A lower bound on the solutions of Kapustin-Witten equations

TL;DR

This paper establishes an L^2 lower bound for solutions of the Kapustin-Witten equations on closed 4-manifolds, highlighting a key analytic property distinguishing non-ASD connections, and also extends results to Vafa-Witten equations.

Contribution

It provides the first known L^2 lower bound for solutions of the Kapustin-Witten equations on closed 4-manifolds, advancing understanding of their solution space.

Findings

L^2 lower bound for Kapustin-Witten solutions on closed 4-manifolds

Extension of lower bound results to Vafa-Witten equations

Distinction between ASD and non-ASD connections based on bounds

Abstract

In this article, we consider the Kapustin-Witten equations on a closed $4$-manifold. We study certain analytic properties of solutions to the equations on a closed manifold. The main result is that there exists an $L^{2}$-lower bound on the extra fields over a closed four-manifold satisfying certain conditions if the connections are not ASD connections. Furthermore, we also obtain a similar result about the Vafa-Witten equations.