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Parent field theory and unfolding in BRST first-quantized terms

Glenn Barnich, Maxim Grigoriev, Alexei Semikhatov, Ilya Tipunin

TL;DR

This work unifies BRST first-quantized methods with the unfolded approach to free field theories by formulating a parent, first-order BRST system whose eliminations yield either the original BRST-based theory or its unfolded formulation. It introduces generalized auxiliary fields at the first-quantized level as contractible BRST pairs and proves the invariance of BRST cohomology under their elimination. The construction is illustrated with Klein-Gordon and Fronsdal higher-spin theories, and extended to a Fedosov-inspired framework that yields a covariant unfolded description, potentially on curved backgrounds. Overall, the paper provides a robust, covariant bridge between BRST quantization and Vasiliev-style unfolding, with implications for global symmetries and interactions in higher-spin systems.

Abstract

For free-field theories associated with BRST first-quantized gauge systems, we identify generalized auxiliary fields and pure gauge variables already at the first-quantized level as the fields associated with algebraically contractible pairs for the BRST operator. Locality of the field theory is taken into account by separating the space--time degrees of freedom from the internal ones. A standard extension of the first-quantized system, originally developed to study quantization on curved manifolds, is used here for the construction of a first-order parent field theory that has a remarkable property: by elimination of generalized auxiliary fields, it can be reduced both to the field theory corresponding to the original system and to its unfolded formulation. As an application, we consider the free higher-spin gauge theories of Fronsdal.

Parent field theory and unfolding in BRST first-quantized terms

TL;DR

This work unifies BRST first-quantized methods with the unfolded approach to free field theories by formulating a parent, first-order BRST system whose eliminations yield either the original BRST-based theory or its unfolded formulation. It introduces generalized auxiliary fields at the first-quantized level as contractible BRST pairs and proves the invariance of BRST cohomology under their elimination. The construction is illustrated with Klein-Gordon and Fronsdal higher-spin theories, and extended to a Fedosov-inspired framework that yields a covariant unfolded description, potentially on curved backgrounds. Overall, the paper provides a robust, covariant bridge between BRST quantization and Vasiliev-style unfolding, with implications for global symmetries and interactions in higher-spin systems.

Abstract

For free-field theories associated with BRST first-quantized gauge systems, we identify generalized auxiliary fields and pure gauge variables already at the first-quantized level as the fields associated with algebraically contractible pairs for the BRST operator. Locality of the field theory is taken into account by separating the space--time degrees of freedom from the internal ones. A standard extension of the first-quantized system, originally developed to study quantization on curved manifolds, is used here for the construction of a first-order parent field theory that has a remarkable property: by elimination of generalized auxiliary fields, it can be reduced both to the field theory corresponding to the original system and to its unfolded formulation. As an application, we consider the free higher-spin gauge theories of Fronsdal.

Paper Structure

This paper contains 19 sections, 13 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Local field theory associated with a BRST first-quantized system
  3. BRST first-quantized system
  4. Local field theory and master action
  5. Generalized auxiliary fields in the Lagrangian context
  6. Non-Lagrangian BRST field theory
  7. Equations of motion and gauge symmetries from a BRST differential
  8. Generalized auxiliary fields in the non-Lagrangian context
  9. Elimination of generalized auxiliary fields at the first-quantized level
  10. The action for an equivalence class of equations of motion
  11. First-quantized description of the Fronsdal Lagrangian
  12. The first-order parent system
  13. Extended first-quantized system and parent field theory
  14. Reduction to the original system
  15. Reduction to the unfolded formalism
  16. ...and 4 more sections

Key Result

Proposition 3.1

Let $v^a$ and $w^a$ be generalized auxiliary fields. Let also $\mathrm{gh}(v^a)=k$ and $\mathrm{gh}(w^a)=k-1$.

Theorems & Definitions (21)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • ...and 11 more