Linking number of vortices as baryon number

Sven Bjarke Gudnason,Muneto Nitta

doi:10.1103/physrevd.101.065011

Sven Bjarke Gudnason, Muneto Nitta

Open Access

https://doi.org/10.1103/physrevd.101.065011

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Abstract

We show that the topological degree of a Skyrmion field is the same as the Hopf charge of the field under the Hopf map and thus equals the linking number of the preimages of two points on the 2-sphere under the Hopf map. We further interpret two particular points on the 2-sphere as vortex zeros and the linking of these zero lines follows from the latter theorem. Finally we conjecture that the topological degree of the Skyrmion can be interpreted as the product of winding numbers of vortices corresponding to the zero lines, summing over clusters of vortices.

Highlights

Skyrmions are topological solitons [1] of the texture type, i.e., they are maps from one-point compactified 3-space, X 1⁄4 R3 ∪ f∞g ≃ S3 to a target space N 1⁄4 S3 with a nonvanishing topological degree π3ðS3Þ 1⁄4 Z ∋ B ≠ 0
We show that the topological degree of a Skyrmion field is the same as the Hopf charge of the field under the Hopf map and equals the linking number of the preimages of two points on the 2-sphere under the Hopf map
We would like to associate the zero lines of each complex scalar field with vortex rings. It proves convenient for our calculations as we will be using the Hopf map, which is naturally written in terms of two complex scalar fields

Summary

INTRODUCTION

Skyrmions are topological solitons [1] of the texture type, i.e., they are maps from one-point compactified 3-space, X 1⁄4 R3 ∪ f∞g ≃ S3 to a target space N 1⁄4 S3 with a nonvanishing topological degree π3ðS3Þ 1⁄4 Z ∋ B ≠ 0. It proves convenient to write the SU(2) field as two complex scalar fields, ψ1;2, living on the complexified 1-sphere (jψ1j2 þ jψ2j2 1⁄4 1). We would like to associate the zero lines of each complex scalar field with (deformed) vortex rings. It proves convenient for our calculations as we will be using the Hopf map, which is naturally written in terms of two complex scalar fields. We conjecture that we can interpret the topological degree of a Skyrmion map as the product of winding numbers of two vortex lines, summing over clusters of wound vortices.

Theorem and conjecture

The rational map

EXAMPLES

Rational map Skyrmions

Findings

DISCUSSION AND OUTLOOK