A variation of the reduced Vafa-Witten equations on 4-manifolds

TL;DR

This paper introduces a perturbed version of the reduced Vafa-Witten equations on 4-manifolds, enabling transversality, a priori estimates, and the removal of singularities to better understand the moduli space structure.

Contribution

It proposes a novel perturbation of the reduced Vafa-Witten equations that achieves transversality and allows for the construction of the Uhlenbeck closure of the moduli space.

Findings

Established transversality of the perturbed equations

Derived a priori estimates for solutions

Constructed the Uhlenbeck closure of the moduli space

Abstract

In this paper we consider a variation of the Vafa-Witten equations on compact, oriented and smooth 4-manifolds, and construct a set of perturbation terms to establish the transversality of that equations. The new perturbed equations provide us a priori estimates of the solutions, while the original reduced Vafa-Witten equations does not. By applying the a priori estimates we show that the singular of the solutions can be removed, and then construct the Ulhenbeck closure of the moduli spaces.