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

TL;DR

This paper extends self-dual Yang-Mills equations to conformally Kahler 4-manifolds, explores hidden symmetries, and finds continuous symmetries only in conformally half-flat geometries, without assuming isometries.

Contribution

It generalizes flat space Yang-Mills equations to complex surfaces and analyzes hidden symmetries in these geometries, revealing new symmetry structures.

Findings

Continuous hidden symmetries exist only in conformally half-flat cases.

The generalization applies to Einstein metrics beyond self-dual curvature.

No isometries are required for the symmetry analysis.

Abstract

We derive a generalization of the flat space Yang's and Newman's equations for self-dual Yang-Mills fields to (locally) conformally Kahler Riemannian 4-manifolds. The results also apply to Einstein metrics (whose full curvature is not necessarily self-dual). We analyse the possibility of hidden symmetries in the form of Backlund 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.