Classification of Nahm pole solutions of the Kapustin-Witten equations on $S^1\times \Sigma\times \mathbb{R}^+$

TL;DR

This paper classifies all solutions to the SU(n) Kapustin-Witten equations with Nahm pole singularities on a specific 3-manifold, extending to solutions with generalized singularities along knots, advancing understanding in gauge theory and geometric analysis.

Contribution

It provides a complete classification of Nahm pole solutions and their generalizations on a product manifold, a novel result in the study of Kapustin-Witten equations.

Findings

Classified all Nahm pole solutions on S^1×Σ×R^+

Extended classification to solutions with generalized Nahm pole singularities along knots

Enhanced understanding of boundary conditions in gauge theory

Abstract

In this note, we classify all solutions to the $\mathrm{SU(n)}$ Kapustin-Witten equations on $S^1\times\Sigma \times \mathbb{R}^+$, where $\Sigma$ is a compact Riemann surface, with Nahm pole singularity at $S^1\times\Sigma \times \{0\}$. We provide a similar classification of solutions with generalized Nahm pole singularities along a simple divisor (a "knot") in $S^1\times\Sigma \times \{0\}$.