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Physics Letters B

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Ambitwistors, oscillators and massless fields on AdS5

D.V. Uvarov

NSC Kharkov Institute of Physics and Technology, 61108 Kharkov, Ukraine A R T I C L E I N F 0 A B S T R A C T

Article history: Positive energy unitary irreducible representations of SU(2, 2) can be constructed with the aid of

Received 19 August 2016 bosonic oscillators in (anti)fundamental representation of SU(2)L x SU(2)R that are closely related to

Received in rewsed form 6 September 2016 Penrose twistors. Starting with the correspondence between the doubleton representations, homogeneous

functions on projective twistor space and on-shell generalized Weyl curvature SL(2, C) spinors and their low-spin counterparts, we study in the similar way the correspondence between the massless representations, homogeneous functions on ambitwistor space and, via the Penrose transform, with the gauge fields on Minkowski boundary of AdS5. The possibilities of reconstructing massless fields on AdS5 and some applications are also discussed.

© 2016 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

Accepted 8 September 2016 Available online xxxx Editor: N. Lambert

1. Introduction

The problem of characterization of irreducible unitary representations of SU(2, 2) has rather long history in mathematical physics (see [1] and [2], where references to earlier literature can be found). In the mid 80-s the interest in positive energy unitary representations of corresponding supergroups SU(2, 2|N) [3,4] was stimulated mainly by the development of supersymmetry and su-pergravity, in particular the necessity to describe the spectrum of D = 10 IIB supergravity compactified on AdS5 x S5 that was shown [3,5] to be given by the infinite-tower of massless and massive representations of SU(2, 2|4). At the end of 90-s the interest in positive energy unitary representations of SU(2, 2|4) renewed (see, e.g., [6,7]) in the context of AdS5/CFT4 gauge/string duality, on both sides of which the SU(2, 2|4) supergroup constitutes finite-dimensional part of the infinite-dimensional symmetry related to integrable structure. More recently in the framework of vectorial AdS5/CFT4 duality it was shown [8] that the spectrum of D = 5 Vasiliev-type higher-spin gauge theories,1 dual to free 4d scalar, spinor or Maxwell theories, is described by the infinite set of the positive energy unitary representations of SU(2, 2) corresponding to massless fields.

Lowest-weight (positive energy) irreducible unitary representations of SU(2, 2) and SU(2, 2|N), as well as those of other (su-

E-mail address: d_uvarov@hotmail.com.

1 Higher-spin theories involved include like already known ones [9,10], as well as

those yet to be identified.

per)groups can be constructed using quantized oscillators carrying fundamental representation labels of the maximal compact subgroup [11,12]. In this approach (super)group generators are realized as bilinear combinations of oscillators and part of them, that contains only raising oscillators, is used to produce the whole representation by acting on the lowest-weight vector annihilated by the lowering oscillators. In the case of SU(2, 2) and SU(2, 2|N) such SU(2) x SU(2) oscillators are naturally combined into Penrose twistors and supertwistors [7,13]. (Super)twistor theory [14] in its turn has long been known to provide interesting alternative to traditional space-time description of 4d massless gauge fields that after the construction of the twistor-string models [15, 16] have got significant attention and allowed to unveil remarkable features of Yang-Mills/gravity amplitudes written in the spinorial form.

Superalgebras SU(2, 2| N) can be extended to infinite-dimensional superalgebras [17,9,19] that admit realizations in terms of above mentioned quantized supertwistors (oscillators) that play the role of auxiliary variables in the construction of 4d conformal higher-spin theories [20] and 5d higher-spin theories [9,10,21]. The spectrum of constituent gauge fields fits into the representation of underlying higher-spin superalgebra and decomposes into an infinite sum of massless representations of SU(2, 2| N) with spins ranging from zero to infinity (see [22] and references therein). More recently it was shown that these superalgebras admit the realization in terms of deformed twistors as enveloping algebras of SU(2, 2|N) [23].

In view of the important role played by massless gauge fields in the bulk of AdS5 and on its D = 4 Minkowski boundary, in this pa-

http://dx.doi.org/10.1016/j.physletb.2016.09.065

0370-2693/© 2016 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

D.V. Uvarov / Physics Letters B ••• (••••) •

per we examine the possibilities to reconstruct bulk (Fang-)Frons-dal fields [24,25],2 starting from the corresponding positive-energy irreducible unitary representations of SU(2, 2) and using the isomorphism between oscillators and twistors and the properties of homogeneous functions on the (ambi)twistor space.

Positive energy irreducible unitary representations of SU(2, 2) can be labeled by positive (half-)integers (E, ji, j2), where the AdS5 energy E is the eigenvalue of the u(l) generator and ji,2 are the representation labels of SU(2)L(R) factors of the maximal compact subgroup SU(2)L x SU(2)R x U(l). Values of AdS5 energy E are bounded from below and the form of the bound depends on the spin s = j1 + j2 and the representation [l]. The simplest doubleton representations (s + l, s, 0) and (s + l, 0, s), by tensoring which all other representations can be constructed and for which associated fields are localized on the D = 4 Minkowski boundary of AdS5, saturate the bound

E > s + 1.

For massless fields on AdS5 the bound has the following form

E > s + 2, jl, j2 > 0. (2)

Note that bounds (l), (2) provide the simplest instances of generic relation between the AdSD energy of the irreducible unitary representation of SO(2, D — l) and the labels of the corresponding SO(D — l) representation that was derived in [27,28] from the requirement of the positive definiteness of scalar product in the Fock space of SO(2, D — l) oscillators. Since these oscillators are the SO(2, D — l) vectors, part of them satisfy 'wrong sign' commutation relations so that the norm of a state in such a Fock space can be positive, negative or null. Then the condition of the norm posi-tivity leads to the above discussed energy bound in similarity with the derivation of the values of critical dimension and intercept in the old covariant quantization of (super)strings. On the contrary the oscillator approach of Refs. [ll,l2] applied to the description of positive energy unitary representations of SU(2, 2) relies on the introduction of (a number of copies of) bosonic oscillators transforming in the fundamental representation of the two SU(2)L(R) factors in the maximal compact subgroup of SU(2, 2) that obey positive-definite commutation relations. This approach thus resembles light-cone gauge (super)string quantization scheme.

In the next section taking doubletons as the simplest example we confront known (but generically considered independently) oscillator and twistor approaches to their description and then apply gained experience to AdS5 massless fields starting from the corresponding representations. For them the oscillator description is also familiar starting from [3]. Twistor description naturally involves ambitwistors and we consider how the ambitwistor data applies to the construction of the (Fang-)Fronsdal fields.

2. Doubletons and twistors

Let us remind the relationship between the SU(2) oscillators and Penrose twistors (see, e.g., [7,l3]). The former correspond to diagonalization of the 'metric'

12x2 0

used to contract indices of fundamental (twistor) and antifundamental (dual twistor) representations of SU(2, 2). The twistor is defined by its primary ¡a and secondary üa SL(2, C) spinor parts

2 The issue of deriving non-linear extensions of the Fronsdal equations for bosonic fields on AdS4 starting from the Vasiliev equations was addressed recently in [26].

and similarly the dual twistor Za = (ua, fa). Since in transition to the oscillator basis only SU(2) covariance is retained, following [l8] we replace dotted indices of the spinor parts of the twistor and its dual by undotted ones in the opposite position that are identified with the SU(2) spinor indices in accordance with the uniqueness of the SU(2) spinor representation. Then the oscillator variables are defined by the linear combinations of the twistor components

aa = —= (-ßa + it"), aa = — (-iia + ua)

ba = -L (fa + ua), ba =-L (fa + Ua) (6)

that can be viewed as a kind of the Bogulyubov transform (for further discussion on that point see, e.g., [19]). Inverse relations express spinor parts of the twistor and its dual via the oscillators3

fa = -= (-aa + ba), ua = —= (aa + ba) (7)

Ua = -—= (aa + ba), / a = —= (-aa + ba). (8)

In quantum theory introduced above oscillators can be shown to satisfy commutation relations

[aa,aß] = sß, [ba,bß] =

that allows interpret aa and ba as annihilation operators and their conjugates aa = (fla)\ ba = (ba)^ as creation operators acting on the unitary vacuum |0). Important invariant - the twistor norm then transforms into the difference between the occupation numbers of b- and a-oscillators

zz ^ -Na + Nb,

N b = b a b a

and su(2, 2) algebra relations can be realized by the bilinears of quantized a- and b-oscillators.

Positive energy unitary representations one can build using just one copy of a- and b-oscillators (9) are called doubletons [3,29]. They correspond to 4d massless fields 'living' on the Minkowski boundary of AdS5. The lowest-weight vectors corresponding to definite doubleton representations are constructed out of the product of creation aa or ba oscillators4

| lwv} = aa(2sL)| 0} or ba(2sR)| 0}

acting on the oscillator vacuum annihilated by the aa and ba operators. The whole representation in the basis corresponding to the maximal compact subalgebra SU(2)L x SU(2)R x U(l) is constructed by applying to (ll) the raising operators L+ap = aab$. They commute with —Na + Nb, thus in any representation fixed

3 The oscillators with upper/lower indices are complex conjugate of each other but bars like in the case of SL(2, C) spinors with dotted indices are not placed conventionally. This does not cause a confusion since changing the position of indices is not required in constructing SU(2, 2) positive energy unitary representations.

4 Throughout the paper we adhere to the notation that a number in round brackets following an index stands for the group of indices equal to that number that are symmetrized with unit weight. Accordingly a number in square brackets following an index denotes the group of indices antisymmetrized with unit weight.

Na = aaa

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D.V. Uvarov/Physics Letters B ••• (••••) 3

Table 1

Description of D = 4 massless gauge fields by homogeneous functions on PT* and PT*.

Irrep Helicity Hom. degree on PT* D = 4 field Hom. degree on PT. D = 4 field

(1 — s, —s, 0) s < 0 -2 - 2s raá (—2s—1)(x) -2 + 2s Wa (—2s)(x)

(3, 2,0) —2 2 raá (3)(x) -6 Wa (4) (x)

(5/2,3/2, 0) —3/2 1 Aail (2) (x) -5 (x)

(2, 1, 0) —1 0 Aaà (x) -4 fa (2) (x)

(3/2,1/2, 0) —1/2 -1 la(x) -3 Xa(x)

(2, 0, 0) 0 -2 V(x) -2 V(x)

(3/2, 0,1/2) 1/2 -3 Xi (x) -1 X1 (x)

(2, 0,1) 1 -4 f l (2 ) ( x) 0 A àa (x)

(5/2, 0, 3/2) 3/2 -5 (3) (x) 1 Aàa (2) (x)

(3, 0, 2) 2 -6 Wl (4) (x) 2 Tàa (3) (x)

(s + 1,0, s) s > 0 -2 - 2s Wl (2s) (x) -2 + 2s r 1 a (2s—1) (x)

is the integer -2sL or +2sR. The SU(2)L x SU(2)R labels j12 are given by half the eigenvalues of Na and Nb on the lowest-weight vectors and the energy equals E = j1 + j2 + 1 = sL(R) + 1.

In the twistor picture doubleton representations are described by homogeneous functions f (Z) on the twistor space PT* or homogeneous functions f (Z) on the dual twistor space PT*. Such a description is based on the quantized twistors

[Z<*, Zp] = 8* (12)

realization as the multiplication and differentiation operators

Za ^ Za, za ^--—

or vice versa

Z a ^ Z a ■

Operator realization (13) is adapted to the action on the twistor space functions and (14) - on the dual twistor space functions.

In the twistor approach doubleton representations built upon the lowest-weight vectors (11) are described by the homogeneous functions on the twistor space with the homogeneity degrees 2sL — 2 or —2sR — 2 as follows from (10). The twistor helicity operator

s = 1 (Z Z + Z Z) (15)

in the realization (13) acquires the form 1 d

s = —Z--1 (16)

2 d Z y '

so that the function f(2sL—2)(Z) homogeneous of degree 2sL — 2 > —2 corresponds to the field with negative helicity —sL that describes left-polarized massless particles, whereas the function f—2sR—2)(Z) homogeneous of degree —2sR — 2 < —2 corresponds to the field with positive helicity +sR and right-polarized particles [14]. In the case of positive helicity fields reconstruction of the on-shell curvatures (linearized Weyl curvature SL(2, C) spinors) on the D = 4 Minkowski space-time proceeds using the contour integral representation

Wi (2s R )(x) =

jußduß Uá

a i uá 2sR

x f(-2sR-2)(iUixil, Uk) : 9Wà(2SR)(x) = 0,

where the incidence relations \xa = iiiaxaa that express primary spinor part of the twistor via the coordinates xaa = Xaci'aa of the

(complexified conformally compactified) D = 4 Minkowski spacetime are assumed to hold. For the negative helicity fields coho-mological arguments suggest a description in terms of the spinor form raa(2sl—1)(x) of linearized Christoffel-type connections [30] modulo the gauge transformations. Accordingly Penrose transform of the dual-twistor-space function for negative helicity yields generalized Weyl curvature SL(2, C) spinor of opposite chirality Wa(2sl)(x), while for positive helicity it produces Christoffel-type connection rda(2sr—1)(x) (see Table 1). General relation between the homogeneity degrees of functions on the twistor space hpT* and on the dual twistor space hpT* that correspond to the field of helicity s is

= —hpx^ - 4.

The correspondence between respective cohomology groups of homogeneous functions is known as the twistor transform.

3. Massless field representations and ambitwistors

Description of the positive energy unitary irreducible representations associated with massless fields on AdS5 necessitates introduction of two copies of a- and b-oscillators [3,29]

[àa (p), aß (r)l = SprSß, [ba (p), bß (r)] = SprSap,

p, r = 1, 2.

In the twistor approach this corresponds to dealing with two Penrose twistors and their duals. Associate in accordance with (7), (8) to the first set of oscillators the twistor Za and its dual Za, and analogously to the second set of oscillators - another pair of twistors WWa = (va, Vd) and Wa = (va, va). Similarly to the doubleton case, in any representation fixed is the difference between the occupation numbers of a- and b-oscillators of the first and the second sets

(—Na(1) + Nb(1))| lwv> = —2s11 lwv), (—Na(2) + Nb (2))| lwv) = 2s21 lwv).

For the representations that correspond to massless fields on AdS5 s1,2 are positive (half-)integers modulo relabeling the oscillators. So one is led to consider homogeneous functions on the am-bitwistor space A:

z jzF(2S1-2|2S2-2)(Z, WW) = (2S1 - 2)F(2s,-2|2S2-2)(Z, WW),

WW -W F (2S,-2|2S3-2)(Z, WW ) = (2S2 - 2) F (2S,-2|2S3-2)(Z, WW ).

D.V. Uvarov / Physics Letters B ••• (••••) <

We have chosen Za and Wa to parametrize A, while the operators corresponding to Za and Wa have been traded for the derivatives of Za and Wa (cf. (l3), (l4)). The condition WaZa = 0 on the ambitwistor space coordinates is imposed via the 8-function

F(2si —212$2 —2) (Z, W) = 8(WZ) f(2sl— l|2s2— l)(Z, W). (22)

In the oscillator approach it translates into the constraint

(b a (2)ba (l) — a a (2)aa (l)) | lwv) = 0. (23)

For homogeneous functions f(s—i|s—1)(Z, WW) = f(s—l)(Z, WW) on A with s non-negative integer details of the Penrose transform can be found, e.g., in [3l] or in Ref. [32]. On-shell of the incidence relations

a aa a aa

fa = m a xa a, va = — ixa a v a

the function /(s— i) satisfies

va ua daa f(s—i)(f, u, V, v) = 0.

f(s—l) is cohomologically trivial since H1 (CP1 x CP1 , O(s — l)) = 0 implying that it can be globally defined as a polynomial of the respective degree in u00 and va, homogeneous coordinates on

CP1 x CP1,

va Ua daa f(s—1) = va(s)Ua(s)ba(s)a(s) (x),

where the symmetric multispinor field ba(s)a (s) is defined modulo the gauge symmetry

Sba(s)a (s)(x) = da(1)a (1)^a(s—1)ci (s—1)(x)

with the symmetric multispinor parameter %a(s—1)a(s—i). In 4d vector notation it corresponds to symmetric traceless rank-s tensor field ba(s)(x)5 and the gauge parameter is given by the symmetric traceless rank-(s — 1) tensor field %a(s—1)(x) so that

Sba(s)(x) = da(1)|a(s—1)(x) — 1 Va(2)d %a(s—2)(x).

The discussion of the last paragraph corresponds to the simplest case of totally symmetric massless bosonic fields. In the oscillator description the solution to the constraints

(—N a (1) + N b (1))| lwv) = s| lwv),

(—Na(2) + Nib(2))| lwv) = s| lwv), s > 0 (29)

and (23) is given by the lowest-weight vector (cf. [29]) | lwv) = a(l)a(s)b(2)p(S) | 0). (30)

Respective representation labels are jl = j2 = s/2 and E = s + 2.

Generalizing the above consideration to the functions f(s—1|s) and f(s|s— l) on A allows to obtain also fermionic fields fa(s+l)a(s)(x) and Xa(s)a(s+l)(x) defined modulo the gauge transformations

5|a(s+1)ii(s) (x) = da(\)à(\)Sa(s)à(s—1)(x) and

&la(s)à (s+1) (x) = da(\)à (\)^a(s—\)à (s)(x).

In vector form these fields are given by the totally symmetric a -traceless tensor-spinors i>a(s)a(x) and Xa(s)a (x), for which the gauge variations read

Sfa(s)(x) = da(1)la(s—1)(x) +

2(s + 1)

&a(1)&b dbla(s—1)(x)

na(2)d blba(s—2)(x)

SXa(s) (x) = da(1)la(s—1)(x) +

2(s + 1)

&a(1)&b dbla(s—1)(x)

na(2)d blba(s—2)(x).

Associated lowest-weight vectors in the oscillator approach

aa(s) (1 )bp(S+1) (2) 10), aa(s+1) (1 )bm (2)| 0>

correspond to representations with j1 = s/2, j2 = (s + 1)/2 and j1 = (s + 1)/2, j2 = s/2 that both have E = s + 5/2.

Generic ambitwistor functions f(2s1—1|2s2—1)(Z, W) with s1, s2 > 0, |s1 — s2| > 1/2 give rise to multispinor bosonic fields la(2s2)a (2s1 ) (x) if + s2 is integer, or, if s1 + s2 is half-integer, fermionic fields |a(2s2)a(2s1)(x) (s2 > sO or la(2s2)a(2s1)(x) (s1 > s2). They are defined modulo the gauge freedom

8ba(2s2)a (2si )(x) = da(l)a (l)^a(2s2—l)0t (2si—1) and analogous transformations for

fa(2s2)0(2sl)(x) and Xa(2s2)a(2sl)(x). Respective lowest-weight vectors are

(27) aa(2s1)(1)b^(2S2)(2)| 0)

and representation labels are (s + 2, s1,s2) with s = s1 + s2. Together with the conjugate representations (s + 2, s2, s1) they make up completely traceless tensor fields J^a(s1+s2)b(|s1 —s2|)(x) and tensor-bispinor fields |a(s1+s2—1/2)b(|s1— s21—1/2)(x). These are mixed-symmetry fields associated with two-row Young tableaux. Above considered representations are the only massless ones. Other SU(2, 2) representations that can be constructed using the two sets of a- and b-oscillators [29], in particular those arising in the limit s1 = 0 or s2 = 0 of (37), correspond to the so called massive self-dual fields [33].

Symmetric traceless bosonic fields (28) and a-traceless fermi-onic fields (33) and (34) can naturally be identified with bosonic and fermionic D = 4 shadow fields [34,35], and treated as boundary values of AdS5 totally symmetric massless gauge fields corresponding to the non-normalizable solutions of the Dirichlet problem for the Fronsdal equations. AdS/CFT adapted description of massless mixed-symmetry is much more involved.6 These results are summarized in Table 2, where a', b' = 0, ..., 3, 5 are D = 5 tangent-space vector indices.

The direct reconstruction of AdS5 fields requires modification of the Penrose incidence relations (24) to accommodate the contribution of the fifth space-time coordinate.7 It is possible to unify secondary spinor parts of ambitwistors va and Ua into four-component D = 5 spinor

5 Tilde is used to indicate tracelessness of a tensor w.r.t. Minkowski metric.

6 To date available is light-cone gauge formulation of AdS5 massless mixed-symmetry fields [33], SO(1, D — l)-covariant formulation for three-cell hook-type Young diagram field [36], as well as the ambient-space one [37].

7 For the space-time of dimension D = 6, for which the space-time coordinate matrices span the space of 4 x 4 antisymmetric matrices, natural generalization of Penrose twistors and on-shell contour integral relations does exist [38,39].

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Table 2

SU(2,2) massless representations, respective ambitwistor functions and space-time fields.

Irrep (E, j1: j2) f on A SL(2, C) multispinors AdS5 field

(s + ~2,s/2, s/2) f(s—1) Xa(s)a(s) Ba'(s)

(s + 5/2, s/2,(s + 1)/2) ® (s + 5/2,(s + 1)/2, s/2) f(s—,|s) ® f№—1) fa(s+1)a(s) ® Xa(s)a(s+1) Xa'(s)

(s1 + s2 + 2, s1, s2) ® (s1 + s2 + 2, s2, s1) f(2s1—1l2s2 — 1) ® f(2s2 —1 |2s1 —1) s1 + s2 mtejjen f)a(2s2)a(2s1) ® Xa(2S1)a(2s2) Ba'(s1+s2)b'(|s1— s2|)

s1,2 > 0, |s1 — s21 > 1/2 s1 + s2 half-integer: iV(s,+s2—1/2)b'to —s21—1/2)

fa(2max{S1 ,s2})à(2min{S1 ,s2}) ® Xa(2min{sj ,s2})à(2max{S1 ,s2})

and combine it with the Dirac conjugate into SU (2) -symplectic Majorana spinor. Such an SU(2)-symplectic Majorana spinor can be treated as the 4 x 2 rectangular block of the D = 5 spinor Lorentz-harmonic matrix. Then by generalizing the construction of [40], an integral of homogeneous functions that belong to definite SU(2) irreducible representations over the spinor harmonics produces the spinor form of D = 5 Yang-Mills, Rarita-Schwinger field strengths, linearized Weyl tensor and its higher-spin counterparts that satisfy Dirac-type equations [41]. Alternatively D = 5 symmetric traceless gauge fields can be obtained via the Penrose transform for D = 5 ambitwistors [32].

4. Conclusions

In this note we discussed the correspondence between the positive energy (lowest-weight) unitary irreducible representations of SU(2, 2) and the space-time fields on AdS5 taking as an example doubleton and massless representations that saturate the bounds E = s + 1 and E = s + 2 respectively. Isomorphism between bosonic oscillators, that can be used to construct positive energy unitary irreducible representations of SU(2, 2), and Penrose twistors allows to establish one-to-one correspondence between the doubleton representations and homogeneous functions of the single argument on projective twistor space (or dual pro-jective twistor space) that via the Penrose transform yield on-shell linearized Weyl curvature spinors and their low-spin counterparts in 4 dimensions. We sought for possible extension of this twistor description to the case of massless representations. Since to construct massless representations it is necessary to use twice more oscillators compared to the doubletons, their natural twistor counterparts are ambitwistors. We have shown that Penrose transform for homogeneous functions on ambitwistor space yields shadow fields on D = 4 Minkowski space-time that admit interpretation as boundary values of the non-normalizable solutions to the Dirich-let problem for (Fang-)Fronsdal equations for the corresponding AdS5 massless gauge fields. This establishes one-to-one correspondence between homogeneous ambitwistor functions and SU(2, 2) massless representations. The fact that one arrives at the boundary values of AdS5 massless gauge fields rather than the bulk fields themselves is encoded in the form of the incidence relations that take into account only the contributions of D = 4 Minkowski space coordinates. Direct obtention of the AdS5 mass-less gauge fields requires extension of Penrose incidence relations to account for the contribution of the fifth space-time coordinate.

Natural generalization of the results reported in this note is to introduce supersymmetry and also consider massive representations pertinent to the adjoint version of the AdS5/CFT4 correspondence. Potentially interesting applications can be also in twistor-string theory. Ambitwistor string models have already been used to reproduce D = 4 Yang-Mills and Einstein gravity tree amplitudes in [42]. It is tempting to speculate that their appropriate ramifications can produce tree amplitudes of D = 5 gauge theories.

While finalizing this paper we learned about Ref. [43] that to some extent overlaps with our results and clarifies some points that we discuss here.

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