Superconformal Gravity And The Topology Of Diffeomorphism Groups

Twisted four-dimensional supersymmetric Yang-Mills theory famously gives a useful point of view on the Donaldson and Seiberg-Witten invariants of four-manifolds. In this paper we generalize the construction to include a path integral formulation of generalizations of Donaldson invariants for smooth families of four-manifolds. Mathematically these are equivariant cohomology classes for the action of the oriented diffeomorphism group on the space of metrics on the manifold. In principle these cohomology classes should contain nontrivial information about the topology of the diffeomorphism group of the four-manifold. We show that the invariants may be interpreted as the standard topologically twisted path integral of four-dimensional $\mathcal{N}=2$ supersymmetric Yang-Mills coupled to topologically twisted background fields of conformal supergravity.