Massless conformal fields, AdS_{d+1}/CFT_d higher spin algebras and their deformations

TL;DR

This paper generalizes the connection between minimal unitary representations of SO(d,2) and massless conformal fields, establishing a correspondence with higher spin algebras and their deformations across arbitrary dimensions.

Contribution

It extends previous work to all dimensions, linking minimal unitary reps to higher spin algebras and classifying their deformations, including spinor singletons and Casimir eigenvalue labels.

Findings

Minimal unitary reps correspond to massless conformal fields.

Higher spin algebras are obtained from enveloping algebras of minimal reps.

Deformations of higher spin algebras are classified by Casimir eigenvalues.

Abstract

We extend our earlier work on the minimal unitary representation of $SO(d,2)$ and its deformations for $d=4,5$ and $6$ to arbitrary dimensions $d$. We show that there is a one-to-one correspondence between the minrep of $SO(d,2)$ and its deformations and massless conformal fields in Minkowskian spacetimes in $d$ dimensions. The minrep describes a massless conformal scalar field, and its deformations describe massless conformal fields of higher spin. The generators of Joseph ideal vanish identically as operators for the quasiconformal realization of the minrep, and its enveloping algebra yields directly the standard bosonic $AdS_{(d+1)}/CFT_d$ higher spin algebra. For deformed minreps the generators of certain deformations of Joseph ideal vanish as operators and their enveloping algebras lead to deformations of the standard bosonic higher spin algebra. In odd dimensions there is a unique deformation of the higher spin algebra corresponding to the spinor singleton. In even dimensions one finds infinitely many deformations of the higher spin algebra labelled by the eigenvalues of Casimir operator of the little group $SO(d-2)$ for massless representations.