Canonical quantization of lattice Chern-Simons theory

TL;DR

This paper develops a canonical quantization framework for lattice U(1) Chern-Simons theory, revealing a finite-dimensional physical Hilbert space with ground state degeneracy depending on the level and spin structure.

Contribution

It introduces a detailed lattice quantization method for U(1) Chern-Simons theory, including bosonic and fermionic cases, and clarifies the role of gauge constraints and spin structures.

Findings

Finite-dimensional Hilbert space with k ground states on a torus

Quantization of both bosonic and fermionic theories with spin structure dependence

Implementation of gauge constraints including non-local Wilson loop projections

Abstract

We discuss the canonical quantization of $U(1)_k$ Chern-Simons theory on a spatial lattice. In addition to the usual local Gauss law constraints, the physical Hilbert space is defined by 1-form gauge constraints implementing the compactness of the $U(1)$ gauge group, and (depending on the details of the spatial lattice) non-local constraints which project out unframed Wilson loops. Though the ingredients of the lattice model are bosonic, the physical Hilbert space is finite-dimensional, with exactly $k$ ground states on a spatial torus. We quantize both the bosonic (even level) and fermionic (odd level) theories, describing in detail how the latter depends on a choice of spin structure.