furthermore, it leads to an approach toward geometric Langlands duality for algebraic surfaces, reproducing and extending the recent results of Braverman and Finkelberg.