TOPOLOGICAL GRAVITY IN 1+1 DIMENSIONS

We propose a topological action for 1 + 1 dimensional gravity based on the gauge symmetry SO(1,2) or its contraction ISO(1,1). The construction employs, besides the gauge multiplet, a scalar multiplet and is closely related to the Liouville action and induced gravity action. The theory is solvable at both the classical and quantum levels and has a renormalizable perturbation expansion. The only contributions that could be non-vanishing are those which are one-loop. The partition function is computed exactly and is related to the moduli space measure, implying a cancellation among the one-loop contributions. It is argued that if the theory is formulated on space-times admitting classical solutions then the quantum theory escapes the planckian domain to the classical domain. We give the generalizations to include matter interactions and supersymmetrization. Matter fields of any dimensions can be quantized and the interactions introduced are renormalizable. © 1990.