Quantum Moduli Spaces of Flat Connections

Using the formalism of discrete quantum group gauge theory, one can construct the quantum algebras of observables for the Hamiltonian ChernSimons model. The resulting moduli algebras provide quantizations of the algebra of functions on the moduli spaces of flat connections on a punctured 2-dimensional surface. In this note we describe some features of these moduli algebras with special emphasis on the natural action of mapping class groups. This leads, in particular, to a closed formula for representations of the mapping class groups on conformal blocks. DESY 96-261 q-alg/yymmxxx 1 Introduction Moduli spaces of flat connections on a Riemann surface \Sigma enter physics as phase spaces of the so-called Chern-Simons gauge theory on a three dimensional manifold \Sigma \Theta R. They possess a famous Poisson structure that has been studied extensively in the mathematical literature (see e.g. [1]). The information about a particular flat connection on \Sigma can be encoded in its holonom..