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Higher order singletons, partially massless fields and their boundary values in the ambient approach

Xavier Bekaert, Maxim Grigoriev

TL;DR

This work develops a gauge- and $\mathfrak{o}(d,2)$-covariant ambient-space framework to relate AdS$_{d+1}$ bulk fields with boundary higher-order singletons, shadow fields, and partially massless (PM) fields. By combining ambient methods with BRST and the parent formulation, the authors derive a comprehensive boundary–bulk dictionary, construct reducible PM multiplets, and prove a generalized Flato–Fronsdal theorem for higher-order singletons. They show that Fradkin–Tseytlin-type equations arise as obstructions to extending boundary data into the bulk, and provide manifestly conformal descriptions of higher-order singletons, shadows, and their currents. These results motivate a generalized HS holography where the $O(N)$ Lifshitz multicritical fixed point could be dual to a bulk theory built from partially massless tensors, with the higher-order singleton symmetry algebra organizing the bulk spectrum and boundary symmetries.

Abstract

Using ambient space we develop a fully gauge and o(d,2) covariant approach to boundary values of AdS(d+1) gauge fields. It is applied to the study of (partially) massless fields in the bulk and (higher-order) conformal scalars, i.e. singletons, as well as (higher-depth) conformal gauge fields on the boundary. In particular, we identify the corresponding Fradkin-Tseytlin equations as obstructions to the extension of the off-shell boundary value to the bulk, generalizing the usual considerations for the holographic anomalies to the partially massless fields. We also relate the background fields for the higher-order singleton to the boundary values of partially massless fields and prove the appropriate generalization of the Flato-Fronsdal theorem, which is in agreement with the known structure of symmetries for the higher-order wave operator. All these facts support the following generalization of the higher-spin holographic duality: the O(N) model at a multicritical isotropic Lifshitz point should be dual to the theory of partially massless symmetric tensor fields described by the Vasiliev equations based on the higher-order singleton symmetry algebra.

Higher order singletons, partially massless fields and their boundary values in the ambient approach

TL;DR

This work develops a gauge- and -covariant ambient-space framework to relate AdS bulk fields with boundary higher-order singletons, shadow fields, and partially massless (PM) fields. By combining ambient methods with BRST and the parent formulation, the authors derive a comprehensive boundary–bulk dictionary, construct reducible PM multiplets, and prove a generalized Flato–Fronsdal theorem for higher-order singletons. They show that Fradkin–Tseytlin-type equations arise as obstructions to extending boundary data into the bulk, and provide manifestly conformal descriptions of higher-order singletons, shadows, and their currents. These results motivate a generalized HS holography where the Lifshitz multicritical fixed point could be dual to a bulk theory built from partially massless tensors, with the higher-order singleton symmetry algebra organizing the bulk spectrum and boundary symmetries.

Abstract

Using ambient space we develop a fully gauge and o(d,2) covariant approach to boundary values of AdS(d+1) gauge fields. It is applied to the study of (partially) massless fields in the bulk and (higher-order) conformal scalars, i.e. singletons, as well as (higher-depth) conformal gauge fields on the boundary. In particular, we identify the corresponding Fradkin-Tseytlin equations as obstructions to the extension of the off-shell boundary value to the bulk, generalizing the usual considerations for the holographic anomalies to the partially massless fields. We also relate the background fields for the higher-order singleton to the boundary values of partially massless fields and prove the appropriate generalization of the Flato-Fronsdal theorem, which is in agreement with the known structure of symmetries for the higher-order wave operator. All these facts support the following generalization of the higher-spin holographic duality: the O(N) model at a multicritical isotropic Lifshitz point should be dual to the theory of partially massless symmetric tensor fields described by the Vasiliev equations based on the higher-order singleton symmetry algebra.

Paper Structure

This paper contains 39 sections, 8 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Higher-order singletons: boundary, bulk, and ambient descriptions
  3. Boundary values in terms of intrinsic AdS geometry
  4. Boundary values in terms of ambient geometry
  5. Group theoretical preliminaries
  6. Generalized Verma $\mathfrak{o}(d,2)$-modules
  7. Group-theoretical catalogue of $\mathfrak{o}(d,2)$-modules
  8. Generalized Flato--Fronsdal theorem
  9. Partially massless fields
  10. Ambient representation of partially massless fields
  11. Parent formulation of partially massless fields
  12. Global reducibility parameters
  13. Reducible multiplet of PM fields
  14. Boundary values in the (gauge-) covariant approach
  15. Boundary data for partially massless fields
  16. ...and 24 more sections

Key Result

Proposition 1

The tensor product of two singletons of order $\ell$($\geqslant 1$) decomposes as the following sum of the irreducible $\mathfrak{o}(d,2)$-modules describing the partially conserved currents of all ranks $s\in\mathbb N$ and all odd depths $t$ (= $2k-1$) ranging from $1$ to $2\ell-1$.

Theorems & Definitions (8)

  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proposition 5
  • Lemma 6
  • Lemma 7
  • Lemma 8