Anton Yu. Alekseev, Volker Schomerus
Representation Theory of Chern-Simons Observables
The paper is published: Duke Math. J. 85, 2 (1996) 447-510

MSC:

58D27 Moduli problems for differential geometric structures

81T40 Two-dimensional field theories, conformal field theories, etc.

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

81S10 Geometric quantization, symplectic methods, See Also {

Abstract: In \cite{AGS1}, \cite{AGS2} we suggested a new quantum algebra,
the moduli algebra, which
is conjectured to be a quantum algebra of observables of
the Hamiltonian Chern-Simons theory.
This algebra provides the quantization
of the algebra of functions on the moduli space of flat connections
on a 2-dimensional surface. In this paper we classify unitary representations
of this new algebra and identify the corresponding
representation spaces with
the spaces of conformal blocks of the WZW model. The mapping class
group of the surface is proved to act on the moduli algebra by
inner automorphisms. The generators of these automorphisms are
unitary elements of the moduli algebra. They are constructed
explicitly and proved to satisfy the relations of the (unique)
central extension of the mapping class group.

Keywords: Poisson structures, quantization, Chern-Simons theory, moduli space, mapping class group, conformal field theory, geometric quantization