CFT adapted gauge invariant formulation of arbitrary spin fields in AdS and modified de Donder gauge

Using Poincare parametrization of AdS space, we study totally symmetric arbitrary spin massless fields in AdS space of dimension greater than or equal to four. CFT adapted gauge invariant formulation for such fields is developed. Gauge symmetries are realized similarly to the ones of Stueckelberg formulation of massive fields. We demonstrate that the curvature and radial coordinate contributions to the gauge transformation and Lagrangian of the AdS fields can be expressed in terms of ladder operators. Realization of the global AdS symmetries in the conformal algebra basis is obtained. Modified de Donder gauge leading to simple gauge fixed Lagrangian is found. The modified de Donder gauge leads to decoupled equations of motion which can easily be solved in terms of Bessel function. Interrelations between our approach to the massless AdS fields and the Stueckelberg approach to massive fields in flat space are discussed.


Introduction
Further progress in understanding AdS/CFT correspondence [1] requires, among other things, better understanding of field dynamics in AdS space. Conjectured duality of conformal SYM theory and superstring theory in AdS 5 × S 5 has lead to intensive and in-depth study of various aspects of AdS field dynamics. Although many interesting approaches to AdS fields are known in the literature (for review see [2]- [4]), analysis of concrete dynamical aspects of such fields is still a challenging procedure. One of ways to simplify analysis of field and string dynamics in AdS space is based on use of the Poincaré parametrization of AdS space 1 . Use of the Poincaré coordinates simplifies analysis of many aspect of AdS field dynamics and therefore these coordinates have extensively been used for studying the AdS/CFT correspondence. In this paper we develop a formulation which is based on considering of AdS field dynamics in the Poincaré coordinates. This is to say that using the Poincaré parametrization of AdS space we discuss massless totally symmetric arbitrary spin-s, s ≥ 1, bosonic field propagating in AdS d+1 space of dimension d + 1 ≥ 4. Our results can be summarized as follows.
i) Using the Poincaré parametrization of AdS, we obtain gauge invariant Lagrangian for free massless arbitrary spin AdS field. The Lagrangian is explicitly invariant with respect to boundary Poincaré symmetries, i.e., manifest symmetries of our Lagrangian are adapted to manifest symmetries of boundary CFT. We show that all the curvature and radial coordinate contributions to our Lagrangian and gauge transformation are entirely expressed in terms of ladder operators that depend on radial coordinate and radial derivative. Besides this, our Lagrangian and gauge transformation are similar to the ones of Stueckelberg formulation of massive field in flat d-dimensional space. General structure of the Lagrangian we obtained is valid for any theory that respects Poincaré symmetries. Various theories are distinguished by appropriate ladder operators.
ii) We find modified de Donder gauge that leads to simple gauge fixed Lagrangian. The surprise is that this gauge gives decoupled equations of motion 2 . Note that the standard de Donder gauge leads to coupled equations of motion whose solutions for s ≥ 2 are not known in closed form so far. In contrast to this, our modified de Donder gauge leads to simple decoupled equations which are easily solved in terms of the Bessel function. Application of our approach to studying the AdS/CFT correspondence may be found in Ref. [10].
Motivation for our study of higher-spin AdS fields in Poincaré parametrization which is beyond the scope of this paper may be found at the end of Section 5.

Lagrangian and gauge symmetries
We begin with discussion of field content of our approach. In Ref. [11], the massless spin-s field propagating in AdS d+1 space is described by double-traceless so(d, 1) algebra totally symmetric tensor field Φ A 1 ...As . 3 This tensor field can be decomposed in scalar, vector, and totally symmetric tensor fields of the so(d − 1, 1) algebra: The fields in (2.1) subject to constraints (2.2) constitute a field content of our approach. To simplify presentation we use a set of the creation operators α a , α z , and the respective set of annihilation operators,ᾱ a ,ᾱ z . Then, fields (2.1) can be collected into a ket-vector |φ defined by 5 (2.4), we see that the ket-vector |φ is degree-s homogeneous polynomial in the oscillators α a , α z , while the ket-vector |φ s ′ is degree-s ′ homogeneous polynomial in the oscillators α a , i.e., these ket-vectors satisfy the relations 6 In terms of the ket-vector |φ , double-tracelessness constraint (2.2) takes the form 7 Action and Lagrangian we found take the form φ| ≡ (|φ ) † , where operator E is given by 10) 4 Note that so(d − 1, 1) tensorial components of the Fronsdal field Φ A1...As are not double-traceless. Using appropriate transformation (see (5.22)) those tensorial components can be transformed to our fields in (2.1). 5 We use oscillator formulation [12]- [14] to handle the many indices appearing for tensor fields (see also [15]). It can also be reformulated as an algebra acting on the symmetric-spinor bundle on the manifold M [16]. 6 Throughout this paper we use the following notation for operators constructed out the oscillators and derivatives: N α ≡ α aᾱa , N z ≡ α zᾱz , α 2 = α a α a ,ᾱ 2 =ᾱ aᾱa , 2 = ∂ a ∂ a , α∂ = α a ∂ a ,ᾱ∂ =ᾱ a ∂ a . 7 We adapt the formulation in terms of the double-traceless gauge fields [11]. Adaptation of approach in Ref. [11] to massive fields may be found in Refs. [17,18]. Discussion of various formulations in terms of unconstrained gauge fields may be found in Refs. [19]- [24]. Study of other interesting approaches which seem to be most suitable for the theory of interacting fields may be found e.g. in Refs. [25]- [27].
14)   20) and subscript n in E (n) (2.8) tells us that E (n) is degree-n homogeneous polynomial in the flat derivative ∂ a . We note that gauge invariance requires ε 2 = 1. Because ε depends on N z , this leaves two possibilities ε = ±1 at least.
The following remarks are in order. i) Operator E (2) (2.9) is the symmetrized Fronsdal operator represented in terms of the oscillators. This operator does not depend on the radial coordinate and derivative, z, ∂ z , and it takes the same form as the one of massless field in d-dimensional flat space. ii) Dependence of operator E (2.8) on the radial coordinate and derivative, z, ∂ z , is entirely governed by the operators e 1 andē 1 which are similar to ladder operators appearing in quantum mechanics. Sometimes, we refer to the operators e 1 andē 1 as ladder operators 8 . iii) Representation for the Lagrangian in (2.7) -(2.13) is universal and is valid for arbitrary Poincaré invariant theory. Various Poincaré invariant theories are distinguished by ladder operators entering the operator E. This is to say that the operators E of massive and conformal fields in flat space depend on the oscillators α a ,ᾱ a and the flat derivative ∂ a in the same way as the operator E of AdS fields (2.8). In other words, the operators E for massless AdS fields, massive and conformal fields in flat space are distinguished only by the operators e 1 andē 1 . For example, all that is required to get the operator E for massive spin-s field in d-dimensional flat space is to make the substitutions where m is mass parameter of the massive field and f is given in (2.17). Note also that our field content (2.1) is similar to the one of Stueckelberg formulation of massive field in d-dimensional space [17]. Expressions for e 1 ,ē 1 appropriate for conformal fields may be found in Refs. [31]. Gauge symmetries. We now discuss gauge symmetries of Lagrangian in (2.7). To this end we introduce the following set of gauge transformation parameters: The gauge parameters ξ 0 , ξ a 1 , and ξ a 1 ...a s ′ s ′ , s ′ ≥ 2 in (2.22), are the respective scalar, vector, and rank-s ′ totally symmetric tensor fields of the so(d − 1, 1) algebra. The gauge parameters ξ We now, as usually, collect gauge transformation parameters in ket-vector |ξ defined by The ket-vectors |ξ , |ξ s ′ satisfy the algebraic constraints which tell us that |ξ is a degree-(s − 1) homogeneous polynomial in the oscillators α a , α z , while |ξ s ′ is degree-s ′ homogeneous polynomial in the oscillators α a . In terms of the ket-vector |ξ , tracelessness constraint (2.23) takes the form Gauge transformation can entirely be written in terms of |φ and |ξ . We find the following gauge transformation: where e 1 ,ē 1 are given in (2.14),(2.15). From (2.28), we see that the flat derivative ∂ a enters only in α∂-term in (2.28), while the radial coordinate and derivative, z, ∂ z , enter only in the operators e 1 ,ē 1 . Thus, all radial coordinate and derivative contributions to gauge transformation (2.28) are entirely expressed in terms of the ladder operators e 1 andē 1 9 . We finish this Section with the following remark. Introducing new mass-like operator and using explicit expressions for operators e 1 andē 1 (2.14),(2.15) we find We make sure that the operators M 2 , e 1ē1 satisfy the following commutators:

Global so(d, 2) symmetries
Relativistic symmetries of AdS d+1 space are described by the so(d, 2) algebra. In our approach, the massless spin-s AdS d+1 field is described by the set of the so(d − 1, 1) algebra fields (2.1). Therefore it is reasonable to represent the so(d, 2) algebra so that to respect manifest so(d − 1, 1) symmetries. For application to the AdS/CFT correspondence, most convenient form of the so(d, 2) algebra that respects the manifest so(d − 1, 1) symmetries is provided by nomenclature of the conformal algebra. This is to say that the so(d, 2) algebra consists of translation generators P a , conformal boost generators K a , dilatation generator D, and generators J ab which span so(d −1, 1) algebra. We use the following normalization for commutators of the so(d, 2) algebra generators:

3)
[J ab , J ce ] = η bc J ae + 3 terms . Requiring so(d, 2) symmetries implies that the action is invariant with respect to transformation δĜ|φ =Ĝ|φ , where the realization of so(d, 2) algebra generatorsĜ in terms of differential operators takes the form

7)
x∂ ≡ x a ∂ a , x 2 ≡ x a x a . In Operator R a appearing in K a (3.7) is given by where e 1,1 ,ē 1,1 are given in (2.16). We see that realization of Poincaré symmetries on bulk AdS fields (3.5) coincide with realization of Poincaré symmetries on boundary CFT operators. Note that realization of Dand K a -symmetries on bulk AdS fields (3.6),(3.7) coincides, by module of contributions of operators ∆ and R a , with the realization of Dand K a -symmetries on boundary CFT operators. Realizations of the so(d, 2) algebra on bulk AdS fields and boundary CFT operators are distinguished by ∆ and R a . The realization of the so(d, 2) symmetries on bulk AdS fields given in (3.5)-(3.7) turns out to be very convenient for studying AdS/CFT correspondence [10].

Modified de Donder gauge condition is then defined to bē
where the operatorC is given in (4.3). Because of double-tracelessness of |φ (2.6), operatorC (4.3) satisfies the relationᾱ 2C |φ = 0, i.e., gauge condition (4.14) respects constraint for gauge transformation parameter |ξ , (2.27). Using the modified de Donder gauge condition in gauge invariant equations of motion (4.10) leads to the following gauge fixed equations of motion: where M 2 is defined in (2.30). In terms of fields (2.1), equation (4.15) can be represented as s ′ = 0, 1, . . . , s. Thus, our modified de Donder gauge condition (4.14) leads to decoupled equations of motion (4.16) which can easily be solved in terms of the Bessel function 12 . For spin-1 field, gauge condition (4.14), found in [9], turns out to be a modification of the Lorentz gauge. We note that equations of motion (4.15) have on-shell leftover gauge symmetries. These onshell leftover gauge symmetries can simply be obtained from generic gauge symmetries (2.28) by the substituting |ξ → |ξ lf ov , where the |ξ lf ov satisfies the following equations of motion: (4.17)

Comparison of standard and modified de Donder gauges
Our approach to the massless spin-s field in AdS d+1 is based on use of double-traceless so(d−1, 1) algebra fields (2.1). One of popular approaches to the massless spin-s field in AdS d+1 is based on use of double-traceless so(d, 1) algebra field Φ A 1 ...As [11]. The aim of this Section is twofold. First we explain how our modified de Donder gauge is represented in terms of the commonly used field Φ A 1 ...As . Also, we compare the modified de Donder gauge and commonly used standard de Donder gauge 13 . Second we show explicitly how our fields (2.1) are related to the field Φ A 1 ...As . We begin with discussion of modified de Donder gauge-fixing procedure at the level of Lagrangian. First we present gauge invariant Lagrangian for the field Φ A 1 ...As . To simplify presentation we introduce, as before, the following ket-vector where (5.2) tells us that the Φ A 1 ...As is double-traceless, and the scalar products like α A α A are decomposed as α A α A = α a α a + α z α z . In terms of |Φ , gauge invariant Lagrangian takes the 12 Interesting method of solving AdS field equations of motion which is based on star algebra products in auxiliary spinor variables is discussed in Ref. [32]. 13 Recent applications of the standard de Donder gauge to the various problems of higher-spin fields may be found in Refs. [33,34].
where e = det e A µ , e A µ stands for vielbein of AdS d+1 space, and D A are covariant derivatives (for details of notation, see Appendix). Lagrangian (5.4) can be represented as We now ready to discuss the modified de Donder gauge. To make our study more useful we discuss both the modified and standard de Donder gauges. Note that our formulas for standard de Donder gauge are valid for arbitrary parametrization of AdS, while the ones for modified de Donder gauge are adapted to the Poincaré parametrization. Gauge-fixing term is defined to be where operator E g.f ix corresponding to the standard de Donder gauge fixing and the modified de Donder gauge fixing is given by C modCmod , modified gauge, (5.10) and we use the notation We now make sure that the gauge fixed Lagrangian L total takes the form L total ≡ L + L g.f ix , (5.14) 14 Since Ref. [11], various approaches to massless totally symmetric AdS fields were developed in the literature (see e.g. [12,9,35,16]). We use setup discussed in Ref. [9]. Formulas in Ref. [11] are adapted to AdS 4 with mostly negative metric tensor, while our formulas are adapted to AdS d+1 with mostly positive metric tensor. Taking this into account and plugging d = 3 in (5.5) we make sure that our operator E matches with the operator L 0 in Eq.(2.7) in Ref. [11].
Alternatively, the operator E total corresponding to the modified de Donder gauge in (5.16) can be represented as where ν is given in (2.30).
We proceed with discussion of gauge-fixing procedure at the level of equations of motion. To this end we note that gauge invariant Lagrangian (5.4) leads to the following equations of motion: where E total is given in (5.16). We note that, because of C z ⊥ -andC z ⊥ -terms, the modified de Donder gauge breaks some of the so(d, 2) symmetries. In the conformal algebra nomenclature, these broken symmetries correspond to broken conformal boost K a -symmetries.
From E total (5.16), we see that, because of α 2ᾱzᾱz -term, the modified de Donder gauge for |Φ does not lead to decoupled equations for the ket-vector |Φ when 15 s ≥ 2. It turns out that in order to obtain decoupled equations of motion we should introduce our set of fields in (2.1). We remind that |Φ is a double-traceless field (5.2) of the so(d, 1) algebra, while |φ describes double-traceless fields (2.6) of the so(d − 1, 1) algebra. This is to say that to get decoupled equations of motion we have to make transformation from the so(d, 1) ket-vector |Φ to so(d − 1, 1) ket-vector |φ . We find the following transformation from the ket-vector |Φ to our ket-vector |φ : 15 For spin-1 field, gauge condition (5.20) and the corresponding decoupled equations of motion were found in [9].
Finally we compare realization of so(d, 2) symmetries on the ket-vectors |φ and |Φ . To this end we note that on space of |Φ realization of the so(d, 2) algebra transformations takes the form Comparing (3.5) and (5.34), we see that the realizations of Poincaré symmetries on |φ and |Φ match from the very beginning. Taking into account z-factor in (5.28), it is easily seen that D-transformations for |φ (3.6) and |Φ (5.35) also match. All that remains to do is to match conformal boost K a -transformations given in (3.7) and (5.35). Choosing ε = −1 in (2.17), we make sure that realizations of the operator K a on |φ (3.7) and on |Φ (5.35) match. To summarize, using the Poincaré parametrization of AdS space, we have developed the CFT adapted formulation of massless arbitrary spin AdS field. In our approach, Poincaré symmetries of the Lagrangian are manifest. As is well known string theory solutions like AdS d+1 ×S d+1 and Dpbrane backgrounds supported by RR-charges have the respective the dand (p + 1)-dimensional Poincaré symmetries. We note that the structure of the Lagrangian we obtained for AdS field is valid for any theory that respects Poincaré symmetries. Various theories are distinguished by appropriate ladder operators. Therefore we think that our approach might be a good starting point for formulation of higher-spin gauge fields theory in AdS d+1 × S d+1 and Dp-brane backgrounds. For the case of AdS d+1 field, the ladder operators depend on the radial coordinate and the radial derivative. It would be interesting to unravel a structure and role of ladder operators in AdS d+1 × S d+1 and Dp-brane backgrounds 16 . The AdS d+1 ×S d+1 and Dp-brane backgrounds play important role in studying string/gauge theory dualities. Developing a theory of higher-spin gauge fields in these backgrounds might be useful for better understanding string/gauge theory dualities. ∂ µ = ∂/∂x µ , where e µ A is inverse vielbein of AdS d+1 space, D µ is the Lorentz covariant derivative and the base manifold index takes values µ = 0, 1, . . . , d. The ω AB µ is the Lorentz connection of AdS d+1 space, while M AB is a spin operator of the Lorentz algebra so(d, 1). Note that AdS d+1 coordinates x µ carrying the base manifold indices are identified with coordinates x A carrying the flat vectors indices of the so(d, 1) algebra, i.e., we assume x µ = δ µ A x A , where δ µ A is Kronecker delta symbol. AdS d+1 space contravariant tensor field, Φ µ 1 ...µs , is related with field carrying the flat indices, Φ A 1 ...As , in a standard way Φ A 1 ...As ≡ e A 1 µ 1 . . . e As µs Φ µ 1 ...µs . Helpful commutators are given by where Ω ABC = −ω ABC + ω BAC is a contorsion tensor and we define ω ABC ≡ e Aµ ω BC µ . For the Poincaré parametrization of AdS d+1 space, vielbein e A = e A µ dx µ and Lorentz connection, de A + ω AB ∧ e B = 0, are given by With choice made in (A.5), the covariant derivative takes the form D A = z∂ A +M zA , ∂ A = η AB ∂ B .