Growth of the Higgs Field for Kapustin-Witten solutions on ALE and ALF gravitational instantons

Abstract

The $θ$-Kapustin-Witten equations are a family of equations for a connection $A$ on a principal $G$-bundle $E \to W^4$ and a one-form $ϕ$, called the Higgs field, with values in the adjoint bundle $\operatorname{ad} E$. They give rise to second-order partial differential equations that can be studied more generally on Riemannian manifolds $W^n$ of dimension $n$. For $G=SU(2)$, we report a dichotomy that is satisfied by solutions of the second-order equations on Ricci-flat ALX spaces with sectional curvature bounded from below. This dichotomy was originally established by Taubes for $W^n=\mathbb{R}^n$; the alternatives are: either the asymptotic growth of the averaged norm of the Higgs field over geodesic spheres is larger than a positive power of the radius, or the commutator $[ϕ\wedgeϕ]$ vanishes everywhere. As a consequence, we are able to confirm a conjecture by Nagy and Oliveira, namely, that finite energy solutions of the $θ$-Kapustin-Witten equations on ALE and ALF gravitational instantons with $θ\neq 0$ are such that $[ϕ\wedgeϕ]=0$, $\nabla^A ϕ=0$, and $A$ is flat.