The Kapustin-Witten equations and nonabelian Hodge theory
Abstract
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 parametrized 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 K\"ahler 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 K\"ahler 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 \.