Mixed-symmetry fields in de Sitter space: a group theoretical glance

TL;DR

This work builds a group-theoretical bridge between unitary irreducible representations of $\mathfrak{so}(1,d+1)$ and mixed-symmetry fields in de Sitter space, detailing principal, complementary, exceptional, and discrete series and their characters via generalised Verma modules and BGG resolutions. It clarifies how massless and partially massless de Sitter fields arise as quotients in AdS/dS, and develops a framework for a flat limit that mirrors the Brink-Metsaev-Vasiliev mechanism, yielding a spectrum of flat-space Poincaré irreps through controlled branching rules. While a consistent BMV-type flat limit is achieved in odd dimensions for exceptional and discrete series, even dimensions exhibit subtleties suggesting a separation between gauge potentials and curvature content, and raising questions about a possible singleton/Flato-Fronsdal-like structure in de Sitter. Overall, the paper provides a refined toolkit for understanding higher-spin content in dS via representation theory, with implications for dS/CFT and higher-spin reconstruction.

Abstract

We rederive the characters of all unitary irreducible representations of the $(d+1)$-dimensional de Sitter spacetime isometry algebra $\mathfrak{so}(1,d+1)$, and propose a dictionary between those representations and massive or (partially) massless fields on de Sitter spacetime. We propose a way of taking the flat limit of representations in (anti-) de Sitter spaces in terms of these characters, and conjecture the spectrum resulting from taking the flat limit of mixed-symmetry fields in de Sitter spacetime. We comment on a possible equivalent of the scalar singleton for the de Sitter (dS) spacetime.

Figures (15)

• Figure 1: Unitary (blue and green), and non-unitary (red) regions for the squared mass of a scalar field in de Sitter (left) and anti-de Sitter space (right).
• Figure 2: Young diagrams corresponding, from left to right, to a mixed-symmetry partially massless field of depth-$t$, its gauge parameter acting in the isolated middle block (with crosses indicating removed cells), and finally its curvature built by acting with derivatives in the same block. The symbols $\mathbb{Y}_u$ and $\mathbb{Y}_d$ represent arbitrary Young diagrams that can be glued respectively above (up) and below (down) the middle block.
• Figure 3: Repartition of massive and massless fields in dS$_{d+1}$ (left) and AdS$_{d+1}$ (right) as a function of the conformal weight $\Delta_c / \Delta$, for a fixed diagram $\mathbb{Y}$ of total height $p-1$, and first block of height $h_1$ and length $s$. On the left / de Sitter side, massive field in the principal and complementary series are depicted respectively by a red and a green line, the massless field is represented by a blue dot. On the right / anti-de Sitter side, massive fields correspond to the green line and the massless fields are the blue dots. Massless fields in the discrete series appear as discrete points on the green line.
• Figure 4: Repartition of the massive scalar fields in dS$_{d+1}$ discussed above, as a function of the conformal weight $\Delta_c$. Massive field in the principal and complementary series are depicted respectively by a red and a green line, whereas the blue dots indicate a discrete collection of representations in the complementary series with $\Delta_c = \tfrac{d}2 \pm \ell\,$ that would correspond to higher order singletons and their shadows in AdS$_{d+1}$.
• Figure :

Theorems & Definitions (10)

• proof
• proof
• Definition C.1: Verma module
• Theorem C.1: Bernstein-Gel'fand-Gel'fand
• Definition C.2: Length of a Weyl group element
• Theorem C.2: Bernstein-Gel'fand-Gel'fand resolution
• Definition C.3: Generalised Verma module
• Lemma F.1
• proof
• proof