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 \({\mathbb {R}}^4\) or \({\mathbb {R}}^3 \times {\mathbb {S}}^1\) and the structure group is \(\text {SU}(2)\).

中文翻译：

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ALE 和 ALF 引力瞬子上的 Kapustin-Witten 方程

我们研究了 ALE 和 ALF 引力瞬子上 Kapustin-Witten 方程的解。在任何这样的空间和任何紧凑结构群上，我们证明了希格斯场的渐近估计。然后我们用它来证明当基础流形为\({\mathbb {R}}^4\)或\({\mathbb {R}}^3 \times {\mathbb {S} }^1\)并且结构组是\(\text {SU}(2)\)。