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Parent form for higher spin fields on anti-de Sitter space

Glenn Barnich, Maxim Grigoriev

TL;DR

The paper develops a first-order parent theory for free higher-spin gauge fields on AdS by embedding AdS into a flat space and applying a Fedosov-like BRST construction. It shows how both Fronsdal’s metric-like and Vasiliev’s unfolded descriptions arise as consistent reductions from this parent framework, clarifying the algebraic structure (notably an $sp(4)$-type subalgebra) and enabling intermediate formulations. The approach provides a geometrically transparent setting in which generalized auxiliary fields and Lagrangian formulations can be systematically managed, potentially aiding the compatibility of Lagrangian and interaction structures in higher-spin theory. Overall, the work unifies BRST, embedding-space, and unfolded formalisms into a cohesive, covariant picture with explicit reduction rules to standard and unfolded descriptions on AdS.

Abstract

We construct a first order parent field theory for free higher spin gauge fields on constant curvature spaces. As in the previously considered flat case, both Fronsdal's and Vasiliev's unfolded formulations can be reached by two different straightforward reductions. The parent theory itself is formulated using a higher dimensional embedding space and turns out to be geometrically extremely transparent and free of the intricacies of both of its reductions.

Parent form for higher spin fields on anti-de Sitter space

TL;DR

The paper develops a first-order parent theory for free higher-spin gauge fields on AdS by embedding AdS into a flat space and applying a Fedosov-like BRST construction. It shows how both Fronsdal’s metric-like and Vasiliev’s unfolded descriptions arise as consistent reductions from this parent framework, clarifying the algebraic structure (notably an -type subalgebra) and enabling intermediate formulations. The approach provides a geometrically transparent setting in which generalized auxiliary fields and Lagrangian formulations can be systematically managed, potentially aiding the compatibility of Lagrangian and interaction structures in higher-spin theory. Overall, the work unifies BRST, embedding-space, and unfolded formalisms into a cohesive, covariant picture with explicit reduction rules to standard and unfolded descriptions on AdS.

Abstract

We construct a first order parent field theory for free higher spin gauge fields on constant curvature spaces. As in the previously considered flat case, both Fronsdal's and Vasiliev's unfolded formulations can be reached by two different straightforward reductions. The parent theory itself is formulated using a higher dimensional embedding space and turns out to be geometrically extremely transparent and free of the intricacies of both of its reductions.

Paper Structure

This paper contains 31 sections, 11 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Gauge field theories associated to first-quantized systems
  3. BRST differential and equations of motion
  4. Reductions
  5. Lagrangians
  6. Scalar particle on AdS
  7. (A)dS space as an embedding
  8. BRST operator
  9. Reduction to the surface
  10. Reduction to standard description
  11. Reduction to unfolded form
  12. Free higher spin gauge fields on AdS
  13. The first-quantized model
  14. Reduction to the surface and algebraic structure
  15. Reduction to standard description
  16. ...and 16 more sections

Key Result

Proposition 2.1

Let $\boldsymbol{\mathcal{H}}$ decompose into a direct sum of vector bundles $\boldsymbol\mathcal{H}=\boldsymbol\mathcal{E}\oplus\boldsymbol\mathcal{G}\oplus\boldsymbol\mathcal{F}$ and the component $\overset{\boldsymbol\mathcal{G}\boldsymbol\mathcal{F}}{\mathsf{\Omega}}=\mathscr{{P}}_{\boldsymbol\m In this case, the gauge field theories associated with $(\mathsf{\Omega},\Gamma(\boldsymbol{\mathca

Theorems & Definitions (13)

  • Proposition 2.1
  • Proposition 3.1
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Proposition 4.5
  • Proposition 4.6
  • Proposition A.1
  • Lemma C.1
  • ...and 3 more