WARD IDENTITIES IN 2-DIMENSIONAL GRAVITY

We study the decoupling of null states in two-dimensional gravity, using methods of critical string theory. We identify a family of null states which fail to decouple due to curvature and boundary terms. This gives relations involving amplitudes at different genus. At genus zero, these determine certain operator product coefficients. At genus one, they determine the partition function. At higher genus. we obtain a relation similar in form to the Painleve equation. but due to an incomplete understanding of a certain ghost/curvature term we do not have a closed relation for the partition function. Our results appear to correspond to the L0 and L1 equations in the topological and matrix model approaches.