A solution of two-dimensional topological quantum gravity

We give a formulation of two-dimensional topological gravity without matter in terms of a supersymmetric conformally invariant field theory and derive a path integral expression for the physical amplitudes. A careful analysis of the contact terms of the physical operators reveals the presence of a non-commutative algebra, isomorphic to the Virasoro algebra. We show that this algebra completely determines all the amplitudes at arbitrary genus, which coincide with those of the one-matrix model at the k = 1 critical point.