Toroidal solitons in 3+1 dimensional integrable theories

TL;DR

This paper explores the integrability of 3+1 dimensional models on SU(2)/U(1), introduces a new relativistic field theory with toroidal solitons and infinite conservation laws, and extends methods to the Skyrme-Faddeev model.

Contribution

It proposes a novel 3+1D integrable field theory with toroidal solitons and infinite conserved currents, expanding the understanding of higher-dimensional integrable models.

Findings

Exact construction of a 3+1D relativistic field theory with toroidal solitons.

Identification of key ingredients for infinite local conservation laws.

Application of the method to the Skyrme-Faddeev model with proposed integrable submodels.

Abstract

We analyze the integrability properties of models defined on the symmetric space SU(2)/U(1) in 3+1 dimensions, using a recently proposed approach for integrable theories in any dimension. We point out the key ingredients for a theory to possess an infinite number of local conservation laws, and discuss classes of models with such property. We propose a 3+1-dimensional, relativistic invariant field theory possessing a toroidal soliton solution carrying a unit of topological charge given by the Hopf map. Construction of the action is guided by the requirement that the energy of static configuration should be scale invariant. The solution is constructed exactly. The model possesses an infinite number of local conserved currents. The method is also applied to the Skyrme-Faddeev model, and integrable submodels are proposed.