The Hilbert space of Chern-Simons theory on the cylinder. A Loop Quantum Gravity approach

TL;DR

This paper applies loop quantum gravity methods to the canonical quantization of 3D Chern-Simons theory on a cylindrical space, successfully constructing a gauge-invariant physical Hilbert space.

Contribution

It provides a complete loop quantum gravity quantization of Chern-Simons theory on a cylinder, explicitly solving constraints and characterizing the physical Hilbert space.

Findings

Constructed a gauge and diffeomorphism invariant Hilbert space

Solved all quantum constraints explicitly

Found the physical Hilbert space to be infinite dimensional but separable

Abstract

As a laboratory for loop quantum gravity, we consider the canonical quantization of the three-dimensional Chern-Simons theory on a noncompact space with the topology of a cylinder. Working within the loop quantization formalism, we define at the quantum level the constraints appearing in the canonical approach and completely solve them, thus constructing a gauge and diffeomorphism invariant physical Hilbert space for the theory. This space turns out to be infinite dimensional, but separable.