Stationary ring solitons in field theory — Knots and vortons

We review the current status of the problem of constructing classical field theory solutions describing stationary vortex rings in Minkowski space in 3+1 dimensions. We describe the known up to date solutions of this type, such as the static knot solitons stabilized by the topological Hopf charge, the attempts to gauge them, the anomalous solitons stabilized by the Chern–Simons number, as well as the non-Abelian monopole and sphaleron rings. Passing to the rotating solutions, we first discuss the conditions ensuring that they do not radiate, and then describe the spinning Q$Q$-balls, their twisted and gauged generalizations reported here for the first time, spinning skyrmions, and rotating monopole–antimonopole pairs. We then present the first explicit construction of global vortons as solutions of the elliptic boundary value problem, which demonstrates their non-radiating character. Finally, we describe the analogs of vortons in the Bose–Einstein condensates, analogs of spinning Q$Q$-balls in the non-linear optics, and also moving vortex rings in superfluid helium and in ferromagnetics.