Chern-Simons Theories on Noncommutative Plane

TL;DR

This paper studies U(N) Chern-Simons theories on a noncommutative plane, revealing quantization conditions for the Chern-Simons coefficient and fractional spin phenomena in different phases.

Contribution

It demonstrates the quantization of the Chern-Simons coefficient in noncommutative U(N) theories and explores fractional spin of particles and vortices.

Findings

Quantization of the Chern-Simons coefficient $oxed{ ext{for consistency, } oxed{ ext{} ext{}} ext{ }oxed{ ext{}} ext{ } ext{must be } rac{n}{2 extpi}$}

Modified quantization rule with background charge density: $oxed{ ext{ } ext{ ho}_e heta + ext{ extkappa} = rac{n}{2 extpi}$}

Charged particles in the symmetric phase carry fractional spin $1/2n$, vortices in the broken phase carry half-integer or integer spin $-n/2$.

Abstract

We investigate U(N) Chern-Simons theories on noncommutative plane. We show that for the theories to be consistent quantum mechanically, the coefficient of the Chern-Simons term should be quantized $\kappa = n/2\pi$ with an integer $n$. This is a surprise for the U(1) gauge theory. When uniform background charge density $\rho_e$ is present, the quantization rule changes to $\kappa +\rho_e\theta = n/2\pi$ with noncommutative parameter $\theta$. With the exact expression for the angular momentum, we argue in the U(1) theory that charged particles in the symmetric phase carry fractional spin $1/2n$ and vortices in the broken phase carry half-integer or integer spin $-n/2$.