Model knot solutions for the twisted Bogomolny equations

TL;DR

This paper proves the existence of model knot solutions for the twisted Kapustin-Witten equations on half-space, extending explicit solutions at a specific parameter to all positive parameters, confirming a theoretical prediction.

Contribution

It establishes the existence of solutions for all twisting parameters by continuity, building on explicit solutions at a specific parameter value.

Findings

Existence of model knot solutions for all positive twisting parameters.

Continuity argument from explicit solutions at t=1.

Confirmation of Gaiotto and Witten's prediction.

Abstract

In this paper we prove existence for model knot solutions to the dimensionally reduced twisted Kapustin-Witten equations on $\mathbb{R}^3_+$ for any twisting parameter $t\in(0,\infty)$. We start with the explicit solutions for $t = 1$ derived in \cite{WFivebranes} and perform a continuity argument in $t$. This corroborates a prediction of Gaiotto and Witten \cite[][p. 961]{GW}.