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

TL;DR

This paper demonstrates the existence of $ ext{R}$-invariant instanton solutions to the Kapustin-Witten equations on a specific 4-manifold, which interpolate between model solutions with related integer labels, supporting Witten's gauge theory approach to knot invariants.

Contribution

It proves the existence of interpolating solutions between model solutions with integer label differences that are positive even numbers, clarifying the structure of solutions relevant to Witten's knot theory program.

Findings

Existence of interpolating solutions between model solutions with specific label differences.

The difference in labels must be a positive even integer for solutions to exist.

Moduli space of solutions characterized by complex parameters depending on label differences.

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 $\mathbb{C}^{(m_0 -m_\infty - 2)/2}\times \mathbb{C}^*$ moduli space of these interpolation solutions.)