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

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.Comment: AMSLatex, 15 pages, no figures. Corrected in line with the referee's comments. In particular, restriction to simply-connected open sets now explicitly stated. Version to appear in Communications in Mathematical Physic