Chern-Simons Theories on Noncommutative Plane

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 $-n/2$.