ON THE BACKGROUND INDEPENDENCE OF 2-DIMENSIONAL TOPOLOGICAL GRAVITY

We formulate two-dimensional topological gravity in a background covar iant Lagrangian framework. We derive the Ward identities which charact erize the dependence of physical correlators on the background world-s heet metric defining the gauge-slice. We point out the existence of an ''anomaly'' in Ward identities involving correlators of observables w ith higher ghost number. This ''anomaly'' represents an obstruction fo r physical correlators to be globally defined forms on moduli space wh ich could be integrated in a background independent way. Starting from the anomalous Ward identities, we derive ''descent'' equations whose solutions are cocycles of the Lie algebra of the diffeomorphism group with values in the space of local forms on the moduli space. We solve the descent equations and provide explicit formulas for the cocycles, which allow for the definition of background independent integrals of physical correlators on the moduli space.