Deformed Twistors and Higher Spin Conformal (Super-)Algebras in Four Dimensions

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 AdS_4 where the 3d conformal group Sp(4,R) admits only two massless representations (minreps), namely the scalar and spinor singletons.


Introduction
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]. Minimal unitary representation of a noncompact Lie group is defined over an Hilbert space of functions depending on the minimal number of variables possible. They have been studied extensively in the mathematics literature [2][3][4][5][6][7][8][9][10][11][12][13][14]. A unified approach to the construction and study of minimal unitary representations of noncompact groups was developed after the discovery of novel geometric quasiconformal realizations of noncompact groups in [15]. Quasiconformal realizations exist for different real forms of all noncompact groups as well as for their complex forms [15,16] 1 .
The quantization of geometric quasiconformal action of a noncompact group leads directly to its minimal unitary representation as was first shown explicitly for the split exceptional group E 8 (8) with the maximal compact subgroup SO(16) [17]. The minimal unitary representation of three dimensional U-duality group E 8(−24) of the exceptional supergravity [18] was similarly obtained in [19]. In [20] a unified formulation of the minimal unitary representations of noncompact groups based on the quasiconformal method was given and it was extended to the minimal representations of noncompact supergroups G whose even subgroups are of the form H × SL(2, R) with H compact 2 . These supergroups include G(3) with even subgroup G 2 × SL(2, R), F (4) with even subgroup Spin(7) × SL(2, R), D (2, 1; σ) with even subgroup SU (2) × SU (2) × SU (1, 1) and OSp (N |2, R). These results were further generalized to supergroups of the form SU (n, m|p + q) and OSp(2N * |2M ) in [21][22][23] and applied to conformal superalgebras in 4 and 6 dimensions. In particular, the construction of the minreps of 5d anti-de Sitter or 4d conformal group SU (2, 2) and corresponding supergroups SU (2, 2|N ) was given in [21]. One finds that the minimal unitary representation of the group SU (2, 2) obtained by quantization of its quasiconformal realization is isomorphic to the scalar doubleton representation that describes a massless scalar field in four dimensions. Furthermore the minrep of SU (2, 2) admits a one parameter family (ζ) of deformations that can be identified with helicity, which can be continuous. For a positive (negative) integer value of the deformation parameter ζ, the resulting unitary irreducible representation of SU (2, 2) corresponds to a 4d massless conformal field transforming in 0 , ζ 2 − ζ 2 , 0 representation of the Lorentz subgroup, SL(2, C). These deformed minimal representations for integer values of ζ turn out to be isomorphic to the doubletons of SU (2, 2) [24][25][26].
The minrep of 7d AdS or 6d conformal group SO(6, 2) = SO * (8) and its deformations were studied in [22]. One finds that the minrep admits deformations labelled by the eigenvalues of the Casimir of an SU (2) T subgroup of the little group, SO(4), of massless particles in six dimensions. These deformed minreps labeled by spin t of SU (2) T are positive energy unitary irreducible representations of SO * (8) that describe massless conformal fields in six dimensions. Quasiconformal construction of the minimal unitary supermultiplet of OSp(8 * |2N ) and its deformations were given in [22,23]. The minimal unitary supermultiplet of OSp(8 * |4) is the massless conformal (2, 0) supermultiplet whose interacting theory is believed to be dual to M-theory on AdS 7 × S 4 . It is isomorphic to the scalar doubleton supermultiplet of OSp(8 * |4) first constructed in [27].
For symplectic groups Sp(2N, R) the construction of the minimal unitary representation using the quasiconformal approach and the covariant twistorial oscillator method coincide [20]. This is due to the fact that the quartic invariant operator that enters the quasiconformal construction vanishes for symplectic groups and hence the resulting generators involve only bilinears of oscillators. Therefore the minreps of Sp(4, R) are simply the scalar and spinor singletons that were called the remarkable representations of anti-de Sitter group by Dirac [28]. Unitary supermultiplets of general spacetime superalgebras were first constructed using the oscillator method developed in [29,30]. The singleton supermultiplets of AdS 4 superalgebras , in particular those of N = 8 superalgebra OSp(8|4) were first constructed, using the oscillator method, in [31,32]. Scalar and spinor singletons as representation of the N = 1 , AdS 4 super algebra OSp(1/4, R) were studied by Fronsdal [33] who called it a Dirac supermultiplet. The oscillator construction of the general unitary representations of OSp(N |4, R) was further developed in [27,34]. The singleton supermultiplets of OSp(N/4, R) were also studied in [35].
The Kaluza-Klein spectrum of IIB supergravity over the AdS 5 × S 5 space was first obtained via the twistorial oscillator method by tensoring of the CPT self-conjugate doubleton supermultiplet of SU (2, 2 | 4) with itself repeatedly and restricting to the CPT self-conjugate short supermultiplets [24]. Authors of [24] also pointed out that the CPT self-conjugate doubleton supermultiplet SU (2, 2 | 4) does not have a Poincaré limit in five dimensions and its field theory lives on the boundary of AdS 5 on which SU (2, 2) acts as a conformal group and that the unique candidate for this theory is the four dimensional N = 4 super Yang-Mills theory that is conformally invariant. Similarly the Kaluza-Klein spectra of the compactifications of 11 dimensional supergravity over AdS 4 × S 7 and AdS 7 × S 4 were obtained by tensoring of singleton supermultiplet of OSp(8 | 4, R) [36] and of scalar doubleton supermultiplet of OSp(8 * | 4) [27], respectively. The authors of [36] and [27] also pointed out that the field theories of the singleton and scalar doubleton supermultiplets live on the boundaries of AdS 4 and AdS 7 as conformally invariant field theories, respectively 3 . As such these works represent some of the earliest work on AdS/CF T dualities within the framework of Kaluza-Klein supergravity theories. Their extension to the superstring and M-theory arena [37][38][39] started the modern era of AdS/CF T research. The fact that the scalar doubleton supermultiplets of SU (2, 2|4) and OSp(8 * | 4) and the singleton supermultiplet of OSp(8 | 4, R) turn out to be the minimal unitary supermultiplets show that they are very special from a mathematical point of view as well.
Tensor product of the two singleton representations of the AdS 4 group Sp(4, R) decomposes into infinitely many massless spin representations in AdS 4 as was shown in [40]. These higher spin theories were studied by Fronsdal and collaborators [33,[41][42][43]. In the eighties Fradkin and Vasiliev initiated the study of higher spin theories involving fields of all spins 0 ≤ s < ∞ [44,45]. A great deal of work was done on higher spin theories since then and for comprehensive reviews on higher spin theories we refer to [46][47][48][49][50] and references therein. The work on higher spin theories has intensified in the last decade since the conjectured duality between Vasiliev's higher spin gauge theory in AdS 4 and O(N ) vector models in [51,52]. The three point functions of higher spin currents were computed directly and matched with those of free and critical O(N ) vector models in [53,54]. Substantial work has also been done in higher spin holography in the last few years and for references we refer to the review [55].
In the early days of higher spin theories it was pointed out in [56] that the Fradkin-Vasiliev higher spin algebra in AdS 4 [44] corresponds simply to the infinite dimensional Lie algebra defined by the enveloping algebra of the singletonic realization of Sp(4, R) and that this can be extended to construction of HS algebras in higher dimensions. The oscillator construction of the singleton representations [31,32] of AdS 4 superalgebras were used in the study of higher spin (super)algebras in [45], where the admissibility condition was formulated. Conformal higher spin superalgebras were studied shortly thereafter in [57]. Again in [56] it was pointed out that the supersymmetric extensions of the higher spin algebras in AdS 4 , AdS 5 and AdS 7 could be similarly constructed as enveloping algebras of the singletonic or doubletonic realizations of the super algebras OSp(N/4, R), SU (2, 2|N ) and OSp(8 * |2N ). Higher spin algebras and superalgebras in AdS 5 and AdS 7 were studied along these lines in [58][59][60][61] using the doubletonic realizations of underlying algebras and superalgebras given in [25][26][27]62]. Higher spin superalgebras in dimensions d > 3 were also studied by Vasiliev in [63]. However, they do not have the standard finite dimensional AdS superalgebras as subalgebras except for the case of AdS 4 . The relation between higher spin algebras and cubic interactions for simple mixed-symmetry fields in AdS space times using Vasiliev's approach was studied in [64].
Mikhailov showed the connection between AdS 5 /Conf 4 higher spin algebra and the algebra of conformal Killing vectors and Killing tensors in d = 4 and their relation to to higher symmetries of the Laplacian [65] . The connection between conformal Killing vectors and tensors and higher symmetries of the Laplacian was put on a firm mathematical foundation by Eastwood [66] who gave a realization of the AdS (d+1) /CF T d higher spin algebra as an explicit quotient of the universal enveloping algebra U (g), (g = so(d, 2)), by a two-sided ideal J (g) . 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] 4 . This result agrees with the proposal of [56] for AdS 4 /CF T 3 higher spin algebras since singletons are simply the minreps of SO (3,2) and in its singletonic twistorial oscillator realization the Joseph ideal vanishes identically as discussed in section 4.1. The covariant twistorial oscillators have been used extensively in the formulation and study of higher spin AdS 4 algebras since the early work of Fradkin and Vasiliev. For the doubletonic realization of SO(4, 2) and SO (6,2) in terms of covariant twistorial oscillators the two sided Joseph ideal does not vanish identically as operators and must be quotiented out. However, as will be shown explicitly in section 4.2.2, the Joseph ideal vanishes identically as operators for the minimal unitary realization of SU (2, 2) obtained via quasiconformal approach. The same result holds true for the AdS 7 /Conf 6 algebras [68].
One of the key results of this paper is to show that the basic objects for the construction of irreducible higher spin AdS 5 /Conf 4 algebras are not the covariant twistorial oscillators but rather the deformed twistorial oscillators that transform nonlinearly under the Lorentz group SL(2, C). The minimal unitary representation of SO(4, 2) and its deformations obtained by quasiconformal methods [21] can be written as bilinears of these deformed twistors. One parameter family of higher spin AdS 5 /Conf 4 algebras and superalgebras can thus be realized as enveloping algebras involving products of bilinears of these deformed twistors 5 . We shall also review the AdS 4 /CF T 3 algebras and their supersymmetric extensions so as to highlight the differences with the higher dimensional algebras. AdS 4 group SO(3, 2) is isomorphic to the symplectic group Sp(4, R) and as was shown in [20] the quasiconformal realization of symplectic groups reduce to realization in terms of bilinears of covariant oscillators.
The plan of the paper is as follows: In section 2 we review the covariant twistorial oscillator (singleton) construction of the conformal group in three dimensions SO(3, 2) ∼ Sp(4, R) and its superextension OSp(N |4, R). Then we review the covariant twistorial oscillator (doubleton) construction for the four dimensional conformal group SO(4, 2) ∼ SU (2, 2) in section 3.1. In section 3.2, we present the minimal unitary representation of SU (2, 2) obtained by the quasiconformal approach [21] in terms of certain deformed twistorial oscillators that transform nonlinearly under the Lorentz group. We then define a one-parameter family of these deformed twistors, which we call helicity deformed twistorial oscillators and express the generators of a one parameter family of deformations of the minrep given in [21] as bilinears of the helicity deformed twistors. They describe massless conformal fields of arbitrary helicity which can be continuous. In section 3.4, we use the deformed twistors to realize the superconformal algebra P SU (2, 2|4) and its deformations 4 We should note that in the work of references [58][59][60][61] on higher spin algebras in AdS5 and in AdS7 the authors mod out the infinite algebra generated by the twistorial oscillators by an ideal. In the case of AdS5 this ideal is generated by a singlet operator that commutes with SU (2, 2) and in AdS7 it is generated by a triplet of operators that form a SU (2) algebra that commutes with SO(6, 2). On the other hand the Joseph ideal is generated by an operatorJABCD that transforms like a tensor of rank four under the corresponding AdS d group SO(d − 1, 2) as will be discussed in section 4. Therefore the ideals considered in [58][59][60][61] are not the Joseph ideals in the respective dimensions. 5 Our results for deformed twistorial oscillators extend to higher spin superalgebras in d = 6 and their deformations [68].
in the quasiconformal framework. In section 4, we review the Eastwood's formula for the generator J of the annihilator of the minrep (Joseph ideal) and show by explicit calculations that it vanishes identically for the singletons of SO (3,2) and the minrep of SU (2, 2) obtained by quasiconformal methods. We then present the generator J of the Joseph ideal in 4d covariant indices and use them to define the deformations J ζ that are the annihilators of the deformations of the minrep. In section 4.4, we use the fact that annihilators vanish identically to identify the AdS 5 /Conf 4 higher spin algebra (as defined by Eastwood [66]) and define its deformations as the enveloping algebras of the deformations of the minrep within the quasiconformal framework. In section 4.4 we discuss the extension of these results to higher spin superalgebras. Finally in section 5 we discuss the implications of our results for higher spin theories of massless fields in AdS 5 and their conformal duals in 4d.
2 3d conformal algebra SO(3, 2) ∼ Sp(4, R) and its minimal unitary realization In this section 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 also called the singleton (scalar and spinor singleton) representations of Dirac [28]. The quasiconformal and covariant oscillator rconstructions of symplectic groups Sp(2N, R) coincide [20] and thus we will only review the oscillator construction of Sp(4, R) following [36,69].

Twistorial oscillator construction of SO(3, 2)
The covering group of the three (four) dimensional conformal (anti-de Sitter) group SO(3, 2) is isomorphic to the noncompact symplectic group Sp(4, R) with the maximal compact subgroup U (2). Commutation relations of its generators can be written as where η AB = diag(−, +, +, +, −) and A, B = 0, 1, . . . , 4. Spinor representation of SO (3,2) can be realized in terms of four-dimensional gamma matrices γ µ that satisfy where η µν = diag(−, +, +) and µ, ν = 0, 1, . . . , 3 and γ 5 = γ 0 γ 1 γ 2 γ 3 as follows: We adopt the following conventions for gamma matrices in four dimensions: where σ m (m = 1, 2, 3 are Pauli matrices. Consider now a pair of bosonic oscillators a i , a † i ( i = 1, 2) that satisfy a i , a † j = δ ij . (2.4) and define a twistorial (Majorana) spinor Ψ and its Dirac conjugate in terms of these oscillators Ψ = Ψ † γ 0 Then the bilinears M AB = 2ΨΣ AB Ψ satisfy the commutation relations (2.1) of SO(3, 2) Lie algebra. The Fock space of these oscillators decompose into two ireducible unitary representations of Sp(4, R) that are simply the two remarkable representations of Dirac [28] which were called Di and Rac in [41]. These representations do not have a Poincaré limit in 4d and their field theories live on the boundary of AdS 4 which can be identified with the conformal compactification of three dimensional Minkowski space [43].

SO(3, 2) algebra in conformal three-grading and 3d covariant twistors
The conformal algebra in d dimensions can be given a three graded decomposition with respect to the noncompact dilatation generator ∆ as follows: We shall call this conformal 3-grading. The commutation relations of the algebra in this basis are given as follows: where M µν (µ, ν = 0, 1, ..., (d − 1)) are the Lorentz groups generators. P µ and K µ are the generators of translations and special conformal transformations. In d = 3 dimensions the Greek indices µ, ν, ... run over 0, 1, 2 and dilatation generator is simply D = −M 34 (2.8) and translations P µ and special conformal transformations K µ are given by: In order to make connection with higher spin (super-)algebras it is best to write the algebra in SO(2, 1) covariant spinorial oscillators. Let us now introduce linear combinations of a i , a † i which we shall call 3d twistors 6 : They satisfy the following commutation relations: Using these we can write (spinor conventions for SO(2, 1) are given in appendix A): Similarly we can define the Lorentz generators and the dilatation generator In this basis the conformal algebra becomes: The conformal group Sp(4, R) in three dimensions admits extensions to supergroups OSp(N |4, R) with even subgroups Sp(4, R) × O(N ). We review the minimal unitary realization of OSp(N |4, R) in Appendix B.

Conformal and superconformal algebras in four dimensions
In this section, we present two different realizations of the conformal algebra and its supersymmetric extensions in d = 4. We start by reviewing the doubleton oscillator realization [24][25][26] and its reformulation in terms of Lorentz covariant twistorial oscillators [25,69]. We then present a novel formulation of the quasiconformal realization of the minimal unitary representation and its deformations first studied in [21] in terms of deformed twistorial oscillators.

Covariant twistorial oscillator construction of the doubletons of SO(4, 2)
The covering group of the conformal group SO(4, 2) in four dimensions is SU (2, 2). Denoting its generators as M AB the commutation relations in the canonical basis are where η AB = diag(−, +, +, +, +, −) and A, B = 0, . . . , 5. The spinor representation of SO(4, 2) can be realized in terms in four-dimensional gamma matrices γ µ that satisfy where η µν = diag(−, +, +, +) ( µ, ν = 0, . . . , 3) as follows: Consider now two pairs of bosonic oscillators a i , a † j (i, j = 1, 2) and b r , b † s (r, s = 1, 2) that satisfy We form a twistorial Dirac spinor Ψ and its conjugate Ψ = Ψ † γ 0 in terms of these oscillators: Then the bilinears M AB = ΨΣ AB Ψ (A, B = 0, . . . , 5) generate the Lie algebra of SO(4, 2): which was called the doubleton realization [24][25][26] 7 . The Lie algebra of SU (2, 2) can be given a three-grading with respect to the algebra of its maximal compact subgroup SU which is referred to as the compact three-grading. For the doubleton realizations one has where the creation operators are denoted with upper indices, i.e a † i = a i . Under the SU (2) L × SU (2) R subgroup of SU (2, 2) generated by the bilinears L i j and R r s oscillators a i (a † i ) and b r (b † r ) transform in the (1/2, 0) and (0, 1/2) representation. In contrast to the situation in three dimensions, the Fock space of these bosonic oscillators decomposes into an infinite set of positive energy unitary irreducible representations (UIRs), called doubletons of SU (2, 2). These UIRs are uniquely determined by a subset of states with the lowest eigenvalue (energy) of the U (1) generator and transforming irreducibly under the SU (2) L × SU (2) R subgroup. The possible lowest energy irreps of SU (2) L × SU (2) R for positive energy UIRs of SU (2, 2) are of the form It is worth mentioning that the doubleton representations are massless in four dimensions and their tensor products decompose into an infinite set of massless spin representations in AdS 5 [24][25][26]. The tensoring procedure is straightforward in oscillator construction and it just corresponds to taking two copies ( colors) of oscillators a i (ξ), The resulting representations are multiplicity free 8 . Tensoring more that two copies of doubleton irreps decomposes into an infinite set of massive representations in AdS 5 which are also multiplicity free [24][25][26].
To relate the oscillators transforming covariantly under the maximal compact subgroup SO(4) × U (1) to twistorial oscillators transforming covariantly with respect to the Lorentz group SL(2, C) with a definite scale dimension one acts with the intertwining operator [26,69] T = e π 4 M 05 .
(3.11) 7 The term doubleton refers to the fact that we are using oscillators that decompose into two irreps under the action of the maximal compact subgroup. For SU (2, 2) that is the minimal set required. For symplectic groups the minimal set consists of oscillators that form a single irrep of their maximal compact subgroups. 8 The explicit formulas for the tensor product decompositions of two irreducible doubleton representations were given in [70].
which intertwines between the compact and the noncompact pictures where L a and R a denote the generators of SU (2) L and SU (2) R , respectively. M a and N a are the generators of SU (2) M and SU (2) N given by the following linear combinations of the Lorentz group generators M µν They satisfy where a, b, .. = 1, 2, 3. The oscillators that transform covariantly under the compact subgroup SU (2) L × SU (2) R get intertwined into the oscillators that transform covariantly under the Lorentz group SL(2, C) . More specifically the oscillators a i (a i ) and b i (b i ) that transform in the (1/2, 0) and (0, 1/2) representation of SU (2) L × SU (2) R go over to covariant oscillators transforming as Weyl spinors (1/2, 0) and (0, 1/2) of the Lorentz group SL(2, C). Denoting the components of the Weyl spinors with undotted (α, β, . . . = 1, 2) and dotted Greek indices (α,β, . . . = 1, 2) one finds : where α, β,α,β, .. = 1, 2 and the covariant indices on the left hand side match the indices i, j.. on the right hand side of the equations above. They satisfy They lead to the standard twistor relations 9 .
The dilatation generator in terms of covariant twistorial oscillators takes the form: The Lorentz generators M µν in a spinorial basis can also be written as bilinears of Lorentz covariant twistorial oscillators: In this basis the conformal algebra becomes: which shows that 1 2 Z is the helicity operator. Denoting the lowest energy irreps in the compact basis as |Ω(j L , j R , E) one can show that the coherent states of the form transform exactly like the states created by the action of conformal fields with exact numerical coincidence of the compact and the covariant labels (j L , j R , E) and (j M , j N , −l), respectively, where l is the scale dimension [26]. The doubletons correspond to massless conformal fields transforming in the (j L , j R ) representation of the Lorentz group SL(2, C) whose conformal (scaling dimension) is ℓ = −E where E is the eigenvalue of the U (1) generator which is the conformal Hamiltonian (or AdS 5 energy) [24][25][26].
3.2 Quasiconformal approach to the minimal unitary representation of SO(4, 2) and its deformations The construction of the minimal unitary representation (minrep) of the 4d conformal group SO(4, 2) by quantization of its quasiconformal realization and its deformations were given in [21], which we shall reformulate in this section in terms of what we call deformed twistorial oscillators which transform nonlinearly under the Lorentz group. The group SO(4, 2) can be realized as a quasiconformal group that leaves invariant light-like separations with respect to a quartic distance function in five dimensions. The quantization of this geometric action leads to a nonlinear realization of the generators of SO(4, 2) in terms of a singlet coordinate x, its conjugate momentum p and two ordinary bosonic oscillators d, d † and g, g † satisfying [21]: The nonlinearities can be absorbed into certain "singular" oscillators which are functions of the coordinate x, momentum p and the oscillators d, g, d † , g † : They satisfy the following commutation relations: The realization of the minrep of SO(4, 2) obtained by the quasiconformal approach is nonlinear and "interacting" in the sense that they involve operators that are cubic or quartic in terms of the oscillators in contrast to the covariant twistorial oscillator realization, reviewed in section 3.1 [24], which involves only bilinears. The algebra so(4, 2) can be given a 3-graded decomposition with respect to the conformal Hamiltonian, which is referred to as the compact 3-grading and the generators in this basis are reproduced in Appendix C following [21].
The Lie algebra of SO(4, 2) has also a noncompact (conformal) three graded decomposition determined by the dilatation generator ∆ as well One can write the generators of quantized quasiconformal action of SO(4, 2) as bilinears of deformed twistorial oscillators Z α , Zα, Y α , Yα (α,α = 1, 2) which are defined as: Using (σ µ ) αα = (½ 2 , σ) and (σ µ )α α = (−½ 2 , σ) one finds that the generators of translations and special conformal transformations can be written as follows 10 : We see that the operators Z and Y in the quasiconformal realization play similar roles as covariant twistorial oscillators λ and η in the doubleton realization. However, they transform nonlinearly under the Lorentz group and their commutations relations are given in Appendix D The dilatation generator in terms of deformed twistorial oscillators takes the form: The Lorentz group generators M µν in a spinorial basis can also be written as bilinears of 10 Note that in our conventions P 0 is positive definite.
these deformed twistorial oscillators: We should stress the important point that even though the deformed twistorial oscillators transform nonlinearly under the Lorentz group, their bilinears P αβ , Kα β , M β α andMαβ transform covariantly and satisfy the commutation relations given in equations 3.22. (2, 2) As was shown in [21], the minimal unitary representation of SU (2, 2) that corresponds to a conformal scalar field admits a one-parameter, ζ, family of deformations that correspond to massless conformal fields of helicity ζ 2 in four dimensions, which can be continuous. For non-integer values of the deformation parameter ζ they correspond, in general, to unitary representations of an infinite covering of the conformal group.

Deformations of the minimal unitary representation of SU
The generators of the deformed minrep take the same form as given in section 3.2 with the simple replacement of the singular oscillators A L and A † L by "deformed" singular oscillators: Since ζ/2 labels the helicity we define "helicity deformed twistors" as follows: The realization of the minimal unitary representation in terms of deformed twistors carry over to realization in terms of helicity deformed twistors: The dilatation generator then takes the form: and the Lorentz generators M µν also take the same form in terms of helicity deformed twistors: The realization of the generators of SU (2, 2) in terms of helicity deformed twistors describe positive energy unitary irreducible representations which can best be seen by going over to the compact three-grading reviewed in Appendix C.
Since the quasiconformal realization of the minrep and its deformations are nonlinear the tensoring procedure in quasiconformal framework is a non-trivial and open problem. However since the representations with integer values of ζ are isomorphic to the doubleton representations, the result of tensoring is already known and was discussed in section 3.1.
The helicity deformed twistors for the superalgebra SU (2, 2|4) are obtained from the deformed twistors of SU (2, 2) by replacing L ζ in the corresponding deformed singular oscillators A L ζ with L s ζ : where N ξ = ξ J ξ J is the number operator of four fermionic oscillators ξ I (ξ J ) (I, J = 1, 2, 3, 4) that satisfy The expressions for the generators of SU (2, 2) given in 3.2 get modified as follows in going over to SU (2, 2|4): where the "supersymmetric" helicity deformed twistors are defined as: The supersymmetry generators of SU (2, 2|4) are given by the bilinears of deformed twistorial oscillators and fermionic oscillators: The generators R I J of R-symmetry group SU (4) are given by: They satisfy the following anti-commutation relations: where C = ζ 2 is the central charge. The commutators of conformal group generators with supersymmetry generators are as follows: The su(4) R generators satisfy the following commutation relations: They act on the R-symmetry indices I, J of the supersymmetry generators as follows: The minimal unitary representation of P SU (2, 2|4) is obtained when the deformation parameter ζ, which is also the central charge, vanishes. The resulting minimal unitary su-permultiplet of massless conformal fields in d = 4 is simply the N = 4 Yang-Mills supermultiplet [21]. For each value of the deformation parameter ζ one obtains an irreducible unitary representation of SU (2, 2|4). For integer values of the deformation parameter these unitary representations are isomorphic to doubleton supermultiplets studied in [24][25][26].
The unitarity of the representations of SU (2, 2|N ) may not be manifest in the Lorentz covariant noncompact five grading. It is however manifestly unitary in compact three grading with respect to the subsupergroup SU (2|N − M ) × SU (2|M ) × U (1) as was shown for the the doubletons in [24,25] and for the quasiconformal construction in [21]. The Lie algebra of SU (4) can be given a 3-graded structure with respect to the Lie algebra of its subgroup SU (2) × SU (2) × U (1). Similarly the Lie superalgebra SU (2, 2|4) can be given a 3-graded decomposition with respect to its subalgebra SU (2|2) × SU (2|2) × U (1). This is the basis that was originally used by Gunaydin and Marcus [24] in constructing the spectrum of IIB supergravity over AdS 5 × S 5 using twistorial oscillators. In this basis, choosing the Fock vacuum as the lowest weight vector leads to CPT-self-conjugate supermultiplets and it is also the preferred basis in applications to integrable spin chains. The corresponding compact 3-grading of the quasiconformal realization of SU (2, 2|4) was given in [21], to which we refer for details.

Higher spin (super-)algebras, Joseph ideals and their deformations
In this section we start by reviewing Eastwood's results [66,71] on defining HS(g) algebras as the quotient of universal enveloping algebra U (g) by its Joseph ideal J (g). We will then explicitly compute the Joseph ideal for SO(3, 2), SO(4, 2) and its deformations, using the Eastwood formula [71] and recast it in a Lorentz covariant form.
The universal enveloping algebra U (g), g = so(d − 1, 2) is defined as follows: where G is the associative algebra freely generated by elements of g, and I is the ideal of G generated by elements of form gh − hg − [g , h] (g, h ∈ g). The enveloping algebra U (g) can be decomposed into standard adjoint action of g which by Poincare-Birkhoff-Witt theorem is equivalent to computing symmetric products M AB ∼ . In particular, 2 so(d − 1, 2) decomposes as: It was already noted in [47] that the higher spin algebra HS(g) must be a quotient of U (g) because the higher spin fields in AdS d are described by traceless two row Young tableaux. Thus the relevant ideal should quotient out all the diagrams except the first one in the above decomposition. This ideal was identified in [66] to be the Joseph ideal or the annihilator of the minimal unitary representation (scalar doubleton). The uniqueness of this quadratic ideal in U (g) was proved in [71] and an explicit formula for the generator J ABCD of the ideal was given as : where the dot · denotes the symmetric product of the generators and M AB , M CD is the Killing form of SO(n − 2, 2). η AB is the SO(n − 2, 2) invariant metric and the symbol ⊚ denotes the Cartan product of two generators, which for SO(n − 2, 2), can be written in the form [72]: The Killing term is given by where h = 2(n−2) n(4−n) is a c-number fixed by requiring that all possible contractions of J ABCD with the metric vanish. We shall refer to the operator J ABCD as the generator of the Joseph ideal.
The generator of the Joseph ideal defined in equation 4.3 contains exactly the operators that correspond to the "unwanted" diagrams described in equation 4.2 and quotienting the enveloping algebra by this ideal guarantees that the resulting algebra will contain only two row traceless diagrams and thus correctly describe massless higher spin fields.
In the following sections we will compute the generator J ABCD for d = 3 and 4 conformal algebras SO(3, 2) and SO(4, 2) in various realizations discussed in previous sections. We shall also decompose the generators of the Joseph ideal of SO(3, 2) and SO(4, 2) with respect to the corresponding Lorentz groups SO(2, 1) and SO(3, 1), respectvely. The Lorentz covariant decomposition makes the massless nature of the minimal unitary representations explicit along with certain other identities that must be satisfied within the representation in order for it to be annihilated by the Joseph ideal. This will also allow us to define the annihilators of the deformations of the minrep of SO(4, 2) and the corresponding deformations of the Joseph ideal. These deformations define a one parameter family of AdS 5 /CF T 4 HS algebras.

Joseph ideal for SO(3, 2) singletons
We will now use the twistorial oscillator realization for SO(3, 2) described in section 2.1. For Sp(4, R) = SO(3, 2), the generator J ABCD of the Joseph ideal is Substituting the realization of Sp(4, R) = SO(3, 2) in terms of a twistorial Majorana spinor Ψ one finds that the operator J ABCD vanishes identically. Considered as the the three dimensional conformal group the minreps of Sp(4, R) (Di and Rac) correspond to massless scalar and spinor fields which are known to be the only massless representations of the Poincaré group in three dimensions [4].
If instead of a twistorial Majorana spinor one considers a twistorial Dirac spinor corresponding to taking two copies (colors) of the Majorana spinor one finds that the generators J ABCD of the Joseph ideal do not vanish identically and hence they do not correspond to minimal unitary representations. The corresponding Fock space decomposes into an infinite set of irreducible unitary representations of Sp(4, R), which correspond to the massless fields in AdS 4 [43]. Taking more than two colors in the realization of the Lie algebra of Sp(4, R) as bilinears of oscillators leads to representations corresponding to massive fields in AdS 4 [36].

Joseph ideal of SO(3, 2) in Lorentz covariant basis
To get a more physical picture of what the vanishing of the Joseph ideal means we shall go to the conformal 3-graded basis defined in equation 2.6. Evaluating the Joseph ideal in this basis, we find that the vanishing ideal is equivalent to the linear combinations of certain quadratic identities, full set of which hold only in the singleton realization. First we have the masslessness conditions: The remaining set of quadratic relations that define the Joseph ideal are 6∆ · ∆ + 2M µν · M µν + P µ · K µ = 0 (4.9) P µ · (M µν + η µν ∆) = 0 (4.10) M [µν · P ρ] = 0 (4.14) The ideal generated by these relations is completely equivalent to equation 4.7 but it sheds light on the massless nature of these representations. The scalar and spinor singleton modules for SO (3,2) are the only minreps and there are no other deformations. This is a general phenomenon for all symplectic groups Sp(2n, R) (n > 2) and within the quasiconformal approach it can be explained by the fact that the corresponding quartic invariant that enters the minimal unitary realization vanishes for symplectic groups. The Casimir invariants for the singleton or the minrep of SO(3, 2) are as follows: Computing the products of the generators of the above singleton realization corresponding to the Young tableaux and one finds that they vanish identically and the resulting enveloping algebra contains only the operators whose Young tableaux have two rows.

Joseph ideal of SO(4, 2)
In this subsection we shall first evaluate the generator J ABCD of the Joseph ideal of SO (4,2) in the covariant twistorial operator realization and then in the quasiconformal realization of its minrep to highlight the essential differences.

Joseph ideal in the covariant twistorial oscillator or doubleton realization
We will use the doubleton realization [24,25] reviewed in section 3.1 to compute the generator J ABCD of the Joseph ideal for SO(4, 2) : Substituting the expressions for the generators in the covariant twistorial realization one finds that it does not vanish identically as an operator in contrast to the situation with the singletonic realization of SO (3,2). However one finds that J ABCD has only 15 independent non-vanishing components which turn out to be equal to one of the following expressions (up to an overall sign): Similarly, computing the products of the generators corresponding to the the Young tableaux explicitly in the covariant twistorial realization one finds that they do not vanish but we have the following relation: Thus all non-vanishing four-row diagrams can be dualized to two-row diagrams and the resulting generators of the universal enveloping algebra in doubleton realization are described by two-row diagrams. The operator Z = (N a − N b ) commutes with all the generators of SU (2, 2) and its eigenvalues label the helicity of the corresponding massless representation of the conformal group [21,24,25]. All the components of the generator J ABCD of Joseph ideal vanish on the states that form the basis of an UIR of SU (2, 2) whose lowest weight vector is the Fock vacuum |0 since Z|0 = 0 (4.28) The corresponding unitary representation describes a conformal scalar field in four dimensions (zero helicity) and is the true minrep of SU (2, 2) annihilated by the Joseph ideal [21]. The Casimir invariants for SO (4,2) in the doubleton representation are as follows: Thus we see that all the higher order Casimir invariants are functions of the quadratic Casimir C 2 which itself is given in terms of Z = N a − N b .

Joseph ideal and the quasiconformal realization of the minrep of SO(4, 2)
To apply Eastwood's formula to the generator of Joseph ideal in the quasiconformal realization it is convenient to go from the conformal 3-graded basis to the SO(4, 2) covariant canonical basis where the generators M AB satisfy the following commutation relations: where the metric η AB = diag(−, +, +, +, +, −) is used to raise and lower the indices A, B = 0, 1, . . . , 5 etc. In addition to Lorentz generators M µν (µ, ν, .. = 0, 1, 2, 3) , we have the following linear relations between the generators in the canonical basis and the conformal 3-graded basis Substituting the expressions for the quasiconformal realization of the generators of SO(4, 2) given in subsection 3.2 into the generator of the Joseph ideal in the canonical basis: one finds that it vanishes identically as an operator showing that the corresponding unitary representation is indeed the minimal unitary representation. We should stress the important point that the tensor product of the Fock spaces of the two oscillators d and g with the state space of the singular oscillator A L (A † L ) form the basis of a single UIR which is the minrep. In contrast, the Fock space of the covariant twistorial oscillators reviewed in section 3.1 decomposes into infinitely many UIRs (doubletons) of which only the irreducible representation whose lowest weight vector is the Fock vacuum, which is annihilated by J ABCD , is the minimal unitary representation of SO(4, 2).

4d Lorentz Covariant Formulation of the Joseph ideal of SO(4, 2)
Above we showed that the generator of the Joseph ideal given in equation (4.36) vanishes identically as an operator for the quasiconformal realization of the minrep of SO(4, 2) in the canonical basis. To get a more physical picture of what the vanishing of generator J ABCD means we shall go to the conformal three grading defined by the dilatation generator ∆. Evaluating the generator of the Joseph ideal in this basis, we find that the vanishing condition is equivalent to linear combinations of certain quadratic identities, full set of which hold only in the quasiconformal realization of the minrep. First we have the conditions: which hold also for the twistorial oscillator realization given in section 3.1. The remaining set of quadratic relations that define the Joseph ideal are 4∆ · ∆ + M µν · M µν + P µ · K µ = 0 (4.38) In four dimensions, using the Levi-Civita tensor one can define the Pauli-Lubanski vector, W µ and its conformal analogue, V µ as follows: where ǫ 0123 = +1, ǫ 0123 = −1 and the indices are raised and lowered by the Minkowski metric. For massless fields, W µ and V µ are proportional to P µ and K µ respectively with the proportionality constant related to helicity of the fields [73,74]. Equations (4.44) imply that for the minrep both the W µ and V µ vanish implying that it describes a zero helicity (scalar) massless field 12 .
Computing the products of the generators in the above quasiconformal realization corresponding to Young tableaux and explicitly one finds that they vanish identically and the resulting enveloping algebra contains only operators with two row Young tableaux. The Casimir operators of the minrep of SO(4, 2) are as take on the following values (using the definitions given in equations 4.29 -4.31):

Deformations of the minrep of SO(4, 2) and their associated ideals
As was shown in [21], the minimal unitary representation of SU (2, 2) that corresponds to a conformal scalar field admits a one-parameter (ζ) family of deformations corresponding to massless conformal fields of helicity ζ 2 in four dimensions, which can be continuous. For non-integer values of the deformation parameter ζ they correspond, in general, to unitary representations of an infinite covering of the conformal group 13 .
The generators of the deformed minreps were reformulated in terms of deformed twistorial oscillators in section 3.3. Substituting the expressions for the generators of the deformed minreps of SO(4, 2) into the generator J ABCD of the Joseph ideal one finds that it does not vanish for non-zero values of the deformation parameter ζ. One might therefore ask if there exists deformations of the Joseph ideal that annihilate the deformed minimal unitary representations labelled by the deformation parameter ζ. Remarkably, this is indeed the case. The quadratic identities that define the Joseph ideal in the conformal basis discussed in the previous section go over to identities involving the deformation parameter ζ and define the deformations of the Joseph ideal. One finds that the helicity conditions are modified as follows: The identities (4.41), (4.42) and (4.43) get also modified as follows: The other quadratic identities remain unchanged in going over to the deformed minimal unitary representations . The Casimir invariants for the deformations of the minrep of SO(4, 2) depend only the deformation parameter ζ and are given as follows (using the definitions given in equations 4.29 -4.31): The products of generators corresponding to the Young tableaux do not vanish for the deformed minimal unitary representations and depend on the deformation parameter ζ as follows: = ζ (4.55) Hence all non-vanishing four-row diagrams can be dualized to two-row diagrams and the resulting generators of the universal enveloping algebra in deformed realization are described by two-row diagrams. For ζ = 0 all four row diagrams vanish and one obtains the standard high spin algebra of Vasiliev type whose generators transform in representations of the underlying AdS group corresponding to Young tableaux containing only two row traceless diagrams. For non-zero ζ the corresponding enveloping algebras describe deformed higher spin algebras as discussed in the next subsection. We saw earlier in section 4.2.1 that the generator J ABCD of the Joseph ideal did not vanish identically as an operator for the covariant twistorial oscillator realization of SO (4,2). It annihilates only the states belonging to the subspace that form the basis of the true minrep of SO(4, 2). By going to the conformal three grading, one finds that the generator J ABCD of the Joseph ideal can be written in a form similar to the deformed quadratic identities above with the deformation parameter replaced by the linear Casimir The Fock space of the oscillators decompose into an infinite set of unitary irreducible representations of SU (2, 2) corresponding to massless conformal fields of all integer and half-integer helicities labelled by the eigenvalues of Z/2.

Higher spin algebras and superalgebras and their deformations
We shall adopt the definition of the higher spin AdS (d+1) /CF T d algebra as the quotient of the enveloping algebra U (SO(d, 2)) of SO(d, 2) by the Joseph ideal J (SO(d, 2)) and denote it as HS(d, 2) [66]: We shall however extend it to define deformed higher spin algebras as the enveloping algebras of the deformations of the minreps of the corresponding AdS d+1 /Conf d algebras.
For these deformed higher spin algebras the corresponding deformations of the Joseph ideal vanish identically as operators in the quasiconformal realization as we showed explicitly above for the conformal group in four dimensions. We expect a given deformed higher spin algebra to be the unique infinite dimensional quotient of the universal enveloping algebra of an appropriate covering 14 of the conformal group by the deformed ideal as was shown for the undeformed minrep in [76]. Similarly, we define the higher spin superalgebras and their deformations as the enveloping algebras of the minimal unitary realizations of the underlying superalgebras and their deformations, respectively 15 . In four dimensions (d = 4) we have a one parameter family of higher spin algebras labelled by the helicity ζ/2: HS(4, 2; ζ) = U (SO(4, 2)) J ζ (SO(4, 2)) (4.62) where J ζ (SO(4, 2)) denotes the deformed Joseph ideal of SO(4, 2) defined in section 4.3 16 On the AdS d+1 side the generators of the higher spin algebras correspond to higher spin gauge fields, while on the Conf d side they are related to conserved tensors including the conserved stress-energy tensor. The charges associated with the generators of conformal algebra SO(d, 2) are defined by conserved currents constructed by contracting the stressenergy tensor with conformal Killing vectors. Similarly, the higher conserved currents are obtained by contracting the conformal Killing tensors with the stress-energy tensor. These higher conformal Killing tensors are obtained simply by tensoring the conformal Killing vectors with themselves. Though implicit in previous work on the subject [78,79], explicit use of the language of conformal Killing tensors in describing higher spin algebras seems to have first appeared in the paper of Mikhailov [65] 17 . This connection was put on a rigorous foundation by Eastwood in his study of the higher symmetries of the Laplacian [66] who showed that the undeformed higher spin algebra can be obtained as the quotient of the enveloping algebra of SO(d, 2) generated by the conformal Killing tensors quotiented by the Joseph ideal.
The connection between the doubleton realization of SO(4, 2) in terms of covariant twistorial oscillators and the corresponding conformal Killing tensors was also studied by Mikhailov [65]. He pointed out that the higher conformal Killing tensors correspond to the products of bilinears of oscillators that generate SO (4,2) in the doubleton realization of 14 In d = 4 deformed minreps describe massless fields with the helicity ζ 2 . For non-integer values of ζ one has to go to an infinite covering of the 4d conformal group. 15 We should note that the universal enveloping algebra of a Lie group as defined in the mathematics literature is an associative algebra with unit element. Under the commutator product inherited from the underlying Lie algebra it becomes a Lie algebra. 16 After the main results of this paper was announced at the GGI Conference on higher spin theories in May 2013, we became aware of the work of [64] where possible deformations of purely bosonic higher spin algebras in arbitrary dimensions were studied and it was shown that the deformations can depend at most on one parameter. In a subsequent work [77] it was shown that the AdS d higher spin algebra is unique in d = 4 and d > 6 under their assumptions on the spectrum of generators. They also imply that the one parameter family of deformations of [64] must be the same as the one parameter family discussed in this paper which is based on earlier work [21] that they cite. The results of [64] are based on Young tableaux analysis of gauge fields in AdS5. Whether and how their deformation parameter is related to helicity in 4d is not known at this point. Furthermore, we find a discrete infinite family of higher spin algebras and superalgebras in AdS7 [68]. 17 The generators of the higher spin algebra shs E (8|4) of reference [45] in terms of Killing spinors in AdS4 were obtained in the work of [80] on higher spin N = 8 supergravity in d = 4.
The supersymmetric extension of the higher spin algebras HS(4, 2; ζ) is given by the enveloping algebra of the deformed minimal unitary realization of the N-extended conformal superalgebras SU (2, 2|N ) ζ with the even subalgebras SU (2, 2) ⊕ U (N ). We shall denote the resulting higher spin algebra as HS [SU (2, 2|N ); ζ]. These supersymmetric extensions involve odd powers of the deformed twistorial oscillators and the identities that define the Joseph ideal get extended to a supermultiplet of identities obtained by the repeated actions of Q and S supersymmetry generators on the generators of Joseph ideal. On the conformal side the odd generators correspond to the products of the conformal Killing spinors with conformal Killing vectors and tensors. If we denote the resulting deformed super-ideal as As discussed in previous sections, the bosonic higher spin gauge fields are described by two-row Young tableaux of SO (4,2). However in order to extend the SO(4, 2) Young tableaux to super Young tableaux , one needs to represent the corresponding representations in terms of SU (2, 2) Young tableaux since the relevant superconformal algebras are SU (2, 2|N ) with the even subalgebra SU (2, 2) ⊕ U (N ) and not superalgebras of the orthosymplectic type. The identifications of Young tableaux can be easily checked by calculating the dimensions of corresponding irreps. In going from SU (2, 2) to SU (2, 2|N ) one simply replaces the Young tableaux of SU (2, 2) by super Young tableaux of SU (2, 2|N ) 18 . The Young diagram of the adjoint representation of SO(4, 2) = SU (2, 2) goes over to the following super tableau of SU (2, 2|N ) which involves one dotted and one undotted "superboxes" : as follows: The window diagram appearing in the tensor product of two adjoint representations can be supersymmetrized as follows: Thus the higher spin gauge fields are described by the following SU (2, 2) diagram and can be consequently supersymmetrized in a straightforward manner: The situation is much simpler for AdS 4 /Conf 3 higher spin algebras. There are only two minreps of SO (3,2), namely the scalar and spinor singletons. The higher spin algebra HS(3, 2) is simply given by the enveloping algebra of the singletonic realization of Sp(4, R) [45,56]. Singletonic realization of the Lie algebra of Sp(4, R) describes both the scalar and spinor singletons. They form a single irreducible supermultiplet of OSp(1|4, R) generated by taking the twistorial oscillators as the odd generators [56]. The odd generators correspond to conformal Killing spinors , which together with conformal Killing vectors in d = 3 , generate the Lie super-algebra of OSp(1|4, R). Its enveloping algebra leads to the higher spin super-algebra of Fradkin-Vasiliev type involving all integer and half integer spin fields in AdS 4 . One can also construct N-extended higher spin superalgebras in AdS 4 as enveloping algebras of the singletonic realization of OSp(N |4, R) [45,56].

Discussion
The existence of a one-parameter family of AdS 5 /Conf 4 higher spin algebras and superalgebras raises the question as to their physical meaning. Before discussing the situation in four dimensions, let us summarize what is known for AdS 4 /Conf 3 higher spin algebras. In 3d, there is no deformation of the higher spin algebra except for the super extension corresponding to Sp(4; R) → OSp(N |4; R). For the bosonic AdS 4 higher spin algebras one finds that the higher spin theories of Vasiliev are dual to certain conformally invariant vector-scalar/spinor models in 3d [51,52] in the large N limit. More recently, Maldacena and Zhiboedov studied the constraints imposed on a conformal field theory in d = 3 three dimensions by the existence of a single conserved higher spin current. They found that this implies the existence of an infinite number of conserved higher spin currents. This corresponds simply to the fact that the generators of SO(d, 2) get extended to an infinite spin algebra when one takes their commutators with an operator which is bilinear or higher order in the generators, except for those operators that correspond to the Casimir elements. They also showed that the correlation functions of the stress tensor and the conserved currents are those of a free field theory in three dimensions, either a theory of N free bosons or a theory of N free fermions [83], which are simply the scalar and spinor singletons.
The distinguishing feature of 3d is the fact that there exists only two minimal unitary representations corresponding to massless conformal fields which are simply the Di and Rac representations of Dirac. However in 4d, we have a one-parameter, ζ, family of deformations of the minimal unitary representation of the conformal group corresponding to massless conformal fields of helicity ζ/2. The same holds true for the minimal unitary supermultiplet of SU (2, 2|N ). From M/superstring point of view the most important interacting and supersymmetric CFT in d = 4 is the N = 4 super Yang-Mills theory. It was argued in references [58,84,85] that the holographic dual of N = 4 super Yang-Mills theory with gauge group SU (N ) at g 2 Y M N = 0 for N → ∞ should be a free gauge invariant theory in AdS 5 with massless fields of arbitrarily high spin and this was supported by calculations in [65]. Moreover the scalar sector of N = 4 super Yang-Mills theory at g 2 Y M N = 0 with N → ∞ should be dual to bosonic higher spin theories in AdS 5 which provides a non-trivial extension of AdS/CF T correspondence in superstring theory [37,38] to nonsupersymmetric large N field theories. The Kaluza-Klein spectrum of IIB supergravity over AdS 5 × S 5 was first obtained by tensoring of the minimal unitary supermultiplet (scalar doubleton) of SU (2, 2|4) with itself repeatedly and restricting to CPT self-conjugate sector [24]. The massless graviton supermultiplet in AdS 5 sits at the bottom of this infinite tower. In fact all the unitary representations corresponding to massless fields in AdS 5 can be obtained by tensoring of two doubleton representations of SU (2, 2) which describe massless conformal fields on the boundary of AdS 5 [24][25][26]56]. As was argued by Mikhailov [65], in the large N limit the correlation functions in the CFT side become products of two point functions which correspond to products of two doubletons. As such they correspond to massless fields in the AdS 5 bulk. At the level of correlation functions the same arguments suggest that corresponding to a one parameter family of deformations of the N=4 Yang-Mills supermultiplet there must exists a family of supersymmetric massless higher spin theories in AdS 5 . Turning on the gauge coupling constant on the Yang-Mills side leads to interactions in the bulk and most of the higher spin fields become massive.
The fact that the quasiconformal realization of the minrep of SU (2, 2|4) is nonlinear implies that the corresponding higher spin theory in the bulk must be interacting. Since the same minrep can also be obtained by using the doubletonic realization [24], which corresponds to free field realization, suggests that the interacting supersymmetric higher spin theory may be integrable in the sense that its holographic dual is an integrable conformal field theory with infinitely many conserved currents , just like the classical N = 4 super Yang-Mills theory in four dimensions [86]. There are other deformed higher spin algebras corresponding to non-CPT self-conjugate supermultiplets of SU (2, 2|4) that contain scalar fields and are deformations of the minrep. The above arguments suggest that they too should correspond to interacting but integrable supersymmetric higher spin theories in AdS 5 . One solid piece of the evidence for this is provided by the fact that the symmetry superalgebras of interacting (nonlinear) superconformal quantum mechanical models of [87] furnish a one parameter family of deformations of the minimal unitary representation of the N = 4 superconformal algebra D(2, 1; α) in one dimension. This was predicted in [88] and shown explicitly in [89].
Most of the work on higher spin algebras until now have utilized the realizations of underlying Lie (super)algebras as bilinears of oscillators which correspond to free field realizations. The quasiconformal approach allows one to give a natural definition of super Joseph ideal and leads directly to the interacting realizations of the superextensions of higher spin algebras. The next step in this approach is to reformulate these interacting quasiconformal realizations in terms of covariant gauge fields and construct Vasiliev type nonlinear theories of interacting higher spins in AdS 5 .
Another application of our results will be to reformulate the spin chain models associated with N = 4 super Yang-Mills theory in terms of deformed twistorial oscillators and study the integrability of corresponding spin chains non-perturbatively. In fact, a spectral parameter related to helicity and central charge was introduced recently for scattering amplitudes in N = 4 super Yang-Mills theory [90]. This spectral parameter corresponds to our deformation parameter which is helicity and appears as a central charge in the quasiconformal realization of the super algebra SU (2, 2|4). We hope to address these issues in future investigations.
Acknowledgements: Main results of this paper were announced in GGI Workshop on Higher spin symmetries (May 6-9, 2013) and Summer Institute (Aug. 2013) at ENS in Paris by MG and at Helmholtz Internations Summer School (Sept. 2013) at Dubna by KG. We would like to thank the organizers of these workshops and institutes for their kind hospitality where part of this work was carried out. We enjoyed discussions with many of their participants. We are especially grateful to Misha Vasiliev, Eugene Skvortsov and Massimo Taronna for stimulating discussions regarding higher spin theories and Dmytro Volin regarding quasiconformal realizations. This research was supported in part by the US National Science Foundation under grants PHY-1213183, PHY-08-55356 and DOE Grant No: DE-SC0010534.

B Minimal unitary supermultiplet of OSp(N|4, R)
In this section we will formulate the minimal unitary representation of OSp(N |4, R) which is the superconformal algebra with N supersymmetries in three dimensions. The superalgebra osp(N |4) can be given a five graded decomposition with respect to the noncompact dilatation generator ∆ as follows: We shall call this superconformal 5-grading. The bosonic conformal generators are the same as given in previous section. In order to realize the R-symmetry algebra SO(N ) and supersymmetry generators, we introduce Euclidean Dirac gamma matrices γ I (I, J = 1, 2, . . . , N ) which satisfy γ I , γ J = δ IJ (B. 2) The R-symmetry generators are then simply given as follows: and supersymmetry generators are the bilinears of 3d twistorial oscillators and γ I : They satisfy the following commutation relations: The action of conformal group generators on supersymmetry generators is as follows: C The quasiconformal realization of the minimal unitary representation of SO(4, 2) in compact three-grading In this appendix we provide the formulas for the quasiconformal realization of generators of SO(4, 2) in compact 3-grading following [21]. Consider the compact three graded decomposition of the Lie algebra of SU (2, 2) determined by the conformal Hamiltonian so(4, 2) = C − ⊕ C 0 ⊕ C + where C 0 = so(4) ⊕ so (2). Folllowing [21] we shall label the generators in C ± and C 0 subspaces as follows: where L ±,0 and R ±,0 denote the generators of SU (2) L × SU (2) R and H is the U (1) generator.
The generators of so(4, 2) in the compact 3-grading take on very simple forms when expressed in terms of the singular oscillators introduced in section 3.2: (H + 1) (C.6)