Mirror Symmetry, Hitchin's Equations, And Langlands Duality

TL;DR

This paper explores the deep connections between geometric Langlands duality, mirror symmetry, and Hitchin's equations, emphasizing the importance of complex structures and gauge theory in understanding these dualities.

Contribution

It provides a topological formulation of mirror symmetry related to Langlands duality and discusses the role of Hitchin's equations and four-dimensional gauge theory in this context.

Findings

Formulation of mirror symmetry statements for unramified and ramified cases

Highlighting the necessity of complex structures and Hitchin's equations

Indications of the role of four-dimensional gauge theory in the duality

Abstract

Geometric Langlands duality can be understood from statements of mirror symmetry that can be formulated in purely topological terms for an oriented two-manifold $C$. But understanding these statements is extremely difficult without picking a complex structure on $C$ and using Hitchin's equations. We sketch the essential statements both for the ``unramified'' case that $C$ is a compact oriented two-manifold without boundary, and the ``ramified'' case that one allows punctures. We also give a few indications of why a more precise description requires a starting point in four-dimensional gauge theory.