Reducible connections and non-local symmetries of the self-dual Yang-Mills equations

TL;DR

This paper classifies all reducible solutions to the self-dual Yang-Mills equations on flat four-dimensional space, showing they are related to flat connections via non-local symmetries, and introduces a broader class of solutions akin to finite-type harmonic maps.

Contribution

It constructs the most general reducible self-dual Yang-Mills connection and demonstrates their relation to flat connections through non-local symmetries, expanding the solution space.

Findings

All reducible solutions are in the orbit of flat connections under non-local symmetries.

Such solutions form a larger class analogous to finite-type harmonic maps.

The work provides a comprehensive classification of reducible self-dual Yang-Mills connections.

Abstract

We construct the most general reducible connection that satisfies the self-dual Yang-Mills equations on a simply connected, open subset of flat $\mathbb{R}^4$. We show how all such connections lie in the orbit of the flat connection on $\mathbb{R}^4$ under the action of non-local symmetries of the self-dual Yang-Mills equations. Such connections fit naturally inside a larger class of solutions to the self-dual Yang-Mills equations that are analogous to harmonic maps of finite type.