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

TL;DR

This paper investigates the structure and finite-dimensional truncations of the symmetry algebras associated with partially-massless higher-spin fields in (A)dS spaces, revealing new algebraic relations and ideal structures.

Contribution

It introduces a detailed construction of these algebras via quotienting ideals and explores their finite-dimensional truncations related to specific Young diagram representations.

Findings

Explicit formula for the trace in these algebras

Identification of additional ideals at specific parameter values

Finite-dimensional coset algebra corresponds to certain Young diagram symmetries

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.