Title: On a topology property for moduli space of Kapustin-Witten equations,
Abstract: In this article, we study the Kapustin-Witten equations on a closed,
simply-connected, four-manifold. We using a compactness theorem due to Taubes
to prove that if $(A,\phi)$ is a 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 $generic$.
As one application, we extend the ideas of Kapustin-Witten equations to other
equations on gauge theory-- Hitchin-Simpson equations and Vafa-Witten on
compact Kähler surface with a Kähler metric $g$.