Gribov problem, contact terms and Cech-De Rham cohomology in 2D topological gravity

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 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 Cech-De Rham 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 Cech-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.