From Liouville Theory to the Quantum Geometry of Riemann Surfaces

• Quantum Liouville theory provides the simplest example for a two-dimensional conformal field theory with continuous spectrum . It can therefore be regarded as a paradigm for a whole new class of two-dimensional conformal field theories which are neither rational nor quasi-rational. • The quantized Liouville theory is related to quantized spaces of Riemann surfaces. This interpretation should provide the basis for a deeper understanding of two-dimensional quantum gravity 3 as well as a future theory of three-dimensional quantum gravity (see e.g. 4,5 and references therein).