Ambitwistors, oscillators and massless fields on $AdS_5$

Positive energy unitary irreducible representations of $SU(2,2)$ can be constructed with the aid of bosonic oscillators in (anti)fundamental representation of $SU(2)_L\times SU(2)_R$ that are closely related to Penrose twistors. Starting with the correspondence between the doubleton representations, homogeneous functions on projective twistor space and on-shell $SL(2,\mathbb C)$ generalized Weyl curvature 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 $AdS_5$. The possibilities of reconstructing massless fields on $AdS_5$ and some applications are also discussed.


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 supergravity, in particular the necessity to describe the spectrum of D = 10 IIB supergravity compactified on AdS 5 × S 5 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 AdS 5 /CF T 4 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 AdS 5 /CF T 4 duality it was shown [8] that the spectrum of D = 5 Vasiliev-type higher-spin gauge theories 2 , 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 (super)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 associated lowest-weight vector annihilated by the lowering oscillators. In the case of SU(2, 2) and SU(2, 2|N) such SU(2) × 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 spacetime 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], [18] 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 [19] and 5d higher-spin theories [9], [10], [20]. 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 [21] 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) [22].
In view of the important role played by massless gauge fields in the bulk of AdS 5 and on its D = 4 Minkowski boundary, in this paper we examine the possibilities to reconstruct bulk (Fang-)Fronsdal fields [23], [24], 3 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, j 1 , j 2 ), where the AdS 5 energy E is the eigenvalue of the u(1) generator and j 1,2 are the representation labels of SU(2) L(R) factors of the maximal compact subgroup SU(2) L × SU(2) R × U(1). Values of AdS 5 energy E are bounded from below and the form of the bound depends on the spin s = j 1 + j 2 and the representation [1]. The simplest doubleton representations (s + 1, s, 0) and (s + 1, 0, s), the tensor products of which contain all other positive energy unitary irreducible representations and for which associated fields are localized on the D = 4 Minkowski boundary of AdS 5 , saturate the bound For massless fields on AdS 5 the bound has the following form Note that bounds (1), (2) provide the simplest instances of generic relation between the AdS D energy of the irreducible unitary representation of SO(2, D − 1) and the labels of the corresponding SO(D − 1) representation that was derived in [26], [27] from the requirement of the positive definiteness of scalar product in the Fock space of SO(2, D − 1) oscillators. Since these oscillators are the SO(2, D −1) 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 positivity 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. [11], [12] 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 the 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 AdS 5 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.

Doubletons and twistors
Let us remind the relationship between the SU(2) oscillators and Penrose twistors (see, e.g., [7], [13]). The former correspond to diagonalization of the 'metric' used to contract indices of fundamental (twistor) and antifundamental (dual twistor) representations of SU(2, 2). The twistor is defined by its primary µ α and secondaryūα SL(2, C) spinor parts and similarly the dual twistorZ α = (u α ,μα). Since in transition to the oscillator basis only SU(2) covariance is retained, following [28] we replace dotted indices of the SL(2, C) 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 and that can be viewed as a kind of the Bogulyubov transform (for further discussion on that point see, e.g., [18]). Inverse relations express spinor parts of the twistor and its dual via the oscillators 4 and In quantum theory introduced above oscillators can be shown to satisfy commutation relations that allows interpretâ α andb α as annihilation operators and their conjugatesâ α = (â α ) † , b α = (b α ) † as creation operators acting on the unitary vacuum |0 . Important invariantthe twistor norm then transforms into the difference between the occupation numbers of band a-oscillatorsZ 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 boscillators (9) are called doubletons [3], [29]. They correspond to 4d massless fields 'living' on the Minkowski boundary of AdS 5 . The lowest-weight vectors corresponding to definite doubleton representations are constructed out of the product of creationâ α orb α oscillators 5 | lwv =â α(2s L ) | 0 orb α(2s R ) | 0 (11) acting on the oscillator vacuum annihilated by theâ α andb α operators. The whole representation in the basis corresponding to the maximal compact subalgebra SU(2) L ×SU(2) R ×U (1) is constructed by applying to (11) the raising operatorsL + α β =â αb β . They commute with −N a +N b , thus in any representation fixed is the integer −2s L or +2s R . The SU(2) L ×SU(2) R labels j 1,2 are given by half the eigenvalues ofN a andN b on the lowest-weight vectors and the energy equals E = j 1 + j 2 + 1 = s L(R) + 1.
In the twistor picture doubleton representations are described by homogeneous functions f (Z) on the twistor space PT • or homogeneous functionsf (Z) on the dual twistor space PT • . Such a description is based on the quantized twistors realization as the multiplication and differentiation operatorŝ 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 2s L − 2 or −2s R − 2 as follows from (10). The twistor helicity operator in the realization (13) acquires the form so that the function f (2s L −2) (Z) homogeneous of degree 2s L − 2 > −2 corresponds to the field with negative helicity −s L that describes left-polarized massless particles, whereas the function f (−2s R −2) (Z) homogeneous of degree −2s R − 2 < −2 corresponds to the field with positive helicity +s R 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 irrep helicity hom. degree on PT • D = 4 field hom. degree on PT • D = 4 field where the incidence relations µ α = iūαxα α that express primary spinor part of the twistor via the coordinatesxα α = x aσαα a of the (complexified conformally compactified) D = 4 Minkowski space-time are assumed to hold. For the negative helicity fields cohomological arguments suggest a description in terms of the spinor form Γ αα(2s L −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 W α(2s L ) (x), while for positive helicity it produces Christoffel-type connection Γα α(2s R −1) (x) (see Table 1). General relation between the homogeneity degrees of functions on the twistor space h PT • and on the dual twistor space h PT • that correspond to the field of helicity s is The correspondence between respective cohomology groups of homogeneous functions is known as the twistor transform.
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 twistorẐ α and its dualẐ α , and analogously to the second set of oscillators -another pair of twistorŝ W α = (ν α ,vα) andŴ α = (v α ,να). 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 For the representations that correspond to massless fields on AdS 5 s 1,2 are positive (half-)integers modulo relabeling the oscillators. So one is led to consider homogeneous functions on the ambitwistor space A: We have chosen Z α andW α to parametrize A, while the operators corresponding toZ α and W α have been traded for the derivatives of Z α andW α (cf. (13), (14)). The condition W α Z α = 0 on the ambitwistor space coordinates is imposed via the δ-function In the oscillator approach it translates into the constraint For homogeneous functions f (s−1|s−1) (Z,W ) ≡ f (s−1) (Z,W ) on A with s non-negative integer details of the Penrose transform can be found, e.g., in [31] or in Ref. [32]. On-shell of the incidence relations f (s−1) is cohomologically trivial since H 1 (CP 1 × CP 1 , O(s − 1)) = 0 implying that it can be globally defined as a polynomial of the respective degree inūα and v α , the homogeneous coordinates on where the symmetric multispinor field b α(s)α(s) is defined modulo the gauge symmetry with the symmetric multispinor parameter ξ α(s−1)α(s−1) . In 4d vector notation it corresponds to symmetric traceless rank-s tensor field b a(s) (x) 6 and the gauge parameter is given by the symmetric traceless rank-(s − 1) tensor field ξ a(s−1) (x) so that 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 , (−N a (2) +N b (2))| lwv = s| lwv , s ≥ 0 (29) and (23) is given by the lowest-weight vector (cf. [29]) Respective representation labels are j 1 = j 2 = s/2 and E = s + 2. 6 Tilde is used to indicate tracelessness of a tensor w.r.t. Minkowski metric.

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 AdS 5 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 projective twistor space) that via the Penrose transform yield on-shell linearized Weyl curvature SL(2, C) 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 Dirichlet problem for (Fang-)Fronsdal equations for the corresponding AdS 5 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 AdS 5 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 AdS 5 massless 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 AdS 5 /CF T 4 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.