Transversality of the perturbed reduced Vafa-Witten moduli spaces on 4-manifolds

TL;DR

This paper completes the proof of transversality for the reduced Vafa-Witten moduli spaces on 4-manifolds by constructing perturbations that ensure the moduli space is a smooth, zero-dimensional manifold.

Contribution

It introduces a perturbation method to establish transversality of the reduced Vafa-Witten moduli spaces on closed 4-manifolds with structure groups SU(2) or SO(3).

Findings

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

The paper extends previous work by handling the reduced part of the Vafa-Witten moduli space.

Perturbation techniques are effective in achieving transversality in this setting.

Abstract

Previously we finish the establishment of the transversality of the general part of the Vafa-Witten moduli spaces, in this paper, we deal with the rest, i.e., the reduced part. We consider Vafa-Witten equation on closed, oriented and smooth Riemann 4-manifolds with $C\equiv0$, and construct perturbation to establish the transversality of the perturbed equation. Then we show that for a generic choice of the perturbation terms, the moduli space of solutions to the perturbed reduced Vafa-Witten equation for the structure group $SU(2)$ or $SO(3)$ on a closed 4-manifold is a smooth manifold of dimension zero.