Chern-Simons Theories on the Noncommutative Plane

We investigate U(N) 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 k = n/2 π 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 k + ρ_eθ = n/2 π with the 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.