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Physics Letters B

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

R.R. Metsaev

Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky prospect 53, Moscow 119991, Russia

ARTICLE INFO ABSTRACT

Using Poincaré 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 the Bessel function. Interrelations between our approach to the massless AdS fields and the Stueckelberg approach to massive fields in flat space are discussed.

© 2008 Elsevier B.V. All rights reserved.

Article history:

Received 2 September 2008

Received in revised form 1 December 2008

Accepted 1 December 2008

Available online 6 December 2008

Editor: L. Alvarez-Gaume

1. 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 AdS5 x S5 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 Poincare parametrization of AdS space.1 Use of the Poincare 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 Letter we develop a formulation which is based on considering of AdS field dynamics in the Poincare coordinates. This is to say that using the Poincare parametrization of AdS space we discuss massless totally symmetric arbitrary spin-s, s > 1, bosonic field propagating in AdSd+1 space of dimension d + 1 > 4. Our results can be summarized as follows.

(i) Using the Poincare 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 La-grangian 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].

E-mail address: metsaev@Ipi.ru.

1 Studying AdS5 x S5 superstring action [5] in Poincare parametrization may be found in Ref. [6]. Recent interesting application of Poincare coordinates to studying AdS5 x S5 string T-duality may be found in [7] (see also [8]).

0370-2693/$ - see front matter © 2008 Elsevier B.V. AII rights reserved. doi:10.1016/j.physIetb.2008.12.002

2 Our modified de Donder gauge seems to be unique first-derivative gauge that leads to decoupled equations of motion. Light-cone gauge [9] also leads to decoupled equations of motion, but the light-cone gauge breaks boundary Lorentz symmetries.

Motivation for our study of higher-spin AdS fields in Poincare parametrization which is beyond the scope of this Letter may be found at the end of Section 5.

2. Lagrangian and gauge symmetries

We begin with discussion of field content of our approach. In Ref. [11], the massless spin-s field propagating in AdSd+1 space is described by double-traceless so(d, 1) algebra totally symmetric tensor field &Al "As .3 This tensor field can be decomposed in scalar, vector, and totally symmetric tensor fields of the so(d — 1,1) algebra:

, s'= o, 1,...,s — 1, s. The fields $aJ...as' with s' > 3 are double-traceless4

,aabba5...asi „ ts' s = 0,

s' = 4, 5,...,s - 1, s.

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 aa, az, and the respective set of annihilation operators, aa, az. Then, fields (2.1) can be collected into a ket-vector defined by5

« it*:

s'i^/isSy.

«ai ■ ■ ■ aas'C^'IO).

From (2.3), (2.4), we see that the ket-vector is degree-s homogeneous polynomial in the oscillators aa, az, while the ket-vector \$si) is degree-s' homogeneous polynomial in the oscillators aa, i.e., these ket-vectors satisfy the relations6

(N« + Nz - s)t) = 0, (N« - s'

1 = 0.

In terms of the ket-vector it), double-tracelessness constraint (2.2) takes the form7

{a2)2 |0) = o.

Action and Lagrangian we found take the form

S = j ddxdzL, L = 2<0|E№),

(0| = (|0))^, where operator E is given by

E = E (2) + E(1) + E (0), E (2) = □ — ada d + 2 (ad)2 a2 + 1 a2(a d)2

3 A, B, C = 0,1,...,d and a, b, c = 0,1,...,d — 1 are the respective flat vector indices of the so(d, 1) and so(d — 1,1) algebras. In Poincare parametrization of AdSd+1 space, ds2 = (dxadxa + dzdz)/z2. We use the conventions: da = d/dX, dz = d/dz. Vectors of so(d, 1) algebra are decomposed as XA = (Xa, Xz).

4 Note that so(d — 1,1) tensorial components of the Fronsdal field &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 Letter we use the following notation for operators constructed out the oscillators and derivatives: Na = aaaa, Nz = azaz, a2 = aaaa, a2 = aaaa, □ = dada, ad = aada, a3 = aada.

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].

1 2 - 2 1 2 — — 2

--a2 a « 2--« «d« da ,

E (1) = + e1Ä,

E (0) = m1 + a2a 2m2 + m 3a2 + m3«2

A = ad - a2«d + -a2ad«2, 4

Ä = ad - ada2 +1 a2«da2, 4

2s + d - 5 - 2Nz

e1 = eu dz +

(2.2) f = s

( 2s + d - 5 - 2Nz

e1 = (d--2Z-J^1,

e1,1 = «zf, e1,1 = f «z, 2s + d - 4 - Nz ^ 1/2

2s + d - 4 - 2Nz

m1 = e1e1 - 2

2s + d - 3 - 2Nz 2s + d - 4 - 2N-

e = ±1,

1. 1 2s + d - 2Nz

m2 = — ei ei +---ei ei,

2 2 1 1 4 2s + d - 4 - 2Nz

m3 = 2 e1e1,

m3 = 2 e1e1,

(2.10) (2.11) (2.12)

(2.13)

(2.14)

(2.15)

(2.16) (2.17) (2.18) (2.19) (2.20)

and subscript n in E(n) (2.8) tells us that E(n) is degree-n homogeneous polynomial in the flat derivative da. We note that gauge invariance requires e2 = 1. Because s depends on Nz, this leaves two possibilities e = ±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, dz, 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, dz, is entirely governed by the operators e1 and e1 which are similar to ladder operators appearing in quantum mechanics. Sometimes, we refer to the operators e1 and e 1 as ladder operators.8

(iii) Representation for the Lagrangian in (2.7)-(2.13) is universal and is valid for arbitrary Poincare invariant theory. Various Poincare 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 aa, aa and the flat derivative da 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 e1 and e1. 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

e1 ^ mazf,

-mf a z

(2.21)

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 e1, e 1 appropriate for conformal fields may be found in Refs. [31].

8 Interesting application of other ladder operators to studying AdS/QCD correspondence may be found in [28]. We believe that our approach will also be useful for better understanding of various aspects of AdS/QCD correspondence which are discussed e.g. in [28-30].

Gauge symmetries. We now discuss gauge symmetries of La-grangian in (2.7). To this end we introduce the following set of gauge transformation parameters:

d1' ^, s' = 0, 1.....s - 1.

(2.22)

The gauge parameters £0, , and "'as', 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 ^a1--"*' with s' > 2 are subjected to the tracelessness constraint

fs' =0,

s' > 2.

(2.23)

We now, as usually, collect gauge transformation parameters in ket-vector ) defined by

s— 1

s'=0 |fs'> = ■

«Z)s—1—s |fs'>,

,aa1 ...aas' f s'1 .as' |0>.

s'!V (s — 1 — s')! The ket-vectors |f >, |fs'> satisfy the algebraic constraints

(Na + Nz — s + 1) |f > = 0, (Na — s')|f5'> = 0,

(2.24)

(2.25)

(2.26)

which tell us that |£) is a degree-(s — 1) homogeneous polynomial in the oscillators aa, az, while |£s') is degree-s' homogeneous polynomial in the oscillators aa. In terms of the ket-vector |£), tracelessness constraint (2.23) takes the form

a 2|£) = 0.

Gauge transformation can entirely be written in terms of |£). We find the following gauge transformation:

= ad — ei —

2s + d — 6 — 2N

a2e1 )|f >,

(2.28)

where e1, e 1 are given in (2.14), (2.15). From (2.28), we see that the flat derivative 9a enters only in ad-term in (2.28), while the radial coordinate and derivative, z, dz, enter only in the operators e1, e1. Thus, all radial coordinate and derivative contributions to gauge transformation (2.28) are entirely expressed in terms of the ladder operators e1 and e 1.9

We finish this Section with the following remark. Introducing new mass-like operator

2 2s + d — 2 — 2Nz

M = — e 1e1 + ———-—— e1e1, 2s + d — 4 — 2Nz

(2.29)

and using explicit expressions for operators e1 and e1 (2.14), (2.15) we find

M2 = — d2 + -1 ( V2 — 1

d—4 v = s +---Nz.

(2.30)

We make sure that the operators M2, e1 e1 satisfy the following commutators:

[e1, M2] = 0, [e 1, M2] = 0.

(2.31)

Because the operators e1, e 1 enter gauge transformation (2.28), relations (2.31) can be considered as requirement for gauge invariance of the operator M2. Therefore, M2 in (2.29) can be considered as a definition of gauge invariant mass operator. We note that making substitutions (2.21) in (2.29) gives M2 = m2. Thus, we see that our definition of mass operator M2 (2.29) gives desired result for massive field in flat space and provides interesting generalization of notion of mass operator to the case of massless AdS field (2.30).

9 Making substitutions (2.21) in (2.8) and (2.28) one can make sure that our La-grangian and gauge transformation match with those of flat limit of AdS massive field theory in Ref. [17].

3. Global so(d, 2) symmetries

Relativistic symmetries of AdSd+1 space are described by the so(d, 2) algebra. In our approach, the massless spin-s AdSd+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 Pa, con-formal boost generators Ka, dilatation generator D, and generators Jab which span so(d — 1,1) algebra. We use the following normalization for commutators of the so(d, 2) algebra generators:

[~D, Pa] = —Pa, [ Pa, jbcj = nabPc — nacPb,

[d, Ka] = Ka, [Ka, jbc] = nabKc — nacKb, [ Pa, Kb ] = nabD — Jab, [ Jab, Jce] = nbcJae + 3 terms.

Requiring so(d, 2) symmetries implies that the action is invariant with respect to transformation S^fi) = G|0), where the realization of so(d, 2) algebra generators G in terms of differential operators takes the form

(2.27) Pa = da, Jab = xadb — xb da + M'

D = xd + 4, 4 = zdz + ■

Ka = —22 x23a + xaD + Mabxb + Ra,

xd = xada, x2 = xaxa. In (3.5), (3.7), Mab is spin operator of the so(d — 1 , 1 ) algebra. Commutation relations for Mab and representation of Mab on space of ket-vector |0> (2.3) take the form

[.Mab, Mce] = nbcMae + 3 terms, Mab = aaab — abaa. Operator Ra appearing in Ka (3.7) is given by

na -,ra~ I __ r.a 1 a

R = —zC e1 1 + ze1 1<x--z d ,

Ca = aa — a

2Na + d — 2

(3.9) (3.10)

where e11, e 11 are given in (2.16). We see that realization of Poincare symmetries on bulk AdS fields (3.5) coincide with realization of Poincare symmetries on boundary CFT operators. Note that realization of D- and Ka-symmetries on bulk AdS fields (3.6), (3.7) coincides, by module of contributions of operators A and Ra, with the realization of D - and Ka-symmetries on boundary CFT operators. Realizations of the so(d, 2) algebra on bulk AdS fields and boundary CFT operators are distinguished by A and Ra. 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].

4. Modified de Donder gauge

We begin with discussion of gauge-fixing procedure at the level of Lagrangian (2.7). We find that use of the following modified de Donder gauge-fixing term

Lg.fix = 1 <0| fg.flxh

E g. fix = CC,

C = ad — - a2a d — e1n[1,2] + - e 1a2,

C = a d--ada2 +— e1û'2 — e1n

n [1,2] = 1 — a2

2(2Na + d)

leads to the surprisingly simple gauge fixed Lagrangian Ltotal:

L total = L + L

g.fix,

L total = ^ 0E total\0),

E total = (l — 4 a2â 2) (□ — M2),

where M2 is given in (2.30).10 We note that our gauge-fixing term (4.1) respects the Poincare and dilatation symmetries but breaks the conformal boost Ka-symmetries, i.e., the simple form of gauge fixed Lagrangian (4.5) is achieved at the cost of the Ka-symmetries.

We now discuss gauge-fixing procedure at the level of equations of motion. To this end we note that gauge invariant Lagrangian (2.7) leads to the following equations of motion11:

(E (2) + n [2,3]( E (1) + E(0)))\0) = 0, n [2,3] = 1 — (a2)2 1

8(2Na + d)(2Na + d + 2)"

These equations can be represented as

(£(2) + £(1) + £(o))\0) = O, £(2) = □ — ada d + 2 (ad)2 a2, £(1) = e^ a d — ada2) + e A ad + a7

2Na + d — 2

-ad — a2ad

2Na + d

£(0) = m — (m1 + 4m2)a2

22 + m3a — m 3a

2(2Na + d — 2) 2

2Na + d — 2

-n [1,2].

Modified de Donder gauge condition is then defined to be

C\0) = 0,

(4.12)

(4.13)

(4.14)

where the operator C is given in (4.3). Because of double-tracelessness of \0) (2.6), operator C (4.3) satisfies the relation â2 C \ 0) = 0, i.e., gauge condition (4.14) respects constraint for gauge transformation parameter \£), (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:

(□ — M2)\0) = 0, (4.15)

where M2 is defined in (2.30). In terms of fields (2.1), Eq. (4.15) can be represented as

□ + ®z — M v2 — 4) W = 0, V,= s' +

(4.16)

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 on-shell leftover gauge symmetries can simply be obtained from generic gauge symmetries (2.28) by the substituting |£) ^ |/f0v), where the |£jfov) satisfies the following equations of motion:

(□ — M2)^tfov) = 0. (4.17)

5. Comparison of standard and modified de Donder gauges

Our approach to the massless spin-s field in AdSd+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 AdSd+1 is based on use of double-traceless so(d, 1) algebra field &A1-.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 &A1...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 &A1".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 &A1".As. To simplify presentation we introduce, as before, the following ket-vector

|&) = 1 & A1 -As a A1 s!

(4.10)

(4.11) (a 2)> )=0

•a As\0),

a2 = aA aA, a2 = aA aA, (5.3)

where (5.2) tells us that the &A1...As is double-traceless, and the scalar products like aA aA are decomposed as aA aA = aaaa + azaz. In terms of |&), gauge invariant Lagrangian takes the form14

L = -e{$\\ 1--a a £)

£ = □mS — aDa D + ^ (aD)2 a — s(s + d — 5) + 2d — 4 — a2â2

m _ nA nA 1 s*AABnB

OAdS = D D + W D

where e = deteA, eA stands for vielbein of AdSd+1 space, and DA are covariant derivatives (for details of notation, see Appendix A). Lagrangian (5.4) can be represented as

L = 2 e(&|E |&), (5.7)

E = □AdS — aDa D + 1 (aD)2a2 + 1 a2 (a D)2

— 2 a2^AdSa2 — 1 a2aDa Da2

— s(s + d — 5) + 2d — 4 + 2 (s(s + d — 3) — d)a2a2. (5.8)

We now ready to discuss the modified de Donder gauge. To make our study more useful we discuss both the modified and

aD = a ADA

aD = a aDa

10 Making substitutions (2.21) in (4.2), (4.3) gives gauge fixed Lagrangian for massive field of the form (4.5)-(4.7) with M2 = m2.

11 Appearance of the projector n[2,3] in equations of motion (4.8) is related to the fact that the operators E(1 ) , E(0) , in contrast to the symmetrized Fronsdal operator E(2), do not respect double tracelessness constraint (2.6). Note that the ket-vectors E(1)|$), E(0)|^) are triple-traceless, (a2)3E(1)|^) = 0, (a2)3E(0)|^) = 0.

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].

14 Since Ref. [11], various approaches to massless totally symmetric AdS fields were developed in the literature (see e.g. [9,12,16,35]). We use setup discussed in Ref. [9]. Formulas in Ref. [11] are adapted to AdS4 with mostly negative metric tensor, while our formulas are adapted to AdSd+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 L0 in Eq. (2.7) in Ref. [11].

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 Poincare parametrization. Gauge-fixing term is defined to be

Lg. fix = 1 e(*\E g.flx|0),

where operator £g.nx corresponding to the standard de Donder gauge fixing and the modified de Donder gauge fixing is given by

CstandCstand, standard gauge, CmodCmod, modified gauge,

Eg.fix =

and we use the notation

Cstand = aD - 2a aD,

Cmod = Cstand — 2Ci,

Cstand = a D - 2 a Da

C mod = C stand + 2Ci

CZ = az--a2ä z,

CZ = âz--aza

(5.10)

(5.11)

(5.12)

(5.13)

We now make sure that the gauge fixed Lagrangian Aotal takes the form

L total = L + Lg.fix,

Ltotal = ~e(*\ 1 --a2a2 £totai\*)

IoAdS — s(s + d — 5) + 2d — 4 — a2a2, □0 AdS — s(s + d — 4) + 2d — 4 — a2âl + (2s + d — 5)Nz, modified gauge,

(5.14)

(5.15) standard gauge,

(5.16)

where n0AdS = z2(n + 3|) + (1 — d)zdz. Alternatively, the operator ¿total corresponding to the modified de Donder gauge in (5.16) can be represented as

¿total = □ 0Ads - a2äzäz - v2 +

(5.17)

where v 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:

£\*) = 0.

(5.18)

We now define the standard and modified de Donder gauge conditions as

Cstand|&> = 0, standard de Donder gauge, Cmod | & > = 0, modified de Donder gauge,

(5.19)

(5.20)

where Cstand, CCmod are given in (5.11), (5.12). Using (5.19), (5.20) in (5.18) we get gauge fixed equations of motion

¿total ) = 0,

(5.21)

where £total is given in (5.16). We note that, because of C^-and Cz_ -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 Ka-symmetries.

From £total (5.16), we see that, because of a2âzaz-term, the modified de Donder gauge for & > does not lead to decoupled equations for the ket-vector |&> when15 s > 2. It turns out that

15 For spin-1 field, gauge condition (5.20) and the corresponding decoupled equations of motion were found in [9].

in order to obtain decoupled equations of motion we should introduce our set of fields in (2.1). We remind that |№) is a double-traceless field (5.2) of the so(d, 1) algebra, while 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 ) to so(d — 1,1) ket-vector We find the following transformation from the ket-vector ) to our ket-vector

\0) =z LTd N n\*), n= n}1 + a2

2(2Na + d)

x nHH â2 +

2Na + d . 2Na + d - 2C

n,[1]= n[1](a, 0, Na,â, 0, d),

Kr '2NzT(Na + Nz + d-3)F(2Na + d - 3)\ 1/2

N = (-^-nr ,

Na + d-T)F(2Na + Nz + d - 3)

Na= aaâa

Nz = a z az

(5.22)

(5.23)

(5.24)

(5.25)

(5.26)

where T is Euler gamma function and operator n^ in (5.24) is obtained from the function

n[1](a, az, X,ä,äz, Y)

= y^ (a2 + az az

„ (-)"r(X + ^ + n) 4nn\F(X + Y-2 + 2n)

a 2 + a z a z)n, (5.27)

by equating az = az = 0, X = N a, Y = d. We introduce the z-factor in r.h.s. of (5.22) to obtain canonically normalized ket-vector Inverse transform of (5.22) takes the form

\*) =zV n*0N\0),

n*0 = nai]+ a2

2(2Na + d + 1)

x nal] à2 -

2Na + d - 1

ni] = n[1](a, az, Na,a, az, d + 1), Na = Na + Nz

(5.28)

(5.29)

(5.30)

where a2 is given in (5.3) and n^ is obtained from n[1] (5.27) by equating X = Na, Y = d + 1.

We now ready to compare modified de Donder gauges for (4.14) and |№) (5.20). Inserting (5.28) in (5.20) and choosing s = —1 in (2.17), we make sure that modified de Donder gauge for |$) (5.20) amounts to modified de Donder gauge for (4.14) i.e., modified de Donder gauges for (4.14) and |$) (5.20) match. Also we make sure that inserting (5.28) in Eq. (5.21) leads to Eq. (4.15), i.e., equations of motions for and №) match. Finally, one can make sure that gauge invariant Lagrangian for |№) (5.4) and the one for (2.7) match.

We now compare gauge transformation of the ket-vector (2.28) and gauge transformation of ) which takes the form

) =aD\i

(s - 1)!

'\0), (5.31)

where gauge transformation parameter E A1-..As-1 is traceless, EAAA3...As—1 = 0, i.e., a2|e) =0. To this end we note that gauge transformation parameters |£) and |E) are related as

\ç )=z N 'n[1]\£ N '= N \ Na ^Na + 1,

\S )=zd-3 n™ N '\H ),

(5.32)

(5.33)

where n[1\ , N are given in (5.24), (5.30), (5.25), respectively.

We note that n™ and n^1 are projectors on traceless ket-vectors,

A]...A

i.e., if |0trf) and \&trf) are tracefull ket-vectors, a2|0trf>=0

Vtrf ) = 0, -*2n m

then on has the relations a2Пa^|0trf) = 0, a2naiJ|&trf) = 0. Using (5.28), (5.32), and e = —1 in (2.17), we make sure that gauge transformations (2.28) and (5.31) match.

Finally we compare realization of so(d, 2) symmetries on the ket-vectors |0) and |&). To this end we note that on space of |&) realization of the so(d, 2) algebra transformations takes the form

Spa |&) = дa|&), Sjab |&) = (xaдb — xbda + Mab)|&), (5.34) Sd & )=xB d B &),

SKa) = ( - 2 XBXB da + xaxB dB + MaBXB )

■a„BnB

aaB„B

(5.35)

where xBxB = xbxb + z2, xBdB = xbdb + zdz, MaBxB = Mabxb — Mzaz. Comparing (3.5) and (5.34), we see that the realizations of Poincare symmetries on |0) and |&) match from the very beginning. Taking into account z-factor in (5.28), it is easily seen that D-transformations for |0) (3.6) and |&) (5.35) also match. All that remains to do is to match conformal boost Ka-transformations given in (3.7) and (5.35). Choosing e = —1 in (2.17), we make sure that realizations of the operator Ka on |0) (3.7) and on |&) (5.35) match.

To summarize, using the Poincare parametrization of AdS space, we have developed the CFT adapted formulation of massless arbitrary spin AdS field. In our approach, Poincare symmetries of the Lagrangian are manifest. As is well known string theory solutions like AdSd+1 x Sd+1 and Dp-brane backgrounds supported by RR-charges have the respective the d- and (p + 1)-dimensional Poincare symmetries. We note that the structure of the Lagrangian we obtained for AdS field is valid for any theory that respects Poincare 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 AdSd+1 x Sd+1 and Dp-brane backgrounds. For the case of AdSd+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 AdSd+1 x Sd+1 and Dp-brane backgrounds.16 The AdSd+1 x

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.

Acknowledgements

This work was supported by the RFBR Grant No.08-02-00963, RFBR Grant for Leading Scientific Schools, Grant No. 1615.2008.2, by the Dynasty Foundation and by the Alexander von Humboldt Foundation Grant PHYS0167.

Appendix A. Notation

Vector indices of the so(d — 1, 1) algebra take the values a, b, c = 0, 1,...,d — 1, while vector indices of the so(d, 1) algebra take the values A, B, C = 0,1,...,d — 1, d. We use mostly positive flat metric tensors nab, nAB. To simplify our expressions we drop nab, nAB in the respective scalar products, i.e., we use XaYa = nabXaYb, XAYA = nABXAYB. Using the identification Xd = Xz gives the following decomposition of the so(d, 1) algebra vector: XA = Xa, Xz. This implies XAYA = XaYa + XzYz.

16 It would also be interesting to unravel the ladder operators in the tension-less limit of AdS strings [36,37]. Also we think that formalism developed in this paper might be useful for the study of (A)dS massive fields [17] and (A)dS partial-massless fields [38-41].

We use the creation operators aa, az, and the respective annihilation operators aa, az,

[àa ,ab] = nab, [â z,az] = 1, â a|0) = 0, âz |0)=0.

These operators are referred to as oscillators in this Letter. The

oscillators aa, aa and az,

transform in the respective vector

and scalar representations of the so(d — 1 , 1 ) algebra and satisfy the hermitian conjugation rules, aa^ = aa, az = az. Oscillators aa, az and aa, az are collected into the respective so(d, 1) algebra oscillators aA = aa,az and aA = aa,az.

xA = xa, z denote coordinates in (d + 1)-dimensional AdSd+1 space,

ds2 = z2 (dxadxa + dzdz),

while dA = da,dz denote the respective derivatives, da = d/dxa, dz = d/dz. We use the notation = da da, ad = aada, a d = aa da, a2 = aaaa, ä2 = äaäa.

The covariant derivative DA is given by DA = nAB DB,

CAU ß,

D ß = dß + 2

MAB = a A a B — a B a A,

dp = d/dxwhere e^ is inverse vielbein of AdSd+1 space, D is the Lorentz covariant derivative and the base manifold index takes values p = 0,1,..., d. The is the Lorentz connection of AdSd+1 space, while MAB is a spin operator of the Lorentz algebra so(d, 1). Note that AdSd+1 coordinates xp carrying the base manifold indices are identified with coordinates xA carrying the flat vectors indices of the so(d, 1) algebra, i.e., we assume xp = SpxA, where Sp is Kronecker delta symbol. AdSd+1 space contravariant tensor field, &p1...ps, is related with field carrying the flat indices, &A1.As, in a standard way &A1".As = eP1 ...eApss&p1...ps. Helpful commutators are given by

[da, d^ = aabcdc - mab,

[a D, aD] = aAdS + 2 MABMAB,

where aABC = —wABC + û)bac is a contorsion tensor and we define wABC = eApwB,C.

For the Poincaré parametrization of AdSd+1 space, vielbein eA = ePdxp and Lorentz connection, deA + wAB A eB = 0, are given by

eA = 1SA

ß z A ,

= 1 (SA S B - SB S A).

With choice made in (A.5), the covariant derivative takes the form

Da = zdA + MzA, dA = nABdB.

References

[1] J.M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231, Int. J. Theor. Phys. 38 (1999) 1113, hep-th/9711200.

[2] X. Bekaert, S. Cnockaert, C. lazeolla, M.A. Vasiliev, hep-th/0503128.

[3] D. Sorokin, AIP Conf. Proc. 767 (2005) 172, hep-th/0405069.

[4] A. Fotopoulos, M. Tsulaia, arXiv: 0805.1346 [hep-th].

[5] R.R. Metsaev, A.A. Tseytlin, Nucl. Phys. B 533 (1998) 109, hep-th/9805028.

[6] R.R. Metsaev, Class. Quantum Grav. 18 (2001) 1245, hep-th/0012026.

[7] N. Beisert, R. Ricci, A. Tseytlin, M. Wolf, arXiv: 0807.3228 [hep-th].

[8] N. Berkovits, J. Maldacena, JHEP 0809 (2008) 062, arXiv: 0807.3196 [hep-th].

[9] R.R. Metsaev, Nucl. Phys. B 563 (1999) 295, hep-th/9906217.

[10] R.R. Metsaev, Phys. Rev. D 78 (2008) 106010, arXiv: 0805.3472 [hep-th].

[11] C. Fronsdal, Phys. Rev. D 20 (1979) 848.

[12] V.E. Lopatin, M.A. Vasiliev, Mod. Phys. Lett. A 3 (1988) 257.

[13] M.A. Vasiliev, Nucl. Phys. B 301 (1988) 26.

[14] J.M.F. Labastida, Nucl. Phys. B 322 (1989) 185.

[15] X. Bekaert, N. Boulanger, Commun. Math. Phys. 271 (2007) 723, hep-th/ 0606198.

[16] K. Hallowell, A. Waldron, Nucl. Phys. B 724 (2005) 453, hep-th/0505255.

[17] Yu.M. Zinoviev, hep-th/0108192.

[18] R.R. Metsaev, Phys. Lett. B 643 (2006) 205, hep-th/0609029.

[19] D. Francia, A. Sagnotti, Phys. Lett. B 543 (2002) 303, hep-th/0207002.

[20] A. Sagnotti, M. Tsulaia, Nucl. Phys. B 682 (2004) 83, hep-th/0311257.

[21] I.L Buchbinder, V.A. Krykhtin, Nucl. Phys. B 727 (2005) 537, hep-th/0505092.

[22] I.L. Buchbinder, V.A. Krykhtin, P.M. Lavrov, Nucl. Phys. B 762 ( 2007 ) 344, hep-th/0608005.

[23] J. Engquist, O. Hohm, Nucl. Phys. B 786 (2007) 1, arXiv: 0705.3714 [hep-th].

[24] I.L. Buchbinder, A.V. Galajinsky, V.A. Krykhtin, Nucl. Phys. B 779 (2007) 155, hep-th/0702161.

[25] K.B. Alkalaev, O.V. Shaynkman, M.A. Vasiliev, Nucl. Phys. B 692 (2004) 363, hep-th/0311164;

K.B. Alkalaev, O.V. Shaynkman, M.A. Vasiliev, hep-th/0601225.

[26] E.D. Skvortsov, JHEP 0807 (2008) 004, arXiv: 0801.2268 [hep-th]; E.D. Skvortsov, arXiv: 0807.0903 [hep-th].

[27] C. Iazeolla, P. Sundell, JHEP 0810 (2008) 022, arXiv: 0806.1942 [hep-th].

[28] S.J. Brodsky, G.F. de Teramond, arXiv: 0802.0514 [hep-ph].

[29] O. Andreev, Phys. Rev. D 67 (2003) 046001, hep-th/0209256.

[30] H.R. Grigoryan, A.V. Radyushkin, Phys. Lett. B 650 (2007) 421, hep-ph/0703069.

[31] R.R. Metsaev, arXiv: 0707.4437 [hep-th]; R.R. Metsaev, arXiv: 0709.4392 [hep-th].

[32] K.I. Bolotin, M.A. Vasiliev, Phys. Lett. B 479 (2000) 421, hep-th/0001031.

[33] S. Guttenberg, G. Savvidy, SIGMAP Bull. 4 (2008) 061; arXiv: 0804.0522 [hep-th].

[34] R. Manvelyan, K. Mkrtchyan, W. Ruhl, Nucl. Phys. B 803 (2008) 405, arXiv: 0804.1211 [hep-th].

[35] I.L. Buchbinder, A. Pashnev, M. Tsulaia, Phys. Lett. B 523 (2001) 338, hep-th/0109067.

[36] A.A. TseytIin, Theor. Math. Phys. 133 (2002) 1376, hep-th/0201112.

[37] G. BoneIIi, JHEP 0311 (2003) 028, hep-th/0309222.

[38] S. Deser, R.I. Nepomechie, Ann. Phys. 154 (1984) 396.

[39] S. Deser, A. WaIdron, NucI. Phys. B 662 (2003) 379, hep-th/0301068;

S. Deser, A. WaIdron, Phys. Rev. Lett. 87 (2001) 031601, hep-th/0102166.

[40] S. Deser, A. WaIdron, NucI. Phys. B 607 (2001) 577, hep-th/0103198; S. Deser, A. WaIdron, Phys. Lett. B 603 (2004) 30, hep-th/0408155.

[41] E.D. Skvortsov, M.A. Vasiliev, Nucl. Phys. B 756 (2006) 117, hep-th/0601095.