Light-cone gauge massive and partially-massless fields in AdS(4)

Using light-cone gauge approach, bosonic and fermionic massive and partially-massless fields in AdS(4) space are considered. For such fields, light-cone gauge action is presented. Considering the massive and partially-massless fields in helicity basis and CFT adapted basis, two simple representations of spin operators entering the light-cone gauge action are discussed. The simple representations for the spin operators are obtained by using bosonic spinor-like oscillators. The bosonic spinor-like oscillators allow also us to treat the bosonic and fermionic fields on an equal footing.

Light-cone gauge formulation in AdS space was developed in Refs. [8,9] (see also Refs. [22]). Here we adapt the formulation in Refs. [9,22] for the study of massive and partially-massless fields. Light-cone gauge action. To discuss light-cone gauge formulation of arbitrary spin field we introduce a ket-vector |φ = |φ(x, z, u, v) . The arguments x, z, where x ≡ x + , x − , x 1 are used for the coordinates of AdS 4 space, 1 while the arguments u, v stand for creation operators. For integer spin-s field, s ∈ N 0 , the ket-vector |φ describes a collection of bosonic fields, while for half-integer spin-s field, s ∈ N 0 + 1 2 , the ket-vector |φ describes a collection of fermionic fields. Ordinary light-cone gauge AdS fields which depend on the space-time coordinates x, z are obtainable by expanding the ket-vector |φ(x, z, u, v) into u and v. Explicit expansions of the |φ into u and v will be given below. Light-cone gauge actions for the bosonic and fermionic fields in AdS 4 can be presented as for bosonic field, for fermionic field, (2.1) where an operator A appearing in (2.1) depends only on the oscillators. For massless arbitrary spin fields in AdS 4 , we note the equality A = 0 (see Ref. [8]). Relativistic symmetries of fields in AdS 4 . Relativistic symmetries of fields propagating in AdS 4 are described by the so (3,2) algebra. The use of light-cone gauge spoils manifest symmetries of the so (3,2) algebra. In order to show that symmetries of the so (3,2) algebra are maintained we should to present the Noether charges (generators) which generate them. For free bosonic and fermionic fields, Noether generators take the following respective forms: for bosonic field, where G diff stand for differential operators acting on the ket-vectors. Actions (2.1) are invariant under the transformations δ|φ = G diff |φ , where the operators G diff are given by 2

5)
(2.10) (2.12) Operators A, B, M z1 appearing in (2.4)-(2.11) depend only on the oscillators. The operator B is expressed in terms of the operator A. The operator A is AdS cousin of a flat space mass operator, while the operator M z1 is AdS cousin of a flat space helicity operator. For massive and partially-massless fields, the operators A and M z1 are not commuting when s = 0. Therefore they cannot be diagonalized simultaneously. This motivates us to introduce two bases for AdS fields which we refer to as helicity basis and CFT adapted basis. These two bases are defined as M z1 is diagonal for helicity basis; A is diagonal for CFT adapted basis. (2.13) As the basis with the diagonalized operator A turned out to be convenient for the study of lightcone gauge AdS/CFT correspondence in Ref. [23] we refer to such basis as CFT adapted basis. The bra-vector φ| in (2.1), (2.3) and hermitian properties of the operators A, M z1 , B are given by where an operator µ depends only on the oscillators and satisfies the relations µ † = µ, µ 2 = 1.
For massive field we use µ = 1, while, for partially-massless field, the µ will be given below. Throughout this paper we use bosonic spinor-like creation operators u, v and the respective annihilation operatorsū,v which we refer to as oscillators, Generators of the so(3) algebra S R , S L , and S and ket-vectors |n are defined by the relations For two integers a, b ∈ Z (or half-integers a, b ∈ Z + 1 2 ), a ≤ b, we use the convention n ∈ [a, b] ⇐⇒ n = a, a + 1, . . . , b − 1, b . (2.18)

Massive and partially-massless fields in helicity basis
Massive fields in helicity basis. For light-cone gauge description of spin-s massive field we use ket-vector |φ given by where, for spin-s bosonic field, s ∈ N 0 , while, for fermionic spin-s massive field, s ∈ N 0 + 1 2 . We should provide a realization of the operators A, B and M z1 on ket-vector (3.1). Here we present our result for the operators A, M z1 , and B, 3) where S R,L , S are given in (2.16). An energy parameter E 0 (3.4) stands for lowest eigenvalue of the energy operator of the so(3, 2) algebra irrep that is associated with the massive spin-s field, while C 2 (3.4) stands for the corresponding eigenvalue of the 2nd order Casimir operator. For the brief derivation of (3.2), (3.3), see Sec.5. The following remarks are in order. i) For scalar field (s = 0), spin-s bosonic field (s ∈ N), and spin-s fermionic field (s ∈ N 0 + 1 2 ), the E 0 is expressed in terms of mass parameter m by the well known relations (see, e.g. Refs. [9,11]) Accordingly, in terms of the mass parameter m, the respective values of C 2 (3.4) take the form Comparing (3.4) and (3.6), we see that it is the use of the E 0 that provides us the universal expression for C 2 (3.4) which is valid for arbitrary spin massive fields.
ii) We recall that, in the massless limit, the mass parameters and C 2 are given by iii) We recall the unitarity restriction in Ref. [25], 3 E 0 > s + 1 , for spin-s massive field. (3.8) Using (3.8), we verify that eigenvalues of the operator F (3.3) on space of ket-vector |φ (3.1) are real-valued. Therefore, in (2.14), we can use the simplest choice µ = 1. With this choice, the operators A, M z1 , B (3.2) satisfy the hermitian conjugation rules given in (2.14). iv) In view of the relation S|n = n|n (see (2.16) and (3.1)), the operator M z1 (3.2) is diagonal on space of the fields φ n (x, z) (3.1).
Partially-massless fields in helicity basis. Consider irreps of the so(3, 2) algebra with the following values of the energy parameter E 0 : Field in AdS 4 associated with the so(3, 2) algebra irrep having E 0 as in (3.9) is referred to spin-s and depth-t partially-massless field. For t = 0, we get E 0 corresponding to spin-s massless field. 4 For light-cone gauge description of spin-s and depth-t partially-massless field, we introduce the ket-vector |φ defined as Helicity of the component field φ n (x, z) is equal to n. Therefore the ket-vectors |φ ⊕ and |φ ⊖ (3.10) describe the respective positive and negative helicity component fields related by the hermicity conjugation rule (3.11). The ket-vector |φ (3.10) consists of 2t + 2 component fields as it should be for depth-t partially-massless field in AdS 4 . 5 Using E 0 (3.9), we note that realization of the operators A, M z1 , and B on ket-vectors |φ ⊕,⊖ (3.10) takes the same form as in (3.2), where operators f andf are given by while, in the expression for C 2 (3.4), we use E 0 given in (3.9). The bra-vector φ| entering action of partially-massless field (2.1) is expressed in terms of the ket-vectors |φ ⊕,⊖ as , we verify that the operators A, M z1 , B satisfy the hermitian conjugation rules given in (2.14). For example (µ ⊕ A) † = µ ⊕ A, and so on. Using the notation S(φ) for the action in (2.1), we note that the action (2.1) is factorized as is constructed in terms of the ket-vector |φ ⊕ and the bra-vector φ ⊕ |. The ket-vector |φ ⊕ (3.10) is decomposed into the fields with the positive helicities λ ∈ [s − t, s], while the bra-vector φ ⊕ | ≡ |φ ⊕ † , in view of (3.11), is decomposed into the fields with the negative helicities λ ∈ [−s, −s + t]. Therefore the action S(φ ⊕ ) is built from the pairs of the fields of opposite helicities. The same holds true for the action S(φ ⊖ ) which is constructed in terms of the ket-vector |φ ⊖ and the bra-vector φ ⊖ |. The ket-vector |φ ⊖ (3.10) is decomposed into the fields with the negative helicities λ ∈ [−s, −s + t], while the bra-vector φ ⊖ | ≡ |φ ⊖ † , in view of (3.11), is decomposed into the fields with the positive helicities λ ∈ [s − t, s]. Therefore the action S(φ ⊖ ) is also built from the pairs of the fields of opposite helicities. Moreover, in view of (3.11), one has the equality S(φ ⊕ ) = S(φ ⊖ ). Note that the action S(φ) = S(φ ⊕ ) + S(φ ⊖ ) appears naturally upon consideration of the partial-massless limit in the action for massive fields (see relations (5.15), (5.16) in Sec.5.) Partially-massless fields in helicity basis. Alternative formulation. We introduce the ket-vector |ψ related to the ket-vectors |φ ⊕ , |φ ⊖ (3.10) as 6 14) where (3.17) are obtained from (3.14)-(3.16) and (3.11). The operators A, M z1 , B are found to be where the realization of the operators M RL , f andf on the ket-vectors |ψ ⊕,⊖ is given by while C 2 in (3.18) is obtained by using (3.4) and (3.9). The bra-vector ψ| is defined as where ψ ⊕ | ≡ |ψ ⊕ † , ψ ⊖ | ≡ |ψ ⊖ † . Using µ ⊕ , µ ⊖ (3.21), we verify that the operators A, M z1 , B satisfy the rules given in (2.14). For example (µ ⊕ A) † = µ ⊕ A, and so on. Action for |ψ is obtained by using the replacement φ → ψ in (2.1). The action S(ψ) is factorized as S(ψ) = S(ψ ⊕ )+S(ψ ⊖ ). Moreover, in view of (ψ ⊕ n ) † = ψ ⊖ −n (3.17) one has the equality S(ψ ⊕ ) = S(ψ ⊖ ).

Massive and partially-massless fields in CFT adapted basis
Massive fields in CFT adapted basis. For light-cone gauge description of massive bosonic and fermionic fields in CFT adapted basis we use ket-vector |φ defined as where all component fields are real-valued and |n is defined in (2.17). Realization of the operators A, M z1 , and B on the ket-vector |φ (4.1) is given by where S R,L , S are given in (2.16), while the energy parameter E 0 (4.2) satisfies the unitary restriction (3.8). In view of the relation S|n = n|n , the operator A (4.2) is diagonal on |φ (4.1). Using (3.8), we verify that eigenvalues of the operator F (4.3) on space of ket-vector |φ (4.1) are real-valued. Therefore, in (2.14), we can use the simplest choice µ = 1. With such choice the operators A, M z1 , B (4.2) satisfy the hermitian conjugation rules given in (2.14).
In the CFT adapted basis, the equations of motion take simple form and therefore this basis is convenient for study of AdS/CFT correspondence for massive and partially-massless fields [23]. For massless fields, we have A = 0. Therefore, for massless fields, the helicity basis and the CFT adapted basis can be used on an equal footing for study of light-cone gauge AdS/CFT correspondence (see, e.g., Refs. [28,29]). The study of AdS/CFT correspondence for arbitrary spin massive and partially-massless fields by various covariant methods may be found in Refs. [30,31].

Basic equations of light-cone gauge approach in AdS 4
In this section, we present basic equations of light-cone gauge approach in AdS 4 and briefly outline procedure of the derivation of our results in Secs. 3,4. 7 Basic equations in helicity basis. Basic equations are formulated in terms of the operators M z1 and B 1 , B z . In terms of these operators, the operators A and B take the form (see Refs. [9,22]), where C 2 is given in (3.4), while the operators B 1 , B z , M z1 satisfy the basic equations given by and the restrictions (µB 1 ) † = µB 1 , (µB z ) † = µB z , (µM z1 ) † = −µM z1 . We also find the following representation of the 4th-order Casimir operator of the so(3, 2) algebra: 8 Using helicity basis operators B R , B L , and M RL defined by the relations In the helicity basis, the operators A and B (5.1) take the form We note also the restrictions (µB R ) † = µB L , (µM RL ) † = µM RL . The C 4 (5.4) takes the form Commutators given in (5.6) motivate us to look for the following solution for the operators B R , B L and M RL , 7 For AdS d+1 , d ≥ 3, the basic equations in CFT adapted basis were obtained in Ref. [23]. 8 For all solutions to spin operators obtained in this paper, we find C 4 = (E 0 − 1)(E 0 − 2)s(s + 1) as it should be.
where f = f (S),f =f (S). The S R,L , S are given in (2.16). What is required is to find f andf . Using equation (5.7), we find the following solution for ff : To fix f andf , we use µ = µ(S) and note that the hermicity conditions (2.14) amount to the equation Introducing the decomposition of ff (5.11), we note that equation (5.12) can be represented as (5.14) For massive fields, using (3.8), (5.11), (5.13) , we find ϕ S = 0. Therefore solution to equation (5.14) can be chosen to be µ = 1. The corresponding solution for f ,f given in (3.3) is obtained by using equations (5.11), (5.12).
We now consider a partially-massless field. Using the notation |φ m,s for the ket-vector of massive field in (3.1), we decompose the |φ m,s as Note that the Noether charges (2.3) are also decomposed as in (5.16). This is to say that, for E 0 given in (3.9), the ket-vector |φ m,s is decomposed into three decoupled systems -one massive spin-s msv field |φ msv , one helicity λ = s and depth-t partially-massless field |φ ⊕ and one helicity λ = −s and depth-t partially-massless field |φ ⊖ . However plugging E 0 (3.9) into (3.2)-(3.4), we get non-hermitian actions for the partially-massless fields (5.16). Hermitian actions and the Noether charges for φ ⊕,⊖ are obtained by choosing a suitable µ. Namely, plugging E 0 (3.9) into (5.11) and using (5.13), we find ϕ S = 1 This implies that, for the partially-massless fields, equation (5.14) takes the form µ(S) = −µ(S − 1). Solutions to such equation can be chosen as in (3.13).
The corresponding solutions for f ,f given in (3.12) are found by using equations (5.11), (5.12). Basic equations in CFT adapted basis. The basic equations are now formulated in terms of operators ν, W 1 ,W 1 . In terms of these operators, the operators A, B, and M z1 are expressed as where C 2 is given in (3.4). The basic equations for the operators ν, W 1 ,W 1 take then the form We note also the restrictions (µν) † = µν, (µW 1 ) † = µW 1 . In the CFT adapted basis, the 4thorder Casimir operator (5.4) is represented as General results in Ref. [23] suggest the following representation for the operators W 1 ,W 1 and ν, where f = f (S),f =f (S), while κ is given in (4.2). Equation (5.19) leads to the solution for ff : To fix f andf , we use equations (5.12)-(5.14) and (5.22). For massive fields, using (3.8), (5.13), (5.22), we find ϕ = 0. Using equation (5.14), we find then the solution µ = 1 , while the corresponding f ,f given in (4.3) are obtained by using equations (5.12), (5.22). Now consider partially-massless field. Using the notation |φ m,s for ket-vector of massive field in (3.1) and the notation |φ pms for ket-vector (4.4), we decompose the |φ m,s as |φ m,s = |φ msv + |φ pms , Plugging E 0 corresponding to partially-massless field (3.9) into expressions for massive field in (4.2), (4.3), we verify that the ket-vectors |φ msv , |φ pms (5.23) form invariant subspaces under action of operators A, B, and M z1 and the Noether charges for massive field (2.3) are decomposed as 9 G field (φ m,s ) = G field (φ msv ) + G field (φ pms ) .
Conclusions. In this paper, we applied light-cone gauge approach for the study of Lagrangian formulation of massive and partially-massless fields in AdS 4 . We studied both the bosonic and fermionic fields. In our approach, we used bosonic spinor-like oscillators. In Ref. [32], we demonstrated that the use of the bosonic spinor-like oscillators allows us to find the simple solution for all cubic vertices for massive fields in flat space, while, in Ref. [33], we developed the method for study of interacting fields in AdS 4 . We expect therefore that the results in this paper as well as the methods in Refs. [32,33] will provide us new interesting possibilities for building interaction vertices of the massive and partially-massless fields in AdS 4 . For the extensive study of interacting partially-massless fields, see Ref. [21] and Refs. [34]- [38]. We note that the formalism we used in this paper seems to be closely related to twistor approach. Recent interesting application of