Vortices in (abelian) Chern-Simons gauge theory

TL;DR

This paper reviews various vortex solutions in Abelian Chern-Simons gauge theories, including relativistic, non-relativistic, fermionic, and topological vortices, highlighting their explicit solutions and underlying symmetries.

Contribution

It provides a comprehensive review of vortex solutions in Abelian Chern-Simons theories, including new explicit solutions and symmetry analyses.

Findings

Explicit solutions to the Liouville equation for vortices

Fermionic vortices with spinor matter fields

Conformal symmetry explains vortex dynamics

Abstract

The vortex solutions of various classical planar field theories with (Abelian) Chern-Simons term are reviewed. Relativistic vortices, put forward by Paul and Khare, arise when the Abelian Higgs model is augmented with the Chern-Simons term. Adding a suitable sixth-order potential and turning off the Maxwell term provides us with pure Chern-Simons theory with both topological and non-topological self-dual vortices, as found by Hong-Kim-Pac, and by Jackiw-Lee-Weinberg. The non-relativistic limit of the latter leads to non-topological Jackiw-Pi vortices with a pure fourth-order potential. Explicit solutions are found by solving the Liouville equation. The scalar matter field can be replaced by spinors, leading to fermionic vortices. Alternatively, topological vortices in external field are constructed in the phenomenological model proposed by Zhang-Hansson-Kivelson. Non-relativistic Maxwell-Chern-Simons vortices are also studied. The Schroedinger symmetry of Jackiw-Pi vortices, as well as the construction of some time-dependent vortices, can be explained by the conformal properties of non-relativistic space-time, derived in a Kaluza-Klein-type framework.