Long, partial-short, and special conformal fields

In the framework of metric-like approach, totally symmetric arbitrary spin bosonic conformal fields propagating in flat space-time are studied. Depending on the values of conformal dimension, spin, and dimension of space-time, we classify all conformal field as long, partial-short, short, and special conformal fields. An ordinary-derivative (second-derivative) Lagrangian formulation for such conformal fields is obtained. The ordinary-derivative Lagrangian formulation is realized by using double-traceless gauge fields, Stueckelberg fields, and auxiliary fields. Gauge-fixed Lagrangian invariant under global BRST transformations is obtained. The gauge-fixed BRST Lagrangian is used for the computation of partition functions for all conformal fields. Using the result for the partition functions, numbers of propagating D.o.F for the conformal fields are also found.


Introduction
A study of arbitrary spin conformal fields was initiated in ref. [1], where a Lagrangian description of totally symmetric conformal fields in space-time R 3,1 (Fradkin-Tseytlin fields) was developed. A Lagrangian formulation of totally symmetric conformal fields in spacetime R d−1,1 for arbitrary d was developed in ref. [2]. Throughout this paper conformal fields studied in refs. [1,2] will be referred to as short conformal fields. For the reader's convenience, we recall that, in the framework of AdS/CFT correspondence, the short conformal fields in R d−1,1 are dual to non-normalizable modes of massless fields in AdS d+1 . Namely, for spin-2 and spin-s, s ≥ 2, fields it was demonstrated in the respective ref. [5] and ref. [6] that ultraviolet divergence of an action of bulk AdS field evaluated on a solution of the Dirichlet problem coincides with an action of the short conformal field. Besides the short conformal fields, there are conformal fields which we will refer to as long, partial-short and special conformal fields in this paper (for definition, see below). We note then that a minimal Lagrangian formulation of long, short, partial-short and special conformal fields may be found in ref. [3]. 1

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In this paper, we study the long, partial-short and special conformal fields. Such conformal fields are also interesting, among other things, in the context of AdS/CFT correspondence. This is to say that, for arbitrary spin fields, it was demonstrated explicitly in ref. [7], that the long conformal field in R d−1,1 is dual to non-normalizable modes of massive field in AdS d+1 having some discrete value of mass parameter. Our speculation on string theory interpretation of a conjectural model that involves long higher-spin conformal fields and short low-spin conformal fields may be found in the conclusions of this paper. Before we formulate our main aim in this paper let us discuss a terminology we use throughout this paper.
Consider a free totally symmetric conformal bosonic field propagating in R d−1,1 . If a Lagrangian of the conformal field is built in terms of one traceless totally symmetric rank-s tensor field of the Lorentz algebra so(d − 1, 1) then the conformal field will be referred to as spin-s conformal field, while the Lagrangian will be referred to as minimal Lagrangian. If the minimal Lagrangian of free conformal field involves 2κ derivatives, where κ ≥ 1 is arbitrary integer, then, as well known, a conformal dimension of the conformal field is given by the expression The use of the labels κ, s, d, and our ordinary-derivative approach allows us to classify all conformal fields propagating in R d−1,1 . Namely, depending on values of the arbitrary integer κ ≥ 1 and the arbitrary integer s ≥ 1, conformal fields in R d−1,1 with arbitrary d ≥ 3 and ∆ as in (1.1) will be referred to as long, short, partial-short, and special conformal fields. Result of our classification of conformal fields is summarized in table 1 (see next page).
For the reader's convenience, we now recall the references devoted to the study of the minimal Lagrangian formulation of conformal fields given in table 1.
i) For d = 4 and d ≥ 4, the minimal Lagrangian of the totally symmetric arbitrary spin short conformal field in table 1 was obtained in the respective ref. [1] and ref. [2] (see also ref. [6]).
ii) For d = 4 and d ≥ 4, the minimal Lagrangian of the totally symmetric arbitrary spin conformal fields with κ = 1 in table 1 was obtained in the respective ref. [8] and ref. [9].
iii) Minimal Lagrangian for all totally symmetric conformal fields given in table 1 can be found in ref. [3].
With the exception of the particular case κ = 1, the minimal Lagrangian of the totally symmetric conformal fields involves higher-derivatives. Also we note that, with the exception of the short and partial-short conformal fields, the minimal Lagrangian turns out to be gauge variant. odd d ≥ 7, s ≥ 1 Table 1. Classification of conformal fields in R d−1,1 . The integer s indicates spin of conformal field, while the integer κ is related to conformal dimension as ∆ = d 2 − κ.
Our main aim in this paper is to construct the second-derivative (ordinary-derivative) gauge invariant Lagrangian for all conformal fields given in table 1. We note that the ordinary-derivative description of the short conformal fields was obtained in refs. [10,11]. In other words, in this paper we extend approach in refs. [10,11] to the cases of the long, partial-short and special conformal fields. Our Lagrangian formulation of conformal fields has the following two attractive features.
i) For spin-1, spin-2, and spin-s, s > 2, conformal fields, two-derivative contributions to our ordinary-derivative Lagrangian take the form of the standard Maxwell, Einstein-Hilbert, and Fronsdal kinetic terms.

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ii) In our approach, all vector and tensor fields are supplemented by appropriate gauge transformations which do not involve higher than first order terms in derivatives. Also we note that the one-derivative contributions to the gauge transformations of all vector and tensor fields take the form of standard gradient gauge transformations. This paper is organized as follows.
In section 2, we summarize conventions and notation we use in this paper. In section 3, we briefly review a minimal Lagrangian formulation of spin-s conformal field in R d−1,1 . We present the minimal Lagrangian for arbitrary values of κ, s, and d and then we discuss the Lagrangian for some particular values of κ, s, and d. Also we discuss how the minimal Lagrangian can be obtained in the framework of AdS/CFT correspondence. Section 4 is devoted to an ordinary-derivative Lagrangian formulation of conformal fields. First, we discuss a field content entering our approach. Second, we present our gauge invariant Lagrangian for all conformal fields in table 1.
In section 5, we describe gauge symmetries of our ordinary-derivative Lagrangian. We start with a discussion of gauge transformation parameters entering our approach and then we present gauge transformations of conformal fields. In our ordinary-derivative approach, only symmetries of the Lorentz algebra so(d − 1, 1) are realized manifestly. Therefore, in section 6, in order to complete our ordinary-derivative formulation, we discuss a realization of the conformal algebra symmetries on a space of the gauge fields entering our approach.
In section 7, using the Faddeev-Popov procedure, we obtain various representations for ordinary-derivative gauge-fixed BRST Lagrangian. Excluding auxiliary gauge fields and auxiliary Faddeev-Popov fields we obtain a higher-derivative BRST Lagrangian and use such Lagrangian for a computation of partition functions for all conformal fields.
In section 8, we discuss directions for future research.

Notation and conventions
Our notation and conventions are as follows. Coordinates of the space-time R d−1,1 are denoted by x a , while derivatives with respect to x a are denoted by ∂ a , ∂ a ≡ ∂/∂x a . We use vector indices a, b, c, e of the Lorentz algebra so(d − 1, 1) which take the following values a, b, c, e = 0, 1, . . . , d − 1. Our flat metric tensor η ab is mostly positive. In scalar products, to simplify our expressions we drop the metric tensor η ab . In other words, we use the convention X a Y a ≡ η ab X a Y b . Throughout this paper a set of creation operators α a , α z , ζ, α ⊕ , α ⊖ and the respective set of annihilation operatorsᾱ a ,ᾱ z ,ζ,ᾱ ⊖ ,ᾱ ⊕ are referred to as oscillators. We adopt the following conventions for commutation relations, the vacuum, and hermitian conjugation rules

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The oscillators α a ,ᾱ a transform in the vector representation of the Lorentz algebra so(d − 1, 1), while the oscillators α z ,ᾱ z , ζ,ζ, α ⊕ ,ᾱ ⊖ , α ⊖ ,ᾱ ⊕ transform in the scalar representation of the Lorentz algebra. A hermitian conjugation rule for the derivatives is given by ∂ a † = −∂ a . We use the following shortcuts for operators constructed out of the oscillators, the derivatives ∂ a and the coordinates x a : Throughout this paper we adopt the following conventions and notation: λ ∈ [p, q] 2 ⇐⇒ λ = p, p + 2, . . . , q − 2, q . (2.13)

Global conformal symmetries
In the space-time R d−1,1 , the so(d, 2) algebra is realized as algebra of conformal symmetries. In a basis of the Lorentz algebra so(d − 1, 1), the generators of the so(d, 2) algebra are decomposed into the translation generators P a , the dilatation generator D, the conformal boost generators K a , and the generators of Lorentz algebra so(d − 1, 1) denoted by J ab . We use the following commutators of the so(d, 2) algebra: (2.14) Consider conformal fields propagating in R d−1,1 . Let us collect all scalar, vector and tensor fields required for a Lagrangian description of the conformal fields into a ket-vector |φ . If a Lagrangian is invariant with respect to conformal algebra transformation (invariance of a Lagrangian is assumed to be up to total derivatives)

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then we can present a realization of the conformal algebra generators G in terms of differential operators acting on |φ in the following way In general, this operator depends on the derivatives ∂ a . 3 In the framework of minimal Lagrangian formulation of conformal fields, the operator R a is equal to zero, while, in the framework of our ordinary-derivative approach, the operator R a turns out to be non-trivial. From relations (2.16)- (2.19), we see that all that is required for the complete description of conformal symmetries is to find a realization of the operators ∆, M ab , and R a on a space of the ket-vector |φ .

Review of minimal Lagrangian formulation of conformal fields
To discuss the minimal Lagrangian formulation of a conformal field with arbitrary integer spin s ≥ 1 and arbitrary integer κ ≥ 1 we use a field φ a 1 ...as which is totally symmetric traceless rank-s tensor field of the Lorentz algebra so(d − 1, 1), Conformal dimension of the field φ a 1 ...as is given in (1.1). The minimal Lagrangian found in ref. [3] can be presented as

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We recall that (p) q defined in (3.5) is the Pochhammer symbol. From (3.2) we see that the minimal Lagrangian involves 2κ derivatives. For the reader's convenience, we note that the leading terms entering minimal Lagrangian (3.2) are given by The following remarks are in order.
i) For the short and partial-short conformal fields given in where a gauge transformation parameter ξ a t+2 ...as is a rank-(s − 1 − t) traceless totally symmetric tensor field of the Lorentz algebra so(d − 1, 1) and we use a projector Π tr to respect the tracelessness constraint (3.1).
ii) For the long and special conformal fields, minimal Lagrangian (3.2) is gauge variant.
iii) With the exception of the particular case κ = 1, minimal Lagrangian (3.2) involves higher-derivatives.
iv) Lagrangian (3.2) is invariant under the conformal algebra transformations presented in (2.16)- (2.19). We note then that a realization of spin operator M ab (2.20) on a space of the traceless tensor field φ a 1 ...as of the Lorentz algebra so(d−1, 1) is well known. Realization of the operator ∆ on a space of the traceless field φ a 1 ...as is given by (1.1). We note also that the operator R a is trivially realized on a space of the traceless field φ a 1 ...as , i.e., R a = 0.
We now discuss minimal Lagrangian (3.2) for some particular values of κ, s, and d.
Conformal spin-s field with arbitrary integer s and κ = s + d−4 2 , d-even. According to table 1 such conformal field is referred to as short conformal field. For the short conformal field, Lagrangian (3.2) takes the form where we use the notation as in (3.3), (3.5). Lagrangian (3.8) is invariant under gauge transformations given in (3.7) with t = 0. Representation for the minimal Lagrangian of the short conformal field given in (3.8) was obtained in ref. [6]. Alternative representations for the minimal Lagrangian of the short conformal field may be found in refs. [1,2].
Conformal spin-1 field with arbitrary integer κ ≥ 1. For this case, Lagrangian (3.2) takes the following form Alternatively, Lagrangian (3.10) can be represented the form similar to the Proca, Lagrangian , (3.14) where we introduce formally coupling constant g (3.13) and dimensionless mass parameter m (3.14). For κ = 1, . . . , (d − 4)/2, d ≥ 6, we have m 2 = 0 and Lagrangian (3.11) is gauge variant. According to our classification in table 1, for κ = 1 and d ≥ 6, we refer to the field φ a as the special conformal field, while, for κ = 2, . . . , (d − 4)/2 and d ≥ 8, the field φ a is referred to as the secondary long conformal field. For κ = (d − 2)/2, we get m 2 = 0 and this case corresponds to a spin-1 short conformal field with gauge invariant Lagrangian in (3.11). For κ > s − 2 + [d/2] and d ≥ 3, the Lagrangian given in (3.11) is gauge variant and the field φ a is referred to as the long conformal field.
Minimal Lagrangian of conformal field from AdS/CFT correspondence. Minimal Lagrangian (3.2) can be obtained in the framework of AdS/CFT correspondence. We recall that, in the framework of AdS/CFT correspondence, conformal field that propagates in R d−1,1 and has conformal dimension as in (1.1) is dual to a non-normalizable mode of bulk field that propagates in AdS d+1 and has lowest eigenvalue of an energy operator equal to E 0 = κ + d 2 . Let us refer to an action of AdS field evaluated on a solution of the Dirichlet problem as effective action. We will denote the effective action as S eff . For arbitrary values of κ, s, and d, the effective action for spin-s field in AdS d+1 was found in ref. [19] and is given by where a 2-point function Γ stand appearing in (3.19) is defined by the relations From (3.20), (3.21), we see that Γ stand is a standard 2-point CFT function for a boundary shadow field φ a 1 ...as which has the conformal dimension given in (1.1). For massless and massive spin-1 and spin-2 fields, the normalization factor appearing in front of Γ stand in (3.19) is in agreement with the results obtained in the earlier literature (see refs. [5,[20][21][22]) For integer values of κ, the 2-point function (3.21) is not well defined (see, e.g., refs. [23,24]). Using the regularization κ → κ − ǫ, ǫ ∼ 0, and the well known expression for UV divergence of the regularized kernel entering 2-point function Γ stand ,

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we verify that UV divergence of the 2-point function Γ stand (3.20) takes the form where Lagrangian appearing in (3.23) is nothing but the minimal Lagrangian given in (3.2). Plugging (3.23) into (3.19) we see that the UV divergence of the effective action is proportional to the minimal action of conformal field From relation (3.24), we see that UV divergence of the effective action for field in AdS d+1 with integer value of κ is indeed realized as the minimal action for conformal field in R d−1,1 .

Ordinary-derivative gauge invariant Lagrangian of conformal field
Field content for long, partial-short and special conformal fields. In order to develop an ordinary-derivative gauge invariant metric-like formulation of totally symmetric arbitrary spin-s conformal field that propagates in R d−1,1 and has conformal dimension given in (1.1), we introduce the following set of real-valued scalar, vector, and tensor fields of the Lorentz algebra so(d − 1, 1): where labels s ′ , λ, k ′ take the following values κ − 1 + λ ≥ 0, for long and secondary long conformal fields; (4.2) for special conformal fields; (4.3) for type I partial-short conformal fields; (4.4) for type II partial-short conformal fields; (4.5) for short conformal fields. (4.6)

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Relation p ∈ [q] 2 appearing in (4.2)-(4.6) is defined in (2.11). We recall also that values of s and κ for the various conformal fields are defined in table 1.
The following remarks are in order.
i) In the catalogue (4.1)-(4.6), fields φ a 1 ...a s ′ λ,k ′ with s ′ = 0 and s ′ = 1 are the respective scalar and vector fields of the Lorentz algebra so(d − 1, 1), while fields φ are totally symmetric rank-s ′ tensor fields of the Lorentz algebra. By definition, the tensor fields φ ii) Conformal dimensions of the fields φ a 1 ...a s ′ λ,k ′ (4.1) are given by the relation iii) Taking into account the restrictions on the label λ given in the second lines in (4.3)-(4.5), we see that the domains of values of the label λ for the special and partial-short fields (4.3)-(4.5) are obtained by decreasing the domain of values of the label λ for the long fields in (4.2). In other words, the field contents of the special, type I partial-short and type II partial-short conformal fields are obtained from the field content of the long conformal fields by setting to zero those fields in (4.2) whose values of λ do not respect constraints in the second lines in the respective relations (4.3), (4.4), and (4.5). It is the restrictions on λ appearing in the second lines in (4.3)-(4.5) and AdS/CFT dictionary that motivate us to classify fields into the special, types I and II partial-short conformal fields (see below). Also we see that the field content of the short conformal field is obtained by setting to zero all fields in (4.2) with λ = s ′ − s. The field content entering the ordinaryderivative formulation of the short conformal fields has been found in ref. [11].
iv) The terminology we use in this paper is inspired by AdS/CFT dictionary. Namely, a conformal field that propagates in R d−1,1 and has conformal dimension as in (1.1) is dual to a non-normalizable mode of bulk field that propagates in AdS d+1 and has lowest eigenvalue of an energy operator equal to E 0 = κ + d 2 . Our long and secondary long conformal fields (4.2) are dual to AdS massive fields associated with the respective unitary and nonunitary irreps of the so(d, 2) algebra. Conformal field in (4.3) is related to AdS massive field associated with non-unitary irrep of the so(d, 2) algebra. In view of the restriction on λ in the second line in (4.3) we refer to such conformal field as special conformal field. Conformal fields in (4.4), (4.5) are dual to AdS partial-massless fields. 5 In view of the restrictions on λ in the second lines in (4.4) and (4.5) we refer to such conformal fields as the respective type I and type II partial-short conformal fields. Conformal field in (4.6) is dual to massless AdS field. Therefore we refer to such conformal field as short conformal field. 5 Arbitrary spin partial-massless fields in AdS4 were first studied in refs. [26,27]. Generalization of results in the latter references to AdS d+1 , d ≥ 3, may be found in refs. [28,29]. Discussion of various aspects of free and interacting partial-massless AdS fields may be found in the respective refs. [30][31][32][33] and refs. [34][35][36].

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v) For the special and partial-short conformal fields, the restriction κ − 1 + λ ≥ 0 appearing in (4.2)-(4.5) is satisfied automatically, while, for some long conformal field in R 2,1 , this restriction leads to a constraint on the field content. Namely, using expression for κ corresponding to the long conformal field in R 2,1 with N = 0 (see table 1), we find the relation κ = s and note that, for s ′ = 0 and λ = −s, the restriction κ − 1 + λ ≥ 0 is not satisfied. This implies that the fields with s ′ = 0 (scalar fields) and λ = −s do not enter the field content of the spin-s long conformal field in R 2,1 that has κ = s, (the case of N = 0 in table 1). We note then that, for a short conformal field in R 3,1 , the restriction k s ′ ≥ 0 appearing in (4.6) also leads to some constraint on the field content. Namely, using expression for k s ′ in (4.6), we see that for d = 4 and s ′ = 0, the constraint k s ′ ≥ 0 is not satisfied. This implies that fields with s ′ = 0 (scalar fields) do not enter the field content of the ordinary-derivative formulation of the short conformal field in R 3,1 .
vi) For d = 4, it is easy to see that the restrictions on λ in the second line in (4.5) lead to the restriction s ′ ≥ 1. This implies scalar fields do not enter field content of the ordinary-derivative formulation of the type II partial-short conformal fields in R 3,1 .
Generating form of field content. In order to obtain a gauge invariant description of a conformal field in an easy-to-use form, we use oscillators α a , α z , ζ, α ⊕ , α ⊖ and collect all fields given in (4.2)-(4.6) into the following ket-vector: where basis ket-vectors |φ s ′ λ,k ′ appearing in (4.9) take the form and, depending on the type of the conformal field, the summation indices s ′ , λ, k ′ in (4.9) run over values given in (4.2)-(4.6). Note that, for the short conformal field, the λ is fixed (see (4.6)). From relations (4.9), (4.10), it is easy to see that the ket-vector |φ satisfies the following algebraic constraints while the basis ket-vectors |φ s ′ λ,k ′ (4.10) satisfy the algebraic constraints where a definition of the operators N α , N ζ , etc., may be found in relations (2.6), (2.7).

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From relation (4.11), we learn that the ket-vector |φ is a degree-s homogeneous polynomial in the oscillators α a , α z , ζ. Relation (4.13) tells us that the basis ket-vector |φ s ′ λ,k ′ is a degree-s ′ homogeneous polynomial in the oscillators α a . From relations (4.14) and (4. 15) we learn that the ket-vector |φ s ′ λ,k ′ is an eigenvector of the respective operators N ζ − N α z and N α ⊖ − N α ⊕ . Also we note that, in terms of the ket-vector |φ , constraint given in (4.7) takes the following form Remark on the short conformal fields. As we have already said, the field content entering the ordinary-derivative formulation of the short conformal fields given in (4.1), (4.6) has been found in ref. [11]. For the reader's convenience and in order to match result in ref. [11] with the presentation in this paper we now write down the explicit form of the ket-vector |φ for the short conformal field. Such explicit form is obtained by plugging κ = s + d−4 2 and the labels s ′ , λ, k ′ given in (4.6) into ket-vector (4.9), (4.10). Doing so, we get where k s ′ is defined in (4.6).
Comparing the basis ket-vectors in (4.10) and (4.18), we see that the basis ket-vectors of the short conformal field (4.18) do not depend on the oscillator ζ. 6 This implies that ket-vector |φ (4.17) satisfies the following algebraic constraints while basis ket-vectors |φ s ′ s ′ −s,k ′ (4.18) satisfy the algebraic constraints Gauge invariant Lagrangian. We now discuss an ordinary-derivative gauge invariant Lagrangian for all conformal fields given in table 1. We find the following representation for 6 We note the clash of the notation and conventions in this paper and in ref. [11]. Namely, the fields in ref. [11]. Also we note that the ket-vector |φ of the short conformal field in ref. [11] is obtained from (4.17), (4.18) by using the replacements α z → ζ,

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action and ordinary-derivative Lagrangian in terms of the ket-vector |φ above discussed: for secondary long and special fields in R d−1,1 , d-odd; Note that, on a space of the ket-vector |φ , one has the relation β 2 = 1. Appearance of the operator β in (4.30) is related to the fact that our Lagrangian is constructed out of real-valued fields of the Lorentz algebra so(d − 1, 1). Only for the long and short conformal fields, eigenvalues of the operator β are strictly positive (4.31). For the secondary long, special, and partial-short conformal fields, we see that, depending on eigenvalues of the operator N ζ on a space of ket-vector |φ (4.9), a spectrum of the operator β involves both the positive and negative eigenvalues. For the case of the long and short fields, the strictly positive spectrum of the operator β can intuitively be explained by the fact that, in the framework of AdS/CFT, the long and short conformal fields in R d−1,1 are related to unitary massive and massless fields in AdS d+1 . Accordingly, the appearance of the both positive and negative eigenvalues of β for the secondary long, special, and partial-short conformal JHEP05(2016)096 fields can intuitively be explained by the fact that, in the framework of AdS/CFT, the secondary long and special conformal fields in R d−1,1 are related to non-unitary massive fields in AdS d+1 , while the partial-short conformal fields in R d−1,1 are related to partialmassless fields in AdS d+1 which are also non-unitary.
iii) The quantity |Lφ appearing in (4.25) is defined as |Lφ ≡L|φ , while L φ| is defined as L φ| ≡ (|Lφ ) † β. We note that if, in expression forL (4.27), we set e 1 = 0,ē 1 = 0, then the quantity |Lφ becomes the standard de Donder divergence entering Lagrangian of a massless field in R d−1,1 . 7 For this reason, the quantity |Lφ withL as in (4.27) we refer to as modified de Donder divergence. Obviously, it is the use of the modified de Donder divergence that allows us to simplify significantly our representation for the gauge invariant Lagrangian given in (4.25).
iv) If, in Lagrangian (4.25), we set M 2 = 0, e 1 = 0,ē 1 = 0, then we are left with two derivative contributions to the Lagrangian. It is easy to see that, for spin-1, spin-2, and spin-s, s > 2, fields, those two-derivative contributions take the form of the respective Maxwell, Einstein-Hilbert, and Fronsdal kinetic terms.
v) Using the above-given explicit expressions for the operators e 1 ,ē 1 , and β, we check that, on a space of the ket-vector |φ , the following hermitian conjugation rules hold true. For the derivation of relations (4.40), we use the fact that the ket-vector |φ satisfies the relation e 2πiN ζ |φ = |φ which, in turn, implies the relation β 2 |φ = |φ . 7 Interesting applications of the standard de Donder divergence for studying various aspects of higherspin field theory may be found in refs. [37][38][39][40][41]. We think that our modified de Donder gauge could be useful for the computations in conformal higher-spin field theory discussed in ref. [42].

JHEP05(2016)096 5 Gauge symmetries of conformal fields in ordinary-derivative approach
Gauge transformation parameters for long, partial-short, short, and special conformal fields. In order to discuss gauge symmetries of the ordinary-derivative Lagrangian (4.25), we introduce the following set of scalar, vector, and tensor gauge transformation parameters: ξ where labels s ′ , λ, k ′ take the following values κ − 1 + λ ≥ 0, for long and secondary long conformal fields; (5.2) for special conformal fields; for type I partial-short conformal fields; (5.4) for type II partial-short conformal fields; (5.5) In order to simplify a presentation of gauge symmetries we use the oscillators α a , α z , ζ, α ⊕ , α ⊖ and collect all gauge transformation parameters given in (5.2)-(5.6) into the following ket-vector where basis ket-vectors appearing in (5.11) take the form and, depending on the type of the conformal field, the summation indices s ′ , λ, k ′ in (5.11) run over values given in (5.2)-(5.6).
Using relations (5.11), (5.12), it is easy to see that ket-vector |ξ (5.11) satisfies the following relations while the basis ket-vectors |ξ s ′ λ,k ′ (5.12) satisfy the relations From relation (5.13), we learn that the ket-vector |ξ is a degree-(s − 1) homogeneous polynomial in the oscillators α a , α z , ζ. Relation (5.15) tells us that the basis ket-vector |ξ s ′ λ,k ′ is a degree-s ′ homogeneous polynomial in the oscillators α a . From relations (5.16) and (5.17) we learn that the ket-vector |ξ s ′ λ,k ′ is an eigenvector of the respective operators N ζ − N α z and N α ⊖ − N α ⊕ . Also we note that, in terms of the ket-vector |ξ , constraint given in (5.7) takes the following formᾱ

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Remark on gauge symmetries of short conformal fields. Gauge transformation parameters (5.1), (5.6) entering the ordinary-derivative formulation of the short conformal fields have been found in ref. [11]. In order to match result in ref. [11] with the presentation in this paper we now write down an explicit form of the ket-vector |ξ for gauge transformation parameters of the short conformal field. Such explicit form is obtained by plugging κ = s + d−4 2 and labels s ′ , λ, k ′ given in (5.6) into ket-vector |ξ (5.11), (5.12). Doing so, we get where k s ′ is given in (5.6).
Comparing the basis ket-vectors in (5.12) and (5.20), we see that the basis ket-vectors for gauge transformation parameters (5.20), which are related to gauge symmetries of the short conformal field, do not depend on the oscillator ζ. 8 This implies that ket-vector |ξ (5.19) satisfies the following algebraic constraints while basis ket-vectors |ξ s ′ λ,k ′ (5.20) satisfy the algebraic constraints Gauge transformations of conformal fields. The use of the ket-vector |φ for the description of the conformal fields and the ket-vector |ξ for the description of the gauge transformation parameters allows us to present gauge transformations of all conformal fields on an equal footing. Namely, gauge transformations of the long, partial-short, short and special conformal fields can entirely be presented in terms of the ket-vectors |φ , |ξ in the following way: where the operators e 1 ,ē 1 appearing in (5.26) are given by relations (4.36)-(4.39). From (5.26), we see the following two characteristic features of the gauge transformations in our approach to all conformal fields listed in table 1. 8 We note the clash of the notation and conventions in this paper and in ref. [11]. Namely the gauge transformation parameters ξ a 1 ...a s ′ s ′ +1−s,k ′ appearing in (5.20) are denoted by ξ a 1 ...a s ′ k ′ −1 in ref. [11]. Also we note that the ket-vector |ξ in ref. [11] is obtained from (5.19), (5.20) by using the following replacements:

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i) The gauge transformations of fields do not involve higher than first order terms in derivatives.
ii) The one-derivative contributions to the gauge transformations of fields take the form of standard gradient gauge transformations.

Realization of conformal symmetries in ordinary-derivative approach
The dynamics of conformal fields propagating in R d−1,1 should respect the conformal algebra so(d, 2) symmetries. Note however that, in our ordinary-derivative approach to conformal fields, only the Lorentz algebra so(d − 1, 1) symmetries are realized manifestly. This implies that in order to complete our ordinary-derivative formulation of the conformal fields we should provide a realization of the conformal algebra symmetries on a space of conformal fields. As we have already said, from relations (2.15)- (2.19), we see that all that is required to complete a description of the conformal symmetries is to find a realization of the operators ∆, M ab , and R a on a space of the ket-vector |φ . Our ket-vector |φ is built in terms of the oscillators (see relations (4.9), (4.10)). For such ket-vector, a realization of the spin operators M ab of the Lorentz algebra is well known and is given by the following relation: A realization of the conformal dimension operator ∆ on a space of the |φ can be read from relations (4.8), (4.15), (4.23) and is given by Realization of the operator R a on a space of the ket-vector |φ we find is given by R a = r 0,1ᾱ a + A ar 0,1 + r 1,1 ∂ a , (6.3)
The following remarks are in order.
i) We verify that the conformal boost operator K a (2.19) with the operator R a given in (6.3)-(6.6) satisfies the commutator [K a , K b ] = 0.

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iii) Using relations for the operator R a presented in (6.3)-(6.6), (4.36)-(4.39) and (2.9), (2.10) we verify that the operator R a is indeed acting on space of ket-vector |φ (4.9) with the values of the labels s ′ , λ, k ′ given in (4.2)-(4.6). Note also that, using the explicit expressions for the operators r ζ , r z (2.9), (2.10) one can check that the field contents of the special, partial-short, and short conformal fields in (4.3)-(4.6) can be realized as invariant subspaces in the field content of the long conformal field in (4.2). In this respect there is full analogy with massive and massless fields. As is well known a field content of a massless field can be realized as an invariant subspace in a field content of a massive field when a mass parameter tends to zero. In our case, the field contents of the special, partial-short, and short conformal fields in (4.3)-(4.6) are realized as invariant subspaces in the field content of the long conformal field in (4.2) when the parameter κ takes the respective values given in table 1.
iv) A complete ordinary-derivative Lagrangian formulation of a conformal field implies finding a Lagrangian, gauge symmetries and the operator R a . We note that gauge symmetries taken alone do not admit to fix an ordinary-derivative Lagrangian uniquely. It turns out that in order to determine an ordinary-derivative Lagrangian, gauge symmetries, and the operator R a uniquely we should analyse restrictions imposed by both gauge symmetries and the conformal algebra so(d, 2) symmetries. The general procedure for finding an ordinary-derivative Lagrangian by using restrictions imposed by gauge symmetries and the conformal algebra so(d, 2) symmetries has been developed in appendix B in ref. [11]. In the latter reference, we used our general procedure for finding a Lagrangian formulation for the short conformal fields. Our general procedure in ref. [11] is applied to the cases of the long, partial-short, and special conformal fields in a rather straightforward way.
v) Our ordinary-derivative approach involves Stueckelberg and auxiliary fields. The Stueckelberg fields can be removed by using the gauge symmetries in our approach, while the auxiliary fields can be removed by using equations of motion. Doing so, one can make sure that our ordinary-derivative Lagrangian leads to the minimal higher-derivative Lagrangian given in (3.2). The general procedure for matching ordinary-derivative and higher-derivative Lagrangian formulations was developed in section 5 in ref. [11]. Our general procedure in ref. [11] can straightforwardly be used for the cases of the long, partialshort, and special conformal fields. For the reader's convenience, in section 7.2 in this paper, by using example of BRST Lagrangian, we will demonstrate how higher-derivative BRST Lagrangian is obtained from the ordinary-derivative BRST Lagrangian.

BRST Lagrangian and partition functions of conformal fields
In this section, by using the gauge invariant Lagrangian obtained in section 4 and the standard Faddeev-Popov procedure, we obtain a gauge-fixed Lagrangian of conformal fields which is invariant under global BRST transformations. After this we use the gauge-fixed BRST Lagrangian for a derivation of partition functions of conformal fields.

BRST Lagrangian of conformal fields
A general structure of our ordinary-derivative gauge invariant Lagrangian (4.25) and gauge transformations (5.26) for conformal fields is similar to the one for massive fields. 9 For the case of arbitrary spin massive fields, gauge-fixed BRST Lagrangian was obtained in ref. [47]. Result in the latter reference is straightforwardly extended to the case of conformal fields.
Let us now to discuss our result for gauge-fixed BRST Lagrangian of conformal fields.
To discuss BRST Lagrangian of conformal fields we introduce a set of Faddeev-Popov fields and Nakanishi-Lautrup fields, where, depending on the type of the conformal field, the labels s ′ , λ, k ′ take the same values as the ones for gauge transformation parameters given in (5.2)-(5.6). In (7.1), (7.2), the fields with s ′ = 0 and s ′ = 1 are the respective scalar and vector fields of the Lorentz algebra so(d − 1, 1), while the fields with s ′ > 1 are traceless totally symmetric rank-s ′ tensor fields of the Lorentz algebra.
To describe the Faddeev-Popov fields and the Nakanishi-Lautrup fields in an easyto-use form we collect the Faddeev-Popov fields (7.1) into ket-vectors |c , |c , while the Nakanishi-Lautrup fields (7.2) are collected into a ket-vector |b . We note then that the ket-vectors |c , |c , |b are obtained by making the respective replacements in the expressions for the ket-vector |ξ given in (5.11), (5.12). Using the ket-vectors above-described, gauge-fixed BRST Lagrangian L tot can be presented as where the gauge invariant Lagrangian L is given in (4.25), while the operator M 2 and the modified de Donder operatorL are given in (4.26) and (4.27) respectively. We now note that gauge-fixed Lagrangian L tot (7.4) is invariant not only under BRST but also under anti-BRST transformations. Namely, gauge-fixed Lagrangian L tot (7.4) is invariant under BRST and anti-BRST transformations which take the following form: where an operator G appearing in (7.5), (7.6) is defined in (5.26). Using (7.5), (7.6), we verify that the BRST and anti-BRST transformations (7.5), (7.6) are off-shell nilpotent: Representation of gauge invariant Lagrangian of massive fields in terms of modified de Donder operator was found in ref. [12]. For review, see section 2 in ref. [43].

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For a computation of partition functions of the conformal fields, it is convenient to use the ξ = 1 gauge. Doing so, and integrating out the Nakanishi-Lautrup fields, we find that the BRST Lagrangian L tot (7.4) takes the following form: where the operator M 2 is given in (4.26). Gauge-fixed Lagrangian (7.8) is also invariant under BRST and anti-BRST transformations which take the following form s|φ = G|c , s|c = 0 , s|c =L|φ , (7.9) s|φ = G|c ,s|c = −L|φ ,s|c = 0 , (7.10) where the operatorsL and G appearing in (7.9), (7.10) are defined in (4.27) and (5.26) respectively. BRST and anti-BRST transformations given (7.9), (7.10) are also nilpotent (7.7). However, in contrast to the transformations given in (7.5), (7.6), transformations (7.9), (7.10) are nilpotent only for on-shell Faddeev-Popov fields. Gauge-fixed BRST Lagrangian (7.8) can be represented in terms of traceless fields which sometimes turn out to be more convenient for computations. To this end we use the well known decomposition of the double-traceless ket-vector |φ into two traceless ketvectors |φ I , |φ II , Plugging decomposition (7.11) into BRST Lagrangian (7.8), we get To conclude this section, we note that it is the use of the modified de Donder operator L in gauge-fixed Lagrangian (7.4) that allows us to get the simple representations for gauge-fixed BRST Lagrangian given in (7.8), (7.13).

Partition functions and number of D.o.F for conformal fields
Partition functions for long and special conformal fields. For the arbitrary spin-s long, secondary long, and special conformal fields in R d−1,1 , we find that partition functions take one and same form and are given by 14) where, in (7.15) and below, a quantity D n stands for a determinant of the Laplace operator evaluated on space of rank-n traceless tensor field. From (7.14), we see that numbers of JHEP05(2016)096 propagating D.o.F for the long, secondary long, and special conformal fields take one and same form and are given by where n so(d) s given in (7.16) is nothing but the dimension of the totally symmetric spin-s irrep of the so(d) algebra. It is well known, that the n so(d) s describes a number of D.o.F for a spin-s totally symmetric massive field propagating in (d + 1) dimensional spacetime. In other words, we come to the conclusion that numbers of D.o.F for the spin-s totally symmetric long, secondary long, and special conformal fields that propagate in ddimensional space-time are equal to κ times the number of D.o.F for massive spin-s totally symmetric field that propagates in (d + 1) dimensional space-time. For the case of the long conformal field, the same conclusion was achieved by using light-cone gauge formulation of the long conformal field in ref. [7].
Partition function for partial-short conformal fields. For the arbitrary spin-s partial-short conformal field in R d−1,1 with arbitrary d, we find the following partition function where values of the κ for the types I and II partial-short conformal fields are given in table 1. Using (7.17), we find that a number of propagating D.o.F for the partial-short conformal field is given by the relation where (p) q stands for the Pochhammer symbol Γ(p + q)/Γ(p). We note that the number of D.o.F given in (7.18) is found by using the relation where n so(d) s is defined in (7.16). For type II partial-short conformal field in R 3,1 with κ = 1 (maximal-depth partialshort conformal field), partition function (7.17) and number of D.o.F (7.18) take the form for maximal-depth partial-short conformal field in R 3,1 . (7.20) The partition function and number of D.o.F given in (7.20) were first obtained in ref. [44].

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Partition function for short conformal field. Partition function for the arbitrary spin-s short conformal field in R d−,1 with arbitrary d is well known, For d = 4 and d ≥ 4, the partition function (7.21) was obtained in the respective ref. [1] and [45,46]. Derivation of partition function (7.21) by using the gauge-fixed BRST Lagrangian of the short conformal field may be found in ref. [47]. As a side remark, we note that partition function (7.21) can also be obtained by equating t = 0 in the partition function of the partial-short conformal field (7.17).
Using (7.21), we find that the number of propagating D.o.F for the spin-s short conformal field in R d−1,1 is given by the well known relation We note that the number of D.o.F given in (7.22) is found by using relation (7.19), where we set t = 0 and use n so(d) s defined in (7.16).
For the case of the short conformal field in R 3,1 , partition function ( For d = 4 and d ≥ 4, the numbers of D.o.F (7.23) and (7.22) were found first in the respective ref. [1] and ref. [11]. In ref. [11], the n D.o.F (7.22) was obtained by counting D.o.F that enter the ordinary-derivative approach of the short conformal field. For d ≥ 4, the computation of n D.o.F by using a partition function may be found in refs. [45,46]. As a side remark, we note that, according to (7.22), there are no local D.o.F for short conformal fields in R 2,1 . Interesting recent discussion of conformal fields in R 2,1 may be found in ref. [48].
Comparison of D.o.F for short conformal field and maximal-depth partial-short conformal field in R 3,1 . In ref. [44], it was noticed that, in R 3,1 , the numbers of D.o.F for the short conformal field and the maximal-depth partial-short conformal field coincide (see relations (7.20), (7.23)). Here we would like to demonstrate how our ordinaryderivative approach provides a simple and transparent explanation for this interesting fact. To this end, using a shortcut φ s ′ k ′ for the so(3, 1) algebra vector and tensor fields φ a 1 ...a s ′ s ′ −s,k ′ , we note that, for the arbitrary spin-s short conformal field in R 3,1 , the field content appearing JHEP05(2016)096 in (4.6) can be presented as follows: Field content for spin-s short conformal field in R 3,1 Accordingly, using a shortcut φ s ′ k ′ for the so(3, 1) algebra vector and tensor fields φ a 1 ...a s ′ s−s ′ ,k ′ , we note that, for the arbitrary spin-s maximal-depth partial-short conformal field in R 3,1 , the field content in (4.5) can be presented as the follows: Field content for spin-s maximal-depth partial-short conformal field in R 3,1 , we see that, in R 3,1 , scalar fields do not appear in the field contents of the short conformal field and the maximal-depth partial-short conformal field. Using a light-cone gauge formulation, we verify then that light-cone gauge field contents of the short and maximal-depth partial-short conformal fields take the same respective forms as in (7.24) and (7.25), where all vector and totally symmetric tensor fields of the Lorentz algebra so(3, 1) should be replaced by the respective vector and totally symmetric traceless tensor fields of the so(2) algebra. Now taking into account that, for arbitrary s ≥ 1, a dimension of the totally symmetric spin-s irrep of the so(2) algebra is equal to 2, we see that total numbers of D.o.F for light-cone gauge fields in (7.24) and (7.25) coincide and equal to s(s + 1).
Higher-derivative Lagrangian. For the reader's convenience, we now explain how the partition functions and the numbers of D.o.F above-discussed can be obtained by using gauge-fixed BRST Lagrangian. To this end it is convenient to exclude auxiliary fields and cast the ordinary-derivative BRST Lagrangian (7.8) into a higher-derivative form. We now explain some details of the derivation of a higher-derivative BRST Lagrangian.

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For the illustration purposes, we note that Lagrangian (7.33) can be expressed in terms of the scalar, vector, and traceless tensor fields as The summation indices s ′ , λ for the L s ′ I,λ -terms in (7.34) take values given in (4.2)-(4.6), while the summation indices s ′ , λ for the L s ′ II,λ -terms and the L s ′ FP,λ -terms in (7.34) take values which are obtained by making the respective replacements s → s − 2, d → d + 4 and s → s − 1, d → d + 2 in (4.2)-(4.6). Quantities β I s ′ ,λ , β II s ′ ,λ , and β FP s ′ ,λ appearing in (7.35), (7.36) are obtained by using the respective substitutions N ζ → 1 2 (s − s ′ + λ), N ζ → 1 2 (s−2−s ′ +λ), and N ζ → 1 2 (s−1−s ′ +λ) in expressions for β given in (4.31)-(4.33). Using Lagrangian (7.34) allows us to obtain a general representation for a partition function. Namely, we see that Lagrangian (7.34) leads to the following representation for a partition function where, depending on the type of the conformal field, the product indices s ′ , λ appearing in (7.38) take values given in (4.2)-(4.6). We now demonstrate how the general expressions given in (7.37), (7.38) can be used for the computation of partition functions of various conformal fields.
Computation of Z for long conformal field. For the case of the long conformal field, the labels s ′ , λ take values shown in (4.2). This implies that Z s,d (7.38) takes the form where the relations k ∈ [p, q] 1 , k ∈ [p] 2 are defined in (2.11)-(2.13). Using the general formula where the notation λ ∈ [p, q] 2 is defined in (2.13), we see that Z s,d (7.39) can be represented as Plugging (7.41) in (7.37), we obtain Z given in (7.14). Note that for the long conformal field the Z s,d (7.41) does not depend on d explicitly. Repeating the above-described computation for the case of the secondary long and special conformal fields, we get the same Z as in (7.14).

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Computation of partition functions for partial-short conformal fields. Partition functions for the type I and type II partial-short conformal fields take one and same form given in (7.17). Computation of a partition function turns out to be more involved for the case of the type II partial-short conformal fields in R d−1,1 , d ≥ 6. Therefore, for the reader's convenience, we consider this case. To this end we note that, for the type II partial-short conformal field, one has the relation t > (d − 6)/2, where t is defined in (7.17). We verify then that the domain of values of the labels s ′ , λ given in (4.5) can be represented as a direct sum of three domains of values of labels denoted by (1), (2), and (3) and defined by the following relations: (1) : (2) : where the relations k ∈ [p] 2 , k ∈ [p, q] 1 , and k ∈ [p, q] 2 are defined in (2.11)-(2.13), while the parameter t is defined in (7.17). Accordingly, the general expression for the partition function Z given in (7.37), (7.38) can be represented as Using general formula (7.40), we straightforwardly compute the products over indices λ appearing in (7.47)-(7.49),

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Using the definition of Z (k) given in (7.46) and relations (7.53), we find (7.54) Relations (7.45) and (7.54) lead to the partition function for the partial-short conformal fields given in (7.17). For the derivation of (7.17), the interrelation between κ and t given in (7.17) also should be used.

Conclusions
In the framework of AdS/CFT correspondence, conformal field that propagates in R d−1,1 and has conformal dimension as in (1.1) is dual to a non-normalizable mode of bulk field that propagates in AdS d+1 and has lowest eigenvalue of an energy operator equal to E 0 = κ + d 2 . 10 This implies that the short, partial-short and long conformal fields in R d−1,1 are dual to the respective massless, partial-massless and massive fields in AdS d+1 . Taking this into account, we speculated [52] on some special regime in AdS superstring theory when parameters κ for all massive higher-spin fields take integer values. One can expect that such conjectured regime in the AdS superstring theory should be related via AdS/CFT correspondence to stringy theory of conformal fields that involves low-spin short conformal fields and higher-spin long conformal fields. 11 In fact it is such conjectured regime in AdS superstring theory that triggered our interest to the study of long conformal fields.
In this paper, we developed the ordinary-derivative Lagrangian formulation for all totally symmetric conformal fields propagating in R d−1,1 . Though, at the present time, the minimal Lagrangian formulation of conformal fields, which with exception of some particular cases involves higher derivatives, is more popular, we think that the ordinary-derivative approach is more perspective. This is to say that in our ordinary-derivative approach the gauge symmetries of conformal fields are realized, among other things, by using Stueckelberg fields. Stueckelberg fields turned out be useful for the study of string theory. Namely, all Lorentz covariant formulations of string theory available in the literature have been built by exploiting Stueckelberg fields. Stueckelberg fields turn also to be helpful for study of field theoretical models of interacting massive AdS fields (see, e.g., refs. [58][59][60]). We think therefore that the use of gauge symmetries involving Stueckelberg fields might also be useful for the study of various problems of conformal fields. The use of gauge symmetries involving Stueckelberg fields for building ordinary-derivative Lagrangian of the interacting spin-2 conformal field in six dimensions may be found in ref. [61]. In this respect it would be interesting to extend result in the latter reference to higher space-time dimensions along the lines in refs. [62,63].
We now discuss various potentially interesting generalizations of our approach and review some related studies in the literature. 10 For arbitrary values of κ, s, and d, action for massive spin-s AdS field evaluated on a solution of the Dirichlet problem was first found in refs. [19,49]. Interesting discussion of group theoretical aspects of AdS/CFT correspondence may be found in refs. [50,51]. 11 One can speculate on the similar regime in the higher-spin gauge theory [53,54] considered in the perspective of broken higher-spin symmetries. The recent interesting discussion of broken higher-spin symmetries may be found in ref. [55] (see also refs. [56,57]).