On the general part of the perturbed Vafa-Witten moduli spaces on 4-manifolds

TL;DR

This paper develops perturbation techniques for the Vafa-Witten equations on 4-manifolds, proving that for generic perturbations, the moduli space is a smooth, zero-dimensional manifold, advancing understanding of gauge theory solutions.

Contribution

It introduces a new set of perturbation terms that achieve transversality for the Vafa-Witten equations, ensuring smoothness of the moduli space for generic parameters.

Findings

The perturbed moduli space is smooth and zero-dimensional for generic perturbations.

The method applies to structure groups SU(2) and SO(3).

Establishes transversality at the general part of solutions.

Abstract

In this paper we consider the Vafa-Witten equations on closed, oriented and smooth 4-manifolds, and construct a set of perturbation terms to establish the transversality of the perturbed Vafa-Witten equations at the general part of the solutions. Then we show that for a generic choice of the perturbation terms, this part of the moduli space for the structure group $SU(2)$ or $SO(3)$ on a closed 4-manifold is a smooth manifold of dimension zero.