S-duality of boundary conditions and the Geometric Langlands program

TL;DR

This paper explores how S-duality in four-dimensional supersymmetric gauge theories relates boundary conditions to the Geometric Langlands program, revealing dual pairs of objects that deepen the mathematical-physical connection.

Contribution

It demonstrates how S-duality acts on half-BPS boundary conditions and constructs dual pairs of objects relevant to the Geometric Langlands correspondence.

Findings

S-duality extends to boundary conditions in supersymmetric gauge theories.

Pairs of dual boundary conditions correspond to dual objects in Geometric Langlands.

The framework provides a physical perspective on the Langlands duality.

Abstract

Maximally supersymmetric gauge theory in four dimensions admits local boundary conditions which preserve half of the bulk supersymmetries. The S-duality of the bulk gauge theory can be extended in a natural fashion to act on such half-BPS boundary conditions. The purpose of this note is to explain the role these boundary conditions can play in the Geometric Langlands program. In particular, we describe how to obtain pairs of Geometric Langland dual objects from S-dual pairs of half-BPS boundary conditions.