BRST cohomology of Yang-Mills gauge fields in the presence of gravity in Ashtekar variables

TL;DR

This paper explores the BRST cohomology of Yang-Mills fields coupled with gravity in Ashtekar variables, revealing simplified algebraic structures and deriving key physical quantities like Lagrangians and anomalies.

Contribution

It introduces a simplified algebraic framework for BRST cohomology in gravity-coupled Yang-Mills theory using Maurer-Cartan conditions and Sorella's operator.

Findings

Derived Yang-Mills Lagrangians and Chern-Simons terms.

Solved gauge anomaly conditions.

Simplified algebraic structure for BRST cohomology.

Abstract

The BRST transformations for the Yang-Mills gauge fields in the presence of gravity described by Ashtekar variables are obtained by using the so-called Maurer-Cartan horizontality conditions. The BRST cohomology group expressed by the Wess-Zumino consistency condition is solved with the help of an operator $\delta$ introduced by S.P. Sorella which in our case has a very simple form and generates, together with the differential $d$ and the BRST operator $s$, a simpler algebra than in the pure Yang-Mills theory. In this way we shall find the Yang-Mills Lagrangians, the Chern-Simons terms and the gauge anomalies.