The existence of instanton solutions to the $\mathbb{R}$-invariant Kapustin-Witten equations on $(0,\infty )\times {\mathbb{R}}^2\times \mathbb{R}$

Abstract

A non-negative integer labeled set of model solutions to the $\mathbb{R}$-invariant Kapustin-Witten equations on $(0,\infty )\times {\mathbb{R}}^2\times \mathbb{R}$ plays a central role in Edward Witten's program to interpret the colored Jones polynomial or a knot in the context of SU(2) gauge theory. This paper explains why there are $\mathbb{R}$-invariant solutions to these equations on $(0,\infty )\times {\mathbb{R}}^2\times \mathbb{R}$ that interpolate between two model solutions as the $(0,\infty )$ parameter increases from 0 to $\infty$ while respecting the ${\mathbb{R}}^2$ factor asymptotics. The only constraint on the limiting pair of model solutions is this: Letting $m_0$ and $m_{\infty }$ denote their non-negative integer labels, then $m_0-m_{\infty }$ must be a positive, even integer. (As explained in the paper, there is a ${\mathbb{C}}^{(m_0-m_{\infty }-2)/2}\times {\mathbb{C}}^{\ast }$ moduli space of these interpolation solutions.)

References in corpus (1)

• Classification of Nahm pole solutions of the Kapustin-Witten equations on $S^1\times \Sigma \times {\mathbb{R}}^{+}$