Mixed-symmetry fields in AdS(5), conformal fields, and AdS/CFT

Mixed-symmetry arbitrary spin massive, massless, and self-dual massive fields in AdS(5) are studied. Light-cone gauge actions for such fields leading to decoupled equations of motion are constructed. Light-cone gauge formulation of mixed-symmetry anomalous conformal currents and shadows in 4d flat space is also developed. AdS/CFT correspondence for normalizable and non-normalizable modes of mixed-symmetry AdS fields and the respective boundary mixed-symmetry anomalous conformal currents and shadows is studied. We demonstrate that the light-cone gauge action for massive mixed-symmetry AdS field evaluated on solution of the Dirichlet problem amounts to the light-cone gauge 2-point vertex of mixed-symmetry anomalous shadow. Also we show that UV divergence of the action for mixed-symmetry massive AdS field with some particular value of mass parameter evaluated on the Dirichlet problem amounts to the action of long mixed-symmetry conformal field, while UV divergence of the action for mixed-symmetry massless AdS field evaluated on the Dirichlet problem amounts to the action of short mixed-symmetry conformal field. We speculate on string theory interpretation of a model which involves short low-spin conformal fields and long higher-spin conformal fields.


Introduction
Conjectured string/gauge theory duality in Ref. [1] and light-cone gauge approach to string theory in AdS space in Ref. [2] triggered our interest to light-cone gauge formulation of field dynamics in AdS space. This is to say that the light-cone approach to string theory in AdS space implies the corresponding light-cone formulation for target space fields in AdS. It is expected that arbitrary spin totally symmetric and mixed-symmetry massive AdS fields together with low-spin massless fields form the spectrum of states of AdS string. Therefore a better understanding of the light-cone formulation for arbitrary spin massive AdS fields might be useful for discussion of various aspects of string theory in AdS space. This is long-term motivation of our research in this paper.
Before describing the particular problem we investigate in this paper, let us discuss group theoretical interpretation of fields in AdS 5 we are going to study. Massive and massless fields in AdS 5 space are associated with unitary lowest weight representations of the so(4, 2) algebra. A unitary positive-energy lowest weight irreducible representation of the so(4, 2) algebra is denoted by D(E 0 , h), where a label E 0 stands for the lowest eigenvalue of the energy operator, while a label h = (h 1 , h 2 ) is the highest weight for representation of the so(4) algebra. For bosonic fields, the highest weights h 1 , h 2 are integers, while for fermionic fields, the highest weight h 1 , h 2 are half-integers. These weights satisfy the well known restriction (1.1) We recall that the field with h = (0, 0) is a scalar bosonic field, while the field with h = (1/2, ±1/2) is a spin one-half fermionic field. 1 The labels E 0 and h associated with the remaining fields in AdS 5 satisfy the following restrictions [3], 2 Often, in place of the labels E 0 , h 1 , h 2 , we use the respective labels κ, j 1 , j 2 defined by the relations For fields in (1.5),(1.6), light-cone gauge formulation was developed in Refs. [5,6]. 3 Lightcone gauge actions in Refs. [5,6] lead to coupled equations of motion for fields in (1.5),(1.6). Light-cone gauge actions leading to decoupled equations of motion for fields in (1.5),(1.6) were obtained recently in Ref. [8]. For fields in (1.2),(1.3), light-cone gauge formulation was developed 1 For the reader's convenience, we note that the field with h = (1, 0) is spin-1 field, while the field with h = (2, 0) is a spin-2 field. Sometimes, the labels h i are referred to as Gelfand-Zeitlin labels. The Gelfand-Zeitlin labels are connected with the Dynkin labels h D i by the relations (h D 1 , h D 2 ) = (h 1 − h 2 , h 1 + h 2 ). 2 Generalization of restrictions (1.2)-(1.6) to the case of so(d, 2) algebra with arbitrary d may be found in Ref. [4]. 3 In Refs. [5,6], light-cone gauge formulation was developed for totally symmetric fields in AdS d+1 with arbitrary d, d ≥ 3. Light-cone gauge formulation for fields in AdS 3 was investigated in [7]. in Ref. [9], while, for fields in (1.4), light-cone gauge formulation was studied in Ref. [10]. 4 For fields in (1.2),(1.3), light-cone gauge actions in Ref. [9] lead to decoupled equations of motion, while, for fields in (1.4), light-cone gauge actions in Ref. [10] lead to coupled equations of motion.
Two main aims of this paper are as follows. The first aim is to develop light-cone gauge formulation which leads to decoupled equations of motion for mixed-symmetry massive fields (1.4). The second aim is to apply such light-cone formulation to study of AdS/CFT correspondence. In this paper, our primary interest are the massive mixed-symmetry fields. The massless and selfdual massive fields are realized as appropriate limits of the massive fields. This allows us to study a Lagrangian formulation and the AdS/CFT correspondence for the mixed-symmetry massive, mixed-symmetry massless, and self-dual massive fields on an equal footing.
Before proceeding to main theme of this paper let us briefly review various Lorentz covariant approaches in the literature which have been developed for the description of mixed-symmetry fields in AdS space. At the level of equations of motion given in Lorentz gauge, massless and massive mixed-symmetry fields in AdS d+1 , with arbitrary d were studied in Refs. [4,6,11]. Onshell BRST invariant formulation of massless and massive mixed-symmetry fields may be found in Ref. [12]. On-shell formulation of massless mixed-symmetry fields in terms of generalized connections of AdS symmetry algebras is studied in Refs. [13]. Lagrangian frame-like description of massless and massive mixed-symmetry fields may be found in the respective Refs. [14] and Refs. [15]. Lagrangian metric-like formulation for some particular mixed-symmetry massless field is considered in Ref. [16]. Important aspects of mixed-symmetry massless fields were studied in Refs. [17]. Maxwell-like Lagrangians for massless mixed-symmetry fields are considered in Ref. [18]. Discussion of Lagrangian formulation of massive fields corresponding to two-column Young tableaux may be found in Ref. [19]. BFV-BRST Lagrangian formulation of mixed-symmetry massive fields is discussed in Ref. [20]. In the framework of world-line spinning particle approach, the self-dual massive fields are studied in Ref. [21]. Interesting recent discussion of mixed-symmetry fields may be found in Ref. [22]. Interacting mixed-symmetry fields were studied in Refs. [23]. 5 This paper is organized as follows. In Sec.2, we develop light-cone gauge Lagrangian formulation which leads to decoupled equations of motion for mixed-symmetry massive fields in AdS 5 . Also we describe light-cone gauge realization of relativistic symmetries of actions for such fields. In Secs.3,4, mixed-symmetry massless and self-dual massive fields are considered. We demonstrate how light-cone gauge Lagrangian for such fields can be obtained from the Lagrangian for the mixed-symmetry massive field. In Sec.5, light-cone gauge formulation for mixed-symmetry anomalous conformal currents and shadows is developed. Two-point vertices for such currents and shadows are discussed. Section 6 is devoted to study of the AdS/CFT correspondence for mixedsymmetry AdS fields and boundary mixed-symmetry conformal currents and shadows. Mixedsymmetry conformal fields are considered in Sec.7. We demonstrate how the action for such fields can be obtained from 2-point vertex of mixed-symmetry shadows. Our conventions and notation are explained in Appendix A. 4 We note that it is light-cone gauge approach in Refs. [9,10] that made it possible to develop a Lagrangian formulation of AdS fields in (1.2)-(1.4) for the first time. Also, for the reader's convenience, we recall that, in Lorentz covariant approaches, the mixed-symmetry fields in (1.2)-(1.4) are described by tensor (or tensor-spinor) fields whose so(4, 1) space-time tensor indices have the structure of the Young tableaux with two rows. Namely, mixed-symmetry bosonic fields are associated with Young tableaux having row lengths h 1 , |h 2 |, while mixed-symmetry fermionic fields are associated with Young tableaux having row lengths h 1 − 1 2 , |h 2 | − 1 2 . 5 Interacting mixed-symmetry fields in flat space were studied by using light-cone gauge approach in Refs. [24]- [26] and by BRST method in Refs. [27,28] (see also Ref. [29]). Taking into account successful applications of these two approaches in string theory (see, e.g., Ref. [30]), the light-cone gauge and BRST approaches seem to be promising for developing Lagrangian formulation of interacting mixed-symmetry AdS fields.
2 Mixed-symmetry massive fields in AdS 5 Field content. To develop a light-cone gauge formulation of bosonic mixed-symmetry massive fields in AdS 5 we introduce the following set of complex-valued fields In order to obtain the light-cone gauge description in an easy-to-use form, we introduce oscillators u τ , v τ , τ = 1, 2 and collect fields (2.1) into ket-vector |φ defined by the relation Commutation relations for the oscillators, hermitian conjugation rules, and the vacuum |0 are defined by the relations where δ 11 = δ 22 = 1, δ 12 = δ 21 = 0. Light-cone gauge action. To discuss light-cone gauge action we use Poincaré parametrization of AdS 5 space given by (details of our notation may be found in Appendix A) General representation for light-cone gauge action and Lagrangian found in Ref. [5] takes the form where, in light-cone frame, the D'Alembertian operator ✷ in R 3,1 is given by Bra-vector φ| in (2.7) is defined as φ| ≡ (|φ ) † . The operator A appearing in (2.7) is independent of space-time coordinates and their derivatives. This operator depends only on the oscillators. We refer to the operator A as AdS mass operator. From (2.7), we see that all that is required to fix the Lagrangian is to find the operator A. Solution for the operator A we found is given by 10) In terms of fields (2.1), Lagrangian (2.7) takes the form From (2.13), we see that our Lagrangian leads to decoupled equations of motion for the fields φ m 1 ,m 2 , 14) The following remarks are in order. i) Using field transformation rules given below (see formulas (2.55), (2.56)), we verify that if a set of fields (2.1) transform in the representation of the so(4, 2) algebra labeled by κ, j 1 , j 2 , then a set of the hermitian conjugated fields φ † −m 2 ,−m 1 with m 1 and m 2 as in (2.1) transform in the representation of the so(4, 2) algebra labeled by κ, j 2 , j 1 . From (2.13), we see that our Lagrangian involves the fields φ m 1 ,m 2 and their hermitian conjugated φ † m 1 ,m 2 . This implies that our Lagrangian (2.13) describes dynamics of fields associated with a direct sum of two representations of the so(4, 2) algebra which are labeled by κ, j 1 , j 2 and κ, j 2 , j 1 . ii) Lagrangian (2.13) gives light-cone gauge description of the massive mixed-symmetry field. This Lagrangian can also be used for the description of the totally symmetric massive field. To get Lagrangian for the totally symmetric fields we set h 2 = 0 in (2.13), .i.e., j 1 = j 2 . We can impose then the reality condition on the fields in (2.1), φ † −m 2 ,−m 1 = φ m 1 ,m 2 . This leads to the standard totally symmetric massive field having number of real-valued D.o.F equal to (2j 1 + 1) 2 . iii) Our study demonstrates that AdS space admits decoupled equations of motion for light-cone gauge fields. In this respect it will be interesting to find other gravitational backgrounds which admit decoupled equations of motion for massive fields. Discussion of massive fields in gravitational backgrounds may be found, e.g., in Refs. [31,32]. Mixed-symmetry massive fermionic fields. Our results for bosonic fields are easily generalized to the case of fermionic fields. To this end, we introduce the following set of anticommuting complex-valued fields ψ m 1 ,m 2 , m 1 = −j 1 , −j 1 + 1, . . . , j 1 , m 2 = −j 2 , −j 2 + 1, . . . , j 2 .
(2.15) By analogy with (2.2), fermionic fields in (2.15) can be collected into a ket-vector defined by Lagrangian for fermionic field is given by where AdS mass operator A entering Lagrangian (2.17) takes the same form as the one for bosonic field in (2.9)-(2.12).

Light-cone gauge realization of relativistic symmetries
Relativistic symmetries of fields in AdS 5 are described by the so(4, 2) algebra. This algebra contains the Lorentz sublagebra so (3,1). The choice of the light-cone gauge spoils the manifest so(3, 1) Lorentz symmetries. This is to say that, in the framework of the light-cone approach, complete description of field dynamics in AdS 5 implies that we have to work out realization of the so(3, 1) algebra symmetries and the remaining relativistic symmetries as well. We now turn to the discussion of the so(4, 2) algebra symmetries of the light-cone gauge action (2.6).
In light-cone approach, the so(4, 2) algebra generators can be separated into two groups: where vector indies of the so(2) algebra take values i, j = 1, 2 (for more details, see Appendix A). Field theoretical representation of so(4, 2) algebra generators G field in (2.18), (2.19) takes the form where G diff stands for a realization of the generators in terms of differential operators acting on the bosonic ket-vector |φ (2.2). We now present the explicit expressions for the differential operators G diff . Kinematical generators,

27)
Dynamical generators, Also, we note the following interesting commutator Operator A is presented in (2.9). All that remains to complete a description of generators in (2.27)-(2.30) is to find the operators M zi , M ⊕i , B. In order to present these operators explicitly, we introduce a frame of complex coordinates x R , x L defined by In such frame, a vector X i is decomposed as X i = X R , X L , while a scalar product of two vectors This is to say that, in a frame of the complex coordinates, the spin operator M zi is decomposed as M zi = M zR , M zL , while the operator M ⊕i is decomposed as M ⊕i = M ⊕R , M ⊕L . Note also that, in a frame of the complex coordinates, the so(2) algebra generator M ij = −M ji is represented as M RL . In a frame of the complex coordinates, we find the following representation for the operators entering generators of the so(4, 2) algebra in (2.27)-(2.30): where we use the notation The following remarks are in order. i) Spin operators S R,L τ defined in (2.45),(2.46) and the ones in (2.11) satisfy commutation relations of two su(2) algebras (2.50) In this paper, we use ket-vector (2.2) which depends on the oscillators u τ , v τ . Realization of the operators S τ , S R,L τ on space of such ket-vector is given in (2.11),(2.45), (2.46). We note however that the above-given representation for the operators A, B, M zi , M ij in terms of the operators S τ , S R,L τ does not depend on the particular form for the operators S τ , S R,L τ given in (2.11),(2.45), (2.46). What is important for our practical computations is that the operators S τ , S R,L τ satisfy the relations given in (2.50).
ii) The light-cone gauge action (2.6) is invariant under the following so(4, 2) algebra transformations: where a dependence of the ket-vectors on the coordinates of AdS 5 space is shown explicitly and the notation | | stands for the corresponding unit operator on space of ket-vector (2.2). iv) Using (2.52), we verify that equal time commutator of the ket-vector |φ with the field theoretical generator (2.20) takes the form as it should be. v) Defining an action of the operators M zi on the component fields by the relation we get the following transformation rules of the component fields φ m 1 ,m 2 in (2.1), vi) Expressions for operators A, B, M zi above given are obtained by solving the defining equations (A1)-(A4) given in Appendix A in Ref. [6]. Relativistic symmetries of fermionic fields. Lagrangian for fermionic field (2.17) is invariant under transformations of the so(4, 2) algebra given by where differential operators of fermionic fields G ferm diff can be obtained from those of bosonic fields (2.21)-(2.30) by making there the following substitution where the notation | | stands for the corresponding unit operator on space of ket-vector (2.16).
For the fermionic field, field theoretical representation of the so(4, 2) algebra generators G field in (2.18), (2.19) takes the form

Massless mixed-symmetry fields in AdS 5
In Sec.2, we found the light-cone gauge action for the mixed-symmetry massive fields. For the massive field, the lowest eigenvalue of the energy operator should satisfy the restriction E 0 > h 1 +2 (see (1.4)). To realize the limit of the mixed-symmetry massless field, we take In terms of labels κ, j 1 , j 2 , (1.7), limit in (3.1) can equivalently be represented as Below, we will demonstrate, that limit (3.1) leads to appearance of invariant subspace in |φ (2.2). Namely we state that, in the limit (3.1), the following set of fields are invariant under action of so(4, 2) algebra generators given in (2.21)-(2.30). In other words, in limit (3.1), fields (3.3) transform into themselves under the action of the generators given in (2.21)-(2.30). Before proving this statement let us present our result for Lagrangian of mixed-symmetry massless field. Lagrangian for mixed-symmetry massless field. Equating to zero all component fields φ m 1 ,m 2 in (2.2) with the exception of ones appearing in (3.3), we note that ket-vector which describes massless mixed-symmetry field is given by Plugging (3.4) into Lagrangian (2.7) and using κ given in (3.2), we find the following Lagrangian: where S 1 , S 2 are given in (2.11). In terms of component fields (3.3), expressions for L 0 , L 1 , L 2 (3.7) can be represented as Transformations of mixed-symmetry massless fields. Now we demonstrate that, in limit (3. From (3.9), (3.11), we see that, for arbitrary κ, fields in (3.3) are not invariant under the action of the operators M zR , M zL . However, we note that, in the massless limit, relations (3.9)-(3.12) take the form From (3.14)-(3.17), we see that, in massless limit, fields in (3.3) are indeed invariant under the action of the operators M zR , M zL . Lagrangian for totally symmetric massless fields. For totally symmetric bosonic fields, we have the condition j 1 = j 2 . Using the notation j ≡ j 1 , we can impose the reality condition Introducing then the real-valued field φ 0 , φ † 0 = φ 0 , and a set of the complex-valued fields φ m by the relations we see that Lagrangian (3.6),(3.8) takes the form Fermionic mixed-symmetry and totally symmetric massless fields. To get Lagrangian for mixed-symmetry massless fermionic field we use a ket-vector |ψ which is obtained from (3.4) by substituting anticommuting fields ψ −j 1 ,j 2 , ψ m,j 2 , ψ −j 1 ,m in the respective ket-vectors in (3.5).

Self-dual massive fields in AdS 5
The case of self-dual massive fields (3.2) is realized by considering Using labels κ, j 1 , j 2 (1.7) and considering h 2 > 1/2, we represent (4.1) as Using the relation j 2 = 0 in (2.2), we see that ket-vector of the bosonic field (2.2) simplifies as Ket-vector (4.3) describes the massive self-dual field. Plugging this ket-vector into (2.7), we obtain Lagrangian of the massive self-dual field with the operator A given by where S 1 is given in (2.11). In terms of the component fields φ m,0 , Lagrangian of the massive self-dual field takes the form Note that the relation j 2 = 0 implies the relations S 2 |φ = 0, S R,L 2 |φ = 0. Using these relations in (2.39)-(2.44), we get operators entering generators of the so(4, 2) algebra (2.21)-(2.30) corresponding to the self-dual massive field, Fermionic self-dual massive fields. Our result for bosonic fields above presented can be extended to the case of fermionic fields is a rather straightforward way. For this case, we use the fermionic ket-vector given by while the operator A takes the form given in (4.4). Plugging (4.9) and A (4.4) into (2.17), we get Lagrangian of fermionic self-dual massive field Operators for bosonic fields in (4.6)-(4.8) take the same form for fermionic self-dual massive fields.

Mixed-symmetry anomalous currents and shadows in R 3,1
Let us recall some basic notions of CFT. Fields of CFT can be separated into two groups: currents and shadows. Currents and shadows in R 3,1 transform in representations of conformal algebra so (4,2). Currents transform in representations of so(4, 2) labelled by ∆ cur , h 1 , h 2 , where ∆ cur is conformal dimension, while h 1 and h 2 are highest weights for representations of the so(4) algebra.
In this paper, currents having conformal dimensions ∆ cur = E 0 with E 0 as in (1.2), (1.3), and (1.4) are referred to as self-dual current, mixed-symmetry canonical current, and mixed-symmetry anomalous current respectively. Shadows transform in representations of so(4, 2) labelled by ∆ sh , h 1 , h 2 , where ∆ sh is conformal dimension, while h 1 and h 2 are highst weights for representations of the so(4) algebra. In this paper, shadows having conformal dimensions ∆ sh = 4 − E 0 with E 0 as in (1.2), (1.3), and (1.4) are referred to as self-dual shadow, mixed-symmetry canonical shadow, and mixed-symmetry anomalous shadow respectively. Often, in place of labels E 0 , h 1 , h 2 , we will use the labels κ, j 1 , j 2 defined in (1.7). We start our light-cone gauge formulation of currents and shadows with the description of field contents. 6 Field content of mixed-symmetry anomalous current and anomalous shadow. To discuss light-cone gauge formulation of bosonic mixed-symmetry anomalous current and anomalous shadow we use the following complex-valued fields in R 3,1 : (5.1) In order to streamline the presentation of light-cone gauge description, we use oscillators u τ , v τ (2.3),(2.4) and collect fields (5.1) into ket-vector |φ cur , |φ sh defined by the relation Conformal dimensions of the fields in (5.1) are given by

Light-cone gauge 2-point vertices of mixed-symmetry anomalous currents and shadows.
For currents and shadows, one can construct two 2-point vertices. The first 2-point vertex, which we denote by Γ cur−sh , is a local functional of current and shadow. Using notation |φ cur and |φ sh for the respective ket-vectors of currents and shadows given in (5.2), we note the following expression for the local vertex: The second 2-point vertex, which we denote by Γ sh−sh , is a non-local functional of shadows. Using notation |φ sh for the ket-vector of shadow (5.2), we note the following expression for the non-local vertex: In terms of component fields (5.1), light-cone gauge 2-point vertex (5.5) can be represented as Light-cone gauge 2-point vertex for mixed-symmetry canonical shadow. All that is required to get 2-point vertex L sh−sh for canonical shadow is to set κ = j 1 + j 2 and plug ket-vector corresponding to canonical shadow into (5.6). For canonical shadow, an expansion of ket-vector into component fields takes the same form as for massless AdS field in (3.4). This is to say that ketvector of the mixed-symmetry canonical shadow can be presented in terms of the component fields as |φ sh = |φ sh, 0 + |φ sh, 1 + |φ sh, 2 , (5.11) f m ≡ 4 m Γ(m + 2)Γ(m + 1) 4 j 1 +j 2 Γ(j 1 + j 2 + 2)Γ(j 1 + j 2 + 1) .
Expression for L sh−sh in (5.13) provides us light-cone gauge 2-point vertex for the canonical mixed-symmetry shadow in R 3,1 with arbitrary labels j 1 , j 2 . Light-cone gauge 2-point vertex of self-dual shadow. In order to get 2-point vertex L sh−sh for the self-dual shadow, we set j 2 = 0 and plug ket-vector corresponding to the self-dual shadow into (5.6). For the self-dual shadow, expansion of ket-vector into component fields takes the same form as for the massive self-dual AdS field in (4.3). This is to say that ket-vector of the self-dual shadow can be presented in terms of the component fields as Plugging (5.16) into (5.6) and using the self-duality restrictions j 2 = 0, S 2 = 0, we find the following 2-point vertex: Expression for L sh−sh in (5.17) provides us light-cone gauge 2-point vertex of the self-dual shadow in R 3,1 with arbitrary labels κ and j 1 .

Light-cone gauge 2-point vertex for totally symmetric canonical shadow.
For totally symmetric bosonic shadow, we have the condition j 1 = j 2 . Using the notation j ≡ j 1 , we can impose the reality condition on the shadows as in (3.18), (3.19). We introduce then real-valued field φ sh, 0 , φ † sh, 0 = φ sh, 0 , and a set of complex-valued fields φ sh, m by the relations Using (5.19) in (5.13) and setting j 1 = j, j 2 = j, we see that vertex (5.13) takes the form We note that light-cone gauge vertices for totally symmetric arbitrary spin canonical and anomalous shadows in R d−1,1 , with arbitrary d were obtained in Ref. [36] and Ref. [37] respectively. In Refs. [36,37], gauge invariant and manifestly Lorentz invariant representation of vertices for totally symmetric arbitrary spin canonical and anomalous shadows were also obtained.

Light-cone gauge symmetries of anomalous currents and shadows.
To complete the lightcone gauge description of conformal currents and shadows we have to work out realization of the so(4, 2) algebra symmetries on space of the ket-vectors |φ cur , |φ sh . Let us present transformation rules of currents and shadows under action of the so(4, 2) algebra transformations in the following way: δ|φ cur = G cur |φ cur , δ|φ sh = G sh |φ sh , where G cur and G sh stand for representation of generators of the so(4, 2) algebra in terms of differential operators acting on the respective light-cone gauge ket-vectors of currents and shadows. Expressions for G cur and G sh we found can be presented on an equal footing as where expressions for g 1 , g 2 are given in (2.47), (2.48), while the operators S τ and S R,L τ , τ = 1, 2 are defined in (2.11) and (2.45), (2.46). We note the following interesting commutator (5.48)

Light-cone gauge symmetries of canonical mixed-symmetry (and self-dual) currents and shadows.
Relations in (5.23)-(5.47) provide light-cone gauge realization of the so(4, 2) algebra symmetries on space of mixed-symmetry anomalous currents and shadows. From these expressions, we can easily obtain realization of the so(4, 2) algebra symmetries on space of mixedsymmetry canonical currents and shadows and self-dual currents and shadow. To get realization on space of the mixed-symmetry canonical currents and shadows, we use ket-vectors as in (5.11) and set κ = j 1 + j 2 . To get realization on space of the self-dual currents and shadows we use ket-vectors as in (5.16) and set j 2 = 0, S 2 = 0, S R,L 2 = 0. Note that for the self-dual fields the quantity g 1 (2.47) is simplified as g 1 = 1.

AdS/CFT correspondence
We now study the AdS/CFT correspondence for mixed-symmetry AdS fields and boundary mixedsymmetry conformal currents and shadows. To study the AdS/CFT correspondence we use the light-cone gauge formulation for mixed-symmetry AdS fields, mixed-symmetry anomalous currents, and shadows we developed in the preceding sections in this paper. We recall that our lightcone approach to mixed symmetry AdS fields leads to the decoupled equations of motion and surprisingly simple light-cone gauge Lagrangian. Owing this property of our approach a study of AdS/CFT correspondence for arbitrary spin mixed-symmetry AdS field becomes similar to the one for scalar AdS field. We note also that the computation of action of mixed-symmetry AdS fields on solution of the Dirichlet problem is considerably simplified in our approach. Perhaps, this is the main advantage of our light-cone gauge approach.
In our study of AdS/CFT correspondence, we deal with light-cone gauge formulation not only at AdS side but also at the boundary CFT. This is to say that, in the framework of our light-cone gauge formulation, the study of AdS/CFT correspondence implies the matching of: i) on-shell so(4, 2) light-cone gauge symmetries of bulk mixed-symmetry field and the corresponding so(4, 2) conformal light-cone gauge symmetries of mixed-symmetry boundary conformal current and shadow; ii) Action of mixed-symmetry AdS field evaluated on the solution of equations of motion with the Dirichlet problem corresponding to the boundary mixed-symmetry shadow and the boundary two-point vertex for the shadow.

AdS/CFT for normalizable modes of massive AdS field and anomalous conformal current
We now consider the AdS/CFT correspondence for mixed-symmetry massive AdS field and mixedsymmetry anomalous conformal current. We begin with the discussion of the normalizable solution of equations of motion for massive AdS field. To this end we note that equations of motion obtained from action in (2.6)-(2.9) takes the form The normalizable solution of Eq.(6.1) is given by where J ν in (6.3) is the Bessel function. From (6.2), we see that solution for |φ is governed by the operator U ν . We note the following relations which are helpful for the study of AdS/CFT correspondence: where T ν is defined in (2.49). Relations (6.5)-(6.10) are valid on space of ket-vectors which depend on boundary coordinate x a and do not depend on the radial coordinate z. We note also that relations (6.5)-(6.10) can easily be proved by using the following well known formulas for the Bessel functions The asymptotic behavior of solution (6.2) takes the form From (6.12), we see that, up to overall normalization factor, the ket-vector |φ cur is indeed boundary value of the normalizable solution.
Matching of bulk and boundary so(4, 2) symmetries. In the framework of AdS/CFT correspondence, we expect that the boundary value |φ cur is realized as mixed-symmetry anomalous conformal current. To make sure that |φ cur is indeed realized as mixed-symmetry anomalous conformal current we prove the following statement: Bulk light-cone gauge so(4, 2) symmetries of the normalizable solution (6.2) amount to boundary light-cone gauge so(4, 2) conformal symmetries of the anomalous conformal current.
Thus our purpose is to demonstrate the matching of the so(4, 2) algebra generators for bulk mixed-symmetry massive AdS field given in (2.21)-(2.30) and the ones for the boundary mixedsymmetry anomalous conformal current given in (5.23)-(5.31) and (5.34)- (5.40). To this end we note that all that is required is to match the following generators of the so(4, 2) algebra Using notation G AdS and G CF T for the respective bulk and boundary generators given in (2.21)-(2.30) and (5.23)-(5.31), we note that the matching implies that we should verify the following relation for the solution in (6.2) and for the bulk and boundary generators G AdS |φ = U ν G CF T |φ cur . (6.14) We now verify relation (6.14) for generators of the so(4, 2) algebra given in (6.13) in turn. i) We start with matching the following generators: From (2.21)-(2.24) and (5.23)-(5.26), we see that bulk and boundary generators given in (6.15) coincide. As generators (6.15) commute with the operator U ν (6.3), we conclude that generators (6.15) satisfy relation (6.14). ii) We now match generators P − AdS and P − CF T given in (2.28) and (5.23) respectively. To this end we note that operator M 2 (2.31) entering P − AdS (2.28) can be represented in terms of operator T ν (2.49) as Using (6.5),(6.6) and (6.16), we find the following remarkable relation Relation (6.17) is valid on space of ket-vectors which are independent of the radial coordinate z. Taking into account (2.8), we note the simple relation From (6.17),(6.18), we see that P − AdS and P − CF T satisfy the relation (6.14). Note that, because operator ∂ − commutes with U ν , relation (6.14) implies that on the space of solution (6.2) one has the relation In (6.19) and below, the notation ≈ is used to indicate the fact that relation (6.19) is valid on the space of solution |φ in (6.2). iii) Next we match the generators J +− AdS , D AdS (2.22) (2.25) and J +− CF T , D CF T (5.24), (5.28). To math these generators we use relation (6.19) which implies that on space of solution (6.2) expressions for J +− AdS , D AdS can be represented as 20) From (6.20) and (5.24), we see that generators J +− AdS and J +− CF T coincide. As these generators commute with operator U ν (6.3), we conclude that generators J +− AdS , J +− CF T satisfy relation (6.14). Using (6.21) and (5.28),(5.34), we verify that D AdS and D CF T also satisfy relation (6.14). iv) We proceed with matching of generators J −i AdS (2.29), J −i CF T (5.27). To match these generators we use relation (6.19) which implies that on space of solution (6.2) expressions for J −i AdS can be represented as where M −i AdS is given in (2.32),(2.42), (2.43). Comparing (6.22) with (5.27),(5.33), we see that all that remains is to match M ⊕i AdS in (2.42), (2.43) and M ⊕i CF T in (5.36), (5.37). In other words, we should prove the relation M ⊕i AdS |φ = U ν M ⊕i CF T |φ cur . (6.23) Using (6.5)-(6.8), we verify that relation (6.23) holds true. v) Finally, we match generators K + AdS (2.26), K + CF T (5.29). To this end we note the following useful relation where x 2 ≡ x a x a , x∂ ≡ x a ∂ a . Relation (6.24) can straightforwardly be proved by using (6.9), (6.10). Using (6.24) for a = +, we see that K + AdS (2.26) and K + CF T (5.29) satisfy relation (6.14). Thus the generators K + AdS and K + CF T match. As a side remark we note that, by using (6.24), we checked explicitly that the remaining generators K i , K − also satisfy relation (6.14).

AdS/CFT for non-normalizable modes of massive AdS field and anomalous shadow
In this Section, we consider the AdS/CFT correspondence for mixed-symmetry massive AdS field and mixed-symmetry anomalous shadow. We begin with the discussion of the non-normalizable solution of equations of motion for massive AdS field. To this end we note that non-normalizable solution of equations of motion (6.1) with the Dirichlet problem corresponding to anomalous shadow takes the form |φ(x, z) = σ ν d 4 y G ν (x − y, z)|φ sh (y) , (6.26) where G ν in (6.27) is the Green function. From (6.26), we see that solution for |φ is governed by the σ ν G ν . The asymptotic behavior of solution (6.26) takes the form From (6.30), we see that, up to overall normalization factor, the ket-vector |φ sh is indeed boundary value of the non-normalizable solution.
In the framework of AdS/CFT correspondence, we expect that the boundary value |φ sh is realized as mixed-symmetry anomalous shadow. To make sure that |φ sh is indeed realized as mixed-symmetry anomalous shadow we prove the following two statements: i) Bulk light-cone gauge so(4, 2) symmetries of the non-normalizable solution (6.26) amount to boundary light-cone gauge so(4, 2) conformal symmetries of the mixed-symmetry anomalous shadow. ii) action of mixed-symmetry AdS field evaluated on solution (6.26)

coincides, up to normalization factor, with boundary 2-point light-cone gauge vertex for the mixed-symmetry anomalous shadow.
We now prove these statements in turn. Matching of bulk and boundary so(4, 2) symmetries. Our purpose is to demonstrate the matching of the so(4, 2) algebra generators for bulk mixed-symmetry massive AdS field given in (2.21)-(2.30) and the ones for the boundary mixed-symmetry anomalous shadow given in (5.23)-(5.31) and (5.41)-(5.47). Let us use the notation G AdS and G CF T for the respective bulk and boundary generators given in (2.21)-(2.30) and (5.23)-(5.31). All that is required then is to verify that these generators satisfy the following relation Procedure for checking relation (6.31) is the same as the one we used for the case of conformal current in Sec.6.1. Therefore to avoid the repetition we just present the following relations which are helpful in the studying of AdS/CFT correspondence for the case of shadow: where the operator T ν is defined in (2.49). Effective action for mixed-symmetry anomalous shadow. In this paper, action of AdS field evaluated on non-normalizable solution in (6.26) is referred to as effective action. In order to find the effective action, we should plug the solution given in (6.26) into light-cone gauge action (2.6) with Lagrangian given in (2.7). Note however that we should add to the light-cone gauge action an appropriate boundary term. Using the method for finding boundary term developed in Ref. [38], we verify that a light-cone gauge Lagrangian which involves an appropriate boundary term can be presented in the following way Also note that, to adapt our expressions to commonly used Euclidean signature, we have changed sign of Lagrangian, L → −L, when passing from (2.7) to (6.38). It is easy to see that light-cone gauge action (2.6), (6.38) considered on solution of equations of motion can be represented in the following way Plugging solution (6.26) into (6.39), we get the following effective action for mixed-symmetry anomalous shadow − S eff = 2κc κ Γ sh−sh , (6.40) where Γ sh−sh takes the same form as in (5.5), while c κ is defined in (6.28). Thus we see that, up to overall normalization factor given by 2κc κ , the action of the mixed-symmetry massive AdS field evaluated on the non-normalizable solution amounts to the 2-point vertex of mixed-symmetry anomalous shadow. Our analysis allows us to find the normalization factor which might be important for the systematical study of the AdS/CFT correspondence. Effective actions for canonical mixed-symmetry shadow, self-dual shadow, and canonical totally symmetric shadow. Expression for S eff in (6.40) gives effective action for mixed-symmetry anomalous shadow. From this expression, we can easily obtain effective action for mixed-symmetry canonical shadow, self-dual shadow, and canonical totally symmetric shadow. To get S eff for canonical mixed-symmetry shadow we use Γ sh−sh given in (5.5), (5.13) and set κ = j 1 + j 2 in (6.40). S eff for canonical totally symmetric shadow is obtained by using Γ sh−sh given in (5.5), (5.20) and setting κ = 2j 1 , j 1 = j, j 2 = j in (6.40). To get S eff for the self-dual shadow we use Γ sh−sh given in (5.5), (5.17).
The following remarks are in order. i) In the Lorentz covariant framework, effective action for mixed-symmetry canonical shadow in R d−1,1 with arbitrary d and particular values j 1 = 3/2, j 2 = 1/2 was found in Ref. [39]. ii) In Lorentz covariant framework, effective action for self-dual shadow in R 3,1 with arbitrary κ and the particular value of the label j 1 = 1 was discussed in Ref. [40].
iii) The intertwining operator realization of AdS/CFT correspondence was studied in Refs. [41].
In conclusion of this section, we note that we applied our light-cone gauge formulation for the study of the AdS/CFT correspondence for mixed-symmetry AdS fields and boundary mixedsymmetry currents and shadows. Beyond this, we expect that our light-cone gauge approach might have other interesting applications to studying AdS/CFT correspondence along the lines in Refs. [42]- [45].

Mixed-symmetry conformal fields in R 3,1
For shadows in R 3,1 , a kernel of the effective action in (5.6) is not well defined when κ (1.7) takes integer values (see, e.g., Ref. [46]). We note however that the kernel becomes well defined for shadows having non-integer κ. Taking this into account, we use non-integer κ to regularize the kernel of effective action. When removing the regularization, we are left with a logarithmic divergence of the effective action. This logarithmic divergence of the effective action for shadow turns out to be light-cone gauge action of mixed-symmetry conformal field.
For the discussion of short totally symmetric conformal fields, j 1 = j 2 , see below.
Self-dual conformal field. For the case of self-dual conformal fields (7.5), one of the labels j 1 , j 2 should be equal to zero. Let us set j 2 = 0. Then, in terms of component fields, Lagrangian for self-dual conformal field is obtained by using relations (7.3), (7.5) and plugging a ket-vector |φ , which takes the same form as in (4.3), into (7.4). Doing so, we get Lagrangian (7.17) describes conformal fields associated with direct sum of two non-unitary representation of the so(4, 2) algebra labelled by ∆, j 1 , 0 and ∆, 0, j 1 , where ∆ is given in (7.8),(7.5).
Simplest self-dual conformal field described by Lagrangian (7.17) corresponds to the case j 1 = 1, N = 0. Such self-dual conformal field appears in the field content of N = 4, 4d conformal supergravity (for review, see Ref. [47]). Also we note that, for the self-dual conformal field with j 1 = 1, N = 0, ordinary-derivative light-cone gauge Lagrangian was obtained in Ref. [48]. Short totally symmetric conformal field. For totally symmetric bosonic fields, we have the condition j 1 = j 2 . Using the notation j ≡ j 1 , we can impose the reality condition φ † m,j = φ −j,−m , m = −j + 1, −j + 2, . . . , j . Introducing then a set of complex-valued fields φ m by the relations φ j+m ≡ √ 2φ m,j , m = −j + 1, −j + 2, . . . j , (7.20) we see that Lagrangian (7.15),(7.13),(7.14) takes the form L = 2j m=1 φ † m ✷ m φ m . (7.21) Using the notation s ≡ 2j, we note that Lagrangian (7.21) gives light-cone gauge description of totally symmetric spin-s conformal field which, in the framework of Lorentz covariant approach, was discussed for first in Ref. [47]. From (7.21), it is easy to see that number of real-valued on-shell D.o.F. described by Lagrangian (3.21) is equal to s(s + 1). Thus we see that our light-cone gauge Lagrangian leads to the number of D.o.F found in Ref. [47]. Let us conclude with following remarks. i) All conformal unitary representations of the so(d + 1, 2) algebra which are associated with fields in R d, 1 and AdS d+1 were classified in Refs. [49,50]. Our Lagrangian of conformal fields involves higher-derivatives. From the expressions for Lagrangian given in (7.9), (7.11), (7.17) is is clear that our conformal fields are related to non-unitary representation of the so(4, 2) algebra. This is to say that conformal fields considered in this paper should not be confused with the ones studied in Refs. [49,50]. ii) In the framework of Lorentz covariant formulation, a wide class of mixed-symmetry conformal fields was studied in Refs. [51,52]. We expect that our short mixed-symmetry conformal fields might be related to the ones studied in Refs. [51,52]. We note however that precise relation of our mixed-symmetry conformal fields to the ones in Refs. [51,52] is still to be understood. iii) Our study leads to simple expressions for the light-cone gauge Lagrangian of conformal fields. Therefore we believe that our light-cone gauge formulation of conformal fields might provide us new interesting possibilities for the studying partition functions of conformal fields along the lines in Refs. [53]. In the framework of BRST approach, discussion of partition functions of conformal fields may be found in Refs. [54,55]. iv) In the context of AdS/CFT correspondence, scalar long conformal fields were discussed in Ref. [56]. v) In the framework AdS/CFT correspondence, short conformal fields are dual to massless AdS fields, while long conformal fields are dual to massive AdS fields having integer values of lowest eigenvalue of the energy operator E 0 . Taking this into account, it is difficult to overcome a desire to speculate on some special regime in the type IIB superstring theory in AdS 5 × S 5 background when E 0 for all massive higher-spin fields take integer values. Such conjectured regime in the superstring theory should lead via AdS/CFT to stringy theory of conformal fields which involves low-spin short conformal fields and higher-spin long conformal fields.
The coordinate x + is considered as the light-cone evolution parameter. For derivatives with respect to x i , i = 1, 2, we use the notation ∂ i = ∂ i ≡ ∂/∂x i . Also, for derivatives, we adopt the following conventions: ∂ ± = ∂ ∓ ≡ ∂/∂x ∓ , ∂ R = ∂ L ≡ ∂/∂x L , ∂ L = ∂ R ≡ ∂/∂x R , ∂ z = ∂/∂z , (A. 4) and the D'Alembertian operator ✷ in R 3,1 takes the form ✷ = 2∂ + ∂ − + 2∂ R ∂ L . In frame of the coordinates x ± , x R , x L , a product of two so(3, 1) algebra vectors X a , Y a is decomposed as where X a Y a ≡ η ab X a Y b and η ab stands for the flat metric η ab = (−, + + +). so(4, 2) algebra consists of translation generators P a , conformal boost generators K a , dilatation generator D, and generators of the Lorentz so(3, 1) algebra J ab . We use the following commutations relations of the so(4, 2) algebra [P a , J bc ] = η ab P c − η ac P b , [P a , K b ] = η ab D − J ab , [J ab , J ce ] = η bc J ae + 3 terms , In frame of the coordinates x ± , x R , x L , the generators are decomposed as P a = P ± , P R , P L , K a = K ± , K R , K L ,