Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space

We formulate the geometric quantization of Teichmüller space by using its relation with SL(2, R) Chern-Simons gauge theory and show that the physical state conditions arising in this formalism are equivalent to the Virasoro Ward identities satisfied by the conformal blocks in CFT. We further show that transition amplitudes between the physical states of this quantum system have a direct correspondence with covariant amplitudes of two-dimensional induced quantum gravity. Possible applications of these results to Virasoro modular geometry and (2 + 1)-dimensional quantum gravity are indicated.