On the relation between quantum Liouville theory and the quantized Teichm"uller spaces

Abstract

We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichmüller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verification of this conjecture.

Contribution to the proceedings of the 6th International Conference on CFTs and Integrable Models, Chernogolovka, Russia, September 2002; v2: Typos corrected, typographical changes

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• Liouville theory revisited
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• A lecture on the Liouville vertex operators
• R-operator, co-product and Haar-measure for the modular double of U_q(sl(2,R))
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