The Yang-Mills equation near instanton-anti-instanton configurations

TL;DR

This paper investigates the limits of Yang-Mills connections near instanton-anti-instanton configurations, revealing obstructions and classifying solutions with low energy on .

Contribution

It establishes conditions under which instantons are the only solutions below a certain energy threshold and proves energy spectrum discreteness.

Findings

Instantons are the only solutions with energy below a specific threshold.

Obstructions from deformations prevent certain bubbling configurations.

Energy spectrum on the trivial -bundle is discrete below 16 7 4.

Abstract

We study the question of whether a sequence of non-instanton Yang-Mills connections can limit to a bubbling configuration composed only of instantons. In the case that the Uhlenbeck limit and the bubbles are of opposite charge, we determine an obstruction coming from deformations of the Uhlenbeck limit. As an application, we prove that instantons are the only solutions of the $\mathrm{SU}(2)$ Yang-Mills equation on $\mathbb{R}^4$ with energy less than $4\pi^2 \left( |\kappa| + 2 \right) + \varepsilon_\kappa,$ where $\kappa$ is the charge. We also prove discreteness of the energy spectrum on the trivial $\mathrm{SU}(2)$-bundle in the range $\left[ 0, 16 \pi^2 \right).$