An analytic superfield formalism for tree superamplitudes in D=10 and D=11

Tree amplitudes of 10D supersymmetric Yang-Mills theory (SYM) and 11D supergravity (SUGRA) are collected in multi-particle counterparts of analytic on-shell superfields. These have essentially the same form as their chiral 4D counterparts describing ${\cal N}=4$ SYM and ${\cal N}=8$ SUGRA, but with components dependent on a different set of bosonic variables. These are the D=10 and D=11 spinor helicity variables, the set of which includes the spinor frame variable (Lorentz harmonics) and a scalar density, and generalized homogeneous coordinates of the coset $\frac{SO(D-2)}{SO(D-4)\otimes U(1)}$ (internal harmonics). We present an especially convenient parametrization of the spinor harmonics (Lorentz covariant gauge fixed with the use of an auxiliary gauge symmetry) and use this to find (a gauge fixed version of) the 3-point tree superamplitudes of 10D SYM and 11D SUGRA which generalize the 4 dimensional anti-MHV superamplitudes.


Analytic superamplitudes in D=10 and
6. Convenient parametrization of spinor frame variables (convenient gauge fixing of the auxiliary gauge symmetries) 29 6.1 Reference spinor frame and minimal parametrization of spinor frames 29 6.2 Generic parametrization of spinor frame variables and K #I = 0 gauge 30 6.3 Internal frames and reference internal frame 31

Introduction
An impressive recent progress in calculation of multi-loop amplitudes of d=4 supersymmetric Yang-Mills (SYM) and supergravity (SUGRA) theories, especially of their maximally supersymmetric versions N = 4 SYM and N = 8 SUGRA [1,2,3,4,5], was reached in its significant part with the use of spinor helicity formalism and of its superfield generalization [6,7,9,10,11,12,13], working with superamplitudes depending on additional fermionic variables and unifying a number of different amplitudes of the bosonic and fermionic fields from the SYM or SUGRA supermultiplet. The spinor helicity formalism for D=10 SYM was developed by Caron-Huot and O'Connel in [14] and for D=11 supergravity in [15,16]. The progress in the latter was reached due to the observation that the 10D spinor helicity variables of [14] can be identified with Lorentz harmonics or spinor moving frame variables used for the description of massless D=10 superparticles in [17,18,19]. (Similar observation was made and used in D=5 context in [20]). The spinor helicity formalism of [15,16] uses the 11D spinor moving frame variables of [21,22,23,24].
As far as the generalization of D=4 superamplitudes is concerned, in [14] a kind of Clifford superfield representation of the amplitudes of 10D SYM was constructed. However, as this happens to be quite nonminimal and difficult to apply, the subsequent papers [25,26,27,28] used the D=10 spinor helicity formalism of [14] in the context of type II supergravity where the natural complex structure helps to avoid the use of the above mentioned Clifford superfields. An alternative, constrained superfield formalism was proposed for 11D SUGRA amplitudes in [15,16] and for 10D SYM in [16]. In it the superamplitudes carry the indices of 'little groups' SO(D − 2) i of the light-like momenta k a(i) of i-th scattered particles and obey a set differential equations involving fermionic covariant derivatives D + q(i) . This formalism is quite different from the 4D superamplitude approach; some efforts on development of the necessary technique and on deeper understanding of its structure are still required to be accomplished to make possible its efficient application to physically interesting problems.
In this paper we develop a simpler, 'almost unconstrained' analytic superfield formalism for the description of 11D SUGRA and 10D SYM amplitudes. In it the superamplitudes are multiparticle counterparts of an on-shell analytic superfields which depend on the fermionic variable in exactly the same manner as the chiral superfields describing N = 8 SUGRA and N = 4 SYM, but containing component fields depending on another set of bosonic variables, including some 'harmonic variables' (in terminology of [29,30,31]) w A q ,w qA parametrizing the coset Spin(D−4)⊗U (1) . These are used to split the set of (2N ) real spinor fermionic coordinates θ − q of the natural on-shell superspaces of 11D SUGRA and 10D SYM on the set of N complex spinor coordinates η − A and its complex conjugateη −A . The analytic on-shell superfields describing 11D SUGRA and 10D SYM depend on η − A but notη −A , and in this sense are similar to the chiral onshell superfields describing N = 8 SUGRA and N = 4 SYM. However, as in higher dimensional case η − A = θ − qwqA is formed with the use of 'harmonic variables', we call these superfields analytic rather than chiral.
We show how the analytic superamplitudes are constructed from the basic constrained superamplitudes, of 10D SYM and 11D supergravity and the set of complex (D − 2) component null-vectors U I i related to the internal frame associated to i-th scattered particle. We describe the properties of analytic superamplitudes and present a convenient parametrization of the spinor frame variables (gauge fixing with respect to a set of auxiliary symmetries acting on spinor frame variables) which allows to establish the relation between D=10, 11 superamplitudes and their 4d counterparts. Using this relation we have found a gauge fixed expressions for the on-shell 3-point tree superamplitudes. This can be used as basic elements of the analytic superamplitude formalism based on a generalization of the BCFW recurrent relations. The development of these latter and also the use of analytic superamplitudes to gain new insight in the development of the constrained superamplitude formalism will be the subject of future papers.
The rest of this paper has the following structure.
In the remaining part of the Introduction, after a resume of our notation, we briefly review the D=4 spinor helicity and on-shell superfield formalism. In sec. 2 we describe the D=10 and D=11 spinor helicity formalism as developed in [16,15]. In sec. 3 we review briefly the on-shell superfield description of 10D SYM and 11D SUGRA [21,16,15]. Analytic on-shell superfield formalism for 10D SYM and 11D SUGRA is developed in sec. 4. In sec. 5 we introduce the anlalytic superamplitudes and describe their poperties and their relation with constrained superamplitude formalism. The real supermomentum, which is supersymmetric invariant due to the momentum conservation, is introduced there. The convenient parametrization of the spinor frame variables (convenient gauge fixing of the auxiliary symmetries) is described in sec. 6 and used there to show the analyticity of a complex supermomentum which is constructed from the real supermomentum using the internal frame variables. The gauge fixed expression for 3-point analytic superamplitudes are presented in sec. 7. We conclude by discussion in sec. 8. Appendix A is devoted to spinor frame re-formulation of 4D spinor helicity formalism which is useful for comparision of 4D and 10/11D (super)amplitudes. Appendix B shows how to obtain the BCFW-like deformation of the 10/11D spinor helicity and complex fermionic variables from the deformation of real spinor frame and real fermionic variables found in [14,16,15].

Notation
As we will use many different types of indices, for reader convenience we resume the index notation here.
To avoid doubling of the equations which have similar structure in 10D and 11D cases, it is convenient to introduce the parameters N and s which take the values N = 4, 8 and s = 1, 2 for the case 10D SYM and 11D SUGRA theories, respectively, Notice that when we consider D=4 counterparts of our theory, we use the complex Weyl spinor indices α, β = 1, 2 andα,β = 1, 2 so that the above equations do not apply.
In the case of D=11 the dotted Spin (9) indices are identical to undotted,q = q, while for D=10 they are transformed by different (although equivalent) 8s and 8c representations of SO (8). The latter notation also applies to the 4D dimensional reduction of 11D and 10D theories, where A, B, C, D denote the indices of the fundamental representation of SU (N ) R-symmetry group. Finally a, b, c, d = 0, 1, ..., (D − 1) are D-vector indices. In D=4 we also use µ, ν, ρ = 0, 1, 2, 3 to stress the difference from D = 10 and D = 11.
The symbols i, j = 1, .., n are used to enumerate the scattered particles described by n-point (super)amplitude.

D=4 superamplitudes and on-shell superfields
A superamplitude of N = 4 SYM or N = 8 supergravity depends, besides n sets of complex bosonic spinors, on n sets of complex fermonic variables η A (i) ((η A (i) ) * =η A(i) ) carrying the index of fundamental representation of the SU (N ) R-symmetry group A, B = 1, ..., N , it obeys n super-helicity constraints, (1.8) It is important that the dependence of amplitude on fermionic variables is holomorphic: it depends on η A i but is independent ofη A(i) = (η A (i) ) * . Furthermore, according to (1.8), the degrees of homogeneity in these fermionic variables is related to the helicity h i characterizing dependence on bosonic spinors. Thus decomposition of superamplitude on the fermionic variables involves amplitudes of different helicities.
These superamplitudes can be regarded as multiparticle generalizations of the so-called onshell superfields which obey the super-helicity constraint The chiral superfields on a real superspace Σ (4|2N ) = {λ,λ, η,η} obeying Eq. (1.11) describe the on-shell states of N = 4 SYM and N = 8 SUGRA. Eq. (1.11) just fixs the charge of superfield with respect to a phase transformations of its arguments 1 , so this can be considered as almost unconstrained superfield on chiral on-shell superspace Such on-shell superfields can be obtained by quantization of D = 4 Brink-Schwarz superparticle with N -extended supersymmetry in its Ferber-Shirafuji formulation [38,39] (see also [40,41] as well as [42] and [43]). This observation has served us as an important guide: in [16] we show how to obtain the 10D and 11D on-shell superfield formalism from D=10 and D=11 superparticle quantization. Here we will not consider superparticle quantization but describe briefly the resulting constrained on-shell superfields and constrained superamplitude formalism of [15,16] and use these as a basis to search for the analytic on-shell superfields and analytic superamplitude formalism.
To conclude our brief review, let us present the expressions for its basic building blocks of the 4D superamplitude formalism, the 3-point superamplitudes of D=4 N = 4 SYM theory. These are two: the anti-MHV (MHV) The relative charges of bosonic and fermionic coordinates of this phase transformations can be restored from the relation between supertwistors and standard superspace coordinates [38]. In superamplitude context these relations can be found e.g. in [2]. and the MHV superamplitude A M HV (1, 2, 3) = 1 [12] [23] [31] δ 8 λα 1 η A1 +λα 2 η A2 +λα 3 η A3 . (1.14) Here we set the SYM coupling constant to unity and use the standard notation for the contraction of 4D Weyl spinors (1. 15) 2. Spinor helicity formalism in D=10 and D=11 As we have already mentioned in the Introduction, the D=10 spinor helicity formalism [14] and its D=11 generalization [15,16] can be constructed using the spinor (moving) frame or Lorentz harmonic variables. In this section we describe these, and, to start with, we introduce the vector frame variables (called light-cone harmonic variables in [44,45]).

Vector frame
Let us consider a vector frame associated with D-dimensional (D=10, 11) light-like momentum k a(i) , k a(i) k a (i) = 0, by the condition that one of the light-like vectors of the frame, say The additional index i will enumerate particles scattered in the process described by an on-shell amplitude. Below in this section, to lighten the equations, we will omit this index when this does not lead to a confusion. The condition (2.1) imply [44,45] u = a u a= = 0 , and also Notice that the sign indices = and # of two light-like elements of the vector frame may be considered as indicating their weight under the transformations of SO(1, 1) subgroup of the Lorentz group SO(1, D − 1), It is convenient to consider splitting of the vector frame matrix (2.1) on two light-like and (D −2) orthogonal vectors [44] This is manifestly invariant under the direct product SO(1, 1) ⊗ SO(D − 2) of the above scaling symmetry (2.7) and the rotation group SO(D − 2) mixing the spacelike vectors u I ai , If only one light-like vector u = ai of the frame is relevant, as it will be the case in our discussion below, the transformations mixing u # a(i) and u I ai can be also considered as a symmetry. These are so-called K (D−2) transformations (identified with Eucledean version of conformal boosts in [17,18] (2.10). This is the Borel subgroup of SO(1, D − 1) so that SO(1, D − 1)/H B coset is compact; actually it is isomorphic to the sphere S (D−2) . If we use H transformations as identification relation on the set of vector frame variables, these can be considered as a kind of homogeneous coordinates of such a sphere [17,18] Such a treatment as constrained homogeneous coordinates of the coset makes the vector frame variable similar to the internal coordinate of harmonic superspaces introduced in [29,30], and stays beyond the other names, vector harmonics or vector Lorents harmonics, which we also use for them (in [44,45] Emeri Sokatchev called these 'light-cone harmonics' ).
In the context of (2.2), S (D−2) in (2.12) can be identified with the celestial sphere of a Ddimensional observer. Notice that this is in agreement with the fact that a light-like D-vector defined up do a scale factor can be considered as providing homogeneous coordinates for the S (D−2) sphere The usefulness of seemingly superficial construction with the complete frame (2.12) becomes clear when we consider spinor frame variables, which provide a kind of square root of the light-like vectors of the Lorentz frame.

Spinor frame in D=10 and D=11
To each vector frame u we can associate a spinor frame described by , (2.14) and also of the charge congugation matrix if such exists in the minimal spinor representation of D-dimensional Lorentz group, if C exists f or given D .  Γ a αβ = σ a αβ = σ a βα andΓ a αβ =σ a αβ =σ a βα are 16×16 generalized Pauli matrices which obey σ aσb + σ bσa = 2η ab I 16×16 . We prefer to write this relation in the universal form which also describes the properties of symmetric 32×32 11D Dirac matrices. The charge conjugation matrix does not exist in 10D Majorana -Weyl spinor representation, so that there is no way to rise or to lower the spinor indices. The elements of the inverse of the spinor frame matrix are introduced as additional variables which obey the constraints V (2.20) In D = 11 α is the index of 32 dimensional Majorana spinor representation, and the spinor frame variables v ± αq carry the indices of the same Majorana spinor representation of SO(9), The symmetric 32 × 32 matrices Γ a αβ = Γ a βα = Γ a α γ C γβ andΓ a αβ =Γ a αβ = C αγ Γ a β γ , obeying (2.18), are constructed as the products of 11D Dirac matrices Γ a α γ = −(Γ a α γ ) * obeying the Clifford algebra, Γ a Γ b +Γ b Γ a = 2η ab I 32×32 , and of 11D charge conjugation matrix C γβ = −C βγ = −(C γβ ) * . This latter can be used to define the elements of inverse spinor frame matrix in terms of the same spinor frame variables For shortness below we will tend to write one equation describing both D=10 and D=11 variables/quantities when this is possible and cannot result in a confusion. Using such equations, one should keep in mind that in its 11D versionq ≡ q. For instance, the constraints (2.14) can be split on the following set of SO(1, When we use these to describe D = 10 case, I = 1, ..., 8 and 8×8 matrices γ I qṗ =γ Iṗ q are SO(8) Klebsh-Gordan coefficients. To describe D=11 case we consider I = 1, ..., 9,q = q = 1, ..., 16; the nine 16×16 matrices γ I qp = γ I pq are SO(9) Dirac matrices. In the Majorana spinor representation of SO(9) the charge conjugation matrix is symmetric and we identify it with δ qp .
In D=10 we have to keep in mind also the constraints for the inverse spinor frame variables [SO(1,1)⊗Spin(D−2)]⊂ ×K D−2 = S D−2 and K D−2 acts on the complementary spinor frame variables v + αq by In a model with [SO(1, 1) ⊗ Spin(D − 2)] ⊂ ×K D−2 gauge symmetry, v + αq does not carry any degree of freedom: any v + αq forming Spin(1, D − 1) matrix with given v − αq can be obtained from some reference solution of this condition, v + αq0 , by K D−2 transformations (2.30). This justifies the simplified form of (2.29) where only v − αp are presented as the constrained homogeneous coset coordinates: it indicates that, although the K D−2 is realized trivially on this minimal set of variables, we are allowed to choose an arbitrary representative of the set of v + αq 's complementing this till Spin(1, D − 1) valued matrix.

D=10 and D=11 spinor helicity formalism
When the vector frame is attached to a light-like momentum as in (2.2), In D=10 we should also mention the existence of the similar relations for the inverse spinor frame variables, and v − αq , and using (2.20) we easily find that these obey the massless Dirac equations (or, better to say, D = 10, 11 Weyl equations) Thus they can be identified, up to a scaling factor, with D=10 spinor helicity variables of [14] and with its D=11 generalization, Polarization spinor of the D=10 and D=11 fermionic fields can be associated with the elements of the inverse D=10 and D=11 spinor moving frame matrix, (2.36) For D = 11 (whereq = q) Eq. (2.36) is equivalent to (2.35) (2.37)

D=10 SYM multiplet in spinor frame/spinor helicity formalism
The polarization vector of the vector field can be identified with spacelike vectors u I a of the frame adapted to the light-like momentum of the particle by (2.31) (cf. [14]) so that the on-shell field strength of the D=10 gauge field can expressed by in terms of one SO(8) vector w I . It is easy to check that both Bianchi identities and Maxwell equations in momentum representations are satisfied, k [a F bc] = 0 = k a F ab . As we have already said, the polarization spinor can be identified with the spinor frame variable v −α q . Hence in the linear approximation, the on-shell states of spinor superpartner of the gauge field can be described by in terms of a fermionic SO(8) c-spinor ψq. Indeed, due to (2.34), the field (2.39) solves the free Dirac equation. When the formalism is applied to external particles of scattering amplitudes, the bosonic w I and fermionic ψq are considered to be dependent on ρ # and on the set constrained spinors v − αq (constrained homogeneous coordinates of the celestial sphere S 8 (2.29)) related to the momentum of the particle through (2.32), When describing the on-shell states of the SYM multiplet, it is suggestive to replace ρ # by its conjugate coordinate and consider the field on the nine-dimensional space R ⊗ S 8 . The supersymmetry acts on these 9d fields by where 8 component fermionic ǫ −q is the construction of the constant fermionic spinor ǫ α and the spinor frame variable, Let us stress that x = and v − αq are independent coordinates so that ∂ = v − αq ≡ 0 and, the parameter ǫ −q in (2.42) is also x = -independent, ∂ = ǫ −q ≡ 0 as far as ǫ α in (2.43) is a constant fermionic spinor parameter.

Linearized D=11 SUGRA in spinor frame/spinor helicity formalism
The linearized on-shell field strength of 3-form gauge field of 11D SUGRA (called 'formon' in [46]) can be expressed by in terms of light-like momentum (2.31), spacelike vectors u b I of the frame adapted to the momentum by (2.31), and an antisymmetric SO(9) tensor A IJK = A [IJK] (in 84 of SO (9)). The linearized on-shell expression for the Riemann tensor reads where the second rank SO(9) tensor h IJ is symmetric and traceless (in 44 of SO (9)) Finally the gravitino field strength solving the Rarita-Schwinger equations is expressed in terms of γ-traceless SO(9) vector-spinor Ψ Iq (128 of SO (9)) The set of on-shell fields h IJ , A IJK , Ψ Ip can be used to describe the supergravity mulitplet in light-cone gauge [47]. In our spinor helicity/spinor frame description, which can be deduced from the on-shell superfield formslism of [21], these fields depend on the density ρ # and spinor frame variables v − αq (homogeneous coordinates of S 9 , (2.29)) related to the momentum by (2.32), In the next section we will use the (superfield generalization) of the Fourier images of the above fields defined on R ⊗ S 9 space, 3. Constrained on-shell superfield description of the 10D SYM and 11D SUGRA The above described fields of the spinor helicity formalism of 10D SYM and 11D SUGRA can be collected in on-shell superfields, which can be considered as one-particle prototypes of tree superamplitudes. A constrained on-shell superfield formalism for linearized D=11 SUGRA and 10D SYM was proposed in [21] and generalized for the case of superamplitudes in [15,16]. We briefly describe this in this section and, in the next sec. 4, use it as a basis to obtain an almost unconstrained analytic superfield description of 10D SYM and 11D SUGRA.

On-shell superspaces for 10D SYM and 11D SUGRA
The constrained superfields describing 10D SYM and 11D SUGRA in the approach of [21] are defined on the real on-shell superspace with bosonic coordinates x = and v − αq , and fermionic The 10D and 11D supersymmetry acts on the coordinates of the corresponding on-shell superspace by This specific form indicates that our on-shell superspaces Σ (9|8) and Σ (10|16) can be regarded as invariant subspaces of the D=10 or D=11 Lorentz harmonic superspaces, i.e. of the direct product of standard 10D and 11D superspaces and of the internal sector parametrized by [16]) for details).

considered as homogeneous coordinates of the coset
The generic unconstrained superfield on Σ (D−1|2N ) (3.1) contains too many component fields so that it is not surprising that on-shell superfields on Σ (9|8) and Σ (10|16) describing D=10 SYM and D=11 SUGRA should obey some superfield equations. Such equations have been proposed in [21]. To write them in a compact form we will need the fermionic derivatives covariant under (3.2) These carry the indices of Spin(D − 2) group and obey d = 1 N = 2N extended supersymmetry algebra (3.4) The one particle counterpart of the superamplitude is actually given by Fourier images of a superfield in (3.1) with respect to x = . These will depend on the set of coordinates The fermionic covariant derivative acting on such Fourier-transformed on-shell superfields reads and obeys

On-shell superfields and superfield equations of 10D SYM
The basic superfield equations of D=10 SYM [21] D = 10 : are imposed on the fermionic superfield Ψq = Ψq( The superfield V I is defined by eq. (3.7) itself which also imply that it obeys This equation shows that there are no other independent components in the constrained on-shell superfield Ψq.

On-shell superfields and superfield equations of 11D SUGRA
The linearized 11D supergravity was described in [21] by a bosonic antisymmetric tensor superfield which obeys the superfield equation [21] D The consistency of Eq. (3.10) requires and then The tree superamplitude generalization of this constrained on-shell superfield formalism was proposed in [15,16]. Here we will develop an alternative unconstrained on-shell superfield approach, more similar to 4D one, and build on this basis an alternative superamplitudes formalism, which is potentially simpler to develop and to apply.

An analytic on-shell superfield description of 10D SYM and 11D SUGRA
In this section we present an almost unconstrained, analytic superfield formalism for the on-shell D=10 SYM and D=11 supergravity which is alternative to both the Clifford superfield approach (to 10D SYM) in [14] and to the constrained superfield formalism in [15,16]. We begin by solving the equations of the constrained on-shell superfields of 10D SYM from [21] in terms of one analytic on-shell superfield. Then we generalize this for 11D supergravity and describe the almost unconstrained analytic superamplitudes for both these theories.

From constrained to unconstrained on-shell superfield formalism for 10D SYM
To arrive at our unconstrained superfield formalism it is convenient to write the superspace equations (3.8) and (3.7) for on-shell superfield describing 10D SYM [21] in the form of The superfield V I in (3.7) and (3.8) is related to W I by V I = ∂ = W I . After such a redefinition, we can discuss the bosonic superfield W I as fundamental and state that Ψq is defined by the γ-trace part of (4.1).
We are going to show that, after breaking SO(8) symmetry down to its SO(6)=SU(4) subgroup, Eq. (4.1) splits into a chirality condition for a single complex superfield (Φ = W 7 + iW 8 ) and other parts which, together with split equation (4.2), allow to determine Ψq and all the remaining components of W I in terms of this single chiral superfield.

SU(4) invariant solution of the constrained superfield equations.
Breaking (6), we find that (4.1) implies It is important to notice that the matrices are orthogonal projectors and hance that (4.5) implies As, according to (4.8), the projectors P + and P − are complementary and complex conjugate, we can introduce complex 8×4 matrix w q A and its complex conjugatew qA such that In terms of these rectangular blocks Eqs. (4.9) can be written as chirality (analyticity) conditions The remaining parts of Eqs. (4.5) determine the fermionic superfield Ψq, Eq. (4.2) allows us to find also the derivatives of the remaining 6 components WǏ of the SO (8) vector superfield W I , (4.14) To conclude, we have solved the equations for constrained on shell superfields of 10D SYM, proposed in [21], in terms of one chiral (analytic) on-shell superfield Φ and its c.c.Φ (4.4).

The on-shell superfields are analytic rather than chiral
as well as that it is orthogonal to six mutually orthogonal real vectors U IǏ Now we can easily define SO(8) covariant counterparts of the projectors in (4.6) I U I U The elements of these real matrices can be combined in two rectangular 8 × 4 complex conjugate and factorize the orthogonal projectors (4.18) With a suitable choice of representation of 8d Clebsch-Gordan coefficients γ I qq =γ Iq q in terms of 6d ones the first equation in (4.19) can be split into which are analytic and anti-analytic, (4.11).
The expression for fermionic superfield Ψq can be written in the form of (4.13), but now with w andw factorizing the covariant projectors (4.23). It is also not difficult to write the covariant counterpart of the expression (4.14) for other 6 projections WǏ = W J UǏ J of the 8-vector superfield W I .

Analytic superfields and harmonic on-shell superspace
Thus we have solved the superfield equations for constrained on-shell superfields of D = 10 SYM in term of the analytic superfield Φ obeying the chirality-type equation (4.11) with complex fermionic derivatives (4.12) defined with the use of (4.22), 'harmonic variables' in terminology of [29,30,31].
These analytic superfields are actually defined on an 'harmonic on-shell superspace' which can be understood as direct product of the on-shell superspace (3.1) and the Spin( Supersymmetry acts on the coordinates of the harmonic on-shell superspace by (cf. (3.2)) and leaves invariant the covariant derivatives (3.3) as well asD + A =w qA D + q used to define analytic superfields Φ byD + A Φ = 0, (4.11). To see the analytic superfields as unconstrained superfields on a sub-superspace of (4.29), we have to pass to the analytic coordinate basis.

Analytic subsuperspace of the harmonic on-shell superspace
The presence of additional harmonic variables allows to change the coordinate basis of the harmonic on-shell superspace Σ (3(D−3)|2N ) to the following analytical basis The supersymmetry acts on the coordinates of this basis by This is generated by the differentail operators and leaves invariant the covariant derivatives 2 (4.36) It is not difficult to see that supersymmetry (4.33) leaves invariant analytical on-shell superspace Σ The above defined analytic superfields are unconstrained superfield on this analytic sub-superspace, The supersymmetry transformation of the analytical superfields are defined by Φ ...), or equivalently, For our discussion of the amplitudes, it will be useful to consider a Fourier image of an analytic superfield with respect to x = L , The supersymmetry acts on the Fourier image of the analytic superfield as This analytic on-shell superfield of 10D SYM can be decomposed in series on complex fermionic variable, To be rigorous, one might want to write the L symbol also on the fermionic derivatives in (4.36),∂ + A →∂ + A L , ∂ +A → ∂ +A L . We, however, prefer to make the formulae lighter and write this symbol on the bosonic derivative ∂ L = only.
The sign and numerical superscripts of the fields describe their charge with respect to U (1) group acting on η A = η − A . The origin of the analytic superfield in components of SO(8) vectors suggests that its charge is equal to -2, Φ = Φ (−2) . This can be expressed by the differential equation is the 10D counterpart of the helicity operator (more precisely, the counterpart is given by − 1 2ĥ (10D) ). 3 The spectrum of the above component fields coincide with fields of N = 4 D=4 SYM. However, these fields depend on different set of bosonic variables: on 1+8+12=21 instead of (λ,λ) = CP 1 in 4D case. The indices A, B = 1, ..., 4 of the fermionic coordinates and of some of the on-shell component fields of 10D SYM are transformed by SU (4). However, in distinction to the rigid SU (4) Rsymmetry group of N = 4 D=4 SYM, in ten dimensional theory SU (4) is a gauge symmetry: it is used as identification relation on the set of constrained w A q ,w Aq variables (4.22) making them generalized homogeneous coordinates of the SO (8) SU (4)⊗U (1) coset ('harmonic variables' or 'harmonics' in terminology of [29,30,31]).

Dynamical realization (spontaneous breaking) of SO(9)
To find the general solution of such a type, we need to break spontaneously the SO(9) little group symmetry down to SO(7) × SO(2) by introducing the bridge or harmonic variables providing a kind of constrained homogeneous coordinates for the  The condition that the matrix is orthogonal, (4.49), is equivalent to the set of relations which are described by (4.16) and (4.17), but now with I, J = 1, ..., 9 andǏ,J = 1, ..., 7. Eqs.  Hence U /Ū //4 and its complex conjugateŪ /U //4 are orthogonal projectors and thus can be factorized in terms of complex 16×8 matrices w q A = (w pA ) * which obey Eqs. (4.21) and (4.22).
Another useful observation is that the second equation in (4.54) with J = 8, 9 can be written in the form where the complex symmetric matrices U AB andŪ AB = (U AB ) * obeȳ In this sense one can roughly state that w A q is a square root of the complex nilpotent matrix U / qp .

Analytic on-shell superfield from constrained on-shell superfield
Using the above bridges it is not difficult to extract an analyticity (chirality-type) condition from the equations (3.10)-(3.12) of the constrained superfield formalism. To this end, let us define a complex superfield Multiplying (3.12) on U I U J we find that this obeys which, in the light of (4.51), implies (Ū /U /) pq D + q Φ = 0. Using the factorization of the projector (4.53) and Eqs. (4.22) we write this as an analyticity (chirality-type) conditions As in the case of 10D SYM, the remaining parts of (4.60) and (3.12), as well as (3.10) and (3.11), can be used to obtain the expression for other components of the constrained superfields in terms of Φ and its complex conjugateΦ.
Eq. (4.61) defines an analytic (chiral-type) superfield, which depends on η A but not on its c.c.η A = (η A ) * . Notice that this complex fermionic variable η A is almost identical with the one used in the description of N = 8 4D supergravity; the 'almost' refers to the fact that (with the assumption that (w A q ,w qA ) parametrize the coset SO(9) SO(7)⊗SO(2) ) only SO (7) subgroup of SU (8) acts on its index A 4 . The decomposition of our analytic superfield in η A looks very much the same as chiral superfield (1.9) describing the linearized N = 8 supergravity. However, all the fields in its decomposition depend on a different set of variables: The set of 24 bosonic variables our on-shell fields depend on includes 'energy' ρ # , spinor frame variables v − αq which are considered as (a kind of) homogeneous coordinates of the celestial sphere S 9 realized as a coset of Lorentz group Spin(1,10) [SO(1,1)×Spin(9)]⊂ ×K 9 , (2.29), and a set of Spin(9) Spin(7)⊗Spin (2) harmonic variables w A q ,w Aq (4.56). The signs and numerical superscribes of the component fields in (4.63) and (4.62) indicate their charges under U (1) symmetry transformations acting on η A ,w and w, in the assumption 4 Probably, to observe the SU(8) symmetry, one has to consider (w,w) as parametrizing the coset SO(16)/[SU (8) ⊗ H] with some H ⊂ SO (16). Thus a hidden SO(16) symmetry of 11D SUGRA might be relevant in this problem. It is tempting to speculate that E8 hidden symmetry might also happen to be useful in this context. 5 To streamline the presentation at this stage we prefer to pass to the Fourier image of the superfields with respect to x = (actually x = L = x = + 2iηAη A ) coordinate ((2.48) vs (2.49)).
that η A = η − A and that the overall charge of the superfield is equal to −4 6 . With this convention these sign indices also indicate the weight of the fields and of the fermionic coordinate under SO(1, 1) subgroup of SO(1, 10).

Supersymmetry transformation of the analytic superfields
As in the case of 10D SYM, we can find that supersymmetry transformations acts on the analytic on-shell superfield of 11D supergravity by and ǫ α is constant fermionic parameter of rigid 11D supersymmetry. The supersymmetry generator defined by Φ ′ = e −ǫ α Qα Φ, is given by the sum of the algebraic partq α and of the differentail operatorq α , However, in distinction to the D=4 case, to split the parameter of rigid supersymmetry ǫ α on the parts corresponding toq α andq α we need to use the bosonic spinor variables v +α q w A q and v +α qwqA (while in D=4 the splitting appears automatically becauseq To show that the algebra of supersymmetry generators (4.70) is closed on the momentum, we have to use (4.21), 2w A (qw p)A = δ qp .

Properties of analytic superamplitudes
The simplest superamplitudes are multiparticle counterparts of the chiral superfields (4.62) and (4.43) Notice that here, in distinction to sec. 2 devoted to D=4 case, we prefer to write explicitly the momentum preserving delta function δ D ( n i k ai ) = δ D ( n i ρ # i u = ai ) and denote by A the amplitudes with arguments obeying the overall momentum conservation The representations of the variables and superamplitude with respect to the symmetry groups are summarized in Table 1, where the parameter s = N /4 distinguish the cases of D=10 SYM (s=1) and D=11 SUGRA (s=2).  (They also can be written as a differential equations with the use of covariant derivatives in the space of Lorentz harmonic and internal harmonic variables, similar to (4.47) and (4.65), but we will not discuss these here).
More complicated superamplitudes, which do carry the nontrivial representations of SO(D − 4) i and different charges under SO(2) i = U (1) i can be obtained by acting on the analytic superamplitude (5.1) by covariant derivatives D + A(i) .

From constrained to analytic superamplitudes
Let us discuss the relation of the above described analytic superamplitude (5.1) with the constrained superamplitude formalism [15,16]. The basic constrained superamplitude of 10D SYM theory where (5.7) To express the analytic superamplitude (5.1) through the constrained superamplutide, let us first contract their SO(8) i vector indices with the complex null-vectors U I i of the internal frames In these equations Our analytic 10D SYM superamplitude is related to (5.8) by  Due to (5.16) and (5.17),Ã n of (5.14) obeys (5.10), and the 11D superamplitude (5.12) is analytic, i.e. it depends on η − Ai but is independent on its complex conjugateη −A i .

Supersymmetry transformations of the analytic superamplitudes
The supersymmetry acts on our analytical superamplitudes as and As in the case of the on-shell superfields, the supersymmetry generator acting on superamplitude splits onto the purely algebraic part and the differential operator It is easy to check (using 2w (q|(i) Aw |p)A(i) = δ qp (4.21)) that the generators (5.21) obey the supersymmetry algebra, and actually anti-commute as far as the momentum is conserved, Like in D = 4 case (see [11]), the supersymmetry can be used to set to zero two of the complex fermionic coordinates η A(i) the amplitude depend on, η A(1) = 0 = η A(2) , if the light-like momenta of the first and the second particles are not parallel. This cannot be the case for the 3-point on-shell amplitude, in which we can set to zero one fermionic argument only.

Supermomentum in D=10 and D=11
Although we have succeed in writing the supersymmetry generator in terms of complex η − Ai and its derivative, the simplest way to write a supersymmetric invariant linear combination of fermionic variables uses real fermionic θ − qi of the constrained superfield formalism (see (4.32)). Indeed, the real fermionic spinor which can be called supermomentum, is transformed into the momentum by supersymmetry and, hence is supersymmetric invariant when momentum is conserved, 6. Convenient parametrization of spinor frame variables (convenient gauge fixing of the auxiliary gauge symmetries)

Reference spinor frame and minimal parametrization of spinor frames
It looks convenient to fix the gauge under the defining gauge symmetries of the spinor frame variables [SO (1,1) is an auxiliary reference spinor frame the components of which can be identified with homogeneous coordinates of an auxiliary coset (or reference coset)
Eq. (6.1) provides the explicit parametrization of the spinor frame (v − αqi , v + αqi ) describing a celestial sphere of a D-dimensional observer by SO(D − 2) vector K =I i . This is manifestly invariant under one set of [SO(1, 1) ⊗ SO(D − 2)] ⊂ ×K D−2 gauge symmetries acting on the reference spinor frame variables; these should also act on Spin(D − 2) indices q, p = 1, ..., 8s of 'internal frame' variables w A qi ,w qAi . Notice that (6.1) implies the vector frames relation The momentum of i-th particle is expressed through K =I i and density ρ # i , Thus the gauge (6.1) the n-point amplitude (5.1) is a function of energies ρ # i , of SO(D − 2) vectors K I i as well as of the fermionic η Ai the constrained complex bosonic w i ,w i variables, 6.2 Generic parametrization of spinor frame variables and K #I = 0 gauge A generic parametrization of the spinor frame variables (2.16) is In D = 10 case we have to complete (6.7), (6.8) by (6.10) while in D = 11, whereq = q, these equations are equivalent to (6.7) and (6.8).
Eq. (6.7) and (6.8) imply For our discussion below it is even more useful to notice that In the gauge (6.1) this simplifies to (6.15) and becomes antisymmetric in i, j. This latter fact suggests to search the 10D (and 11D) counterparts of the 4D expression < ij > (1.15) on the basis of (6.15). The complete parametrization of the vector frame variables corresponding to (6.7), (6.8) is given by (6.16) and quite complicated expressions for u # ai and u I ai . The light-like momentum of i-th particle has the form of (6.5), but with a redefined ρ # i , The expressions for u # ai and u I ai simplify essentially if we use the K (D−2)i symmetry to fix the gauge The spinor frame parametrization with K #I i = 0, (6.18), is given by the same Eqs. (6.7) and (6.9), while Eqs. (6.8) and (6.10) simplifies essentially: Then v +α pi v − αqj = e α i −α j (O i O T j ) pq and we find the following equivalent form of (6.13) The advantage of this formula is that both its non-contracted indices are transformed by the same

Internal frames and reference internal frame
As SO(D − 2) i auxiliary gauge symmetry acts not only on i-th spinor frame but also on i-th internal frame (w qA i , w A q i ), the introduction of the reference spinor frame in (6.7), (6.8) should be accompanied by the introduction of the reference 'internal frame', the constrained variables (w qA , w A q ) parametrizing the coset SO(D−4)⊗SO(2) (reference coset). The i-th internal frame variables can be decomposed on this reference frame, In the case of D=10, we must also introduce the internal reference frame with c-spinor SO (8) indices, (wq A , w Ȧ q ) and relate it to i-th internal frame bȳ  26) or, equivalently by Tracing these latter matrix with (6.21) we find an important relation v +κ (6.29) where N = 4 for D=10 SYM case we are working with. The equations with N = 8 andq = q are also valid and are relevant for 11D SUGRA.
The above definitions allow to express the complex spinor frame variiables v − in terms of contractions of reference spinor frame and reference internal frame, In particular, one finds When deriving these equations the following consequences of (4.21), (4.26) and (4.27) are useful

3-point analytic superamplitudes in 10D and 11D
One of the important ingredient to construct D=4 superamplitudes is the supermomentum, the counterpart of (5.25) which in D=4 can be split onto chiral (analytic) and anti-chiral parts. The delta function of the chiral part of supermomentum enters the basic MHV superamplitude (1.14). This suggests to begin our search for 3-point amplitudes of 10D SYM and 11D SUGRA by looking for a possibility to define a projection of supermomentum (5.25) which is analytic or becomes analytic when the D=10 or D=11 amplitude includes the particles all the momenta of which lay in the same 4-dimensional subspace. After finding such a projection of supermomentum in sec. 7.1 we pass, in sec. 7.2, to the study of three particle kinematics which then allows us, in sec. 7.3 to write the explicit form of the gauge fixed expression for the 3-point superamplitudes of 10D SYM and 11D SUGRA.

Searching for an analytic projection of supermomentum
The reference spinor frame of sec. 6.1 can be used to split the real spinorial supermomentum (5.25) on two real 2N (= 8s) component fermionic invariants, In the gauge (6.1) and it becomes clear that only q − q depends on K =I i defining the direction of the i-th particle momentum through (6.17) 7 . 7 The generic expressions for the components of supercharge in the parametrization (6.7), (6.9) and (6.10) read Now we can use the reference internal frame (wq A , w Ȧ q ) to replace the real 2N (= 8s) component supersymmetric invariant q − q by the complex N (= 4s) component and its complex conjugateq −A . An immediate important observation is that if the SO(D − 2) vectors K =I i characterizing the momentum of i-th scattered particle is situated in a plane spanned by the reference internal frame vectors U I ,Ū I , then q − A depends on corresponding η − Ai , but not on its complex conjugateη −B i , Then, if all n SO(D − 2) vectors K =I i are located in the U I -Ū I , plane, i.e., when the supermomentum q − A is analytic. We will use this fact in discussing the form of analytic 3-point superamplitude.
Similar conclusion can be made for the projection of supermomentum made with the use of i-th spinor frame and internal frame variables instead of the reference ones, To make shorter the second line of this equation we have used the notatioñ Again, it is not difficult to check that (7.7) is analytic if K =I ji UJ I = 0 , ∀ i = 1, ..., n , (7.9) this is to say = 0 . (7.10)

Three particle kinematics and supermomentum
Let us study 3-particle kinematics in the vector frame formalism. With (2.2) we can write the momentum conservation as Then, using (6.2)-(6.4) we split (7.11) into Eq. (7.12) makes (7.13) equivalent to 14). Using (7.15) and (7.12) we find that (7.14) implies The solution of Eq. (7.16) for real vectors K =I ji are trivial. Thus a nontrivial on-shell 3-particle amplitude can be defined only for complexified K =I ji which implies that the light-like momenta k a i of the scattered particles are complex. The general solution of the momentum conservation conditions can be written in terms of say K =I 1 and complex null vector K I as Notice that, to make the above equations valid for arbitrary parametrization (6.16), it is sufficient just to rescale the scalar densities (see (7.8)) In particular, are valid for a generic parametrization of the spinor frame. Now, using the SO(D − 2) gauge symmetry of the reference spinor frame we can chose the light-like vector U I of the reference internal frame to be proportional to the null-vector K I so that K I ∝ U I and (7.19) implies K =I ji UJ I = 0 ∀ i, j = 1, 2, 3 (7.20) and the N -component projection of supermomentum ((7.7) with n=3) on such reference frame is analytic, For our discussion below it will be useful to calculate the explicit form of these analytic projections of the supermomentum, or better their redefined versions All the three projections are supersymmetric invariant, and so is any of its linear combinations, for instanceq (7.26) where in the second equality we have used the momentum conservation conditions (7.12). Inverting (7.26) and its cousins we find which will be used below to find the expression for the 3-point superamplitude in D=10, 11 from its D=4 counterpart. Notice that one can also obtain a complementary, K =I ji ↔ρ # k =i,j relation

3-points analytical superamplitudes in 10D SYM and 11D SUGRA
A suggestion about the structure of 10D and 11D tree superamplitudes may be gained from the observation that, when the external momenta belong to a 4d subspace of the D-dimensional space, they should reproduce the known answer for a 4-dimensional tree superamplitudes of N = 4 SYM and N = 4 SUGRA, respectively.
When all the momenta of higher dimensional scattered particles belong to one 4-dimensional subspace of D=10 or D=11 momentum space, we can choose the reference spinor frame and reference internal frame in such a way that all the SO(D − 2) vectors K =I ji , characterizing the spinor frames associated to the scattered particles by (6.7) and (6.14), belong to the same 2dimensional space the basis of which is provided by U I andŪ I . Then all K =I ji obey (7.9). To resume, in the parametrization (6.7) of spinor frames, the quasi-4D configuration of the momenta of scattered particles is characterized by (7.9). Importantly, this is just the condition for the supermomentum projections (7.4) to be analytic, (7.5).

Gauge fixed form of the 3-points analytical superamplitudes
We chose as a D=4 reference point the anti-MHV superamplitude of N = 4 SYM (1.13). As we show in the Appendix A, its gauge fixed form, covariant under only one of three U (1) symmetries, reads (see (A.1.43)) (7.30) Here K = 21 is a complex number, which can be considered as 2-vector, K = 21 = K =1 21 + iK =2 21 , a D=4 counterpart of the generic (D − 2)-vector K =I 21 in (6.14). It is tempting to identify it with K =I 21Ū I of the previous section, The argument of the fermionic delta function in (7.29) also has a strightforward 10D counterpart (7.23). These observations suggest to propose the following 'gauge fixed' expression for the 3-point amplitude of D = 10 SYM theory The multiplier e −2i(β 1 +β 2 +β 3 ) is strictly necessary to make amplitude invariant under U (1) symmetry acting on the reference internal frame variables and to supply it instead with charges −2 with respect to all U (1) i groups, i = 1, 2, 3, related to scattered particles. All other variables we have used in (7.32) are redefined in such a way that they are inert under The 1-2-3 symmetry, although non-manifest, is actually present in (7.32): using (7.15), we can express the first multiplier in terms of any combination the argument of the fermionic delta function and (7.32) can be expressed in terms of (q

A more generic expression for analytic 3-point superamplitude of 10D SYM
A bit less parametrization-dependent expression for D=10 3-point superamplitude reads (for simplicity we omit the overall constant coefficient) Notice that the symmetry in 1-2-3 permutations, although not manifest, is actually present in it due to (7.28). In (7.35) we have introduced the notation (cf. (6.28), (6.29)). This expression serves as a counterpart of the D=4 < ij > (see (1.15)), but it is not antisymmetric in ij. Moreover, it is not a unique analog of < ij >: we can also attribute this property to the expression One can check that in the special parametrization (6.18) (see (6.7), (6.20), (6.9) and (6.28)) ij] =ρ # j e 2iβ i K =I jiŪI = − [ij (7.38) holds. In the generic parametrization with nonvanishing K #I ij the relation between ij] and [ij looks more complicated. However, as we have stressed many times, K #I i parametrize the freedom in choosing a reference spinors and vectors complementary to the spinor frame variables related to the momentum and 'helicity' degrees of freedom, so that to fix it by K #I i = 0 is not a restriction of generality. Substituting (7.38) into (7.35) brings us back to (7.32).

Analytic 3-point superamplitude of D = 11 supergravity
Similarly, the form of 3-point N = 8 4D supergravity superamplitude, which is essentially the square of the N = 4 4D SYM one (see e.g. [11,12]), suggests the following form of the basic 3-point superamplitude of 11D supergravity, In the second line of this equation, (7.40), we just have written explicitly the expression for Grassmann delta function. Of couse the same can be done in the case of 10D amplitudes. Notice that both (7.35) and (7.39) still heavily depend on the explicit parametrization of the spinor frame and on the internal frame variables (6.7), (6.20) and also on the choice of reference frames. In particular, it includes The further development of the formalism requires a deeper comprehension of the structure of superamplitudes, which can be reached either through finding a completely covariant (more parametrization independent) counterparts of the above (7.35) and (7.39), or by development of the gauge fixed formalism for superamplitudes. Both these possibilities are presently under investigation. The second way is actually quite convenient for using the experience of the 4D superamplitude calculus as far as setting U B A i = δ B A we reduce i SO(D − 4) i gauge symmetry to single SO(D − 4) ⊂ SU (N ) symmetry, and this latter can be recognized as R-symmetry of D=4 N = 4 SYM and N = 8 SUGRA.

Conclusion and discussion
In this paper we have constructed the basis of an analytic superfield formalism to calculate (super)amplitudes of 10D SYM and 11D SUGRA theories. This is alternative to the constrained superamplitud formalism of [15,16] and also to the 'Clifford superfield' approach of [14].
We have begun by solving the equations of the constrained on-shell superfield formalism of 10D SYM and 11D SUGRA [21,15,16] in terms of single analytic superfield, depending holomorphically on N = 4 and N = 8 complex coordinates, respectively. These 2N complex coordinates η − A are related to 2N real fermionic coordinates of the constrained superfield formalism θ − q by complex rectangular matricesw qA (= (w A q ) * ) which obey some set of constraints allowing to treat them as homogeneous coordinates of the coset SO(D−2) SO(D−2)⊗SO (2) or 'internal' harmonic variables (see [31]).
Similarly, the constrained n-point superamplitudes of the n i=1 SO(D − 2) i covariant constrained superfield formalism can be expressed in terms of analytic superamplitudes which depend, besides the n sets of 10D or 11D spinor helicity variables, also on n sets (w qA,i , w A q i ) of (2) i internal frame variables (internal harmonic). The sets of 10D and 11D spinor helicity variables, which include densities ρ # i and spinor frame variables v − αq i or (Lorentz harmonics), describe the light-like momenta and the "polarizations" (SO(D − 2) i small group representations) of the scattered particles. The constrained superamplitude, depending on these spinor helicity variables and (2N )-component real fermionic variables θ − qi , carry indices of the small groups SO(D − 2) i . In contrast, the analytic superamplitude does not carry indices but only charges −2s = −N /2 under the SO(2) i = U (1) i subgroups of SO(D − 2) i which act on the internal frame variables (w qA i , w A q i ) and on the complex coordinates η − Ai = θ − qiw qA i . We have found a parametrization of spinor frame variables which is especially convenient for the analysis of the analytic superamplitudes. This allowed us to establish the correspondence between basic building blocks of 4D superamplitudes and use it to find the expressions for analytic 3-point superamplitudes of D=10 SYM and D=11 SUGRA theories. These are the necessary ingredients of the basis for calculation of n-point superamplitudes, the problem which we intend to address in the forthcoming paper.
The first stages in this direction should include a better understanding of the structure of the 3-point analytic superamplitudes, in particular the search for its more convenient, more parametrization independent form, as well as the derivation of the BCFW-type relations for analytic superamplitudes 8 . These should be more closely related to the BCFW-type relations for D=4 superamplitudes, which are chiral and almost unconstrained, than the BCFW-type relations for real constrained 11D and 10D superamplitudes presented in [15] and [16].
In particular, one expects the BCFW deformations used in such recurrent relation to have an intrinsic complex structure, similar to the one of the D=4 relations [7]. As we show in Appendix B starting from BCFW deformations of spinor frame and fermionic variables in [15,16], which are essentially real, this is indeed the case. The resulting BCFW-like deformations of the complex spinor frame variables (6.30) and complex fermionic variables (4.32) has the structure quite similar to that of the 4D super-BCFW transformations [11] (presented in (B.2.1)-(B.

2.3) in Appendix B).
An alternative direction is to use the structure of analytic 3-point superamplitude for deriving the expression for its cousin from the real constrained superamplitudes formalism, and then to calculate the higher point unconstrained superamplitudes with the use of the BCFWtype recurrent relations presented in [15,16]. Both this and the further elaboration of the analytic superamplitude approach with the use of the above described BCFW-like deformations are presently under study.
It will be also interesting to reproduce the analytic superamplitude from an appropriate formulation of the ambitwistor string [50,51,52]. Notice that, although original ambitwistor string model [50] had been of NSR-type and had been formulated in D=10, quite soon [53] it was appreciated its relation with null-superstring [43] (see [54] for related results and [55] for more references on null-string) and with twistor string [6,56,57,55]. This suggested its existence in spacetime of arbitrary dimension, including D=11 and D=4, and the last possibility was intensively elaborated in [58,59,60]. An approach to derive the analytic superamplitudes from the Green-Schwarz type spinor moving frame formulation of D=10 and D=11 ambitwistor superstring [53] looks promising and we plan to address it in the future publications.
the set of such harmonic variables parametrize the sphere S 2 [17,18], When the spinor frame is associated with a light-like momenta by the generalized Cartan-Penrose relation In the scattering problem we can associate the spinor frame to each of n light-like momenta and to express the corresponding helicity spinors of (1.2) in terms of these spinor frames As we have used only v − α(i) , the complementary spinors v + α(i) remains arbitrary up to the constraint Actually, this is the statement of K 2 symmetry (parametrized by k # andk # in (A.1.2), (A.1.3)), which can be used as an identification relation on the set of harmonic variables (as indicated in (A.1.4)), and in this sense is the gauge symmetry. We can fix these K 2(i) gauge symmetries by identifying (up to a complex multipliers) all the complementary spinors of the spinor frames associated to the momenta of the scattered particles It is convenient to reformulate this statement by introducing an auxiliary spinor frame (v ± α ), not associated to any of the scattered particles, and to state that any of the spinor frames (v ± α(i) ) is related to that by (cf.
In this gauge the contractions of the spinors from different frames read Of course, we can use the SO(1, 1) i × SO(2) i gauge symmetries to fix also α i = 0 and β i = 0 ∀i = 1, ..., n, but the multipliers β i might be useful as they actually indicate the helicity of the field or amplitude, while α i can be 'eaten' by the 'energy' variables ρ # i . Indeed, the i-th light-like momentum (A.1.6) can be now written as whereρ # i = e −2α i ρ # i (7.18) and are two real and two complex conjugate (u +− a = (u −+ a ) * ) vectors of Newman-Penrose light-like tetrade (see [37] and refs. therein).
For our multidimensional generalization it is important that . (see [48,49] for the definition of t abc thensor). Notice that the last term in the second line of this equation vanishes as a result of (A. 1.19) and that at the last stage of transformations of this equation we have used the consequence of the momentum conservation in 3-particle process which we are going to discuss now.
A.1 Momentum conservation in a 3-point 4D amplitude In our notation the momentum conservation in the 3-particle process is expressed by  1.25). From now on we will denote these complex nonvanishing K = (1,2,3) restricted by 3-particle kinematics by Similarly, the fermionic delta function in (1.14) can be written as i.e. that K =İ 21 is a complex null 2-vector. To reconstruct the (gauge fixed) expression for 10D or 11D amplitudes from the known 4D expression we need to embed this 2-vector into a complex null (D − 2)-vector. Roughly speaking, when the external momenta of D-dimensional 3-point amplitude have a quasi 4d configuration, this is achieved by

B. BCFW-like deformations of complex frame and complex fermionic variables
An important tool to reconstruct tree D = 4 (super)amplitudes from the basic 3-point (super)amplitude is given by BCFW recurrent relation [7] and their superfield generalization [11]. The counterparts of these latter 4D relations for constrained superamplitudes of 11D SUGRA and 10D SYM have been presented in [15] and [16]. They use the real BCFW deformations of real bosonic and fermionic variables of the constrained superamplitude formalism. In contrast, in the case of the BCFW-type recurrent relations for analytic superamplitudes (which are still to be derivaied), one expects the BCFW deformations used in such recurrent relation to have an intrinsic complex structure, similar to the one of the D=4 relations [7,11] λ A (n) → λ A (n) = λ A (n) + zλ A (1) ,λȦ (n) → λȦ (n) =λȦ (n) , (B.2.1) Let show see how this can be reached starting from the BCFW deformations of spinor frame variables [15,16] which are essentially real, Here α = 1, .., 4N and q, p = 1, ..., 4N (we set N = 8 and 4 for 11D SUGRA and 10D SYM, respectively) and z is an arbitrary number. In principle this can be considered to be real z ∈ R [14], although z ∈ C is neither forbidden and actually more convenient in amplitude calculations. The above shift of spinor moving frame variables results in shifting the momentum of the first and of the n-th particle, k a (1) = k a (1) − zq a , k a (n) = k a (n) + zq a , (B.2.8) on a light-like vector q a orthogonal to both k a (1) and k a (n) , q a q a = 0 , q a k a (1) = 0 , q a k a (n) = 0 , (B.2.9) provided we choose We can also write the expression for light-like complex vector in terms of deformation matrix,