Yang-Mills instantons in Kaehler spaces with one holomorphic isometry

TL;DR

This paper explores self-dual Yang-Mills instantons in 4D Kaehler spaces with a holomorphic isometry, deriving generalized Bogomol'nyi equations and finding explicit solutions in various geometries.

Contribution

It generalizes the Bogomol'nyi equations for instantons in Kaehler spaces with isometries and provides explicit solutions in specific geometries.

Findings

Derived a generalized Bogomol'nyi equation for Kaehler spaces with isometry.

Reduced the problem to a simple radial differential equation with universal solutions.

Explicitly constructed instanton solutions in selected Kaehler geometries.

Abstract

We consider self-dual Yang-Mills instantons in 4-dimensional Kaehler spaces with one holomorphic isometry and show that they satisfy a generalization of the Bogomol'nyi equation for magnetic monopoles on certain 3-dimensional metrics. We then search for solutions of this equation in 3-dimensional metrics foliated by 2-dimensional spheres, hyperboloids or planes in the case in which the gauge group coincides with the isometry group of the metric (SO(3), SO(1,2) and ISO(2), respectively). Using a generalized hedgehog ansatz the Bogomol'nyi quations reduce to a simple differential equation in the radial variable which admits a universal solution and, in some cases, a particular one, from which one finally recovers instanton solutions in the original Kaehler space. We work out completely a few explicit examples for some Kaehler spaces of interest.