Combinatorial quantization of the Hamiltonian Chern-Simons theory I

TL;DR

This work introduces a combinatorial quantization of Hamiltonian Chern--Simons theory by formulating a lattice model with quantum group gauge symmetry that reproduces the continuum theory. It builds a finite-dimensional lattice algebra B of quantum-group valued link variables together with a ribbon Hopf-*-algebra of local gauge transformations, and extracts an invariant *-algebra A of observables through a regular representation on a function space F with a positively defined inner product. The construction relies on quadratic exchange relations governed by R-matrices, a consistent *-structure, and eyelash data to manage vertex ordering; it is later generalized to weak quasi-Hopf algebras to accommodate truncation at roots of unity. The outlook indicates a program to impose quantum flatness (moduli algebra), establish graph-independence, and connect to conformal blocks and Knizhnik–Zamolodchikov theory, thereby tying the lattice model to the Hamiltonian CS theory and WZNW/CFT structures. This framework provides a concrete, finite-dimensional, gauge-invariant route to quantize CS theory and study its observables via a moduli-space–oriented, algebraic approach.

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.

Theorems & Definitions (20)

• Definition 1
• Proposition 2
• Definition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7
• Lemma 8
• Theorem 9
• Definition 10