Energy bound for Kapustin-Witten solutions on $S^3\times\mathbb{R}^+$

TL;DR

This paper proves a uniform energy bound for Nahm pole solutions of the Kapustin-Witten equations on the manifold $S^3 imes R^+$, establishing a key estimate for these solutions.

Contribution

It establishes a universal $L^2$ energy bound for Nahm pole solutions of the Kapustin-Witten equations on $S^3 imes R^+$, a significant step in understanding their behavior.

Findings

Existence of a uniform $L^2$ bound for $F_A$ in Nahm pole solutions.

Bound applies to all solutions with Nahm pole boundary conditions.

Provides a foundational estimate for further analysis of Kapustin-Witten solutions.

Abstract

We consider solutions of Kapustin-Witten equation with Nahm pole boundary on $S^3\times \mathbb{R}^+$. These solutions are usually called Nahm pole solutions. In this paper, we will prove that there exists a constant $C>0$ such that $\|F_A\|_{L^2}\leq C$ for any Nahm pole solution $(A,\phi)$.