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 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 AdS7. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS7 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 AdS7. Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS7 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 AdS7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.
Highlights
Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation of the conformal group SO(6, 2)
One of our main results in this paper is the reformulation of the minimal unitary representation of SO∗(8) and its deformations in terms of deformed twistors that transform nonlinearly under the Lorentz group in six dimensions
Their enveloping algebras lead to a discrete infinite family of AdS7/CF T6 higher spin algebras labelled by the spin of an SU(2) symmetry for which certain deformations of the Joseph ideal vanish
Summary
Substituting the generators MAB of SO∗(8) realized as bilinears of covariant twistorial oscillators (doubletons) (section 2.1) in equation (3.6) to compute the generator of the Joseph ideal one finds that it does not vanish identically. The USp(2P ) invariant (singlet) subspace transforms in the spin t = P/2 representation of SU(2)G Restricting to this invariant subspace we get a deformed minrep corresponding to a 6d massless conformal field transforming as a totally symmetric tensor of rank P in the spinor indices with respect to the Lorentz group SU∗(4). We shall define the 2N extended AdS7/CF T6 higher spin superalgebra as the enveloping algebra of the minimal unitary realization of the super algebra OSp(8∗|2N ) obtained via the quasiconformal approach For this algebra there are only 2N supersymmetry fermions and the minimal supermultiplet of OSp(8∗|2N ) consists of the following massless conformal fields: Φ[A1A2...AN ]| ⊕ Ψ[αA1A2...AN−1]| ⊕ Φ[(Aα1βA) 2...AN−2]| ⊕ · · ·. Deformed AdS7/CF T6 higher spin superalgebras HS(6, 2|2N ; t) algebras are defined as enveloping algebras of the deformed minimal unitary realizations of the super algebras OSp(8∗|2N ) with the even subalgebra SO∗(8) ⊕ USp(2N ) involving deformation fermions 2.4.
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