THE DILATON EQUATION IN SEMIRIGID STRING THEORY

We show how to obtain explicit integration measures on ordinary moduli space corresponding to the correlation functions of pure two-dimensional topological gravity. In particular. our prescription tells how to remove the zero modes of the beta-gamma-system. We then use our formula to derive the "dilaton equation" introduced by E. Verlinde and H. Verlinde, a relation between the N-point and ( N - 1)-point correlations of this theory. Just as in critical string theory we use the fact that certain BRST-exact states fail to decouple. Instead they build up Cech classes, in this instance the Euler class of an N-times punctured surface. Throughout we use the "semirigid' formulation of topological gravity. Thus the Liouville sector of other approaches never enters.