Conservation laws in Skyrme-type models

TL;DR

This paper explores the integrability of Skyrme-type models by constructing theories with infinitely many conserved currents linked to volume-preserving diffeomorphisms, revealing new integrable subsectors within these models.

Contribution

It introduces a geometric method to identify integrable and partially integrable subsectors in Skyrme models based on target space symmetries.

Findings

Identified a class of theories with infinitely many conserved currents.

Established a connection between conserved currents and volume-preserving diffeomorphisms.

Discovered weak and strong integrability conditions for the Skyrme model.

Abstract

The zero curvature representation of Zakharov and Shabat has been generalized recently to higher dimensions and has been used to construct non-linear field theories which either are integrable or contain integrable submodels. The Skyrme model, for instance, contains an integrable subsector with infinitely many conserved currents, and the simplest Skyrmion with baryon number one belongs to this subsector. Here we use a related method, based on the geometry of target space, to construct a whole class of theories which are either integrable or contain integrable subsectors (where integrability means the existence of infinitely many conservation laws). These models have three-dimensional target space, like the Skyrme model, and their infinitely many conserved currents turn out to be Noether currents of the volume-preserving diffeomorphisms on target space. Specifically for the Skyrme model, we find both a weak and a strong integrability condition, where the conserved currents form a subset of the algebra of volume-preserving diffeomorphisms in both cases, but this subset is a subalgebra only for the weak integrable submodel.