Canonical Chern-Simons Theory and the Braid Group on a Riemann Surface

TL;DR

This paper explores how wave functions in Abelian Chern-Simons theory on Riemann surfaces realize braid group representations, extending previous models to rational levels, arbitrary genus, and non-zero total charge, with explicit solutions and constraints.

Contribution

It generalizes the quantization of Chern-Simons theory to rational levels, arbitrary genus surfaces, and non-zero total charge, providing explicit wave function solutions and identifying key constraints.

Findings

Wave functions carry braid group and projective gauge transformation representations.

Explicit solutions to the Schrödinger equation are obtained.

A fundamental charge and parameters constraint is established.

Abstract

We examine the problem of determining which representations of the braid group on a Riemann surface are carried by the wave function of a quantized Abelian Chern-Simons theory interacting with non-dynamical matter. We generalize the quantization of Chern-Simons theory to the case where the coefficient of the Chern-Simons term, $k$, is rational, the Riemann surface has arbitrary genus and the total matter charge is non-vanishing. We find an explicit solution of the Schr\"odinger equation. We find that the wave functions carry a representation of the braid group as well as a projective representation of the discrete group of large gauge transformations. We find a fundamental constraint which relates the charges of the particles, $q_i$, the coefficient $k$ and the genus of the manifold, $g$.