Light-cone AdS/CFT-adapted approach to AdS fields/currents, shadows, and conformal fields

Light-cone gauge formulation of fields in AdS space and conformal field theory in flat space adapted for the study of AdS/CFT correspondence is developed. Arbitrary spin mixed-symmetry fields in AdS space and arbitrary spin mixed-symmetry currents, shadows, and conformal fields in flat space are considered on an equal footing. For the massless and massive fields in AdS and the conformal fields in flat space, simple light-cone gauge actions leading to decoupled equations of motion are found. For the currents and shadows, simple expressions for all 2-point functions are also found. We demonstrate that representation of conformal algebra generators on space of currents, shadows, and conformal fields can be built in terms of spin operators entering the light-cone gauge formulation of AdS fields. This considerably simplifies the study of AdS/CFT correspondence. Light-cone gauge actions for totally symmetric arbitrary spin long conformal fields in flat space are presented. We apply our approach to the study of totally antisymmetric (one-column) and mixed-symmetry (two-column) fields in AdS space and currents, shadows, and conformal fields in flat space.


Introduction
In spite of its Lorentz noncovariance, the light-cone gauge approach to relativistic field and string dynamics [1] simplifies considerably the study of various problems of quantum field and string theories. This is to say that a number of important problems of modern quantum field and string theories have successfully been solved by using the light-cone gauge formalism. Perhaps the most attractive example of application of this formalism is the building of the light-cone gauge formulation of superstring field theory [2].
Motivated by desire to provide the light-cone gauge framework for the study of the conjectured string/gauge duality [3] and its cousins, the light-cone gauge formulation of AdS field theory in Ref. [4] and light-cone gauge formulation of AdS superstring theory in Refs. [5,6] were developed. Also, in Ref. [4], light-cone gauge approach to CFT was developed. AdS/CFT correspondence for arbitrary spin massless AdS fields was studied in Ref. [4] and it was demonstrated that the lightcone approach simplifies considerably the study of AdS/CFT correspondence.
To explain our terminology let us discuss briefly the group theoretical interpretation of fields in AdS d+1 and related currents, shadows, and conformal fields in R d−1,1 we are going to study in this paper. Massless and massive fields propagating in AdS d+1 space are associated with unitary positive-energy lowest weight irreducible representations of the so(d, 2) algebra. We will denote such representations by D(E 0 , h), where the E 0 is the lowest eigenvalue of the energy operator, while the h = (h 1 , . . . , h r ), r = [ d 2 ] stands for the highest weight of the so(d) algebra representation. For bosonic fields, the highest weights h i are integers. As shown in Ref. [7], the E 0 and h satisfy the following restriction: 1 where k is defined from the relation 2 (1.2) Using the notation E m=0 we note that, for massless and massive fields, restriction in (1.1) takes the form 3 for massless fields in AdS d+1 , (1.4) for massive fields in AdS d+1 . (1.5) Conformal currents in R d−1,1 are also associated with the representations D(∆ cur , h) of the so(d, 2) algebra, where ∆ cur stands for the conformal dimension of the currents. In this paper, conformal currents with ∆ cur = E m=0 0 and ∆ cur > E m=0 0 will be referred to as short and long currents respectively. Conformal shadows in R d−1,1 are associated with non-unitary representations 1 For d = 3, 4, the restriction (1.1) was obtained in Refs. [8,9]. The study of unitary representations of various superalgebras may be found, e.g., in Refs. [10,11]. 2 For d-even, the weight h r in (1.1), (1.2) should be replaced by |h r |. Bosonic irrep of the so(d) algebra having highest weight h can be described by tensor field whose so(d) space tensor indices have the structure of the Young tableaux labelled by h. In such Young tableaux, h i is equal to length of the i-th row. 3 As known, the case with E 0 = E m=0 0 , and k = d/2, d-even is not associated with local field propagating in AdS d+1 . This case will be ignored in this paper. of the so(d, 2) algebra labeled by ∆ sh , h, where ∆ sh = d − ∆ cur . In this paper, conformal shadows having ∆ sh = d − E m=0 0 are referred to as short shadows, while conformal shadows having ∆ sh = d − E 0 with E 0 as in (1.5) are referred to as long shadows. 4 The short shadows in R d−1,1 can be used to build the conformal invariant Lagrangian dynamics of fields propagating in R d−1,1 . Such fields will be referred to as short conformal fields. The long shadows with some particular values of ∆ sh can also be used to build the conformal invariant Lagrangian dynamics of fields propagating in R d−1,1 . Such fields will be referred to as long conformal fields. Most of conformal fields enter higher-derivative Lagrangian dynamics and are associated with non-unitary representations of the so(d, 2) algebra labeled by ∆, h. For the short conformal fields, ∆ = d − E m=0 0 , while for the long conformal fields ∆ = d − E 0,long , where E 0,long takes integer or half integer values which satisfy the restriction (1.5). 5 Throughout this paper, fields with h = (h 1 , 0, . . . , 0) are referred to as totally symmetric fields, while fields with h = (h 1 , . . . , h r ), h 1 > h 2 ≥ 1 are referred to as mixed-symmetry fields. 6 The light-cone gauge description of arbitrary spin totally symmetric short currents (shadows) and long currents (shadows) was developed in the respective Ref. [4] and Ref. [16]. The general lightcone formalism for arbitrary spin mixed-symmetry short currents (shadows) and long currents (shadows) was developed in Ref. [17]. 7 We note however that the light-cone gauge formulation of CFT in Ref. [17] has the following unsatisfactory feature. Representation for generators of the conformal symmetries obtained in Ref. [17] is non-local. 8 One of the aims of this paper is to obtain the local representation for generators of the conformal symmetries in CFT. Doing so, we obtain not only the desired light-cone gauge formulation of currents and shadow but also new light-cone gauge formulation of fields in AdS space and conformal fields in flat space. The new formulations of AdS field theory and CFT are based on the use of one and the same spin operators satisfying some equations which take the form of deformed commutation relations of so(d) algebra. Finding a solution to these equations for the spin operators implies immediately the complete light-cone description of both the AdS field theory and CFT. Therefore, in the framework of our new light-cone gauge approach, the study of the AdS/CFT correspondence is realized in a rather straightforward way. This is why we refer the new approach as AdS/CFT-adapted light-cone approach.
The reason why it is possible to develop the light-cone formulations of AdS field theory and CFT in terms of one and the same spin operators is related to the following fact. In the light-cone gauge formulations of AdS field theory and CFT, we deal with fields which are not subject to any differential constraints. For the light-cone gauge field theory in AdS d+1 , it is convenient to decompose light-cone gauge massless and massive fields into irreps of the so(d − 2) algebra. For the light-cone gauge CFT in R d−1,1 , it is also convenient to decompose light-cone gauge short currents (shadows) and long currents (shadows) into irreps of the so(d − 2) algebra. In the framework of AdS/CFT correspondence, massless AdS field is related to short current (shadow). Namely, considering massless field in AdS d+1 and the corresponding short current (shadow) in R d−1,1 , we note that irreps of the so(d − 2) algebra which enter the light-cone gauge formulation of the massless field in AdS d+1 coincide with irreps of the so(d − 2) algebra which enter the light-cone gauge formulation of the short current (shadow) in R d−1,1 . In other words, there is precise matching between irreps of the so(d − 2) algebra involved in the light-cone gauge formulation of the massless field in AdS d+1 and irreps of the so(d − 2) algebra involved in the light-cone gauge formulation of the corresponding short current (shadow) in R d−1,1 . The same matching happens for massive fields and the corresponding long currents (shadows). Namely, irreps of the so(d − 2) algebra which enter the light-cone gauge formulation of the massive field in AdS d+1 coincide with irreps of the so(d − 2) algebra which enter the light-cone gauge formulation of the long current (shadow) in R d−1,1 . In fact, it is the remarkable matching of bulk and boundary irreps of the so(d − 2) algebra that explains why it is possible to develop the light-cone gauge formulations of fields in AdS d+1 and currents (shadows) in R d−1,1 in terms of one and the same spin operators.
For the reader convenience, let us mention briefly some results concerning the Lorentz covariant description of mixed-symmetry AdS fields. General mixed-symmetry massless fields in AdS d+1 , d-arbitrary, were studied in Refs. [7], [18]- [24]. Two-column mixed-symmetry massless fields in AdS d+1 were considered in Ref. [25], while two-row mixed-symmetry massless fields in AdS 5 were investigated in Refs. [26]. General mixed-symmetry massive fields in AdS d+1 were studied in Refs. [23,27]. Two-column and two-row mixed-symmetry massive fields in AdS d+1 were investigated in Ref. [28] and Refs. [29] respectively. Interacting hook mixed-symmetry massive fields in AdS d+1 were considered in Refs. [30], while interacting two-row and two-column mixed-symmetry massless fields in AdS d+1 were studied in Refs. [31] and Refs. [32] respectively. 9 The paper is organized as follows. In Sec.2, we start with brief review of light-cone approach to AdS fields developed in Ref. [4]. After that, we present our new light-cone formulation which is adapted for the study of AdS/CFT. We present light-cone gauge action which leads to decoupled equations of motion for light-cone gauge AdS fields. Realization of relativistic symmetries of light-cone gauge AdS fields is also discussed. In Sec.3, we discuss our new light-cone gauge formulation of currents and shadows in flat space. Two-point functions for light-cone gauge currents and shadows are presented. Realization of conformal symmetries on space of currents and shadows is discussed. Sec.4 is devoted to the study of AdS/CFT. We demonstrate that, in the framework of AdS/CFT, the normalizable modes of AdS fields are related to currents, while the non-normalizable modes of AdS fields are related to shadows. Also we show that light-cone gauge action of AdS field computed on solution of Dirichlet problem coincides with light-cone gauge two-point function of shadow. In Sec.5, we study the light-cone gauge formulation of conformal fields in flat space. We present the action which leads to decoupled equations of motion for lightcone gauge conformal fields. Simple examples of short and long totally symmetric arbitrary spin conformal fields are discussed with some details. In Secs.6,7, to illustrate our approach, we deal with totally antisymmetric and two-column fields which, in the earlier literature, have not been studied in the framework of light-cone gauge approach. For such fields, we demonstrate how finding a solution to our defining equations for the spin operators leads immediately to the complete light-cone description of fields in AdS space and currents, shadows, and conformal fields in flat space. Various technical details are collected in three Appendices.

Light-cone gauge formulation of fields in AdS d+1 space
We begin with the discussion of light-cone gauge formulation of field dynamics in AdS space. To this end we use Poincaré parametrization of AdS d+1 space given by (for details of our notation, see Appendix A) To simplify the presentation, we collect all scalar, vector, tensor, and arbitrary spin fields into a ket-vector |φ . By definition, in light-cone gauge approach, all fields entering the ket-vector |φ are not subject to any differential constraints. In terms of the ket-vector |φ , light-cone action and Lagrangian can be cast into the following form [4]: . Note however that the choice of the light-cone gauge spoils the manifest so(d − 1, 1) algebra symmetries. This is to say that, in the framework of the light-cone approach, in order to complete the description of relativistic dynamics of fields in AdS d+1 space we have to work out an explicit realization of the so(d − 1, 1) algebra symmetries and the remaining relativistic symmetries as well. To this end we now discuss the so(d, 2) algebra symmetries of the light-cone gauge action given in (2.2).
In the light-cone formulation, generators of the so(d, 2) algebra can be separated into the following two groups: where the vector indices of the so(d − 2) algebra take values i, j = 1, . . . , d − 2 (for details of our notation see Appendix A). For the case of free fields, the field theoretical realization of so(d, 2) algebra generators G field in (2.5), (2.6) takes the following form: Quantity G diff appearing on right hand side (2.7) stands for the realization of generators (2.5), (2.6) in terms of differential operators defined on space of the ket-vector |φ . Light-cone gauge action (2.2) is invariant under the transformation δ|φ = G diff |φ . In Ref. [4], we found the following explicit expressions for the differential operators G diff : Kinematical generators,

13)
Dynamical generators, In (2.20) and below, the quantity C so(d,2) stands for an eigenvalue of the second order Casimir operator of the so(d, 2) algebra. For the representation D(E 0 , h), the C so(d,2) is given by The spin operators M ij , i, j = 1, . . . , d − 2, form commutation relations of the so(d − 2) algebra, while the spin operators M ij and M zi form commutation relations of the so(d − 1) algebra (see relations in (B.5)-(B.7)). Hermitian conjugation rules are given by From (2.8)-(2.20), we see that a knowledge of the operators A, M ij , M zi allows us to fix the realization of the kinematical and dynamical generators on space of the ket-vector |φ . Thus problem of the light-cone gauge description of AdS fields amounts to finding an explicit expressions for the operators A, M ij , M zi which we refer to as basic operators. Closed system of equations for these basic operators was found in Ref. [4] (see Eqs.(B.1)-(B.7) in Appendix B in this paper).
The presentation above-given summarizes the light-cone formulation of AdS fields developed in Ref. [4]. We now proceed to the discussion of AdS/CFT-adapted light-cone formulation which is based on the use of new basic operators. Such new basic operators turn out to be very convenient for the study of AdS/CFT correspondence and conformal fields. New basic operators. In this paper, we introduce new basic operators denoted by ν, W i ,W i . As before, the so(d − 2) algebra spin operator M ij also enters the game. In terms of the new operators and the operator M ij , the operators A, B, M zi and M ⊕i take the form We note the following Defining equations for the new basic operators ν, W i ,W i , M ij : where the operator B appearing in (2.37) is defined by relations (2.26), (2.21). We assume the hermitian conjugation rules, Equations (2.28)-(2.37) together with relations in (2.26), (2.21) constitute the defining equations of AdS/CFT-adapted light-cone gauge formulation of the relativistic dynamics in AdS space we suggest in this paper. These equations are equivalent to the equations for operators A, M zi , M ij we obtained in Ref. [4]. Derivation of Eqs.(2.28)-(2.37) from the ones in Ref. [4] may be found in Appendix B.
The following remarks are in order. i) For AdS field associated with the representation D(E 0 , h), the operator ν can be presented as The defining equations (2.28)-(2.37) take the form of some special deformation of commutation relations of the so(d) algebra. In the flat space limit which is realized as R → ∞, where R is radius of AdS space, the defining equations (2.28)-(2.37) become the usual commutators of the so(d) algebra. To see this we note the following relations in the limit as R → ∞: where m in (2.44) stands for mass parameter of a massive field. 10 Using (2.44), (2.45), we see that, in the flat limit, commutators in (2.28), (2.30), (2.31), (2.37) take the form vi) The explicit form of the operator ν is known for the following dynamical systems: a) totally symmetric arbitrary spin massless and massive fields in AdS d+1 , d ≥ 3, [4,16]; b) mixed symmetry arbitrary spin massless and self-dual massive fields in AdS 5 [41]; c) mixed symmetry arbitrary spin massive fields in AdS 5 [42]; d) type IIB supergravity in AdS 5 × S 5 and AdS 3 × S 3 backgrounds [43,44].
Just to illustrate our approach we consider arbitrary spin massless fields in AdS 4 . Massless arbitrary spin field in AdS 4 . For AdS 4 , one gets d = 3 and hence the vector index of the so(d − 2) algebra i takes only one value, i = 1. In order to describe field content entering dynamics of massless field in AdS 4 we use oscillators α 1 , α z . Commutation relations for the oscillators, the vacuum |0 , and hermitian conjugation rules are defined as (2.48) Using such oscillators, we introduce the ket-vector |φ to discuss massless spin-s field, where two real-valued fields φ s = φ s (x, z) and φ s−1 = φ s−1 (x, z) appearing in (2.49) stand for two propagating physical D.o.F which describe spin-s massless field in AdS 4 . Operators A, ν, W 1 ,W 1 , B can be read from results in Ref. [16], A a side remark we note that all unitary representations of the so(d + 1, 2) algebra which are associated with fields propagating in R d, 1 and AdS d+1 were found in Refs. [45,46]. For fields in AdS d+1 which respect not only the relativistic so(d, 2) symmetries but also the conformal so(d + 1, 2) symmetries, the operator A is equal to zero and the light-cone gauge action (2.2) is invariant under transformations of the so(d + 1, 2) algebra. 11 We note then, that, for arbitrary spin massless fields in AdS 4 , the light-cone gauge action (2.2),(2.53) is invariant under the conformal so(4, 2) symmetries.

Light-cone gauge approach to currents and shadows in R d−1,1
Light-cone gauge formulation of currents (shadows) can be developed by starting with Lorentz covariant approach to currents (shadows). Doing so, we use differential constraints appearing in Lorentz covariant formulation of currents (shadows) to solve all fields entering Lorentz covariant approach in terms of unconstrained fields. 12 It is such unconstrained fields that constitute a field content entering the light-cone gauge formulation of currents (shadows). For the case of totally symmetric fields, the demonstration of how the Lorentz covariant approach can be used for the derivation of the light-cone gauge formulation of currents (shadows) in R d−1,1 , d-arbitrary, may be found in Ref. [4]. 13 However this strategy is difficult to realize in the case of mixed-symmetry currents (shadows) because, in general, Lorentz covariant formulations of mixed-symmetry currents (shadows) are complicated or not available. One of the attractive features of the light-cone approach is that this approach admits to develop light-cone gauge formulation of currents and shadows without the use of Lorentz covariant formulation. Namely, in Ref. [17], we developed the general light-cone formulation of arbitrary spin mixed-symmetry currents and shadows in R d−1,1 .
The light-cone gauge formulation of CFT worked out in Ref. [17] has the following unsatisfactory feature. Representation for generators of the so(d, 2) algebra obtained in Ref. [17] is non-local. 14 In Ref. [17], we noticed that the local representation can be obtained by choosing a suitable basis of currents (shadows). However an explicit local realization of the so(d, 2) algebra generators on such suitable basis of currents (shadows) has not been worked out in Ref. [17]. In this section, we fill this gap and find the following interesting result. It turns out that the local representation of the so(d, 2) algebra generators is entirely formulated in terms of basic operators ν, W i ,W i which we used to develop the light-cone gauge formulation of arbitrary spin AdS fields in Sec.2. In other words, it is the basic operators ν, W i ,W i discussed in Sec.2 that allows us to develop the local representation of the so(d, 2) algebra generators for currents and shadows. We start our presentation of new light-cone gauge approach to CFT with the discussion of light-cone gauge 2-point vertices for currents and shadows. Light-cone gauge 2-point vertices of currents and shadows. To simplify the presentation we collect all scalar, vector, tensor, and arbitrary spin fields entering the light-cone gauge formulation of currents and shadows into the respective ket-vectors |φ cur and |φ sh . By definition, in light-cone 11 As shown in Ref. [46], for fields in AdS d+1 which respect the conformal so(d + 1, 2) symmetries, one has the following relations and restrictions for E 0 , h i , and d: Recent interesting notes on conformal symmetries of massless and partial-massless fields may be found in Refs. [47]. 12 Study of differential constraints for short currents may be found in Refs. [48]- [51]. 13 The use of Lorentz covariant approach for the derivation of canonical formulation of totally symmetric arbitrary spin currents in R 3,1 may be found in Ref. [52]. Canonical formulation of totally symmetric arbitrary spin massless and massive fields in AdS d+1 , d-arbitrary, may be found in Ref. [53]. In the framework of canonical formulation, interesting recent discussion of AdS particle/string may be found in Ref. [54]. 14 We recall that if a representation for generators of the conformal symmetries is non-polynomial in the derivatives ∂ i , ∂ − , then such representation is referred to as non-local. Accordingly, if a representation for the generators is polynomial in the ∂ i , ∂ − , then such representation is referred to as local. We recall also that appearance of terms non-polynomial in the derivative ∂ + is unavoidable in light-cone gauge formulation. gauge approach, all fields entering the ket-vector |φ cur , |φ sh are not subject to any differential constraints.
For currents and shadows, one can construct three 2-point vertices. The first 2-point vertex, which we denote by Γ cur−sh , is a local functional of current and shadow. We now note the following expression for the local vertex: The second 2-point function denoted by Γ sh−sh is a nonlocal functional of two shadows, while the third 2-point function, denoted by Γ cur−cur , is a nonlocal functional of two currents. The 2-point functions Γ sh−sh and Γ cur−cur are given by where expression for the operator ν appearing in (3.4), (3.7) is given in (2.39). It the literature, sometimes one prefers to present the 2-point vertices in the momentum space. Therefore, for the reader convenience, we represent our above-given expressions for Γ sh−sh ,Γ cur−cur in the momentum space. To this end we make the Fourier transform for currents and shadows, Now it is easy to check that expressions for Γ sh−sh (3.2) Γ cur−cur (3.5) take the following respective forms where the momentum-space bra-vectors φ cur (p)| and φ sh (p)| are defined according the rule φ(p)| = (| φ(p) ) † . Note that expressions for Γ sh−sh (3.2) and Γ cur−cur (3.5) are equal to the respective expressions for Γ sh−sh (3.10) and Γ cur−cur (3.11) up to overall normalization factors. so(d, 2) symmetries of light-cone gauge currents and shadows. In the light-cone approach to currents and shadows, the Lorentz so(d−1, 1) algebra symmetries are not realized manifestly. This is to say that, in the framework of the light-cone approach, the complete description of currents and shadows implies that we have to work out an explicit realization of the so(d − 1, 1) algebra symmetries and the remaining symmetries of the so(d, 2) algebra as well. To this end we now discuss the so(d, 2) algebra symmetries of the light-cone gauge 2-point vertices above given. The so(d, 2) algebra transformations of light-cone gauge currents and shadows can be presented as δ|φ cur = G cur |φ cur , δ|φ sh = G sh |φ sh , (3.13) where the quantities G cur and G sh stand for realization of generators of the so(d, 2) algebra in terms of differential operators defined on the respective ket-vectors of currents and shadows. Expressions for the differential operators G cur and G sh we found admit the following representation: 14) Expressions for generators given in (3.14)-(3.23) are valid for both currents and shadows. Note however that explicit expressions for the operators ∆, M ⊖i , M ⊕i corresponding to the currents and shadows are different. Using the subscript 'cur' and 'sh' we note that the operators ∆, M ⊖i , M ⊕i corresponding to the currents and shadows are given by The following remarks are in order. i) The representation for the generators of conformal symmetries we obtained is local, i.e., generators (3.14)-(3.31) are polynomial in the derivatives ∂ i , ∂ − . In fact, the main advantage of the light-cone approach to currents (shadows) in this paper as compared the one in Ref. [17] is that the representation for generators of conformal symmetries obtained in this paper is local. ii) Operators ν, W i ,W i , M ij , which enter 2-point functions in (3.2), (3.5) and generators of so(d, 2) algebra symmetries of currents and shadows in (3.14)-(3.31), satisfy the same defining equations as the ones in light-cone gauge AdS field theory in (2.28)-(2.37). In other words, in our light-cone approach, the so(d, 2) algebra symmetries of bulk AdS field theory and the so(d, 2) algebra symmetries of boundary CFT are governed by one and the same operators ν, W i ,W i , M ij . Also, in our light-cone gauge approach, the Lagrangian of bulk AdS fields and 2-point functions of boundary currents and shadows are governed by the same operator ν. To summarize, finding a solution to defining equations given in (2.28)-(2.37) leads immediately to the complete description of light-cone gauge AdS field theory and light-cone gauge currents and shadows in CFT.
iii) The quantity C so(d,2) appearing in (3.27), (3.31) stands for the second order Casimir operator of the so(d, 2) algebra. For current labelled by E 0 , h and for shadow labelled by d − E 0 , h, the C so(d,2) takes the form given in (2.21). iv) Operators B cur (3.27) and B sh (3.31) coincide and take the same form as in light-cone gauge AdS field theory (2.26). v) Realization for generators of so(d, 2) algebra symmetries on space of currents and shadows given in (3.14)-(3.31) can be derived by using the general light-cone gauge approach to CFT developed in Ref. [17]. For some comments, see Appendix B.

AdS/CFT correspondence
In the framework of AdS/CFT, boundary currents are related to normalizable solution of equations of motion for bulk AdS fields, while boundary shadows are related to non-normalizable solution of equations of motion for bulk AdS fields. In Sec.2, we obtained the light-cone gauge formulation for bulk AdS fields, while, in Sec.3, we obtained the light-cone gauge formulation for boundary currents and shadows. We ready therefore to demonstrate the AdS/CFT explicitly. In the light-cone approach, the study of AdS/CFT implies, firstly, the matching of bulk and boundary symmetries of the so(d, 2) algebra and, secondly, the matching of effective action of bulk AdS field and boundary 2-point function of shadow. As a side remark we note that, in the framework of our light-cone gauge approach, the study of AdS/CFT is essentially simplified in view of the following two reasons. i) In the light-cone gauge approach, we deal with unconstrained fields on both the AdS and CFT sides. This implies that on both the AdS and CFT sides we deal with the same spin degrees of freedom. As we have already said, it is the matching of bulk and boundary spin degrees of freedom that explains why the generators of the so(d, 2) algebra in the bulk AdS theory and on the boundary CFT are governed by one and the same operators ν, W i ,W i , M ij which we discussed in Sections 2,3. ii) Light-cone gauge bulk action turns out be surprisingly simple and leads to the decoupled equations of motion. 15 Therefore, finding a solution to the Dirichlet problem and the computation of the bulk action on solution of the Dirichlet problem is considerably simplified.
We now study the AdS/CFT duality for normalizable and non-normalizable modes of bulk AdS fields and corresponding boundary currents and shadows in turn.

AdS/CFT for normalizable modes of AdS field and boundary current
In this section, we study the AdS/CFT for bulk AdS field and boundary current. As we have already said, in the framework of AdS/CFT, boundary current is related to normalizable solution of equations of motion for bulk AdS field. Therefore we start with analysis of the normalizable solution. To this end we note that the light-cone gauge action in (2.2) leads to the following equations of motion for AdS field The normalizable solution of Eq.(4.1) is well known where J ν in (4.3) stands for the Bessel function. From relation (4.2), we see that the normalizable solution for bulk AdS field |φ is entirely governed by the operator U ν . To proceed we note the following helpful interesting relations for the operator U ν which streamline the study of AdS/CFT: 15 It will be interesting to find Lagrangian formulation of massive fields in non-conformal gravitational backgrounds admitting decoupled equations of motion and apply such formulation for the study of holography along the lines in Refs. [55]. For recent discussion of massive fields in gravitational backgrounds, see, e.g., Refs. [56] (see also Refs. [57]).
where the operator T ν is defined in (2.27). Relations in (4.4)-(4.9) should be understood in weak sense. Namely, those relations hold true on space of ket-vectors which depend on the boundary coordinate x a and do not depend on the radial coordinate z. We note also that relations (4.4)-(4.9) can easily be proved by using the following well known formulas for the Bessel function: In fact, it is concise form of relations for the Bessel function in (4.10) that motivates us to use the operator T ν . Using textbook formulas for the Bessel function, we find that the asymptotic behavior of the normalizable solution given in (4.2) takes the following form The asymptotic behaviour in (4.11) tells us that, up to normalization factor, the ket-vector |φ cur is nothing but the boundary value of the normalizable solution of equations for bulk AdS field. From AdS/CFT dictionary, we expect that the ket-vector |φ cur appearing in (4.11) describes boundary current. The fact that |φ cur (4.11) is indeed realized as boundary current can be checked by matching of bulk and boundary symmetries of the so(d, 2) algebra. To check that |φ cur appearing on right hand side (4.11) is realized as boundary current we should prove the following statement: Representation of the light-cone gauge symmetries of the so(d, 2) algebra on space of bulk normalizable solution (4.2) amounts to representation of the light-cone gauge symmetries of the so(d, 2) algebra on space of boundary current.
To this end let us use the notation G AdS for the representation of the so(d, 2) algebra generators on space of bulk AdS field given in (2.8)-(2.20) and the notation G cur for representation of the so(d, 2) algebra generators on space of boundary current given in (3.14)- (3.27). With the use of such notation we note that all that is required is to demonstrate that the following relation holds true, where |φ is the normalizabe solution given in (4.2). The relation in (4.12) can be proved by following the procedure we described in Section 6.1 in Ref. [42]. For the reader convenience we present some relations which are helpful for analysis of relation (4.12). We note that the following relations are helpful to verify relation (4.12) for the generators P − and J −i . Other relations given by are helpful to verify relation (4.12) for the generators K a . Relations (4.13)-(4.15) are easily proved by using relations given in (4.4)-(4.9). Also note that, as in (4.4)-(4.9), relations in (4.13)-(4.15) should be understood in weak sense. Namely, relations (4.13)-(4.15) hold true on space of ket-vectors which depend on the boundary coordinate x a and do not depend on the radial coordinate z.
Representation of bulk fields in terms of currents (4.2) is referred sometimes as holographic representation of bulk fields. Our relation (4.2) is valid for massless and massive arbitrary spin totally symmetric and mixed-symmetry fields. Thus, we see that our light-cone approach allows us us to study the holographic representation for all just mentioned fields on an equal footing. 16

AdS/CFT for non-normalizable modes of AdS field and boundary shadow
This section is devoted to the study of AdS/CFT for bulk AdS field and boundary shadow. As we have already said, in the framework of AdS/CFT, boundary shadow is related to non-normalizable solution of equations of motion for bulk AdS field. 17 Therefore we start with the analysis of the non-normalizable solution of bulk equations of motion.
The bulk equations of motion obtained from light-cone gauge action (2.2) are given in (4.1). Non-normalizable solution to equations in (4.1) with the Dirichlet problem corresponding to shadow is well known, where G ν appearing on right hand side in (4.16) is the bulk to boundary Green function. The asymptotic behaviors of the Green function and solution in (4.16) are well known Relation (4.20) tells us that, up to normalization factor, the ket-vector |φ sh is nothing but the boundary value of the non-normalizable solution of the Dirichlet problem for bulk equations of motion.
Using the AdS/CFT dictionary, we expect that the ket-vector |φ sh appearing in (4.16) describes boundary shadow. To make sure that ket-vector |φ sh (4.16) is indeed realized as boundary shadow we should prove the following two statements: i) Representation of the light-cone gauge symmetries of the so(d, 2) algebra on space of bulk nonnormalizable solution (4.16) amounts to the representation of the light-cone gauge symmetries of the so(d, 2) algebra on space of boundary shadow.
ii) The light-cone gauge action of bulk AdS field evaluated on the solution of the Dirichlet problem (4.16) amounts to the boundary 2-point vertex for the shadow.
We outline proof of these statements.
Matching of bulk and boundary representations of the so(d, 2) algebra. We use the notation G AdS for the representation of the so(d, 2) algebra generators on space of bulk AdS field given in (2.8)-(2.20) and the notation G sh for the representation of the so(d, 2) algebra generators on space of boundary shadow given in (3.14)-(3.23), (3.28)- (3.31). With the use of such notation we note that all that is required is to demonstrate that G AdS and G sh respect the relation where |φ is given in (4.16). Relation given in (4.21) can be proved by following the procedure we described in Section 6.1 in Ref. [42]. For the reader convenience, we present the list of relations which are helpful for the analysis of relation (4.21) 2) should be supplemented with appropriate boundary term. Using the method in Ref. [62], we verify that a light-cone gauge Lagrangian which involves an appropriate boundary term takes the form Note also that, to respect the commonly used Euclidean signature, the overall sign of Lagrangian (2.3) has been changed, L → −L, when passing from expression (2.3) to expression (4.31).
Now it is easy to check that action (2.2), (4.31) considered on the solution of the Dirichlet problem admits the following general representation: Plugging solution to the Dirichlet problem (4.16) into (4.32), we obtain the effective action where c κ is given in (4.18), while the 2-point vertex of shadow Γ sh−sh takes the form given in (3.2), (3.3). We recall also that κ is defined in (2.39). Relation (4.33) tells that the effective action and the 2-point vertex of shadow indeed match. By product, we find the normalization factor 2κc κ entering (4.33). This normalization factor might be important for the systematical investigation of the AdS/CFT duality. Our relation for effective action (4.33) is valid for massless and massive arbitrary spin totally symmetric and mixed-symmetry fields. Thus we see that the light-cone gauge approach allows us to study the AdS/CFT correspondence for all just mentioned fields on an equal footing. 18 To summarize, we used new light-cone gauge formulation of bulk AdS field theory and boundary CFT for the investigation of the AdS/CFT correspondence of AdS fields and boundary currents (shadows). We believe that our new light-cone gauge approach to AdS field theory and CFT might have other interesting applications along the lines in Refs. [66,67].

Light-cone gauge approach to conformal fields in R d−1,1
Our expression for 2-point vertex of shadow given in (3.2) provides interesting opportunity for finding a light-cone gauge Lagrangian of conformal field. To explain what has just been said we note that a kernel of the 2-point vertex of shadow in (3.2), (3.3) is not well-defined for the cases when ν given in (2.39) takes integer values (see, e.g., Ref. [68]), while for the cases when ν takes non-integer values the kernel turns out to be well-defined. Therefore 2-point vertex of shadow with integer ν can be regularized by using non-integer ν. Removing the regularization, we are left with a logarithmic divergence of the 2-point vertex for shadow which turns out to be light-cone gauge action of conformal field.
Let us describe the regularization procedure we are going to use. Taking into account (2.39) and recalling that eigenvalues of the operator M z take integer values, we see that non-integer values of κ imply non-integer values of ν. Denoting the integer part of κ by κ int , we define the regularization parameter ε by the following relation: Taking into account (5.1) and (2.39), we use the textbook asymptotic behavior for the kernel: . (5.2) Plugging expression (5.2) into Γ sh−sh given in (3.2), we get where κ int in (5.4) is arbitrary integer which satisfies (5.5). Conformal field which is related to massive AdS field is referred to as long conformal field. As there are no restrictions on space-time dimension d ≥ 4, we conclude that the long conformal field propagates in space-time of arbitrary dimension d ≥ 4. Conformal dimension of the long conformal field is given by ∆ = d − E 0,long , where E 0,long is given in (5.4), (5.5). Thus, the short conformal fields propagate in space-time of even dimension d ≥ 4 and these fields are related to massless fields, while the long conformal fields propagate in space-time of arbitrary dimension d ≥ 4 and are related to massive AdS fields having E 0 defined by (5.4), (5.5).
Light-cone gauge action of conformal field (5.3) is invariant under symmetries of the so(d, 2) algebra. Realization of light-cone gauge symmetries of the so(d, 2) algebra on space of conformal field |φ is described by the same relations as for shadow (see (3.13), (3.14)-(3.23) (3.28)-(3.31)). Operators ν, W i ,W i , M ij , which enter light-cone gauge action (5.3) and generators of the so(d, 2) algebra symmetries of conformal fields (3.14)-(3.31), satisfy the same defining equations as the ones in light-cone gauge AdS field theory in (2.28)-(2.37). In other words, in our light-cone approach, the so(d, 2) algebra symmetries of massless (massive) fields in AdS d+1 and the so(d, 2) algebra symmetries of short (long) conformal fields in R d−1,1 are governed by one and the same operators ν, W i ,W i , M ij . Also, in our approach, the Lagrangian of massless (massive) fields in AdS d+1 and Lagrangian of conformal short (long) fields in R d−1,1 are governed by the same operator ν. To summarize, finding a solution to defining equations given in (2.28)-(2.37) leads immediately to the complete description of light-cone gauge theory of massless (massive) fields in AdS d+1 and light-cone gauge theory of short (long) conformal fields in R d−1,1 . We note also that irreps of the so(d − 2) algebra which enter the light-cone gauge formulation of massless (massive) field in AdS d+1 coincide with irreps of the so(d − 2) algebra which enter the light-cone gauge formulation of short (long) conformal field in R d−1,1 . In fact, it is matching of irreps of the so(d − 2) algebra entering the massless (massive) light-cone gauge fields in AdS d+1 and the irreps of the so(d − 2) algebra entering the light-cone gauge short (long) conformal fields that explains why the light-cone gauge formulations of massless (massive) AdS fields and conformal short (long) fields are governed by one and the same operators ν, W i ,W i , M ij .
For the illustration purposes, we now discuss totally symmetric conformal fields. The lightcone gauge Lagrangian of totally symmetric short conformal fields has already been presented in Ref. [69], while light-cone gauge Lagrangian of totally symmetric long conformal fields has not been discussed in the earlier literature. For the reader convenience we discuss the both short and long conformal fields.
spin-1 short conformal field in R d−1,1 . Field content we use for the light-cone gauge description of spin-1 short conformal field in R d−1,1 involves vector field φ i and scalar field φ of the so(d − 2) algebra. These fields can be collected into the following ket-vector: 20 Realization of the operator ν on space of ket-vector (5.6) takes the form Plugging (5.6) and (5.7) into (5.3), we get Lagrangian for spin-1 short conformal field The number of propagating D.o.F described by the Lagrangian (5.8) is given by spin-1 long conformal field in R d−1,1 . Field content we use for the light-cone gauge description of spin-1 long conformal field in R d−1,1 involves one vector fields φ i and two scalar fields φ −1 , φ 1 of the so(d − 2) algebra, We collect fields (5.10) into the ket-vector defined by Realization of the operator ν on space of ket-vector (5.11) takes the form 12) where N z = α zᾱz , N ζ = ζζ. Plugging (5.11) and (5.12) into (5.3), we get Lagrangian for spin-1 long conformal field, (5.14) Using the notation n for the number of propagating D.o.F described by the Lagrangian (5.13) we note the relation where n so(d) 1 in (5.15) stands for dimension of the spin-1 irreps of the so(d) algebra. spin-2 short conformal field in R d−1,1 . Field content we use for the light-cone gauge description of spin-2 short conformal field in R d−1,1 involves traceless tensor field φ ij , vector field φ i and scalar field φ. All these fields transform as irreps of the so(d − 2) algebra. We introduce ket-vector by the relation Realization of the operator ν on space of ket-vector (5.16) takes the form where κ is given in (5.17). Note that number of propagating D.o.F described by the Lagrangian (5.18) is given by spin-2 long conformal field in R d−1,1 . Field content we use for the light-cone gauge description of spin-2 long conformal field in R d−1,1 involves one traceless rank-2 tensor field, two vector fields, and three scalar fields All fields in (5.20) transform as irreps of the so(d − 2) algebra. We collect fields (5.20) into ket-vector defined by Realization of the operator ν on space of ket-vector (5.21) takes the form  in (5.25) stands for dimension of the totally symmetric spin-2 irrep of so(d) algebra. spin-s totally symmetric short conformal field in R d−1,1 . Field content we use for the lightcone gauge description of spin-s totally symmetric short conformal field in R d−1,1 involves scalar, vector, and traceless tensor fields of the so(d − 2) algebra, φ i 1 ...i s ′ , s ′ = 0, 1, . . . , s. We collect these fields into ket-vector defined by Realization of the operator ν on space of ket-vector (5.26) takes the form Note that number of propagating D.o.F described by the Lagrangian (5.28) is given by spin-s totally symmetric long conformal field in R d−1,1 . Field content we use for the light-cone gauge description of spin-s totally symmetric long conformal field in R d−1,1 involves scalar, vector and traceless tensor fields of the so(d − 2) algebra, φ Here and below the notation λ ∈ [n] 2 implies that λ = −n, −n + 2, . . . , n − 2, n. We collect the fields into ket-vector defined by Realization of the operator ν on space of ket-vector (5.30) takes the form 31) where N ζ = ζζ, N z = α zᾱz . Plugging (5.30) and (5.31) into (5.3), we get Lagrangian for spin-s long conformal field For the case of short conformal fields, the expression for n given in (5.29) agrees with expression for n when d = 4 in Ref. [13] and d ≥ 4 in Ref. [70]. 21 To our knowledge, for the case of long conformal fields, expression for n given in (5.34) has not been discussed in the earlier literature.

Antisymmetric (one-column) fields
In the framework of Lorentz covariant approach, Lagrangian formulation of antisymmetric massless and massive fields in AdS is well known. Using such formulation, the AdS/CFT for totally antisymmetric AdS fields and related boundary shadows was studied in Ref. [72]. To our knowledge, the AdS/CFT for totally antisymmetric AdS fields and related boundary currents has not been considered in the literature.
In this section, we discuss light-cone gauge formulation of massless and massive antisymmetric AdS fields. In the framework of our approach, equations of motion turn out to be decoupled and this considerably simplifies the study of AdS/CFT. Also, we recall that finding a solution to the defining equations for the spin operators (2.28)-(2.37) leads immediately to the complete lightcone gauge description of AdS field theory and boundary currents, shadows, and conformal fields. This is to say that using the antisymmetric fields we demonstrate how our general approach works.

Antisymmetric (one-column) massive fields in AdS
Restriction on E 0 (6.2) is the standard unitarity constraint (1.5) represented in terms of the label s. For even d and s = d/2, we deal with a sum of self-dual and anti self-dual massive fields associated with the respective unitary irreps D(E 0 , h + ) and D(E 0 , h − ), where for even d , Field content of massive field. In order to describe field content entering dynamics of antisymmetric massive field it is convenient to introduce anti-commuting oscillators α i , α z , ζ. The oscillators α i , i = 1, . . . , d − 2, transform as vector of the so(d − 2) algebra, while the oscillators α z and ζ transform as scalars of the so(d − 2) algebra. Anti-commutation relations for the oscillators, the vacuum |0 , and hermitian conjugation rules, are defined as All the remaining anti-commutators for the oscillators are equal to zero. Using such oscillators we introduce the following ket-vector to discuss the antisymmetric rank-s massive fields: where the ket-vector, by definition, satisfies the following constraint Finite number of ordinary light-cone gauge fields depending of space-time coordinates x a , z are obtained by expanding ket-vector (6.8) into the oscillators α i , α z , ζ, |φ = |φ 1 + |φ 0 + |φ −1 , (6.10) where field φ i 1 ...i K λ in (6.14) is antisymmetric rank-K tensor field of the so(d − 2) algebra. Thus we see that ket-vector |φ (6.10) is expanded into finite set of antisymmetric tensor fields of the so(d − 2) algebra. 22 To label ket-vectors (6.11)-(6.13) we use the subscript λ which is eigenvalue of the operator M z (see (6.16)). For the illustration purposes, we use the shortcut φ K λ for rank-K antisymmetric tensor field φ i 1 ...i K λ to represent the tensor fields entering ket-vector in (6.10) as Lagrangian of massive field. From (2.3), (2.23), we see that the Lagrangian is determined by the operator ν. We find the following realization of the operator ν on space of ket-vector (6.10): Plugging then ket-vector (6.10) into (2.3), we see that Lagrangian takes the form where ket-vectors |φ λ , λ = 0, ±1, are defined in (6.11)-(6.13).
According to the general setup in Sec.2, to complete the light-cone gauge description of the antisymmetric field we should provide a realization of the operators M ij , W i ,W i , B on space of ket-vector (6.10). The realization of these operators on space of ket-vector |φ (6.10) is given by We note that the realization for the spin operator of the so(d − 2) algebra M ij given in ( where the ket-vectors |φ sh,λ take the same form as in (6.11)-(6.14). Long conformal field in R d−1,1 . Antisymmetric rank-s long conformal field in R d−1,1 is associated with non-unitary representation of the so(d, 2) algebra with labels ∆, h, where h is given in (6.1), while the conformal dimension ∆ is given by For s = d/2, d-even, we deal with a sum of self-dual and anti self-dual conformal fields having h ± as in (6.3).
Ket-vector of long conformal field |φ in R d−1,1 takes the same form as the one for massive field in AdS d+1 (6.10). Relations in (6.18)-(6.23) with the substitution κ → κ int provide solution for the operators ν, W i ,W i , M ij entering the light-cone gauge description of the long conformal field. Plugging such operators ν, W i ,W i , M ij into (3.14)-(3.23), (3.28)-(3.31) provides the realization of the conformal so(d, 2) symmetries on space of the ket-vector |φ , while plugging the ket-vector |φ into (5.3), we get Lagrangian for the long conformal field where the ket-vectors |φ λ take the same form as in (6.11)-(6.14). In terms of tensor fields (6.14), Lagrangian (6.28) takes the form where s and κ int satisfy restrictions in (6.1), (6.27). Using the notation n for the number of propagating D.o.F described by the Lagrangian (6.29), we note the relation  , we see that in order to realize the limit of antisymmetric massless field we have to consider the limit 23 In the framework of Lorentz covariant approach, it is well-known that, in the massless limit, antisymmetric rank-s massive AdS field is decomposed into rank-s and rank-(s − 1) massless AdS fields. We now demonstrate how this result is obtained in the framework of light-cone gauge approach. To this end we note that the ket-vector of massive field in (6.10) can be represented as |φ = |φ + ζ|ϕ , (6.32) where the ket-vectors |φ K λ appearing in (6.33),(6.34) are defined in (6.14). It turns out that, in the massless limit, it is the ket-vectors |φ and |ϕ that describe the respective rank-s and rank-(s − 1) massless AdS fields. Namely, considering the spin operators (6.16), (6.18)-(6.23) in the limit (6.31), we verify that ket-vectors |φ , |ϕ (6.33), (6.34) are invariant under action of the so(d, 2) algebra generators given in (2.8)- (2.16). 24 In other words, in limit (6.31), the ket-vector |φ is decomposed into two invariant sub-spaces described by the ket-vectors |φ , |ϕ . The ket-vectors |φ and |ϕ describe the respective antisymmetric rank-s and rank-(s − 1) irreps of so(d − 1) algebra. The ket-vectors |φ and |ϕ are associated with the unitary irreps of the so(d, 2) algebra with E 0 = d − s and E 0 = d + 1 − s respectively. This implies that, in the massless limit, the antisymmetric rank-s massive field is decomposed into antisymmetric rank-s and rank-(s − 1) massless fields described by the respective ket-vectors |φ and |ϕ .
Plugging ket-vector (6.33) into (2.3), we get Lagrangian of rank-s antisymmetric massless field To complete the light-cone gauge description of the antisymmetric massless field, we provide the realization of the operators M ij , W i ,W i , B on space of ket-vector |φ (6.33), We note that the spin operators in (6.36)-(6.38) are simply obtained by taking the limit (6.31) in expressions for spin operators in (6.18)-(6.23). Short currents and shadows in R d−1,1 . Antisymmetric rank-s short current in R d−1,1 is related to unitary irrep D(∆ cur , h) of the so(d, 2) algebra, where ∆ cur = d − s, while h is given in (6.1) and s ≤ (d − 1)/2. Antisymmetric rank-s short shadow in R d−1,1 is associated with representation of the so(d, 2) with ∆ sh = s and h as in (6.1), where s ≤ (d − 1)/2. For short currents and shadows in R d−1,1 , ket-vectors |φ cur , |φ sh and the operators ν, W i , W i , M ij take the same form as the ones for massless field in AdS d+1 (see (6.33), (6.36)-(6.38)). Plugging the ket-vectors |φ cur , |φ sh and the operators ν, W i ,W i , M ij into expressions for 2-point functions and generators of so(d, 2) algebra presented in Sec.3, we get the complete lightcone gauge description of the short currents and shadows. Short conformal field in R d−1,1 . Antisymmetric rank-s short conformal field in R d−1,1 is associated with representation of the so(d, 2) algebra with labels ∆, h, where h is given in (6.1), while the conformal dimension ∆ is given by For rank-s antisymmetric short conformal field in R d−1,1 , ket-vector |φ and operators ν, W i , W i , M ij take the same form as the ones for massless field in AdS d+1 (see (6.33), (6.36)-(6.38)).
Plugging the operators ν, W i ,W i , M ij into (3.14)-(3.23), (3.28)-(3.31) provides realization of the conformal so(d, 2) symmetries on space of the ket-vector |φ , while plugging the ket-vector |φ into (5.3), we get Lagrangian for the short antisymmetric conformal field In terms of tensor fields (6.14), Lagrangian (6.40) takes the form (6.42) Using the notation n for the number of propagating D.o.F described by the Lagrangian (6.41) we note the relation

Mixed-symmetry (two-column) fields
In the framework of Lorentz covariant approach, Lagrangian formulation of two-column massless and massive fields in AdS was developed in Ref. [25] and Ref. [28] respectively. AdS/CFT for such fields and related boundary currents (shadows) has not been considered in the literature. Lorentz covariant description of two-column short conformal field can be obtained from Ref. [15]. Long conformal fields have not been discussed in the framework of Lorentz covariant approach.
In this section, we discuss the light-cone gauge description of two-column massless and massive AdS fields. We recall that, in our approach, the complete light-cone description of massless AdS fields leads immediately to the complete light-cone description of short currents, shadows, and conformal fields, while the complete light-cone description of massive AdS fields leads immediately to the complete light-cone description of long currents, shadows, and conformal fields. We discuss AdS fields and CFT in turn. Massive field in AdS d+1 . We now consider mixed-symmetry (two-column) massive fields in AdS d+1 . Physical D.o.F of such fields are described by irreducible tensor fields of the so(d) algebra whose so(d) space tensor indices have the structure of two-column Young tableaux. We use integers s 1 , s 2 , s 1 ≥ s 2 , to indicate of height of the first and second columns of the two-column Young tableaux. Two-column field described by such Young tableaux will be referred to as type 25 For s = (d − 2)/2, the field φ can be decomposed into self-dual and anti self-dual irreps of the so(d − 2) algebra.

Mixed-symmetry (two-column) massive fields in AdS
[s 1 , s 2 ] field. The type [s 1 , s 2 ] massive field in AdS d+1 is associated with the unitary representation D(E 0 , h) of the so(d, 2) algebra, where E 0 , h are given by h = (2, 2, · · · 2, 2 s 2 times , 1, 1, · · · 1, 1 s 1 −s 2 times , 0, 0, · · · 0, 0 r−s 1 times Field content of massive field with s 1 > s 2 ≥ 1. In order to describe field content entering dynamics of two-column massive field we introduce anti-commuting oscillators α i n , α z n , ζ n , n = 1, 2. The oscillators α i n , i = 1, . . . , d − 2, transform as vector of the so(d − 2) algebra, while the oscillators α z n and ζ n transform as scalars of the so(d − 2) algebra. Anti-commutation relations for the oscillators, the vacuum |0 , and hermitian conjugation rules are defined as We note also that all remaining anti-commutators for the oscillators are equal to zero. Using such oscillators we introduce the ket-vector |φ , which, by definition, satisfies the following algebraic constraints: (N αn + N α z n + N ζn − s n )|φ = 0 , n = 1, 2 , (7.8) Constraints (7.8) tell us that the ket-vector |φ is degree-s n homogeneous polynomial in the oscillators α i n , α z n , ζ n . Constraints (7.9) and (7.10) are the respective Young symmetry constraint and tracelessness constraint. Finite number of ordinary light-cone gauge fields depending of spacetime coordinates x a , z are obtained by expanding ket-vector (7.7) into the oscillators α i n , α z n , ζ n , , K 1 ≥ K 2 , appearing in (7.19) is mixed-symmetry (two-column) irreducible tensor field of the so(d − 2) algebra. By definition, ket-vector (7.19) satisfies constraints (7.9), (7.10). Note that we assume the convention |φ K 1 K 2 λ = 0 for K 1 < K 2 . To label ket-vectors (7.14)-(7.18) we use the subscript λ which is eigenvalue of the operator M z (see (7.21)). For the illustration purposes, we use the shortcut φ K 1 K 2 λ for two-column tensor field φ i 1 ...i K 1 ,j 1 ...j K 2 λ to represent the tensor fields entering ket-vector in (7.13) as Lagrangian of massive field. We note the following realization of the operator ν on space of ket-vector (7.13): where the operators N α z n , N ζn are defined in (7.12). Plugging ket-vector (7.13) into (2.3), we see that Lagrangian takes the form where ket-vectors |φ λ , λ = 0, ±1, ±2, are defined in (7.14)- (7.18). To complete the light-cone gauge description of the two-column massive field we provide a realization of the operators M ij , The realization of these operators on space of ket-vector (7.13) is given by

25)
where the operator M z appearing in (7.32) is defined in (7.21). The spin operator of the so(d − 2) algebra M ij given in (7.23) is well known, while the realization of the operators ν, W i ,W i in (7.24), (7.25) is found by solving the defining equations given in (2.30)-(2.37). Explicit expressions for the operators W i ,W i may be found in Appendix C. Expression for the operator B in (7.32) can be obtained by using (2.40) and well know expression for eigenvalue of the second order Casimir operator of the so(d) algebra for two-column irrep, For two-column long currents and shadows in R d−1,1 , ket-vectors |φ cur , |φ sh and the operators ν, W i ,W i , M ij take the same form as the ones for two-column massive field in AdS d+1 (see (7.21), (7.23)-(7.32)). Plugging the ket-vectors |φ cur , |φ sh and the operators ν, W i ,W i , M ij into expressions for 2-point functions and generators of the so(d, 2) algebra presented in Sec.3, we get the complete light-cone gauge description of the two-column long currents and shadows. Long conformal field in R d−1,1 . Type [s 1 , s 2 ] long conformal field in R d−1,1 is associated with non-unitary representation of the so(d, 2) algebra with labels ∆, h, where h is given in (7.1), while the conformal dimension ∆ is given by where the ket-vectors |φ λ take the same form as in (7.14)-(7.18).
7.2 Mixed-symmetry (two-column) massless fields in AdS d+1 and short currents, shadows, and conformal fields in R d−1,1 Massless field in AdS d+1 . From (1.3), (1.4), we see that in order to realize the limit of two-column massless field we have to consider the limit We note that the ket-vectors of massive field in (7.13) can be presented in terms of two new ketvectors |φ , |ϕ in the following way |φ = |φ + ζ 2 |ϕ (7.38) where ket-vectors |φ K 1 K 2 λ appearing in (7.41)-(7.48) are defined in (7.19). Now, considering the massless limit (7.37), we are led to the following conclusion: for s 2 > 1, the ket-vectors |φ and |ϕ describe the respective type [s 1 , s 2 ] and type [s 1 , s 2 − 1] massless AdS fields, while, for s 2 = 1, the ket-vectors |φ and |ϕ describe the respective type [s 1 , 1] massless and type [s 1 , 0] massive AdS fields (see also Refs. [75,28]). The type [s 1 , 0] massive field is associated with unitary irreps of the so(d, 2) algebra with E 0 = d + 1. On space of type [s 1 , s 2 ] massless AdS field |φ (7.39), the operator ν takes the form as in (2.39) with M z = N ζ 1 − N α z 1 − N α z 2 and E 0 as in (7.37). Plugging ket-vector |φ (7.39) into (2.3), we get Lagrangian of type [s 1 , s 2 ] massless AdS field For massless AdS field, the spin operators W i ,W i are obtained by taking limit (7.37) in the respective spin operators of the massive field (see Appendix C).

Conclusion.
In conclusion, we make the following comment. Our presentation for the (one)column and (two)column AdS fields demonstrates how the field content and explicit form of the operator M z which are required for the light-cone gauge formulation of mixed-symmetry arbitrary spin AdS field can be specified. This is to say that, by analogy with the light-cone gauge formulation of massive field in R d,1 , in order to develop the light-cone gauge formulation of massive field in AdS d+1 we can use field content which is realized as irreducible representation of the so(d) algebra. Note that, in this respect, the light-cone gauge formulation of massive field is similar to the Lorentz covariant formulation of massive field. Namely, we recall that the field content entering the Lorentz covariant formulation of massive field in R d,1 coincides with the field content entering the Lorentz covariant formulation of massive field in AdS d+1 . Use of the oscillators provides the easy way to describe the field content and the operator M z . For example, in order to describe field content entering the light-cone gauge formulation of mixed-symmetry arbitrary spin massive AdS field (1.5), we introduce a finite set of anti-commuting oscillators α i n , α z n , ζ n , n = 1, . . . , h k , which satisfy the same relations as in (7.4)-(7.6). Also, by analogy with (7.7), we can introduce a ket-vector |φ which, by definition, depends on all just mentioned oscillators and satisfies the well-known algebraic constraints needed to select the irreducible representation of the so(d) algebra. Those constraints for mixed-symmetry arbitrary spin field are the counterparts of the ones for two-column field in (7.8)-(7.10). For such oscillator realization of |φ , the operator M z takes the form M z = h k n=1 M z n , where we use M z n ≡ N ζn − N α z n and notation as in (7.12). On space of |φ , eigenvalues of such operator M z take values −h k , −h k + 1, . . . , h k , as it should be. For massless AdS field, the field content is realized as an invariant subspace in space of massive AdS field. so(d, 2) algebra is realized. Then introducing a current by the relation q ν+ 1 2 |O = |φ cur , q ≡ √ ✷, we verify that the non-local representation on space of the conformal field |O given by relations (3.1)-(3.15) in Ref. [17] is realized as the local representation on the space of |φ cur given by relations in (3.14)- (3.27) in this paper, while introducing a shadow by the relation q −ν+ 1 2 |O = |φ sh , q ≡ √ ✷, we verify that the non-local representation on space of conformal field |O given by relations (3.1)-(3.15) in Ref. [17] is realized as the local representation on space of |φ sh given by relations (3.14)-(3.23), (3.28)-(3.31) in this paper. σ ζ 2 = κ − g ζ 1 (1) = r ζ 1 (1) sin ψ ζ 1 , sin(ψ ζ 2 + ψ α z 2 ) = t − h + 2 (t + 2)h − 1 (κ 2 − 1)σ ζ 1 σ α z 1 κ 2 ρ ζ 2 ρ α z 2 1/2 , (C. 47) sin(ψ α z 1 + ψ α z 2 ) = − κ + t + 2 κ (h 2 − 1)σ ζ 1 σ ζ 2 ((t + 2)h + 1)((t + 2)h − 1)ρ α z 1 ρ α z 2 1/2 .