Algebraic structure of gravity in Ashtekar variables

TL;DR

This paper explores the algebraic structure of gravity formulated with Ashtekar variables by analyzing BRST cohomology and differential invariants, providing insights into the geometric and topological aspects of four-dimensional manifolds.

Contribution

It introduces a method to compute BRST cohomology in Ashtekar variables using Maurer-Cartan horizontality and Sorella's operator, revealing new differential invariants.

Findings

BRST cohomology in Ashtekar variables is explicitly calculated.

Differential invariants for four-dimensional manifolds are derived.

A novel approach using Maurer-Cartan horizontality conditions is presented.

Abstract

The BRST transformations for gravity in Ashtekar variables are obtained by using the Maurer-Cartan horizontality conditions. The BRST cohomology in Ashtekar variables is calculated with the help of an operator $\delta$ introduced by S.P. Sorella, which allows to decompose the exterior derivative as a BRST commutator. This BRST cohomology leads to the differential invariants for four-dimensional manifolds.