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

TL;DR

This paper investigates how rotation symmetry influences abelian Chern-Simons field theory and anyon equations on a sphere, revealing the impact of SO(3) and SO(2) invariances on the Hilbert space, quantization, and ground states.

Contribution

It provides a detailed analysis of the effects of rotation symmetry on Chern-Simons theory and anyon equations on a sphere, including boundary conditions and monopole harmonic solutions.

Findings

SO(3) invariance restricts Hilbert space to fixed charge states

Dirac quantization condition arises from SO(3) symmetry

Ground states with monopole sources relate to monopole harmonics

Abstract

We analyze the Chern-Simons field theory coupled to non-relativistic 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 Dirac quantization condition to the Chern-Simons coefficient, whereas SO(2) invariance does not. The corresponding Schr\"odinger equation for many anyons (and for multispecies) on the sphere are presented with appropriate boundary condition. 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.