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Nuclear Physics B 885 (2014) 734-771

www. elsevier. com/locate/nuclphysb

Arbitrary spin conformal fields in (A)dS

R.R. Metsaev

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

Received 10 June 2014; accepted 12 June 2014 Available online 18 June 2014 Editor: Stephan Stieberger

Abstract

Totally symmetric arbitrary spin conformal fields in (A)dS space of even dimension greater than or equal to four are studied. Ordinary-derivative and gauge invariant Lagrangian formulation for such fields is obtained. Gauge symmetries are realized by using auxiliary fields and Stueckelberg fields. We demonstrate that Lagrangian of conformal field is decomposed into a sum of gauge invariant Lagrangians for massless, partial-massless, and massive fields. We obtain a mass spectrum of the partial-massless and massive fields and confirm the conjecture about the mass spectrum made in the earlier literature. In contrast to conformal fields in flat space, the kinetic terms of conformal fields in (A)dS space turn out to be diagonal with respect to fields entering the Lagrangian. Explicit form of conformal transformation which maps conformal field in flat space to conformal field in (A)dS space is obtained. Covariant Lorentz-like and de-Donder like gauge conditions leading to simple gauge-fixed Lagrangian of conformal fields are proposed. Using such gauge-fixed Lagrangian, which is invariant under global BRST transformations, we explain how the partition function of conformal field is obtained in the framework of our approach. © 2014 Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

1. Introduction

In view of aesthetic features of conformal symmetries, conformal field theories have attracted considerable interest during long period of time (see e.g., Ref. [1]). One of characteristic features of conformal fields propagating in space-time of dimension greater than or equal to four is that Lagrangian formulations of most conformal fields involve higher derivatives. Often, higherderivative kinetic terms entering Lagrangian formulations of conformal fields make the treatment

E-mail address: metsaev@lpi.ru. http://dx.doi.Org/10.1016/j.nuclphysb.2014.06.013

0550-3213/© 2014 Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.Org/licenses/by/3.0/). Funded by SCOAP3.

of conformai field theories cumbersome. In Refs. [2,3], we developed ordinary-derivative La-grangian formulation of conformai fields. Attractive feature of the ordinary-derivative approach is that the kinetic terms entering Lagrangian formulation of conformal fields turn out to be conventional well known kinetic terms. This is to say that, for spin-0, spin-1, and spin-2 conformal fields, the kinetic terms in our approach turn out to be the respective Klein-Gordon, Maxwell, and Einstein-Hilbert kinetic terms. For the case of higher-spin conformal fields, the appropriate kinetic terms turn out to be Fronsdal kinetic terms. Appearance of the standard kinetic terms makes the treatment of the conformal fields easier and we believe that use of the ordinary-derivative approach leads to better understanding of conformal fields.

In Refs. [2,3], we dealt with conformal fields propagating in flat space. Although, in our approach, the kinetic terms of conformal fields turn out to be conventional two-derivative kinetic terms, unfortunately, those kinetic terms are not diagonal with respect to fields entering a field content of our ordinary-derivative Lagrangian formulation. On the other hand, in Refs. [4,5], it was noted that, for the case of conformal graviton field in (A)dS4 space, the four-derivative Weyl kinetic operator is factorized into product of two ordinary-derivative operators. One of the ordinary-derivative operators turns out to be the standard two-derivative kinetic operator for massless transverse graviton field, while the remaining ordinary-derivative operator turns out be, as noted in Refs. [6,7], two-derivative kinetic operator for spin-2 partial-massless field. This remarkable factorization property of the four-derivative operator for the conformal graviton field can also be realized at the level of Lagrangian formulation. Namely, in Ref. [8], it was noted that, by using appropriate field redefinitions, the ordinary-derivative Lagrangian of the conformal graviton field in (A)dS4 can be presented as a sum of Lagrangians for spin-2 massless field and spin-2 partial-massless field.

Recently, in Ref. [9], these results were considered in the context of higher-spin conformal fields.1 Namely, in Ref. [9], it was conjectured that higher-derivative kinetic operator of arbitrary spin-s conformal field propagating in (A)dSd+1 space can be factorized into product of ordinary-derivative kinetic operators of massless, partial-massless, and massive fields.2 Note that the partial-massless fields appear when s > 1, while the massive fields appear when d > 3. This conjecture suggests that ordinary-derivative Lagrangian of conformal field in (A)dS can be represented as a sum of ordinary-derivative Lagrangians for appropriate massless, partial-massless, and massive fields. In this paper, among other things, we confirm the conjecture in Ref. [9]. Namely, for arbitrary spin-s conformal field propagating in (A)dS, we find the ordinary-derivative gauge invariant Lagrangian which is a sum of ordinary-derivative and gauge invariant Lagrangians for spin-s massless, spin-s partial-massless, and spin-s massive fields. To obtain ordinary-derivative Lagrangian of conformal field in (A)dS, we start with our Lagrangian of conformal field in flat space obtained in Ref. [3]. Applying conformal transformation to conformal field in flat space, we obtain ordinary-derivative Lagrangian of conformal field in (A)dS. We note also that, in contrast to conformal fields in flat space, the kinetic terms of conformal fields in (A)dS space turn out to be diagonal with respect to fields entering a field content of our ordinary-derivative Lagrangian formulation.

This paper is organized as follows.

1 Up-to-date reviews of higher-spin field theories may be found in Ref. [10].

2 Discussion of factorized form of higher-derivative actions for higher-spin fields may be found in Ref. [11].

In Section 2, we start with the simplest example of spin-0 conformal field in (A)dSd+1, d-arbitrary. For this example, we briefly discuss some characteristic features of ordinary-derivative approach.

In Section 3, we study the simplest example of conformal gauge field which is spin-1 con-formal field in (A)dS6. We demonstrate that ordinary-derivative Lagrangian of spin-1 conformal field in (A)dS6 is a sum of Lagrangians for spin-1 massless field and spin-1 massive field. We note that, for the case of spin-1 conformal field, there are no partial-massless fields. For completeness, we also present our results for spin-1 conformal field in (A)dSd+1 for arbitrary odd d.

In Section 4, we deal with spin-2 field. We start with the most popular example of spin-2 conformal field in (A)dS4. For this case, Lagrangian is presented as a sum of gauge invariant Lagrangians for spin-2 massless field and spin-2 partial-massless field. Novelty of our discussion, as compared to the studies in earlier literature, is that we use a formulation involving the Stueckelberg vector field. After this we proceed with discussion of other interesting example of spin-2 conformal field in (A)dS6. For this case, Lagrangian is presented as a sum of gauge invariant Lagrangians for spin-2 massless, spin-2 partial-massless, and spin-2 massive fields. Also we extend our consideration to the case of spin-2 conformal field in (A)dSd+i for arbitrary odd d.

In Section 5, we discuss arbitrary spin conformal field in (A)dSd+1, for arbitrary odd d. We demonstrate that ordinary-derivative and gauge invariant Lagrangian of conformal field in (A)dS can be presented as a sum of gauge invariant Lagrangians for massless, partial-massless, and massive fields. We propose de Donder-like gauge condition which considerably simplifies the Lagrangian of conformal field. Using such gauge condition, we introduce gauge-fixed Lagrangian which is invariant under global BRST transformations and present our derivation of the partition function of conformal field obtained in Ref. [9].

In Section 6, we demonstrate how a knowledge of gauge transformation allows us to find gauge invariant Lagrangian for (A)dS field in a straightforward way.

In Section 7, we review ordinary-derivative approach to conformal fields in flat space.

Section 8 is devoted to the derivation of Lagrangian for conformal fields in (A)dS. For scalar conformal field, using the formulation in flat space and applying conformal transformation which maps conformal field in flat space to conformal field in (A)dS, we obtain Lagrangian of scalar conformal field in (A)dS. For arbitrary spin conformal field, using gauge transformation rule of the conformal field in flat space and applying conformal transformation which maps the conformal field in flat space to conformal field in (A)dS we obtain a gauge transformation rule of the arbitrary spin conformal field in (A)dS. Using then the gauge transformation rule of the arbitrary spin conformal field in (A)dS and our result in Section 6, we find the ordinary-derivative gauge invariant Lagrangian of arbitrary spin conformal field in (A)dS.

Our notation and conventions are collected in Appendix A.

2. Spin-0 conformai field in (A)dS

In ordinary-derivative approach, spin-0 conformal field is described by k + 1 scalar fields

, k = 0, 1,___,k, k-arbitrary positive integer. (2.1)

Fields are scalar fields of the Lorentz algebra so(d, 1). Lagrangian we found takes the form

Lv = 1 e0t(V2 - m20, (2.3)

mk< = p(J: - (k +1 -k') )' (k.4)

where e = det eA, eA stands for vielbein of (A)dS, while V2 stands for the D'Alembert operator of (A)dS space. We use p = e/R2, where e = 1(-1) for dS (AdS) and R is radius of (A)dS. For notation, see Appendix A. From (k.k), we see that Lagrangian of spin-0 conformal field is the sum of Lagrangians for scalar fields having square of mass parameters given in (2.4). The following remarks are in order.

i) From field content in (2.1), we see that, in our ordinary-derivative approach, the spin-0 conformal field is described by k + 1 scalar fields and the corresponding Lagrangian involves two derivatives. We recall that, in the framework of higher-derivative approach spin-0 conformal field is described by single field, while the corresponding Lagrangian involves 2k + 2 derivatives. For the illustration purposes, let us demonstrate how our approach is related to the standard higher-derivative approach. To this end consider a simplest case of higher-derivative Lagrangian for spin-0 conformal field in (A)dS4 with k = 1 (see Refs. [4,12]),

1L = 10(V2)20 - p0V20. (2.5)

Introducing an auxiliary field 01, Lagrangian (2.5) can be represented in ordinary-derivative form as

1L = J2\P\0V201 - p0V20 -|p 102. (2.6)

Using, in place of the field 0, a new field 00 defined by the relation

0 =-¡=((00 + 01), (2.7)

it is easy to check that Lagrangian (2.6) takes the form

L =-eL0 + eL1, (2.8)

-L0 = -00V 00, (2.9)

-L1 = -01V 01 - p0101. (2.10)

Plugging the values k = 1 and d = 3 in (2.2), we see that our ordinary-derivative Lagrangian for these particular values of k and d coincides with the one in (2.8)-(2.10).

ii) In the framework of higher-derivative approach, conformally invariant operator in Sd+1, d-arbitrary, which involves 2k + 2 derivatives, was found in Ref. [13]. Our values for square of mass parameter in (2.4) coincide with the ones in Ref. [13]. We note that it is use of the field content in (2.1) that allows us to find Lagrangian formulation in terms of the standard second-order D'Alembert operator. To our knowledge, for arbitrary k, ordinary-derivative Lagrangian (2.2) has not been discussed in the earlier literature.

3. Spin-1 conformal field in (A)dS

We now discuss a spin-1 conformal field in (A)dS. A spin-1 conformal field in (A)dS4 is described by the Maxwell theory which is well-known and therefore is not considered in this

paper. In (A)dSd+1 with d > 3, Lagrangian of spin-1 conformal field involves higher derivatives. Ordinary-derivative Lagrangian formulation of spin-1 conformal field in R1,1, d > 3, was developed in Ref. [2]. Our purpose in this section is to develop a ordinary-derivative Lagrangian formulation of spin-1 conformal field in (A)dSd+1, d > 3. Because spin-1 conformal field in (A)dS6 is the simplest example allowing us to demonstrate many characteristic features of our ordinary-derivative approach we start our discussion with the presentation of our result for spin-1 conformal field in (A)dS6.

3.1. Spin-1 conformal field in (A)dS6

Field content. To discuss ordinary-derivative and gauge invariant formulation of spin-1 conformal field in (A)dS6 we use two vector fields denoted by fit, fit and one scalar field denoted by fi1,

fi0t fit

The vector fields fit, fit and the scalar field fi1 transform in the respective vector and scalar representations of the Lorentz algebra so(5, 1). Now we are going to demonstrate that the vector field fit enters description of spin-1 massless field, while the vector field fit and the scalar field fi1 enter Stueckelberg description of spin-1 massive field. To this end we consider Lagrangian and gauge transformations.

Gauge invariant Lagrangian. Lagrangian we found can be presented as

£ =-e £0 + €Ù1, (3.2)

£0 = L0, (3.3)

£1 = £ + £ (3.4) where we use the notation

1 £1 = 1 fitD2 - 5p)fit + 1L0L0, (3.5)

1 £1 = 1 fif(V2 - m21 - 5p)fif + 1L1L1, (3.6) e 2 2

!£0 = 2fi1(D2 - m2)fi1, (3.7)

L0 = VBfiB, (3.8)

L1 = VBfif + |m1|fi1, (3.9)

m1 = 2p. (3.10)

From (3.3), we see that Lagrangian £ is formulated in terms of the vector field , while the Lagrangian £1 is formulated in terms of the vector field and the scalar field

Gauge transformations. We now discuss gauge symmetries of the Lagrangian given in (3.2). To this end we introduce the following gauge transformation parameters:

£0, £1.

(3.11)

The gauge transformation parameters in (3.11) are scalar fields of the Lorentz algebra so(5, 1). We note the following gauge transformations:

80$ = VAÇo, (3.12)

80$ = VA^1, (3.13)

801 = -e\m1\^1. (3.14)

The following remarks are in order.

i) Lagrangian L0 in (3.3) is invariant under f0 gauge transformations given in (3.12), while the Lagrangian L1 in (3.4) is invariant under f1 gauge transformations given in (3.13), (3.14). This implies that the Lagrangian L0 describes spin-1 massless field, while the Lagrangian L1 describes spin-1 massive field having square of mass parameter m1 given in (3.10).

ii) From (3.14), we see that the scalar field transforms as a Stueckelberg field. In other words, the scalar field is realized as Stueckelberg field in our description of spin-1 conformal field.

iii) Taking into account signs of the kinetic terms in (3.2) it is clear that Lagrangian (3.2) describes fields related to non-unitary representation of the conformal algebra.3

Summary. Lagrangian of spin-1 conformal field in (A)dS6 given in (3.2) is a sum of Lagrangian L0 (3.3) which describes dynamics of spin-1 massless field and Lagrangian L1 (3.4) which describes dynamics of spin-1 massive field.

Lorentz-like gauge. Representation for Lagrangians in (3.5), (3.6) motivates us to introduce gauge condition which we refer to as Lorentz-like gauge,

L0 = 0, L1 = 0, Lorentz-like gauge. (3.15)

3.2. Spin-1 conformal field in (A)dSd+1

To discuss ordinary-derivative and gauge invariant approach to spin-1 conformal field in (A)dSd+1, for arbitrary odd d > 5, we use k + 1 vector fields denoted by 0$, and k scalar fields denoted by 0k<,

0A, k' = 0, 1,...,k,

k = d—. (3.16)

0V, k' = 1, 2,...,k,

For the illustration purposes it is helpful to represent field content in (3.16) as follows.

Field content of spin-1 conformal field in (A)dSd+1 for arbitrary odd d > 5, k = (d — 3)/2

< ...... 1

(3.17)

01 02 ......0k—1 0k

The vector fields 0A and the scalars fields 0k (3.16) transform in the respective vector and scalar representations of the Lorentz algebra so(d, 1). Our purpose is to demonstrate that the vector field 0AA enters description of spin-1 massless field, while the vector field 0A and the scalar field

3 By now, arbitrary spin unitary representations of the conformal algebra that are relevant for elementary particles are well understood (see, e.g., Refs. [14,15]). In our opinion, non-unitary representations of the conformal algebra deserve to be understood better.

Cv = Ci + C, (3.20)

enter Stueckelberg description of spin-1 massive field. To this end we consider Lagrangian and gauge transformations.

Gauge invariant Lagrangian. Lagrangian we found can be presented as

(-€)k L = Lo +J2 (-)k' Lv, (3.18) k'=1

Lo = L0, (3.19)

-k' + tLk' where we use the notation

1 l0 = I <P£(d2 - pd)$A +1l2, (3.21)

1L1' = 1(D2 - m2' - Pd)0* + 1 Lk'Lk', (3.22)

!L0' = 20k' (D2 - m2 , (3.23)

Lo = DB0B, (3.24)

Lk' = DB0B, + \mk'\0k', (3.25)

2= pk'(d - 2 - k'), k= 1,...,k, k = —-. (3.26)

Gauge transformations. We now discuss gauge symmetries of the Lagrangian given in (3.18). To this end we introduce the following gauge transformation parameters:

, k' = 0, 1,...,k. (3.27)

The gauge transformation parameters fk, in (3.27) are scalar fields of the Lorentz algebra so(d, 1). We note the following gauge transformations:

80$ = (3.28)

80$ = , (3.29)

80k' = -e|mk'|fk', k / = 1,...,k. (3.30)

The following remarks are in order.

i) Lagrangian C0 in (3.19) is invariant under f0 gauge transformations given in (3.28), while the Lagrangian Ck in (3.20) is invariant under fk, gauge transformations given in (3.29), (3.30). This implies that the Lagrangian C0 describes spin-1 massless field, while the Lagrangian Ck/ describes spin-1 massive field having square of mass parameter m2, given in (3.26).

ii) From (3.30), we see that the scalar fields transform as Stueckelberg fields. In other words, the scalar fields are realized as Stueckelberg fields in our description of spin-1 conformal field.

Summary. Lagrangian of spin-1 conformal field in (A)dSd+1 given in (3.18) is a sum of Lagrangian C0 (3.19) which describes dynamics of spin-1 massless field and Lagrangians Ck, (3.20), k' = 1, 2, ..., k, which describe dynamics of spin-1 massive fields.

Lorentz-like gauge. Representation for Lagrangians in (3.21), (3.22) motivates us to introduce gauge condition which we refer to as Lorentz-like gauge for spin-1 conformal field,

L0 = 0, Lk, = 0, k' = 1,...,k, Lorentz-like gauge. (3.31)

4. Spin-2 conformai field in (A)dS

In this section, we study a spin-2 conformai field in (A)dS. In (A)dSd+1, d > 3, Lagrangian of spin-2 conformai field involves higher derivatives. Ordinary-derivative Lagrangian formulation of spin-2 conformal field in Rd'1, d > 3, was developed in Ref. [2]. Our purpose in this section is to develop a ordinary-derivative Lagrangian formulation of spin-2 conformal field in (A)dSd+1, d > 3. Because spin-2 conformal fields in (A)dS4 and (A)dS6 are the simplest and important examples of spin-2 conformal field theories, we consider them separately below. These two cases allow us to demonstrate some other characteristic features of our ordinary-derivative approach which absent for the case of spin-1 field in Section 3. Namely, the spin-2 conformal field in (A)dS4 is the simplest example involving partial-massless field, while the spin-2 conformal field in (A)dS6 is the simplest example involving both the partial-massless and massive fields.

4.1. Spin-2 conformal field in (A)dS4

Field content. To discuss ordinary-derivative and gauge invariant formulation of spin-2 conformal field in (A)dS4 we use two tensor fields denoted by 0qB, and one vector field denoted by ;

The fields 0\B and the field are the respective tensor and vector fields of the Lorentz algebra so(3, 1). The tensor fields 0\B are symmetric and traceful. Now we are going to demonstrate that the tensor field enters description of spin-2 massless field, while the tensor field §\B and the vector field enter gauge invariant Stueckelberg description of spin-2 partial-massless field. To this end we consider Lagrangian and gauge transformations.

Gauge invariant Lagrangian. Lagrangian we found can be presented as

L =-e£o + eCi, (4.2)

Lo - L2o, (4.3)

Li = L2 + eLj, (4.4) where we use the notation

eL0 - 4O2 - 2p)^ - 8<^0>2 + 2p+ 2laLa, (4.5)

1L2 - 1<B (V2 - m2 - 2p)4? - 1 < (V2 - m2 + 2p)tfB + 1 LALA, (4.6)

e 4 8 2

1 Li - 14t(D2 - m2 + 3p)4A + 1 LiLi, (4.7)

La = VB$AB -1 VA4BOB, (4.8)

LA = VB4f - 1 Da4BB + \m1 \4t, (4.9)

L1 = VB4B + 2\m1\4BB, (4.10)

m2 = 2p. (4.11)

Gauge transformations. We now discuss gauge symmetries of the Lagrangian given in (4.2). To this end we introduce the following gauge transformation parameters:

frA tA

(4.12)

The gauge transformation parameters frAA, frA and fr1 in (4.12) are the respective vector and scalar fields of the Lorentz algebra so(3, 1). We note the following gauge transformations:

= VAf£ + (4.13)

8$f = VAfB + VB^A + \mi\nABh, (4.14)

5$A = - e\mi\^A. (4.15) The following remarks are in order.

i) Lagrangian C0 in (4.3) is invariant under gauge transformations given in (4.13), while the Lagrangian C\ in (4.4) is invariant under fA and gauge transformations given in (4.14), (4.15). This implies that the Lagrangian C0 describes spin-2 massless field, while the Lagrangian C1 describes spin-2 partial-massless field having square of mass parameter m1 given in (4.11).

ii) From (4.14), (4.15), we see that the vector field $A and a trace of the tensor field ^ transform as Stueckelberg fields. In other words, just mentioned fields are realized as Stueckelberg fields in our description of the spin-2 conformal field. Gauging away the vector field we end up with the Lagrangian obtained in Ref. [8].4

Summary. Lagrangian of spin-2 conformal field in (A)dS4 given in (4.2) is a sum of Lagrangian C0 (4.3) which describes dynamics of spin-2 massless field and Lagrangian C1 (4.4), which describes dynamics of spin-2 partial-massless field. Square of mass parameter of the spin-2 partial-massless field is given in (4.11).

de Donder-like gauge. Representation for Lagrangians in (4.5)-(4.7) motivates us to introduce gauge condition which we refer to as de Donder-like gauge for spin-2 conformal field,

LA = 0, LA = 0, L1 = 0, de Donder-like gauge. (4.16)

4.2. Spin-2 conformal field in (A)dS6

Field content. To discuss ordinary-derivative and gauge invariant formulation of spin-2 conformal field in (A)dS6 we use three tensor fields denoted by $0^, $\B, $AB, two vector fields denoted by $A, $A and one scalar field denoted by $2,

^ $AB $AAB

$A $A (4.17)

4 In four-dimensions, the ordinary-derivative description of the interacting conformal gravity involving the vector Stueckelberg field was obtained in Ref. [2] by using gauge approach in Ref. [16]. Discussion of uniqueness of the interacting conformal gravity in four-dimensions may be found in Ref. [17]. Gauge invariant description of interacting massive fields via Stueckelberg fields turns out to be powerful (see e.g., Refs. [18-20]). Therefore we think that use of Stueckelberg fields for the study of interacting conformal fields might be very helpful.

The tensor fields 0OB, 01^, 0AB, the vector fields 0A, 0A , and the scalar field 02 are the respective tensor, vector, and scalar fields of the Lorentz algebra so(5, 1). The tensor fields , , 0AB are symmetric and traceful. Now we are going to demonstrate that the field enters description of spin-2 massless field, the fields 00B, 0A enter gauge invariant Stueckelberg description of spin-2 partial-massless field, while the fields 0A, 02 enter gauge invariant Stueckelberg description of spin-2 massive field. To this end we consider Lagrangian and gauge transformations.

Gauge invariant Lagrangian. Lagrangian we found can be presented as

L = Co — Li + L2, Lo = CO, Li = C2 + eCi, L2 = L2 + eLi + L

-2 + eL2 + L2, where we use the notation

1 l2 = 4 <pf(v2 — 2^

1 L2 = 4 $f(V2 — m2 —

;*AA( D2 +

,Lo Lo,

— 8 D2 — ,

+ 6p)0

_i_ -jaja + 2 ,

Li = ~D — m2' + 5p)0$ + -Lk'Lk', k' = 1, 2,

1L2 = 102 ( D2 — m2 + 1Op)02,

0 = db0AAb — i Da0Bb,

LA'= DB0AB — 2 DA0BB

\mk'\0A,

k' = 1, 2,

Li = DB0B + ^ \ m 1 \ 01

L2 = DB 0B + ^\m2\0BB + /02, m2 = 4p, m^ = 6p,

/ = V5\p\ .

(4.18)

(4.19)

(4.20)

(4.21)

(4.22)

(4.23)

(4.24)

(4.25)

(4.26)

(4.27)

(4.28)

(4.29)

(4.30)

(4.31)

Gauge transformations. We now discuss gauge symmetries of the Lagrangian given in (4.18). To this end we introduce the following gauge transformation parameters:

èA SO

èA è2

(4.32)

The gauge transformation parameters , and & in (4.32) are the respective vector

and scalar fields of the Lorentz algebra so(5, 1). We note the following gauge transformations:

80^ = DAèBB + DBèoA,

80AB = DAèB+ DBèA + -\mk'\nABèk',

k' = 1, 2,

(4.33)

(4.34)

= VAfk,- €\mk,\fA, k'= 1, 2, (4.35)

502 = -€/&• (4.36)

The following remarks are in order.

i) Lagrangian L0 in (4.19) is invariant under fA gauge transformations given in (4.33). This implies that the Lagrangian L0 describes spin-2 massless field.

ii) Lagrangian L1 in (4.20) is invariant under fA and f 1 gauge transformations given in (4.34), (4.35). This implies that the Lagrangian L1 describes spin-2 partial-massless field having square of mass parameter m12 given in (4.30).

iii) Lagrangian L2 in (4.21) is invariant under fA and f2 gauge transformations given in (4.34)-(4.36). This implies that the Lagrangian L2 describes spin-2 massive field having square of mass parameter m22 given in (4.30).

iv) From (4.33)-(4.36), we see that the scalar field, the vector fields, and trace of the tensor field transform as Stueckelberg fields. In other words, just mentioned fields are realized as Stueckelberg fields in our description of spin-2 conformal field.5

Summary. Lagrangian of spin-2 conformal field in (A)dS6 given in (4.18) is a sum of Lagrangian L0 (4.19), which describes dynamics of spin-2 massless field, Lagrangian L1 (4.20), which describe dynamics of spin-2 partial-massless fields, and Lagrangian L2, which describes dynamics of spin-2 massive field. Squares of mass parameter for partial-massless field, m12, and the one for massive field, m22, are given in (4.30).

de Donder-like gauge. Representation for Lagrangians in (4.22)-(4.24) motivates us to introduce gauge condition which we refer to as de Donder-like gauges for spin-2 conformal field,

la,= 0, kk = 0, 1,2; L1 = 0, L2 = 0, de Donder-like gauge. (4.37) 4.3. Spin-2 conformal field in (A)dSd+1

To discuss ordinary-derivative and gauge invariant formulation of spin-2 conformal field in (A)dSd+1, for arbitrary odd d, we use k + 1 tensor fields denoted by k vector fields denoted by , and k - 1 scalar fields denoted by ,

0AB, k' = 0, 1,...,k, , k '= 1,2,...,k,

0v, k = 2, 3,...,k,k . (4.38)

For the illustration purposes it is helpful to represent field content in (4.38) as follows

Field content of spin-2 conformal field in (A)dSd+1 for arbitrary odd d > 5, k = (d - 1)/2

^ tf ^ ...... 0A-1 0Ab

tA tA tA ...... tA-1 tA (4.39)

02 03 ...... 0k-1 tk

5 For the first time, ordinary-derivative description of six-dimensional gravity involving Stueckelberg fields was developed in Refs. [2,21]. Recent discussion of various aspects of conformal gravity in six-dimensions may be found in Ref. [22].

The tensor fields 0kf, the vector fields 0A, and the scalar fields 0k' are the respective tensor, vector, and scalar fields of the Lorentz algebra so(d, 1). The tensor fields 0AP are symmetric and traceful. Now we are going to demonstrate that the field 0AB enters description of spin-2 massless field, the fields , 0A enter gauge invariant Stueckelberg description of spin-2 partial-massless field, while the fields 0AP, 0a, 0k', k' = 2, ..., k enter gauge invariant Stueckelberg description of spin-2 massive fields. To this end we consider Lagrangian and gauge transformations.

Gauge invariant Lagrangian. Lagrangian we found can be presented as

(-e)kL = Co - Li + J2 (-)k'Lk', (4.40)

Co = L2, (4.41)

Li = Li + eCi, (4.42)

where we use the notation

Lt = L2 + L + L0', (4.43)

i L2 = 4 - m2- 2p ^

- 1- ml + 2p(d - 2))0BB + 1 LA LA, (4.44)

1 Li' = i0A (D2 - m2k' + dp)0A + iLk'Lk', (4.45)

iL0' = 20k' (D2 - m2k' + 2dp)0k', (4.46)

LA = DB0AB - 1DA0BB + \mk'\0A, (4.47)

Lt = DB0Bk, + 2\mk'\0BkB + fk' 0k', (4.48)

( 2d 2 \1/2 f*={ JZTJ\mk' \ - 2d\P \)

2d(k! - 1)(d - 1 - k') ^1/2

\P M , (4.49)

m2k, = pk'(d - k'), k ' = 0, 1,...,k, k = d--1. (4.50)

Gauge transformations. To discuss gauge symmetries of the Lagrangian given in (4.40), we introduce the following gauge transformation parameters:

tA tA tA tA tA

50 51 ?2 ............5k-1 ?k

(4.51)

tl tl ......tk-1 tk

The gauge transformation parameters 5a, 5a in (4.51) are the respective vector and scalar fields of the Lorentz algebra so(d, 1). We note the following gauge transformations:

= DAk + + 2^ nABHk', (4.52)

d — 1

VAfv— e\mv\f A, (4.53)

80k ' = —efk ' Hk ' ■ (4.54)

The following remarks are in order.

i) Lagrangian L0 in (4.41) is invariant under H A gauge transformations given in (4.52) when k' = 0. This implies that the Lagrangian L0 describes spin-2 massless field.

ii) Lagrangian L1 in (4.42) is invariant under HA and H1 gauge transformations given in (4.52), (4.53) when k' = 1. This implies that the Lagrangian L1 describes spin-2 partial-massless field having square of mass parameter m1 given in (4.50).

iii) Lagrangian in (4.43) is invariant under HA and Hv gauge transformations given in (4.52)-(4.54) when k' = 2, ..., k. This implies that the Lagrangian Ck describes spin-2 massive field having square of mass parameter m2k, given in (4.50).

iv) From (4.52)-(4.54), we see that all scalar and vector fields as well as trace of the tensor field $AB transform as Stueckelberg fields. In other words, just mentioned fields are realized as Stueckelberg fields in our description of spin-2 conformal field.

Summary. Lagrangian of spin-2 conformal field in (A)dSd+1 (4.40) is a sum of Lagrangian L0 (4.41), which describes dynamics of spin-2 massless field, Lagrangian L1 (4.42), which describes dynamics of spin-2 partial-massless field with square of mass m2 in (4.50), and Lagrangians Cki, k' = 2, ■■■, k, which describe dynamics of spin-2 massive fields with square of masses m2' (4.50).6

de Donder-like gauge. Representation for Lagrangians in (4.44), (4.45) motivates us to introduce gauge condition which we refer to as de Donder-like gauge for spin-2 conformal field,

La = 0, k' = 0, 1,...,k;

de Donder-like gauge. (4.55)

Lk = 0, k' = 1,...,k, 5. Arbitrary spin-s conformai field in (A)dS^+1

Field content. To develop ordinary-derivative and gauge invariant formulation of spin-i conformal field in (A)dSd+1, for arbitrary odd d > 3, we use the following scalar, vector, and tensor fields of the Lorentz algebra so(d, 1):

1...As', k! = 0, 1,...,ks, s'= max(0,s — k'),...,s; (5.1) d — 5

ks = s . (5.2) Tensor fields l".As' are totally symmetric and, when s' > 4, are double-traceless7

0AABBA5...AS' = 0, s '> 4. (5.3)

6 For arbitrary d > 7, little is known about interacting conformal gravities. Study of local Weyl invariants in eight-dimensions may be found in Ref. [23]. Discussion of conformal supergravity in ten-dimensions may be found in Ref. [24].

7 Discussion of higher-spin field dynamics in terms of unconstrained fields can be found in Refs. [25,26]. Study of mixed-symmetry conformal field my be found in Ref. [27] (see also Ref. [28]). For interesting discussions of various aspects of mixed-symmetry fields see Ref. [29].

The following remarks are in order.

i) For (A)dS4, fields in (5.1) can be divided into two groups

фМ.^

A\...AS, k'

massless

k'= 1,...,s — 1, s ' = s — k' ,...,s; partial-massless

Below, we demonstrate that 0A1..As (5.4) enters spin-s massless field, while fields 0Ain (5.5) with k'-fixed and s' = s - k',..., s enter gauge invariant Stueckelberg description of spin-s partial-massless field having square of mass parameter m2' = pk'(2s - 1 - k'). To illustrate the

' A1 A '

field content given in (5.4), (5.5), we use the shortcut 0SV for the field 0a s and note that, for (A)dS4 and arbitrary s, fields in (5.4), (5.5) can be presented as in (5.6). Field content for spin-s conformal field in (A)dS4, s-arbitrary

К—2

s — 1

s — 1

Ф2—1

ss——21

ФГЛ

Фs2—2

ф2— 1

ii) For (A)dSd+1, d > 5, fields in (5.1) can be divided into the following three groups:

A1...As

фо1 s

A1...As'

фк' s

фА 1...As', k! = s,...,ks, s ' = 0, 1,

massless

k'= 1,...,s — 1, s ' = s — k' ,...,s; partial-massless

massive

A, A Ai A '

Below we show that a) the field ф0 " s (5.7) enters spin-s massless field; b) the fields фк, s (5.8) with k'-fixed and s' = s — k',..., s enter gauge invariant Stueckelberg description of spin-s partial-massless field with square of mass parameter m2e = pk'(2s + d — 4 — k'); c) the fields A1 ... A

фlc;... s (5.9) with k'-fixed and s' = 0, 1,..., s enter gauge invariant Stueckelberg description of

spin-s massive field having square of mass parameter m2, = pk' (2s + d — 4 — k'). To illustrate

s' A1... A

the field content in (5.7)-(5.9), we use the shortcut фк, for the field фк' s and note that, for d > 5 and arbitrary s, fields in (5.7)-(5.9) can be presented as in (5.10).

Field content of spin-s conformal field in (A)dSd+1 for odd d > 5, s-arbitrary, ks = s + d-—-

К — 1

s — 1

is — 1

kss——11

(.s —1

(5.10)

фs1—1 ф1

ф1—1 ф1

iii) The lowest value of d when the scalar fields appear in the field content is given by d = 5. Namely, for (A)dS6 and arbitrary s, the field content in (5.10) is simplified as

Field content for spin-s conformal field in (A)dS6, s-arbitrary

00 0 ■■■ 1 0s

0s-1 02-1 ■■■ 0ss-i 0ss-1

(5.11)

01-1 0s1

Generating form of field content. To streamline the presentation of our gauge invariant ordinary-derivative formulation we use the oscillators aA, Z, &, x, and collect scalar, vector and tensor fields (5.1) into the ket-vector |0) defined by8

fc &ks-k'

|0)=k?O vr-wl0k'), (5.12)

|0k') = i Xk'+s'-sZs-s'aA1 ■■■aAs' 0A-As'■ (5.13)

s-=Jks-krs(k' + s'- s)!(s - s')!0k ( )

From (5.3), (5.12), (5.13), we see that the ket-vectors |0), |0k') satisfy the constraints

(Na + Nz)l0)=sl0), (5.14)

(Nz + N& + Nx)l0)=ks |0), (5.15)

(Nz + Nx)|0k')=k'|0k'), (5.16)

(a 2)2|0)=O■ (5.17)

From (5.14), we learn that the ket-vector |0) is degree-s homogeneous polynomial in the oscillators aA, Z, while constraint (5.15) tells us that |0) degree-ks homogeneous polynomial in the oscillators Z, &, x. Constraint (5.16) implies that the ket-vectors |0k') is the degree-k' homogeneous polynomials in the oscillators Z, x. Constraint (5.17) is just the presentation of the double-tracelessness constraints (5.3) in terms of the ket-vector |0).

We now proceed with the discussion of gauge invariant Lagrangian in the framework of our ordinary-derivative approach. We would like to discuss generating form and component form of the Lagrangian. We discuss these two representations for the Lagrangian in turn.

Gauge invariant Lagrangian. Generating form. The Lagrangian we found is given by

L = 2 e(0|E|0), (5.18)

E = - 4a2«^ (□(A)dS + mx + P*2«2) - LlL, (5.19)

8 In earlier literature, extensive use of oscillator formalism may be found in Ref. [30] (see also Ref. [31]).

m1 = -m2 + p(s(s + d - 5) - 2d + 4 + Nz(2s + d - 1 - Nz)), (5.20)

m2 = pNzx(2s + d - 4 - Nzx), Nzx = Nz + Nx, (5.21)

1L = a D - i a Da2 - e1 П [1Д + i e1a2, (5.22)

L = aD - ia2aD - e1 П[1-2] + ie1a2, (5.23)

ei = ZeiX, ei = -ye\K, (5.24)

e1 = (|p|(2s + d - 5 - 2Nz - N-x))1/2ez, (5.25)

( 2s + d - 3 - Nz \1/2

ez =-+-M . (5.26)

z \2s + d - 3 - 2NzJ

Deflnition of operators appearing in (5.19)-(5.26) may be found in Appendix A. We note that two-derivative part of the operator E (5.19) coincides with the standard Fronsdal operator represented in terms of the oscillators. Operator E (5.19) can also be represented as

E = Q(A)dS + M1 - 1 a2a2(D(A)dS + M2) - LL, (5.27)

M1 = -m2 + p(s(s + d - 5) - 2d + 4 + Nz(2s + d - 1 - Nz)), (5.28)

M2 = -m2 + p(s(s + d - 1) - 6 + Nz(2s + d - 5 - Nz)). (5.29)

Gauge symmetries. To discuss gauge symmetries of the Lagrangian in (5.18) we use the gauge transformation parameters given by

1 ...As', k'= 0,1,...,ks, s ' = max(0,s - 1 - kr),...,s - 1; (5.30)

where ks is defined in (5.2). The gauge transformation parameters in (5.30) are scalar, vector,

A1 A '

and tensor fields of the Lorentz algebra so(d, 1). The gauge transformation parameters 6k' "' s are totally symmetric and, when s' > 2, are traceless

6BBA3...As'= 0, s'> 2. (5.31)

As usually, to simplify the presentation, we collect the gauge transformation parameters into a

ket-vector defined by

§ks-k'

16 )=E ТШ-k)^ (5-32)

s-1 xk'+s'-s+1Zs-1-s' aA1 ...aAs' ax...as,

l6k'} = > ,, , .... , = Ну s . (5.33)

s'=max(fe-1-k')s'V(k + s - s + 1)!(s - s - 1)! k

From (5.31)-(5.33), we see that the ket-vectors l6), l6k') satisfy the constraints

(Na + Nz)l6) = (s - 1)l6), (5.34)

(Nz + Nfi + Nx)l6 )=ks 16), (5.35)

(Nz + Nx)l6k')=k'l6k'), (5.36)

a. 2l6) = 0. (5.37)

From (5.34), we learn that the ket-vector ) is a degree-(s - 1) homogeneous polynomial in the oscillators aA, Z, while constraint (5.35) tells us that the ) is a degree-ks homogeneous polynomial in the oscillators Z, &, X. Constraints (5.36) implies that the ket-vector |fk/) is a degree-V homogeneous polynomial in the oscillators Z, X. Constraint (5.37) is just the presentation of the tracelessness constraints (5.31) in terms of the ket-vector ).

Using ket-vectors |V), ), gauge transformation for spin-s conformal field can be presented

5|V) = G|£), G — aD - ei - a2 ¿1, (5.38)

2Na + a — 1

where operators e1, e1 are given in (5.24).

Component form of Lagrangian and gauge transformations. For deriving the component form of Lagrangian it is convenient to use representation for the Lagrangian with the operator E given in (5.27). By plugging ket-vector (5.12) into (5.18) we obtain the component form of the La-grangian,

L = (-€)ks J2(-f Lv, (5.39)

k/=0 s

Lk/= J2 "s'-s Li, (5.40)

s/=max (0,s-k')

where we use the notation

1 l£ — ¿y 1-A" D2 + M(kr)

^3-As< D + M2 k/Uf^ 8(s' - 2)! 1 2,k! Vk

1 LA1-AS/—1T A1-As/—1 2(s/ - 1)! v

+ LA/1"'A -1 LArA"-1, (5.41)

A1...As/-1 _ B A1...AS/-1B s' - ^(A A2...As/-1)BB

Lv — D VV 2 D Vk/

+k C^-1 + 2 fS/+141...As/-1BB, (5.42)

Vi^' — VAkrAs' - 2(2S( -) 3) n(A1AA2 *rAs')BB, (5.43)

M^ — -m2/ + p(2(s - 1)(s + d - 2) - s'(s' + d - 1)), (5.44)

M2 k/ — -m2k, + p(2s(s + d - 3) - 6 - s'(s' + d - 5)), (5.45)

= pi (2s + d - 4 - k'), (5.46)

s/ ( {s - s' + 1 )(s + s' + d - 3) ( 2 ( ,)( 1/2

fs — (—2j+d -3—- m - ^s - so (s+s'+d - ^ . (5.47)

We recall that e = det e£, e^ stands for vielbein of (A)dS, D2 — DAVA, where DA stands for the

covariant derivative in (A)dS. We use p = e/R2, where e = 1(-1) for dS (AdS) and R is radius

A1... A /

of (A)dS. The quantities Lv s (5.42) are referred to as de Donder divergences in this paper.

Note that using m2k, given in (5.46) allows us to represent f^ in the following factorized form:

, ( (s - s' + 1)(s + s' + d - 3)(k' + s'- s)(s + s' + d - 4 - k') \l/k

fs-K---kkd-r^-l) ■ (5.48)

Component form of gauge transformations. Component form of gauge transformations is easily found by plugging ket-vectors \0}, } into (5.38). Doing so, we get

s^rAs> = sD(Ai ^) - 1-As' + s'(s, -f n(AlAk 3"'As'). (5.49)

ks + d — 5

The following remarks are in order.

Ai A -1

i) Lagrangian L0 in (5.40) is invariant under "' s gauge transformations given in (5.49) when k' = 0. This implies that the Lagrangian L0 describes spin-s massless field.

ii) For k' = 1, .'., s - 1, Lagrangian Lk' (5.40) is invariant under ^A 1'''As', s' = s - k' -1, .'., s - 1, gauge transformations given in (5.49). This implies that the Lagrangian Lk' describes spin-s partial-massless field having square of mass parameter m2k, given in (5.46).

iii) For k! = s,..., ks, Lagrangian Lk' (5.40) is invariant under ^ 1'''As', s' = 0, .s - 1, gauge transformations given in (5.49). This implies that the Lagrangian Lk' describes spin-s massive field having square of mass parameter mkk' given in (5.46).

iv) Using (5.49), one can make sure, that all scalar fields, all vector fields, traceless parts of tensor fields 0p'"As', s' = k,. .., s - 1, k' = s - s', „', ks, and traces of tensor fields , s' = k,..., s, transform as a Stueckelberg fields. In other words, just mentioned fields are realized as Stueckelberg fields in our description of spin-s conformal field.

v) Note that we express our gauge invariant Lagrangian in terms of de Donder divergences given in (5.4k). Obviously it is the use of de Donder divergences that allows us to simplify considerably the gauge invariant Lagrangian.9

Summary. Lagrangian of spin-s conformal field in (A)dSd+1 given in is a sum of Lagrangian L0 (5.40), which describes dynamics of spin-s massless field, Lagrangians Lk' (5.40), k' = 1,..., s - 1, which describe dynamics of spin-s partial-massless fields, and Lagrangians Ck> (5.40), k' = s,..., ks, which describe dynamics of spin-s massive fields. Square of mass parameter for massless, partial-massless, and massive fields is described on equal footing by mkk' given in (5.46). Our result for m2, confirms the conjecture about m2, made in Ref. [9].

de Donder-like gauge. Representation for Lagrangians in (5.18), (5.19) motivates us to introduce gauge condition which we refer to as de Donder-like gauge for spin-s conformal field,

L#} = 0, de Donder-like gauge, (5.50)

where the operator L is given in (5.kk). It is easy to see that gauge (5.50) considerably simplifies the Lagrangian in (5.18).10 In terms of tensor fields, gauge (5.50) can be presented as

9 Representation of gauge invariant Lagrangian for massless, partial-massless, and massive (A)dS fields in terms of de Donder-like divergences was found in Ref. [32,33]. Alternative representation of gauge invariant Lagrangian for partial-massless and massive (A)dS fields was obtained for the first time in Ref. [34]. Study of partial-massless fields in frame-like approach may be found in Ref. [35]. In the framework of tractor and BRST approaches, partial-massless fields were studied in Ref. [36]. Interacting partial-massless fields are studied in Ref. [37] (see also Refs. [8,38]).

10 For massless fields, discussion of the standard de Donder-Feynman gauge may be found in Ref. [39]. Extensive use of Donder-like gauge conditions for studying the AdS/CFT correspondence may be found in Refs. [40-42]. We believe that our de Donder-like gauge might also be useful for the study of AdS/FT correspondence along the lines in

Lp-As, = ^ = 0, 1,...,ks, s' = max(0,s - 1 -k'),...,s - 1; (5.51)

Ai A /

where Lk, s is given in (5.42).

Left-over gauge symmetries of de Donder-like gauge. We note that de Donder-like gauge has left-over gauge symmetry. This symmetry can easily be obtained by using the following relation

L G|£ > = (D(A)dS + Mpp)|f ), (5.52)

Mfp = -m2 + p((s - 1)(s + d - 2) + Nz(2s + d - 3 - Nf)), (5.53)

where m2 is given in (5.21), while G stands for operator entering gauge transformation in (5.38). From (5.52), we see gauge condition (5.50) is invariant under left-over gauge transformation provided the gauge transformation parameter satisfies the equation

(□(A)dS + MfpM ) = 0. (5.54)

In terms of tensor fields, Eq. (5.54) can be represented as

(D2 + Mb l...As'= 0,

k = 0, 1,...,ks, s ' = max(0,s - 1 - k'),...,s - 1; (5.55)

where Msx k, is given in (5.44). We note that the kinetic operators appearing in (5.55) are the kinetic operators of Faddeev-Popov fields when we use de Donder-like gauge given in (5.50).

Partition function of conformai field via de Donder-like gauge. We now explain how the partition function of conformal field obtained in Ref. [9] arises in the framework of our approach. Using gauge-fixed Lagrangian, partition function of conformal field, denoted by Ztotal, can be presented as (for details of the derivation (5.56), see below):

Ztotal = n Zk/, (5.56)

s-1 , s s-2

Zk— n Dk/ D(, n Dk/ n Dk/, (5.57)

s/=max(0,s-1-V) s/=max(0,s-k') s/=max(0,s-2-k')

Dv — (det(-D2 - MS/,/))1/2, (5.58)

where, in (5.58), the determinant is evaluated on space of traceless rank-s' tensor field. In (5.56), the Zk/ is partition function of spin-s (A)dS field having square of mass parameter m2k,. It is easy to see that Zk/ (5.57) take the form

DV-1DV-1-k/

Zk/ = —-k^fT-, for V = 0,1,...,s - 2; (5.59)

k Ds Ds-2-k/

Df;-1D0/

Zk/= kDs k , for V = s - 1; (5.60)

Zk/ = -v—, for V = s,s + 1,...,ks. (5.61) k Dks/ s

Refs. [9,43]. Also we note that our approach to conformal (A)dS fields streamlines application of general methods for the computation of one-loop effective action developed in Refs. [44,45].

We now note the well-known relation for Dsk, in (5.58)

Dsk' = D(±Dk'-1, (5.6k)

where, in (5.6k), the Dsk,L takes the same form as in (5.58), while the determinant is evaluated on space of divergence-free (VA0AAk~'As' = 0) traceless rank-s' tensor field. Using (5.6k), the Zk' in (5.59)-(5.61) can be represented as

Di:-1-k'±

Zk' = k s±—, fork! = 0, 1,...,s - 1; (5.63)

Zk, = for k' = s,s + l,...,ks. (5.64)

Plugging (5.63), (5.64) into (5.56), we get

s-i —s-i-k ± ks i

Zt0tal = П . (5.65)

k'=0 -k' k'=s -k'

Taking into account values of the mass parameters entering —sk7i-k , —sk, (5.58),

Ml-k}-k' = pkk + p(s - i)(s + d - 2), (5.66)

Mshk' = -ps + p (s - 2 - k')(s + d - 2 - k'), (5.67)

we verify that our expression for Ztotal in (5.65) coincides with the one in Ref. [9].i1

We now present some details of the derivation of partition function (5.56). To this end we note that the de Donder-like gauge condition implies the use of the following gauge fixing term:

e £g.flx = 2 + (¿i(D(A)dS + MFp)\c), (5.68)

where |c) and \c) stand for ket-vectors of the Faddeev-Popov fields. Decompositions of the

-Ai A / Ai A /

ket-vectors \c), \c) into the respective tensor fields ck, s , ck, s take the same form as the

A i A /

decomposition of the ket-vector \%) into the tensor fields "' s in (5.32), (5.33). Also, the ket-vectors \c), \c) satisfy the same algebraic constraints as the ket-vector ) in (5.34)-(5.37). Use of (5. i 8) and (5.68), leads to the following gauge-fixed Lagrangian:

Aotal = L + Lg.fix, (5.69)

eLtotal = 2(Ф\( i - 4a2<*2)(n(A)dS + mi + m2a20i2)\ф)

+ <^\(D(A)dS + Mfp)\c). (5.70)

Gauge-fixed Lagrangian (5.70) can alternatively be represented as

e Aotal = 2 <ф \ (n(A)dS + Mi - 4 а2й 2(D(A)dS + M2^j\ ф)

+ <c \ (D(A)dS + Mfp) \ c). (5.7 1 )

1 1 Note that our labels differ from the ones in Ref. [9].

Another representation of gauge-fixed Lagrangian (5.70) is obtained by using the decomposition of double-traceless ket-vector |0) (5.12) into two traceless ket-vectors defined by the relation

|0) = |0i) + a2n |0ii), à 2|0i) = 0, à 2|0ii) = 0, (5.72)

N = ((2s + d - 3 - 2Nz)(2s + d - 5 - 2Nz))~1'2. (5.73)

Using (5.72), we represent gauge-fixed Lagrangian (5.71) as 1 1 1

- Aotal = ^ <0i|(D(A)dS + Mi)|0i) - ^ <0ii|(D(A)dS + M2 + 4p)|0n)

+ <C|(l=l(A)dS + Mfp)|c). (5.74)

it is the representation given in (5.74) that leads immediately to partition function (5.56)-(5.58). To see this it is convenient to write down a component form of Lagrangian in (5.74). For that

A1 A '

purpose we note that a decomposition of the ket-vector |0i) into tensor fields s takes the

Ai A /

same form as the decomposition of the ket-vector |0) into tensor fields 7 s in (5.12), (5.13),

Ai A /

while a decomposition of the ket-vector |0ii) into tensor fields ïk7 s takes the form as the

A1 A /

decomposition of ket-vector |0) into tensor fields 77 s in (5.12), (5.13) with the replacement s ^ s - 2 in (5.13). This is to say that component form of gauge-fixed Lagrangian (5.74) takes the form

Aotal - ks = (-e)k^(-)k' Aotal k', k'=0 (5.75)

Aotal k' s s-2 \ 1 ^s'-s /»s' \ 1 ^s'-s z»s' = z^ fc k' Z^ fc Ai k' s'=max (0,s-k') s'=max (0,s-2-k')

s-1 _i_ Y^ /+1-s/*s' + Z^ £ LFPk', s'=max (0,s-1-k') (5.76)

rs' — A k' = ^ 0iA1'...As' (D2+Ms'k' )0iAk'...As', (5.77)

=' M" ^ (D2+Ms'k' )<C'.A', (5.78)

s' lfp k' - = e CA1"^ (D2 + Ms'k' ^. (5.79)

From (5.77)—(5.79), we see that Aotal (5.76) leads to the partition function given in (5.56)-(5.58).

As remark, we note that gauge-fixed Lagrangian (5.70) is invariant under global BRST transformations given by

sbrst|0>=xg|c>, sbrst|c>=^L|0>, 5brst|c>=0, (5.80)

where X is odd parameter of the global BRST transformations. We assume the following hermi-tian conjugation rules for X and various ket-vectors:

<C| = |C>t, <c| = -|c>f, Xt = —X. (5.81)

6. General form of Lagrangian and gauge symmetries for massless, partial-massless, massive, and conformal fields in (A)dS^+1

In this section, we formulate our result concerning the general structure of Lagrangian and gauge transformations for spin-s massless, partial-massless, massive, and conformal fields in (A)dSrf+1. Namely, we demonstrate that how a knowledge of gauge transformations allows us to fix a gauge invariant Lagrangian for just mentioned fields. Below, in Section 8.k, by using conformal transformation which maps fields in flat space to fields in (A)dS, we obtain gauge transformation of conformal field in (A)dS. Using then the gauge transformation of field in (A)dS and our result in this section we find the gauge invariant Lagrangian of conformal field in (A)dS in Section 8.k.

Field content. In order to formulate our result, it is convenient to use language of generating functions. We note that, by using the oscillators aA, Z, and various other oscillators, fields appearing in gauge invariant formulation of massless, partial-massless, massive, and conformal fields can be collected into a ket-vector ) defined by

where the ket-vector \$) should satisfy the constraints

(Na + Ntm=s\4), (6.k)

(a k)k|0) = 0. (6.3)

Ket-vectors \$s) (6.1) depend on the oscillators aA and various other oscillators. Note however that the ket-vectors \$s) are independent of the oscillator Z. Constraint (6.k) tells us that the ket-vector \$) should be degree-s homogeneous polynomial in the oscillators aA, Z. Constraint (6.3) implies that we are using double-traceless tensor fields. Obviously, for the case of s > 1, we assume the relation ot 2\$) = 0.

Gauge symmetries. To discuss gauge symmetries we should introduce a set of gauge transformation parameters which we use for a formulation of gauge transformations. We note that, using the oscillators aA, Z, gauge transformation parameters appearing in gauge invariant formulation of massless, partial-massless, massive, and conformal fields can be collected into a ket-vector \%) defined by

s-1 Zs-1-s'

where the ket-vector ) satisfies the constraints

(Na + Nz)\%) = (s - 1)£), (6.5)

a. 2\H) = 0. (6.6)

Ket-vectors s) (6.4) depend on the oscillators aA and various other oscillators. Note however that the ket-vectors s) are independent of the oscillator Z. Constraint (6.4) tells us that the ket-vector ) should be degree-(s - 1) homogeneous polynomial in the oscillators aA, Z. Constraint (6.6) implies that gauge transformation parameters we are using are traceless tensor fields.

In terms of the ket-vectors |0), ), a general one-derivative gauge transformation for the ket-vector |0) can be presented as

5|0) = G|f), G = aD - ex - a2 ¿1, (6.7)

2Na + d - 1

where operators e1 and e.1 depend on the oscillators Z, Z and other oscillators entering ket-vectors ) (6.1) and s) (6.4). In view of constraint (6.6), gauge transformation (6.7) respects constraint in (6.3). Also note that, to respect constraint given in (6.2), the operators ex and ex should satisfy the commutation relations

[Nz,ex] = ex, [Nf,ex] = -ei. (6.8)

With the restrictions in (6.8), transformation given in (6.7) is the most general gauge transformation of the ket-vector |0) which does not contain higher than first order terms in derivatives. Now we are ready to formulate our main statement in this section.

Statement. General Lagrangian which does not contain higher than second order terms in derivatives and which respects gauge transformation in (6.7) is given by

L = I ei^EM), (6.9)

E = - 4a2a^ (n(A)dS + mi + m2a2a2) - LL, (6.10)

L = . D - 1 a Da2 - e1n [1,2] + 1 e1u2, (6.11)

L ^ aD - 1 a2aD - e1n[1,2] + 1 e1a2, (6.12)

where operators m1, m2 are given by

m1 = -M2 + p(s(s + d - 5) - 2d + 4 - Nz(2s + d - 5 - Nz)), (6.13)

m2 = p, (6.14)

2 2s + d - 1 - 2NZ

M2 = -^1e1 + Z e1e1, (6.15)

2s + d - 3 - 2Nz

while the operators e1, e1, and M2 (6.15) should satisfy the following equations:

[M2,e1] = -2pex(2s + d - 4 - 2NK), (6.16)

[M2,51] = 2p(2s + d - 4 - 2NZ)e1, (6.17)

e\ = -e1. (6.18)

The statement is proved by direct computation. The computation is straightforward but tedious. We omit the details of the computation. The following remarks are in order.

i) From (6.10)—(6.15), we see that all operators appearing in Lagrangian (6.9) are expressed in terms of the operators e1, e1 which enter gauge transformation in (6.7). In other words, a knowledge of the operators e1 and e.1 allows us immediately to fix the gauge invariant Lagrangian.

ii) From (6.15), we see that operator M2 is expressed in terms of the operators e1 and e1. This implies that equations (6.16), (6.17) should be considered as restrictions imposed on the operators e1, e1. Eqs. (6.16)-(6.18) provide the complete list of restrictions imposed by gauge symmetries.

iii) For massless, partial-massless, and massive fields, Eqs. (6.16)-(6.18) constitute a complete system of equations which allow us to fix the operators e1, e1 uniquely.

iv) For conformal fields, Eqs. (6.16)-(6.18) alone do not allow to fix the operators e1, e1 uniquely. In Refs. [2,3], we have demonstrated that in order to determine uniquely the operators ei, e1 for conformal fields in r1,1, we should solve, in addition to restrictions imposed by gauge symmetries, the restrictions imposed by conformal so(d +1, 2) symmetries. In Section 8.2, using the operators e1, e1 for conformal field in r1,1 and conformal transformation which maps field in flat space to field in (A)dS we obtain operators e1, e1 for conformal field in (A)dSd+1. The knowledge of operators e1, e1 for conformal field in (A)dSd+1 allow us then to find Lagrangian of conformal field in (A)dSd+1 by using relations in (6.13)-(6.15).

v) Let us introduce a new operator m2 defined by the relation

m2 = M2 + 2pNz(2s + d - 3 - Nz). (6.19)

Using (6.19), it is easy to check that Eqs. (6.16), (6.17) can be represented as

[m2,e1] = 0, [m2,e1] = 0. (6.20)

Using (6.15), we see that the operator m2 can be presented in terms of the operators e1 and e1 as

2 2s + d - 1 - 2NZ

m2 = -e1e1 + J eë + 2pN^(2s + d - 3 - Nz), (6.21)

2s + d - 3 - 2NZ

while the operator m1 appearing in E (6.10) takes the form

m1 = -m2 + p(s(s + d - 5) - 2d + 4 + Nz(2s + d - 1 - Nz)). (6.22)

7. Conformai fields in Rd'1

In Section 8, we derive our Lagrangian for conformal fields in (A)dS by using the ordinary-derivative Lagrangian formulation of conformal fields in Ri1,1 and applying an appropriate conformal transformation which maps conformal fields in R1,1 to conformal fields in (A)dSd+1. The ordinary-derivative Lagrangian formulation of conformal fields in R1,1 was developed in Refs. [k,3]. In this section, we review briefly some our results in Refs. [k,3] which, in Section 8, we use to construct a map of conformal fields in R1,1 to conformal fields in (A)dSd+1.

7.1. Spin-0 conformal field in R1,1

In the framework of ordinary-derivative approach, spin-0 conformal field (scalar field) is described by k + 1 scalar fields

fit', k'e[k]k, k-arbitrary positive integer. (7.1)

Here and below, the notation k' e [n]k implies that k' = -n, -n + k, -n + 4,..., n - 4, n - k, n: k'e[n]k k' = -n, -n + k, -n + 4,...,n- 4,n - k,n. (7.k)

Conformal dimensions of the scalar fields are given by d - 1

Hfik') = — + k'. (7.3)

To simplify the presentation we use oscillators u®, ue and collect fields into the ket-vector defined by

|0> — E

1 , , k±*L , , k-kL

(u®) 2 (u®) 2 0^O>,

(Nv - k)|0>=O.

Constraint (7.5) tells us that ket-vector |0) (7.4) is degree-k homogeneous polynomial in the oscillators u®, ue. As found in Ref. [2], ordinary-derivative Lagrangian can be presented as

L = I<0|(d^,i - m2)|0>, - dAdA,

m2 — u ®u®,

9a — nABd/dxB. The component form of Lagrangian (7.6) takes the form

L = Lk, Lk' = 10-k'ÜRd.i 0k'- 20-k'0k'±2.

k's[k]2

7.2. Arbitrary spin conformal fields in Rd,x

Field content. To discuss the ordinary-derivative gauge invariant formulation of totally symmetric arbitrary spin-s conformal field in R1,1, for arbitrary odd d > 3, we use the following set of scalar, vector, and tensor fields of the Lorentz algebra so(d, 1):

ks' — s' ±

, I 0, 1,..., s ; for odd d > 5; ,, ri , 1,2,...,s; ford = 3;" k & [ks']2;

A1 A ,

Tensor fields 0,, ... s are totally symmetric and, when s' > 4, are double-traceless

0 v s = 0, s '> 4.

.A1...A,.

We note that the conformal dimension of the field 0V "' s is given by d-1

Ai...As' k'

± k'.

(7.10)

(7.11)

s, A1 ... A ,

To illustrate field content (7.8), we use the shortcut for the field " s and note that, for arbitrary spin-s conformal field in Rd,1, d > 5, the field content in (7.8) can be presented as

Field content of spin-s conformal field in R1, 1, for odd d > 5, s-arbitrary

C- 0C-11

(7.12)

0s1-1-ks 0s1

s± 1 ks

0s ks ±2

0k1s-s-1 0k1s-

^fe-s-2 0ks-s

For R31, the scalar fields do not enter the field content. Namely, for arbitrary spin-s conformai field in R3'1, the field content in (7.8) can be presented as

Field content of spin-s conformal field in R3'1, s-arbitrary

s-1 s-4

(7.13)

t-1 t2

To simplify the presentation, we use the oscillators aA, Z, u®, ue, and collect fields (7.8) into ket-vector \$) defined by

t ) = H

>/(s - s') !

= \ts 'M

for Rd'1, d > 5'

It ) = H

^^iT-Ty!

\t )= ks'+k'x.

— s )!

aA1 ...aAs' (ue) ^ (u№) ^ tt1"As' |0).

k'e[ks']j ■

From (7.10), (7.14), (7.15), we see that the ket-vector It) satisfies the relations

(Na + Nz - s)It) = 0, (Nz + Nu - ks)It)=0,

( e j)2It ) = 0,

(7.14)

(7.15)

(7.16)

(7.17)

where ks is defined as in (7.9). Relations (7.16) tell us that the ket-vector It) is degree-s homogeneous polynomial in the oscillators aA, Z and degree-ks homogeneous polynomial in the oscillators Z, u®, ue. Constraint (7.17) is just the presentation of the double-tracelessness constraints (7.10) in terms of the ket-vector It).

Gauge invariant Lagrangian found in Ref. [3] takes the form

L = j{tI^(ORd,1 - mj)It) + 1 Lt ILt), QRd'1 = dAdA, mj = u(7.18)

L = ud - ^ada2 - enlia] + 1 e1a2, 2 1 + 2

(7.19)

(7.20)

e1 = Ze^u , e1 = -u wezZ,

dA = nABd/dxB, where operators i, nl1,2], and ez are given in (A.17)-(A.19).

Gauge symmetries of conformal field in R11,1. To discuss gauge symmetries of Lagrangian (7.18), we introduce the following gauge transformation parameters:

ïlA-A, s' = 0, 1,...,s - 1, k'e [ks' + 1]2,

(7.21)

where ks' is given in (7.9). In (7.21), the gauge transformation parameters are scalar, vector, and

Ai A /

tensor fields of the Lorentz algebra so(d, 1). Tensor fields _'1' s are totally symmetric and, when s' > 2, are traceless,

= 0, s' > 2. (7.22)

A_ A '

Conformal dimension of the gauge transformation parameter f v_._ s is given by

/ A_ A I\ d — 1 ,

a^—_ s') = —+k' — _ №)

Now, as usually, we collect the gauge transformation parameters into ket-vector jf) defined

s-1 £s-1-s'

if ) = V |fs' ), (7.24)

1 ks,+1+k' V+1-k'

№ E -TJT+TW-y1 ...aAs' (ue)"^ (u®)^^^ (7.25)

Ket-vector if ) (7.24) satisfies the algebraic constraints,

(Na + Nf - s + 1)|f ) = 0, (Nz + Nu - ks)if ) = 0, (7.26)

«2|f )=0, (7.27)

where ks is defined as in (7.9). Relations (7.26) tell us that |f ) is a degree-(s - 1) homogeneous polynomial in the oscillators aA, Z and degree-ks homogeneous polynomial in the oscillators Z, u®, ue. Constraint (7.27) is a presentation of the tracelessness constraints (7.22) in terms of the ket-vector |f ).

Gauge transformations can entirely be written in terms of |0) and |f ) and take the form

5|0) = G|f ), G = ad - e1 - a2 ¿1, (7.28)

2Na + a - 1

where operators e1, e1 are defined in (7.20).

Realization of conformal symmetries in R1,1 . The conformal algebra so(d + 1, 2) considered in basis of the Lorentz algebra so(d, 1) consists of translation generators PA, dilatation generator D, conformal boost generators KA, and generators JAB which span so(d, 1) Lorentz algebra. We assume the following normalization for commutators of the conformal algebra:

[D, PA] = -PA, [PA, JBC] = nABPC - nACPB,

[D, KA] = KA, [KA, JBC] = nABKC - nACKB, (7.29)

[PA, KB] = nABD - JAB, [JAB, JCE] = nBCJAE + 3 terms,

A,B,C,E = 0, 1,...,d, nAB = (-, +,...,+). (7.30)

Let |0) denotes a free conformal field propagating in R1,1, d > 3. Let a Lagrangian for the field |0) be conformal invariant. This implies that the Lagrangian is invariant under transformation (invariance of the Lagrangian is assumed to be up to total derivatives)

5g№) = G№), (7.31)

where realization of the generators G in terms of differential operators takes the form

PA = dA, JAB = xAdB - xBdA + MAB, (7.32)

D = xBdB + A, (7.33)

KA = KAAM + RA, (7.34)

KAM = -1 xBxBdA + xAD + MABxB. (7.35)

In (7.33), A is operator of conformal dimension, while MAB appearing in (7.32), (7.35) is a spin operator of the Lorentz algebra so(d, 1). Operator RA appearing in (7.34) depends on space-time derivatives dA = nABd/dxB, and does not depend on space-time coordinates xA, [PA,RB ] = 0. Thus we see that in order to find a realization of conformal symmetries we should fix the operators A, MAB, and Ra. Realization of these operators on the space of ket-vector ) (7.14) is given by (for scalar field, |0) is given in (7.4))

MAB = aAaB - aBaA, (7.36)

A = + A', A' = Nve - Nvm. (7.37)

Ra = RA) + RA), (7.38)

RAo) = r0,xaA + AAro,i, RA) = rhiBA, (7.39)

r0,1 = 2ZezUe, r0,1 = -2u eezZ, r1,1 = -2u eUe, (7.40)

where operators a4a and are given in (A.18) and (A.19) respectively. Note that, for scalar field, we get MAB = 0, RA = r1t1BA, and A takes the same form as in (7.37).

so(d + 1, 2) algebra in bases of so(d) and so(d - 1, 1) subalgebras. In order to relate conformal fields in R^1 to the ones in (A)dSd+1, we use the Poincare parametrization of (A)dSd+1 given by

ds2 = — (nabdxadxb - edzdz), (7.41)

nab = (+, +,...,+), e = 1, a,b = 1,2,...,d, fordS (7.42)

nab = (-, +,...,+), e = -1, a,b = 0, 1,...,d - 1, forAdS (7.43)

Manifest symmetries of line element (7.41) are described, for the case of dS, by so(d) algebra and, for the case of AdS, by so(d -1, 1) algebra. Therefore it is reasonable to represent conformal symmetries in the bases of the respective so(d) and so(d - 1, 1) algebras. To this end we note that the Cartesian coordinates xA in Rd,x can be related to the Poincare coordinates z, xa in (7.41) by using the following identification of the radial coordinate z and Cartesian coordinates x0 and xd:

x0 = z, for dS

xd = z, for AdS (7.44)

For dS, the remaining Cartesian coordinates xA, A = 1, 2, ..., d, are identified with the Poincare coordinates xa, a = 1, 2, ..., d, while, for AdS, the remaining Cartesian coordinates xA, A = 0, 1, ..., d - 1, are identified with the Poincare coordinates xa, a = 0, 1, 2, ..., d - 1. In other words, taking into account the identifications in (7.44), we use the following spitting of the Cartesian coordinates xA into the Poincare coordinates z, xa:

xA = z,xa, a = 1, 2,...,d, fordS

xA = z,xa, a = 0, 1,...,d - 1, forAdS (7.45)

We note that the decomposition of the coordinate given in (7.45) implies the following decomposition of flat metric tensor and scalar products:

nAB = nzz,nab (7.46)

nzz = —e, nab = (e, +,...,+) (7.47)

nABXAYB = —eXzYz + XaYa, XaYa = nabXaYb. (7.48)

Also, the decomposition of coordinates in (7.45) implies the following decomposition of generators of the conformal algebra so(d + 1, 2) into generators of (A)dS space isometry symmetries and generators of (A)dS space conformal boost symmetries

Pa, D, Ka, Jab (A)dSd+1 space isometry symmetries (7.49)

Pz, Kz, Jza, (A)dSd+1 space conformal boost symmetries (7.50)

We note that, for dS, generators Jab in (7.49) span so(d) algebra, while, for AdS, generators Jab in (7.49) span so(d — 1, 1) algebra. Realization of generators (7.49), (7.50) in terms of differential operators is obtained from (7.32)-(7.35),

pa = da, Jab = xadb — xbda + Mab, (7.51)

D = xada + zdz + A, (7.52)

Ka = KaAM + Ra, (7.53)

KaAM (xbxb — ez2)da + xa D + Mabxb + eMzaz, (7.54)

Pz = -€dz, (7.55)

jza = zda + exa ^ + Mza, (7.56)

Kz = KZAM + Rz, (7.57)

KZA M = 2(exbxb - z2)dz + zD + Mzaxa, (7.58)

where expressions for the spin operators MAB = Mza, Mab, the conformal dimension operator A, and operator RA = Rz, Ra can be read from (7.36)-(7.40).

7.3. Relativistic symmetries of fields in (A)dSd+1

Relativistic symmetries of fields in (A)dSd+1 are described by the so(d + 1, 1) algebra for the case of dS and by so(d, 2) algebra for the case of AdS. For the description of field dynamics in (A)dSd+1, we used tensor fields of so(d, 1) algebra. However, as we prefer to realize the algebra of (A)dSd+1 space symmetries as subalgebra of conformal symmetries considered, for the case of dS, in the basis of so(d) algebra and, for the case of AdS, in the basis of so(d - 1, 1) algebra (see (7.49)), it is reasonable to represent the so(d + 1, 1) and so(d, 2) algebras in the respective bases of so(d) and so(d - 1, 1) algebras. We recall that so(d + 1, 1) and so(d, 2) algebras considered in the respective bases of so(d) and so(d - 1, 1) algebras consist of translation generators Pa, conformal boost generators Ka, dilatation generator D, and generators of the respective so(d)

and so(d - 1, 1) algebras, Jab. Commutators of generators of so(d + 1, 1) and so(d, 2) algebras take the form

D paj = —pa ^pa jbcj = ^abpc __ ^acpb

[D,Ka] = Ka, [Ka,Jbc] = nabKc - nacKb, (7.59)

[pa,Kb] = nabD - Jab, [Jab,Jce] = nbc Jae + 3 terms,

where nab is given in (7.42), (7.43). Realization of the generators in terms of differential operators acting on field propagating in (A)dS is well-known,

pa = da, Jab = xadb - xbda + Mab, (7.60)

D = xada + z9z, (7.61)

Ka = -1 (xbxb - ez2)da + xaD + Mabxb + eMzaz, (7.62)

where MAB = Mza, Mab is spin operator of the Lorentz algebra so(d, 1),

[MAB,MCE] = nBCMAE + 3 terms, (7.63)

MAB = 5 - aBaA, [aA,aB] = nAB■ (7.64)

8. Conformal transformation from fields in to fields in (A)dSrf+j

In this section, we demonstrate our method for the derivation of Lagrangian for conformal field in (A)dS. For scalar field, using the formulation of conformal field in Rd,x described in Section 7.1 and applying conformal transformation which maps conformal field in Rd,x to conformal field in (A)dSd+ 1, we obtain Lagrangian of scalar conformal field in (A)dSd+1. For arbitrary spin field, using the gauge transformation rule of conformal field in R<1,1 described in Section 7.2 and applying conformal transformation which maps the conformal field in R(1,1 to conformal field in (A)dSd+i, we obtain gauge transformation rule of the arbitrary spin conformal field in (A)dSd+i. Using then the gauge transformation rule of the arbitrary spin conformal field in (A)dS and our result in Section 6, we find the gauge invariant Lagrangian of the arbitrary spin conformal field in (A)dS. We consider the scalar and arbitrary spin conformal fields in turn.

8.1. Conformal transformation for scalar field

In this section, we discuss conformal transformation which maps scalar field in R^1 to the one in (A)dSd+1. It is convenient to realize the conformal transformation in two steps. We now discuss these steps in turn.

Step 1. Derivation of intermediate form of Lagrangian for conformal field in (A)dS. The conformal transformation which allows us to find intermediate Lagrangian for conformal field in (A)dS is found by matching generators of relativistic symmetries for (A)dS fields given in (7.60)-(7.62) and the ones for fields in flat space given in (7.51)-(7.53). Let us use the notation |0Rd,1} and I^A1^+j) for the respective ket-vectors of scalar fields in R^1 and (A)dSd+1. The ket-vectors are related by the conformal transformation given by

I0Rd1 ) = U intmKATdSd+1 >, (8.1)

where Umtm stands for operator of conformal transformation. This operator is found by matching the generators in (7.51)-(7.53) and the respective generators in (7.60)-(7.62). In other words, the operator Uintm is found by solving the following equations:

Pa\фкЛЛ ) = U iAdsd+1), (8.2)

ГЬ\фк*,1) = U intmJ аЬ|ф^+1), (8.3)

mRd,i ) = U "^ф^), (8.4)

Ka Wrj , i ) = U intmKa kiA)dsd+1 )■ (8.5)

Comparing (7.51) with (7.60), we see that Eqs. (8.2), (8.3) are already satisfied. All that remains is to solve Eqs. (8.4), (8.5). Solution to Eqs. (8.4), (8.5) is found to be

rintm i „i1-^ ет v \ „-А

Umtm = |p| Tex^—Xjz~a , (8.6)

r = 4u eUв, X = Щ, 9zj. (8.7)

Plugging (8.1), (8.6) into Lagrangian for conformai field in flat space (7.6), we get Lagrangian for conformal field in (A)dS,

£ = 1 dPlz2)* (0iAmdSd+i l(□ - ^ + i-U) - z2(v2 + kiAmdSd+i).

П = dada, V2 = V®V®- JNv + i) . (8.8)

We now recall that, in the Poincare coordinates (7.41), the D'Alembert operator for scalar field in (A)dSd+! takes the form

D2 =\p\z2(^ □ — e(af + (8.9)

Using (8.9), we see that Lagrangian (8.8) is the presentation of the following covariant La-grangian in terms of Poincare parametrization of (A)dS (7.41),

L = 2 ^iATdSd+: ID2 — -2ntm) kiAtmdSd+1), (8.10)

d2 ( n2

™2ntm = lP IV 9V 9 + pi — - N + - , (8.11)

intm i'- i ■ 1 \ 4 i - v ' 2

where we use the "deformed oscillators" V®, V® defined by the relations

V ® = (1 + ет)1/4и ®(1 + €T)1/4, V® = (1 + ет)1/4д ®(1 + ет)1'4, (8.12)

which can also be represented as

v ® = 1 {VT+7T,v®}, V ® = 1 {VTT7T,u ®}. (8.13)

We refer to Lagrangian (8.10) as intermediate Lagrangian. The intermediate Lagrangian describes conformal field in arbitrary parametrization of (A)dS space. Note that mass operator m2alm (8.11) entering the intermediate Lagrangian is not diagonal on space of ket-vector Ф^А^в^).

The mass operator can be diagonalized by an appropriate transformation of ^("A^i^+j). To this end we proceed to Step 2 of our procedure.

Step 2. Derivation of factorized form of Lagrangian for conformal field in (A)dS. In order to diagonalize the mass operator m2ntm (8.11) we make the following transformation12:

I0(A)dSd+1) = ^<№+1 ^ UU = 1 (8.14)

U = ]T uln#lxk-1 |0)(0|(ue)k(y®)n(v®)n,

(-e)n(l - n + k)!

uin =-ui, for l + n > k,

(l + n - k)!

uin = 0, for l + n < k,

(-)l-k ( 2l + 1 ^1/2

ul = ——- -:- . (8.15)

l k!(k - l)!\l!(k + l + 1)! /

Note that the intermediate ket-vector I^A'Jd^+j) defined in (8.1) depends on the oscillator v®, ve, while the new ket-vector |0(A)dSd+1) defined in (8.14) depends on new oscillators ft, x. For our oscillator algebra, see Appendix A. In terms of the new ket-vector |0(A)dSd+1), the intermediate Lagrangian given in (8.10) takes the same form as in (8.10) with the desired diagonalized mass operator,

L = 1 e(0(A)dSd+1 |(^2 - mLg)|0(A)dSd+1 ), (8.16)

m^iag - P^ - (n + 2)). (8.17)

In terms of |0(A)dSd+1) (8.14), constraint (7.5) takes the form

(Nft + Nx - k)|0(A)dSd+1) = 0. (8.18)

Constraint (8.18) implies that the ket -vectors |0(A)dSd+1) is decomposed into oscillators as

A ftk-k' xk'

^+1 )=j2o7ftrf(=)M0). (8.19)

Plugging (8.19) into (8.16) and using (A.9), we get component form of Lagrangian given in (2.2). 8.2. Conformal transformation for arbitrary spin field

In this section, we discuss conformal transformation which maps arbitrary spin conformal field in R4,1 to the one in (A)dSd+1. As in the case of scalar field, it is convenient to realize the conformal transformation in two steps. We now discuss these steps in turn.

Step 1. Derivation of intermediate form of Lagrangian for conformal field in (A)dS. The con-formal transformation which allows us to find intermediate Lagrangian for conformal field in (A)dS is found by matching generators of relativistic symmetries for fields in (A)dS given in

12 Note that Eqs. (8.2)-(8.5) do not fix the operator Uintm uniquely. This is to say that the Uintm is defined up to unitary

operator that is independent of space-time coordinates and derivatives.

(7.60)-(7.62) and the ones for fields in flat space given in (7.51)-(7.53). Using the notation \ф^,г) and ф®1^+j) for the respective ket-vectors of fields in Rd,1 and (A)dSd+b we consider the conformal transformation given by

\фRdл ) = U intm|фiAtmdSd+1), (8.20)

where Umtm stands for the operator of conformal transformation. This operator is found by matching the generators in (7.51)-(7.53) and the respective generators in (7.60)-(7.62). In other words, the operator Uintm is found by solving the equations given in (8.2)-(8.5). Comparing (7.51) with (7.60), we see that Eqs. (8.2), (8.3) are already satisfied. All that remains is to solve Eqs. (8.4), (8.5). Solution to Eqs. (8.4), (8.5) is found to be

Uintm = Ui U0, (8.21)

1 —d ( ет \ «

U1 = \p \ — expi — YX)Z , (8.22)

U0 = exp(—tcRf0)), (8.23)

т = 4ueUe, X = ij1,8z\. (8.24)

2s + d -5 — 2NZ'

RZ0) = r0,1 az + Azr0,1, Az = az — a2^ t ^ c „ лг az, (8.25)

sin rntc

!--Mil UJlc 2

cos rntc = V 1 + ет, -= 1, or = —ет, (8.26)

where relations (8.26) are considered as a definition of tc. For definition of r0,1, r0,1, see (7.40). Note that, in terms of Ф^^в^), constraints (7.16), (7.17) are represented as

(Na + N — s)^AmiSd+1) = 0, (N + Nu — ks) kiA7dSd+1) = 0,

(<* 2)>iA7dSd+1) = 0. (8.27)

Now our purpose is to find how gauge transformation of ф^д) given in (7.28) is realized on space Ф^)^!). To this end, adopting the notation \^Rd,1) for the gauge transformation parameter ) appearing in (7.24), (7.28), we make the following conformal transformation:

\^Rd1 ) = Uintm Р!1^^^). (8.28)

Using (8.20), (8.28) and adopting notation GRd,1 for operator G appearing in (7.28), we find the following relation:

GRd,1 \^Rd,1 ) = Uintm\p\1/2zGfAmdSd+1 I^ASSdH), (8.29)

where a new gauge transformation operator G'A)^ + appearing in (8.29) takes the form

G(AmdS = aD — e1ntm — a2-1-¿'ntm, (8.30)

(A)dSd+1 1 2Na + d — 1 1 ,

e1ntm = ZezV ®, gi1ltm = —V ®ezl, (8.31)

where the "deformed oscillators" V®, V® are defined in (8.12), while the ez is given in (A.19).

Thus we see that gauge transformation operator G(A)dSd+1 (8.30) takes the form we discussed in Section 6. This is to say that a knowledge of the explicit form of operators e1, e1 given in (8.31)

and relations in (6.9)-(6.12), (6.14), (6.21), (6.22) allows us to find explicit form of the gauge invariant Lagrangian. Plugging e1, e1 (8.31) into (6.21), we get the following two equivalent realizations of the operator m2ntm on space of I^Ads^ + ):

1 \2 ( d-4 x 2

<tm = IPV®V®- p(Nv + 2) + p[s +■

= |pIV®V® + pNZ(2s + d - 4 - NZ). (8.32)

Note that, for the derivation of representations for m?ltm in (8.32), one needs to use the second constraint in (8.27). Finally, we verify that operators e11ntm, (8.31), m2ntm (8.32) satisfy equations (6.18), (6.20).

To summarize, relations (8.30)-(8.32) together with the ones in (6.9)-(6.12), (6.14), (6.22) provide us the description of intermediate Lagrangian for arbitrary spin-s conformal field in (A)dSd+i. Note however that mass operator m^tm (8.32) entering the intermediate Lagrangian is not diagonal on space of ket-vector I^A'jdS+j). In other words, the intermediate Lagrangian

with mass operator m2ntm in (8.32) does not provide the factorized description of conformal field in (A)ds. The mass operator can be diagonalized by an appropriate unitary transformation of I^A'JdSi+i). To this end we proceed to Step 2 of our procedure.

Step 2. Derivation of factorized form of Lagrangian for conformal field in (A)dS. In order to diagonalize the operator m2ntm (8.32) we make the following transformation:

I0(A)dSd+1) = ^KATdSd+1) - UUt = 1, (8.33)

U =EH u(n)ftlxks'-ln(s-/)(D^ ks' (V ®) "(V®)

s'=0 l,n=0 (s) (-e)n(l - n + ks')! (s')

w,v =-u , for l + n > ks',

ln (l + n - ks')! l - - s-

u" = 0, for l + n<ks',

is') (-)l-ks' ( 2l +1 u) =- -

l ks'!(ks' - l)!\ l!(ks'+1 + 1)!,

f zs—s zs-s

n(s-s) = ,Z I0)/0I Z — (8.34)

Z \J(s - s0! \J(s - s')!-

where ks' is given in (7.9). Note that the intermediate ket-vector I^A^^+j) defined in (8.20) depends on the oscillators aA, Z, u®, ue, while the new ket-vector I0(A)dSd+1) defined in (8.33)

depends on oscillators aA, Z, $, X. In terms of I0(A)dSd+1), constraints (8.27) take the form

(Na + Nz - s)I0(A)dSd+1 ) = 0, (Nz + N$ + Nx - ks)I0(A)dSd+1 ) = 0,

(« 2)2I0(A)dSd+1 ) = 0. (8.35)

Constraints (8.35) are easily obtained from the ones in (8.27) by using (8.33). Note that the first and second constraints in (8.35) imply that I0(A)dSd+1) can be represented as in (5.12), (5.13).

Now our purpose is to find how gauge transformation of ^(Ads^ +) given in (8.30) is realized on space |0(A)dSd+1) given in (8.33). To this end, we make the following transformation of gauge transformation parameter:

£(A)dsd+i ) = ^|f(Atmsd+i )■ (8.36)

Using (8.30), (8.33), (8.36), we find the following relation:

G(A)dSd+i ^(A)dSd+i ) = ^(A)^ |^(At)dSd+J' (8.37)

where a new gauge transformation operator G(A)dSd+1 appearing in (8.37) takes the form

G(A)dSd+i = «D - ,1 - + d - 1 ¿1, (8.38)

ei = Zeix, ei = -xei£, (8.39)

ei = + d - 5 - 2Nz - Nx))1/2ez■ (8.40)

Gauge transformation operator (8.38)—(8.40) coincides with the one we presented in (5.38) in Section 5. All that remains to get the Lagrangian of conformal field in Section 5 is to find a realization of operator m2 on space of |^(A)dSd+1) in (8.33). To this end we should plug operators e1, e1 (8.39) into the general formula for m2 given in (6.21). Doing so, we get expression for m2 given in (5.21). Note that, for derivation of m2 via the general formula in (6.21), one needs to use the second constraint in (8.35).

Acknowledgement

This work was supported by the RFBR Grant No. 14-02-01172. Appendix A. Notation

The vector indices of the Lorentz algebra so(d, 1) take the values A, B, C, E = 0, 1, ■ ■■, d. We use the mostly positive flat metric tensor nAB. To simplify expressions we drop nAB in scalar products, i.e., we use the convention XAYA = nAB XAYB.

A covariant derivative DA is defined by the relations DA = nABDB,

Da = e'pDp, Dp = dp + 2mfM^, MAB = aAaB - aBaA, (A.1)

dp = d/dxp, where xp are the coordinates of (A)dSd+1 space carrying the base manifold indices, e'A is inverse vielbein of (A)dSd+1 space, Dp is the Lorentz covariant derivative and the base manifold index takes values p = 0, 1, ■ ■■, d. The mpB stands for the Lorentz connection of (A)dSd+1 space, while MA^ stands for a spin operator of the Lorentz algebra so(d, 1). Fields in (A)dSd+1 space carrying the flat indices, ■ ■ As, are related to contravariant tensor field, ■ ■ ps, in a standard way, ■ ■ As = eA ■ ■ ■ eAAss0p1 ■ ■ps. The D'Alembert operator of (A)dS space is defined as

,-, _ nA nA i rAABnB s+ABC_ „Aa BC _ Jat „A /1

□ (A)dS = D D + m D , m = e mp , e = dete(A.2)

Operator Dp given in (A.1) is acting on the generating function constructed out of the oscillators aA. Using the notation Vp for a realization of this operator on fields carrying flat indices we get

Vp0A = Bp$A + mf(e)^B. (A.3)

Instead of DA, we prefer to use a covariant derivative with the flat indices DA,

Da = eA D^, DA = nABDs, (A.4) [DA, Db= Rabce(A.5) where the Riemann tensor of (A)dS space is given by

TfABCE / AC BE AE BC\ ^

R = P\n n — n n ), (A.6)

e f 1 fordS „

P = R2, e = |—1 forAdS (A.7)

For the Poincare parametrization of (A)dSd+x space given in (7.41), the flat metric nAB takes

the form as in (7.46), (7.47), while vielbein eA = eA.dxA and Lorentz connection, deA + wAB A

eB = 0, are given by

^ = ZA < = 1 (nZX - ^ (A.8)

When using the Poincaré parametrization, the coordinates of (A)dSd+1 space xA 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 xA = S'^xA, where S1^ is the Kronecker delta symbol. With choice made in (A.8), the covariant derivative DA in (A.1) takes the form DA = R(zdA + MzA), dA = nABdB.

The Cartesian coordinates in Rd,x are denoted by xA, while derivatives with respect to xA are denoted by dA, dA = d/dxA.

Creation operators aA, Z, u®, ue, û, x and the respective annihilation operators aA, Z, u®, u®, û, x are referred to as oscillators. Commutation relations of the oscillators, the vacuum |0), and hermitian conjugation rules are defined as

[¿< A,aB ] = nAB, [Z ,Z] = 1, [u®,u®] = 1,

[u®,u®] = 1, [û ,û] = -e, [x,x] = e, (A.9)

a A|0) = o, Z|0) = 0, uu® |0) = 0,

uu® |0) =0, #|0) = 0, x. |0) = 0, (A.10)

aAt = aA, Zt = Z, u®t = u®,

uet = u®, ût = â, xt = X ■ (A.11)

The oscillators aA, aA and Z, Z, u®, u®, u®, u®, û, , x, XX transform in the respective vector and scalar representations of the Lorentz algebra so(d, 1). Throughout this paper we use operators constructed out of the derivatives and the oscillators,

aD = aADA, . D = aADA, (A.12)

a d = aAdA, â d =aAdA, (A.13)

a2 = aAaA, â2 = aAaA, (A.14)

Na = aAaA, NZ = ZZ, Nû = -û, Nx = exx, (A.15)

Nu ® = u ®u®, Nu® = u ®u®, Nu = Nu ®+ Nu®, (A.16)

a = 1 - 1a2â2, n[1,2] = 1 - a2-1-â2, (A.17)

4 2(2N +d +1)

2Na +d - 1

AA = aA - a2-äA, (A.18)

(2s + d - 3 - NK\1/2

M o. , , a J • (A.19)

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