Partially-massless higher-spin algebras and their finite-dimensional truncations

TL;DR

This work classifies partially-massless higher-spin (PM HS) global symmetry algebras in (A)dS$_{d+1}$ by two interrelated approaches: cosets of the universal enveloping algebra of $\mathfrak{so}_{d+2}$ and an oscillator/Howe duality framework. It identifies a one-parameter family $\mathcal{A}_{\lambda}$ that generically contains PM generators of all even depths, and, for integer $\lambda=\ell$, a natural splitting into a finite-depth coset $\mathfrak{p}_{\ell}$ and an ideal $\mathfrak{q}_{\ell}$, with $\mathfrak{p}_{\ell}$ corresponding to algebras spanned by depths $<2\ell$. The authors compute a trace and an invariant bilinear form using a covariant projector (and a Gaussian surrogate) to reveal degeneracies that signal truncations, identify finite-dimensional truncations $\mathfrak{f}_{k}$ isomorphic to $\mathfrak{gl}_{M_k}$, and relate these to symmetric one-row $\mathfrak{so}_{d+2}$ representations; they also connect even-depth PM subalgebras to CHS symmetry, highlighting a potential unitary truncation pathway. Overall, the paper advances the structural understanding of PM HS algebras, their truncations, and their relation to CHS, with implications for holography and higher-spin symmetry realizations.

Abstract

The global symmetry algebras of partially-massless (PM) higher-spin (HS) fields in (A)dS$_{d+1}$ are studied. The algebras involving PM generators up to depth $2\,(\ell-1)$ are defined as the maximal symmetries of free conformal scalar field with $2\,\ell$ order wave equation in $d$ dimensions. We review the construction of these algebras by quotienting certain ideals in the universal enveloping algebra of $(A)dS_{d+1}$ isometries. We discuss another description in terms of Howe duality and derive the formula for computing trace in these algebras. This enables us to explicitly calculate the bilinear form for this one-parameter family of algebras. In particular, the bilinear form shows the appearance of additional ideal for any non-negative integer values of $\ell-d/2\,$, which coincides with the annihilator of the one-row $\ell$-box Young diagram representation of $\mathfrak{so}_{d+2}\,$. Hence, the corresponding finite-dimensional coset algebra spanned by massless and PM generators is equivalent to the symmetries of this representation.