Are Vortex Numbers Preserved?

TL;DR

This paper investigates noncommutative vortex solutions in the Abelian Higgs model, demonstrating that certain vortex numbers are preserved under noncommutative deformation and exploring solutions via Fock space representation.

Contribution

It constructs noncommutative vortex solutions that retain their vortex numbers and introduces a Fock space approach to these solutions.

Findings

Vortex numbers can be preserved under noncommutative deformation.

Constructed explicit solutions with unchanged vortex numbers.

Explored solutions using Fock space representation.

Abstract

We study noncommutative vortex solutions that minimize the action functional of the Abelian Higgs model in 2-dimensional noncommutative Euclidean space. We first consider vortex solutions which are deformed from solutions defined on commutative Euclidean space to the noncommutative one. We construct solutions whose vortex numbers are unchanged under the noncommutative deformation. Another class of noncommutative vortex solutions via a Fock space representation is also studied.