  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 rearrangement 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 to 2D topological field-theories.
As it stands the theory we discuss may prove useful in describing gravitational
instantons moduli-spaces.
