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

Yuuji Tanaka

Published 2014-10-07Version 1

We formulate 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 equation on smooth closed four-manifolds. We introduce a set of perturbation terms to the Vafa-Witten equations, but not depending on connections, and prove that the moduli space of solutions to the perturbed Vafa-Witten equations for the structure group being SU(2) or SO(3) on a closed symplectic four-manifold is a smooth manifold of dimension zero for a generic choice of the perturbation parameters.