Massless conformal fields, AdS(+1)/CFT higher spin algebras and their deformations

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) /CF T 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.

Extended dynamical symmetries of Landau levels in higher dimensions

“…can be constructed using the three-graded decomposition of the Lie algebra so(d, 2) w.r.t its maximally compact subalgebra [45,46]…”

A Note on Rectangular Partially Massless Fields

“…In this section, we start by reviewing the definition of the usual (unitary) higher-spin singletons (about which more details can be found in the pedagogical review [102], and in [103] where they were studied from the point of view of minimal representations), before moving on to the proposed higher-order extension which is the main focus of this paper.…”

Super-Laplacians and their symmetries

“…The minimal multiplets are those whose maximal spin (or helicity) components are the lowest possible for each value of D and N . They have been considered from a somewhat different point of view in the supersymmetric higher-spin context in [24][25][26].…”

“…Some examples are the hypermultipet for N = 2, the vector multiplet for N = 4 and the maximal supergravity multiplet for N = 8. For N = 4 we have 26) where all of the spinor indices are symmetrised and all of the internal indices are antisymmetrised. A similar analysis of the N = 8 case leads to the expected spectrum, 70 + 56 + 28 + 8 + 1, of component fields with spins 0, 1/2, 1/3/2 and 2, respectively, subject to their equations of motion and Bianchi identities.…”

“…Note that trace terms in A and B do not contribute to T 2 because of antisymmetry. The first term in the last line can be evaluated straightforwardly and one finds 26) with the antisymmetry indicated. Here…”