On self-dual Yang–Mills fields on special complex surfaces

Abstract

We derive a generalization of the flat space equations of Yang and Newman for self-dual Yang–Mills fields to (locally) conformally Kähler Riemannian four-manifolds. The results also apply to Einstein metrics (whose full curvature is not necessarily self-dual). We analyze the possibility of hidden symmetries in the form of Bäcklund transformations, and we find a continuous group of hidden symmetries only for the case in which the geometry is conformally half-flat. No isometries are assumed.