New modular representations and fusion algebras from quantized SL(2,R) Chern-Simons theory

We consider the quantum-mechanical algebra of observables generated by canonical quantization of $SL(2,R)$ Chern-Simons theory with rational charge on a space manifold with torus topology. We produce modular representations generalizing the representations associated to the $SU(2)$ WZW models and we exhibit the explicit polynomial representations of the corresponding fusion algebras. The relation to Kac-Wakimoto characters of highest weight $\widehat{sl}(2)$ representations with rational level is illustrated. Talk given at the Como Conference on ``Integrable Quantum Field Theories'', September 1992.