Conserved momenta of ferromagnetic solitons through the prism of differential geometry

TL;DR

This paper employs differential geometry, specifically Cartan's theory of differential forms, to clarify the conserved momenta of ferromagnetic solitons like skyrmions, accounting for gyroscopic forces and gauge dependencies.

Contribution

It introduces a differential geometric framework to derive conserved momenta of ferromagnetic solitons, resolving gauge dependence issues in the Lagrangian description.

Findings

Derived conserved momenta for the Belavin--Polyakov skyrmion.

Clarified the role of symmetries including translation, spin rotation, and dilation.

Applied Cartan's theory to address gauge-dependent canonical momenta.

Abstract

The relation between symmetries and conservation laws for solitons in a ferromagnet is complicated by the presence of gyroscopic (precessional) forces, whose description in the Lagrangian framework involves a background gauge field. This makes canonical momenta gauge-dependent and requires a careful application of Noether's theorem. We show that Cartan's theory of differential forms is a natural language for this task. We use it to derive conserved momenta of the Belavin--Polyakov skyrmion, whose symmetries include translation, global spin rotation, and dilation.