Axially Symmetric Solutions for SU(2) Yang-Mills Theory

TL;DR

This paper maps SU(2) Yang-Mills equations to Ernst equations from general relativity, deriving axially symmetric solutions like the Kerr analogue, which exhibit surfaces where fields diverge, potentially linked to confinement mechanisms.

Contribution

It introduces a novel method to generate Yang-Mills solutions from known Einstein solutions, specifically deriving Kerr-like solutions with unique properties.

Findings

Yang-Mills solutions analogous to known Einstein metrics are constructed.

The Kerr-like solution exhibits surfaces where gauge fields diverge.

Potential connection between these surfaces and confinement phenomena.

Abstract

By casting the Yang-Mills-Higgs equations of an SU(2) theory in the form of the Ernst equations of general relativity, it is shown how the known exact solutions of general relativity can be used to give similiar solutions for Yang-Mills theory. Thus all the known exact solutions of general relativity with axial symmetry (e.g. the Kerr metric, the Tomimatsu-Sato metric) have Yang-Mills equivalents. In this paper we only examine in detail the Kerr-like solution. It will be seen that this solution has surfaces where the gauge and scalar fields become infinite, which correspond to the infinite redshift surfaces of the normal Kerr solution. It is speculated that this feature may be connected with the confinement mechanism since any particle which carries an SU(2) color charge would tend to become trapped once it passes these surfaces. Unlike the Kerr solution, our solution apparently does not have any intrinsic angular momentum, but rather appears to give the non-Abelian field configuration associated with concentric shells of color charge.