Noncommutative Chern-Simons Soliton

TL;DR

This paper explores the effects of spatial noncommutativity on the Chern-Simons-Higgs model, revealing universal corrections to magnetic fields and highlighting ambiguities in defining energy-momentum tensors in noncommutative theories.

Contribution

It provides the first-order noncommutative extension of the relativistic Chern-Simons-Higgs model, analyzing soliton solutions and identifying a universal correction to magnetic fields dependent only on the noncommutative parameter.

Findings

Universal noncommutative correction to magnetic field depends only on θ.

Other observables are influenced by both θ and model parameters.

Identifies a mismatch between BPS equations and equations of motion at O(θ).

Abstract

We have studied the noncommutative extension of the relativistic Chern-Simons-Higgs model, in the first non-trivial order in $\theta$, with only spatial noncommutativity. Both Lagrangian and Hamiltonian formulations of the problem have been discussed, with the focus being on the canonical and symmetric forms of the energy-momentum tensor. In the Hamiltonian scheme, constraint analysis and the induced Dirac brackets have been provided. The spacetime translation generators and their actions on the fields are discussed in detail. The effects of noncommutativity on the soliton solutions have been analysed thoroughly and we have come up with some interesting observations. Considering the {\it{relative}} strength of the noncommutative effects, we have shown that there is a universal character in the noncommutative correction to the magnetic field - it depends {\it{only}} on $\theta$. On the other hand, in the cases of all other observables of physical interest, such as the potential profile, soliton mass or the electric field, $\theta$ as well as $\tau$, (comprising solely of commutative Chern-Simons-Higgs model parameters), appear on similar footings. This phenomenon is a new finding which has come up in the present analysis. Lastly, we have pointed out a generic problem in the NC extension of the models, in the form of a mismatch between the BPS dynamical equation and the full variational equations of motion, to $O(\theta)$. This mismatch indicates that the analysis is not complete as it brings in to fore the ambiguities in the definition of the energy-momentum tensor in a noncommutative theory.