# The Kapustin--Witten equations on ALE and ALF gravitational instantons

## TL;DR

This paper investigates solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons, providing asymptotic estimates and a vanishing theorem for specific manifolds and structure groups.

## Contribution

It offers new asymptotic estimates for the Higgs field and establishes a vanishing theorem on certain gravitational instantons with SU(2) structure group.

## Key findings

- Derived asymptotic estimates for the Higgs field on ALE and ALF spaces.
- Proved a vanishing theorem for solutions on R^4 and R^3 x S^1 with SU(2) group.

## Abstract

We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 \times \mathbb{S}^1$ and the structure group is $\mathrm{SU} (2)$.