Noncommutative Chern-Simons Solitons

TL;DR

This paper studies noncommutative Chern-Simons theories, deriving exact multi-vortex solutions and analyzing their properties, including size, position, and angular momentum, in both nonrelativistic and relativistic contexts.

Contribution

It introduces matter fields coupled to noncommutative Chern-Simons theory and finds explicit multi-vortex solutions, including their moduli dependence and relativistic counterparts.

Findings

Exact multi-vortex solutions in noncommutative Chern-Simons theory

Explicit determination of vortex size, position, and angular momentum

Identification of solution branches and noncommutativity scale effects in relativistic case

Abstract

The Chern-Simons theories on a noncommutative plane, which is shown to be describing the quantum Hall liquid, is considered. We introduce matter fields fundamentally coupled to the noncommutative Chern-Simons field. Exploiting BPS equations for the nonrelativistic Chern-Simons theory, we find the exact solutions of multi vortices that are closely packed and exponentially localized. We determine the position, the size and the angular momentum explicitly. We then construct the solutions of two spatially separated vortices and determine the moduli dependence of the size and the angular momentum. We also consider the relativistic Chern-Simons theory and find nontopological solutions whose properties are similar to the nonrelativistic counterpart. However, unlike the nonrelativistic case, there are two branches of solutions for a given magnetic field and they cease to exist below certain noncommutativity scale.