Topological and Hopf charges of a twisted Skyrmion string

TL;DR

This paper analyzes the topological and Hopf charges of a twisted Skyrmion string within nonlinear sigma models, revealing their relation to vortex winding numbers and demonstrating charge conservation principles in non-compact solutions.

Contribution

It introduces a detailed calculation of topological and Hopf charges for twisted Skyrmion strings and extends charge conservation principles to non-compact vortex solutions.

Findings

Topological charge equals the vortex winding number n.

Hopf charge is proportional to nmk.

Charge conservation applies to non-compact twisted solutions.

Abstract

We examine nonlinear sigma models, in particular the Skyrme model with a twist (the twisted Skyrmion string), which comprises a vortex solution with an added dependence on a twist term $mkz$, where $z$ is the vertical coordinate. The topological and Hopf charges of a twisted Skyrmion string are calculated and are shown to be equivalent to the winding number $n$ of the vortex and proportional to $nmk$, respectively. The principle of conservation of the Hopf charge, which is a standard property of the baby Skyrmion model, is generalised to cover the non-compact twisting solutions studied here.