Quantum uniformisation and conformal field theory

The deformation theory of Riemann surfaces naturally has two faces, one being represented by hyperbolic geometry, the other being complex-analytic in nature. Both pictures can be quantized in a natural way. The result is naturally related to the conformal field theory called Liouville theory. Conformal blocks with fixed intermediate representations coincide with the kernels relating the complex-analytic and the hyperbolic picture on the quantum level