Supertwistor formulation for massless superparticle in $AdS_5\times S^5$ superspace

Starting with the first-order formulation of the massless superparticle model on the $AdS_5\times S^5$ superbackground and presenting the momentum components tangent to $AdS_5$ and $S^5$ subspaces as bilinear combinations of the constrained $SU(2)$-Majorana spinors allows to bring the superparticle's Lagrangian to the form quadratic in supertwistors. The $SU(2,2|4)$ supertwistors are assembled into a pair of $SU(2)$ doublets, one of which has even $SU(2,2)$ and odd $SU(4)$ components, while the other has odd $SU(2,2)$ and even $SU(4)$ components. They are subject to the first-class constraints that generate the $psu(2|2)\oplus u(1)$ gauge algebra. This justifies previously proposed group-theoretic definition of the $AdS_5\times S^5$ supertwistors and allows to derive the incidence relations with the $(10|32)$ supercoordinates of the $AdS_5\times S^5$ superspace. Whenever superparticle moves within the $AdS_5$ subspace of the $AdS_5\times S^5$ space-time, twistor formulation of its Lagrangian involves just one $SU(2)$ doublet of $SU(2,2|4)$ supertwistors with even $SU(2,2)$ and odd $SU(4)$ components. If in addition particle's 5-momentum is null, four first-class constraints which are the $su(2)\oplus u(1)$ generators single out upon quantization the states of $D=5$ $N=8$ gauged supergravity multiplet in the superambitwistor formulation.

Since SU(2, 2) is not only the covering of the conformal group of 4-dimensional Minkowski space-time but also is the isometry group of 5-dimensional anti-de Sitter space and SU(2, 2|4) is the superisometry of the AdS 5 × S 5 superspace as well as N = 4 extended superconformal symmetry providing the crucial symmetry argument in support of the AdS 5 /CF T 4 duality it is natural to seek for relevant (super)twistors. Because AdS 5 space-time is conformally flat, the incidence relations for Penrose twistors (actually ambitwistors) admit natural extension to accommodate extra bosonic coordinate(s) [32], [33] (see also [34]). It is also quite natural to add odd coordinates for D = 4 N-extended Poincare supersymmetry [35], [36]. The case of AdS 5 × S 5 superspace isomorphic to the P SU(2, 2|4)/(SO(1, 4) × SO(5)) supercoset manifold [37], [38] is not so easy to deal with as it is not superconformally flat [39]. In [40] there was given the group-theoretical reasoning for the definition of AdS 5 × S 5 supertwistors as two SU(2) doublets of SU(2, 2|4) supertwistors: one with even SU(2, 2) components and odd SU(4) components and another with odd SU(2, 2) components and even SU(4) ones. These constitute (4|4) × (2|2) rectangular block of the P SU(2, 2|4)/(SO(1, 4) × SO(5)) supermatrix. 4 We call such supertwistors c-and a-type supertwistors by analogy with c-and a-type supermatrices (see, e.g. [42]). Their SU(2, 2|4)-invariant products with dual supertwistors were shown to satisfy seven bosonic and eight fermionic real constraints which number matches that of psu(2|2) ⊕ u(1) generators. 5 However, the incidence relations with the AdS 5 × S 5 supercoordinates, necessary, for instance, to elaborate on the Penrose transform, have not been obtained. Previous experience suggests that supertwistor reformulation of the point-particle model is a proper tool to find such relations. Additionally one of the advantages of the supertwistor approach is that the superparticle's Lagrangian is quadratic in supertwistors that facilitates quantization and can give the supertwistor description of the D = 10 N = 2 chiral supergravity spectrum compactified on AdS 5 × S 5 [48], [49].
Thus the aim of this note is to derive these AdS 5 × S 5 supertwistors starting from the first-order formulation of the D = 1+9 massless superparticle model in AdS 5 ×S 5 superspace [50], [51], [52], [53], [54]. We note that the superparticle's momentum components tangent to AdS 5 and S 5 subspaces can be realized as the bilinears of the constrained SU(2)-Majorana spinors in D = 1 + 4 and D = 5 dimensions respectively. Contraction of their SU (2) indices gives two 4 × 4 traceless antisymmetric matrices that, when contracted with the (4|4) × (4|4) supermatrix of the psu(2, 2|4) Cartan forms, extract bosonic components of the supervielbein tangent to AdS 5 and S 5 subspaces. This allows to bring the kinetic term in the superparticle's Lagrangian to the form appropriate for the introduction of the supertwistors. The details are worked out in Section 2. Section 3 is devoted to the quantization of the superparticle propagating in the AdS 5 subspace of the AdS 5 × S 5 superspace both in the oscillator and ambitwistor approaches that complement each other. In the simplest case, when the particle's momentum tangent to AdS 5 is null, we find how the D = 5 N = 8 gauged supergravity multiplet is embedded into the ambitwistor superfield of homogeneity degree zero in each of the arguments. Two appendices supply relevant details of the spinor algebra and supermatrix realization of the generators of psu(2, 2|4) isometry superalgebra of the AdS 5 × S 5 superspace.
2 Supertwistor mechanics of massless particle in AdS 5 × S 5 superspace 2.1 Superparticle moving on AdS 5 subspace: definition of c-type supertwistors Let us start with the group-theoretic consideration of the AdS 5 × S 5 supervielbein components tangent to the AdS 5 space-time and SO(1, 4) connection 1-form. As is well known they can be identified with the Cartan forms associated with the generators of the su(2, 2) subalgebra of psu(2, 2|4). (5)) supercoset representative then the left-invariant Cartan forms can be defined as where restriction to the su(2, 2) subalgebra amounts to focusing on the upper diagonal block of the (4|4) × (4|4) supermatrix of Cartan 1-forms, E mn (d) (m, n = 0 ′ , 0, 1, 2, 3, 5) are the su(2, 2) Cartan 1-forms, 3,5) are vielbein components tangent to the anti-de Sitter space and E m ′ n ′ (d) is the SO(1, 4) connection 1-form. D = 1+4 γ−matrices γ m ′ α β (α, β = 1, 2, 3, 4) represent the su(2, 2) generators from the g (2) eigenspace of the Z 4 outer automorphism of the psu(2, 2|4) superalgebra and their antisymmetrized products γ m ′ n ′ α β -those from the g (0) eigenspace. In the considered realization important distinction between these generators is that matrices acts as a projector onto the g (2) eigenspace. Promoting this SU(2) doublet of Spin(1, 4) spinors to 'superspinors' This relation is the key to the twistor formulation of the superparticle on the AdS 5 subspace of the AdS 5 × S 5 superspace. 1-form on the r.h.s. of (2.4) can be presented in terms of c−type AdS 5 supertwistors and that on the l.h.s. of (2.4) defines the kinetic part of the superparticle's Lagrangian in the first-order form of the space-time formulation Vector p m ′ can be identified with the 5-momentum of the massless or massive particle provided Λ equals 0 or m. 6 In the former case λ i α can be identified with the 4 × 2 rectangular block of the D = 1 + 4 spinor Lorentz-harmonic matrix that parametrizes the coset-space Spin(1, 4)/(SO(1, 1) × ISO(3)) and p m ′ with the light-like vector-column from the respective vector Lorentz-harmonic matrix [56]. For m non-zero λ i α can be related to the spinor Lorentz-harmonics parametrizing the coset Spin(1, 4)/SO(4) ∼ Spin(1, 4)/(SU(2) × SU(2)) and p m ′ with the vector-column of the respective vector Lorentz-harmonic matrix [44]. The net result is for p m ′ satisfying (2.8) and supertwistors constrained by the relations that provide the supertwistor realization of the generators of the su(2) ⊕ u(1) gauge algebra. So the superparticle's Lagrangian can be reformulated in terms of the SU(2) doublet of c−type SU(2, 2|4) supertwistors where g, a ij = a ji and t are the Lagrange multipliers. It can be checked that the number of the physical degrees of freedom in both formulations is the same. To see how the incidence relations between the supertwistor components and the coordinates of the AdS 5 × S 5 superspace are encoded in (2.6) consider definite P SU(2, 2|4)/(SO(1, 4) × SO(5)) representative, e.g. that discussed in [57], [50], [58] (2.14) corresponding to the isomorphic realization of the psu(2, 2|4) superalgebra as the D = 4 N = 4 superconformal algebra, and In the supermatrix realization of the relevant generators of D = 4 N = 4 superconformal algebra given in Appendix B the first factor in (2.13) acquires the form and the second is The c−type supertwistor incidence relations can be read off from (2.6) Accordingly for the dual supertwistor we havē where G −1 (2.21) and the conjugation rules of the supertwistor components are In the bosonic limit introduced supertwistors reduce to those proposed in [32] modulo the overall rescaling. Penrose-Ferber supertwistors arise in the boundary limit ϕ → −∞ and we also call them boundary supertwistors.

Definition of a-type supertwistors
Now the above discussion can be generalized to the case of superparticle moving also on the S 5 part of the superbackground. Cartan forms associated with the generators of the su(4) subalgebra of psu(2, 2|4) decompose into the sum (4), (2.23) where restriction to the su(4) subalgebra amounts to considering the lower diagonal block of the (4|4)×(4|4) supermatrix of Cartan forms, E IJ (d) (I = 1, 2, 3, 4, 5, 6) are the su(4) Cartan are antisymmetric in the spinor indices and belong to the g (2) eigenspace under the Z 4 automorphism of psu(2, 2|4) superalgebra, while the so(5) generators realized by are symmetric in the spinor indices and belong to the g (0) eigenspace. So that similarly to the AdS 5 case taking it is possible to construct a matrix projector onto the g (2) eigenspace. Presenting this spinor as a 'superspinor' and its dualΨ projected Cartan 1-form on the r.h.s. of (2.26) can be written as Further introducing 5-vector The norm of this vector equals and supertwistors (2.27), (2.28) satisfy four constraints Like for the c-type supertwistors we can give explicit form of the incidence relations for a−type supertwistors (2.27) and their duals (2.28) for the P SU(2, 2|4)/(SO(1, 4) × SO(5)) supercoset representative (2.13) supertwistors depend not only on the S 5 coordinates y I ′ but also onxα β so that in general it is not possible to view c− and a−type supertwistors as related solely to the AdS 5 and S 5 bosonic subspaces of the AdS 5 ×S 5 superspace. Also note that c− and a−type supertwistors are by definition mutually orthogonalΨ 2.3 Classical formulation of massless particle on AdS 5 × S 5 superspace in terms of cand a-type supertwistors The above results can be put together to describe the D = 1 + 9 massless superparticle on the AdS 5 × S 5 superbackground. First its null 10-momentum pm can be assembled from 5-vectors (2.8) and (2.30) The sign ambiguity is resolved by requiring the closure of the algebra of the first-class constraints. Then the first-order Lagrangian (2.11) of the superparticle propagating on the AdS 5 subspace of AdS 5 × S 5 space generalizes to and its supertwistor version acquires the form Lagrange multipliers a ij = a ji , s i ′ j ′ = s j ′ i ′ and t are even and κ i i ′ ,κ i ′ i are odd. They introduce the first-class constraints On the Dirac brackets (D.B.) (2.43) Above D.B. relations show that L i j and M i ′ j ′ generate two copies of the su(2) algebra and with the parameters a i j (τ ) and s i ′ j ′ (τ ). The constraint T + generates phase rotation of the supertwistors where t(τ ) is the local parameter. Odd constraints Φ i ′ i andΦ i i ′ generate the 8-parametric remnant of the κ−symmetry of the superparticle action in the superspace formulation with (2.47) ε(Z) and ε(Ψ) equal 0 for bosonic components and 1 for fermionic components of respective supertwistors.
3 Aspects of quantum theory of massless superparticle on AdS 5 subspace of AdS 5 × S 5 superspace In this section we consider quantum theory of the superparticle moving in the AdS 5 subspace of AdS 5 × S 5 superspace. Quantization is performed both in terms of the oscillators and supertwistors that yields two (complementary) views on the D = 5 N = 8 gauged supergravity multiplet.

Oscillator quantization
There is well-known intimate relation between the SU(2) oscillators and (c-type) supertwistors [59], [60]. For the bosonic components of supertwistors it is based on two forms of the SU(2, 2) 'metric' that connects fundamental and antifundamental representations. It is off-diagonal in the (super)twistor basis and diagonal in the oscillator basis. Thus the SU(2) bosonic oscillators are defined by the linear combinations of the supertwistor bosonic components The indices of internal SU(2) symmetry of the supertwistors have been suppressed for the moment. In quantum theory these oscillators satisfy the commutation relations which can be deduced from the quantum counterpart of the D.B. relations in (2.41). Relations (3.3) suggest interpretation of a α and b α as raising oscillators and of a α and b α as lowering oscillators.

(3.6)
Then the su(2) × u(1) constraints (2.10) can be brought to the following form in terms of (1) and N (β) (1) = βȧ(1)β˙a(1) are the oscillator number operators for bosonic and fermionic oscillators of the first set. Analogous definitions apply to the second set of oscillators. Important feature of the constraints (3.7) is that they commute with the raising supersymmetry operators used to generate lowest-weight vectors corresponding to different SU(2, 2) × SU(4) representations within the same SU(2, 2|4) supermultiplet as well as with the bosonic and fermionic raising operators which applied to a lowest weight vector generate the whole set of basis vectors of the corresponding SU(2, 2) × SU(4) representation [48], [61]. So one can consider the constraints (3.7) to act on the SU(2, 2) × SU(4) lowest-weight vectors. In the simplest case Λ = 0, i.e. when the superparticle does not have non-zero momentum components in the directions tangent to S 5 , it is not hard to verify that the only lowest-weight vector annihilated by the constraints is the oscillator vacuum associated with the D = 5 N = 8 gauged supergravity multiplet [48]. For Λ = 0 the constraints select SU(2) × U(1) invariant lowest-weight vectors which correspond to the supermultiplets discussed in [59] (see also [55]).

Ambitwistor quantization
Quantum counterpart of the D.B. relations (2.41) is consistent with the well-known in twistor theory [62], [4] realizations of quantized (super)twistors as multiplication and differentiation operators. For the ambitwistor space description of the quantized superparticle the supertwistors and can be conveniently realized as (modulo their relabeling) where odd derivatives are defined to act from the left. Then in the simplest case Λ = 0 the su(2) ⊕ u(1) constraints (2.10) applied to the superparticle's wave function acquire the form Eqs. (3.15) imply that F (0,0) has homogeneity degree zero both in Z andW that is indicated by its subscripts and the condition (3.16) can be taken into account by adding a δ-function factor The function f (1,1) (Z,W) has the power series decomposition in odd supertwistor components [ABCD] (Z,W )ζ AζBζCζD + . . . (3.18) where dots stand for higher terms in the series that as we shall see below are irrelevant.
Since the Penrose transform between the homogeneous functions of the introduced twistors and the fields on AdS 5 × S 5 superspace is yet to be elaborated, 7 we can meanwhile consider the Penrose transform of the corresponding functions of the boundary ambitwistors that should produce the fields of D = 4 N = 4 conformal supergravity multiplet [63]. 8 According to the AdS/CFT dictionary these fields of the D = 4 N = 4 conformal supergravity multiplet correspond to the boundary values of non-normalizable solutions of the bulk D = 5 N = 8 gauged supergravity equations linearized around AdS 5 [64], [65]. 9 Thus the function h (1,1) (Z,W ) yields Minkowski space graviton field h α(2)α(2) (x) (see, e.g. [68] and more recent [69] for the details of the ambitwistor transform). The Penrose transform for other functions in the series (3.18) deserves more detailed treatment.
First observe that the SU(4) representations spanned by monomials composed of odd supertwistor components are in general reducible and decompose into a sum of irreducible ones e.g. as 7 See, however, Ref. [33], where, based on the realization of AdS 5 space as a projective manifold, the Penrose transform for the case of spin 0 and 1/2 fields was considered. 8 Hereinafter the same notation introduced above for the bulk ambitwistors is used for the boundary ones. In the Poincare coordinates taking the boundary limit of AdS 5 supertwistors is accompanied by their rescaling but since the superparticle's wave function is scale-invariant in both arguments the relations (3.17) and (3.18) are valid as they stand either for bulk or boundary supertwistors. 9 General discussion of the correspondence between the AdS bulk gauge fields and the boundary conformal (shadow) fields not restricted to the case of low spins may be found in [66], [67].

(3.24)
Monomials with widetilde [BCD] A (Z,W ) should produce scalar fields via the Penrose transform. This can be explained in the following way. Ambitwistor functions with the above homogeneity degrees can be obtained by applying the first-order differential operators to the (−1, −1) ambitwistor functions that correspond to scalar fields and have the following form in terms of the SL(2, C) spinor parts In twistor theory composition of these operators is known to give the space-time d'Alambertian and plays a role in twistor description of massive fields (see e.g. [62]). Let us note that the action of these operators does not change the sum of homogeneity degrees of ambitwistor function. So the Penrose transform of ε EABCφ (Z,W ) should give zero. The argument generalizes one that has been used above for the SU(2, 2) representations corresponding to vector and scalar fields. Using the relation between the twistor components and SU(2) × SU(2) oscillators it is possible to find the SU(2, 2) lowest-weight vectors for the fields on the r.h.s. of (3.27): 1))|0 and (a α (2)a β (1) − a β (2)a α (1))|0 [48] and to verify that they are annihilated by the operator −a α (2)a α (1) + b α (2)b α (1). Note that each antisymmetrized product of bosonic rising oscillators does not change the SU(2) labels of the SU(2, 2) lowestweight vector on which it acts but increases the AdS energy by one unit [59] and so the operators (3.26) are the twistor counterparts of such antisymmetrized products of rising oscillators.
Spin 1/2 fields are obtained by performing the Penrose transform of the cubic terms in (3.18). To this end let us note that similarly to the monomials in (3.19) and (3.21), η A η Bζ C and η Aζ BζC furnish reducible representations of SU(4) that decompose as 6 ×4 = 20 + 4 and 6 × 4 =20 +4: where η A η Bζ C and η Aζ BζC span 20 and20 representations. This allows to decompose respective terms in the expansion (3.18) as

Conclusion and discussion
This note addressed the issue of the definition of supertwistors for the AdS 5 × S 5 superspace, which isometry is described by the P SU(2, 2|4) supergroup that is also the superconformal symmetry of D = 4 N = 4 Minkowski superspace. Twistor reformulation of the massless particle model on the P SU(2, 2|4)/(SO(1, 4) × SO(5)) supermanifold led us to consider SU(2) doublets of c-and a-type SU(2, 2|4) supertwistors subject to seven bosonic and eight fermionic constraints as the AdS 5 × S 5 supertwistors. This justifies proposed earlier on the group-theoretical grounds definition of AdS 5 × S 5 supertwistors [40] and allows to find the incidence relations with the AdS 5 × S 5 supercoordinates via the P SU(2, 2|4)/(SO(1, 4) × SO(5)) supercoset representative.
The superparticle's Lagrangian in the supertwistor formulation is quadratic and yields only the first-class constraints that are generators of the psu(2|2) ⊕ u(1) gauge symmetry. This should facilitate transition to the quantum theory. Here we examined quantization of the superparticle moving in the AdS 5 subspace of the AdS 5 × S 5 superspace. In this case only four bosonic constraints associated with the su(2) ⊕ u(1) gauge symmetry are imposed on the superparticle's wave function that we chose to depend on one c-type supertwistor and one dual c-type supertwistor. This breaks manifest SU(2) symmetry but makes contact with the ambitwistor description of the massless fields on AdS 5 space-time [33], [70]. The constraints imply that the wave function is homogeneous of degree zero in its arguments and can be expanded in the supertwistor odd components. Then the Penrose transform of the component ambitwistor functions gives off-shell fields of the D = 4 N = 4 conformal supergravity multiplet that serve as the boundary values for on-shell fields from the D = 5 N = 8 gauged supergravity multiplet in the AdS 5 bulk. Quantization of the complete model with all the constraints taken into account should give supertwistor description of the whole spectrum of D = 10 N = 2 chiral supergravity compactified on AdS 5 ×S 5 space [48], [49]. To make contact with the field-theoretic description [49] it is necessary to work out the details how the Penrose transform of the homogeneous functions of c-type and a-type supertwistors produces fields in AdS 5 space-time.
The superparticle model discussed in this note can be straight-forwardly generalized to that of a tensionless string with the action S = dτ dσL AdS 5 ×S 5 stwistor T=0 string : extending the ambitwistor string model [71]. Depending on the quantization prescription and the choice of the vacuum [72], [73] we expect it to provide the world-sheet CFT interpretation for the scattering amplitudes in D = 5 N = 8 gauged supergravity or to produce D = 10 N = 2B higher-spin massless supermultiplets compactified on AdS 5 × S 5 . Generalization to the tensile string model is also feasible and should give an interesting reformulation of the P SU(2, 2|4)/(SO(1, 4) × SO(5)) supercoset action [37], [38].
We anticipate that the c-and a-type supertwistors can also be used to construct the supertwistor action for D = 5 N = 8 gauged supergravity and possibly for the whole tower of supermultiplets arising in the compactification of D = 10 N = 2 chiral supergravity on AdS 5 × S 5 in a way similar that of [74], where the super-Yang-Mills action was written in the supertwistor space.
Along the same lines it is possible to consider supertwistors and supertwistor formulations for point-like and extended objects on other supersymmetric supergravity backgrounds, in particular, those with the OSp-type superisometries, some of which were outlined in [40]. and Hermitian conjugation rule where matrices H β γ and H δ α equal They satisfy H α β H β γ = δ γ α , H α β H β γ = δ α γ and are related by the transposition H α β = (H β α ) T . Passing to D = 1 + 4 dimensions, matrices ρ 0 ′ αβ andρ 0 ′ αβ are identified with the charge conjugation matrix and its inverse that are used to raise and lower Spin(1, 4) spinor indices λ α = C αβ λ β λ α = C αβ λ β .
(A.5) D = 1 + 4 γ-matrices are defined as and satisfy For them the conjugation rule (A.2) transforms to and also (C αβ ) † = C βα . Antisymmetrized products of γ-matrices provide realization of the so(1, 4) algebra relations To obtain matrix form of the so(2, 4) generators in conformal basis D = 1 + 4 γ-matrices γ m ′ The charge conjugation matrices are realized in terms of antisymmetric unit rank 2 spinors ε αβ , ε αβ and c.c. ones that are used to raise and lower SL(2, C) spinor indices. 10 In such a realization matrices (A.3) acquire the form conventional in twistor theory (A.14) To calculate the square of the particle's 5-momentum it is used the completeness relation for D = 1 + 4 γ-matrices They are connected by the Hermitian conjugation (ρ I AB ) † =ρ IBA . We adopt the following definition of γ-and charge conjugation matrices in D = 5 dimensions So that D = 5 Clifford algebra relations read Under Hermitian conjugation D = 5 γ-and charge conjugation matrices transform as Antisymmetrized products of γ-matrices satisfy the commutation relations of the so(5) algebra Completeness relation for D = 6 γ-matrices in terms of D = 5 γ-matrices acquires the form and is used to calculate the square of the particle's momentum components in directions tangent to S 5 .  Introducing the s 5 algebra generators relations (B.7) can be written in the form Generators of the Poincare supersymmetry are given by the supermatrices The reasoning behind this definition is that the (4|4) × (4|4) supermatrix g A B : g A B = g α β g α where Explicit supermatrix form of the general psu(2, 2|4) element in SL(2, C) notation is