BRST cohomology and invariants of four-dimensional gravity in Ashtekar variables

We discuss the Becchi-Rouet-Stora-Tyutin (BRST) cohomologies of the invariants associated with the description of classical and quantum gravity in four dimensions, using the Ashtekar variables. These invariants are constructed from several BRST cohomology sequences. They provide a systematic and clear characterization of nonlocal observables in general relativity with unbroken diffeomorphism invariance, and could yield further differential invariants for four-manifolds. The description includes fluctuations of the vierbein fields, but there exists a nontrivial phase which can be expressed in terms of Witten's topological quantum field theory. In this phase, the descent sequences are degenerate, and the corresponding classical solutions can be identified with the conformally self-dual sector of Einstein manifolds. The full theory includes fluctuations which bring the system out of this sector while preserving diffeomorphism invariance.