6 Hopf solitons and Hopf Q - balls on S 3

Field theories with a S 2-valued unit vector field living on S 3 × IR 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 S 3 with arbitrary values of the Hopf index. Using these results, we then find a set of hopfions with topological charge Q H = m 2 , m ∈ Z, in the integrable subsector of the pure CP 1 model. In addition, we show that the CP 1 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.