SCALAR TENSOR QUANTUM-GRAVITY IN 2 DIMENSIONS

We discuss some classical and quantum properties of 2d gravity models involving metric and a scalar field. Different models are parametrized in terms of a scalar potential. We show that a general Liouville-type model with exponential potential and linear curvature coupling is renormalisable at the quantum level while a particular model (corresponding to D = 2 graviton-dilaton string effective action and having a back-hole solution) is finite. We use the condition of a "split" Weyl symmetry to suggest possible expressions for the "effective action" which includes the quantum anomaly term.