The extended Bogomolny equations with generalized Nahm pole boundary conditions, II

TL;DR

This paper establishes a correspondence between solutions of extended Bogomolny equations with generalized Nahm pole boundary conditions on a Riemann surface product and certain stable holomorphic data, extending previous work to higher rank groups.

Contribution

It develops a Kobayashi-Hitchin correspondence for these equations with singular boundary conditions, generalizing earlier results to the $ ext{SL}(n+1, ext{C})$ case.

Findings

Corroborates Gaiotto-Witten's prediction.

Extends previous $ ext{SL}(2, ext{R})$ results to higher rank.

Provides a holomorphic data classification of solutions.

Abstract

We develop a Kobayashi-Hitchin correspondence for the extended Bogomolny equations, i.e., the dimensionally reduced Kapustin-Witten equations, on the product of a compact Riemann surface $\Sigma$ with ${\mathbb R}^+_y$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as $y \to \infty$, and certain holomorphic data consisting of effective triplets $(\mathcal E, \varphi, L)$ where $(\mathcal E, \varphi)$ is a stable $\mathrm{SL}(n+1,\mathbb C)$ Higgs pair and $L \subset \mathcal E$ is a holomorphic line bundle. This corroborates a prediction of Gaiotto and Witten, and is an extension of our earlier paper \cite{HeMazzeo2017} which treats only the $\mathrm{SL}(2,\mathbb R)$ case.