Effect of rotation symmetry on Abelian Chern-Simons field theory and anyon equation on a sphere

We analyze Chern-Simons field theory coupled to a nonrelativistic matter field on a sphere using canonical transformation on the fields with special attention to the role of the rotation symmetry: SO(3) invariance restricts the Hilbert space to the one with a definite number of charges and dictates the Dirac quantization condition to the Chern-Simons coefficient, whereas SO(2) invariance does not. The corresponding Schrödinger equation for many anyons (and for multispecies) on the sphere are presented with appropriate boundary conditions. In the presence of an external magnetic monopole source, the ground state solutions of anyons are compared with monopole harmonics. The role of the translation and modular symmetry on a torus is also expounded.