Multitwistor mechanics of massless superparticle on $AdS_5\times S^5$ superbackground

Supertwistors relevant to $AdS_5\times S^5$ superbackground of IIB supergravity are studied in the framework of the $D=10$ massless superparticle model in the first-order formulation. Product structure of the background suggests using $D=1+4$ Lorentz-harmonic variables to express momentum components tangent to $AdS_5$ and $D=5$ harmonics to express momentum components tangent to $S^5$ that yields eight-supertwistor formulation of the superparticle's Lagrangian. We find incidence relations of the supertwistors with the $AdS_5\times S^5$ superspace coordinates and the set of the quadratic constraints they satisfy. It is shown how using the constraints for the (Lorentz-)harmonic variables it is possible to reduce eight-supertwistor formulation to the four-supertwistor one. Respective supertwistors agree with those introduced previously in other models. Advantage of the four-supertwistor formulation is the presence only of the first-class constraints that facilitates analysis of the superparticle model.


Introduction
To study AdS/CFT correspondence [1], [2], [3], the best explored instance of which is provided by the duality of the Type IIB superstring on AdS 5 × S 5 superbackground and N = 4 supersymmetric Yang-Mills theory on its 4-dimensional conformal boundary, it is commonly used the (super)space setting. Both theories are invariant under the SU(2, 2|4) symmetry supergroup that was one of the primary arguments in favor of the duality. This symmetry supergroup is manifest and linearly realized in supertwistor space [4], [5]. D = 4 N = 4 super-Yang-Mills theory can also be described in supertwistor or superambitwistor space at the level of supermultiplet [6], field equations [7] and Lagrangian [8], [9], [10], [11]. Thus it is natural to wonder whether AdS 5 × S 5 superstring also admits a formulation in terms of supertwistors, what are these supertwistors and what implications such a formulation might have on the study of the AdS 5 /CF T 4 correspondence. In the bosonic limit some of these issues were addressed recently in [12]. 2 In particular, there was given the definition of twistors for AdS 5 space based on its realization as a projective manifold. Another definition of twistors [14], [15] 3 follows from considering point particle mechanics on AdS 5 and is based on the on-shell momentum realization as the product of two constrained spinors. Generalization of these twistors to the AdS 5 × S 5 superbackground appears non-trivial since it is not superconformally flat [19].
Independently I. Bars in the framework of 2T approach [20] proposed superparticle and tensionless superstring models [21], [22] extended local symmetries of which can be fixed in a variety of ways. One of the gauges yields the first-order Lagrangian of the superparticle (tensionless superstring) in AdS 5 × S 5 superspace. In another gauge dynamical variables of the model are supertwistors transforming in the fundamental representation of the SU(2, 2|4) supergroup and fundamental representation of its gauged subgroup. These models provide an interesting alternative to the conventional way of introducing (super)twistors in particle (string, brane) models [5], [6], [23]- [47]. However, it is unclear how the incidence relations that connect supertwistor components and the superspace coordinates can be derived within this approach.
This motivated us to examine in Ref. [48] the traditional root of introducing supertwistors starting with the first-order representation of the massless superparticle' Lagrangian on AdS 5 × S 5 superbackground. Then we expressed momentum components tangent to AdS 5 and S 5 parts of the background in terms of the SU(2)-Majorana spinors for Spin (1,4) and Spin (5) groups so that the 10-momentum null condition was translated into the constraint on the sum of the products of these spinors. Substituting the definition of the supervielbein bosonic components as Cartan 1-forms of the P SU(2, 2|4)/(SO(1, 4) × SO(5)) supercoset allowed to transfer to the supertwistor form of the superparticle's Lagrangian. Respective SU(2, 2|4) supertwistors were assembled into SU (2) doublets that differ from one another by the Grassmann parity of their components and coincide with those found in [22]. The benefit of this approach is that it allows to find the incidence relations that connect the supertwistor components and AdS 5 × S 5 supercoordinates via the P SU(2, 2|4)/(SO(1, 4) × SO(5)) representative. Explicit form of the incidence relations was derived for the particular example of the supercoset representative used to study the AdS 5 × S 5 superstring mechanics [49], [50], [51]. Besides that, in the case when the superparticle's momentum components tangent to 5-sphere vanish, we quantized the model in terms of the superoscillators [52] and superambitwistors and showed how the supermultiplet of D = 5 N = 8 gauged supergravity [53] fits into its spectrum of states.
The geometric way of introducing spinors in the models of point-like and extended (supersymmetric) relativistic objects is based on the formulations of their Lagrangians that involve components of the local orthonormal repere [54]. The repere matrix can be naturally identified with the element of the respective Lorentz group called vector Lorentz-harmonic matrix [55]. It can be expressed as the square of the spinor Lorentz-harmonic matrix [56], [57], [58] in analogy with the relation of the matrices of vector and spinor Lorentz transformations. In the case of D = 1 + 3 dimensions spinor Lorentz harmonics [59], [60] in the literature on twistors [61] were known as the normalized dyad.
The product structure of the bosonic AdS 5 × S 5 background suggests using D = 1 + 4 Lorentz harmonics [62], [63] to express momentum components in directions tangent to AdS 5 and Spin(5) harmonics 4 for the momentum components in directions tangent to S 5 . So one of the goals of the present paper is to derive eight-supertwistor formulation of the D = 10 massless superparticle on the AdS 5 × S 5 superbackground starting with its first-order formulation that involves D = 1+4 spinor Lorentz harmonics and D = 5 harmonics. This explains the origin of the supertwistors that arise in the respective gauge for the 2T tensionless superstring model of Ref. [21] and allows to find their incidence relations with the AdS 5 × S 5 superspace coordinates. We find the set of quadratic constraints that these supertwistors satisfy and calculate their classical algebra. It appears that some of the constraints are the second-class ones. This indicates the presence of the redundant d.o.f. and complicates further steps of the Dirac analysis of the model. Using the defining conditions for the spinor (Lorentz) harmonics we show that it is possible to exclude half of them from the expressions for the momentum components tangent to AdS 5 and S 5 parts of the background. Reduced expressions for the momentum components coincide with those proposed in [48] and allow to obtain four-supertwistor formulation of the superparticle. Since these supertwistors precisely coincide with those found in [22], we actually find the relation between the eight-and four-supertwistor formulations of the massless superparticle on AdS 5 × S 5 superbackground.
The organization of the paper is the following. In the next section as a warm up example we consider the model of the bosonic particle of mass m on AdS 5 . We introduce vector and spinor Lorentz harmonics that parametrize the SO(1, 4)/SO(4) coset and express particle's momentum in terms of them. Then we transfer to the four-twistor formulation and find the set of the constraints that these twistors satisfy. It is shown that among them there are the second-class constraints. In section 3 using the harmonicity conditions we derive the reduced expression for the particle's momentum that depends just on the half of the spinor Lorentz harmonics. Corresponding twistors and two-twistor particle's Lagrangian are those of Ref. [14]. In [15] this model, that is characterized only by the first-class constraints, was quantized in terms of the SU(2) × SU(2) bosonic oscillators and was shown to describe SU(2, 2) irreducible representation with spin zero and AdS energy E = 2 + |m|. We give alternative description of this representation in terms of the ambitwistors.
Section 4 is devoted to study of the massless superparticle on AdS 5 ×S 5 superbackground. Momentum components tangent to AdS 5 and S 5 are expressed in terms of the Spin(1, 4) Lorentz harmonics and Spin (5) harmonics. This allows to pass to the eight-supertwistor formulation of the superparticle's Lagrangian with the supertwistors being the elements of the (4|4) × (4|4) supermatrix. We identify the set of the constraints for supertwistors and calculate their classical superalgebra to show that, similarly to the bosonic particle case, there are present the second-class constraints. Then in section 5 it is shown how proposed in [48] four-supertwistor formulation of the superparticle's Lagrangian arises. We also consider the set of equations for the quantized superparticle's wave function in the superambitwistor space.
Necessary details of the spinor algebra and the algebra of the constraints for the eightsupertwistor formulation for the massless superparticle are collected in the appendices.
2 Massive bosonic particle on AdS 5 As the starting point we choose the first-order form of the action of D = 5 massive bosonic particle where E m ′ τ is the world-line pull-back of the AdS 5 vielbein 1-form. Particle's 5-momentum can be set proportional to the time-like unit-norm vector n that can be chosen as the first column of the D = 5 vector Lorentz-harmonic matrix It is identified with the local orthonormal repere in the tangent space to the particle's world line, the indices in brackets labelling the repere components. Orthonormality of the vector Lorentz harmonics implies that n (0) m ′ n m ′ (0) = −1 so that the particle's mass-shell constraint is satisfied. Vector Lorentz-harmonics transform under left global and right local Lorentz rotations n m ′ is invariant only under the transformations from the subgroup SO(4) R ⊂ SO(1, 4) R that enters the set of the gauge symmetries of the massive particle's action, vector Lorentz harmonics parametrize the coset SO(1, 4)/SO (4).
To transfer to the twistor form of the particle's Lagrangian it is necessary to introduce that can be viewed as the 'square root' of the vector Lorentz harmonics We use letters from the beginning of the Greek alphabet to label Spin(1, 4) L spinor indices, while those from the middle part of the Greek alphabet to label Spin(1, 4) R spinor indices. Dotted and undotted letters from the beginning of the Latin alphabet label fundamental representation indices of the two SU(2) R subgroups of Spin(1, 4) R . Details of the spinor algebra in D = 1 + 4 dimensions are given in [48] and Appendix A of the present paper. In particular, for n (0) Spinor Lorentz harmonics satisfy the reality conditions Orthonormality of the vector Lorentz harmonics then follows from the harmonicity conditions for the spinor Lorentz harmonics that state invariance of the charge conjugation matrices under the left and right Spin(1, 4) transformations since v ν α = C αβ C νλ v β λ and ensure that v α µ ∈ Spin(1, 4). Eq. (2.7) is invariant under local SU(2) R ×SU(2) R ⊂ Spin(1, 4) R symmetry so that spinor harmonics pertinent to the massive particle model parametrize the coset Spin(1, 4)/(SU(2)×SU(2)). This SU(2) R × SU(2) R gauge symmetry will also be the symmetry in the (super)twistor formulation.
Using the definition of su(2, 2) left-invariant Cartan forms where G ∈ SU(2, 2), 1-form, world-line pullback of which defines kinetic term of the particle's Lagrangian (2.1), can be written in terms of SU(2, 2) twistors 5 (2.12) 5 In the first line spinor Lorentz harmonics were rescaled v → m 2 v to bring the twistor kinetic term to the canonical form.

Incidence relations for the introduced twistors read
and dual twistor matrix satisfies (2.14) Matrices H α β and H λ ν are the metrics connecting fundamental and antifundamental representations of SU(2, 2) L and SU(2, 2) R respectively. In Ref. [48] there was chosen such realization of D = 2 + 4 γ-matrices that H α β equals D = 1 + 4 matrix γ 0α β and is offdiagonal as is common in the twistor theory. For In this realization conjugation rules (2.14) for 4 × 2 blocks of the twistor matrix read Using the incidence relations (2.13) and the harmonicity conditions (2.10) one derives the constraints that twistors obeyZ so that 2 m Z α ν ∈ U(2, 2) modulo the gauge symmetries of the particle's action in twistor formulation 19) The structure of the kinetic term of this action is similar to that of the tensionless superstring of Ref. [21] in the twistor gauge. It has manifest global SU(2, 2) L symmetry, whereas SU(2, 2) R symmetry, assumed in the definition of the dual twistor matrix (2.14), is broken by γ (0)ν λ . It is possible to give another definition of the dual twistor matrix that differs from (2.14) by the multiplication by γ (0)ν λ . Since γ (0) γ (0) = 1, the kinetic term of the twistor Lagrangian (2.19) acquires the following form in terms of the dual twistor matrix (2.20) It has manifest SU(2) R ×SU(2) R gauge symmetry but actually is invariant under the SU(4) R symmetry that is broken by the constraint term To analyze the algebra of the constraints let us decompose them into the SU(2) R ×SU(2) R irreducible constituents Basic Dirac bracket (D.B.) relations for the twistor components that follow from (2.19) are Then non-trivial D.B. relations of the constraints acquire the form (2.25) These are relations of the SU(4) R symmetry algebra broken by m. Thus the constraints Z a α Z αḃ andZ αȧ Z α b are the second-class, while other eight constraints are the first-class ones. From the viewpoint of the oscillator realization of the SU(4) R algebraZ a α Z αḃ andZ βȧ Z β b are raising and lowering generators from the +1 and −1 eigenspaces w.r.t. to the three-grading structure introduced by c [70]. Having identified the first-and second-class constraints it is possible to verify that the number of the physical degrees of freedom in the four-twistor formulation equals eight matching that of the space-time formulation of the D = 5 massive particle mechanics.

Reduction of four-twistor formulation to two-twistor formulation
The fact that in the four-twistor formulation there are eight second-class constraints quadratic in twistors essentially complicates further analysis of the Hamiltonian mechanics of the bosonic particle model and requires introduction of the D.B. for these constraints. It is, however, possible to remove some of the redundant degrees of freedom that will simplify the algebra of the remaining constraints without breaking the SU(2, 2) L invariance. The form of the incidence relations (2.13) shows that reduction of the SO(1, 4) Lorentz-harmonic variables underlies reduction of twistors, hence we discuss it on the level of harmonics. To this end decompose spinor Lorentz harmonics into SL(2, when expressed in terms of the SL(2, C) L constituents read Then from the harmonicity conditions we obtain Multiplying the latter relation by v a β we derive Further multiplying by v βċ gives ΥΥ =ῩῩ and in view of (3.5) Then using (3.7) and (3.8) it is possible to show that in the expression for the particle's 5-momentum contributions of the first and the second summands equal so that one can exclude spinor Lorentz harmonics with dotted indices where the factor of two has been absorbed in the definition of Lorentz harmonics with undotted indices that now satisfy Let us remark that using (3.5) and (3.6) it is possible to exclude in a similar way spinor Lorentz harmonics with undotted indices leaving only those with dotted indices. Also note that in the expressions for the space-like vector-columns of the Lorentz-harmonic matrix (2.3) it is impossible to remove contributions of the spinor harmonics with either dotted or undotted indices. Taking expression for the reduced 5-momentum (3.10) one can obtain two-twistor form of the massive particle's Lagrangian following the same steps as in (2.12) that was proposed in [14] (see also [17], [18]). In [15], using the isomorphism of the algebras of quantized twistors and su(2) ⊕ su(2) bosonic oscillators that are employed in constructing positive energy unitary irreps of SU(2, 2) and its superextensions [52], [53], it was shown that this model describes SU(2, 2) unitary irrep with AdS energy E = 2 + |m| and zero spin. One needs at least two copies of su(2) ⊕ su(2) bosonic oscillators to describe such irreps. Their lowest-weight vectors are given by the antisymmetrized products of creation oscillators in the fundamental representation of one of the su(2) algebras [70], [71].
In this section we construct particle's wave function in the ambitwistor space corresponding to such irrep. Let us introduce specific notation for each of the twistors It is readily shown that from (3.12) there follow conventional commutation relations of the quantized twistors (3.14) Hermitian operators corresponding to four first-class constraints of the model can be brought to the form Let Z α andW α be coordinates of the ambitwistor space, then other two quantized twistors are represented as the first-order differential operators Imposing a la Gupta-Bleuler three of the four constraints (3.15) that are given by the utmost first-order differential operators on the particle's wave function F (m−2|−m−2) (Z,W ), we find  These equations imply that particle's wave function is homogeneous in each of its arguments as indicated by the subscript and that its mass should be integer m ∈ Z (cf. [15]). The last constraint can be taken into account by writing When m is zero, it is known that the ambitwistor transform [72], [73] of f (−1|−1) (Z,W ) as a function of Penrose twistors, corresponding to the boundary limit [48] of AdS 5 twistors, gives off-shell scalar field on D = 4 Minkowski space-time. This field can be identified as the D = 4 shadow field [74] that serves as the boundary value of the non-normalizable solution of the Dirichlet problem for AdS 5 scalar field. As discussed in [75], [48] associated SU(2, 2) lowest-weight vector is given by the oscillator vacuum |0 . If m > 0, then the respective SU(2, 2) lowest-weight vector is as was found in [15]. In ambitwistor description each of the antisymmetrized products of a-oscillators maps to the first-order differential operator [48] so that massive particle's wave function in ambitwistor space is Respectively for m < 0 associated SU(2, 2) lowest-weight vector is Antisymmetrized products of b-oscillators map to the operator and particle's wave function equals Note that differential operators ZI ∂ ∂W and W I ∂ ∂Z are used in twistor theory to describe massive fields in four-dimensional Minkowski space [76].
4 Eight-supertwistor formulation of massless superparticle on AdS 5 × S 5 superbackground To derive eight-supertwistor formulation of the D = 10 massless superparticle on the AdS 5 × S 5 superbackground let us introduce diagonal supermatrix while in the lower-diagonal block there is 4 × 4 matrix with Spin(5) ∼ USp(4) indices 6 Reality conditions for Spin(1, 4) variables coincide with those in (2.8) and for Spin(5) variables are In the supermatrix form we have 5) 6 We slightly abuse the notation in this section since, as will be seen below, v α µ contains some additional degrees of freedom compared to the Spin(1, 4) Lorentz harmonics and ℓ A N has more independent degrees of freedom than the U Sp(4) harmonics [64]. with H α β and H λ ν given (2.15) and (2.16). Introduce also the P SU(2, 2|4)/(SO(1, 4) × SO(5)) representative Define twistor supermatrix and its dual bȳ For individual rows of the dual twistor supermatrix this definition translates into the relations Supertwistors Z A b , Z Aḃ and their duals are conventional ones and were named c-type in [48]. Their SU(2, 2) L components Z α b , Z αḃ are even and SU(4) L components η A b , η Aḃ are odd. On the contrary supertwistors Ψ A p , Ψ Aṗ and their duals are a-type since their SU(2, 2) L components ξ α p , ξ αṗ are odd but SU(4) L components L A p , L Aṗ are even. Appearance of such unconventional supertwistors for superparticle model on AdS 5 × S 5 superbackground was discussed in [21] and [22]. Note that the definition of the twistor supermatrix (4.9) assumes that it transforms under left and right SU(2, 2|4) transformations where supermatrices L A B and R M N have SU(2, 2) upper-diagonal blocks and SU(4) lowerdiagonal blocks.
Consider now the first-order form of the massless superparticle action on the AdS 5 × S 5 superbackground E m ′ τ and E I ′ τ are the world-line pullbacks of the D = 10 supervielbein bosonic components tangent to AdS 5 and S 5 . In analogy with the discussion of section 2, momentum components tangent to AdS 5 can be expressed as where Spin(1, 4) spinors v α b and v αḃ satisfy SU(2)-Majorana conditions (2.9) and also v α b v˙b α = 0 (4.15) so that Similarly momentum components tangent to S 5 can be expressed in terms of the Spin(5) spinor variables (4.3) constrained by the relations necessary to obtain the following expression for the square of momentum components tangent to S 5 As a result the null-momentum condition in 10 dimensions translates into Let us note that null 10-momentum (p m ′ , p I ′ ) has 9 independent components, whereas at this point spinors (v α a , v α a ) and (ℓ A q , ℓ Ȧ q ) have 6 + 6 − 1 = 11 independent components. Two additional constraints for the spinor variables will be set below. Now 1-form that enters kinetic term of the superparticle's Lagrangian (4.13) can be expressed in the supertwistor form with the diagonal supermatrix In its upper-diagonal block there is γ (0)ν λ matrix that enters kinetic term of the twistor Lagrangian for the bosonic particle (2.19). As discussed in section 2 it not merely breaks SU(2, 2) R symmetry but actually 'switches' it to SU(4) R symmetry. In the case of superparticle supermatrix Γ interchanges SU(2, 2) R -and SU(4) R -invariant blocks. So the supertwistor 1-form (4.21) is invariant under 'twisted' SU(2, 2|4) R symmetry with SU(4) R parameters in the upper-diagonal block and SU(2, 2) R parameters in the lower-diagonal block. It can be made explicit by adopting another definition of the dual twistor supermatrix where resulting in the following expression for the 1-form (4.21) This 'twisted' SU(2, 2|4) R symmetry, however, will be broken by the supertwistor constraints, that we discuss below, similarly to breaking of the SU(4) R symmetry in the case of massive particle model in section 2. So let us write kinetic term of the superparticle's Lagrangian in the eight-supertwistor formulation in the form with manifest (SU(2) R ) 4 invariance To write down complete superparticle's Lagrangian in the supertwistor form we need to identify the set of constraints for supertwistors. To this end consider SU(2, 2|4) L -invariant supermatrix quadratic in supertwistors (4.27) Using the incidence relations (4.9) and (4.10) it is easy to see that constitute 32 odd constraints. From the relations for spinors (4.15) and (4.18) one finds that diagonal blocks of the supermatrix (4.27) equal . (4.29) This allows to identify 28 bosonic constraints To identify other bosonic constraints consider D.B. relations of the odd constraints. The form of the kinetic term (4.26) allows to find basic D.B. relations of the supertwistors (4.34) With these in mind we obtain necessary D.B. relations of the odd constraints while −Z c Z c +ΨṙΨṙ andZċZ˙c +Ψ r Ψ r are non-zero since the constraints on the l.h.s. of (4.36) constitute 16 odd second-class constraints. Summing up constraints (4.37) and using the incidence relations (4.9), (4.10) gives The constraint on the r.h.s. is the square root of the mass-shell constraint (4.20). This fixes the sign ambiguity in the definition of the square root of the mass-shell constraint. The difference of the constraints (4.37) is also the constraint and equals the difference ofZċZ˙c +Ψ r Ψ r and −Z c Z c +ΨṙΨṙ so one can set As a result the products of supertwistors invariant under four different SU(2) R subgroups of SU(2, 2|4) R can be parametrized as with an arbitrary real q. It is then easy to find another Q-independent linear combination of the supertwistor productsZ c Z c +ZċZ˙c −Ψ r Ψ r +ΨṙΨṙ = 4q. where is the sum of the AdS energy (conformal dimension) generator E =Ψ r Ψ r +ΨṙΨṙ and its counterpart in the su(4) R algebra C =Z c Z c −ZċZ˙c. Generator T ∈ su(2, 2|4) R is T =Z c Z c +ZċZ˙c +Ψ r Ψ r −ΨṙΨṙ ≈ 0 (4. 45) and the constraint (4.42) equals the generator U ∈ u(2, 2|4) R . After taking into account all of the constraints, the supermatrix (4.27) is found to be proportional to the unit supermatrixZ Thus Z A N is an element of U(2, 2|4) × R modulo the gauge symmetries generated by the first-class constraints that we have to identify.
The choice of the first-and second-class fermionic constraints was made after Eq.  where where the kinetic term is given in (4.26) and the second summand is the linear combination of the constraints with Lagrange multipliers L constr = Λ b a L a b + Λ˙b˙aMȧ˙b + Λ p q R q p + ΛṗqSqṗ + Λ E+C (E + C) + Λ T T + Λ U U + iΛ a pΨ p Z a + iΛ p aZ a Ψ p + iΛṗ˙aΨṗZ˙a + iΛ˙aṗZȧΨṗ + Λ˙b aZ a Z˙b +Λ aḃZḃ Z a + Λṗ qΨ q Ψṗ +Λ qṗΨṗ Ψ q + iΛȧ pΨ p Z˙a + iΛ pȧZȧ Ψ p + iΛ aṗΨṗ Z a + iΛṗ aZ a Ψṗ. (4.52) Analysis of the conservation conditions of the constraints yields that the Lagrange multipliers for the second-class constraints in the two last lines in (4.52) turn to zero, while those for the first-class constraints in the first and second lines remain unfixed.

Four-supertwistor formulation of massless superparticle on AdS × S superbackground
Reduction of the Lorentz-harmonic variables was discussed in section 3 and the results obtained there also apply to the D = 10 superparticle model. Superparticle's 10-momentum components tangent to AdS 5 can be brought to the form Reduction of the Spin(5) variables proceeds in the similar way. Resulting expression for the momentum components tangent to S 5 is Expressions for the momentum components (5.1) and (5.3) were the starting point to derive the four-supertwistor formulation of the massless superparticle on AdS 5 × S 5 superbackground from the first-order form of the superspace formulation [48]. Resulting foursupertwistor representation of the superparticle's Lagrangian coincides with that obtained in Ref. [22] by partial gauge fixing 2T superparticle model in 2 + 10 dimensions. There are seven bosonic first-class constraints quadratic in supertwistors. These are the generators of the su(2|2) gauge superalgebra named color superalgebra in [21]. In our previous work [48] massless superparticle model in the four-supertwistor formulation was quantized in the simplest case when it propagates only within the AdS 5 subspace of the AdS 5 × S 5 superspace, i.e. momentum components in directions tangent to S 5 vanish. In supertwistor formulation this amounts to vanishing of Ψ A q andΨ q A supertwistors so that only the c-type supertwistors Z A a andZ a A are dynamical variables. It was demonstrated that the states of quantized superparticle coincide with the D = 5 N = 8 gauged supergravity multiplet [53] both in the supertwistor and superoscillator approaches. In general all four supertwistors contribute. Their (anti)commutation relations are found to be To set up the stage for superambitwistor quantization let us introduce individual notation for the components of c-and a-type supertwistors (5.10) Let us further realize components of quantized supertwistors as the multiplication and differentiation operators and where odd derivatives are defined to act from the left. Then the constraints (5.6) and (5.7) become differential operators acting in superambitwistor space. Let us a la Gupta-Bleuler quantization approach assume that the superparticle's wave function is annihilated by the diagonal elements of the bosonic constraint matrices L 1 1 = −L 2 2 ≈ 0 and R 1 1 = −R 2 2 ≈ 0, as well as off-diagonal elements L 2 1 ≈ 0 and R 2 1 ≈ 0 that are given by the differential operators of utmost the first-order. As a result the superparticle's wave function satisfies the following equations and . Besides that impose on the superparticle's wave function the equations that stem from the odd constraints and are given by the utmost first-order differential operators. In the realization (5.11), (5.12) two odd constraintsW A Ψ A ≈ 0 andΞ A Z A ≈ 0 translate into algebraic equations that together with two algebraic equations in (5.13) can be taken into account by introducing the delta-function factors Additionally two pairs of the conjugate constraintsΨ A Z A ≈ 0,Z A Ψ A ≈ 0 andΞ A W A ≈ 0,W A Ξ A ≈ 0 yield first-order differential equations. Assuming that in each pair of the equations associated with these constraints one equation is imposed on the superparticle's wave function and another on its conjugate gives four possible sets of equations that will be analyzed elsewhere. As an example we present one set of such equations where in the first equation it is assumed that SU(2, 2) L index has parity zero and SU(4) L index has parity one. Since the differential operators in (5.19) annihilate delta-function factors introduced in (5.18) we obtain

Conclusion
In the present paper we addressed the issue of deriving supertwistor formulation for D = 10 massless superparticle on AdS 5 ×S 5 superbackground starting with the first-order representation of the superspace Lagrangian, in which momentum components tangent to AdS 5 and S 5 are expressed via the Spin(1, 4) and Spin(5) variables. Resulting supertwistors coincide with those found in [21] so that we not only established their origin but also derived the incidence relations with the AdS 5 ×S 5 superspace coordinates via the P SU(2, 2|4)/(SO(1, 4) ×SO (5)) supercoset representative. Using the incidence relations we have identified the set of bosonic and fermionic constraints that these supertwistors satisfy and calculated their classical algebra. It appears that among the constraints there are the second-class ones that complicates canonical analysis of the model. We have shown that using the constraints for the spinor variables it is possible to exclude half of them from the expressions for the momentum components reducing the eight-supertwistor form of the superparticle's Lagrangian to the four-supertwistor form. Respective supertwistors were proposed in [22] and the incidence relations with the AdS 5 × S 5 coordinates were obtained in [48]. This not only explains the origin of both kinds of supertwistors pertinent to the AdS 5 × S 5 superparticle model but also establishes the relation between them. The advantage of the four-supertwistor formulation is the presence of only the first-class constraints. As a result we have obtained the set of equations for the superparticle's wave function in the superambitwistor space that generalize those of Ref. [48].
In general the distinctive feature of the twistor formulations is that the Lagrangian of the superparticle models is quadratic in supertwistors as well as the constraints. This is of particular importance for curved superbackgrounds such as AdS 5 × S 5 one, for which, depending on the parametrization, supervielbein components may be highly non-linear in the superspace coordinates. The difficulty with the supertwistor formulation is to provide a space-time interpretation of the states of quantized model. In the case when the superparticle moves within the AdS 5 subspace of the AdS 5 × S 5 superspace we succeeded in [48] to map its quantum states to those of the supermultiplet of D = 5 N = 8 gauged supergravity [53] both in the superoscillator and supertwistor approaches. Next task is to extend these results to the generic case and find a relation between the components of the superparticle's wave function and the towers of Kaluza-Klein states of the IIB supergravity compactified on AdS 5 × S 5 superbackground.
We also plan to examine the spectrum of quantized tensionless supertwistor string on AdS 5 × S 5 sketched in [48] and its relation to the higher-spin supermultiplets emerging on both sides of AdS 5 /CF T 4 duality in the limit of vanishing 't Hooft coupling [77], [78]. Supertwistor formulation of the tensile AdS 5 × S 5 superstring is another direction of the generalization of the results reported here. Such a formulation could be an interesting alternative to the superspace formulation based on the P SU(2, 2|4)/(SO(1, 4) × SO (5)) supercoset sigma-model [79], [80].
Finally in [22] there were considered twistor gauges for superparticle models relevant to other maximally supersymmetric backgrounds. It is of interest to obtain incidence relations for these supertwistors starting with the superspace formulations of conventional superparticle models on such superbackgrounds and study their quantization.

A Details of the spinor algebra
In Appendix A of Ref. [48] there were collected the details of the spinor algebra in D = 2 + 4 and D = 1 + 4 dimensions alongside with the realization of the D = 1 + 4 γ-matrices with manifest SL(2, C) symmetry that was used for the supermatrix realization of the generators of psu(2, 2|4) superalgebra as D = 4 N = 4 superconformal algebra. In order to spell out the relations between the D = 1 + 4 spinor and vector Lorentz harmonics parametrizing the SO(1, 4)/SO(4) coset space another realization of the D = 1 + 4 γ-matrices is used that manifests SO(4) ≃ SU(2) × SU(2) symmetry where ε ab , ε ab : ε ab ε bc = δ c a and ε˙a˙b, ε˙a˙b: ε˙a˙bε˙b˙c = δ˙c˙a are used to change the positions of the fundamental representation indices of the two SU(2) factors of SO(4). γ-matrix realization (A.1) and that used in Ref. [48] are related by the similarity transformation analogous to that connecting D = 1 + 3 γ-matrices in canonical and Weyl bases (see, e.g. [81]).
To write expressions for the D = 5 vector and spinor harmonics we use the following realization of D = 5 γ-matrices with manifest SO(4) covariance  where