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

Abstract

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 Taubes’ compactness theorem [C. H. Taubes, Compactness theorems for SL ⁢ ( 2 ; ℂ ) {\mathrm{SL}(2;\mathbb{C})} generalizations of the 4-dimensional anti-self dual equations, preprint 2014, https://arxiv.org/abs/1307.6447v4] to prove that if ( A , ϕ ) {(A,\phi)} is a smooth solution to the Kapustin–Witten equations and the connection A is closed to a generic ASD connection A ∞ {A_{\infty}} , then ( A , ϕ ) {(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 Kähler surface.