Abstract
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.
Highlights
Motivated by the work of physicists on spectrum generating symmetry groups in the 1960s the concept of minimal unitary representations of noncompact Lie groups was introduced by Joseph in [1]
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 AdS5
We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the higher spin (HS) algebra in AdS5
Summary
Motivated by the work of physicists on spectrum generating symmetry groups in the 1960s the concept of minimal unitary representations of noncompact Lie groups was introduced by Joseph in [1]. This ideal J (g) was identified as the annihilator of the scalar singleton module or the minimal representation and is known as the Joseph ideal in the mathematics literature [67] We shall review the twistorial oscillator construction of the unitary representations of the conformal groups SO(3, 2) in d = 3 dimensions that correspond to conformally massless fields in d = 3 following [36, 69] These representations turn out to be the minimal unitary representations and are called the singleton (scalar and spinor singleton) representations of Dirac [28]. We review the minimal unitary realization of OSp(N |4, R) in appendix B
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