Brst Cohomology and Invariants of 4D Gravity in Ashtekar Variables

TL;DR

This paper explores BRST cohomology in 4D gravity using Ashtekar variables, providing a systematic way to characterize non-local observables and invariants in quantum gravity, with connections to topological quantum field theory.

Contribution

It introduces a cohomological framework for invariants in 4D gravity with Ashtekar variables, linking classical and quantum aspects and identifying special phases related to topological theories.

Findings

Characterization of non-local observables in general relativity

Identification of a topological phase related to Witten's TQFT

Classical solutions correspond to conformally self-dual Einstein manifolds

Abstract

We discuss the 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 non-local observables in general relativity with unbroken diffeomorphism invariance, and could yield further differential invariants for four-manifolds. The theory includes fluctuations of the vierbein fields, but there exits a non-trivial 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.