Minimal unitary representation of 5d superconformal algebra F(4) and AdS_6/CFT_5 higher spin (super)-algebras

We study the minimal unitary representation (minrep) of SO(5,2), obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO(5,2) describes a massless conformal scalar field in five dimensions and admits a unique"deformation"which describes a massless conformal spinor. Scalar and spinor minreps of SO(5,2) are the 5d analogs of Dirac's singletons of SO(3,2). We then construct the minimal unitary representation of the unique 5d superconformal algebra F(4) with the even subalgebra SO(5,2) X SU(2). The minrep of F(4) describes a massless conformal supermultiplet consisting of two scalar and one spinor fields. We then extend our results to the construction of higher spin AdS_6/CFT_5 (super)-algebras. The Joseph ideal of the minrep of SO(5,2) vanishes identically as operators and hence its enveloping algebra yields the AdS_6/CFT_5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a"deformed"higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS_6/CFT_5 superalgebra as the enveloping algebra of the minimal unitary realization of F(4) obtained by the quasiconformal methods.

13 AdS 6 /CF T 5 bosonic higher spin algebra and its deformation 31 14 Unique AdS 6 /CF T 5 higher spin superalgebra as enveloping algebra of the minimal unitary representation of F (4) 35

Comments 37
A Relations between the generators in noncompact 3-grading and compact 3-grading 38 B The "intertwiner" between compact and non-compact 3-graded bases of SO(5, 2) 38 We then study the possible deformations of the minrep and find only a single deformation corresponding to a conformally massless spinor field in five dimensions. This is similar to the situation in three dimensions, where the only conformally massless fields are a scalar and a spinor field corresponding to Dirac's singletons. We then study the minimal unitary realization of the unique simple conformal supergroup F (4) with the even subgroup SO(5, 2) × SU (2). We find that the spinor minrep, together with two copies of the scalar minrep of SO (5,2), form the minimal unitary supermultiplet of F (4). We then extend these results to the construction of AdS 6 /CF T 5 algebras and superalgebra. We show that the Joseph ideal vanishes identically as operators for the minrep of SO(5, 2) and its enveloping algebra yields the bosonic higher spin algebra of Vasiliev-type in AdS 6 . For the deformed minrep a certain deformation of the Joseph ideal vanishes and its enveloping algebra yields a deformed higher spin algebra. The enveloping algebra of the superalgebra F (4) yields the unique higher spin superalgebra in AdS 6 1 .
The plan of the paper is as follows. In section 2, we construct the geometric realization of SO(5, 2) as a quasiconformal group, following the method outlined in [17]. Then in section 3, we obtain the minimal unitary representation of SO(5, 2) via the quantization of the geometric quasiconformal action and show that there is a two-parameter family of degree two polynomials of so (5,2) generators that reduces to a c-number according to Joseph's theorem [18]. We present the 3-grading of so (5,2) with respect to the noncompact subalgebra so(4, 1) ⊕ so (1,1) in section 4, and with respect to the compact subalgebra so(5)⊕so(2) ≈ ups(4)⊕u(1) in section 5. In that section, the results are presented separately in SO(5)-covariant and USp(4)-covariant forms. Then in section 6 we discuss the properties of a distinguished SU (1, 1) subgroup of SO(5, 2) generated by singular (isotonic) oscillators. We then give the K-type decomposition of the minrep of SO(5, 2) in section 7 and show that it corresponds to a massless conformal scalar field in five dimensions. We study the deformations of the minrep in section 8 and show that it admits a unique deformation and give its K-type decomposition in section 9. In section 10, we construct the minimal unitary representation of the unique exceptional superconformal algebra f(4) in five dimensions. A 3-grading of f(4) with respect to the compact subsuperalgebra osp(2|4) ⊕ u(1) is given in section 11, and the supermultiplet of conformal fields corresponding to the minrep of f(4) is given in section 12. Then in section 13, we show that the Joseph ideal vanishes identically as an operator for the minrep of SO(5, 2) and hence its universal enveloping algebra yields directly the bosonic AdS 6 /CF T 5 higher spin algebra. Similarly for the deformed minrep of SO(5, 2) a certain deformation of the Joseph ideal vanishes identically and its enveloping algebra yields a deformed AdS 6 /CF T 5 higher spin algebra. Finally in section 14 we define the unique AdS 6 /CF T 5 higher spin superalgebra as the enveloping algebra of the minrep of F (4) and study some of its properties followed by some concluding comments.

Geometric realization of SO(5, 2) as a quasiconformal group
In this section we shall review the geometric quasiconformal realization of SO(5, 2) following [17]. The Lie algebra so(5, 2) admits the following 5-grading with respect to its subalgebra so(1, 1) ⊕ so(3) ⊕ sp(2, R): where ∆ is the SO(1, 1) generator that determines the 5-grading. The generators of SO (5,2) can be realized as nonlinear differential operators acting on a seven-dimensional space T , whose coordinates we shall denote as X = X i,a , x , where X i,a transform in the (3,2) representation of SU (2) × Sp(2, R) subalgebra, with i = 1, 2, 3 and a = 1, 2, and x is a singlet coordinate. There exists a quartic polynomial of the coordinates X i,a which is an invariant of SU (2) × Sp(2, R) subgroup, where ǫ ab is the symplectic invariant tensor of Sp(2, R) and η ij is the invariant metric of SU (2) in the adjoint representation, which we choose as η ij = −δ ij to agree with the general conventions of [17]. The generators belonging to various grade subspaces will be labelled as follows: where M ij and J ab are the generators of SU (2) and Sp(2, R) subgroups, respectively. In the nonlinear quasiconformal action of SO(5, 2) they are realized as where ǫ ab is the inverse symplectic tensor, such that ǫ ab ǫ bc = δ a c . Substituting the expression for the quartic invariant, one finds the explicit form of the grade +1 generators: These SO(5, 2) generators satisfy the following commutation relations: The quartic norm (length) of a vector X = X i,a , x ∈ T is defined as To see the geometric picture behind the above nonlinear realization, one defines a quartic distance function between any two points X and Y in the seven-dimensional space T as where the "symplectic" difference δ (X , Y) is defined as The lightlike separations between any two points with respect to the quartic distance function are left invariant under the quasiconformal action of SO (5,2). In other words, SO(5, 2) acts as the invariance group of a "light-cone" with respect to a quartic distance function in a seven dimensional space.

Minimal unitary representation of SO(5, 2)
The quantization of the geometric quasiconformal action of a Lie algebra or a Lie superalgebra leads to its minimal unitary realization. For the case of SO(5, 2) this is achieved by splitting the six variables X i,a introduced above into three coordinates X i and three momenta P i , and introducing a momentum p conjugate to the singlet coordinate x: These coordinates and momenta are then treated as quantum mechanical operators satisfying the canonical commutation relations: In the realization that follows, we shall use bosonic oscillators a i and their hermitian conjugates a † i defined in terms of X i and P i as follows: They satisfy the commutation relations: We shall first give the generators of the minimal unitary realization of so(5, 2) in the 5-graded basis: where the generator that defines the 5-grading is simply The single generator in grade −2 subspace is realized in terms of the singlet coordinate x as and the six generators in grade −1 subspace are realized as bilinears of x and the bosonic oscillators a i , a † i as Grade −2 and grade −1 generators form a Heisenberg subalgebra with the generator K − playing the role of the central charge. The generators of su(1, 1) ⊂ g (0) are realized as bilinears of the a-type bosonic oscillators as which satisfy the commutation relations We denote this subalgebra as su(1, 1) M and its quadratic Casimir as M 2 : The su(2) subalgebra, denoted as su(2) L , of grade 0 subspace is also realized as bilinears of the a-type bosonic oscillators as and Σ i are the 3 × 3 adjoint matrices of SU (2) given by They satisfy the commutation relations The quadratic Casimir of su(2) L , denoted as L 2 , is related to that of su(1, 1) M as Upon quantization, the quartic invariant I 4 of SU (2) L × SU (1, 1) M goes over to a linear function of the quadratic Casimir of SU (2) L × SU (1, 1) M . As a consequence one finds that grade +2 generator depends only on the quadratic Casimir of su(2) L (or that of su(1, 1) M ) and can be written as follows: Now the six generators in grade +1 subspace can be obtained by taking the commutators between the respective grade −1 generators and K + : Evaluating the commutators one finds Once again, these grade +2 and grade +1 generators form a Heisenberg algebra with the generator K + playing the role of the central charge. Grade −2 and grade +1 generators close into grade −1 subspace: Grade ±2 generators, together with the generator ∆ from grade 0 subspace, form a subalgebra su(1, 1), which we shall denote as su(1, 1) K : The quadratic Casimir of su(1, 1) K subalgebra, given by is related to the quadratic Casimir of su(2) L (and that of su(1, 1) M ) as The commutators of grade −1 (grade +1) generators with those of SU (2) L ×SU (1, 1) M are given below: The commutators between grade −1 generators and grade +1 generators can be written in terms of grade 0 generators as follows: Finally, we present the quadratic Casimir of so (5,2). Noting that the following combination of bilinears, formed in terms of the generators in grade ±1 subspaces, reduces to the quadratic Casimir of su(1, 1) K modulo some additive and multiplicative constants, one can show that there exists a two-parameter family of degree two polynomials of so (5,2) generators that reduces to a c-number for the minimal unitary realization, according to Joseph's theorem [18,19]: The quadratic Casimir of so(5, 2) corresponds to k 1 = 2 and k 2 = − 1 2 : 4 Noncompact 3-grading of so(5, 2) with respect to the subalgebra so(4, 1)⊕ so(1, 1) ≈ usp(2, 2) ⊕ so(1, 1) We should note that when one goes to the covering group Spin(5, 2) of SO(5, 2), the subgroups SO(5) and SO(4, 1) go over to their covering groups USp(4) and USp(2, 2), respectively. Considered as the five-dimensional conformal group, SO(5, 2) has a natural 3-grading defined by the generator D of dilatations whose eigenvalues determine the conformal dimensions of operators and states. Let us denote the corresponding 3-graded decomposition of so(5, 2) as where N 0 = so(4, 1)⊕so(1, 1) D , with the subalgebra so (4,1) in N 0 representing the Lorentz algebra in five dimensions. The noncompact dilatation generator so(1, 1) D is given by and the generators belonging to N ± and N 0 subspaces are as follows: The Lorentz group generators M µν (µ, ν = 0, 1, 2, 3, 4) are given by and satisfy the commutation relations of so(4, 1): where η µν = diag(−, +, +, +, +). The rotation group SO(4) splits into two SU (2) subgroups, denoted by SU (2) A and SU (2)Å, whose generators are given by satisfying the commutation relations The translation generators P µ (µ = 0, 1, 2, 3, 4) of the conformal group SO(5, 2) are given by and the special conformal generators K µ (µ = 0, 1, 2, 3, 4) are given by (4.9) These generators satisfy the commutation relations of SO(5, 2) as the five-dimensional conformal algebra: We should note that the Poincaré mass operator in five dimensions vanishes identically for the minimal unitary realization.
In appendix A, we present the relation between the generators M AB (A, B = 0, . . . , 6) of SO(5, 2) and the generators in the noncompact three-grading.
5 Compact 3-grading of so(5, 2) with respect to the subalgebra so(5) ⊕ so(2) ≈ usp(4) ⊕ u(1) The Lie algebra so(5, 2) has a 3-grading, with respect to its maximal compact subalgebra and satisfy H , In this decomposition, the generators belonging to C ± and C 0 subspaces are as follows: In the above 3-grading, the operators that belong to C + subspace are the Hermitian conjugates of the operators that belong to C − subspace. In the corresponding minimal unitary realization, one takes only the hermitian linear combinations of these operators as generators of so(5, 2). The generator H is the conformal Hamiltonian or the AdS energy, depending on whether one is considering SO(5, 2) as the five-dimensional conformal group or as the six-dimensional AdS group. We shall refer to this grading as the compact 3grading. We should also note that, in the earlier noncompact 3-grading of so(5, 2) with respect to N 0 = so(4, 1) ⊕ so(1, 1) D , this AdS energy corresponds to 1 2 (P 0 + K 0 ).

SO(5)-covariant basis
The so(5) generators M M N (M, N = 1, 2, 3, 4, 5) in grade zero subspace C 0 are given by and satisfy the commutation relations (4). We shall label the five operators that belong to grade +1 subspace C + as B † M (M = 1, 2, 3, 4, 5) where These C + operators satisfy the following important relation: which corresponds to the masslessness condition in the noncompact picture. We shall label the five operators that belong to grade −1 subspace C − , which are the hermitian conjugates of those in C + , as B M (M = 1, 2, 3, 4, 5) where The commutation relations of the SO(5, 2) generators in this compact basis are:

USp(4)-covariant basis
In this subsection we shall reformulate the compact 3-grading of the Lie algebra so(5, 2) in the USp(4)-covariant form. As gamma-matrices (γ M ) I J in the five-dimensional Euclidean space we choose the following hermitian matrices: where I n are the n × n identity matrices. As the charge conjugation matrix, we choose the antisymmetric matrix where I, J = 1, 2, 3, 4 are the spinor indices of the covering group USp(4) of SO (5). It can be identified with the symplectic metric Ω IJ = Ω IJ of USp(4) and will be used to raise and lower spinorial indices. We find that (C  (4). We find (5.14) Note that they satisfy the symplectic traceless conditions The generators in grade 0 subspace that form the subalgebra usp(4) are realized as where M M N are the so(5) generators given in equation (5.5). They satisfy The commutation relations of these SO(5, 2) generators in this USp(4)-covariant compact basis have the following form: The constraint on grade +1 operators given in equation (5.8) becomes in the USp(4)-covariant basis.
6 Distinguished SU(1, 1) K subgroup of SO(5, 2) generated by the isotonic (singular) oscillators Note that in terms of the oscillators a i (and their respective hermitian conjugates a † i ) and the singlet coordinate x and its conjugate momentum p, the u(1) generator H, as given in equation (5.1), has the following form: This u(1) generator H is the six-dimensional AdS energy operator or the five-dimensional conformal Hamiltonian. H a (= M 0 ) is the contribution to the Hamiltonian from the a-type standard non-singular bosonic oscillators. On the other hand, H ⊙ is the Hamiltonian of a singular harmonic oscillator with a potential function This H ⊙ is exactly of the form of the Hamiltonian of conformal quantum mechanics [20] with G playing the role of the "coupling constant" [21]. In some literature it is referred to as the isotonic oscillator [22,23]. It is also of the form that appears in the Calogero models in [24,25].
Let us now consider this singular harmonic oscillator Hamiltonian The following two linear combinations of the operators B † 4 , B † 5 from the C + subspace and B 4 , B 5 from the C − subspace of so(5, 2): close into H ⊙ under commutation, and they generate the distinguished su(1, 1) K subalgebra: 2 For the positive energy unitary representations of SO(5, 2), the relevant unitary realizations of SU (1, 1) K are also of the positive energy type. Now consider the Fock space of the a-type oscillators, whose vacuum state |0 is annihilated by all a i : a i |0 = 0 (i = 1, 2, 3) (6.8) A "particle basis" in this Fock space is provided by the states of the form where n i are non-negative integers. For a given eigenvalue g of the operator G (as given in equation (6.4)), the state(s) corresponding to the lowest energy eigenvalue of H ⊙ are the superpositions of tensor product states of the form ψ Such solutions are given by [26] ψ (αg ) 0 where C 0 is a normalization constant and The Hermiticity of H ⊙ implies that g ≥ − 1 8 (6.13) and the normalizability of the state given in equation (6.11) imposes the constraint For the minrep of SO(5, 2) given earlier, the lowest possible value of g is zero, i.e. when |Λ g is simply the Fock vacuum |0 of bosonic oscillators a i (i = 1, 2, 3). For g = 0 we have two possible values of α g , namely 0 and 1. It turns out that, even though the state with α g = 0 has lower energy than that with α g = 1, it leads to non-normalizable states under the action of SO(5) when we extend SU (1, 1) K to SO(5, 2). Therefore, we choose the state ψ as the lowest energy "ground state" of H ⊙ . The higher energy eigenstates of H ⊙ can be obtained from this ground state by acting on it repeatedly with the raising operator B † ⊙ : where C n are normalization constants. They correspond to energy eigenvalues We shall denote the corresponding states as | ψ All the states of any given energy level form a single irrep of SO(5) ≈ USp(4). In Table  1, we give the decomposition of the minrep of SO(5, 2) with respect to its maximal compact subgroup by listing the dimension of SO(5) ≈ USp(4) irrep and its USp(4) Dynkin labels as well as their energies. The minrep of SO(5, 2) is the scalar singleton representation of SO(5, 2) in AdS 6 just like the Dirac scalar singleton of SO (3,2) in AdS 4 . It corresponds to a massless conformal scalar field in five-dimensional Minkowski spacetime which can be identified with the boundary of AdS 6 . Table 1: K-type decomposition of the minimal unitary representation of SO(5, 2) with the lowest weight vector |Ω = C 0 x e −x 2 /2 |0 . The AdS energy, dimension of SO(5) ≈ USp(4) irrep and its USp(4) Dynkin labels at each level are given.

States
AdS 6 Dim of Dynkin Labels Energy SO(5) ≈ USp (4) of USp (4)  The little group of massless particles in five dimensions is SU (2) and can be identified with the subgroup SU (2) L of SO(5, 2), considered as the five-dimensional conformal group. In this section we shall study possible "deformations" of the minrep of SO(5, 2), generated by adding a "spin term" S i to the "orbital" generators L i of SU (2) L . We shall first introduce the "spin" generators as bilinears of two fermionic oscillators. This option is the natural one for extending the minrep of SO(5, 2) to that of the conformal supergroup F (4) with the even subgroup SO(5, 2) × SU (2).
Consider the fermionic oscillators α r (r = 1, 2) and their hermitian conjugates α † r that satisfy the usual anti-commutation relations and define the 2-component spinor We shall realize the generators S i of su(2) S as where σ i are the Pauli matrices. The quadratic Casimir of su(2) S , denoted as S 2 , is given by Then the "orbital" generators L i of su(2) L get extended to the "total angular momentum" generators J i by adding the spin terms: The quadratic Casimir of su(2) J , denoted as J 2 , is given by where L · S = L 1 S 1 + L 2 S 2 + L 3 S 3 .
This introduction of fermionic contributions 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(5, 2), defined in section 3. To preserve Jacobi identities, one finds that the quadratic Casimir L 2 appearing in grade +2 generator K + must be replaced by 2 J 2 − L 2 + 2 3 S 2 : Therefore the "coupling constant" of the isotonic (singular) oscillator becomes and the grade +1 generators, also changing according to equation (3.20), become All the commutation relations of SO(5, 2) given in the previous sections are still valid with the above replacements. The quadratic Casimir of the resulting realization of so(5, 2) with the fermionic contributions can be evaluated easily, and one finds where we have used Thus the Casimir of the minrep, extended by fermionic contributions, depends only on the Casimir of SU (2) S generated by fermionic bilinears. The four-dimensional Fock space of the two fermionic oscillators decomposes as a doublet (spin 1/2) and two singlets under the action of SU (2) S . The singlets are the states where |0 F is the fermionic Fock vacuum. The states α † r |0 F (r = 1, 2) also form a doublet. As a consequence, the tensor product space of the Fock spaces of the bosonic oscillators a † i and the fermionic oscillators α † r with the state space of the singular oscillator deformed by the fermionic oscillators decomposes as two copies of the minrep and one copy of the deformed minrep as will be made evident in the next section. Here we should stress that increasing the number of fermionic oscillators so as to generate higher representations of SU (2) S leads to a failure of the Jacobi identities of SO(5, 2). As a consequence we find a single irreducible distinct deformation of the minrep of SO(5, 2), as in the case of SO (3,2). This result is expected in the light of the results of [27], where it was shown that there exist only two conformally massless fields in five dimensions. The 5-dimensional Poincaré mass operator vanishes identically M 2 = η µν P µ P ν = 0 (8.13) for the deformation of the minrep just as it did for the "undeformed" minrep. Thus we shall refer to the minrep as the scalar singleton and its deformation as the spinor singleton. They correspond to the conformally massless scalar and spinor fields in five dimensions. Masslessness property of the momentum generators has a counterpart for the special conformal generators K µ : η µν K µ K ν = 0 (8.14) in both "undeformed" and "deformed" minrep. In four-and six-dimensional conformal algebras, the minrep of the conformal group corresponds to a massless scalar conformal field in the respective dimension, and one can obtain all the irreducible representations that correspond to other massless conformal fields as "deformations" of the minrep [1,2]. In four dimensions one finds a continuous infinity of massless conformal fields labelled by helicity, and in six dimensions one finds a discrete infinity of massless conformal fields labelled by the spin t of an SU (2) T subgroup of the little group SO(4) of massless particles. The five-dimensional conformal group SO(5, 2), on the other hand, admits only two singleton representations corresponding to five-dimensional massless scalar and spinorial conformal fields. The minrep of SO(5, 2) corresponds to the scalar singleton representation (given in section 7), and the "deformation" of the minrep labeled by the spin s = 1/2 of the subgroup SU (2) S corresponds to the spinorial singleton representation. In this section we shall give the K-type decomposition of the deformed minrep.
Consider the Fock space of the a-type bosonic oscillators and the α-type fermionic oscillators introduced in previous sections, whose vacuum state |0 is annihilated by all a i and α r : a i |0 = 0 (i = 1, 2, 3) α r |0 = 0 (r = 1, 2) (9.1) A "particle basis" in this Fock space is provided by the states of the form where n i are non-negative integers, andñ r can be either 0 or 1. The Hilbert space of the deformed minimal unitary representation of SO(5, 2) is spanned by tensor product states of the form: ψ They are the lowest weight vectors of two copies of the minrep of SO(5, 2) with different internal quantum numbers as will become manifest when we study the extension to the superalgebra f(4).
In addition there exist four states that transform in the spinor representation of USp(4) with a definite eigenvalue E and are annihilated by all the grade −1 generators. They are: where the subscripts 0, ± 1 2 in Φ, Φ and Ψ refer to the respective eigenvalues of A 3 andÅ 3 generators of SU (2) A and SU (2)Å subgroups of the rotation group SO(4) in five dimensions as defined in equation (4.6). These four states, which we shall denote as form and irrep of its compact subgroup USp(4) with the lowest eigenvalue of AdS 6 energy.
The corresponding unitary representation of SO(5, 2) is the unique deformation of the minrep. The states of the "particle basis" of the deformed minrep are obtained by repeatedly acting on |Ω I with grade +1 operators in the C + subspace of SO (5, 2).
In Table 2, we give the "deformed" minrep of SO(5, 2), with the corresponding AdS energy, SO(5) ≈ USp(4) irreps and USp(4) Dynkin labels. Clearly, this deformed minrep of SO(5, 2) corresponds to the spinor singleton representation of SO(5, 2), similar to the Dirac spinor singleton of SO (3,2). In the context of the five-dimensional conformal group, it corresponds to a massless spinor field. The superalgebra f(4) with the even subalgebra so(5, 2) ⊕ su (2) is the unique simple conformal superalgebra in five dimensions. To construct the minimal unitary irreducible representation of f(4) via the quasiconformal method, we start from its 5-graded decomposition with respect to its subsuperalgebra d(2, 1; 2) ⊕ so(1, 1): The subsuperalgebra d(2, 1; 2) belongs to the continuous family of 17-dimensional Lie superalgebras d(2, 1; α), and admits a 10-dimensional linear representation with 6 bosonic and 4 fermionic degrees of freedom. Grade ±1 generators of f(4) transform in this 10-dimensional representation as indicated above. In our case the relevant real form of d(2, 1; 2) has the even subalgebra su(2) ⊕ su(2) ⊕ su(1, 1). In a previous section, we constructed the deformation of the minimal unitary representation of SO(5, 2) by using α-type fermionic oscillators. Now we shall realize the SU (2) subgroup outside SO(5, 2), which acts as the R-symmetry group, entirely in terms of the same fermionic oscillators as follows: They satisfy the commutation relations: The raising and lowering operators of this subalgebra will be denoted as T ± = T 1 ± i T 2 .
We shall denote this subalgebra as su(2) T and its quadratic Casimir as T 2 : This quadratic Casimir is related to that of su(2) S as We shall use the generator ∆ = 1 2 (xp + px), first given in equation (3.6), as the operator that defines the 5-grading, and identify the 6 even generators in grade −1 subspace of f(4) with the generators U i = x a i and U † i = x a † i of so(5, 2) given in equation (3.8). Grade −2 generator K − = 1 2 x 2 , given in equation (3.7), will also remain unchanged in the extension to f(4).
The four supersymmetry operators in grade −1 subspace will be defined as follows: Under anti-commutation they form a super-Heisenberg algebra that closes into the grade −2 generator K − .
Grade +2 generator K + also remains unchanged in the extension to f(4), as given in equation (8.7), and the six even generators in grade +1 space are identified with the generators W i and W † i of so(5, 2) given in equation (8.9). Now the four supersymmetry operators in grade +1 subspace can be obtained by taking commutators between the generator K + in grade +2 subspace and the supersymmetry operators in grade −1 subspace: They have the following explicit form: Commutators between these supersymmetry operators in grade +1 subspace and the generator K − in grade −2 subspace produce the respective supersymmetry operators in grade −1 subspace: Under anti-commutation, these four supersymmetry operators in grade +1 subspace generate a super-Heisenberg algebra that closes into the grade +2 generator K + .
The anti-commutators between the supersymmetry operators in grade −1 subspace and those in grade +1 subspace close into even generators in grade 0 subspace: With respect to the compact generators J 3 and T 3 , these eight supersymmetry operators in grade ±1 subspaces have the charges given in Table 3, which show their transfor-mation properties under SU (2) J × SU (2) T .
The remaining eight supersymmetry operators of f(4) reside in grade 0 subspace, and they can be obtained by taking commutators between supersymmetry operators in grade −1 (grade +1) subspace and even generators in grade +1 (grade −1) subspace. These eight supersymmetry operators are bilinears of the a-type bosonic oscillators and the αtype fermionic oscillators and are given as follows: With respect to the compact generators J 3 and T 3 and M 0 , these eight supersymmetry operators in grade 0 subspace have the charges given in Table 4, which show their transformation properties under SU (2) J × SU (2) T × SU (1, 1) M . Table 3: J 3 and T 3 charges of grade ±1 supersymmetry operators Table 4: J 3 , T 3 and M 0 charges of grade 0 supersymmetry operators The eight supersymmetry operators Σ r , Σ † r , Π r and Π † r in grade 0 subspace, along with the generators of SU (2) J , SU (1, 1) M and SU (2) T , form the subsuperalgebra d(2, 1; 2), which is in the grade 0 subspace of f(4) in this 5-graded decomposition with respect to ∆.
Finally, we give below some of the remaining (super-)commutators of f(4) in this 5-grading: 11 Compact 3-grading of f(4) with respect to the subsuperalgebra osp(2|4)⊕ u(1) The Lie superalgebra f(4) can be given a 3-graded decomposition with respect to its com- Note that B M , B † M and M M N are the generators that formed so(5, 2). The u(1) generator H that defines the compact 3-grading of f(4) is given by The subsuperalgebra osp(2|4) in C 0 subspace has the even subalgebra of so(2)⊕ usp(4).
We shall denote this so(2) ≈ u(1) generator as Grade 0 supersymmetry generators, i.e. those that are in the subsuperalgebra osp(2|4), shall be denoted as R I and R I where Their explicit forms are given by The canonical commutation relations of the osp(2|4) subsuperalgebra are as follows: The supersymmetry operators that belong to grade −1 subspace and those that belong to grade +1 subspace shall be denoted by Q I and Q I , respectively, where which have the following explicit expressions: The remaining commutation relations of the superalgebra f(4) are given below: It should be noted that, among the four supersymmetry operators of C ± subspaces, the following relations hold: 11.11) 12 Minimal unitary supermultiplet of F (4) In the Hilbert space spanned by the tensor product states of the form (9.3), there exists a unique lowest weight vector that is annihilated by all grade −1 generators B IJ , T − and Q I in the compact 3-grading and is a singlet of the compact subsuperalgebra osp(2|4) with definite eigenvalue of H, namely the lowest weight vector of the minrep of SO(5, 2) which we labelled as |Φ 0,0 in equation (9.4): where |0 is the Fock vacuum of the a-type bosonic oscillators and the α-type fermionic oscillators.
By acting on |Φ 0,0 with the operators B IJ , T + and Q I in grade +1 subspace C + repeatedly, one obtains an infinite set of states which forms a basis for the minimal unitary irreducible representation of f(4). This infinite set of states can be decomposed into a finite number of irreducible representations of the even subgroup SO(5, 2) × SU (2) T , with each irrep of SO(5, 2) corresponding to a massless conformal field in five dimensions. In Table 5, we present the minimal unitary supermultiplet of f(4) corresponding to this unique lowest weight vector |Φ 0,0 . We should note that the action of B IJ moves one within an irrep of SO(5, 2) and action of T + moves one within the internal symmetry group SU (2) T . On the other hand, with the action of the supersymmetry operators Q I of grade +1 subspace C + , one moves between different irreps of SO(5, 2) that make up the supermultiplet.
Therefore, interpreted as the N = 2 superconformal algebra in five dimensions, the minimal unitary supermultiplet of f(4) corresponds to a supermultiplet of fields consisting of two copies of the scalar singleton transforming as a doublet of the R-symmetry group SU (2) T and a singlet of the spinor singleton of SO(5, 2). 4 Labeling the doublet of massless conformal scalar fields as Φ r (0,0) (x µ ) (r = 1, 2) and the massless spinor field as Ψ (1,0) (x µ ) we have the minimal superconformal multiplet of F (4) as: where the subscripts indicate the Dynkin labels of the Lorentz group USp(2, 2).

13
AdS 6 /CF T 5 bosonic higher spin algebra and its deformation The AdS d /CF T (d−1) higher spin algebra of Fradkin-Vasiliev type corresponds simply to the universal enveloping algebra of SO(d, 2) quotiented by its Joseph ideal [9,10,[13][14][15]28]. The Joseph ideal of a Lie algebra g is a two-sided ideal that annihilates its minimal unitary of the corresponding minimal unitary realization yields directly the AdS 6 /CF T 5 higher spin algebra in complete parallel to the situation for the AdS 5 /CF T 4 and AdS 7 /CF T 6 higher spin algebras [9,10]. The generators of the infinite higher spin algebra then decompose under the finite dimensional subalgebra so(5, 2) as ⊕ ⊕ ... ⊕ · · · · · · ⊕ .... (13.12) For the deformed minrep of SO(5, 2), obtained above via quasiconformal methods, the Joseph ideal does not vanish identically as an operator. More specifically for the deformed minimal unitary representation, the symmetric traceless operator occurring in the tensor product of two generators of so(5, 2) corresponding to the tableau still vanishes: However the operator corresponding to the Young tableaux does not vanish for the deformed minimal unitary representation. To see how the Joseph ideal gets deformed, we rewrite the generators of the Joseph ideal in a 5d Lorentz-covariant form as was done for the AdS 5 /CF T 4 and AdS 7 /CF T 6 higher spin algebras in [9,10]. First we have the conditions: which are valid for both the scalar minrep as well as the spinorial minrep of SO(5, 2). The quadratic relations that define the Joseph ideal in the SO(4, 1)-covariant basis are as follows: 5 D · D + M µν · M µν + 3 2 P µ · K µ = 0 (13.15) P µ · (M µν + η µν D) = 0 (13.16) where the dilatation generator D is identified as M 65 and symmetrizations (round brackets) and anti-symmetrizations (square brackets) are done with weight one. 6 The above identities are all valid for the minimal unitary realization of SO(5, 2). However when we substitute the expressions for the generators of the deformed minrep of SO(5, 2) involving fermionic oscillators, one finds that only the first three equations above remain unchanged. The fourth equation above gets modified as follows: where S 2 is the Casimir of SU (2) S realized as bilinears of the fermionic oscillators. Similarly, the remaining four identities above get modified by spin-dependent terms involving fermionic oscillators. By replacing the fermionic oscillator realization of SU (2) S generators by the two dimensional Pauli matrices S i = σ i /2 above one obtains the spinor singleton realization by itself whose enveloping algebra then defines a deformed AdS 6 /CF T 5 higher spin algebra for which the deformed Joseph ideal vanishes identically.

14
Unique AdS 6 /CF T 5 higher spin superalgebra as enveloping algebra of the minimal unitary representation of F (4) Any definition of a higher spin superalgebra must have, as a subalgebra, the bosonic higher spin algebra based on the even subalgebra of the underlying finite-dimensional subsuperalgebra. Hence shall define the higher spin AdS 6 /CF T 5 superalgebra as the enveloping algebra of the quasiconformal realization of the minimal unitary representation of the superalgebra f(4), which has as subalgebras the bosonic higher spin algebra defined by the scalar singleton as well as the deformed higher spin algebra defined by the spinor singleton in five dimensions. To exhibit the structure of the resulting higher spin superalgebra we shall reformulate the minimal unitary realization of f(4) in an SO(5, 2)-covariant basis.
For this we first choose the SO(5, 2) gamma-matrices (Γ A ) α β (A = 0, . . . , 6) as follows: where γ M are the SO(5) gamma-matrices given in equation (5.11), M = 1, . . . , 5, and α, β = 1, . . . , 8 are the spinor indices of SO(5, 2). The symmetric SO(5, 2) charge conjugation matrix is chosen to be With the above choices, we find that all the matrices (C 7 Γ A ) αβ and (C 7 [Γ A , Γ B ]) αβ are antisymmetric. The spinor representation of SO(5, 2) is then realized by the matrices which satisfy commotion relations where η AB = diag (−, +, +, +, +, +, −). The 16 supersymmetry generators of F (4) transform as two eight dimensional spinors Ξ α , Ξ α of SO(5, 2) and are defined as where I = 1, 2, 3, 4 is the USp(4) spinor index. Under commutation with SO(5, 2) generators M AB they satisfy: Under anticommutation, they close into SO(5, 2) and SU (2) T generators as follows: Defining the SU (2) T doublet of SO(5, 2) supersymmetry generators the above anticommutation relations can be recast in an SU (2) T covariant form: Ξ r α , Ξ s β = iǫ rs M AB Σ AB C 7 αβ + 3i (C 7 ) αβ (iσ 2 σ i ) rs T i (14.9) where ǫ rs is the two dimensional Levi-Civita tensor and r, s = 1, 2 are the SU (2) T spinor indices. The expressions for the SO(5, 2) generators M AB in terms of the generators in the noncompact 3-grading and the compact 3-grading are collected in Appendix A. The higher spin AdS 6 /CF T 5 algebra defined by f(4) has, as a subalgebra, the enveloping algebra of the SO(5, 2) subalgebra spanned by symmetric products of the generators M AB . It also has, as a subalgebra, the enveloping algebra of SU (2) T spanned by the symmetric products of the generators T i , which is finite-dimensional and consists of T i and the quadratic Casimir element T 2 . The additional even elements of the higher spin superalgebra are given by products of the elements of the enveloping algebras of SO(5, 2) and of SU (2) T with antisymmetric products of an even number of supersymmetry generators Ξ r α . The odd elements of the higher spin algebra are given by products of the generators of the enveloping algebras of SO(5, 2) and SU (2) T with antisymmetric products of an odd number of supersymmetry generators Ξ r α .

Comments
The results obtained in this paper on the minimal unitary representations of SO(5, 2) and of the exceptional superalgebra F (4) and the AdS 6 /CF T 5 higher spin (super)algebras can be further developed and applied in several directions. First is the construction of bosonic higher theory in AdS 6 in terms of covariant fields based on the minrep of SO(5, 2) and its supersymmetric extension based on the minrep of F (4). Our results are also relevant to AdS 6 /CF T 5 dualities studied in references [31][32][33][34] and their extensions to higher spin theories. Yet another application is to integrable spin chain models related to conformally invariant (super)-symmetric field theories in five dimensions.

E Indices used in the paper
Here we give a list of indices we used in this paper and their ranges: