Gauge Theory and Langlands Duality

TL;DR

This paper explores the deep connections between gauge theories, geometric Langlands duality, and mirror symmetry, highlighting recent advances linking physics and mathematics in understanding moduli spaces of bundles.

Contribution

It reviews the development of the geometric Langlands program and its relation to S-duality in gauge theories, emphasizing the role of mirror symmetry and recent breakthroughs.

Findings

Link between geometric Langlands and S-duality established

Homological Mirror Symmetry observed in Hitchin moduli spaces

New insights into categories of sheaves on algebraic curves

Abstract

The Langlands Program was launched in the late 60s with the goal of relating Galois representations and automorphic forms. In recent years a geometric version has been developed which leads to a mysterious duality between certain categories of sheaves on moduli spaces of (flat) bundles on algebraic curves. Three years ago, in a groundbreaking advance, Kapustin and Witten have linked the geometric Langlands correspondence to the S-duality of 4D supersymmetric gauge theories. This and subsequent works have already led to striking new insights into the geometric Langlands Program, which in particular involve the Homological Mirror Symmetry of the Hitchin moduli spaces of Higgs bundles on algebraic curves associated to two Langlands dual Lie groups.