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 $κ= n/2π$ with an integer $n$. This is a surprise for the U(1) gauge theory. When uniform background charge density $ρ_e$ is present, the quantization rule changes to $κ+ρ_eθ= n/2π$ with noncommutative parameter $θ$. 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 $-n/2$.