The dilaton equation in semirigid string theory

Abstract

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 βγ system. We then use our formula to derive the “dilation 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 Čech 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.