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.

Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation (minrep) of the conformal group SO(6,2). This minrep admits a family of "deformations" labelled by the spin t of an SU(2) T group, which is the 6d analog of helicity in four dimensions. These deformations of the minrep of SO(6, 2) describe massless conformal fields that are symmetric tensors in the spinorial representation of the 6d Lorentz group. The minrep and its deformations were obtained by quantization of the nonlinear realization of SO(6, 2) as a quasiconformal group in arXiv:1005.3580. We give a novel reformulation of the generators of SO(6, 2) for these representations as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group SO(5, 1) and apply them to define higher spin algebras and superalgebras in AdS 7 . The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS 7 is simply the enveloping algebra of SO(6, 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 7 . Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS 7 for which certain 6d Lorentz covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras OSp(8 * |2N ) and we find a discrete infinite family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a discrete family of (supersymmetric) HS theories in AdS 7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.

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There are many theorems providing annulus containing all the zeros of a polynomial, and it is known that two such theorems cannot be compared, in the sense that one can always find a polynomial for which one theorem gives a sharper bound than the other. It is natural to ask if there is a class of polynomials for which such comparison is possible and in this paper we investigate this problem and provide results which for polynomials with some condition on the degree or absolute range of coefficients, enable us to compare two such theorems.