Anton Yu. Alekseev, Harald Grosse, Volker Schomerus
Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II
The paper is published: Commun. Math. Phys. 174, 3 (1996) 561-604

MSC:

58F06 Geometric quantization (applications of representation theory), See also {22E45, 81S10}

81T70 Quantization in field theory; cohomological methods, See also {58F06}

81R50 Quantum groups and related algebraic methods, See Also {16W30, 17B37}

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 mathematically
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.

Keywords: quantization, Hamiltonian Chern-Simons theory, quantum Wilson lines, Verlinde algebra, deformation, quantized moduli space, quantum gauge group
Notes: second version