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

Abstract

The $\theta$-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 $\phi$, 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 $[\phi\wedge\phi]$ vanishes everywhere. As a consequence, we are able to confirm a conjecture by Nagy and Oliveira, namely, that finite energy solutions of the $\theta$-Kapustin-Witten equations on ALE and ALF gravitational instantons with $\theta\neq 0$ are such that $[\phi\wedge\phi]=0$, $\nabla^A \phi=0$, and $A$ is flat.