On structure and open problems in topological theories coupled to topological gravity

The structure of topological theory coupled to topological gravity is studied. We show that in this theory $Q$-exact terms do not decouple. This nondecoupling in the action of the theory is connected with the existence of boundaries of the moduli space and leads to problems in defining the topological gravity for massive topological theories. Nondecoupling of $Q$-exact observables leads to filtration of the gravitational descendants constructed from matter fields. Two a priori different preferred splittings of this filtration are constructed (one connected with the massive deformation of the theory and the other connected with the flatness of the connection on the space of theories). It is conjectured that they coincide.