Twisted N=2 Supergravity as Topological Gravity in Four Dimensions

TL;DR

This paper demonstrates how four-dimensional N=2 supergravity can be reformulated as a topological gravity theory through BRST twisting, revealing new structures and potential applications in gravitational instanton moduli spaces.

Contribution

It shows the topological twisting of N=2 supergravity and connects it to topological gravity, providing a foundation for future studies of matter coupling and string theory applications.

Findings

Topological BRST complex is a rearrangement of the original complex.

The supergravity action can be obtained by gauge fixing to gravitational instantons.

Explicit descent equations for topological observables are provided.

Abstract

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.