A perturbation and generic smoothness of the Vafa-Witten moduli spaces on closed symplectic four-manifolds

TL;DR

This paper establishes a generic smoothness result for the moduli space of solutions to the Vafa-Witten equations on closed symplectic four-manifolds by introducing perturbations, ensuring the space is a smooth zero-dimensional manifold for generic parameters.

Contribution

It proves a Freed-Uhlenbeck style generic smoothness theorem for Vafa-Witten moduli spaces using perturbation methods adapted from Feehan's work on $PU(2)$-monopoles.

Findings

The moduli space becomes a smooth zero-dimensional manifold for generic perturbations.

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

Perturbations ensure transversality and smoothness of the moduli space.

Abstract

We prove a Freed-Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa--Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the $PU(2)$-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms to the Vafa--Witten equations, and prove that the moduli space of solutions to the perturbed Vafa-Witten equations on a closed symplectic four-manifold for the structure group $SU(2)$ or $SO(3)$ is a smooth manifold of dimension zero for a generic choice of the perturbation parameters.