Hopf solitons and Hopf Q-balls on S^3

TL;DR

This paper constructs and analyzes static and time-dependent Hopf solitons and Q-balls on S^3 within various sigma models, revealing new knotted configurations and exact solutions, including topological Q-balls and stationary hopfions.

Contribution

It provides explicit solutions for knotted Hopf solitons and Q-balls on S^3, expanding the understanding of topological solitons in integrable and non-integrable sigma models.

Findings

Explicit static and time-dependent knotted configurations on S^3.

Existence of topological Q-balls in the $CP^1$ model with potential.

Construction of exact static and stationary Hopf solitons in the Faddeev--Niemi model.

Abstract

Field theories with a $S^2$-valued unit vector field living on $S^3 \times \RR$ 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 $Q_H=m^2$, $m \in \mathbf{Z}$, 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.