M. Menaa et M. Tahiri, BRST-ANTI-BRST SYMMETRY AND OBSERVABLES FOR TOPOLOGICAL GRAVITY, Physical review. D. Particles and fields, 57(12), 1998, pp. 7312-7317
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.