On the structure of the topological phase of two-dimensional gravity

The topological phase of two-dimensional gravity is re-examined. The correlation functions of the naturally occuring operators in the minimal topological model are computed, using topological methods, in genus zero and genus one. The genus-zero results agree with recent results obtained in exact solutions of “matrix models”, suggesting that the two approaches to two-dimensional gravity are equivalent. The coupling of two-dimensional topological gravity to topological sigma models is investigated. The CP1 model appears to be almost as simple as the pure topological gravity theory. General, model-independent properties of the correlation functions are obtained which hold in coupling to arbitrary topological field theories and can serve as a qualitative definition of the topological phase of two-dimensional gravity. A number of facts that are familiar in the usual phase of string theory, such as the relation between vanishing of the canonical line bundle of a Kähler manifold and scale invariance of the corresponding field theory, have simpler echoes in the topological phase.