Geometric Langlands And The Equations Of Nahm And Bogomolny

TL;DR

This paper explores the connection between geometric Langlands duality and gauge theory, specifically using Nahm and Bogomolny equations to explain the appearance of an $SL_2$ subgroup in the dual group context.

Contribution

It provides a gauge-theoretic explanation for the role of an $SL_2$ subgroup in geometric Langlands duality through the equations of Nahm and Bogomolny.

Findings

Gauge theory elucidates the appearance of $SL_2$ in Langlands duality.

Nahm and Bogomolny equations relate to moduli space structures.

The work bridges geometric Langlands and physical gauge theories.

Abstract

Geometric Langlands duality relates a representation of a simple Lie group $G^\vee$ to the cohomology of a certain moduli space associated with the dual group $G$. In this correspondence, a principal $SL_2$ subgroup of $G^\vee$ makes an unexpected appearance. Why this happens can be explained using gauge theory, as we will see in this article, with the help of the equations of Nahm and Bogomolny. (Based on a lecture at Geometry and Physics: Atiyah 80, Edinburgh, April 2009.)