Quantum Liouville theory versus quantized Teichmüller spaces

This note announces the proof of a conjecture of H. Verlinde, according to which the spaces of Liouville conformal blocks and the Hilbert spaces from the quantization of the Teichmüller spaces of Riemann surfaces carry equivalent representations of the mapping class group. This provides a basis for the geometrical interpretation of quantum Liouville theory in its relation to quantized spaces of Riemann surfaces.

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