Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II

TL;DR

This paper advances a combinatorial, quantum-group–based quantization of the Hamiltonian Chern-Simons theory by constructing a graph-independent moduli algebra $\mathcal{A}_{CS}$ of Chern-Simons observables and a positive integration functional $\omega_{CS}$ whose value reproduces Verlinde numbers as the volume of the quantum moduli space. It develops quantum Wilson lines via holonomies $U^I({\cal C})$ and monodromies $M^I({\cal C})$, identifies central Verlinde-type elements $c^I(P)$ and associated projectors $\chi^I(P)$, and shows how flatness constraints emerge from these structures to yield $\mathcal{A}_{CS}^{\{I_\nu\}}$. The framework provides a deformation-quantized description of functions on the moduli space that matches the Verlinde formula and interprets the quantum moduli space volume as a lattice Yang-Mills partition function for a quantum gauge group, thereby linking CS theory, WZW conformal blocks, and lattice gauge theory. The work further extends to quasi-Hopf symmetries, offering substitution rules to adapt the Hopf-algebra results to truncations at roots of unity and outlining directions for comparing with geometric quantization and $G/G$ 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 $ω$ (``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.

Theorems & Definitions (18)

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