Self-gravitating Yang–Mills solitons and their Chern–Simons numbers

Abstract

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 is presented herein. 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 self-gravitating 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.