Self-dual anyons in uniform background fields

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 kappa = n/2pi 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)straight theta = n/2pi with the noncommutative parameter straight 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.