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

Abstract

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 the Taubes' compactness theorem in arXiv:1307.6447v4 to prove that if $(A,\phi)$ is a smooth solution of Kapustin-Witten equations and the connection $A$ is closed to a $generic$ ASD connection $A_{\infty}$, then $(A,\phi)$ must be a trivial solution. We also prove that the moduli space of the solutions of 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 Vafa-Witten on a compact K\"{a}hler surface.