Goldstone models in D + 1 dimensions, D = 3, 4, 5, supporting stable and zero topological charge solutions

Abstract

We study finite energy static solutions to a global symmetry breaking Goldstone model described by an isovector scalar field in D + 1 spacetime dimensions. Both topologically stable multisolitons with arbitrary winding numbers and zero topological charge soliton–antisoliton solutions are constructed numerically in D = 3, 4, 5. We have explored the types of symmetries the systems should be subjected to, for there to exist multisoliton and soliton–antisoliton pairs in D = 3, 4, 5, 6. These findings are underpinned by constructing numerical solutions in the D ⩽ 5 examples. Subject to axial symmetry, only multisolitons of all topological charges exist in even D, and in odd D only zero and unit topological charge solutions exist. Subjecting the system to weaker than axial symmetries results in the existence of all the possibilities in all dimensions. Our findings also apply to finite ‘energy’ solutions to Yang–Mills and Yang–Mills–Higgs systems as well as to sigma models, but we find the numerical work for the Goldstone models more accessible.