Eur. Phys. J. C (2006) 47: 513-524
https://doi.org/10.1140/epjc/s2006-02571-1

Theoretical Physics

Hopf solitons and Hopf Q-balls on S³

¹ Departamento de Fisica de Particulas, Universidad de Santiago and Instituto Galego de Fisica de Altas Enerxias (IGFAE), 15782, Santiago de Compostela, Spain
² Institute of Physics, Jagiellonian University, Reymonta 4, 30-059, Kraków, Poland

^* e-mail: wereszczynski@th.if.uj.edu.pl

Received: 21 February 2006
Revised: 8 April 2006
Published online: 7 June 2006

Abstract

Field theories with a S²-valued unit vector field living on S³×ℝ space-time are investigated. The corresponding eikonal equation, which is known to provide an integrable sector for various sigma models in different spaces, is solved giving static as well as time-dependent multiply knotted configurations on S³ with arbitrary values of the Hopf index. Using these results, we then find a set of hopfions with topological charge Q_H=m², m∈Z, in the integrable subsector of the pure CP¹ model. In addition, we show that the CP¹ model with a potential term provides time-dependent solitons. In the case of the so-called “new baby Skyrme” potential we find, e.g., exact stationary hopfions, i.e., topological Q-balls. Our results further enable us to construct exact static and stationary Hopf solitons in the Faddeev–Niemi model with or without the new baby Skyrme potential. Generalizations for a large class of models are also discussed.