The Kapustin-Witten equations and nonabelian Hodge theory

Arising from a topological twist of N = 4 super Yang-Mills theory are the Kapustin-Witten equations, a family of gauge-theoretic equations on a four-manifold parametrised by t is an element of 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 Miller 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 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 is an element of R \{0} on a smooth, compact Miller 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 is an element of R\{0}.