PM Higher Spin Algebras

• PM higher spin algebras are defined as quotients of the universal enveloping algebra of (A)dS isometries using the Joseph ideal and quadratic Casimir constraints.
• They are constructed via oscillator methods and Howe duality, leading to a spectrum of multiplicity-free two-row Young diagram representations.
• These algebras underpin the symmetry of partially massless fields in AdS/CFT, enabling finite-dimensional truncations and novel gauge theory interactions.

Partially massless (PM) higher spin algebras form a distinguished class of symmetry algebras associated with free or interacting gauge fields of arbitrary spin that, in an (A)dS background, possess gauge invariances with derivative order t (the “depth”) strictly between the strictly massless and massive cases. These algebras generalize both the ordinary higher spin (HS) algebras for massless fields and the global symmetry algebras of higher-order singletons, playing central roles in geometry, representation theory, and gauge theory constructions in AdS/CFT and higher-spin gravity.

1. Algebraic Definition and Structural Framework

A PM higher spin algebra is constructed as a quotient of the universal enveloping algebra (UEA) $U(\mathfrak{so}_{d+2})$ of the (A)dS$_{d+1}$ isometry algebra by a two-sided ideal generated by the so-called Joseph ideal and a quadratic Casimir constraint,

$$\mathcal A_\lambda = U(\mathfrak{so}_{d+2})/\langle J_{ABCD},\ C_2 - \nu_\lambda \rangle,$$

where $J_{ABCD}$ is totally antisymmetric (in the window symmetry $\yng(1,1,1,1)$), $C_2$ is the quadratic Casimir, and $\nu_\lambda= -\frac{(d-2\lambda)(d+2\lambda)}{4}$ (Joung et al., 2015).

The spectrum of $\mathcal A_\lambda$ under $\mathfrak{so}_{d+2}$ is, in the generic case, infinite-dimensional and consists of all traceless (rank-$r+2p$, $r$) two-row Young diagrams with $p\ge0$, corresponding to Killing tensors of even depth $2p$.

There is an alternative realization by Howe duality and oscillators: the centralizer of a suitable symplectic ($\mathfrak{sp}_2$) or orthosymplectic ($\mathfrak{osp}(1|2p)$) algebra inside a bosonic or supersymmetric oscillator algebra, modulo the imposed Casimir constraints (Basile et al., 2024, Joung et al., 2015). This construction naturally explains the multiplicity-free, purely two-row Young diagram content of the PM algebra.

2. Oscillator and Howe Duality Realization

The core oscillator construction for PM HS algebras employs bosonic oscillators $y_{\alpha A}$ ($\alpha=+,-$, $A=1,\dots,d+2$) with

$$[y_{\alpha A},\, y_{\beta B}] = \epsilon_{\alpha\beta} \eta_{AB},$$

and the star-product

$$(f\star g)(y) = \exp\Big[\tfrac12\,\frac{\overleftarrow\partial}{\partial y_{\alpha A}} \epsilon^{\alpha\beta} \eta^{AB} \frac{\vec\partial}{\partial z_{\beta B}} \Big] f(y)g(z)\big|_{z=y}.$$

The two commuting subalgebras are:

• $\mathfrak{so}_{d+2}$: $M_{AB} = y_{+A} y_{-B} + y_{+B} y_{-A}$,
• $\mathfrak{sp}_2$: $K_{\alpha\beta} = y_{\alpha A} y_{\beta}^A$.

The PM algebra $\mathcal A_\lambda$ is then the centralizer of $\mathfrak{sp}_2$, quotiented by the Casimir constraint $C_2(\mathfrak{sp}_2) = (1-\lambda)(1+\lambda)$, which fixes also the $\mathfrak{so}_{d+2}$ quadratic Casimir (Joung et al., 2015).

In $d=4$, a supersymmetric (“oscillator–Clifford”) realization using $4$ bosonic and $8(\ell-1)$ fermionic oscillators with a star-product and Clifford product allows the PM algebra $hs_\ell$ to be realized as the centralizer of $\mathfrak{osp}(1|2(\ell-1))$ inside the combined Weyl–Clifford algebra (Basile et al., 2024, Dhasmana, 31 Jan 2026).

3. Spectrum, Depth Structure, and Classification

The generators of PM HS algebras decompose under $\mathfrak{so}_{d+2}$ into two-row Young tableaux of the form

$\begin{ytableau} r+2p \ r \end{ytableau},$

where $p=0,1,2,\ldots$ counts “depth,” i.e., the number of derivatives in the PM gauge symmetry $\delta \varphi_{s,t} \sim \nabla^t\xi_{s-t}$, and $r\ge 0$. Depths $t=1,3,\ldots,2\ell-1$ (in $d=4$, with $\ell\in\mathbb N$ for type-$A_\ell$ PM algebras) correspond to PM fields of spin $s$ and depth $t$. The spectrum is multiplicity-free for each $(s,t)$, enforced by the Howe dual Casimir constraints (Basile et al., 2024).

For special (“resonant”) values where $\lambda-\frac d2\in\mathbb N$, a further ideal appears, generated by all two-row Young diagrams with first row longer than $2\ell$: the coset $\mathfrak{pms}_\ell$ becomes finite dimensional and coincides with $\mathrm{End}_\mathbb C V$, where $V$ is the irreducible one-row, $\ell$-box representation of $\mathfrak{so}_{d+2}$ (Joung et al., 2015).

4. Commutation Relations, Trace, and Invariant Bilinear Forms

The PM algebras inherit their associative (star) structure from the oscillator algebra, and their Lie bracket is the star-commutator, with structure constants fixed by the enveloping algebra and representation content.

For any quotient $\mathcal A_\lambda$, a trace is constructed via a non-Gaussian projector $\Delta_\lambda$, yielding

$$\mathrm{Tr}_\lambda[f] = (\Delta_\lambda \star f)(0),$$

with $\Delta_\lambda$ given by a hypergeometric integral kernel and normalization $N_\lambda$ that vanishes whenever $\lambda-\frac d2$ is a non-negative integer, leading to the degeneration described above (Joung et al., 2015).

The bilinear form is

$$\langle X, Y\rangle_\lambda = \mathrm{Tr}_\lambda[X\star Y],$$

and closed integral formulas are provided, rendering the structure amenable to explicit computation, including for low-spin or finite-dimensional coset cases (Joung et al., 2015).

5. Physical and Holographic Interpretation

PM higher spin algebras are the symmetry algebras of linearized equations for higher spin fields with partial gauge invariance in (A)dS, generalizing both the strictly massless and massive algebraic cases. In holography, $\mathcal A_\ell$ corresponds to the algebra of global symmetries for the $\ell$th-order singleton, i.e., a boundary scalar field satisfying $\Box^\ell \phi = 0$ (Basile et al., 2024, Joung et al., 2015). The odd-depth restriction matches the spectrum of nontrivial gauge symmetries of the bulk PM fields and the higher-order conservation laws of the boundary theory.

For type-$A_\ell$ PM algebra, the spectrum includes all Killing tensors associated with gauge symmetry transformations of generic depth, organizing the possible partially massless module content of a higher-spin gauge theory in (A)dS. In the context of AdS/CFT, the central charge parameter (or boundary “rank”) is encoded in the trace prescription and is a deformation parameter of the star-algebra (Vasiliev, 2012).

6. Star-Product Algebra, Deformations, and Interactions

The Weyl–Clifford star-product governs the algebraic structure, enabling concrete calculations for commutators, ideals, and construction of field-strengths. The realization via dual pairs ($sp(4)\oplus osp(1|2p)$ in $d=4$) ensures both tractability and a clear identification of the relevant spectrum and trace constraints (Basile et al., 2024).

The existence of nontrivial associative deformations of the PM algebra is supported by the explicit construction of nontrivial Hochschild 2-cocycles associated with cohomological cycles, signaling the possibility of formal interactions in PM higher spin gravity theories (Basile et al., 2024). In particular, such deformations provide nontrivial, interaction-enabling deformations analogous to the familiar Vasiliev higher-spin interactions, with the algebraic cohomology structure controlling the couplings (Basile et al., 2024, Dhasmana, 31 Jan 2026).

7. Finite-Dimensional Truncations and Representation Theory

When $\lambda-\frac d2 = \ell\in\mathbb N$, the PM algebra $\mathcal A_\lambda$ develops a new ideal, and the finite-dimensional quotient $\mathfrak{pms}_\ell$ is isomorphic to the full matrix algebra $\mathfrak{gl}_{M_\ell}$ on the one-row $\ell$-box representation of $\mathfrak{so}_{d+2}$ with dimension

$M_\ell = \frac{(d+1)_{\ell-1}\,(d+2\ell)}{\ell!}.$

Thus, these truncations are characterized by matrix algebra Lie brackets and explicit spectrum/dimension formulas, providing a direct link between PM higher spin algebras and finite-dimensional representation theory (Joung et al., 2015).

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For detailed algebraic constructions, oscillator realizations, invariant forms, and the correspondence with boundary singletons and non-unitary CFTs, see (Joung et al., 2015, Basile et al., 2024, Dhasmana, 31 Jan 2026), and the foundational algebraic framework in (Vasiliev, 2012).