Massless conformal fields, AdS_{d+1}/CFT_d higher spin algebras and their deformations

We extend our earlier work on the minimal unitary representation of $SO(d,2)$ and its deformations for $d=4,5$ and $6$ to arbitrary dimensions $d$. We show that there is a one-to-one correspondence between the minrep of $SO(d,2)$ and its deformations and massless conformal fields in Minkowskian spacetimes in $d$ dimensions. The minrep describes a massless conformal scalar field, and its deformations describe massless conformal fields of higher spin. The generators of Joseph ideal vanish identically as operators for the quasiconformal realization of the minrep, and its enveloping algebra yields directly the standard bosonic $AdS_{(d+1)}/CFT_d$ higher spin algebra. For deformed minreps the generators of certain deformations of Joseph ideal vanish as operators and their enveloping algebras lead to deformations of the standard bosonic higher spin algebra. In odd dimensions there is a unique deformation of the higher spin algebra corresponding to the spinor singleton. In even dimensions one finds infinitely many deformations of the higher spin algebra labelled by the eigenvalues of Casimir operator of the little group $SO(d-2)$ for massless representations.


Introduction
In earlier work we studied the minimal unitary representation (minrep) of SO(d, 2) for d = 4, 5 and 6 and their deformations using the quasiconformal approach. More specifically, in [1] we constructed the minrep of SU (2, 2) and its one-parameter family of deformations which describe all massless conformal fields in four dimensions with the identification of the deformation parameter as helicity. The minrep of the superalgebras su(2, 2|N ) that extend the conformal algebra in four dimensions were also constructed in [1]. The minrep of SU (2, 2|N ) also admits an infinite family of deformation which describe massless Nextended superconformal multiplets in four dimensions. The minimal unitary supermultiplet of the N = 4 superconformal algebra P SU (2, 2|4) is simply the N = 4 Yang-Mills supermultiplet.
ten [12]. From a mathematical point of view their importance lies in the fact that they correspond to the minimal unitary supermultiplets of the respective symmetry superalgebras of P SU (2, 2|4) and OSp(8 * |4), respectively.
The minimal unitary realizations of SU (2, 2|N ) and of OSp(8 * |2N ) and their deformations obtained via quasiconformal methods [1][2][3] were reformulated as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group in [13,14]. Furthermore it was shown that the enveloping algebras of the minimal unitary realizations of SU (2, 2) and SO * (8) thus obtained lead directly to the higher spin algebras in AdS 5 and AdS 7 .
As was first pointed out in [15], the higher spin algebra in AdS 4 as studied by Fradkin, Vasiliev and collaborators [16,17] is simply the enveloping algebra of the scalar singleton of SO (3,2). Later Vasiliev showed that the higher spin algebra in AdS (d+1) , for general d, is given by the enveloping algebra of SO(d, 2) quotiented by the ideal that annihilates the scalar "singleton" representation [18]. That this ideal is simply the Joseph ideal of the minimal unitary representation of SO(d, 2) was established by Eastwood [19]. Undeformed higher spin algebras in d dimensions were further investigated by Vasiliev in [20].
For symplectic groups Sp(2n, R) the minrep is simply the scalar singleton and it admits a single deformation which is the spinor singleton. Quasiconformal realizations of the singletons of Sp(2n, R) coincide with their realizations as bilinears of oscillators transforming covariantly under the maximal compact subgroup U (n) [21]. The Joseph ideal vanishes identically for the singletons [13]. As a consequence the enveloping algebra of the AdS 4 group SO(3, 2) ≡ Sp(4, R) realized as bilinears of covariant twistorial oscillators leads directly to the Fradkin-Vasiliev higher spin algebra as was first pointed out in [15]. On the other hand, for the doubletonic realizations of the minreps of SO(4, 2) and SO(6, 2) as bilinears of twistorial oscillators transforming covariantly under the respective Lorentz groups, the Joseph ideal does not vanish identically as operators. However it was shown in [13,14] that for the minreps of SO(4, 2) and SO(6, 2) obtained by quasiconformal methods, the Joseph ideals vanish identically as operators. Therefore their enveloping algebras provide unitary realizations of the bosonic higher spin algebras in AdS 5 and AdS 7 , respectively. The enveloping algebras of the deformed minreps of SU (2, 2) and of SO * (8) and their supersymmetric extensions lead to infinite families of higher spin algebras and superalgebras in AdS 5 and AdS 7 , respectively [13,14].
Similarly the Joseph ideal for the minrep of SO(5, 2) obtained by the quasiconformal methods vanishes identically as operators and its enveloping algebra yields the bosonic higher spin algebra in AdS 6 [4]. There is a unique deformed higher spin algebra given by the enveloping algebra of the spinor singleton and the enveloping algebra of the minrep of the Lie superalgebra f(4) yields the unique higher spin superalgebra in AdS 6 .
In this paper we extend our previous work to all d-dimensional conformal groups SO(d, 2) and construct their minimal unitary representations and their deformations. We shall first review the quasiconformal approach to the construction of the minreps of SO(d, 2) [21] which describe massless conformal fields. This is followed by the study of possible deformations of these minreps. In odd d dimensions, there exists a single deformation that describes a conformally massless spinor field. In even d dimensions, we find infinitely many deformations of the minrep, corresponding to conformally massless fields that transform nontrivially under the little group SO(d − 2). The generators of the Joseph ideal vanish identically as operators for the quasiconformal realization of the minrep of SO(d, 2) in all dimensions. Therefore the enveloping algebra of the minrep of SO(d, 2) leads directly to the standard bosonic higher spin algebra in AdS (d+1) . For the deformed minreps certain deformations of the Joseph ideal vanish identically as operators and their enveloping algebras lead to deformations of the standard bosonic higher spin algebra. The deformations of higher spin algebras correpond to the quotients of the the enveloping algebra of SO(d, 2) by the deformations of their Joseph ideals.
The plan of the paper is as follows. Following [21], we construct the geometric realization of SO(d, 2) as a quasiconformal group in section 2. Then, by quantizing the geometric quasiconformal action, we obtain the minimal unitary representation (minrep) of SO(d, 2) in section 3. In that section we also show that, according to Joseph's theorem [22], there exists a two-parameter family of degree-two polynomials of so(d, 2) generators which reduces to a c-number. Then in section 4, we give the noncompact 3-grading of so(d, 2) with respect to the subalgebra so(d−1, 1)⊕so(1, 1) and show that the Poincaré mass operator in d dimensions vanishes identically for the minrep of SO(d, 2). Similarly in section 5 we give the compact 3-grading of so(d, 2) with respect to the subalgebra so(d) ⊕ so(2) and obtain the Poincaré-masslessness condition in the compact basis. Then we discuss the properties of a distinguished SU (1, 1) subgroup of SO(d, 2), which is generated by the longest root vector and realized by singular (isotonic) oscillators, in section 6 and present the K-type decomposition of the minrep of SO(d, 2) in section 7. We also show that the minrep of SO(d, 2) corresponds to a massless conformal scalar field in d dimensions. Then in section 8 we introduce the deformations of the minrep of SO(d, 2). In particular, we present a constraint that needs to be satisfied by the generators that introduce deformations. Then we construct the K-type decomposition of the unique deformed minrep for odd d in section 8.1 and the K-type decomposition of the infinitely many deformed minreps for even d in section 8.2. In section 9, after presenting the SO(d, 2)-covariant generators in terms of SO(d) × SO(2)-covariant generators and SO(d − 1) × SO(1, 1)-covariant generators, we show that the Joseph ideal vanishes identically as an operator for the minrep of SO(d, 2) and therefore its universal enveloping algebra yields directly the bosonic AdS (d+1) /CF T d higher spin algebra. We also show that for the deformations of the minrep, a certain deformation of the Joseph ideal vanishes identically and therefore its enveloping algebra yields the deformed AdS (d+1) /CF T d higher spin algebras. Finally we have some concluding comments in section 10. Appendices A and B outline how to realize the "spin" operators that extend the little group SO(d − 2), which allows us to obtain the deformations of the minrep.
so(1, 1) ⊕ so(d − 2) ⊕ sp(2, R) as follows: where the five grading is determined by the so(1, 1) generator ∆. The non-linear quasiconformal group action of SO(d, 2) is generated by nonlinear differential operators acting on a (2d − 3)-dimensional space T corresponding to the Heisenberg subalgebra generated by the elements of the subspace g (−2) ⊕ g (−1) . We shall denote the coordinates of the space T as X = X i,a , x , where X i,a transform in the (d − 2, 2) representation of so(d − 2) ⊕ su(1, 1) subalgebra, with i = 1, . . . , d − 2 and a = 1, 2, and x is a singlet coordinate.
There exists a quartic polynomial of the coordinates X i,a where i, j, k, l = 1, . . . , d−2 and a, b, c, d = 1, 2, which is an invariant of SO(d−2)×SU (1, 1) subgroup. In the above expression, ǫ ab is the symplectic invariant tensor of SU (1, 1) and η ij is the invariant metric of SO(d − 2) in the fundamental representation, which we choose as η ij = −δ ij following the general conventions of [23]. We shall label the generators that belong to various grade subspaces as follows: where L ij and M ab are the generators of SO(d − 2) and SU (1, 1) subgroups, respectively. In the nonlinear quasiconformal action of SO(d, 2), these generators take on the form where ǫ ab is the inverse symplectic tensor, such that ǫ ab ǫ bc = δ a c . The explicit form of the grade +1 generators U i,a can be obtained by substituting the expression for the quartic invariant: The above SO(d, 2) generators satisfy the following commutation relations: The quartic norm (length) of a vector X = X i,a , x ∈ T is defined as In order to see the geometric picture behind the above nonlinear realization, a quartic distance function between any two points X and Y in the (2d − 3)-dimensional space T can be defined as [21,24] d (X , Y) = ℓ (δ (X , Y)) (2.8) where the "symplectic" difference δ (X , Y) is given by The quasiconformal action of SO(d, 2) leaves the lightlike separations between any two points with respect to the quartic distance function invariant, which implies that SO(d, 2) behaves as the invariance group of a "light-cone" with respect to a quartic distance function in a (2d − 3)-dimensional space.
3 Minimal unitary representation of SO(d, 2) from its quasiconformal realization The minimal unitary realization of a Lie algebra can be easily obtained by quantizing its geometric quasiconformal realization [21,[23][24][25]. To achieve this, in the case of SO(d, 2), we split the 2(d − 2) variables X i,a introduced in section 2 into (d − 2) coordinates X i and (d − 2) conjugate momenta P i as and introduce a momentum p conjugate to the singlet coordinate x as well. Treated as quantum mechanical operators, these coordinates and momenta satisfy the canonical commutation relations However, for the rest of this paper, instead of using X i and P i we shall work with bosonic oscillator annihilation operators a i and creation operators a † i , defined as which satisfy the commutation relations The generators of the minimal unitary realization of so(d, 2) has a 5-graded decomposition with respect to the SO(1, 1) generator The generators of the subalgebra su(1, 1) ⊂ g (0) are realized as bilinears of the a-type bosonic oscillators: and satisfy We denote this subalgebra as su(1, 1) M and its quadratic Casimir as M 2 : The subalgebra so(d − 2) ⊂ g (0) corresponding to the little group of massless particles in d dimensions, denoted as so(d − 2) L , is also realized as bilinears of the a-type bosonic oscillators 1 : and satisfy the commutation relations The quadratic Casimir L 2 of so(d − 2) L given by is related to that of su(1, 1) M as The single generator in g (−2) is defined as (3.14) The "quantized generators" (U i , U † i ) in grade −1 subspace are realized as bilinears of x and the above a-type bosonic oscillators as They close into K − under commutation and form a Heisenberg subalgebra with K − playing the role of the central charge.
The quartic invariant I 4 of SO(d−2) L ×SU (1, 1) M subgroup becomes a linear function of the quadratic Casimir of SO(d − 2) L × SU (1, 1) M after quantization. As a result the grade +2 generator K + becomes: where G depends on L 2 : The remaining 2(d − 2) generators in grade +1 subspace can be obtained by taking the commutators between the grade −1 generators and K + : Explicitly one finds (3.20) The grade +2 and grade +1 generators form a Heisenberg algebra as well: with the generator K + playing the role of the central charge. The commutators of grade −2 and grade +1 generators close into grade −1 subspace: Grade ±2 generators, together with the generator ∆ from grade 0 subspace, form a distinguished su(1, 1) subalgebra, which we shall denote as su(1, 1) K : The quadratic Casimir of su(1, 1) K subalgebra, denoted by is related to the quadratic Casimir of so(d − 2) L (and that of su(1, 1) M ) as The commutators between grade −1 generators and grade +1 generators are: Let us now present the quadratic Casimir of so(d, 2). One finds that the quadratic operator constructed out of the generators of grade ±1 subspaces corresponding to the coset space is related to the quadratic Casimir of su(1, 1) K as follows: As a consequence one finds 2 : The quadratic Casimir of so(d, 2) is given by λ 1 = 4 and λ 2 = −1 (for any d) and is therefore given by Considered as the d-dimensional conformal group, SO(d, 2) has a natural 3-grading with respect to the generator D of dilatations, whose eigenvalues determine the conformal dimensions of operators and states. We shall denote the corresponding 3-graded decomposition of so(d, 2) as which symbolically satisfy The subalgebra so(d − 1, 1) in N 0 represents the Lorentz algebra in d dimensions and grade +1 and −1 subspaces are spanned by translation and special conformal generators, respectively. The noncompact dilatation generator so(1, 1) D is given by The Lorentz group generators M µν (µ, ν = 0, . . . , d − 1) are given by and satisfy the commutation relations: where η µν = diag(−, +, . . . , +). The translation generators P µ (µ = 0, . . . , d − 1) are given by and the special conformal generators K µ (µ = 0, . . . , d − 1) are given by (4.7) They satisfy the commutation relations: Finally we note that the Poincaré mass operator in d dimensions vanishes identically for the minimal unitary realization given above. Therefore the minimal unitary representation of SO(d, 2) corresponds to a massless representation in d dimensions. In addition we have η µν K µ K ν = 0 . (4.10) 5 Compact 3-grading of so(d, 2) with respect to the subalgebra so(d)⊕so (2) The Lie algebra so(d, 2) has a 3-grading: with respect to its maximal compact subalgebra C 0 = so(d) ⊕ so (2), determined by the u(1) generator The operators in various grade subspaces satisfy The so(d) generators M M N (M, N = 1, . . . , d) in grade 0 subspace C 0 are given by and satisfy the commutation relations We shall label the d operators that belong to grade +1 subspace C + as B † M (M = 1, . . . , d) where These operators in grade +1 subspace C + satisfy the following important relation: which corresponds to the masslessness condition in the noncompact picture. Similarly, we shall label the d operators that belong to grade The commutation relations of the SO(d, 2) generators in this compact basis can be listed as: The operators in the subspace C + given in equation (5.6) are the Hermitian conjugates of those in the subspace C − given in equation (5.8). The so (2) generator H in the subspace C 0 is simply the conformal Hamiltonian or the AdS energy. In terms of the generators in noncompact 3-grading of so(d, 2) this conformal Hamiltonian (or the AdS energy operator) is simply 1 2 (P 0 + K 0 ).
6 Distinguished SU(1, 1) K subgroup of SO(d, 2) generated by the isotonic (singular) oscillators The u(1) generator H, given in equation (5.2), can be expressed in terms of the a-type bosonic oscillators, singlet coordinate x, and its conjugate momentum p in the following form: As pointed out earlier, this u(1) generator H is the d-dimensional conformal Hamiltonian or the (d + 1)-dimensional AdS energy operator, depending on whether one is treating SO(d, 2) as the d-dimensional conformal group or as the (d + 1)-dimensional AdS group.
The part H a is simply 1/2 the Hamiltonian of standard bosonic oscillators a i and the part H ⊙ is 1/2 the Hamiltonian of a singular harmonic oscillator with a potential function where G was given in equation (3.18). We note that H ⊙ corresponds to the Hamiltonian in the conformal quantum mechanics of [27], with the operator G replacing the "coupling constant" [24]. It also appears in this form in the Hamiltonian of the Calogero models [28,29] as well as in isotonic singular oscillator models [30,31].
Since the singlet coordinate x and its conjugate momentum p enter the minrep as in conformal quantum mechanics, we shall use the coordinate (Schrödinger) picture for the states that form the basis of an irrep of the subgroup SU (1, 1) K . Consider now this conformal quantum mechanics Hamiltonian: Together with the generators B † d−1 and B † d it generates the distinguished SU (1, 1) K subalgebra. In the compact 3-grading determined by H ⊙ , the noncompact generators take the form and satisfy the commutation relations: Now we consider the Fock space of the a-type oscillators spanned by the states of the form where n i are non-negative integers and the vacuum state |0 is annihilated by all bosonic annihilation operators a i : The state(s) with the lowest H ⊙ eigenvalue are obtained by taking the tensor product of states |Λ g with the lowest eigenvalue g of G and the lowest weight vector of SU (1, 1) K determined by g [32]: where C 0 is a normalization constant and These states satisfy The Hermiticity of H ⊙ requires that and the normalizability of the lowest weight vector requires For the minrep of SO(d, 2) given earlier, the lowest possible value of g is which occurs when |Λ g is simply the Fock vacuum |0 of a-type oscillators. For this g we have two possible values of α g , namely (5 − d) /2 and (d − 3) /2. For all d ≥ 6, the state with α g = (5 − d) /2 is non-normalizable. For d = 4, α g = (5 − d) /2 produces the same result as α g = (d − 3) /2. For d = 5, even though the state with α g = (5 − d) /2 is normalizable, it leads to non-normalizable states under the action of SO(5) when SU (1, 1) K is extended to SO(5, 2) [4]. Therefore we need to choose The corresponding tensor product state The higher eigenstates of H ⊙ can be obtained from this "ground state" by repeatedly acting on it with the raising operator B † ⊙ : where C n are normalization constants. They satisfy The states | ψ  (2) which we will refer to as the lowest (energy) K-type. For physical applications it is more convenient to use the labels of the lowest K-type to uniquely designate an irreducible unitary lowest weight representation. Therefore we will label the lowest (energy) K-type as Therefore the lowest K-type of the minrep of SO(d, 2) is the SO(d) singlet state |ψ αg 0 , which will be denotes as All the other states of the particle basis of the minrep with higher energies can be obtained from the lowest K-type by acting on it repeatedly with the operators B † 1 , . . . , B † d in the subspace C + of so(d, 2) given in equation (5.6): 3 All the states that belong to a given energy level form a single irrep of SO(d). In Table  1   States   In the previous section, we studied the minrep of SO(d, 2) using quasiconformal techniques and showed that it describes a massless scalar conformal field in d dimensions. In this section we shall describe how to obtain all possible "deformations" of the minrep of SO(d, 2) by extending the little group of massless states SO(d − 2) L generated by the "orbital" operators L ij by the addition of "spin operators" S ij : We shall denote the subgroup generated by J ij as SO(d − 2) J and its quadratic Casimir as J 2 : where S 2 denotes the quadratic Casimir of so(d − 2) S generated by S ij and L · S = L ij S ij . With the inclusion of spin terms, one can write a general Ansatz for K + , which is invariant under SO(d − 2) J , and impose the constraints due to Jacobi identities. Then one finds that all the Jacobi identities are satisfied if K + has the form Furthermore one finds that replacement of so(d − 2) L by so(d − 2) J does not affect the generators M ±,0 and ∆ in grade 0 subspace, U i and U † i in grade −1 subspace, and K − in grade −2 subspace of so(d, 2). However it leads to modifications of the grade +1 generators W i and W † i involving S ij , according to equation (3.19): Jacobi identities require that the spin generators S ij satisfy the constraint: Remarkably, this constraint (8.5) is precisely the condition that must be satisfied by the little group generators S ij of massless representations of the Poincaré group in d dimensions that extend to the unitary irreducible representations of the conformal group SO(d, 2) [33,34]. That the minrep corresponds to a massless conformal scalar field and its deformations describe higher spin massless conformal fields in d = 4, 5 and 6 dimensions were established in our previous work [1][2][3][4]. The above analysis shows that this is true in all spacetime dimensions, namely there is a one-to-one correspondence between massless conformal fields in d-dimensional Minkowskian spacetimes and the minrep of SO(d, 2) and its deformations. It is interesting to note that each such deformation corresponds to a particular realization of the Calogero model or the conformal quantum mechanics with the distinguished SU (1, 1) K subgroup acting as its symmetry group.
To summarize, of all the generators of so(d, 2): only K + and W i , W † i and J ij involve the spin operators S ij . We shall now give the quadratic Casimir of this deformed minrep of so(d, 2). Noting that the quadratic Casimir of the deformed so(d, 2) takes the form Note that it reduces to the quadratic Casimir of the undeformed minrep of so(d, 2) when S 2 = 0. In our previous work the generators of spin operators S ij were realized as bilinears of fermionic oscillators. For conformal groups in d = 3, 4, 5 and 6 dimensions this is natural since they admit extensions to simple superalgebras OSp(N | 4, R), SU (2, 2 | N ), F (4) and OSp(8 * | 2N ), respectively. Deformations of the minrep can then be fitted into unitary supermultiplets of the corresponding superalgebras. For SU (2, 2 | N ) and OSp(8 * | 2N ) the corresponding supermultiplets turn out to be doubleton supermultiplets that were constructed and studied long time ago [5,8]. Over the field of real and complex numbers there do not exist any simple conformal superalgebras beyond six dimensions that obey the usual spin and statistics connection and do not involve any extra tensorial generators [35]. The construction of the representations of compact orthogonal groups over the Fock spaces of fermionic oscillators is reviewed in appendices A and B. We shall discuss the application of this construction to the spin operators S ij and the study of the relevant representations that satisfy (8.5) in odd and even dimensions separately.

Deformation in odd dimensions
In our previous work on five-dimensional conformal algebra [4], we showed that SO(5, 2) admits only two singleton representations, which are the analogs of Dirac singletons in three dimensions, corresponding to five-dimensional massless scalar and spinorial conformal fields. The minrep of SO(5, 2) is the scalar singleton representation and the unique "deformation" of the minrep, labelled by the spin 1/2 of the little group SO(3), is the spinorial singleton representation. As discussed in appendix A, this property extends to all odd-dimensional conformal algebras so(d, 2) -the minrep of SO(d, 2) for odd d admits a single nontrivial "deformation". Therefore we shall introduce a "deformation parameter" ǫ = 0 or 1, so that the "undeformed" minrep corresponds to ǫ = 0, and the "deformed" minrep corresponds to ǫ = 1. As shown in appendix A, the generators S ij of SO(d − 2) S describing this unique deformation can be realized in terms of a single set of fermionic oscillators transforming in the fundamental representation of u((d − 3)/2) subalgebra of so(d − 2) S for odd d. More specifically the Lie algebra so(d − 2) S has a 5-grading with respect to its subalgebra u((d − 3)/2): and its generators can be realized in terms of fermionic oscillators as follows: where r, s, · · · = 1, 2, . . . (d − 3)/2. Denoting the Fock vacuum as |0 F , such that one finds that the entire fermionic Fock space spanned by the states 4 The spin operators S ij can be recast in terms of 2 n × 2 n (n = (d − 3)/2) Euclidean Dirac gamma matrices γ i as In odd dimensions, the "coupling constant" G of the isotonic (singular) oscillator, which appears in the generator K + , can be written in the form where we have used the value of the quadratic Casimir S 2 from equation (A.7). The parameter ǫ = 0 for the true minrep and ǫ = 1 for the unique deformation.
Let us now give the K-type decomposition of the unique "deformation" of the minrep of SO(d, 2) for all odd d. Consider the tensor product space of the Fock spaces of the atype bosonic oscillators, the ξ-type fermionic oscillators, and the state space of the singular oscillator. Let |0 be the vacuum state annihilated by both a i and ξ r : With respect to the isotonic "coupling constant" G in equation (8.14), this vacuum state has the eigenvalue Therefore, the state of the form is annihilated by all grade −1 operators in C − of so(d, 2), if α g(ǫ) satisfies: More specifically we have The state Ω (0) is the lowest weight vector of the true minrep and is a singlet of SO(d) which we denoted as |ψ αg 0 in section 6. By acting on the state Ω (1) with all the generators of SO(d), one obtains a set of states which we will denote collectively as |Ψ , transforming in the 2 ( d−1 2 ) -dimensional spinor representation of SO(d), with the Dynkin labels (0, 0, . . . , 0, 1). They are eigenstates of the AdS (d+1) energy operator H with the eigenvalue Since the states |Ψ are annihilated by all the operators in C − of so(d, 2), the deformed minrep of SO(d, 2) is also a unitary lowest weight representation, and the states |Ψ α g(1) σ form the lowest K-type of the spinorial singleton. All the other states of the "particle basis" of the spinorial singleton are obtained from the set of states |Ψ α g(1) σ by repeatedly acting with the deformed grade +1 operators in the C + subspace of SO(d, 2): Table 2, we give the K-type decomposition of the unique deformed minrep (spinorial singleton) of SO(d, 2), for odd d. The deformed minrep (ǫ = 1) describes a massless spinor field in d dimensions, and together with the true minrep (ǫ = 0), which describes a massless scalar field, they exhaust the list of conformally massless fields in odd-dimensional spacetimes [33].  States

Deformations in even dimensions
As stated above, in four and six dimensions the minrep of the conformal group admits an infinite set of deformations which correspond to higher spin massless conformal fields [1][2][3].
In four dimensions there exist a continuous infinity of massless conformal fields labelled by helicity. Similarly in six dimensions there exist a discrete infinity of massless conformal fields labelled by the spin of an SU (2) subgroup of the little group SO(4) of massless particles. Let us now show that the minrep of the even conformal group SO(d, 2) admits an infinite set of deformations for all d > 6 as well.
In Appendix B we review the construction of the representations of SO(2n) over the Fock spaces of an arbitrary number of fermionic oscillators transforming in the fundamental representation of the u(n) subalgebra of so(2n) and present their Gelfand-Zetlin as well as Dynkin labels following [36]. We shall apply this construction to the representations of the little group SO(d − 2) S of massless particles, generated by S ij . Using the 3-grading of the Lie algebra so(d − 2) with respect to its subalgebra u((d − 2)/2): one realizes its generators as bilinears of fermionic oscillators as follows: where r, s, · · · = 1, 2, . . . , (d − 2)/2 ; ǫ = 0, 1 and α r · β s = P K=1 α r (K)β s (K) . The expressions of the generators S ij in terms of the above bilinears are given in appendix B (equation (B.3)). The representations of even orthogonal groups SO(d − 2) that satisfy the constraint (8.5) have the Dynkin labels (0, . . . , 0, 0, f ) D or (0, . . . , 0, f, 0) D , where f is a non-negative integer [33,34]. These irreducible representations have the Gelfand-Zetlin labels where f = 2P + ǫ is the number of colors of fermionic oscillators transforming in the fundamental representation of the subgroup U ((d − 2)/2). For d = 4k + 2, where k is a positive integer, these massless irreps have the following lowest weight vectors in the fermionic Fock space: and (r 1 , s 1 , . . . , t) denotes complete symmetrization of indices. 5 For d = 4k, where k is a positive integer, the roles of the lowest weight vectors above are reversed: Since f = 2P + ε, where P = 0, 1, . . . and ε = 0, 1, we have an infinite set of deformations of the minrep that are in one-to-one correspondence with the conformally massless representations of SO(d, 2) for even d, that remain irreducible under reduction to the Poincaré group in d dimensions.
One finds that the quadratic Casimir S 2 have the eigenvalues for the massless irreps of SO(d − 2) in even dimensions in agreement with [33,34]. Note that these eigenvalues depend only on f and enter into the "coupling constant" G that appears in the generator K + : We shall now give the K-type decomposition of the deformations of the minrep of SO(d, 2) for all even d. Let |0 F be the vacuum of the Fock space of fermionic oscillators: where r = 1, . . . , n = 1, . . . , d−2 2 and K = 1, . . . , P . By acting on this vacuum state with the corresponding fermionic creation operators α r (K) † , β r (K) † , ξ † r , one can obtain a basis for this 2 f ( d−2 2 ) -dimensional Fock space. Now consider the tensor product space of the Fock spaces of the a-type bosonic oscillators, α-type, β-type and ξ-type fermionic oscillators, and the state space of the singular oscillator. Let |0 be the tensor product of the vacuum state |0 B annihilated by all a i and the fermionic Fock vacuum |0 F : With respect to the isotonic "coupling constant" G in equation (8.27), the vacuum state has the eigenvalue Therefore, the state Ω (f ) (0) = x α g(f ) e −x 2 /2 |0 (8.31) and the state  2). These eigenvalues turn out to be equal for the two representations: In Tables 3 and 4, we give the lowest K-type of all the unitary lowest weight representations of SO(d, 2) for even d and their K-type decompositions. Table 3: K-type decomposition of the deformed minimal unitary representation of SO(d, 2), for even d, obtained from the lowest weight vector Ω (f ) (0) = | E 0 (f ) , (0, . . . , 0, 0, f ) . The AdS energy and the Dynkin labels of SO(d) irrep at each level are given. States irrep  irrep 9 AdS (d+1) /CF T d bosonic higher spin algebra Before constructing the higher spin algebra in d dimensions, it is useful to write down the so(d, 2) generators in the "canonical basis", that we denote as M AB (A, B = 0, 1, . . . , d+1).
In terms of the generators in the non-compact three grading D, M µν , P µ , K µ (µ, ν = 0, . . . , d − 1), the generators M AB can be expressed as: The AdS (d+1) /CF T d higher spin algebra of Fradkin-Vasiliev type simply corresponds to the universal enveloping algebra of SO(d, 2), quotiented by its Joseph ideal [4, 13-15, 18, 19, 37]. We recall that the universal enveloping algebra U (g) of a Lie algebra g is defined as the quotient: where G is the associative algebra freely generated by elements of g, and L is the ideal of G generated by elements of form denotes the commutator. The Joseph ideal of a Lie algebra g is a two-sided ideal inside its universal enveloping algebra that annihilates its minrep. We shall denote the corresponding higher spin algebra as hs(d, 2): where U (d, 2) is the universal enveloping algebra of so(d, 2) and J (d, 2) is its Joseph ideal. An explicit formula for the generators of the Joseph ideal of SO(d, 2) was given by Eastwood in [37]: In the above expression, the symbol · denotes the symmetric product of two generators of so(d, 2): the symbol ⊚ denotes the Cartan product of two generators of so(d, 2) [38]: where η AB is the SO(d, 2) invariant metric. The decomposition of the enveloping algebra U (d, 2) under the adjoint action of so(d, 2) is equivalent to computing symmetric products of the adjoint representation of so(d, 2) whose Young tableau is M AB ∼ . (9.10) The symmetric tensor product of the adjoint representation of SO(d, 2) decomposes as follows: where • represents the quadratic Casimir of SO(d, 2). As was pointed out by Vasiliev in [18], the higher spin gauge fields in AdS (d+1) are described by traceless two-row Young tableaux: · · · · · · n boxes (9.12) Therefore one has to remove all the representations on the right hand side of equation (9.11) except for the window diagram. This is precisely what one achieves by quotienting the enveloping algebra by the Joseph ideal. After quotienting the enveloping algebra by the Joseph ideal, the generators of the infinite higher spin algebra decompose under the finite dimensional subalgebra so(d, 2) as ⊕ ⊕ ... ⊕ · · · · · · ⊕ . . . (9.13) For the minimal unitary realization of SO(d, 2) obtained via the quasiconformal methods, the generators J ABCD of the Joseph ideal vanish identically as operators. This was shown to be the case for d = 3, 4, 5, 6 dimensions earlier [4,13,14]. We have verified that this is true in all dimensions. Therefore the enveloping algebra of the minimal unitary realization of SO(d, 2) obtained by quantizing its quasiconformal realization leads directly to the bosonic higher spin AdS (d+1) /Conf d algebra for all d.
As we established above in section 8, the minimal unitary representation of SO(d, 2) admits deformations, which are in one-to-one correspondence with the unitary irreducible representations that describe conformally massless fields in d-dimensional Minkowskian spacetimes. As was done in d = 4, 5, 6 dimensions, we shall define deformations of the bosonic higher spin algebras as enveloping algebras of the deformed minreps. For the deformed minreps, the representation in the symmetric tensor product of the adjoint representation does not vanish, while the representation still vanishes: η CD M AC · M DB = 0 (9.14) Therefore the gauge fields of deformed higher spin algebras will not consist only of those that correspond to traceless two-row Young tableaux. The Young tableaux of the representations of higher spin gauge fields defined by the deformed higher spin algebras will be given by the symmetric tensor products of the adjoint of SO(d, 2) subject to the constraint: To understand how the Joseph ideal gets deformed when going from the minrep to a deformed minrep, one decomposes the SO(d, 2)-covariant generators J ABCD of the Joseph ideal with respect to the SO(d − 1, 1) (Lorentz) subgroup.
First we have the lightlike conditions for momentum and special conformal generators: which hold for both the minrep and its deformations. The generators of J ABCD that are quadratic in the generators of SO(d, 2) and vanish for the minrep are as follows: In the above equations, symmetrizations (round brackets) and anti-symmetrizations (square brackets) are done with weight one. However, we should stress that the dot product of two operators A and B is defined as A · B = AB + BA. When the expressions for the generators of the deformed minrep of SO(d, 2), which involve fermionic oscillators, are substituted in the above identities only the first three identities are satisfied and the remaining five identities get modified by "spin"-dependent terms involving fermionic oscillators. For example, the fourth identity becomes: where S 2 is the Casimir of SO(d− 2) S . There are similar changes in the last four identities.

Conclusions
Extending our earlier work we have shown that there is a one-to-one correspondence between the minimal unitary representation of SO(d, 2) and its deformations and conformally massless fields in d-dimensional Minkowskian spacetimes. The minrep describes a massless conformal scalar field, while the deformations describe massless conformal fields that transform nontrivially under the little group SO(d − 2) in d-dimensions. Joseph ideal generators vanish identically as operators for the minrep obtained via the quasiconformal methods.
Similarly for the deformed minreps certain deformations of the Joseph ideal generators involving the spin terms vanish as operators.
Again extending earlier work, we have shown how the enveloping algebras of the deformed minreps lead directly to deformations of the standard bosonic higher spin algebras in all dimensions. These deformed higher spin algebras are defined as the quotient of the universal enveloping algebras of SO(d, 2) by the respective deformed Joseph ideals. Consequently, the higher spin algebras are in one-to-one correspondence with the conformally massless unitary representations of SO(d, 2) in d dimensional Minkowskian spacetimes.
In dimensions d ≤ 6, the minimal unitary representation of the conformal group SO(d, 2) and its deformations can be fitted into the minimal unitary supermultiplet of a Lie superalgebra that extends the Lie algebra of SO(d, 2) and its deformed supermultiplets. For example, the minimal unitary supermultiplet of P SU (2, 2|4) is the N = 4 Yang-Mills multiplet, and the minimal unitary supermultiplet of OSp(8 * |4) is the (2, 0) tensor multiplet in six dimensions. The enveloping algebras of the minreps of the conformal Lie superalgebras in d ≤ 6 and deformations thereof define the higher spin superalgebras and their deformations in the respective dimensions. Since no simple Lie superalgebras over the real or complex field, that extend the conformal algebra so(d, 2) by a compact R-symmetry group [35], exist beyond six dimensions, we do not expect the minrep and its deformations beyond six dimensions to fit into unitary representations of a simple Lie superalgebra that obeys the usual spin and statistics connection.
It was shown in [39] that the three dimensional conformal theories with a unique stress energy tensor and a conserved higher spin current that are dual to Vasiliev type higher spin theories in AdS 4 are free. As was pointed out in [13,14], since the minrep and its deformations obtained via the quasiconformal approach are isomorphic to doubleton representations in d = 4 and d = 6 which are obtained by realizing the generators as bilinears of covariant twistorial oscillators, the conformal field theories that are dual to Vasiliev type theories must be free or interacting but integrable. Since there is a one-to-one correspondence between the massless conformal fields and higher spin algebras and their deformations in all dimensions we expect such a duality to hold for any d. The latter possibility of a duality with an interacting and integrable conformal field theory is suggested by the fact that quasiconformal realizations of the minrep and its deformations are non-linear, involving cubic and quartic terms for d > 3. 6 A concrete result in support of this comes from the results of [40] where the quasiconformal construction of the minrep of D(2, 1; α) and its deformations were shown to describe the spectra of N = 8 supersymmetric interacting quantum mechanical models that were studied using harmonic superspace techniques in [41]. Furthermore the quasiconformal realizations of minreps and their deformations all have a distinguished SL(2, R) subalgebra which is of the form that comes up in Calogero models or conformal quantum mechanical models which are integrable. This suggests the possibility that an interacting but integrable conformal field theory that is dual to an higher spin theory may be a dimensionally reduced CFT.
generators of so(2n + 1) in the 5-graded decomposition were given in [42]: where ε = 0, 1 and the 'dot product' denotes summation over all pairs of oscillators labelled by the "color" or "family" index K, e.g. α r · β s = P K=1 α r (K)β s (K). Hence we have a realization of SO(2n + 1) in terms of f = 2P + ε sets of fermionic creation and annihilation operators transforming in the fundamental and anti-fundamental representation of U (n), respectively. They satisfy the canonical anticommutation relations In the Fock space of all fermionic oscillators, the irreps of SO(2n + 1) = SO(d − 2) are uniquely determined by a set of states, that transforms irreducibly under the subgroup U (n) and are annihilated by the grade −1/2 generators Y r , which we shall refer to as the "lowest weight vector" by an abuse of terminology since they contain a true lowest weight vector.
As was proven in [34] for odd orthogonal groups SO(d − 2) the only irreducible nontrivial representation that satisfies the condition (8.5) ∆ ij = S ik S jk + S jk S ik − 2 (2n + 1) δ ij S 2 = 0 is the 2 n -dimensional spinor representation with 2n + 1 = d − 2. This is simply the irrep obtained by the singletonic realization of SO(2n + 1) in terms of a single set of oscillators, i.e. P = 0 and ε = 1 with the Fock vacuum |0 as the lowest weight vector. The full Fock space of n fermionic oscillators ξ † r form the 2 n -dimensional irreducible spinor representation of SO(2n + 1). Equivalently the action of the generators S ij on the entire Fock space can be represented in terms of 2 n × 2 n gamma matrices γ i , which satisfy the Clifford algebra  Table 5: The U (n) Young-tableau label of a lowest U (n) representation and the Gelfand-Zetlin and Dynkin labels of the corresponding irrep of SO(2n) for even n. The integer f = 2P + ε (P = 1, 2, . . . and ε = 0, 1) designates the number of colors of the fermionic oscillators transforming in the fundamental representation of the U (n) subgroup.
U (n) Young-tableau [l 1 , l 2 , . . . , l n ] Y T of the lowest U (n) rep Gelfand-Zetlin labels  Table 6: The U (n) Young-tableau label of a lowest U (n) representation and the Gelfand-Zetlin and Dynkin labels of the corresponding irrep of SO(2n) for odd n. The integer f = 2P +ε (P = 1, 2, . . . and ε = 0, 1) designates the number of colors of the fermionic oscillators transforming in the fundamental representation of the U (n) subgroup.