BRST–anti-BRST symmetry and observables for topological gravity

Abstract

We consider topological four-dimensional (4D) gravity with an independent spin connection by using a superspace formalism. This gives rise to the basic fields of the quantized theory as well as to two pairs of extra fields, which are needed to close the BRST--anti-BRST algebra off shell. Therefore we build a gauge-fixing action written in BRST--anti-BRST exact form leading to an effective one, which allows us to fix all of the symmetries at once. In particular, the topological symmetries are fixed as in the model of topological 4D self-dual gravity. We construct the observables related to both BRST symmetry and anti-BRST symmetry. We find that the anti-BRST invariant observables are not fundamentally different from the BRST invariant ones, since there is a complete mirror symmetry between them. The obtained observables extend those constructed within the equivariant method in the context of topological 4D self-dual gravity.