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 \({\mathbb {R}}^4\) or \({\mathbb {R}}^3 \times {\mathbb {S}}^1\) and the structure group is \(\text {SU}(2)\).
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Acknowledgements
The authors are grateful to Mark Stern for many helpful conversations about gauge theory. We also thank Bera Gorapada for his valuable comments on the results of this paper. Ákos Nagy would like to thank Siqi He for useful discussions about the Kapustin–Witten equations, and he also thanks the Universidade Federal Fluminense and IMPA for their hospitality during the final stages of the preparation of this paper. We also thank the anonymous referees for their comments and suggestions. Gonçalo Oliveira is supported by Fundação Serrapilheira 1812-27395, by CNPq Grants 428959/2018-0 and 307475/2018-2, and FAPERJ through the program Jovem Cientista do Nosso Estado E-26/202.793/2019.
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Nagy, Á., Oliveira, G. The Kapustin–Witten equations on ALE and ALF gravitational instantons. Lett Math Phys 111, 87 (2021). https://doi.org/10.1007/s11005-021-01426-w
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DOI: https://doi.org/10.1007/s11005-021-01426-w