Notes on Quantum Liouville Theory and Quantum Gravity

We review the status of Liouville field theory and discuss the identification of states and operators. The quantum theory has two kinds of states (and operators). Local operators correspond to non-normalizable states. Normalizable states correspond to non-local operators which create macroscopic holes in the surface. The existence of two kinds of states is important when we combine Liouville theory with a matter conformal field theory to study quantum gravity. If the matter theory has operators with Δ<(c−1)/24 they are dressed by non-local operators in Liouville and lead to an instability of the two-dimensional surface. Such operators exist in the generic CFT for c<1 and are always present for c>1. We discuss several ways to go beyond this, so called, c=1 barrier.