Rotationally invariant singular solutions to the Kapustin–Witten equations

In the present paper, we find a system of non-linear ODEs that gives rotationally invariant solutions to the Kapustin–Witten equations in $4$-dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the Kapustin–Witten equations. The imaginary parts of the solutions are singular. By rescaling, we find some limit behavior for these singular solutions. In addition, for any integer $k$, we can construct a $5 \lvert k \rvert$ dimensional family of $C^1$ solutions to the Kapustin–Witten equations on Euclidean space, again with singular imaginary parts. Moreover, we get solutions to the Kapustin–Witten equation with Nahm pole boundary condition over $S^3 \times (0, + \infty)$.

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Received 12 March 2016

Accepted 11 May 2017

Published 16 November 2018