The Kapustin–Witten equations and nonabelian Hodge theory

Abstract

Arising from a topological twist of ${\mathscr {N}}=4$ super Yang–Mills theory are the Kapustin–Witten equations, a family of gauge-theoretic equations on a four-manifold parametrised by $t\in {\mathbb {P}}^1$. The parameter corresponds to a linear combination of two super charges in the twist. When $t=0$ and the four-manifold is a compact Kähler surface, the equations become the Simpson equations, which was originally studied by Hitchin on a compact Riemann surface, as demonstrated independently in works of Nakajima and the third-named author. At the same time, there is a notion of $\lambda$-connection in the nonabelian Hodge theory of Donaldson–Corlette–Hitchin–Simpson in which $\lambda$ is also valued in ${\mathbb {P}}^1$. Varying $\lambda$ interpolates between the moduli space of semistable Higgs sheaves with vanishing Chern classes on a smooth projective variety (at $\lambda =0$) and the moduli space of semisimple local systems on the same variety (at $\lambda =1$) in the twistor space. In this article, we utilise the correspondence furnished by nonabelian Hodge theory to describe a relation between the moduli spaces of solutions to the equations by Kapustin and Witten at $t=0$ and $t \in {{\mathbb {R}}} \,{\setminus }\, \{ 0 \}$ on a smooth, compact Kähler surface. We then provide supporting evidence for a more general form of this relation on a smooth, closed four-manifold by computing its expected dimension of the moduli space for each of $t=0$ and $t \in {{\mathbb {R}}} \,{\setminus }\, \{ 0 \}$.