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∈ P¹. 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 λ-connection in the nonabelian Hodge theory of Donaldson–Corlette–Hitchin–Simpson in which λ is also valued in P¹. Varying λ interpolates between the moduli space of semistable Higgs sheaves with vanishing Chern classes on a smooth projective variety (at λ= 0) and the moduli space of semisimple local systems on the same variety (at λ= 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∈R\{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∈R\{0}.