Partially Massless Higher-Spin Theory

TL;DR

This work constructs a fully nonlinear partially massless higher‑spin theory by gauging the hs2 algebra, extending Vasiliev’s framework to include third‑order Killing tensors on AdS and dS. The authors perform a detailed linear analysis, computing the masses of the four fields not fixed by gauge symmetry via hs2 bilinear and trilinear forms, and show precise agreement with the dual $ square^2$ CFT predictions across dimensions, including intricate dimension‑dependent phenomena. They also describe a consistent truncation to a minimal even‑spin theory, and highlight novel features in certain dimensions (e.g., dramatic truncations in $D=3,5$ and extended Verma modules in $D=4,7$) together with non‑diagonalizable mixing in $D=4$,7. The results provide substantial evidence for the consistency of PM higher‑spin holography and open avenues for dS/CFT explorations and potential connections to cosmology and quantum gravity.

Abstract

We study a generalization of the D-dimensional Vasiliev theory to include a tower of partially massless fields. This theory is obtained by replacing the usual higher-spin algebra of Killing tensors on (A)dS with a generalization that includes "third-order" Killing tensors. Gauging this algebra with the Vasiliev formalism leads to a fully non-linear theory which is expected to be UV complete, includes gravity, and can live on dS as well as AdS. The linearized spectrum includes three massive particles and an infinite tower of partially massless particles, in addition to the usual spectrum of particles present in the Vasiliev theory, in agreement with predictions from a putative dual CFT with the same symmetry algebra. We compute the masses of the particles which are not fixed by the massless or partially massless gauge symmetry, finding precise agreement with the CFT predictions. This involves computing several dozen of the lowest-lying terms in the expansion of the trilinear form of the enlarged higher-spin algebra. We also discuss nuances in the theory that occur in specific dimensions; in particular, the theory dramatically truncates in bulk dimensions D=3,5 and has non-diagonalizable mixings which occur in D=4,7.

Figures (9)

• Figure 1: Here we show the Higuchi bound on ${\rm dS}_4$ as well as the first few partially massless particles' masses and spins. Above the Higuchi bound, a massive particle is unitary. Below the bound, generically the kinetic term for at least one of the helicity components is of the wrong sign, indicating that some degrees of freedom are ghostly and the particle is non-unitary. However, at the specific partially massless points (represented by the location of the numbers in the figure), the particle develops a gauge symmetry which eliminates the ghostly degrees of freedom, making the particle unitary. This comes at the expense of reducing the number of propagating degrees of freedom; how many degrees of freedom propagate is represented by the number at each partially massless location.
• Figure 2: The AdS one-forms in the master field $W$ which act as unfolding fields for the massless gauge fields in the theory. These are identical to the usual result in the Vasiliev theory.
• Figure 3: Fields in $W$ which correspond to the spin-3 and higher PM fields. The bottom tableau in each column occurs with multiplicity 1, the one to the right of it with multiplicity 2, and all others with multiplicity 3. We box what fields are needed for a given PM spin. Note that mixings may occur between tableaux of the same shape.
• Figure 4: The AdS zero-forms in the master field $C$ which act as unfolding fields and field strengths for the massless gauge fields in the theory. These are identical to the usual result in the Vasiliev theory.
• Figure 5: We show the unfolding fields in $C$ which describe the three new massive particles, as well as the unfolding fields and field strengths for the PM fields. As with the previous figure, these arise from the new generators in $hs_2$ not present in the $hs$ algebra. The bottom tableau in each column occurs with multiplicity 1, the one to the right of it with multiplicity 2, and all others with multiplicity 3. We box what fields are needed for a given spin.