How massless are massless fields in $AdS_d$

TL;DR

This work shows that massless mixed-symmetry fields in $AdS_d$ generically decompose into multiple $o(d-2)$ irreps in the flat limit, rather than remaining irreducible as in Minkowski space. Using unitary representation theory of $o(d-1,2)$ and singular vectors, the authors derive a pattern for which flat-space massless fields can deform to AdS while preserving unitarity, and they provide a concrete field-theoretic example with a three-cell hook diagram. They demonstrate that AdS massless fields are typically reducible in the flat limit, and that maintaining all flat-space gauge symmetries in AdS is not possible without introducing additional fields (notably a graviton-like field) and cross-couplings, which become trivial in the flat limit. A key conjecture links the allowed flat-space massless spectra to the block-structure of Young diagrams, predicting which $o(d-2)$ reps appear in the AdS deformation. The results illuminate structural constraints on higher-spin/gauge-field theories in curved backgrounds and highlight a possible gravity-like sector arising from mixed-symmetry AdS fields.

Abstract

Massless fields of generic Young symmetry type in $AdS_d$ space are analyzed. It is demonstrated that in contrast to massless fields in Minkowski space whose physical degrees of freedom transform in irreps of $o(d-2)$ algebra, $AdS$ massless mixed symmetry fields reduce to a number of irreps of $o(d-2)$ algebra. From the field theory perspective this means that not every massless field in flat space admits a deformation to $AdS_d$ with the same number of degrees of freedom, because it is impossible to keep all of the flat space gauge symmetries unbroken in the AdS space. An equivalent statement is that, generic irreducible AdS massless fields reduce to certain reducible sets of massless fields in the flat limit. A conjecture on the general pattern of the flat space limit of a general $AdS_d$ massless field is made. The example of the three-cell ``hook'' Young diagram is discussed in detail. In particular, it is shown that only a combination of the three-cell flat-space field with a graviton-like field admits a smooth deformation to $AdS_d$.