Selfgravitating Yang-Mills solitons and their Chern-Simons numbers

TL;DR

This paper classifies regular, spherically symmetric Einstein-Yang-Mills solutions, derives their Chern-Simons numbers using bundle theory, and discusses implications for gauge groups like SU(n) with potential physical relevance.

Contribution

It provides a bundle-theoretical classification of Einstein-Yang-Mills solitons and explicit formulas for their Chern-Simons numbers, highlighting their quantized nature for certain gauge groups.

Findings

Chern-Simons numbers are half-integers or integers for SU(n) groups.

Solutions are necessarily magnetic if the magnetic charge vanishes.

Explicit expressions relate Chern-Simons numbers to root systems and asymptotics.

Abstract

We present a classification of the possible regular, spherically symmetric solutions of the Einstein-Yang-Mills system which is based on a bundle theoretical analysis for arbitrary gauge groups. It is shown that such solitons must be of magnetic type, at least if the magnetic Yang-Mills charge vanishes. Explicit expressions for the Chern-Simons numbers of these selfgravitating Yang-Mills solitons are derived, which involve only properties of irreducible root systems and some information about the asymptotics of the solutions. It turns out, as an example, that the Chern-Simons numbers are always half-integers or integers for the gauge groups $SU(n)$. Possible physical implications of these results, which are based on analogies with the unstable sphaleron solution of the electroweak theory, are briefly indicated.