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

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.