We investigate $\mathrm{U}\left(N\right)$ Chern-Simons theories on the noncommutative plane. We show that for the theories to be consistent quantum mechanically, the coefficient of the Chern-Simons term should be quantized $\kappa =n/2\mathrm{}\pi$ with an integer $n$. This is a surprise for the $\mathrm{U}\left(1\right)$ gauge theory. When uniform background charge density ${\rho }_e$ is present, the quantization rule changes to $\kappa +{\rho }_e\theta =n/2\mathrm{}\pi$ with the noncommutative parameter $\theta$. With the exact expression for the angular momentum, we argue in the $\mathrm{U}\left(1\right)$ 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 $-n/2\mathrm{}$.

• Received 28 February 2001

DOI:https://doi.org/10.1103/PhysRevLett.87.030402