Combinatorial quantization of the Hamiltonian Chern-Simons theory I

TL;DR

This paper presents a mathematically rigorous lattice-based quantization of Hamiltonian Chern-Simons theory, capturing its symmetries and algebraic structure, and aims to replicate continuous theory results precisely.

Contribution

It introduces a lattice model for Hamiltonian Chern-Simons theory with quantum gauge symmetry, providing a detailed algebraic framework and a positive inner product.

Findings

Constructed the algebra of observables with *-operation

Established the lattice model reproduces continuous theory results

Demonstrated quantum gauge symmetry in the lattice setting

Abstract

Motivated by a recent paper of Fock and Rosly \cite{FoRo} we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *-operation and a positive inner product.