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Local BRST cohomology in Einstein--Yang--Mills theory

Glenn Barnich, Friedemann Brandt, Marc Henneaux

TL;DR

This paper delivers a comprehensive local BRST cohomology analysis for Einstein–Yang–Mills theory using the antifield formalism, extending beyond standard Lagrangians to general gauge- and diffeomorphism-invariant actions. By decomposing the BRST differential into δ and γ and leveraging descent equations and the Künneth formula, it classifies the equivariant characteristic cohomology, consistent deformations, and candidate anomalies across dimensions n>2. The authors show that for semisimple YM groups antifield dependence can be removed from both consistent deformations and candidate anomalies, reducing possibilities to known chiral and topological anomalies, with extra structures arising only in the presence of abelian factors. They also establish when Noether currents can be covariantized and how global symmetries manifest within the BRST cohomology, providing a robust framework for analyzing quantum consistency of Einstein–Yang–Mills theories in diverse spacetime settings.

Abstract

We analyse in detail the local BRST cohomology in Einstein-Yang-Mills theory using the antifield formalism. We do not restrict the Lagrangian to be the sum of the standard Hilbert and Yang-Mills Lagrangians, but allow for more general diffeomorphism and gauge invariant actions. The analysis is carried out in all spacetime dimensions larger than 2 and for all ghost numbers. This covers the classification of all candidate anomalies, of all consistent deformations of the action, as well as the computation of the (equivariant) characteristic cohomology, i.e. the cohomology of the spacetime exterior derivative in the space of (gauge invariant) local differential forms modulo forms that vanish on-shell. We show in particular that for a semi-simple Yang-Mills gauge group the antifield dependence can be entirely removed both from the consistent deformations of the Lagrangian and from the candidate anomalies. Thus, the allowed deformations of the action necessarily preserve the gauge structure, while the only candidate anomalies are those provided by previous works not dealing with antifields, and by ``topological" candidate anomalies which are present only in special spacetime dimensions (6,9,10,13,...). This result no longer holds in presence of abelian factors where new candidate anomalies and deformations of the action can be constructed out of the conserved Noether currents (if any). The Noether currents themselves are shown to be covariantizable, with a few exceptions discussed as well.

Local BRST cohomology in Einstein--Yang--Mills theory

TL;DR

This paper delivers a comprehensive local BRST cohomology analysis for Einstein–Yang–Mills theory using the antifield formalism, extending beyond standard Lagrangians to general gauge- and diffeomorphism-invariant actions. By decomposing the BRST differential into δ and γ and leveraging descent equations and the Künneth formula, it classifies the equivariant characteristic cohomology, consistent deformations, and candidate anomalies across dimensions n>2. The authors show that for semisimple YM groups antifield dependence can be removed from both consistent deformations and candidate anomalies, reducing possibilities to known chiral and topological anomalies, with extra structures arising only in the presence of abelian factors. They also establish when Noether currents can be covariantized and how global symmetries manifest within the BRST cohomology, providing a robust framework for analyzing quantum consistency of Einstein–Yang–Mills theories in diverse spacetime settings.

Abstract

We analyse in detail the local BRST cohomology in Einstein-Yang-Mills theory using the antifield formalism. We do not restrict the Lagrangian to be the sum of the standard Hilbert and Yang-Mills Lagrangians, but allow for more general diffeomorphism and gauge invariant actions. The analysis is carried out in all spacetime dimensions larger than 2 and for all ghost numbers. This covers the classification of all candidate anomalies, of all consistent deformations of the action, as well as the computation of the (equivariant) characteristic cohomology, i.e. the cohomology of the spacetime exterior derivative in the space of (gauge invariant) local differential forms modulo forms that vanish on-shell. We show in particular that for a semi-simple Yang-Mills gauge group the antifield dependence can be entirely removed both from the consistent deformations of the Lagrangian and from the candidate anomalies. Thus, the allowed deformations of the action necessarily preserve the gauge structure, while the only candidate anomalies are those provided by previous works not dealing with antifields, and by ``topological" candidate anomalies which are present only in special spacetime dimensions (6,9,10,13,...). This result no longer holds in presence of abelian factors where new candidate anomalies and deformations of the action can be constructed out of the conserved Noether currents (if any). The Noether currents themselves are shown to be covariantizable, with a few exceptions discussed as well.

Paper Structure

This paper contains 26 sections, 16 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Method
  4. Main results
  5. Outline of the paper
  6. Assumptions and notation
  7. New basis of generators of ${\cal A}$
  8. Cohomology of $d$ in the algebra ${\cal E}$ of local forms
  9. BRST cohomology in ${\cal A}_1$ and Künneth formula for $H^*(s,{\cal E})$
  10. Descent equations
  11. BRST cohomology in ${\cal A}_2$
  12. Notation
  13. Computation of $H^{*}(s,{\cal A}_2)$
  14. Equivariant characteristic cohomology
  15. Result for $H^{*,n}(s|d)$
  16. ...and 11 more sections

Key Result

Theorem 4.1

: The cohomology $H^k(d,{\cal E})$ of $d$ in the algebra of local $k$-forms is isomorphic to the De Rham cohomology of $E$ in the same degree for $k<n$, Furthermore, in form degree $n$, one has where the $n$-form $\omega = {\cal L} d^nx$ is said to be "variationally closed" iff the Euler-Lagrange derivatives of ${\cal L}$ with respect to all the fields and antifields identically vanish.

Theorems & Definitions (16)

  • Theorem 4.1
  • Corollary 4.1
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Theorem 6.1
  • Corollary 6.1
  • Theorem 6.2
  • Lemma 7.1
  • Theorem 7.1
  • ...and 6 more