TWISTED N=2 SUPERGRAVITY AS TOPOLOGICAL GRAVITY IN 4 DIMENSIONS

We show that the BRST quantum version of pure D = 4 N = 2 supergravity can be topologically twisted, to yield a formulation of topological gravity in four dimensions. The topological BRST complex is just a re-arrangement of the old BRST complex, that partly modifies the role of physical and ghost fields: indeed, the new ghost number turns out to be the sum of the old ghost number plus the internal U(1) charge. Furthermore, the action of N = 2 supergravity is retrieved from topological gravity by choosing a gauge fixing that reduces the space of physical states to the space of gravitational instanton configurations, namely to self-dual spin connections. The descent equations relating the topological observables are explicitly exhibited and discussed. Ours is a first step in a programme that aims at finding the topological sector of matter coupled N = 2 supergravity, viewed as the effective lagrangian of type II superstrings and, as such, already related two-dimensional topological field theories. As it stands the theory we discuss may prove useful in describing gravitational instanton moduli-spaces.