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

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 $\mathrm{SU}(2)$.

12 pages, no figures. A mistake in the main theorem is fixed; supersedes published version

References in corpus (2)

• The Extended Bogomolny Equations and Generalized Nahm Pole Boundary Condition
• Hecke modifications of Higgs bundles and the extended Bogomolny equation