On a topology property for the moduli space of Kapustin-Witten equations

In this article, we study the Kapustin-Witten equations on a closed, simply connected, four-dimensional manifold which were introduced by Kapustin and Witten. We use Taubes' compactness theorem [C. H. Taubes, Compactness theorems for SL(2; C) generalizations of the 4-dimensional anti-self dual equations, preprint (2014), https://arxiv.org/abs/1307.6447v4] to prove that if (A, phi) is a smooth solution to the Kapustin-Witten equations and the connection A is closed to a generic ASD connection A(infinity), then (A, phi) must be a trivial solution. We also prove that the moduli space of the solutions to the Kapustin-Witten equations is non-connected if the connections on the compactification of moduli space of ASD connections are all generic. At last, we extend the results for the Kapustin-Witten equations to other equations on gauge theory such as the Hitchin-Simpson equations and the Vafa-Witten on a compact Kahler surface.