Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II

TL;DR

This paper develops a rigorous combinatorial quantization framework for Hamiltonian Chern-Simons theory, connecting quantum groups, the Verlinde algebra, and moduli space volume calculations.

Contribution

It provides a mathematically rigorous definition of the quantum algebra of observables and links it to deformation quantization and the Verlinde formula.

Findings

The algebra of observables is a *-algebra of functions on the quantum moduli space.

The volume of the quantized moduli space matches the Verlinde number.

The approach offers a rigorous foundation for quantum gauge theory in Chern-Simons models.

Abstract

This paper further develops the combinatorial approach to quantization of the Hamiltonian Chern Simons theory advertised in \cite{AGS}. Using the theory of quantum Wilson lines, we show how the Verlinde algebra appears within the context of quantum group gauge theory. This allows to discuss flatness of quantum connections so that we can give a mathe- matically rigorous definition of the algebra of observables $\A_{CS}$ of the Chern Simons model. It is a *-algebra of ``functions on the quantum moduli space of flat connections'' and comes equipped with a positive functional $\omega$ (``integration''). We prove that this data does not depend on the particular choices which have been made in the construction. Following ideas of Fock and Rosly \cite{FoRo}, the algebra $\A_{CS}$ provides a deformation quantization of the algebra of functions on the moduli space along the natural Poisson bracket induced by the Chern Simons action. We evaluate a volume of the quantized moduli space and prove that it coincides with the Verlinde number. This answer is also interpreted as a partition partition function of the lattice Yang-Mills theory corresponding to a quantum gauge group.