Rotationally Invariant Singular Solutions to the Kapustin-Witten Equations

Abstract

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$|k|$ dimensional family of $C^1$ solutions to the Kapustin-Witten equations on Euclidean space, again with singular imaginary parts. Moreover, we find solutions to the Kapustin-Witten equations over $S^3\times (0,+\infty )$ with the Nahm pole boundary condition.

22 pages, 4 figures, in the 3rd version add the relation of the solutions and the Nahm Pole boundary condition

Cited by in corpus (4)

• Solutions of Kapustin-Witten equations for ADE-type groups
• Energy bound for Kapustin-Witten solutions on $S^3\times {\mathbb{R}}^{+}$
• The Expansions of the Nahm Pole Solutions to the Kapustin-Witten Equations
• Inner Product in Highest-Weight Representation