WHAT IS THE INTRINSIC GEOMETRY OF 2-DIMENSIONAL QUANTUM-GRAVITY

The intrinsic geometry of 2D quantum gravity is discussed within the framework of the semi-classical Liouville Theory. We show how to define local reparametrization-invariant correlation functions in terms of the geodesic distance. Such observables exhibit strong non-logarithmic short-distance divergences. If one regularizes these divergences by a finite-part prescription, there are no corrections to KPZ scaling. the intrinsic fractal dimension of space-time is two, and no cascade of baby universes occurs. However we show that these divergences can be regularized in a covariant way and have a physical interpretation in terms of "pinning" of geodesics by regions where the metric is singular. This raises issues related to the physics of disordered systems (in particular of the 2D random field Ising model), such as the possible occurrence of replica symmetry breaking, which make the interpretation of numerical and analytical results a subtle and difficult problem.