Light-cone gauge cubic interaction vertices for massless fields in AdS(4)

In the framework of light-cone formulation of relativistic dynamics, arbitrary spin massless fields propagating in the four-dimensional AdS space are studied. For such fields, the complete list of light-cone gauge cubic interaction vertices is obtained. Realization of relativistic symmetries on space of light-cone gauge massless AdS fields is also obtained. The light-cone gauge vertices for massless AdS fields take simple form similar to the one for massless fields in the flat space.


Introduction
Light-cone gauge formulation of field dynamics in the flat space developed in Ref. [1] has turned out to be successful for studying many important problems of field/string theory. Perhaps one of the attractive applications of the light-cone formalism is the construction of the light-cone gauge (super)string field theory in Refs. [2,3]. The light-cone gauge superfield formulation of the supersymmetric N = 4 Yang-Mills theory was built in Refs. [4], while the light-cone gauge superfield formulation of supergravity theories in ten and eleven dimensions was studied in Refs. [5,6,7]. Another attractive application of the light-cone formalism is the construction of interaction vertices in the theory of higher spin fields propagating in flat space [8]- [12]. Recent interesting applications of light-cone gauge formalism for studying various dynamical system in flat space may be found in Refs. [13,14].
Light-cone gauge formulation of field dynamics in AdS space was developed in Refs. [15,16]. In Refs. [15,16], we studied free light-cone gauge fields propagating in AdS space. Our aim in this paper is to develop light-cone gauge formulation for interacting massless fields propagating in four dimensional AdS space. We develop the systematic method for building interaction vertices for arbitrary spin massless AdS fields and use this method to find explicit expressions for cubic interaction vertices. We find the complete list of cubic interaction vertices for massless AdS fields. 1 This paper is organized as follows.
In Sec.2, we introduce our notation and describe the light-cone gauge formulation of free arbitrary spin massless fields propagating in AdS 4 space.
In Sec.3, we start with the discussion of n-point interaction vertices for massless AdS fields. We find restrictions imposed on the n-point interaction vertices by the kinematical symmetries of the so(3, 2) algebra. After that we restrict our attention to cubic vertices. Using a particular choice of momentum variables which suits the cubic vertices, we provide the convenient form of restrictions imposed on the cubic vertices by the kinematical symmetries of the so(3, 2) algebra.
In Sec.4, we discuss restrictions imposed on the cubic vertices by dynamical symmetries of the so(3, 2) algebra. After that, requiring the light-cone locality and using field redefinitions, we find the complete list of equations which admit us to determine cubic vertices uniquely.
In Sec.5, we present our method for solving the complete list of equations for the cubic vertices, while in Sec.6 we summarize our final results for the cubic interaction vertices. We discuss various representations for the cubic vertices.
In Sec.7, we summarize our conclusions and suggest directions for future research. Notation, conventions, and various technical details are collected in Appendices. Appendix A is devoted to the basic notation and conventions we use in this paper. In Appendices B and C, we discuss various technical details of our method for solving equations for cubic vertices.
2 Free light-cone gauge arbitrary spin massless fields in AdS 4 space The so (3,2) algebra in light-cone frame. According to the idea in Ref. [1], the problem of finding the new dynamical system amounts to a problem of finding a new (light cone gauge) solution for commutation relations of a basic symmetry algebra. For fields that propagate in the AdS 4 space, basic symmetries are associated with the algebra so (3,2). Light-cone gauge formulation of free fields propagating in AdS space was developed in Ref. [15]. Using the light-cone gauge approach in Ref. [15], we now discuss the light-cone gauge realization of the so(3, 2) algebra symmetries on space of free arbitrary spin massless fields propagating in AdS 4 space.
The so (3,2) algebra is spanned by translation generators P µ , dilatation generator D, conformal boost generators K µ and rotation generators J µν which are generators of the Lorentz algebra so (2,1). The commutation relations of the so(3, 2) algebra take the form [D, P µ ] = −P µ , [P µ , J νρ ] = η µν P ρ − η µρ P ν , where η µν is the mostly positive flat metric tensor. The vector indices of the so(2, 1) Lorentz algebra take values µ, ν, ρ, σ = 0, 1, 2. The generators P µ and K µ are considered to be hermitian while the generators D and J µν are assumed to be anti-hermitian. We use the Poincaré parametrization of AdS 4 space, To discuss the light-cone formulation, we introduce, in place of the Lorentz basis coordinates x 0 , x 1 , x 2 , z, the light-cone basis coordinates x + , x − , x 1 , z, where the coordinates x ± are defined as 3) The coordinate x + is considered as an evolution parameter. Use of the coordinates (2.3) implies that the so(2, 1) Lorentz algebra vector X µ is decomposed as X + , X − , X 1 , while a scalar product of the so(2, 1) Lorentz algebra vectors X µ and Y µ is decomposed as From decomposition (2.4), we conclude, that in light-cone frame, non vanishing elements of the flat metric η µν and its inverse η µν are given by η +− = 1, η −+ = 1, η 11 = 1, η +− = 1, η −+ = 1, η 11 = 1. (2.5) Relations (2.5) imply that the covariant and contravariant components of vectors are related as X + = X − , X − = X + , X 1 = X 1 .
In light-cone approach, generators of the so(3, 2) algebra are separated into the following two groups: For x + = 0, in the field theoretical realization, kinematical generators (2.6) are quadratic in fields 2 , while, dynamical generators (2.7) involve quadratic and higher order terms in fields. In light-cone frame, commutation relations for the generators (2.6),(2.7) are obtained from the ones in (2.1) by using the non vanishing elements of the η µν given in (2.5).
To provide a field theoretical realization for the generators of the so(3, 2) algebra on a space of arbitrary spin massless fields we exploit a light-cone gauge description of the fields. Arbitrary spin-s massless fields. In light-cone gauge, physical degrees of freedom of a massless spin-s field, s > 1, propagating in AdS 4 space are described by two complex-valued fields φ s and φ −s that are hermitian conjugated to each other, while spin-0 field (scalar field) can be described by real-valued field We prefer to deal with fields obtained from the ones in (2.8),(2.9) by using the Fourier transform with respect to the coordinates x − and x 1 , In other words, to discuss spin-s field, s > 0, and spin-0 field we use the following respective fields where, in (2.11),(2.12), the argument p stands for the momenta p 1 , β and the dependence on the evolution parameter x + is implicit. In order to discuss the light-cone gauge formulation of a massless field in an easy-to-use form we introduce the creation operators α R , α L and the respective annihilation operatorsᾱ L ,ᾱ R , Throughout this paper, the creation and annihilation operators will be referred to as oscillators. Sometimes, we prefer to use oscillators with lower case indices defined by the relations (2.14) Using such notation for the oscillators, we introduce the following ket-vectors: and in order to treat arbitrary spin fields on an equal footing we use an infinite chain of massless fields which consists of every spin just once. Such chain of massless fields is described by the ket-vector where the ket-vectors |φ s are defined in (2.15), (2.16). Often, we prefer to use the alternative representation for ket-vector |φ (2.17). Namely, using (2.15), (2.16), it is easy to see that ketvector |φ (2.17) can be represented as where we use a quantity α λ H defined by the relations It is easy to see that ket-vector (2.18) satisfies the algebraic constraint Throughout the paper we use bra-vector φ| defined as φ(p, z, α)| = (|φ(p, z, α) ) † . Using the expansion (2.17), we get where we use a quantityᾱ λ H defined by the relations Field-theoretical realization of the so(3,2) algebra. Now our aim is to provide a field theoretical realization of the so(3, 2) algebra on the space of massless AdS fields. In our approach, massless AdS fields are described by the ket-vector |φ (2.18). A realization of kinematical generators (2.6) and dynamical generators (2.7) in terms of differential operators acting on the ket-vector |φ is given by Kinematical generators :

27)
Dynamical generators : where we use the notation In (2.31) and below, a quantity M z1 stands for a spin operator of the so(2) algebra. On space of ket-vectors (2.17), the operator M z1 is realized as Relations (2.23)-(2.33) provide the realization of the generators of the so(3, 2) algebra in terms of differential operators acting on the ket-vector |φ (2.18). Using these relations we are ready to present a field theoretical realization for the generators of the so(3, 2) algebra in terms of the ket-vectors |φ (2.18). This is to say that, at the quadratic level, a field theoretical realization of the generators given in (2.6),(2.7) takes the form where G [2] stands for the field theoretical generators, while G stands for the differential operators presented in (2.23)-(2.33). By definition, the ket-vector |φ satisfies the Poisson-Dirac commutation relations where Π stands for the projector on space of the ket-vector given in (2.18 [|φ , G [2] ] equal x + = G|φ . (2.36) In terms of the component fields φ λ (p, z), the Poisson-Dirac commutation relations take the form All commutators between fields φ λ (p, z) and their hermitian conjugated φ † λ (p, z) ≡ (φ λ (p, z)) † are obtained from (2.37) by using the hermicity condition φ −λ (−p, z) = φ † λ (p, z) (2.11). For the reader convenience, we note that, in the framework of the Lagrangian approach, the light-cone gauge action takes the form S = dx + dzd 2 p φ(p, z, α)|i β∂ − |φ(p, z, α) + dx + P − , (2.38) where P − is the Hamiltonian. Expressions for the action given in (2.38) is valid both for the free and interacting light-cone gauge fields. In the theory of free light-cone gauge fields, the Hamiltonian P − is obtained by plugging the differential operator P − (2.28) into (2.34).
Incorporation of a internal symmetry for massless AdS fields can be done by analogy with the Chan-Paton method used in string theory [20] and in massless arbitrary spin fields in [21] (see remark at the end of Sec.6 in this paper).
3 Restrictions imposed on n-point interaction vertices by kinematical symmetries of so(3, 2) algebra Our aim in this Section is to discuss restrictions imposed on the dynamical generators (2.7) by the kinematical symmetries (2.6). Namely we are going to find equations obtained from commutators between the kinematical generators given in (2.6) and dynamical generators given in (2.7).
In theories of interacting fields propagating in AdS space, the dynamical generators receive corrections involving higher powers of physical fields. Namely, the dynamical generators (2.7) can be presented in the following way (3.1) is a functional that has n powers of physical fields |φ . Dynamical generators at quadratic approximation are given by expressions (2.28)-(2.30) and (2.34). Now we discuss the general structure of the generators G dyn [n] when n ≥ 3. We discuss restrictions obtained from commutators between the kinematical generators (2.6) and dynamical generators (2.7) in turn. Note that, in what follows, without loss of generality, we study the commutators of the generators of the so(3, 2) algebra for x + = 0. P 1 -and P + symmetries restrictions. Using the commutators of the dynamical generators (2.6) with the kinematical generators P 1 and P + , we find that the dynamical generators G dyn [n] with n ≥ 3 can be presented as where bra and ket-vectors appearing in (3.2)-(3.4) are defined as while the remaining quantities appearing in (3.2)-(3.8) are given by dz a , d 2 p a = dp 1 a dβ a , (3.9) In relations (3.5)-(3.11) and below, we use the indices a, b = 1, . . . , n to label n interacting fields. The Dirac δfunctions appearing in (3.9) imply conservation laws for the momenta p 1 a and β a . Densities p − [n] , j −1 [n] , and k − [n] appearing on r.h.s in relations (3.6)-(3.8) depend on the momenta p 1 a , β a , the radial derivatives ∂ za , the radial coordinate z, and spin variables denoted by α a in this paper. We note that the shortcut α a stands for the oscillators α R a , α L a . Sometimes, the density p − J +1 -symmetry restrictions. Using commutators of the dynamical generators (2.7) with the kinematical generator J +1 , we find the equations K + -symmetry restrictions. Using commutators of the dynamical generators (2.7) with the kinematical generator K + , we get the following equations where for the derivation of equations in (3.21)-(3.23) we use equations (3.18)-(3.20). K 1 -symmetry restrictions. Using commutators of the dynamical generators (2.7) with the kinematical generator K 1 , we get the equations where X 1 is defined in (3.10), while K 1 † is defined as Using relations given in (3.10), (3.27), we find the helpful relation Note that, for the derivation of equations given in (3.24)-(3.26), we use restrictions imposed by D-symmetry (3.15)-(3.17) and restrictions imposed by J +1 -symmetry (3.18)- (3.20). We summarize our consideration in this section by the following two remarks.
i) The use of the commutators of the dynamical generators (2.7) with the kinematical generators P 1 , P + leads to relations (3.2)-(3.4), while the use of the commutators of the dynamical generators (2.7) with the kinematical generators J +− , J +1 , D, K + , K 1 leads to the equations given in (3.12)- (3.26).
, and k − [n] depend on the momenta p 1 a through the new momentum P 1 ab defined by the relation In other words, the densities p − Using the momentum conservation laws we check that not all momenta P 1 ab (3.31) are independent. Namely, it easy to check that, for the n-point vertex, there are n − 2 independent momenta P 1 ab . This implies that, for the case of n = 3, there is only one independent P 1 ab . This considerably simplifies analysis of kinematical symmetry equations for cubic densities p − [3] , j −1 [3] , k − [3] .
4 Restrictions imposed on cubic interaction vertices by dynamical symmetries of so(3, 2) algebra Throughout this paper, restrictions obtained from commutators of the dynamical generators of the so(3, 2) algebra (2.7) are referred to as dynamical symmetry restrictions. The commutators of the dynamical generators of the so(3, 2) algebra (2.7) are given by Therefore our aim in this section is to obtain restrictions on the densities imposed by the commutators (4.1),(4.2). We note then the following important feature of the commutators of the so (3,2) algebra. It turns out that studying the commutators (4.1),(4.2) amounts to studying the commutator (4.1) and the kinematical K 1 symmetry equations (3.50)- (3.52). To see this we note that kinematical K 1 -symmetry equations (3.50)-(3.52) amount to the following commutators: Now by using the Jacoby identities, it is easy to check that, if commutators (4.1) and (4.3) are satisfied, then the commutators (4.2) are satisfied automatically. Thus, if we respect K 1 -symmetry equations (3.50)-(3.52), then we can restrict ourselves to studying commutator in (4.1). We now study the commutator (4.1) in the cubic approximation.
where we use the notation Operators P − and J −1 † (4.6), (4.7) are expressed in terms of the momenta p 1 a , β a . Using definition of P 1 (3.37) and plugging (3.38),(3.39) into (4.5), we can express the operators P − and J −1 † in terms of the momenta P 1 and β a , where N β and β are defined in (3.55).
Plugging |j −1 [3] δ (3.50) into (4.5), we get the equation for the cubic vertex |p − [3] δ , Now we are ready to summarize our study of restrictions imposed on the densities which are obtained from commutators of the so(3, 2) algebra.
In the framework of light-cone approach, the assumption i) is a counterpart of locality condition commonly used in Lorentz covariant approach. We note also that the assumption ii) is related to field redefinitions. Namely, if we ignore requirement (4.18) then we get vertices which can be removed by field redefinitions. As we are interested to deal with the cubic interaction vertices that cannot be removed by field redefinitions, we respect the requirement in (4.18).

Method for solving equations for cubic interaction vertex
Finding solution to the complete system of equations (4.11)-(4.17) turns out to be complicated problem. We note that the most difficult point in analysis of equations (4.11)-(4.17) is related with a proper treatment of the radial derivatives ∂ za . We now describe our procedure for solving equations (4.11)-(4.17). Our procedure is realized in the following eight steps.
Step 1. Our aim at this step is to introduce a convenient basis for the radial derivatives ∂ za . To this end we introduce the following decomposition of the radial derivatives ∂ za , a = 1, 2, 3: whereβ a , β, ∆ β are given in Appendix (A.10)-(A.13). Quantities P z , P z , and P z defined in (5.2)-(5.4) will be referred to as radial momenta. Relations (5.1)-(5.4) describe the one-to-one mapping between the three radial derivatives ∂ za , a = 1, 2, 3, and three radial momenta P z , P z , P z . Comparing (5.3) with (3.37), we see that the radial momentum P z is defined by analogy to the momentum P 1 (3.37). Below we will demonstrate that, in view of various reasons, the radial momenta P z and P z can be eliminated from consideration. First, we consider the radial momentum P z . Using decomposition (5.1) in (3.38), we see that the vertex p − [3] is represented as As the vertex p − [3] is a finite-order polynomial in the radial momentum P z , we can represent the vertex p − [3] as p − [3] = N n=0 P n z V n (P 1 , P z , P z , β a , z, α a ) . Now taking into account definition of the δ z (3.11), we get the relation Using relation (5.7) in (3.6), we see that, up to total derivative, the vertex |p − [3] δ (3.6) having density p − [3] as in (5.6) amounts to the vertex |p − [3] δ having the density given by Relation (5.8) implies that the radial momentum P z (5.1) can be eliminated from our consideration. In other words, without loss of generality, we can restrict our attention to the vertex which does not depend on the P z , p − [3] = p − [3] (P 1 , P z , P z , β a , z, α a ) . (5.9) Using density p [3] (5.9), we now proceed to the next step of our procedure.
Step 2. Our aim at this step is to provide solution to the requirement in (4.18). As in the flat space, this requirement can be solved by using field redefinitions. In the framework of light-cone gauge formulation of field dynamics in flat space, detailed discussion of field redefinitions may be found in Appendix B in Ref. [11]. Analysis of field redefinitions in AdS space follows the pattern of the analysis described in Appendix B in Ref. [11]. Therefore, to avoid repetitions, we briefly describe result of the analysis. Under field redefinitions the vertex p − [3] transforms as p [3] → p − [3] − P − f , (5.10) where vertex f given in (5.11) describes generating function of field redefinitions (see relations B3 and B15 in Appendix B in Ref. [11]). As seen from (5.11), the vertex f depends on the same variables as the cubic interaction vertex p − [3] in (5.8). Operator P − appearing in (5.10) is defined in (4.8). It is easy to check that, on space of the vertex f , operator P − (4.8) is realized as Now we introduce a definition of harmonic vertex. By definition, vertex p − [3] that satisfies the equation is refereed to as harmonic vertex. We recall that, by definition, the vertex p − [3] is a polynomial in the momenta P 1 , P z . As is well known an arbitrary polynomial in two variables P 1 , P z can be made a harmonic polynomial in P 1 , P z by adding a suitable polynomial proportional to P 1 P 1 − P z P z . From (5.10),(5.12), we see that it is the polynomial proportional to P 1 P 1 − P z P z that is generated by field redefinitions. This implies that, by using field redefinitions (5.10), the vertex p − [3] can be made the harmonic vertex (5.13).
Summarizing the two steps above discussed, we note that we are left with vertex (5.8) which satisfies the equation (5.13). Such harmonic vertex obviously satisfies the requirement (4.18). Using (5.9),(5.13) we now proceed to the next step of our procedure.
Step 3. We now study restrictions imposed on the vertex (5.9) by K + symmetry equation (4.13). To this end we note that, for the ket-vector |p − [3] δ (3.6) with p − [3] as in (5.9), the following relation holds true Making use of (5.13) and (5.14), we note that K + -symmetry equation (4.13) amounts to the following equation for p − [3] : For the vertex p − [3] , which by definition is polynomial in the radial momentum P z , equation (5.15) implies that p − [3] is independent of the radial momentum P z , Thus, at this step of our procedure, we obtain vertex (5.16) which is independent of the radial momentum P z . Summarizing the three steps of our procedure above discussed we note that we are left with vertex (5.16) which is harmonic with respect to the momenta P 1 , P z (5.13). Using such vertex we now proceed to the next step of our procedure.
Step 4. Our aim at this step, is to represent J +− -and D-symmetry equations (4.11),(4.12) in terms of the harmonic vertex (5.16). Taking into account the definition of P z (5.3), we verify that in terms of the harmonic vertex p − [3] (5.16), equations (4.11),(4.12) can be represented as Step 5. At this step we consider equation (4.17) for the ket-vector |p − [3] δ which involves deltafunctions δ z (3.6). Our aim is to represent equation (4.17) in terms the ket-vector |p − [3] that does not involve delta functions δ z (3.6), where the vertex p − [3] takes the form as in (5.16) and satisfies harmonic constraint (5.13). To this end we start with the presenting realization of the operators K 1 † (3.53), P − (4.8), J −1 † (4.9) on space of the harmonic ket-vector |p − [3] (3.6), (5.13), (5.16), where β,β, ∆ β are defined in Appendix (A.10)-(A.13) and we use the notation where operators P 1 Th , P Th,z appearing in (5.26) are defined as P Th,z ≡ P z + (P 1 P 1 − P z P z ) 28) while N P 1 , N Pz are defined in (5.19). For harmonic vertex (5.13), we note the following important relations for the operators P 1 Th , P Th,z , Relations (5.29) tell us that the operators P 1 Th , P Th,z respect the harmonic condition (5.13).
Step 7. Our aim at this step is to represent the remaining equations for the harmonic vertex |p − [3] given in (5.17),(5.18) and (5.25), (5.26) in terms of holomorphic momenta and anti-holomorphic momenta. The use of such momenta allows us to introduce holomorphic and anti-holomorphic vertices. It turns out that then the remaining equations (5.17),(5.18) and (5.25),(5.26) become decoupled for holomorphic and anti-holomorphic vertices and this simplifies our procedure for finding the vertices. Holomorphic and anti-holomorphic momenta denoted by P L and P R are defined by the relations In terms of momenta (5.38), equation (5.13) takes the form General solution to equation (5.39) can be presented as where vertices V andV do not involve the respective terms of zero-order in P L and P R . Obviously, for vertices given in (5.41), the J +− -and D-symmetry equations (5.17),(5.18) turn out to be decoupled and take the following form: In Appendix C, we outline the proof of the following two Statements. i) Using equations (5.25),(5.26) and (5.42)-(5.44) we get the following solution for the vertex V 000 : where C 000 is a constant parameter (coupling constant of three scalar fields). ii) Equation (5.25) leads to the following decoupled equations for holomorphic and anti-holomorphic ket-vectors |V = V |0 , |V =V |0 : while the equation (5.26) leads to the following decoupled equations for the holomorphic and anti-holomorphic ket-vectors |V = V |0 , |V =V |0 : where we use the notation Step 8 . Our aim at this step is to find solution to equations for vertices V ,V given in (5.43),(5.44) and (5.47)-(5.50). We see that the equations for the vertices V andV are similar. Therefore to avoid the repetitions we focus on studying the holomorphic vertex V . To this end we introduce a new vertex |V 0 which is related to the vertex |V by the following invertible transformation: where the operator U is given by Thus, we see that dependence of the vertex V 0 on the momentum P L is completely fixed. Using solution in (5.58), we now analyse equations (5.43) and (5.49). To this end we note that plugging (5.58) into (5.49), we find that equation (5.49) leads to the following simple equation for the vertex |V ′ 0 = V ′ 0 |0 (5.58): Note that equations (5.59),(5.60) coincide with the ones for cubic vertex of massless fields in flat space. Introducing a new vertex V ′′ 0 by the relation we find that, in terms of the new vertex V ′′ 0 , equations (5.59)-(5.61) take the form From equations (5.63),(5.64), we learn that the vertex V ′′ 0 is independent of the momenta β 1 ,β 2 ,β 3 , while equation (5.65) tells us that the vertex V ′′ 0 is independent of the coordinate z. In other words, equations (5.63)-(5.65) imply that vertex V ′′ 0 (5.62) depends only on the oscillators, where, in (5.66), we recall that the argument α a stands for the oscillators α R a , α L a . Thus, we exhaust all equations we imposed on the cubic vertex and find that the general solution for the cubic vertex is governed by the vertex C(α a ) which depends only on the oscillators. Collecting relations (5.58), (5.62), (5.66), we see that general solution for the vertex V 0 entering the cubic interaction vertex in (5.40), (5.53) takes the following form: Note that the full expression for the holomorphic vertex V entering cubic vertex p − [3] (5.40) is obtained by using relation given in (5.53).
The procedure above described can be used for the derivation of the explicit representation for the anti-holomorphic vertexV . In next Section, we summarize our results for both the holomorphic and anti-holomorphic vertices.

Cubic interaction vertex for massless AdS fields
We now summarize our result for cubic interaction vertex we obtained in the previous sections. For the reader's convenience, we present the representation for the cubic vertex in terms of generating functions as well as the representation for the cubic vertex in terms of the component fields. Generating form of the cubic vertex. Let us refer to cubic vertex that describes interaction of spin-s 1 , spin-s 2 , and spin-s 3 massless fields as s 1 -s 2 -s 3 cubic vertex. We note then that solution for cubic interaction vertex we found is given by where the C 000 term describes the 0-0-0 cubic vertex, while the holomorphic vertex V and antiholomorphic vertexV describe the s 1 -s 2 -s 3 cubic vertices when s 1 + s 2 + s 3 > 0, s 1 ≥ 0, s 2 ≥ 0, s 3 ≥ 0. Explicit expressions for the holomorphic and anti-holomorphic vertices V ,V are given by where the symbol α λ H is defined in (2.19), while the spin operators M RL , M RL a are defined below in (6.14),(6.15). Definition of the momenta P L , P R is given in (3.37), (5.3), (5.38). In (6.5), (6.6) and below, the quantities C λ 1 ,λ 2 ,λ 3 andC λ 1 ,λ 2 ,λ 3 are constant parameters. These constant parameters are freedom of our solution. As we shall see below, the C λ 1 ,λ 2 ,λ 3 is a coupling constant describing interaction of three massless fields having helicities −λ 1 , −λ 2 , −λ 3 that satisfy the restriction λ 1 + λ 2 + λ 3 ≥ 1, whileC λ 1 ,λ 2 ,λ 3 is a coupling constant describing interaction of three massless fields having helicities −λ 1 , −λ 2 , −λ 3 that satisfy the restriction −λ 1 − λ 2 − λ 3 ≥ 1. In other words, the coupling constants C λ 1 ,λ 2 ,λ 3 andC λ 1 ,λ 2 ,λ 3 satisfy the restrictions C λ 1 λ 2 λ 3 = 0 , for λ 1 + λ 2 + λ 3 ≥ 1 , (6.7) Operators U andŪ appearing in (6.2) are given by where operators u t ,ū t are defined by the relations 10) 12) For definition of N β and ∆ β , see (A.10)-(A.14). We recall also that the spin operators M RL a and various quantities constructed out of M RL a and the momenta β 1 , β 2 , β 3 are defined by the relations Acting with the operator N β on the quantities given in (6.15), we get relations which can be helpful for the computation of the τ -ordered exponentials in (6.9), Our cubic vertices for massless fields in AdS 4 are simply related to cubic vertices for massless field in flat space. Namely, if we multiply the holomorphic vertex V 0 (6.3) by the factor z 1−M RL , then we get holomorphic vertex for massless fields in flat space. Also we note that, if we multiply the anti-holomorphic vertexV 0 (6.3) by the factor z 1+M RL , then we get anti-holomorphic vertex for massless fields in flat space. This is to say that, by module of the overall factor Uz M RL −1 , the holomorphic cubic vertex V (6.2) for massless fields in AdS 4 coincides with the holomorphic cubic vertex for massless fields in flat space, while, by module of the overall factorŪ z −M RL −1 , the antiholomorphic cubic vertexV (6.2) for massless fields in AdS 4 coincides with the anti-holomorphic cubic vertex for massless fields in flat space. For the massless fields in four-dimensional flat space, the full list of cubic vertices was obtained in Ref. [9]. Thus we see that all cubic vertices for massless fields in flat space obtained in Ref. [9] have their counterparts in AdS space. Alternative representation for the operators U,Ū . Operator U (6.9) is realized as differential operators with respect to the momenta P L , β 1 , β 2 , β 3 , and the coordinate z, while, operatorŪ (6.9) is realized as differential operators with respect to the momenta P R , β 1 , β 2 , β 3 , and the coordinates z. Note that the derivatives of the momenta β 1 , β 2 , β 3 enter the operators U,Ū through the operator N β . Using equation (5.49), we see that on space of the vertex V , the operator N β can be replaced by differential operator with respect to momentum P L and the coordinate z, while, using (5.50), we see that, on space of the vertexV , the operator N β can be replaced by differential operator with respect to the momentum P R , and the coordinate z. Doing so, we get the following alternative representation for the holomorphic and anti-holomorphic vertices V ,V , where V 0 ,V are given in (6.3),(6.4), while the operators U P andŪ P are defined by the relations (6.22) and the quantities Y ,Ȳ , and M RL are given in (6.12)-(6.15). Expressions (6.20),(6.21) provide the realization of the operator U in terms of differential operators with respect to the momentum P L and the coordinate z, while expressions (6.20), (6.22) provide the realization of the operatorŪ in terms of differential operators with respect to the momentum P R and the coordinate z. Component form of the cubic vertex. Massless spin-s field in AdS 4 is described by two complexvalued fields φ λ , λ = ±s (2.18). Plugging vertex (6.1) into (3.2) and using representation of the bra-vector φ| and relations (3.5), (3.6), we can work out explicit representation for the cubic Hamiltonian (3.2) in terms of the component complex-valued fields φ † λ . Computation of the component form of the cubic Hamiltonian is simplified by noticing the following relations for the spin operators M RL a : M RL a α λa H,a |0 = λ a α λa H,a |0 , a = 1, 2, 3 . (6.23) Using relations (6.23), we get the following component form for the cubic vertex: where we use the notation where the C λ 1 λ 2 λ 3 andC λ 1 λ 2 λ 3 should satisfy the restrictions in (6.7),(6.8).
iii) Using expressions for the operators u t ,ū t given in (6.10)-(6.12), we see that powers of the momenta P L , P R decrease upon acting on the vertices V 0 ,V 0 with the operators u t ,ū t . This implies that maximal number of powers of momentum P L appearing in the vertex V λ 1 ,λ 2 ,λ 3 (6.26) is equal to λ 1 + λ 2 + λ 3 , while a maximal number of powers of the momentum P R appearing in the vertex V λ 1 ,λ 2 ,λ 3 (6.27) is equal to −λ 1 − λ 2 − λ 3 . Taking this into account we now note that vertices (6.26), (6.27) have the following expansions in the momenta P L , P R , and the coordinate z for where the expansion coefficients V λ 1 ,λ 2 ,λ 3 n andV λ 1 ,λ 2 ,λ 3 n depend only on the momenta β 1 , β 2 , β 3 and the helicities λ 1 , λ 2 , λ 3 . Note that in the expansions (6.35) and (6.36) there are no terms of zero-order in P L and P R respectively. Absence of the terms of zero-order in P L and P R in the respective expansions (6.35) and (6.36) can be verified by using (6.26), (6.27) and the following respective relations for the operators u t ,ū t : where operators u t ,ū t are defined in (6.10)-(6.12). iv) Coupling constants C 000 , C λ 1 λ 2 λ 3 ,C λ 1 λ 2 λ 3 appearing in (6.25)-(6.27) are dimensionless. Obviously, the coupling constants turn out to be dimensionless in view of the particular powers of the radial coordinate z appearing in vertices (6.25)- (6.27) . v) For the case of internal algebra o(N), incorporation of a internal symmetry into the theory of massless AdS fields can be realized in the same way as for massless fields in flat space. Namely, first, in place of fields φ λ , we introduce fields φ ab λ , where the indices a, b are the matrix indices of the o(N) algebra. By definition, the fields satisfy the relation φ ab λ = (−) λ φ ba λ . Hermicity property of fields are as follows (φ ab λ (p, z)) † = φ ab −λ (−p, z). Second, in scalar products, the expressions φ † λ φ λ are replaced by φ ab † λ φ ab λ , while, in cubic vertices, the expressions φ † λ 1 φ † λ 2 φ † λ 3 are replaced by φ ab † λ 1 φ bc † λ 2 φ ca † λ 3 . Finally, in place of commutator (2.37), we use Note also that hermicity of the cubic Hamiltonian leads to the following relations for the coupling constants: C 000 * = C 000 , C λ 1 λ 2 λ 3 * = (−) λ 1 +λ 2 +λ 3C −λ 1 −λ 2 −λ 3 .

Conclusions
Light-cone gauge formulation for free fields propagating in AdS space was developed in Ref. [15].
In this paper, we extended the formulation in Ref. [15] to the case of interacting massless fields propagating in AdS 4 space. Using such light-cone gauge formulation, we built cubic interaction vertices for arbitrary spin massless fields in AdS 4 . We found the full lists of such cubic interaction vertices. We expect that our results have the following interesting applications and generalizations. i) We built the cubic vertices for light-cone gauge massless AdS fields. Extension of our study to quartic vertices might shed light on our understanding of the locality in the framework of light-cone gauge formulation of higher-spin field theory. As our cubic vertices for massless AdS fields are similar to the ones for massless fields in flat space [9], we expect, in view of results in Refs. [22,23], that the solution for the cubic coupling constants for massless fields in flat space found in Ref. [22] will be valid for massless AdS fields too. Recent discussion of various methods for analysis of quartic vertices may be found in Refs. [24,25,26]. ii) We considered interaction vertices for bosonic AdS fields. It is well known that supersymmetry imposes additional constraints on interactions vertices. Such constraints might simplify interaction vertices considerably. Therefore, in this respect, it would be interesting to extend our discussion to the case of supersymmetric theories. Recent investigations of various supersymmetric higher-spin theories may be found in Refs. [27,28,29]. For interesting discussion of arbitrary spin fermionic AdS fields see Ref. [30]. iii) In this paper, we studied fields in AdS 4 which, when considering in the framework of Lorentz covariant formulation, are associated with totally symmetric fields of the Lorentz so(3, 1). It is known string theory involves mixed-symmetry fields. Therefore from the perspective of studying the interrelations between massless higher-spin AdS field theory and string theory it seems reasonable to extend our study to the case of mixed-symmetry fields. Interesting discussion of this theme may be found in Ref. [31,32]. Discussion of various interesting Lorentz covariant formulations of free mixed-symmetry fields may be found, e.g., in Refs. [33]. Full list of cubic vertices for light-cone gauge massless fields in 6d flat space was found in Ref. [34] (see also Refs. [11,12]). Particular examples of Lorentz covariant interaction vertices for mixed-symmetry gauge AdS fields were discussed, e.g., in Ref. [35]. Light-cone gauge free mixed-symmetry AdS fields were studied in Refs. [36]- [39]. We believe therefore that the method developed in this paper will allows us to study light-cone gauge interacting mixed-symmetry AdS fields. iv) We studied tree-level cubic vertices for arbitrary spin massless AdS fields. Recently quantum corrections in theory of higher-spin fields flat space were studied in Refs. [40]. Use of our results for the analysis of quantum corrections for higher-spin AdS fields along the lines in Refs. [40] could be very interesting. Finally, we note that use of our results for studying AdS/CFT correspondence along the lines in Ref. [41] might be helpful for better understanding of AdS/CFT correspondence.
v) Lorentz covariant description of the vertices we obtained in this paper is of some interest. 3 At present time, many promising approaches have been developed for studying Lorentz covariant cubic vertices of AdS fields (see, e.g., Refs. [42,43]). As we noted, our light-cone gauge vertices for massless AdS fields are closely related to the ones for massless fields in flat space obtained in Ref. [9]. For the flat space, the discussion of various approaches for studying Lorentz covariant vertices may be found in Refs. [44]- [46]. However it seems likely that, already for the fields in flat space, covariant description of all light-cone gauge vertices presented in Ref. [9] is not an easy problem. Recent discussion of this theme may found in Ref. [47]. vi) Application of light-cone gauge approach for studying conformal fields propagating in AdS space and general gravitational background could also be of some interest. Recent studies of Lorentz covariant formulation of conformal fields in general gravitational background may be found in Refs. [48]. Ordinary derivative formulation of free conformal fields in AdS background was discussed in Ref. [49]. Discussion of higher-spin conformal fields in the framework of worldline approach may be found in Ref. [50].
Step 1. We use the second equations in (5.42)-(5.44) which tell us that dependence of the vertices in (5.41) on the momenta P L , P R , and the coordinate z can be presented as (P L ) n z n−1 V n ,V = N n=1 (P R ) n z n−1V n , (C.1) V 000 0 = V 000 0 (β a , α a ) , V n = V n (β a , α a ) ,V n =V n (β a , α a ) , where, as displayed in (C.2), the vertices V 000 0 , V n , andV n depend only on the momenta β a and the oscillators α a = α L a , α R a , a = 1, 2, 3. Note also that the first equation in (5.42) leads the following equation for the vertex V 000 0 : a=1,2,3 β a ∂ βa V 000 Step 2. Plugging p − [3]  where we use relations for the spin operators as in (2.33), (5.24). We now note that equation (C.3) and the first equation in (C.5) tell us that V 000 0 is independent of the momenta β 1 , β 2 , β 3 . From equations (C.7), (C.8), we learn that vertex V 000 0 is independent of the oscillators α R a , α L a , a = 1, 2, 3. Thus we conclude that V 000 0 is a constant.