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 $(A,\varphi )$ is a smooth solution of Kapustin-Witten equations and the connection $A$ is closed to a $generic$ ASD connection $A_{\infty }$, then $(A,\varphi )$ 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 Kähler surface.

To appear in Forum Math. I would like to thank the anonymous referee for careful reading of my manuscript and helpful comments

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