# Opers and the twisted Bogomolny equations

## TL;DR

This paper establishes a correspondence between certain solutions to twisted Kapustin-Witten equations on a Riemann surface product and the moduli space of Beilinson-Drinfeld opers, confirming a prediction by Gaiotto and Witten.

## Contribution

It introduces a Kobayashi-Hitchin type correspondence linking tilted Nahm pole solutions to Beilinson-Drinfeld opers, advancing understanding in geometric representation theory.

## Key findings

- Proves the correspondence between solutions and opers.
- Supports Gaiotto and Witten's theoretical prediction.
- Enhances the geometric understanding of twisted equations.

## Abstract

In this paper, we study the dimensionally reduced twisted Kapustin-Witten equations on the product of a compact Riemann surface $\Sigma$ with $\mathbb{R}^+$. The main result is a Kobayashi-Hitchin type correspondence between the space of tilted Nahm pole solutions and the moduli space of Beilinson-Drinfeld opers. This corroborates a prediction of Gaiotto and Witten \cite[p.971]{gaiotto2012knot}.