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

TL;DR

This paper explicitly solves the Schrödinger equation for a Chern-Simons theory on a Riemann surface with rational coupling, revealing how wave functions form a finite-dimensional projective representation of the braid group and large gauge transformations.

Contribution

It provides an explicit wave function solution involving Jacobi theta functions and establishes a novel connection between gauge transformations and braiding in Chern-Simons theory on Riemann surfaces.

Findings

Wave functions form a finite-dimensional representation of the braid group.

Wave functions involve Jacobi theta functions.

A quantization condition relates charges, Chern-Simons level, and genus.

Abstract

We find an explicit solution of the Schr\"odinger equation for a Chern-Simons theory coupled to charged particles on a Riemann surface, when the coefficient of the Chern-Simons term is a rational number (rather than an integer) and where the total charge is zero. We find that the wave functions carry a projective representation of the group of large gauge transformations. We also examine the behavior of the wave function under braiding operations which interchange particle positions. We find that the representation of both the braiding operations and large gauge transformations involve unitary matrices which mix the components of the wave function. The set of wave functions are expressed in terms of appropriate Jacobi theta functions. The matrices form a finite dimensional representation of a particular (well known to mathematicians) version of the braid group on the Riemann surface. We find a constraint which relates the charges of the particles, $q$, the coefficient of the Chern-Simons term, $k$ and the genus of the manifold, $g$: $q^2(g-1)/k$ must be an integer. We discuss a duality between large gauge transformations and braiding operations.