# Hecke modifications of Higgs bundles and the extended Bogomolny equation

## TL;DR

This paper proves a correspondence linking solutions of a specific gauge theory equation with modifications of Higgs bundles, advancing the mathematical understanding of the geometric Langlands program and its physical interpretations.

## Contribution

It establishes a Kobayashi-Hitchin correspondence for the extended Bogomolny equation with singularities, confirming a conjecture by Witten and connecting gauge theory with algebraic geometry.

## Key findings

- Proves the Kobayashi-Hitchin correspondence for the extended Bogomolny equation.
- Links solutions with Hecke modifications of Higgs bundles.
- Supports the physical realization of the geometric Langlands program.

## Abstract

We establish a Kobayashi-Hitchin correspondence between solutions of the extended Bogomolny equation with a Dirac type singularity and Hecke modifications of Higgs bundles. This correspondence was conjectured by Witten and plays an important role in the physical description of the the geometric Langlands program in terms of S-duality for N=4 super Yang-Mills theory in four dimensions.

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Source: https://tomesphere.com/paper/1812.08994