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Comments on higher-spin symmetries

Xavier Bekaert

TL;DR

This work investigates an unconstrained frame-like formulation for an infinite tower of symmetric higher-spin fields in the flat-space limit and derives the corresponding metric-like gauge transformations to first order in a weak-field expansion. It shows that, at this order, the non-Abelian HS gauge algebra is isomorphic to the real Lie algebra of Hermitian differential operators on ${\mathbb R}^n$, with the Moyal bracket encoding the deformed bracket and the spin-2 sector reproducing diffeomorphisms. The analysis connects HS gauge symmetries to deformation quantisation (Fedosov/Moyal) and ambient-space constructions for AdS/CFT, providing a geometric interpretation via Cartan-like connections and torsion constraints. Despite its linear scope, the paper suggests a HS geometry where unitary differential operators extend diffeomorphisms, offering a potential route to HS interactions, though it acknowledges substantial obstacles in extending to higher orders and consistently constructing vertices. Overall, this work advances the understanding of HS symmetries by linking frame-like and metric-like formulations through a concrete algebraic structure, highlighting both the potential and the limits of the unconstrained approach in a flat background.

Abstract

The unconstrained frame-like formulation of an infinite tower of completely symmetric tensor gauge fields is reviewed and examined in the limit where the cosmological constant goes to zero. By partially fixing the gauge and solving the torsion constraints, the form of the gauge transformations in the unconstrained metric-like formulation are obtained till first order in a weak field expansion. The algebra of the corresponding gauge symmetries is shown to be equivalent, at this order and modulo (unphysical) gauge parameter redefinitions, to the Lie algebra of Hermitian differential operators on R^n, the restriction of which to the spin-two sector is the Lie algebra of infinitesimal diffeomorphisms.

Comments on higher-spin symmetries

TL;DR

This work investigates an unconstrained frame-like formulation for an infinite tower of symmetric higher-spin fields in the flat-space limit and derives the corresponding metric-like gauge transformations to first order in a weak-field expansion. It shows that, at this order, the non-Abelian HS gauge algebra is isomorphic to the real Lie algebra of Hermitian differential operators on , with the Moyal bracket encoding the deformed bracket and the spin-2 sector reproducing diffeomorphisms. The analysis connects HS gauge symmetries to deformation quantisation (Fedosov/Moyal) and ambient-space constructions for AdS/CFT, providing a geometric interpretation via Cartan-like connections and torsion constraints. Despite its linear scope, the paper suggests a HS geometry where unitary differential operators extend diffeomorphisms, offering a potential route to HS interactions, though it acknowledges substantial obstacles in extending to higher orders and consistently constructing vertices. Overall, this work advances the understanding of HS symmetries by linking frame-like and metric-like formulations through a concrete algebraic structure, highlighting both the potential and the limits of the unconstrained approach in a flat background.

Abstract

The unconstrained frame-like formulation of an infinite tower of completely symmetric tensor gauge fields is reviewed and examined in the limit where the cosmological constant goes to zero. By partially fixing the gauge and solving the torsion constraints, the form of the gauge transformations in the unconstrained metric-like formulation are obtained till first order in a weak field expansion. The algebra of the corresponding gauge symmetries is shown to be equivalent, at this order and modulo (unphysical) gauge parameter redefinitions, to the Lie algebra of Hermitian differential operators on R^n, the restriction of which to the spin-two sector is the Lie algebra of infinitesimal diffeomorphisms.

Paper Structure

This paper contains 42 sections, 27 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Higher-spins vs spin-two
  3. Non-Abelian symmetries at lowest order
  4. Plan of the paper
  5. Notation
  6. Gravity example
  7. Cartan versus Riemann
  8. Moving frames
  9. Klein & Cartan view of geometry
  10. Vielbeins
  11. Perturbative analysis around a background space-time
  12. Moving frames
  13. Vielbeins
  14. Higher-spin algebras
  15. Anti de Sitter / Conformal algebras
  16. ...and 27 more sections

Key Result

Lemma 1

The centraliser ${\cal C}_{_{A_{n+1}}}(\,\mathfrak{sp}(2)\,)$ of the subspace $\mathfrak{sp}(2)$ in the Weyl algebra $A_{n+1}$ is isomorphic to the subspace of ${\mathbb C}[X^A,P^B]$ of polynomials, the coefficients of which are $\mathfrak{gl}(n+1)$-irreducible tensors described by rectangular two-r

Theorems & Definitions (31)

  • Remark 1
  • Lemma 1
  • Corollary 1
  • Corollary 2
  • Lemma 2: Dirac
  • Lemma 3: Eastwood
  • Corollary 3
  • Lemma 4: Frø nsdal
  • Corollary 4
  • Lemma 5
  • ...and 21 more