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First order parent formulation for generic gauge field theories

Glenn Barnich, Maxim Grigoriev

TL;DR

The paper develops a systematic method to reformulate generic gauge field theories, described by a BRST differential, as a first-order parent system in which spacetime derivatives enter exclusively through the de Rham differential. By enlarging the field space to include derivatives as independent fields and incorporating antifield terms, the construction yields a BRST operator $s^P$ that can reproduce on- and off-shell formulations and, for diffeomorphism-invariant models, can be written as an AKSZ-type sigma model. It then analyzes reductions via generalized auxiliary fields, degree counting, and contractions to produce equivalent formulations, including connections to unfolding and Fedosov quantization. The framework is illustrated through a range of examples—relativistic particle, parametrized mechanics, Yang–Mills, gravity, and a 2D sigma model—demonstrating the versatility of the parent approach for organizing gauge structure and constructing unfolded-like off-shell theories. The results suggest that the parent formulation provides a unifying language that blends BRST cohomology, AKSZ techniques, and unfolding, with potential applications to interacting higher-spin theories and gravity.

Abstract

We show how a generic gauge field theory described by a BRST differential can systematically be reformulated as a first order parent system whose spacetime part is determined by the de Rham differential. In the spirit of Vasiliev's unfolded approach, this is done by extending the original space of fields so as to include their derivatives as new independent fields together with associated form fields. Through the inclusion of the antifield dependent part of the BRST differential, the parent formulation can be used both for on and off-shell formulations. For diffeomorphism invariant models, the parent formulation can be reformulated as an AKSZ-type sigma model. Several examples, such as the relativistic particle, parametrized theories, Yang-Mills theory, general relativity and the two dimensional sigma model are worked out in details.

First order parent formulation for generic gauge field theories

TL;DR

The paper develops a systematic method to reformulate generic gauge field theories, described by a BRST differential, as a first-order parent system in which spacetime derivatives enter exclusively through the de Rham differential. By enlarging the field space to include derivatives as independent fields and incorporating antifield terms, the construction yields a BRST operator that can reproduce on- and off-shell formulations and, for diffeomorphism-invariant models, can be written as an AKSZ-type sigma model. It then analyzes reductions via generalized auxiliary fields, degree counting, and contractions to produce equivalent formulations, including connections to unfolding and Fedosov quantization. The framework is illustrated through a range of examples—relativistic particle, parametrized mechanics, Yang–Mills, gravity, and a 2D sigma model—demonstrating the versatility of the parent approach for organizing gauge structure and constructing unfolded-like off-shell theories. The results suggest that the parent formulation provides a unifying language that blends BRST cohomology, AKSZ techniques, and unfolding, with potential applications to interacting higher-spin theories and gravity.

Abstract

We show how a generic gauge field theory described by a BRST differential can systematically be reformulated as a first order parent system whose spacetime part is determined by the de Rham differential. In the spirit of Vasiliev's unfolded approach, this is done by extending the original space of fields so as to include their derivatives as new independent fields together with associated form fields. Through the inclusion of the antifield dependent part of the BRST differential, the parent formulation can be used both for on and off-shell formulations. For diffeomorphism invariant models, the parent formulation can be reformulated as an AKSZ-type sigma model. Several examples, such as the relativistic particle, parametrized theories, Yang-Mills theory, general relativity and the two dimensional sigma model are worked out in details.

Paper Structure

This paper contains 26 sections, 8 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Parent theory
  3. Original BRST differential
  4. Extended space of fields and basic operations
  5. BRST differential of parent theory
  6. AKSZ-type sigma model
  7. Linear theories and first quantized description
  8. Reductions
  9. Generalized auxiliary fields and algebraically trivial pairs
  10. Degrees
  11. Target space reductions
  12. Going on-shell
  13. Equivalence of parent and original theory
  14. Contractible pairs for $\widetilde{s}$
  15. Diffeomorphism invariant theories
  16. ...and 11 more sections

Key Result

Proposition 2.1

Algebraically trivial pairs for $s_{-1}$ are generalized auxiliary fields of the theory determined by $s$.

Theorems & Definitions (10)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8