BRST-ANTI-BRST SYMMETRY AND OBSERVABLES FOR TOPOLOGICAL GRAVITY

We consider topological four-dimensional (4D) gravity with an independ ent spin connection by using a superspace formalism. This gives rise t o 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 al l of the symmetries at once. In particular, the topological symmetries are fixed as in the model of topological 4D self-dual gravity. We con struct the observables related to both BRST symmetry and anti-BRST sym metry. We find that the anti-BRST invariant observables are not fundam entally different from the BRST invariant ones, since there is a compl ete mirror symmetry between them. The obtained observables extend thos e constructed within the equivariant method in the context of topologi cal 4D self-dual gravity.