  We point out that averages of equivariant observables of 2D topological
gravity are not globally defined forms on moduli space, when one uses the
functional measure corresponding to the formulation of the theory as a 2D
superconformal model. This is shown to be a consequence of the existence of the
Gribov horizon {\it and} of the dependence of the observables on derivatives of
the super-ghost field. By requiring the absence of global BRS anomalies, it is
nevertheless possible to associate global forms to correlators of observables
by resorting to the \v{C}ech-De Rham notion of form cohomology. To this end, we
derive and solve the ``descent'' of local Ward identities which characterize
the functional measure. We obtain in this way an explicit expression for the
\v{C}ech-De Rham cocycles corresponding to arbitrary correlators of
observables. This provides the way to compute and understand contact terms in
string theory from first principles.

High Energy Physics - Theory