Scholarly article on topic 'CFT adapted gauge invariant formulation of massive arbitrary spin fields in AdS'

CFT adapted gauge invariant formulation of massive arbitrary spin fields in AdS Academic research paper on "Physical sciences"

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Abstract of research paper on Physical sciences, author of scientific article — R.R. Metsaev

Abstract Using Poincaré parametrization of AdS space, we study massive totally symmetric arbitrary spin 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 by using Stueckelberg formulation of massive fields. We demonstrate that the mass parameter, curvature and radial coordinate contributions to the gauge transformation and Lagrangian of the AdS massive fields can be expressed in terms of ladder operators. Three representations for the Lagrangian are discussed. 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. New simple representation for gauge invariant Lagrangian of massive ( A ) dS field in arbitrary coordinates is obtained. Light-cone gauge Lagrangian of massive AdS field is also presented.

Academic research paper on topic "CFT adapted gauge invariant formulation of massive arbitrary spin fields in AdS"

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Physics Letters B

www.elsevier.com/locate/physletb

CFT adapted gauge invariant formulation of massive arbitrary spin fields in AdS

R.R. Metsaev

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

ARTICLE INFO

ABSTRACT

Article history:

Received 14 July 2009

Received in revised form 28 October 2009

Accepted 16 November 2009

Available online 24 November 2009

Editor: L. Alvarez-Gaume

Using Poincare parametrization of AdS space, we study massive totally symmetric arbitrary spin 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 by using Stueckelberg formulation of massive fields. We demonstrate that the mass parameter, curvature and radial coordinate contributions to the gauge transformation and Lagrangian of the AdS massive fields can be expressed in terms of ladder operators. Three representations for the Lagrangian are discussed. 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. New simple representation for gauge invariant Lagrangian of massive (A)dS field in arbitrary coordinates is obtained. Light-cone gauge Lagrangian of massive AdS field is also presented.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Further progress in understanding AdS/CFT correspondence requires, among other things, better understanding of field dynamics in AdS space. Although many interesting approaches to AdS fields are known in the literature (for review see Refs. [1-3]), 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.2 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 Ref. [8], we developed a approach which is based on considering of AdS field dynamics in the Poincare coordinates and applied our approach to study of massless AdS fields. We think that our approach might be useful for study of AdS string massive modes. Therefore it is desirable to generalize our approach to the case of massive AdS fields. This is that what we do in this Letter. Namely, using the Poincare parametrization of AdS space we discuss massive totally symmetric arbitrary spin-s, s > 1, bosonic

E-mail address: metsaev@lpi.ru.

1 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.

2 Studying AdS5 x S5 superstring action [4] in Poincare parametrization may be

found in Ref. [5]. Recent interesting application of Poincare coordinates to studying AdS5 x S5 string T-duality may be found in Refs. [6,7].

0370-2693/$ - see front matter © 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.physletb.2009.11.037

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 massive arbitrary spin AdS field. The Lagrangian is explicitly invariant with respect to boundary Poincare symmetries, i.e., manifest symmetries of our Lagrangian are adapted to manifest symmetries of boundary CFT. We show that all the mass parameter, curvature and radial coordinate contributions to our Lagrangian and gauge transformation are entirely expressed in terms of ladder operators that depend on the mass parameter, radial coordinate and radial derivative. General structure of the Lagrangian we use is the same as the one for massless AdS fields. Lagrangian of massive AdS field is distinguished by appropriate ladder operators. We find two new concise expressions for the gauge invariant Lagrangian.

ii) We generalize modified de Donder gauge, found for mass-less AdS fields in Ref. [8], to the case of massive fields. As in the case of massless fields, the modified de Donder gauge leads to simple gauge fixed Lagrangian and decoupled equations of motion.3 Note that the standard de Donder gauge leads to coupled equations of motion whose solutions for s > 2 are not known in closed form even for massless AdS fields. In contrast to this, our modified de Donder gauge leads to simple decoupled equations which are easily solved in terms of the Bessel function.

3 Our modified de Donder gauge seems to be unique 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.

2. Lagrangian and its gauge and global symmetries

We begin with discussion of field content of our approach. In Ref. [10], the massive spin-s field propagating in AdSd+1 space is described by double-traceless so(d, 1) algebra totally symmetric tensor fields &Al ■'V, s' = 0, 1,..., s.4 These tensor fields can be decomposed in real-valued scalar, vector, and totally symmetric tensor fields of the so(d — 1,1) algebra:

,ai...as/

X = [s - s']2, s' = 0,1,..., s - 1, s.

Henceforth, the notation k = [n]2 implies that k = —n, —n + 2, —n + 4,...,n — 4, n — 2, n. To illustrate the field content given in

(2.1) we use shortcut for the field

fields in (2.1) can be represented as

ai...as/

and note that

'(s,0)

>(s-1,-1)

'(s-1,1)

<P(1,1-s)

$(1,3-s) ■■■ <P(1,s-3)

<P(1,s-1)

$(0,-s)

<P(0,2-s)

<P(0,s-2)

$(0,s).

The fields ^..."s' with s' > 4 are double-traceless,5

, aabba5...as'

= 0, X = [s - s'] 2, s' = 4, 5,...,s.

The fields in (2.1) subject to constraints (2.3) constitute a field content of our approach.

To simplify presentation we use creation operators aa, az, Z and the respective annihilation operators, aa, az, Z. Then, fields (2.1) can be collected into a ket-vector defined by6

s-s;+X s-s'-X

|$> = E E

az ' aai... aas a1...a, - / / i — '

From (2.4) we see that the ket-vector \0) is degree-s homogeneous polynomial in aa, az, Z.7 In terms of the ket-vector \0), double-tracelessness constraint (2.3) takes the form8

(a 2)2|0> = 0.

Action and Lagrangian we found take the form

S =1 ddxdzL, L = ^ (0| E |0>,

4 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 Poincaré parametrization of AdSd+1 space, ds2 = (dxadX + dzdz)/z2. We use the conventions: da = 3/3xa, dz = d/dz. Vectors of so(d, 1) algebra are decomposed as XA = (Xa, Xz).

5 Note that so(d - 1,1) tensorial components of the Fronsdal-Zinoviev fields @A1 ..As are not double-traceless. Using appropriate transformation (see (4.28)) those tensorial components can be transformed to our fields in (2.1).

6 We use oscillator formulation to handle the many indices appearing for tensor fields (for review see Refs. [12,13]). In a proper way, oscillators arise in the framework of world-line approach to higher-spin fields (see e.g. Refs. [14,15]).

7 Throughout this Letter we use the following notation for operators constructed out the oscillators and derivatives: Na = aaaa, Nz = azaz, Nz = zZ, a2 = aaaa, a2 = <Zaaa, □ = dada, ad = aada, âd = aada.

8 We adapt the formulation in terms of the double-traceless gauge fields [11]

(see also Refs. [10,16]). Discussion of various formulations in terms of unconstrained gauge fields may be found in Refs. [17-21]. Study of other interesting approaches which seem to be most suitable for the theory of interacting fields may be found e.g. in Refs. [22,23].

{<p\ = (|0))t. We now discuss various representations for operator E and the Lagrangian in turn.

1st representation. This representation is given by

E = E(2) + E (1) + E (0),

E(2) = □ - adZd + 1 (ad)2a2 + 2a2(<Z9)2 - 1 a2na2

- 1 a2adâ daz 2, 4 ,

E(1) = e 1A + e1A,

E(0) = m1 + a2 â 2m2 + rn3a2 + m3â2,

A = ad - a2â d + - a2 ada2, 4

J[ = Zd - adZ2 + 1 a2â3â2, 4

2s + d - 3 - 2Nz - 2Nz m1 = e1e1 - 2---- e1e 1,

1 11 2s + d - 4 - 2Nz - 2Nz

1 1 2s + d - 2Nz - 2Nz m2 = — e1e1 +----— e1e1,

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

m3 = ^ e1e1,

mm 3 = ^ e 1e1,

-e1 = Z rz ^ 1 + a zrzTv_ 1,

1 rz(Z z,

-e 1 = Tv+1 rz~Z+ 2

Tv = dz +-, v = K + Nz - Nz, k = E0 —,

- Nz)(k - s - ^ + Nz)(k + 1 + Nz)

(2.8) (2.9)

(2.10)

(2.11) (2.12)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

2(s + - Nz - Nz)(k + Nz - Nz)(k + Nz - Nz + 1)

(2.18)

(s + d—4 — Nz)(K + s + d—4 — Nz )(k — 1 — Nz) 2(s + d——4 — Nz — Nz)(K + Nz — Nz )(k + Nz — Nz — 1)

(2.19)

where subscript n in E(n) (2.7) tells us that E(n) is degree-n homogeneous polynomial in the flat derivative da. The following remarks are in order.

i) The parameter k (2.17) is expressed in terms of spin-s massive field lowest energy E0. Using result in Ref. [24] we can express k in terms of the standard mass parameter m,

k = ,im2 + I s +

(2.20)

ii) Operator E(2) (2.8) is the symmetrized Fronsdal operator represented in terms of the oscillators. This operator takes the same form as the one of massless field in d-dimensional flat space. Thus, the operator E (2.7) is given by the sum of the standard Fronsdal operator E(2) and new operators E(1), E(0) which depend on the mass parameter m, the radial coordinate and derivative, z, dz.

iii) Dependence of E (2.7) on the mass parameter m, the radial coordinate and derivative, z, dz, is entirely governed by the operators e1 and e 1 (2.16) which we will refer to as ladder operators.9

iv) Representation for the Lagrangian in (2.6)-(2.15) is universal and is valid for arbitrary Poincare invariant theory. Various

9 Interesting application of other ladder operators to studying AdS/QCD correspondence may be found in Ref. [25]. 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 Refs. [25,26].

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 (2.7). In other words, the operators E for massless and massive AdS fields, massive and conformal fields in flat space are distinguished only by the operators e1 and e 1. We note that it is finding the ladder operators that provides real difficulty. Expressions for e1, e1 appropriate for conformal and massive fields in flat space may be found in Refs. [27,28].

2nd representation for the operator E. Lagrangian can be presented in the form given in (2.6) with the following concise expression for the operator E:

s—1—s'+k s— 1— s - к

|*> = £ E

2 aai ... aas'

s'=0 k=[s —1—s']2 s'l^ ( s—1—/+k)!( s—I—^)!

ai...as'

(2.30)

We note that the ket-vector |£) is a degree-(s — 1) homogeneous polynomial in the oscillators aa, az, Z . In terms of the ket-vector ), tracelessness constraint (2.29) takes the form

a2 > =0.

(2.31)

Lagrangian (2.6) is invariant under the following gauge transformation:

E = ß( □ — M2V) — CC, M 3Z2 + v2 —

C = a d — 2 ada2 — e 1П [U] + 2 e1ä2, C = ad — 1 a2a д — e1n [U] + 1 e 1a2,

ß = 1 — 1 a2a2, 4

П [1-2] = 1 — a

2(2Na + d)

(2.21) (2.22)

(2.23)

(2.24)

(2.25)

$1ф> = G |S>, G = ad — ei — a2

2Na + d — 2

(2.32)

where v is given in (2.17). Operator E in (2.21) differs from the one in (2.7) by terms proportional to (a2)2 and (a2)2. Therefore, in view of double-tracelessness constraint (2.5), these two representations for E lead to the same Lagrangian (2.6). We note that operator E in (2.21) respects, in contrast to the one in (2.7), double-tracelessness constraint (2.5). For massless field in d-dimensional flat space, e1 = e 1 = 0, operator C (2.23) coincides with the standard de Donder operator. In terms of the ladder operators, mass operator M2 takes the form

2 2s + d — 2 — 2Nz — 2NZ M = —e 1e1 + ---—-——-—L e1e 1,

2s + d — 4 — 2Nz — 2N,

(2.26)

while the M\ (2.22) is obtained form (2.26) by using ladder operators given in (2.16).

CFT adapted representation of the Lagrangian. Taking into account representation for operator E in (2.21) and noticing the relations

where e1, e1 are given in (2.16). From (2.32), we see that the mass parameter, radial coordinate and derivative contributions to gauge transformation (2.32) are entirely expressed in terms of the ladder operators e1 and e1. We note that use of operator G (2.32) allows us to write new representation for the operator E entering Lagrangian (2.6),

E = fi(n — M2V — GC). (2.33)

Global so(d, 2) symmetries. Relativistic symmetries of AdSd+1 space are described by the so(d, 2) algebra. In our approach, the massive 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, conformal boost generators Ka, dilatation generator D, and generators Jab which span so(d — 1, 1) algebra. Normalization for commutators of the so(d, 2) algebra generators we use may be found in formulas (3.1)-(3.4) in Ref. [8].

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

mv = t

, Tv— 1, C = —CT, where Tv is given in (2.17), we see pa = da, Jab = xadb — xbda + M

that Lagrangian (2.6) can be represented as (up to total derivatives)

L = — 2 (da ф^\д (1ф) — 2 <TV— 1 1 ф> + 1 <C ф || CT ф>.

(2.27)

This form of the Lagrangian turns out to be very convenient for studying AdS/CFT correspondence.

Gauge symmetries. We now discuss gauge symmetries of Lagrangian (2.6). To this end we introduce the following set of gauge transformation parameters:

D = xd + A, A = zdz + , Ka = -2 x2da + xaD + Mabxb + Ra,

xd = xada, x2 = xaxa. In (2.34), (2.36), Mab is spin operator of the so(d — 1,1) algebra. Representation of Mab and operator Ra (2.36) on space of ket-vector (2.4) takes the form

(2.34)

(2.35)

(2.36)

Mab = aa ab — ab aa,

a1 ...as' к

, к = [s — 1 — s'] 2, s' = 0,1,..., s — 1.

(2.28)

Ra = zIa(rzÇ + rz az) — z (f rz + azrz) aa — z2da

The gauge parameters , , and ^ . , s > 2 in (2.28), are the respective scalar, vector, and rank-s' totally symmetric tensor fields of the so(d — 1, 1) algebra. The gauge parameters ^ . "j' with s' > 2 are subjected to the tracelessness constraint,

^aaa3...as, = 0, k = — 1 — s']2, s'= 2, 3,...,s — 1. (2.29)

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

Ia = aa — a2

2Na + d — 2

(2.37)

(2.38)

(2.39)

where rz, rz are given in (2.18), (2.19). We see that realization of Poincare symmetries on bulk AdS fields (2.34) coincide with realization of Poincare symmetries on boundary CFT operators. Note that realization of D- and Ka-symmetries on bulk AdS fields (2.35), (2.36) 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 given in (2.34)-(2.36) turns out to be very convenient for studying AdS/CFT correspondence [28].

3. Modified de Donder gauge

To discuss modified de Donder gauge we use representation for Lagrangian given in (2.6), (2.21). It is easy to see that use of the following modified de Donder gauge-fixing term

£g.nx = 2 <Ф\сс\Ф),

leads to the surprisingly simple gauge fixed Lagrangian Aotai,

Ltotal = L + Lg.flx,

Using the modified de Donder gauge condition in gauge invariant equations of motion (3.6) leads to the following gauge fixed equations of motion:

(□- M2V)\ф) = 0,

where M\ is defined in (2.22). In terms of fields (2.1), Eqs. (3.8) can be represented as

к = [s - s'] 2, s' = 0,1,...,s,

□k +x$Xras'= 0,

where aK+X is given in (3.5). Thus, our modified de Donder gauge condition (3.7) leads to decoupled equations of motion (3.9) which can easily be solved in terms of the Bessel function.11 For spin-1 field, gauge condition (3.7) turns out to be a modification of the Lorentz gauge.

L total = 2 Ф\Е total\Ф), Etotal = (1 - 42

)(□ - Ml)

where M\ is given in (2.22). We note that our gauge-fixing term (3.1) respects the Poincare and dilatation symmetries but breaks the conformal boost Ka-symmetries, i.e., the simple form of gauge fixed Lagrangian (3.2) is achieved at the cost of the Ka-symmetries. In terms of tensorial components, gauge fixed Lagrangian (3.2) takes the form

L total = E E A',^ s'=0 k=[s-s']2

¿иФк

s'(s' - 1) Aaaa3...as

LS',x =

ai ...as , ■

Фк s иК+кФ

a1 ...as' к

'□ К +кФ,

bba3 ...as к

□ к+к = □ + д2 - -If (К + к)2 - 1

We see that the modified de Donder gauge fixing leads to simple gauge fixed Lagrangian.

We now discuss gauge-fixing procedure at the level of equations of motion. Representation for the operator E in (2.33) turns out to be convenient for this purpose.10 This is to say that La-grangian with E in (2.33) leads to the following gauge invariant equations of motion

(□ - m2v - gc)\Ф) = о. (3.6)

Modified de Donder gauge condition is then defined to be

С\Ф) = 0, (3.7)

where С is given in (2.23). The fact that this gauge is accessible with gauge transformation (2.32) can be proved as follows. i) By virtue of (2.5), we have the relation а2С\Ф) = 0 which implies that gauge condition (3.7) respects constraint for gauge transformation parameter \%), (2.31); ii) Gauge variation of С\Ф) is given by 5(С\Ф)) = (□ - MV)\%). Making standard assumption that the operator □ - M2V is invertible, we see that gauge condition (3.7) is indeed accessible.

10 Operators E in (2.21), (2.33) respect, in contrast to operator E in (2.7), double-tracelessness constraint (2.5).

4. Comparison of standard and modified de Donder gauges

Our approach to the massive spin-s field in AdS^+1 is based on use of double-traceless so(d — 1, 1) algebra fields (2.1). One of popular approaches to the massive spin-s field in AdSd+1 is based on use of double-traceless so(d, 1) algebra fields ФA1"As, s' = 0, 1,...,s, [10]. In this section, our aims are as follows. i) Using the fields ФAb"As and arbitrary parametrization of (A)dS space, we find new representation for gauge invariant Lagrangian of massive (A)dS field and standard de Donder gauge condition.12 ii) We explain how our modified de Donder gauge (3.7) is represented in terms of the fields ФA1".As'. iii) We show explicitly how our fields (2.1) are related to the fields ФA1^As.

New representation for gauge invariant Lagrangian of massive field in (A)dSd+1. We begin with discussion of gauge invariant Lagrangian using arbitrary coordinates of (A)dS. To simplify the presentation we introduce ket-vector |Ф),

z s-s' a A1 \ф> = £ Z

a A1 ... a As'

s'lVCs-s7)!

A1... As,

(öi2) \Ф) =0,

a2 = aAaA, a2 = âAâA, where (4.2) tells us that the ФA1".As are double-traceless. We find the following concise expression for gauge invariant Lagrangian of massive spin-s field in (A)dSd+1 :

L = 2\E\Ф), (4.3)

E = - 4a2ct2 j (□(A)ds + m1 + pa2ct2) - CstCst, (4.4)

m1 = -m2 + p(s(s + d - 5) - 2d + 4 + Nz(2s + d - 1 - Nz)),

Cst = aD - -aDa2 - e 1П[1,2] ^e1a

Cst = aD - -a2aD - e1n[1,2] + -e 1a

11 Interesting method of solving AdS field equations of motion which is based on star algebra products in auxiliary spinor variables is discussed in Refs. [29,30]. As a side of remark we note that our modified de Donder gauge can be generalized to conformal flat spaces (see Appendix D in Ref. [28]).

12 To our knowledge, the standard de Donder gauge for arbitrary spin massive (A)dS fields has not been discussed in earlier literature. Study of the standard de Donder gauge for flat arbitrary spin massive fields may be found in Ref. [28]. Recent applications of the standard de Donder gauge to the various problems of massless fields may be found in Refs. [31,32].

n[1'2] = 1 -

2(2Na + d + 1)

Na = a A a

ei = Z ei, e 1 =-ëif,

2s + d - 3 - N,

2s + d - 3 - 2N,

I (m2 - pNZ(2s + d - 4 - NZ))

,-, _ nAnA I ,,AABn

□(A)dS = D D + M D

.AAB nB

a D = a aDa ,

aD = a aDa ,

(4.10)

(4.11)

where e = det e^, eA stands for vielbein of (A)dSd+1 space, and DA is covariant derivative (for details of notation, see Appendix A). We use p = —1 for AdS space, p = 0 for flat space, and p = 1 for dS space. It is the use of operators Cst, Cst (4.6), (4.7) that allows us to write down the concise expression for operator E in (4.4). For massless field in (A)dS, e1 = e 1 = 0, our operator Cst (4.6) coincides with the standard de Donder operator in (A)dS background.

Lagrangian (4.3) is invariant under gauge transformation

5№}=G|S}, G = aD - e1 -

2s + d - 5 - 2N,

— Zs-1-s'aA.

s'V(s - 1 - s')!

7A1... As,

(4.12)

(4.13)

where gauge transformation parameters

EAAA3 -As' = 0, i.e., a2|E) =0. Also we note that Lagrangian (4.3) can alternatively be represented as

L = 1 e<№ 1 — 1 a2 a A E №),

(4.14)

£ = □( A)ds + m1 + p a2 a2 - GCst. (4.15)

Using (4.10) and denoting eigenvalues of by k, we find the critical values of the mass parameter, m^ = pk(2s + d - 4 - k), k = 0, 1,..., s - 1. The case k = 0 corresponds to massless field, while k = 1, 2,...,s - 1 correspond to the partial massless fields [33,34] (see also [10,16,35]).

Standard de Donder gauge. We proceed with discussion of standard de Donder gauge for (A)dS massive field. Representation for Lagrangian given in (4.3), (4.4) is well adopted for this purpose. This is to say that use of the following de Donder gauge-fixing term

Lg.fix = ^ e№|CstC st№},

(4.16)

leads to de Donder gauge fixed Lagrangian Ltotal, Aotal = L + Lg.fix,

L total = - e{№ |( 1 - t a2a2 )£total№},

(4.17)

Etotal = □( A)dS + m1 + p a2ci2. (4.18)

Note that Lagrangian (4.14) leads to the following gauge invariant equations of motion E|№) = 0, where E is given in (4.15). It easy to see that imposing standard de Donder gauge Cst ) = 0 we obtain gauge fixed equations of motion Etotal ) = 0, where Etotal is given in (4.18).

Modified de Donder gauge. We now discuss modified de Donder gauge. From now on we consider fields in AdS, i.e. we set p = —1, and use Poincare parametrization of AdS. The modified de Donder gauge fixing is defined to be

Lg.fix = ^|CC|№),

C = Cst - 2CZ

z z 1 2

C^ = az - - a a

C = C st + 2C^,

2™ z

Cz = az - -aza

(4.19)

(4.20)

(4.21)

We now make sure that gauge fixed Lagrangian Ltotal, Ltotal = L + Lg.fix, takes the form

Ltotal = -e{№^ 1 -7a a2 £total№}

(4.22)

£total = □ 0 AdS - m2 - a2âz az - Is +

- NZ(2s + d - 2 + 2Nz - NZ) + — + 2Ize 1 - 2e1az,

□0Ads = z2(n + 92) + (1 - d)zdz,

Iz = az - a2

2Na + d - 1

(4.23)

(4.24)

We proceed with discussion of gauge-fixing procedure at the level of equations of motion. To this end we note that gauge invariant Lagrangian (4.14) leads to the following equations of motion:

£ |№} = 0,

(4.25)

where £ is given in (4.15). We now define modified de Donder gauge conditions as

are traceless, C|№} = 0,

(4.26)

where C is given in (4.20). Using (4.26) in (4.25) we get gauge fixed equations of motion

£total№}=0,

(4.27)

where £total is given in (4.23). 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 (4.23), we see that, because of terms like a2azaz, Ize 1, and e1câz the modified de Donder gauge itself does not lead automatically to decoupled gauge fixed equations for the ket-vector |№}. It turns out that in order to obtain decoupled gauge fixed equations of motion we should introduce our fields in (2.1). We remind that |№} is a double-traceless field (4.2) of the so(d, 1) algebra, while |0} describes double-traceless fields (2.5) 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 |0}. We find a transformation from the ket-vector |№} to our ket-vector |0} and the corresponding inverse transformation,

|0} =z V V Wn^ №}, № }=zkT V |0},

(4.28)

(4.29)

where V is unitary operator, V t V = 1, and we introduce the z-factor in r.h.s. of (4.28) to obtain canonically normalized ket-vector |0). Operators n^, n№, N, and V are defined in Appendix A.

We now ready to compare modified de Donder gauges for |0) (3.7) and ) (4.26). Inserting (4.29) in (4.26), we make sure that modified de Donder gauge for ) (4.26) amounts to one for |0) (3.7) i.e., modified de Donder gauges for |0) (3.7) and ) (4.26) match. Also one can make sure that gauge invariant Lagrangian for |№) (4.3) and the one for |0) (2.6) match.

We now compare gauge transformation of |0) (2.32) and gauge transformation of ) given in (4.12). To this end we note that gauge transformation parameters |£) and |E) are related as

A1...A

If ) = z ¥ v N 'na\ N '- N N +i,

?), |S )=z¥ V ),

(4.30) (4.3 1)

xA = xa, z denote coordinates in d + 1-dimensional AdSj+1 space,

where П™, П1 are defined in Appendix A. Using (4.29), (4.30), ds2 = (dX dxa + dzdz),

we make sure that gauge transformations (2.32) and (4.12) 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

5Pa |ф) = da |Ф),

5 ;ab |ф )= (xa3b - xb дa + Mah )|Ф), SD |Ф) =xB дB |Ф ),

SKa | ф ) = Г- 1XBXB da + XaXB дB + MaBXB ) | Ф ) ,

(4.32)

(4.33)

where xBxB = xbxb + z2, xBдB = xbdb + zdz, MaBxB = Mabxb -Mzaz. Comparing (2.34) and (4.32), we see that the realizations of Poincare symmetries on \ф) and |Ф) match from the very beginning. Taking into account z-factor in (4.29), it is easily seen that D-transformations for \ф) (2.35) and \Ф) (4.33) match. After this, we make sure that realizations of the operator Ka on \ф) (2.36) and on \Ф) (4.33) also match.

Light-cone Lagrangian. Using gauge invariant action (2.6) and imposing light-cone gauge, we find the light-cone Lagrangian

Ac. = £ E

s'=0 A.=[s—s']2

'□к+хФх'

(4.34)

where □ K+k is defined in (3.5) and transverse indices take values i = 1, 2,...,d — 2. As usually, the light-cone fields 0'x' Js' = 0 are

traceless, ^k"3"^ = 0.

To summarize, using the Poincare parametrization of AdS space, we have developed the CFT adapted formulation of massive totally symmetric arbitrary spin AdS field. In recent years, mixed symmetry fields have attracted considerable interest (see e.g. Refs. [3640]). We think that generalization of our approach to the case of mixed symmetry massless and massive AdS fields might be useful for study of dynamical aspects of such fields. In this respect, it would be interesting to find generalization of the modified de Donder gauge to the case of mixed symmetry fields.

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 =

ПAB = (nab, 1). Using the identifi-

nabXaYb, XaYa = nabXaYb

cation Xd = Xz gives the following decomposition of the so(d, 1 ) algebra vector: XA = Xa, Xz. This implies XAYA = XaYa + XzYz.

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

[a a,ab] = n

[az, az

] = 1, [?,?] = 1,

âa |0)=0, â z|0) = 0, Z |0)=0.

These operators are referred to as oscillators in this Letter. The oscillators aa, aa and az, Z, «z, Z, transform in the respective vector and scalar representations of the so(d — 1 , 1 ) algebra and satisfy the hermitian conjugation rules, aa = aa, azt = az, Z^ = Z. Oscillators aa, az and aa, az are collected into the respective so(d, 1) algebra oscillators aA = aa, az and aA = aa, az.

while dA = da,dz denote the respective derivatives, da = d/dxa, dz = d/dz. We use the notation □ = dada, ad = aada, ad = aada, a2 = aaaa, a2 = aaaa, Na = aaoia, Nz = azoiz, Nz = ZZ. The co-variant derivative DA is given by DA = nABDB,

Da - e%d p nab

Dp - д„ + 2o4BMAB,

MAB - aAâB — aBàA,

d^ = d/dx¡, where el is inverse vielbein of AdSd+1 space, D/x is the Lorentz covariant derivative and the base manifold index takes values i = 0,1,..., d. The a'B 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 xl 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 xl = S^xA, where 81 is Kronecker delta symbol. AdSd+1 space contravariant tensor field, , is related with field carrying the flat indices,

&A1...As, in a standard way &A1".As = e^ ...el. Helpful commutators involving the covariant derivative DA and the oscillators aA, aA may be found in Appendix in Ref. [8].

For the Poincare parametrization of AdSd+1 space, vielbein eA =

el dxl, Lorentz connection aAB = alB dxl, and aABC = eAlaB.C

fj, fj, fj,

are given by

0A 1 о A

ep = zш

. .ABC AC eB AB eC

ш = n 5z — n 5z.

= z 5 5B — s A)

With choice made in (A.4), the covariant derivative takes the form da = zdA + MzA, dA = nABdB.

The operators n^, used in Section 4 are defined by relations

Пфф - + a2-!-я^Ч a2 +

a 2(2Na + d) ^ 2Na + d — 2

2Na + d _z_z

na1]- П[1](a, 0, Na,a, 0, d) -

2Nz r(Na + Nz + d——3 )r(2Na + d — 3) X 1/2

r(Na + d—3 )r(2Na + Nz + d — 3)

Пфф - П1]+ a2

x la2 —

2(2Na + d + 1) а 2

2Na + d — 1

П1] - П[1](a,az, Na,a,az, d + 1), П[1] (a,az, X,a,az, Y)

"" )n (—)nГ(X + iy2 + n)

(A.5) (A.6)

(A.8) (A.9)

^ (a2 +<

4пп!Г( X + ■Y—2 + 2n)

(a2 + a zaz)

(A.10)

where Na = Na + Nz, a2 = a2 + azaz, and r is Euler gamma function. We note that the n¿1] in (A.6) is obtained from (A.10) by equating az = az = 0, X = Na, Y = d, while n™ in (A.9) is obtained from (A.10) by equating X = Na, Y = d + 1.

The operator V used in Section 4 is defined by relations

V = VV (N =

l,n=0,1,...,N

VN = v(yN-'zl!0><0|(zN-nzn, v(N = nnxn,

(-)t (k - N + l)t t!(l - t)!(k + 1)t'

(A.11) (A.12) (A.13)

min n,t

Xnt = E

(n + 1 - p)p (t + 1 - p)p (2s + d - 2 - n - p)p p ! (s + - p)p (k - s - )p

(A.14)

n(N ) = ■

(-)n (k - N + 2l\ 1/2/2s + d - 3 - 2n\

n!(N - n)!\ k J V 2s + d - 3 - nj (s + - Pi(s + - N + l)N-i \ 1/2 (2s + d - 3 - n - N)n

(A.15)

where in (A.13)-(A.15) we use the notation (a)b for the Pochhammer symbol, (a)b = ^^.

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(k - s - !- )n (K - s - !- )l(K + s + i-2 - N + l)N-l (K + 1)l

(K + s + V - n)n(K - N + l)N_l