Generic Spontaneous Symmetry Breaking in SU(n) — Equivariant Bifurcation Problems

Abstract

An important feature of bifurcation problems with symmetry is the occurrence of spontaneous symmetry breaking. A basic state of a bifurcation equation loses stability when a distinguished bifurcation parameter passes through a critical value. Typically, then, new solutions branch off that basic state which possess a lower symmetry. One of the fundamental problems of equivariant (or covariant) bifurcation theory is to determine the symmetry of the bifurcating solution branches, i. e., their isotropy subgroups. The approach to this problem (e. g. [5,6]) was so far mainly based on the fixed-point subspaces corresponding to the maximal isotropy subgroups. In this paper we focus attention on a complementary approach which rests upon the geometry of the orbit strata in the space of the basic invariants. This concept has been introduced in [1,10,11] into field theory, but till now is disregarded in bifurcation theory. Our objective here is to demonstrate in terms of bifurcation problems with SU(n)-symmetry that the dimensions of the strata corresponding to maximal isotropy subgroups are closely related to the number k of basic nonquadratic invariants of lowest order. Specifically for k = 1 generic spontaneous symmetry breaking leads to 1-dimensional strata whereas for k > 1 strata of higher dimensions may occur. The basic group theory is summarized in Section 2. Bifurcation problems with SU(n)-symmetry are analysed in Section 3 and a discussion including consequences for bifurcation theory and particle physics is postponed to Section 4.