Deformed twistors and higher spin conformal (super-)algebras in four dimensions

Abstract

Massless conformal scalar field in d = 4 corresponds to the minimal unitary representation (minrep) of the conformal group SU(2, 2) which admits a one-parameter family of deformations that describe massless fields of arbitrary helicity. The minrep and its deformations were obtained by quantization of the nonlinear realization of SU(2, 2) as a quasiconformal group in arXiv:0908.3624. We show that the generators of SU(2, 2) for these unitary irreducible representations can be written as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group and apply them to define and study higher spin algebras and superalgebras in AdS 5 . The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS 5 is simply the enveloping algebra of SU(2, 2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS 5 . Furthermore, the enveloping algebras of the deformations of the minrep define a one parameter family of HS algebras in AdS 5 for which certain 4d covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras SU(2, 2|N ) and we find a one parameter family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a family of (supersymmetric) HS theories in AdS 5 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 4d. We also discuss the corresponding picture in HS algebras in AdS 4 where the corresponding 3d conformal group Sp(4, R) admits only two massless representations (minreps), namely the scalar and spinor singletons.

Minimal unitary representation of 5d superconformal algebra F(4) and AdS6Fernando1, Günaydın2 2015Nuclear Physics BSelf Cite

“…The quasiconformal realization of the minimal unitary representation of symplectic groups coincides with the realization as bilinear of oscillators [16] and the Joseph ideal vanishes identically [9]. That is why the enveloping algebra of the singletonic realization of Sp(4, R) ≈ SO (3,2) as bilinears of covariant twistorial oscillators leads directly to the Fradkin-Vasiliev higher spin algebra in AdS 4 .…”

“…For the doubletonic realization of the minimal unitary representation of AdS 5 group SO (4,2) and AdS 7 group SO (6,2), in terms of bilinears of covariant twistorial oscillators, the Joseph ideal does not vanish identically as an operator. However for their minreps obtained by quasiconformal methods, the Joseph ideal vanishes identically as an operator [9,10] both in AdS 5 and AdS 7 , respectively. Hence their enveloping algebras yield directly the corresponding bosonic higher spin algebras.…”

“…More recently the results of [1][2][3] were reformulated in terms of deformed twistors in four and six dimensions that transform nonlinearly under the Lorentz group and applied to higher spin theories in AdS 5 and AdS 7 [9,10]. One finds that the enveloping algebras of the minimal unitary realizations of SU (2,2) and SO * (8), obtained by the quasiconformal methods, yield directly the higher spin algebras in AdS 5 and AdS 7 .…”

Notes on higher-spin algebras: minimal representations and structure constants

“…Much progress has been made on the subject of minimal representations, see for example [40][41][42][43][44][45][46][47][48][49][50] and references therein. In physics literature, the minimal representations of isometry algebras are explored to a large extent by Gunaydin and collaborators [51][52][53][54][55].…”

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].…”