Higher spins in AdS_5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT

We consider general-symmetry higher spin fields in AdS_5 and derive expressions for their one-loop corrections to vacuum energy E and the associated 4d boundary conformal anomaly a-coefficient. We a propose a similar expression for the second conformal anomaly c-coefficient. We show that all the three quantities (E, a, c) computed for N=8 gauged 5d supergravity are -1/2 of the values for N=4 conformal 4d supergravity and also twice the values for N=4 Maxwell multiplet. This gives 5d derivation of the fact that the system of N=4 conformal supergravity and four N=4 Maxwell multiplets is anomaly free. The values of (E, a, c) for the states at level p of Kaluza-Klein tower of 10d type IIB supergravity compactified on S^5 turn out to be equal to those for p copies of N=4 Maxwell multiplets. This may be related to the fact that these states appear in the tensor product of p superdoubletons. Under a natural regularization of the sum over p, the full 10d supergravity contribution is then minus that of the Maxwell multiplet, in agreement with the standard adjoint AdS/CFT duality (SU(n) SYM contribution is n^2-1 of one Maxwell multiplet). We also verify the matching of (E, a, c) for spin 0 and 1/2 boundary theory cases of vectorial AdS/CFT duality. The consistency conditions for vectorial AdS/CFT turn out to be equivalent to the cancellation of anomalies in the closely related 4d conformal higher spin theories. In addition, we study novel example of vectorial AdS/CFT duality when the boundary theory is described by free spin 1 fields and is dual to a particular higher spin theory in AdS_5 containing fields in mixed-symmetry representations. We also discuss its supersymmetric generalizations.


Introduction
AdS d+1 /CFT d framework leads to interesting connections between properties of conformal fields in dimension d and their counterparts in d + 1. In particular, there are "kinematic" relations based on symmetries and special properties of AdS type spaces. One set of such relations involves singlet sector of free CFT d , dual higher spin theory in AdS d+1 and "shadow" conformal higher spin theory in d dimensions (see, e.g., [1][2][3][4][5][6][7] for some recent discussions related to the topic of this paper). Here we will be interested in the case of d = 4.
Starting, e.g., with a free massless complex scalar theory d 4 x Φ * r ∂ 2 Φ r one gets a tower of conserved symmetric traceless higher spin currents J s ∼ Φ * r ∂ s Φ r which are primary conformal fields of dimension ∆ = 2+s ≡ ∆ + . Adding these currents to the action with the source or shadow fields ϕ s (x) one observes that φ s has the same dimension ∆ − = 4 − ∆ + = 2 − s and effectively the same algebraic and gauge symmetries as (in general, non-unitary) conformal higher spin (CHS) fields. Integrating out the free fields Φ r gives an "induced" action for ϕ s with the kinetic term K(x, x ) ∼ J s (x)J s (x ) . The leading (logarithmically divergent) local part of this action is the same as the CHS action d 4 x ϕ s ∂ 2s ϕ s + ... (with s = 1 being Maxwell vector, s = 2 being Weyl graviton, etc.). From the dual AdS 5 perspective (implying matching between the correlators of currents and amplitudes for dual AdS fields φ s ) this induced action can be found upon the substitution of the solution of the Dirichlet problem with φ s ∂ = ϕ s into the classical 5d action for a massless spin s field φ s .
In addition to this "tree-level" relation between 5d fields φ s and and 4d conformal higher spin fields ϕ s (or shadow counterparts of the conserved currents J s ) there is also a relation between the corresponding one-loop partition functions, i.e. between the determinant of the 4d kinetic operator K ∼ ∂ 2s δ(x, x ) and the ratio of determinants of 2nd-order 5d operators for the field φ s with Neumann-type (∆ − ) and Dirichlet-type (∆ + ) boundary conditions. This relation has essentially a "kinematic" origin belonging to a general class of bulk-boundary relations discussed in [8]; for scalar operators it was also implicit in mathematics literature as discussed in [9,10]. In the context of AdS/CFT it appeared in the context of the discussion of the bulk counterpart of a "double trace" deformation of the boundary CFT (see [11-14, 9, 10, 1]).
The generalization to higher symmetric tensors was made explicit in [1,3,4]). In the case when the 4d boundary is a sphere S 4 this leads to an expression for the conformal anomaly a-coefficient of the 4d CHS field in terms of the properties of the AdS 5 determinants [1]. 1 In the case of the R × S 3 boundary one gets a relation for the AdS 5 vacuum energy or the Casimir energy on S 3 for totally symmetric CHS fields [7]. For a more general curved 4d boundary one should be able to obtain also a 5d expression for the second conformal anomaly coefficient c.
The point which will be important below is that instead of a 4d CHS field we may consider a generic primary 4d conformal field that will be associated to a particular (in general, massive or massless higher spin) field in AdS 5 which will effectively encode its quantum characteristics. Dimension 4 is the first case when the conformal fields and the dual higher spin fields in AdS 5 are not only totally symmetric, but may also appear in mixedsymmetry representations (described by SO(4) Young tableau with two rows). We shall use the SU (2) × SU (2) weights (j 1 , j 2 ) to label a representation of the Lorentz group (with spin s = j 1 + j 2 ), i.e. a conformal group SO (2,4) representation with scaling dimension ∆ will be denoted as (∆; j 1 , j 2 ).
Our aim will be to determine the expressions for the S 3 Casimir (or vacuum) energy E c and 4d conformal anomaly coefficients a and c corresponding to a generic AdS 5 field for the representation (∆; j 1 , j 2 ). Our results will generalize those for j 1 = j 2 = s 2 found for a in [1,3,5] and for E c in [6,7]. We shall also propose a general expression for the c(∆; j 1 , j 2 ) coefficient which matches all known values in special cases and provides very non-trivial consistency checks of AdS/CFT. We shall then discuss applications of our general relations to the adjoint and vectorial AdS/CFT dualities.

Structure of 4d conformal anomaly
Let us first recall the general expression for the stress tensor trace anomaly in a free 4d CFT defined on a curved space [15,16] Here b 4 is the Seeley coefficient (often called also a 2 ) for the corresponding kinetic operator. There may be several operators in the case of gauge symmetries and they may be of higher order than 2 in general. E = R * R * is the Euler density and C the Weyl tensor (C 2 = E + 2R 2 µν − 2 3 R 2 ). The coefficient g of the total derivative term is a priori ambiguous (regularization-dependent) as it can be changed by adding a local R 2 counterterm. 3 It enters the expression for the Casimir energy on S 3 that can be found from the stress tensor [17] E c = 3 4 a + 1 2 g . (1.2) Like g, the vacuum energy E c also depends on a choice of regularization. 4 Computed in the standard heat kernel or ζ-function scheme the coefficient g happens to vanish in theories with large amount of supersymmetry 5 so that E c and a-coefficient become directly proportional (see also [19,20]). In UV finite theories with (extended) supersymmetry one also finds that c is equal to a, and the two conditions appear to hold at the same time if 2 Our choice of normalisation is such that for a real conformal scalar a = 1 360 , c = 1 120 , g = 1 180 . 3 If one uses dimensional regularisation [15] and defines Weyl tensor in d dimensions then g = 3 2 c. This is the "Q-curvature". This relation is not true in the standard heat kernel (proper time cutoff) [16] or ζ-function regularization that we shall assume. For example, for standard spin ≤ 1 fields one then finds a = 31 180 n1 + 11 720 n 1 2 + 1 360 n0, c = 1 10 n1 + 1 40 n 1 2 + 1 120 n0, g = − 1 10 n1 + 1 60 n 1 2 + 1 180 n0, where ni are the numbers of gauge vectors, Majorana fermions and real conformal scalars. 4 Ec computed from the spectrum of the Hamiltonian is given by a formally divergent sum which may be defined using spectral ζ-function regularization. 5 This was found [18] in N = 4 SYM and also appears to be the case in N = 3, 4 conformal supergravity as we shall see below.
the number of global supersymmetries is N ≥ 3, i.e.
N ≥ 3 susy : E c = 3 4 a , a = c , g = 0 . (1.3) We would like to find the general expressions for the conformal anomaly coefficients a, c and also E c (or g) as functions of the representation labels ∆, j 1 , j 2 by starting with a dual 5d description of a given conformal 4d field. Consider a 2nd-order operator O = −D 2 + X defined on a 5d field φ which corresponds to a representation (∆; j 1 , j 2 ). In the case when 5d space is AdS 5 X is a constant "mass" term (we shall make the definition of O precise below). More generally, we may consider a generalization of AdS 5 to an Einstein space ds 2 = 1 z 2 [dz 2 + g µν (x, z) dx µ dx ν ] which asymptotes to a curved boundary metric g µν (x) ≡ g µν (x, 0). 6 The corresponding one-loop partition function (with Dirichlet-type "+" or Neumann-type "−" boundary conditions) (1.4) will then be a functional of the boundary metric g µν . One may define the associated boundary conformal anomaly A ± as the variation of Z ± under the variation of the conformal factor of the boundary metric: δ log Z ± = − 1 (4π) 2 d 4 x √ g δσ A ± , δg µν = 2 δσ g µν (generalizing the "tree-level" 5d derivation of 4d conformal anomaly [21]). It was argued in [22,23] that one should find where according to [23] andŌ is a 4d operator corresponding to a "restriction" of O to the boundary. In this caseĀ (which should have the same structure as (1.2)) can not depend on ∆.
As we shall see below, while (1.5) is indeed true, i.e. both a and c are proportional to ∆ − 2, the coefficientĀ should have a non-trivial dependence on ∆ (in addition to its dependence on j 1 , j 2 ). 7 Our expressions for a and c will thus be different from the ones proposed in [23] for spins j 1 + j 2 ≤ 2. The individual field contributions to c − a will also disagree with the general ansatz in [24][25][26] based on the prescription of [23], though the agreement (for c − a but not for a in [23]) will be restored when fields are combined into for N = 1 superconformal multiplets.

Relation between 5d and 4d partition functions
To understand the precise relation between the 5d determinants (1.4) and the conformal anomaly of the associated 4d operator let us start with a 5d action S 5 = d 5 x φOφ + ... 6 In general, for a higher spin field φ in a 5d Einstein background the corresponding kinetic operator may contain non-minimal curvature couplings and its consistency may require an existence of a proper embedding into an interacting higher spin theory. 7 To find the a-coefficient it is enough to consider the case of AdS5 with conformally flat boundary, while to determine c one may specialize to the case of Ricci flat boundary metric. That a coefficient inĀ should have 4-th order polynomial dependence on ∆ follows already from the results for a general massive 5d scalar in euclidean AdS5 with boundary S 4 [9, 10, 1]. and evaluate it on a solution of the Dirichlet problem φ ∂ = ϕ, i.e. symbolically S 5 = d 5 x φOφ + ... → S 4 = d 4 x ϕKϕ ∼ log ε d 4 x ϕÕϕ + ... . (1.6) Here ε = R −1 → 0 is an IR cutoff in 5d. In the case of ∆ = 2 + s when φ is a massless higher spin field the 4d field ϕ is the conformal higher spin field andÕ ∼ D 2s + ... is the corresponding Weyl-invariant 4d operator depending on g µν . 8 Let us now consider the following path integral 9 Z(ϕ) = φ ∂ =ϕ dφ e −S 5 (φ) = Z + e −S 4 (ϕ) 1 + ... , (1.7) where in the r.h.s. we considered semiclassical expansion near the solution of the Dirichlet problem. Here Z + is the "free" one-loop 5d partition function in (1.4). Next, let us integrate (1.7) over the 4d field ϕ. As was argued in a similar context in [8], this results in path integral over φ with "free" Neumann boundary conditions, with the leading 1-loop term then being Z − in (1.4) dϕ Combining this with (1.7) we find at the one-loop order Here we assume that ∆ is such that K has leading local termÕ as in (1.6) and the subleading terms can be ignored in the limit. The overall singular constant will not contribute to observables like conformal anomaly. The case of an arbitrary ∆ will be defined by an analytic continuation, which should give consistent results at least for the boundary conformal anomaly parts of the corresponding determinants. We thus get a relation between the 5d and 4d determinants of local operators. In general, for a 5d field corresponding to a massive or massless representation (∆; j 1 , j 2 ) of SO(2, 4) the associated boundary conformal field will have canonical dimension equal to ∆ − = 4−∆. Thus ∆ ≥ 4 cases will correspond to 4d fields with higher 2(∆−2) ≥ 2(j 1 +j 2 ) derivative kinetic operators ∼ D 2(∆−2) + ... which should give a Weyl-invariant action in curved 4d background. This implies, in particular, that the corresponding anomaly should vanish at ∆ = 2 as in (1.5) as then the operator becomes algebraic. One simple case (cf. [10]) is when O = −D 2 + X is the 5d scalar operator with X = ∆(∆ − 4) = 0, i.e. corresponding to the representation (4; 0, 0). ThenÕ is the 4-derivative Weyl invariant scalar operator of [27,28]. 10 As another example, we may consider O being a massless higher spin gauge field operator in (a generalization of) AdS 5 space. ThenÕ will be the kinetic operator of the corresponding 4d CHS field and we will get the following 5d representation for its 1-loop partition This relation was verified for the leading (log divergent) part of Z CHS on S 4 and the corresponding IR divergent parts of Z ± in the euclidean AdS 5 space, i.e. for the conformal anomaly a-coefficient [1,3]. In the case of the "thermal" cover of AdS 5 with S 1 × S 3 boundary eq. (1.10) was demonstrated explicitly (for any value of the length β = − ln q of S 1 ) in [7]. In particular, it then relates the Casimir energy E c of a CHS field on S 3 to the vacuum energy of the corresponding massless higher spin field in AdS 5 space.
The above heuristic argument makes clear the simple kinematic origin of the relation (1.9) or (1.10) and suggests that it should also extend to the case when AdS 5 is deformed to an Einstein space asymptotic to a generic curved 4d boundary. Then the variation over the boundary metric should provide a 5d representation for the 4d conformal anomaly which should apply to all (a, c and g) coefficients in (1.1). It was noticed in the special case of the symmetric tensor representation (2 + s; s 2 , s 2 ) that the a-coefficients corresponding to A ± obey [5] a + = −a − . Then (1.11) implies that a = −2a + . Similar relation is true [7] for the Casimir energy and thus for the g coefficient in (1.1),(1.2).
We shall see below that the same applies also for the general representations (∆; j 1 , j 2 ). This is a consequence of the change of sign of the expressions for a + and E + c under ∆ − → ∆ + , i.e. under ∆ − 2 → −(∆ − 2). It is then natural to assume that the same should be true also for the c-coefficient, 11 i.e. that in general 12 (1.12)

Higher spin operators in AdS 5
Let us now describe the structure of the 5d operators O we will be considering below. Let φ be a massive (∆ > 2 + j 1 + j 2 for j 1 j 2 = 0 or ∆ > 1 + j 1 + j 2 for j 1 j 2 = 0) or massless 10 Weyl-invariant operators are not unique in general: for example, one can add a Weyl-invariant C 2 term to the D 4 + ... Weyl-invariant operator with an arbitrary coefficient [27,18] and the same is true for the 2nd-derivative Weyl-invariant operator defined on symmetric traceless tensor [29][30][31] corresponding to representation (3; 1, 1) and on 4th rank tensor with symmetries of Weyl tensor [32] corresponding to representation (3; 2, 0) + (3; 0, 2). The relation to a consistent 5d operator should fix this ambiguity. This ambiguity is absent in the case of D 4 operator defined on dimension zero tensor or (4; 1, 1) coming out of the expansion of the C 2 Weyl action related [33] to the Einstein gravity action in 5d. 11 A (not directly related) indication that local properties of variations of 5d determinants may have opposite signs for the Dirichlet and Neumann boundary conditions is that this is what happens for the coefficients of E and C 2 in the expression for the boundary b5 Seeley coefficient in [34]. 12 Equivalently, in the notation of (1.5) that meansĀ(∆) =Ā(4 − ∆).
(∆ = ∆ 0 ≡ 2 + j 1 + j 2 , j 1 j 2 = 0) field in AdS 5 corresponding the SO(2, 4) representation (∆; j 1 , j 2 ) (see (A.1)). One may also define the weights 13) which are integer for bosonic fields and half-integer for fermionic fields. In the bosonic case, h 1 and |h 2 | are the lengths of a two-row Young tableau. According to [35][36][37], the covariant equation of motion for such bosonic transverse field φ is (for j 1 ≥ j 2 ) where D 2 is the standard Laplacian in AdS 5 . This equation is also valid not only for the bosonic, but also for the fermionic fields after squaring the 5d Dirac operator. For a generic fermion spinor-tensor field Ψ one has ( / D + ∆ − 2) Ψ = 0 [38]. After squaring, this turns out to be −D 2 + 1 4 R−h 1 −|h 2 |+1+(∆−2) 2 Ψ = 0 (see [39] for details), where R = −20 is the scalar curvature of AdS 5 assumed to have unit scale. This gives the same X as in (1.14). A natural definition of mass of a bosonic field in AdS 5 is such that it vanishes for the massless representation with ∆ = ∆ 0 = 2 + s, i.e. 13 The partition function of a massive higher spin field with standard (Dirichlet) boundary conditions corresponding to ∆ = ∆ + is then given by (1.4) with O defined on transverse fields in representation (j 1 , j 2 ). We shall denote the massive case quantities with in what follows, i.e.
In the massless case of ∆ = ∆ 0 = 2 + s we need to take into account the contribution of the corresponding ghosts that belong to representation (∆ 0 + 1; j 1 − 1 2 , j 2 − 1 2 ) (the gauge transformation parameters ξ in δφ ∼ ∂ξ have one unit of spin and canonical dimension 4 − ∆ less): For example, in the case of the totally symmetric massless higher spin field representation one finds [40,41] Below we will use (1.14),(1.15), (1.16) to compute the corresponding E c and a coefficients. A direct 5d computation of c or c − a would require a generalization of O in (1.14) to an Einstein space which is asymptotically AdS 5 with Ricci flat boundary which is not known in general for s > 2 (cf. [42,43]). However, the expresions for E c and a and known results in special cases will allow us to suggest a unique expression for the c-coefficient which will then pass AdS/CFT consistency checks. 13 In the fermionic case there is a possible alternative definition of mass as the parameter in the Dirac

Summary
Let us summarize the content of the rest of this paper. In section 2 we consider the S 1 × S 3 partition function Z in (1.10) and also find the corresponding S 3 Casimir energy for the case of generic representation (∆; j 1 , j 2 ). The resulting expression for E c will follow the pattern in (1.12). The one-particle partition functions corresponding to Z + will be given directly by the SO(2, 4) characters but the case of Z − will be more subtle, and we will determine it in few special cases.
In section 3 we find the general expression for the a(∆; j 1 , j 2 ) conformal anomaly coefficient in (1.1), (1.12), generalizing the computation of [1,5] done in the totally symmetric (2 + s; s 2 , s 2 ) bosonic case. Combined together, the results for E c and a determine also the form of the coefficient g(∆; j 1 , j 2 ) in (1.1), (1.2).
In section 4 we determine a similar expression for the second conformal anomaly coefficient c. While we are presently unable to give its systematic derivation, we shall make a proposal for c(∆; j 1 , j 2 ) that reproduces all known special cases and leads to non-trivial consistency checks and predictions in the context of AdS/CFT.
In section 5 we apply our general expressions for E c (2.31),(2.32), a (3.3),(3.4) and c (4.10),(4.3) to compute the corresponding quantities for sets of fields forming long or short SU (2, 2|N ) superconformal multiplets. We shall find that the total a and c vanish for long "massive" N = 1 supermultiplets and observe that c − a for short N = 1 supermultiplets agrees with the expressions in [24,25] formally extended to all values of spins j 1 , j 2 ≥ 1. We will also rederive from the 5d approach the values of K = (E c , a, c) for N ≤ 4 Maxwell and conformal supergravity supermultiplets, verifying the relation (1.3) for N = 3, 4 cases. We will demonstrate that all the three quantities vanish when N = 4 conformal supergravity is combined with exactly four N = 4 Maxwell multiplets as in [27,44]. The 5d approach provides a direct relation between the conformal anomaly of N = 4 conformal supergravity and the one-loop contribution of fields of N = 8, d = 5 gauged supergravity as the two theories are described by the equivalent short P SU (2, 2|4) supermultiplet (this generalizes to the one-loop level the known tree-level relation [33]). We will also show that K = 0 for a general long massless supermultiplet of P SU (2, 2|4).
In section 6 we turn to applications of our expressions for K = (E c , a, c) to AdS/CFT dualities. We first consider in section 6.1 the "adjoint" duality between N = 4 SU (N ) SYM and string theory in AdS 5 ×S 5 . We find that the values of K for the states at level p of Kaluza-Klein tower of 10d type IIB supergravity compactified on S 5 are equal to the values of p copies of N = 4 Maxwell multiplets, in line with the fact that these states appear in the tensor product of p superdoubletons [45]. Under a particular regularization of the sum over p, this is consistent with the adjoint AdS/CFT duality with SU (N ) SYM contribution to K being N 2 − 1 times that of one N = 4 Maxwell multiplet. As we explain on the example of the vacuum energy E c , the required regularization of the sum over the KK states is, in fact, a spectral ζ-function one applied to 10d instead of 5d energy states.
In section 6.2 we compute (E c , a, c) on both sides of the vectorial AdS/CFT examples. We consider the earlier studied cases of type A and type B higher spin theories in AdS 5 corresponding to the scalar and spin 1 2 fermion 4d boundary theories and also a novel example of "type C" theory dual to a singlet sector of N Maxwell fields at the boundary. We also discuss supersymmetric generalizations of vectorial AdS/CFT. In section 6.3 we point out that consistency conditions of vectorial AdS/CFT in non-minimal scalar and fermion theory cases implying cancellation of total a and c coefficients are equivalent to the consistency (cancellation of conformal anomalies or UV finiteness) of the corresponding 4d conformal higher spin theories. Some concluding remarks are made in section 7.
In Appendix A we summarize basic representations of SO(2, 4), decompositions of products of two doubleton and superdoubleton representations and present useful relations for their characters that play important role in the discussion of one-particle partition functions in vectorial AdS/CFT examples. Appendix B contains the computation of S 1 × S 3 partition functions of low-spin conformal 4d fields that appear in extended conformal supergravities and provide useful examples for the discussion in section 2. In Appendix C we give details of the derivation of the spectral ζ-function for massive higher spin AdS 5 operator O in (1.14) which is used in section 3. In Appendix D we complement the discussion in section 4 by presenting a more general ansatz for the c-coefficient that contains one free parameter. Appendix E summarizes the spectrum of 5d fields appearing in 10d type IIB supergravity compactified on S 5 which we use in sections 5 and 6.

Partition function on S 1 × S 3 and Casimir energy
In this section we shall consider the expressions for one-particle partition function and S 3 Casimir energy. We shall start with the previously discussed case of totally symmetric (2 + s; s 2 , s 2 ) representation and then turn to the case of mixed representation (∆; j 1 , j 2 ).

Totally symmetric bosonic spin s conformal fields
The canonical partition function of a free CFT in S 1 × S 3 can be computed by direct evaluation of the free QFT path-integral, i.e. by finding the eigenmodes of the quadratic kinetic operator. The same expression can be obtained by the operator counting method [46,47,7]. In radial quantisation, conformal operators in R 4 with dimensions ∆ n are related to eigenstates of the Hamiltonian on R t × S 3 . From the spectrum of eigenvalues ω n = ∆ n and their degeneracies d n one gets the one-particle, or canonical, partition function (2.1) The multi-particle, or grand canonical, partition function is then given, in the bosonic and fermionic cases, by 3) The analysis of the counting of states implies the following structure of Z(q) [7] Z(q) = Z − (q) − Z + (q), Here Z − counts the off-shell components (and their derivative descendants) of a suitable gauge invariant field strength modulo non-trivial gauge identities while Z + counts the components of the equations of motion for the field strength (and their derivatives).
In the case of totally symmetric conformal higher spin gauge field with spin s, canonical dimension 2 − s and generalized s-derivative field strength of dimension ∆ = 2 (with s = 1 being Maxwell vector, s = 2 being Weyl graviton, etc.) one finds [7] The form of Z −,s reflects the fact that the counts of gauge identities and of equations of motion are isomorphic. These expressions can be interpreted also from the AdS 5 perspective. In general, [7] where Z ± (q) are the one-particle partition functions (2.2) for the one-loop partition function Z ± of the corresponding massless higher spin gauge fields in thermal quotient of AdS 5 computed with teh standard ("Dirichlet") or alternative ("Neumann") boundary conditions. This is the special case of the general relation (1.10) with (1.17). Explicitly, one finds from the AdS 5 heat kernel expression [41,48] that Z + s is given by the same expression as Z +,s in (2.5). The full singlet-sector partition function of the boundary CFT is then given by the sum of Z +,s = Z + s contributions over all spins. Massless higher spin s field in AdS 5 with standard boundary condition is dual to the conserved spin s current operator of dimension ∆ + = 2+s in the free complex scalar CFT 4 . Z + s = Z +,s has the interpretation of counting the bilinear current field J s components (and its derivative descendants) modulo the on-shell conservation condition. This counting problem is isomorphic to that of counting the equations of motion for the 4d conformal higher spin field. Similarly, Z − s = Z −,s is counting the components of CHS fields φ s modulo gauge identities and also counting the components of the shadow spin s conformal field of dimension ∆ − = 2 − s (conjugate to J s ) in the 4d scalar CFT modulo gauge identities. The negative term in Z +,s in (2.5) corresponds [7] to the subtraction of the contribution of identities among equations of motion from the 4d CHS theory point of view, of the current conservation condition from the 4d scalar CFT point of view and of the ghost spin s − 1 field contribution from the AdS 5 bulk point of view.

Mixed-symmetry conformal fields
Let us now consider the case of a conformal primary field in SO(2, 4) representation (∆; j 1 , j 2 ). For generic ∆ the character of this long representation of SO(2, 4) should be equal to the one-particle partition function for the massive AdS 5 higher spin field partition function (1.15) which should just count all the components of (derivative descendants of) such field weighted with its dimension ∆ (see (A.8) in Appendix A) In the special case of ∆ = 2 + j 1 + j 2 such primary field should correspond to a conserved current in the boundary CFT or to its dual mixed-symmetry massless AdS 5 higher spin gauge field. In this case Z + = Z + should be given by the character of the associated short representation of SO(2, 4) (A.9), i.e. should correspond to (1.16) where the ghost contribution is included. Taking into account the current conservation condition or, equivalently, subtracting the 5d ghost contribution gives the massless partition function [49,50] (2.9) Note that eq. (2.9) reduces to (2.5) for j 1 = j 2 = s 2 , ∆ = 2 + s. To find the partition function Z in (2.6) corresponding to (1.10) for a 4d conformal spin (j 1 , j 2 ) field of canonical dimension ∆ − = 4 − ∆ it remains to determine the expression for the shadow partition function Z − (∆; j 1 , j 2 ). Let us start with the special case of "matter" conformal fields in SO(4) representation (j, 0) + (0, j) corresponding to massive 5d fields (here the subtraction term in (2.9) is absent as j 1 j 2 = 0 so formally Z + = Z + ). In this non-degenerate case it is natural to expect Z − = Z − to be related to Z + = Z + by the substitution which, according to (2.7), is equivalent to Then using (2.7) we get for Z in (2.6) 14 Examples of such 4d conformal fields are provided by matter fields appearing in extended conformal supergravities [51,44] (see Table 2 below) φ ∼ (3; 0, 0), Φ ∼ (4; 0, 0), T ∼ (3; 1, 0) + (3; 0, 1), ψ ∼ ( 5 2 ; 1 2 , 0) + ( 5 2 ; 0, 1 2 ), Ψ ∼ ( 7 2 ; 1 2 , 0) + ( 7 2 ; 0, 1 2 ) . (2.13) Here φ and ψ are the standard 4d massless scalar and spinor, Φ and Ψ are conformal fields with ∂ 4 and / ∂ 3 kinetic operators and T is conformal antisymmetric 2-tensor with ∂ 2 kinetic term and no gauge invariance. ∆ in (∆; j 1 , j 2 ) stands for ∆ + dimension associated to the corresponding massive 5d field with standard boundary conditions while the canonical dimensions of these 4d fields are ∆ − = 4 − ∆ (i.e. φ has dimension 1, Φ has dimension 0, etc.). The partition functions Z for these fields are derived in Appendix B by the explicit path-integral computation on S 1 × S 3 and also by the operator counting method and the results are consistent with (2.12).
Turning to the massless gauge field case with j 1 j 2 = 0 let us recall the derivation [7] of the expression (2.6) for Z − = Z − in the case of the bosonic totally symmetric field with (∆; j 1 , j 2 ) = (2 + s; s 2 , s 2 ). The presence of gauge degeneracy or ghost contribution implies that in this case the simple relation (2.11) between Z + and Z − is no longer true. The shadow field with dimension ∆ − = 2 − s corresponds to a non-unitary SO(2, 4) representation which in general contains singular states with their associated submodules. The AdS 5 counterpart of this complication is that in the case of the alternative boundary condition one has additional gauge transformations allowed by non-normalizability [1]. These can be put in one-to-one correspondence with the conformal Killing tensors that may be associated to the finite dimensional SO(6) representation (s−1, s−1, 0) labelled by the Young tableau with two rows with s − 1 columns. Then (2.11) is replaced by [7] (same for lower ± labels) where σ s (q) is the character of the representation for the conformal Killing tensors. Computing σ s (q) one then arrives at the expression in (2.5). A similar derivation should be possible in the mixed representation case leading to Below we will demonstrate this on the example of the fermionic conformal higher spin gauge fields described by totally symmetric spinor-tensor with one spinor index and s = 0, 1, 2, ... vector indices. Its total spin is s = s + 1 2 and it is represented by the sum of two mixed SO(4) representations: (2.16) Here ∆ = ∆ + = 2 + j 1 + j 2 = 2 + s. The Z + partition function is given by (2.9), i.e. (2.17) Then by analogy with the bosonic CHS case (2.14) we should find where σ s+ 1 2 (q) is the character for the conformal algebra representation corresponding to the conformal Killing spinor-tensors. The latter may be associated to the SO(6) representation 15 See also footnote 24 of [1].
The relevant character can be found by a specialization of the discussion in [7] σ s+ 1 This gives σ s+ 1 2 (q) as a finite sum 16 obeying the important property which was also true for the bosonic σ s in (2.14). Doing the sum over p in (2.23) gives Then using this in (2.14),(2.15) leads to the final result for Z s+ 1 As a check, for the standard massless spin 1 2 fermion (s = 0) this agrees with (2.12) with j = 1 2 and ∆ = 5 2 . Also, for the conformal gravitino (s = 1) this leads to which is the same expression (B.17) as derived in Appendix (B) by directly computing the conformal gravitino partition function on S 1 × S 3 . 16 Some explicit values are σ 1

General expression for the Casimir energy on S 3
The Casimir energy on S 3 can be extracted from the one-particle partition function Z(q) in (2.1) using the standard relations (see, e.g., [50]) 17 The representation in terms of ζ E (−1) has the advantage that it allows one to show that the Casimir energy vanishes if the partition function obeys Z(q) = Z(q −1 ) [6] (see also [52]). If we start with Z + corresponding to a primary field (∆; j 1 , j 2 ), the associated Casimir energy E + c (∆; j 1 , j 2 ) is then the same as the vacuum energy of a single massless higher spin field in AdS 5 with standard boundary conditions. If we consider a 4d conformal higher spin field, its Casimir energy on S 3 can be found from the corresponding one-particle partition function in (2.15). The Killing tensor character should in general obey the property (2.24), implying that the same should be true forZ(q) in (2.15), and ifZ(q) =Z(q −1 ) then it does not contribute to E c . As a result, we conclude that the Casimir energy of a 4d conformal field in representation (∆; j 1 , j 2 ) is given by -2 of the AdS 5 vacuum energy of the corresponding 5d field with the standard boundary condition In the non-gauge 4d field case (corresponding to a massive 5d field) we thus get from For a gauge conformal field (or a massless 5d field) with ∆ = 2 + j 1 + j 2 we get according to (1.16) As in (2.9), the second term here vanishes if j 1 j 2 = 0. Special cases include fields of extended conformal supergravity with values of E c listed in Table 2. For the general spin s totally symmetric bosonic (2 + s; s 2 , s 2 ) [7] and fermionic 17 Given the data (dn, ωn) the formal sum over n is usually divergent and requires a regularization. A natural regularization is a spectral ζ-function one as above which is also equivalent to computing Ec as the finite part of the → 0 expansion of the following regularized expression (see, e.g., [6]) (2 + s; 2 ) conformal 4d fields we obtain from (2.31),(2.32) (or directly from (2.5) and (2.17)) the following expressions for the Casimir energies Here ν b and ν f are the numbers of dynamical degrees of freedom of the bosonic and fermionic CHS fields [3]. The coefficient 2 in the fermionic case accounts for the equal contributions of the two j 1 ↔ j 2 representations.

Conformal anomaly a-coefficient
Next, let us turn to the computation of the conformal anomaly a-coefficient of 4d conformal field with canonical dimension 4 − ∆ and SO(4) spins (j 1 , j 2 ) corresponding to a generic representation (∆; j 1 , j 2 ).
As follows from (1.1), to find the a-coefficient it is sufficient to consider the case of conformally flat S 4 background (for unit-radius sphere A + = −24a + ). We shall use (1.10),(1.12) to give the AdS 5 derivation of the a-anomaly generalizing the computation of [1,5] in the totally symmetric (2 + s; s 2 , s 2 ) bosonic case. The expressions for a for both bosonic and fermionic totally symmetric conformal higher spin fields were found directly in 4d in [3].
A + in (1.12) is associated with the variation of the one-loop partition function of 5d field corresponding to the representation (∆; j 1 , j 2 ) under a local conformal variation of the boundary metric. In the case of the Euclidean AdS 5 with boundary S 4 (i.e. hyperboloid H 5 ) the conformal anomaly is proportional to the logarithmic IR singular part of the one-loop partition function (see, e.g., [9,1]) Here ζ(z) is the spectral zeta function defined by evaluating the trace of the H 5 heat kernel associated with the "massive" 5d operator O in (1.14) (see [53,1]). The trace is proportional to the regularised volume of H 5 that has a factor log R depending on IR cutoff. The explicit derivation of ζ(z) for the operator O acting on a transverse field in a general representation (∆; j 1 , j 2 ) is given in Appendix C. Using (C.14) the a-coefficient for the 4d conformal field associated to "massive" (∆; j 1 , j 2 ) representation can be represented as (-2 factor is as in (1.12)) This expression is odd under ∆ → 4 − ∆, i.e. under (2.10). This implies that the anomaly corresponding to Z − computed with the alternative boundary condition has the opposite sign, i.e. we have a = a − − a + = −2 a + . This is also the same pattern that was found for the Casimir energy (2.30). 18 In the massless field case one is to subtract the ghost contribution in (1.16), i.e.
As in the case of E c in (2.31),(2.32), the second term in (3.4) vanishes for j 1 j 2 = 0.
It is easy to check that in the special cases of conformal fields appearing in extended conformal supergravity the expressions (3.3),(3.4) reproduce the known values [27,44] of the corresponding a-coefficients (see Table 2). Also, for the totally symmetric bosonic and fermionic conformal higher spin gauge fields we find as in (2.33),(2.34) a s = 2 a s + 2; Eq.(3.5) was first found in the 5d approach in [1]; both expressions were also obtained by direct computation in 4d [3]. Eq.(3.3) is a generalisation of the one in [1] (see also [5]) derived there for the (∆; s 2 , s 2 ) fields Also, a special case of a massive scalar field with m 2 = ∆(∆ − d) in AdS d+1 with even d corresponding to a conformal field (∆; 0, 0) at the boundary was considered in [9,10], where it was found ((...) n is Pochhammer symbol)

9)
18 As was discussed in section 2, in the case of S 1 × S 3 boundary the Z − partition function is not simply given by Z + with ∆+ → ∆− = 4 − ∆+ (eq. (2.14) contains non-trivial σ term) but this relation still holds for Ec in (2.30). Same may be true in the case of S 4 boundary: while the IR divergent parts of log Z − and log Z + proportional to a − and a + are the same up to sign, the relation between the finite (non-universal) parts of the partition functions may be more involved.
in agreement with (3.7). For general (∆; j 1 , j 2 ), it follows from (3.3) that the ∆ derivative of a has a simple factorized structure Let us note in passing that since this expression is an obvious generalization of (3.8),(3.9) (obtained by ∆ → ∆ − j 1 − j 2 in the Pochhammer symbols, etc.), this suggests that the general field bulk-to-bulk propagator can be obtained from the scalar one by a similar replacement (with the prefactor coming from the trace over spin). This is indeed consistent with the known expressions in the case of totally symmetric tensors considered in [54].

Conformal anomaly c-coefficient
In this section we shall propose the general expression for the c(∆; j 1 , j 2 ) coefficient in the 4d conformal anomaly (1.1) which will be the counterpart of the expression for a(∆; j 1 , j 2 ) in (3.3), (3.4). We shall motivate it by imposing various consistency conditions and agreement with known special cases.

Expression for c in low spin cases
Once the value of a is known, to find c it is sufficient to compute c − a by considering the case of Ricci flat 4d space when the conformal anomaly (1. In the case of "massive" low spin 5d fields appearing in supergravity (e.g., in the KK spectrum of 10d type IIB supergravity compactified on S 5 ) ref. [24,26] suggested, following the proposal in [22], a general parametrization of c−a coefficient in the boundary conformal anomaly 19 This expression follows from the ansatz (1.5) withĀ = − 1 2 b 4 (Ō) assuming thatŌ, i.e. the 4d boundary restriction of the 5d massive kinetic operator defined on an Einstein space which is a generalization of AdS 5 space asymptotic to the Ricci-flat boundary, is the standardŌ = −D 2 + U operator defined on 4d field in Lorentz representation (j 1 , j 2 ) with "minimal" curvature coupling. Then applying the standard algorithm to compute its Seeley coefficient b 4 [16] gives (4.1). 20 19 To recall, we use to indicate massive representation and + indicates the one-loop 5d field contribution computed with standard (Dirichlet) boundary conditions. The normalization of c − a in [24,26] is such that that it corresponds to 1-loop contributions of 5d fields dual to composite 4d operators in the AdS/CFT picture; summing over all such contributions should reproduce the conformal anomaly of the boundary CFT. Thus c + − a + for, e.g., a scalar field corresponding the (3; 0, 0) representation is − 1 2 of the standard value 1 180 . 20 Here the 4d operator obtained by restricting the 5d operator defined on transverse fields to the boundary acts on unconstrained 4d fields.
Applying (4.1) together with our result (3.3) for the value of a-coefficient to compute the corresponding 4d conformal field anomaly c-coefficient according to (1.12), we find in the non-gauge 5d massive ∆ > 2 + j 1 + j 2 case To get c for CHS gauge fields corresponding to massless 5d fields with ∆ = 2 + j 1 + j 2 we are to subtract the 5d ghost contribution as in (1.16),(3.4): This expression reproduces the known values of c for spin ≤ 2 N = 4 conformal supergravity fields [27] in Table 2 (which are dual to fields of 5d N = 8 gauged supergravity). If we formally assume (4.2),(4.3) to be valid also for all totally symmetric higher spin fields with j 1 = j 2 = s 2 then we find as in (3.5),(3.6) 21 These are the same expressions as obtained in [3] by the direct computation in 4 dimensions.
The key assumption there was that the factorization of the higher-derivative CHS kinetic operator on Ricci-flat background into a product of standard 2nd derivative operators known to apply for s ≤ 2 continues to be valid also for s > 2.
It is useful to understand the reason for this agreement. Let us consider, for example, the bosonic CHS field on a curved Ricci-flat background. Assuming factorization of the conformal D 2s + ... kinetic operator into a product of s 2nd-derivative massless spin s operators with minimal coupling to curvature the corresponding CHS partition function can be written as [3] whereŌ k = (−D 2 +U ) k , U = −R ab mn Σ mn Σ ab are covariant 2nd-order differential operators defined on traceless rank k tensors and having the standard massless higher spin form that was assumed also in [16]. Then the conformal anomaly β 1 ≡ c − a coefficient for spin s CHS field can be expressed in terms of β 1 coefficients for the operatorsŌ s Here the scaling dimension is ∆ = 2 + s so that (4.8) has exactly the same structure as required for a massless 5d field anomaly (cf. (4.2),(4.3)).
Since β 1 (Ō s ) was computed in [3] from the same expression for b 4 (Ō s ) in [16] as used in (4.1) we conclude that the expressions for c − a should indeed match. As the a coefficients are already known to agree, this implies the agreement of the c coefficients found from the 5d approach based on (4.1) and from the 4d approach based on (4.7). However, there are good reasons to believe that both (4.1) and (4.7) are to be modified for spins j 1 , j 2 > 1. First, the expression for the Seeley coefficient of 4d operator on (j 1 , j 2 ) field used in (4.1) was taken from [16] which formally applies only for spins ≤ 2: for higher spins the consistency of "minimal coupling" operators considered in [16] requires extra constraints on the curvature (in addition to Ricci flatness) invalidating the derivation of c − a. Indeed, kinetic operators of higher spin 4d fields should in general contain terms with non-minimal (e.g., R .... D . D . ) coupling to the curvature [42,43] which does not allow the application of the standard algorithm for computing the b 4 Seeley coefficient used in [16].
Second, the assumption of factorization of the CHS operator on Ricci flat background made in [3] was questioned in [55]. It is likely that c−a for CHS fields may still be computed by assuming that factorization formally applies (extra terms obstructing factorization appear to involve derivatives of the curvature that can not produce non-trivial contribution to conformal anomaly in 4d) but the corresponding 2nd-derivative factor-operators should then also have non-minimal structure rather than being minimal operators as assumed in [3].
While the form of such 2nd-derivative higher spin operators that may appear in factorization of CHS operator on a Ricci-flat background remains to be understood, below we shall present a conjecture for what should be the correct generalization of c in (4.2) to higher spins j 1 , j 2 > 1. Our expression will lead to unique consistency properties when applied in the context of AdS/CFT.

Proposal for general expression for
Our proposal for c that replaces (4.2) in the massive representation case (∆; The corresponding expression for c − a following from (3.3) and (4.9) is then This is different from (4.1) as the dependence on ∆ is not just via the overall ∆ − 2 factor. Eq. (4.9) and its massless representation counterpart (4.3) is consistent with all lowspin data, giving, e.g., the correct values for all the fields of extended conformal supergravity (see Table 2): scalars with ∆ = 3, 4, spin 1 2 fermions with ∆ = 5 2 , 7 2 , non-gauge antisymmetric tensor, conformal gravitino and conformal graviton. Applying (4.3),(4.9) to the cases of totally symmetric bosonic and fermionic CHS fields we find again the expressions in (4.4),(4.5) but now with instead of (4.6). These values of the parameters are precisely the ones that lead to the vanishing of the sum s c s over all totally symmetric CHS fields [3,5], assuming the same regularization that implies the vanishing of s a s [1,3] and s E c,s [7]. The crucial feature of (4.9) is that it leads to important consistency checks of vectorial AdS/CFT duality which are direct analogs of the earlier checks based on the expressions for a-coefficient and E c . 22 These checks will be discussed in detail in section 6. Here we just mention two non-trivial relations in the case of a particular mixed representation satisfied by c (4.3) defined by (4.9) but not by (4.2): In this section we shall compute E c , a, c for collections of primary fields of SO(2, 4) representations (∆; j 1 , j 2 ) forming superconformal multiplets. It turns out that the difference between c − a in (4.1) and our proposal (4.10) disappears once one sums over all fields in the supermultiplet, implying that the resulting c − a is linear in ∆ as in (4.1) (but separate values of the coefficients a and c are still different from the ones implied by the prescription of [23]).

Summary of contributions of a single conformal
It is useful first to summarize the expressions for E c , a, c and c − a in (2.31)), (3.3), (4.9),(4.10) for a non-gauge (massive 5d) field in a compact form using the variables d 1 = 2j 1 + 1, d 2 = 2j 2 + 1: 22 We present a more general ansatz for c that reduces to (4.9) after imposing this consistency constraint in Appendix D.
Note that these expressions are odd under ∆ → 4 − ∆, cf. (2.10). The values in the gauge (massless 5d) field case with ∆ = 2 + j 1 + j 2 follow from (2.32), (3.4), (4.3). Written in terms of the variables they read . 23 These expressions are symmetric under j 1 ↔ j 2 so that in the case of j 1 = j 2 when the physical combination is (j 1 , j 2 ) c = (j 1 , j 2 ) + (j 2 , j 1 ) an extra factor of 2 is to be added (in our notation bosonic j 1 = j 2 fields are real).

N = 1 superconformal multiplets
Let us now find the total contributions of N = 1 superconformal multiplets containing (∆; j 1 , j 2 ) field as the lowest dimension member. The structure of relevant multiplets was given, e.g., in [56]. In addition to long massive multiplets there are shortened ones: chiral and right-handed semi-long (SLII), as well as their CP conjugates -anti-chiral and lefthanded semi-long (SLI). There are also CP self-conjugate ("conserved") multiplets that are the sums of one SLI and one SLII multiplet (thus they need not be considered separately). SO(2, 4) representation content of massive long N = 1 superconformal multiplet is 24 (5.9) 23 Here we wrote the fermionic contribution in terms of ν = s(s + 1) rather than ν f . 24 The term 2 (∆ + 1; j1, j2) comes from two representations with the same SO(2, 4) labels but different R-charge. For the computation of Ec, a, c we do not need to keep track of the R charge (in general, it is constrained by the shortening conditions).
Using the above expressions we find that the total a and c anomalies of a long massive multiplet vanish but the Casimir energy does not: The vanishing of c − a for long multiplets follows also from (4.1) [24,26]. The fact that E c is not proportional to a-coefficient as in (1.3) means that the coefficient g of the D 2 R term in the trace anomaly (1.1) does not vanish (in the heat kernel scheme we are using to define E c ); indeed, g is expected to cancel only in N > 2 extended supersymmetric cases (cf. (1.3)).

N > 1 superconformal multiplets
Next, let us present the expressions for E c , a, c in the case of some N > 1 superconformal multiplets.

Maxwell supermultiplets
Considering massless 4d multiplets with the highest spin 1 we get the values in Table 1.
and thus the coefficient g of the derivative term in (1.1) vanishes [18].

Conformal supergravity multiplets
The case of short multiplets with highest spin value is 2 is that of 4d extended conformal supergravity (CSG) multiplets. The relevant fields are listed in Table 2 together with their individual E c , a, c values. The total values for N ≤ 4 conformal conformal supergravity multiplets are given in Table 3 (the numbers in the central square are multiplicities of the fields, i.e. dimensions of their U (N ) or SU (4) representations).
As in the case of Maxwell supermultiplets, for N = 3, 4 we find the relation (1.3) or (5.15) satisfied, implying g = 0 (cf. (1.2)). The values of E c and g for conformal supergravities were not computed previously.    As was found in [27,44], the conformal anomalies of the combined system of N = 4 conformal supergravity and four N = 4 Maxwell multiplets cancel, i.e. this is a UV finite theory. This is readily seen from the values in Tables 1 and 2: The vanishing of the total E c is a new result (implied by (5.15) which is valid for each of the N = 4 multiplets).
The N = 4 conformal supergravity multiplet 26 is isomorphic to the supercurrent mul- 26 In addition to fields listed in Table 2 this P SU (2, 2|4) short multiplet contains also 20 auxiliary scalars with ∆ = 2 which do not contribute to physical quantities (the total number of helicity 2j1 + 2j2 + 1 states is 256).
tiplet of N = 4 Maxwell theory [58] and also to the short massless multiplet of fields of gauged N = 8 supergravity in 5 dimensions whose AdS 5 vacuum isometry is P SU (2, 2|4) [57,59,60,33]. 27 The field content of the latter is given in p = 2 entry in Table 6 in Appendix E. Indeed, the 5d expression for the conformal anomaly and the Casimir energy for N = 4 CSG is directly given by the one-loop contributions of fields of N = 8 5d supergravity, i.e. K(N = 4 CSG) = −2 K + (N = 8 5d SG) .
This one-loop relation between the two theories generalizes the tree-level one in [33]. In view of (5.17) this also implies that one-loop contribution of N = 8 5d supergravity is the same as of two N = 4 Maxwell multiplets, Remarkably, this non-trivial relation may be interpreted as expressing the fact that the states of N = 8 5d supergravity appear in the product of two N = 4 superdoubletons [45].
We shall return to this observation in section 6.1 below.

General long higher spin massless P SU (2, 2|4) supermultiplet
The general long massless multiplet of P SU (2, 2|4) [61,45] has spin range 4 (8 supercharges). Its conformal representation content is that of Table 4. There j 1 , j 2 ≥ 1 are the labels of the supermultiplet and all states have ∆ = 2+j 1 +j 2 . The members of this multiplet may be viewed as representing massless higher spin AdS 5 fields or the corresponding 4d conformal higher spin gauge fields.

Applications to AdS/CFT
Let us now apply the general expressions for (E c , a, c) to specific examples of AdS/CFT duality. This will require summation of contributions of infinite collections of 5d fields (in the above discussion of supermultiplets the sets of fields were finite), and thus a choice of a regularization that should be consistent with symmetries of the underlying theory.  Table 1. At the leading N 2 order (string tree level or classical type IIB supergravity) this matching was demonstrated in [21] (for the conformal anomalies) and in [65] (for the vacuum energy). To consider the next -string one-loop order it is natural to assume that the contributions of loops of all massive string modes should vanish. Indeed, string modes form long massive P SU (2, 2|4) multiplets 28 and thus should give zero contribution (cf. section 5). Equivalently, string mode masses depend on 't Hooft coupling (m 2 ∼ α −1 ∼ √ λ) and thus a non-trivial contribution from them would contradict the expectred non-renormalization of (6.1).
Assuming this, the subleading O(N 0 ) term in (6.1) should be reproduced just by the loop of massless string modes, i.e. by the one-loop correction in 10d type IIB supergravity compactified on S 5 . The latter is given by the sum of the contributions of the massless N = 8 5d supergravity multiplet and an infinite tower of massive KK multiplets [67]. Thus for consistency with (6.1) one should find that one-loop 10d IIB supergravity on S 5 : 28 KK descendants of massive string excitations sit in long multiplets given by tensoring string primaries with the long Konishi multiplet [66].
Here we put superscript + as we are interested in direct contributions of 5d fields with standard ("Dirichlet") boundary conditions given by This matching of both a and c coefficients at the one-loop supergravity level was earlier claimed in [22,23]. In particular, using (4.1) motivated by the prescription of [23], the vanishing of the type IIB supergravity contribution to c−a implied by (6.2) was interpreted in [24,25] as a consequence of the vanishing of the contributions of each of the long KK multiplet and the separate cancellation of the c−a contributions from states in the massless multiplet. 29 Reproducing the explicit value of a in (6.2) is much more non-trivial, requiring a specific choice of a regularization of the sum over the infinite number of KK modes. While our final conclusion is the same as in [23] the intermediate steps of the derivation disagree. Starting with our general expressions for E c , a, c given in section 5.1 we shall explicitly demonstrate the validity of (6.2) or (6.4). The proportionality (1.3) of E c and a-coefficient is expected due to the maximal supersymmetry, implying, in particular, that E c (i.e. the AdS 5 vacuum energy) does not vanish in the one-loop type IIB supergravity compactified on S 5 . This is different from the vanishing of the vacuum energy in N > 4 gauged supergravities in 4 dimensions [69] and in also in 11d supergravity compactified on S 7 [70][71][72]. 30 The non-vanishing of the vacuum energy in the pure N = 8 5d supergravity was already noted in [50] but the inclusion of the contribution of the KK multiplets leading to the value of E c in (6.2) is a new result.
The P SU (2, 2|4) multiplet content of 10d supergravity compactified on S 5 is recalled in Table 6 in Appendix E (where p is KK level). The degeneracies, i.e. the dimensions of the corresponding SU (4) representations can be found using (E.1). Summing up the elementary 5d field contributions using (5.6)-(5.8) and (6.3) we find for the massless p = 2 supermultiplet in Table 6 The p = 2 multiplet corresponding to the states of pure N = 8 5d gauged supergravity is isomorphic to the N = 4 4d conformal supergravity multiplet. 32 The corresponding values for the conformal anomaly and E c should thus be related as in (6.3): indeed, -2 times the 29 Similar pattern applies to matching of axial anomalies [68]. 30 A possible way to reconcile these different conclusions from the AdS/CFT point of view is to note that Casimir energy should automatically vanish in the case of 3d boundary theory, but need not in the 4d case (see also below). 31 The value of Ec is the same as found in [50]. 32 The full set of states of 10d supergravity compactified on S 5 will then correspond in 4d to N = 4 conformal supergravity coupled to infinite collection of conformal fields with canonical dimensions ∆− = 4 − ∆ corresponding to massive p ≥ 3 states in 5d spectrum in Table 6.
values in (6.5) gives the values in the last line of Table 3, i.e. we get the expression given above in (5.18). The equivalent form of (6.5) was given in (5.19). For both p = 3 and p ≥ 4 massive KK multiplets in Table 6 we obtain Remarkably, despite the different structure of the p = 2, p = 3 and p ≥ 4 multiplets in Table 6, their contributions to K = (E c , a, c) are thus universally described by 33 K + (KK level p of 10d IIB SG on S 5 ) = p K(N = 4 Maxwell) , p = 2, 3, 4, ... (6.7) As the p = 1 level may be interpreted as the N = 4 superdoubleton multiplet, this relation formally applies also for p = 1, becoming (5.16). For p = 2 eq.(6.7) is equivalent to (5.19), while for p > 2 to (6.6). A natural interpretation of this non-trivial identity (which relies on the particular values of E c , a, c we used) 34 is that it expresses the fact that the 5d states at the KK level p appear in the tensor product of p copies of N = 4 superdoubleton [45].
It remains to sum up the supermultiplet contributions (6.7) over the KK level p, i.e. to assign a consistent value to the divergent sum ∞ p=2 p. The prescription that is required to reproduce (6.4) is This can be interpreted as follows. As was noted above, the p = 1 case of (6.7) is the same as the contribution of one N = 4 Maxwell multiplet (5.15) or superdoubleton. The contribution of the p = 1 superdoubleton should not to be included [57] in the list of physical multiplets in Table 6 as it is gauged away [67] but if we would formally include it then under (6.8) the total 10d supergravity contribution would vanish. 35 The condition (6.8) is satisfied if one defines the sum over KK level p with a sharp cutoff and then drops all cutoff-dependent terms. 36 33 Note, in particular, that the relation (1.3) or (5.15) applies level by level, i.e. for each N = 4 supermultiplet. 34 For example, this relation would not be true for the c-coefficient had we used (4.2) instead of (4.9).
The expressions for the contributions of each p level to a and c coefficients found in [23] were very different: they were not linear in p but polynomials of order 5. The reason for this was that the expressions for the individual 5d field contributions to a and c used there (cf. (1.5)) were linear in ∆ − 2 and thus linear in p (cf. Table 6), while the higher powers in p were coming from the multiplicities given by the dimensions (E.1) of the corresponding SU (4) representations. The correct expressions for Ec, a and c found here are instead 5th order polynomials in ∆ − 2 (and thus in p, for the states in Table 6), but, remarkably, the non-linearity in p cancels out after multiplying by the dimensions of SU (4) representations and summing over the members of each supermultiplet. 35 Adding the p = 1 superdoubleton contribution would be equivalent to adding the decoupled U (1) D3brane contribution, i.e. the same as replacing SU (N ) by U (N ) group on the dual SYM side and thus dropping -1 term in (6.1). An alternative interpretation might be in terms of an effective bulk+boundary anomaly cancellation (conformal anomaly analog of "anomaly inflow"). That would also formally imply the cancellation of the total AdS5× S 5 vacuum energy as in in the AdS4× S 7 case. 36 Explicitly, one has P p=1 p = 1 2 P 2 + 1 2 P → 0. The same sharp cutoff regularization of the sum over KK level was assumed in [23]. In such a regularization all sums ∞ p=1 p n with positive integer n are just While the prescription (6.8) may look artificial (e.g., it is not the ubiquitous Riemann ζ-function rule) it is possible, in fact, to justify it by starting with the standard spectral ζ-function regularization. The key point is that a regularization consistent with symmetries of the theory should be applied directly at the 10d rather than 5d level, i.e. it should be based on the spectrum of the original 10d differential operators defined on AdS 5 × S 5 or its generalization.
Let us demonstrate this on the example of the sum of E c contributions. The expression for the contribution of a massive (∆; j 1 , j 2 ) 5d field to the vacuum energy E c can be obtained from the partition function (2.7) which may be written as Then (2.28) implies that a formal (divergent) expression for E c is given by This sum can be computed using the ζ-function prescription (2.28) applied to the full effective energy eigenvalue ∆ + k, or, equivalently, by introducing an exponential cutoff e k → e k e − (∆+k) , (6.12) doing the sum, expanding in → 0 and finally dropping all singular terms. Keeping finite we may find the contribution to the sum (6.10) from all states of the p ≥ 4 massive KK multiplet in Table 6. This gives the total summand e k (p; ). Summing over both k and p we obtain ∞ k=0 ∞ p=4 e k (p; ) = e −2 (95e +120e 3 /2 −220e 2 −420e 5 /2 +50e 3 +420e 7 /2 +210e 4 −6) (e /2 −1) 2 (e /2 +1) 10 = 249 Keeping only the finite part and adding the contributions of the p = 2 and p = 3 multiplets in (6.5),(6.6) gives finally for the total 10d supergravity contribution This is in agreement with (6.2) and thus confirms the prescription in (6.8).
set to zero. This formally explains why a different expression for the summand in [23] still led to the same correct expression for the result in (6.2).

Vectorial AdS 5 /CFT 4
In the case of vectorial AdS d+1 /CFT d correspondence one considers N free fields transforming in a vector (fundamental) representation of U (N ) or O(N ). The restriction to the singlet sector of bilinear conserved higher spin current operators implies duality to massless higher spin fields in AdS d+1 described by Vasiliev-type theories (see, e.g., [73][74][75][76]). The coefficient in front of the classical action in AdS d+1 is proportional to N , with the cubic and higher amplitudes supposed to match free-theory correlators of conserved currents at the boundary in 1/N expansion.
In d = 4 case we will be interested in here in the free fermion case there is a new feature: the corresponding conserved currents belong to particular mixed-symmetry representations of SO(4) [88,49,6]. Another novelty of the d = 4 case is that here one can also use spin 1 fields 37 as building blocks for higher spin conserved currents (free spin 0, 1 2 , 1 are the only options to get a 4d theory with a higher spin symmetry if one assumes unitarity [89][90][91][92]). 38 The corresponding conserved currents appearing in the product of two spin 1 doubletons as in [94,95,61,96,97,88,49,98] are also in specific mixed-symmetry representations of SO (2,4). The singlet sector of a theory of N Maxwell vectors should then be dual to a particular version of higher spin theory in AdS 5 involving mixed-symmetry fields which should exist but was not studied detail so far (we shall call it "type C" theory). 39 The field content of the corresponding dual pairs is summarized in Table 5 where we use the notation (∆; j 1 , j 2 ) c ≡ (∆; j 1 , j 2 ) + (∆; j 2 , j 1 ). Higher spin theory content matches the list of bilinear conserved currents in the boundary theory. It can be obtained by taking the product of two doubleton representations corresponding to the boundary fields (see Appendix A). In addition to conserved currents there are also scalar bilinears dual to (2; 0, 0) AdS 5 scalars in type A theory (see (A.5)) and fermion bilinears dual to (3; 0, 0) AdS 5 scalar and pseudoscalar in type B theory (see (A.6)). 40 Type A theories contain symmetric tensors while type B and type C theories include also particular mixed-symmetry representations 37 In d = 3 Maxwell vector is dual to a scalar. 38 In priniciple, one can also explore the possibility of defining the boundary theory in terms of higher spin singletons which are unitary and conformal when described in terms of field strengths. This possibility was noticed in [93] where the corresponding higher spin algebras were studied. 39 Interacting higher spin theory for totally symmetric fields in AdS5 was considered in [99,100]. Mixedsymmetry fields in AdS5 and the associated currents were discussed in [36,37,[101][102][103][104]39]. Cubic interactions of mixed-symmetry higher spin fields in flat space were studied in [105] and in AdS5 they were considered in [106,98,107,89,108,109]. The question of consistency of an interacting AdS5 theory involving mixed-symmetry fields goes beyond the cubic order and requires, in particular, the closure of the symmetry algebra [89]. Unitarity imposes additional constraints, excluding, e.g., partially massless fields. 40 The massless (2; 0, 0) scalars having ∆ − 2 = 0 will not contribute to the quantities K = (Ec, a, c) discussed below.  of massless higher spin fields in AdS 5 . The second series of massless (2 + s; s 2 , s 2 ) fields in U (N ) type B and type C theories are parity-odd. Restriction to real fields at the boundary implies projecting out some (odd-spin parity even and even-spin parity odd) fields in the bulk that either vanish or become total derivatives (see [6] for discussion of the minimal type B theory case).
Since the content of type C theory dual to (complex or real) 4d Maxwell fields was not explicitly studied in the literature let us comment on it in some detail. It can be obtained by taking the product of two spin 1 doubletons as in (A.7). 41 In the complex Maxwell field case the tower of relevant operators starts with dimension 4 operators appearing in the decomposition of F * µν F κρ into SO(4) irreps: 42 (i) scalar F * µν F µν and pseudoscalar F * µνF µν in massive representation (4; 0, 0); (ii) antisymmetric tensor F * µ[ν F κ]µ which is not conserved on shell and corresponds to massive selfdual + anti-selfdual rank 2 tensor, i.e. representation (4; 1, 0) c = (4; 1, 0)+(4; 0, 1); 43 (iii) spin 2 conserved stress tensor (4; 1, 1) and its parity-odd counterpart with one F µν replaced byF µν ; 41 Related discussions appeared in [110,95,111]; see also [112] for a general construction of higher spin currents as bilinears in higher spin fields in flat space. 42 Here * is complex conjugation, tilde denotes dual tensor and we suppress U (N ) vector index. 43 The corresponding antisymmetric tensor field in AdS5 appears, e.g., in S 5 compactification of type IIB supergravity and was discussed in [67,113]. Its AdS5 Lagrangian has first-derivative topological kinetic term plus the standard mass term.
(iv) conserved current with symmetries of Weyl tensor, i.e. the massless state (4; 2, 0) c described by the Young tableu with 2 rows and 2 columns. In addition, the product (A.7) of two spin 1 doubletons {1, 0}+{0, 1} ⊗ {1, 0}+{0, 1} (where {1, 0} and {0, 1} correspond to selfdual and antiselfdual parts of F µν ) contains also higher spin conserved currents dual to massless AdS 5 fields. The real vector case (minimal type C theory) is found by a projection similar to the one in type B theory case: removing one set (parity-odd) of symmetric tensor states and odd-spin mixed-symmetry states. This results in the spectrum given in Table 5.
The AdS/CFT duality implies the equality of the corresponding partition functions. For example, the singlet-sector partition function Z CFT of U (N ) conformal scalar defined on a curved space M 4 should be equal to the quantum partition function Z HS of the corresponding higher spin theory with coupling constant N −1 defined on an AdS 5 type Einstein space which is asymptotic to M 4 boundary. If M 4 has no non-trivial holonomies log Z CFT should be given just by the free-theory one-loop contribution. 44 It should match the leading classical term in log Z HS that should thus scale as N .
As the full non-linear classical actions for higher spin theories in AdS 5 are presently unknown, one is not able to compare the leading large N terms in the corresponding observables like (E c , a, c). Remarkably, it is still possible [2] to perform non-trivial nextorder checks: as O(N 0 ) term in log Z CFT is absent in the free theory case, the one-loop contribution to ln Z HS should vanish too. This was explicitly demonstrated for the acoefficient of type A theories in [5], and for the Casimir energy of type A and B theories in [6].
In the non-minimal type A and B theories where one sums over all spins one finds the vanishing results for the one-loop corrections to a-coefficient (from Z HS on AdS 5 with S 4 boundary) and to E c (from Z HS on AdS 5 with R × S 3 boundary). In the minimal theories the one-loop HS correction turns out to be non-zero and equal to that of one real 4d scalar (in the minimal type A case) and one Majorana fermion (in the minimal type B case). The proposed interpretation [2] of this fact is that the bulk coupling constant in the minimal HS theory is not N −1 but (N − 1) −1 , so that there is an extra O(N 0 ) contribution that comes from the corresponding N − 1 coefficient of the tree-level term that cancels the non-zero one-loop HS correction.
As for the c-coefficient, its matching was not attempted so far (apart from a remark in [5] that similar conclusions as for a-coefficient may apply in type A theory if one uses the expression (4.4) with the special "finite" choice of r b = −1 [3]). Neither a-nor c-coefficients were discussed previously in type B theories containing mixed-symmetry 5d fields.
The expressions for a and c coefficients corresponding to one-loop corrections of general 44 The singlet constraint may be imposed by integrating over an auxiliary pure-gauge vector field gauging the U (N ) or O(N ) global symmetry. This constraint does not change the leading order N term in the partition function, i.e. is not relevant for computing vacuum energy and conformal anomaly coefficients, but in presence of non-trivial holonomy like in S 1 × S 3 case it leads to an additional O(N 0 ) contribution to the non-trivial β-dependent part of the partition function (see [114][115][116]6] and refs. there). Note that the case of adjoint-representation vector fields (cf. [117] and also [118]) is different from the vector-representation one we consider here.
(∆; j 1 , j 2 ) fields in AdS 5 presented in section 5.1 allow us to complete the picture and explicitly demonstrate that the above matching pattern applies universally not only to E c [6] but also to a and c in all type A and type B cases. The matching of both conformal anomaly coefficients provides further non-trivial test of the consistency of the vectorial AdS/CFT duality. Note that here there is no supersymmetry, so there is no a priori reason to expect a correlation between the values of a and c or a and E c as in (1.3). As we shall see below, the novel case of type C theory appears to require a different matching pattern.
Since HS theories contain infinite number of fields, one needs a prescription of how to regularize the infinite sum of individual contributions. In the computations of the acoefficient and E c (from the partition functions in AdS 5 with S d and R × S d−1 boundaries where the heat kernel is explicitly known) there is a preferred regularization equivalent to the use of the spectral ζ-function [5,6]. Its use should be required by the preservation of symmetries of the theory at the quantum level. This regularization amounts to first doing the sum over spins of individual-field ζ(z)-functions for an arbitrary z and then analytically continuing the result (or its derivative) to the required value of z. As was found in [5], in the case of d-dimensional boundary this regularization is equivalent to introducing a specific exponential cutoff factor exp − (s + d−3 2 ) into the sum over spins s, doing the sum and then dropping all singular terms in the → 0 limit.
Below we shall apply the same prescription also for the summation of the contributions to the c-coefficient where a direct spectral ζ-function regularization is not available. In the present d = 4 case this prescription amounts to Here s = j 1 + j 2 is the total spin and the sum includes summation over all states. Let us denote by K + (∆; j 1 , j 2 ) any of the three quantities E + c , a + , c + corresponding to the one-loop contribution of a 5d field in the representation (∆; j 1 , j 2 ). Then, as in (6.3), K = −2K + will give the quantities for the associated elementary 4d conformal field with the canonical dimension equal to ∆ − = 4 − ∆.
Starting with the non-minimal type A theory and using the expressions in (2.33),(3.5) and (4.4),(4.11) together with the regularization (6.15) one finds that the total one-loop HS contribution to each of the three quantities is indeed zero In the minimal type A theory we get instead ∞ s=2,4,...
i.e. the total AdS 5 HS theory one-loop correction is equal exactly to the one-loop contribution of a single real massless 4d scalar. 45 As the contribution of N such scalars should match the classical plus one loop minimal type A higher spin theory result, this is consistent with the AdS/CFT duality provided the coefficient in front of the classical minimal HS theory action is not N but N − 1.
Similarly, in the non-minimal type B theory we get from (2.34),(3.6) and (4.5),(4.11) Here the first term 2 K + (3; 0, 0) = −K(3; 0, 0) stands for the contribution of the two 5d scalars appearing in the type B spectrum in Table 5. The contribution of the totally symmetric higher spin fields vanishes separately due to (6.16). The contributions of (∆; j 1 , j 2 ) and (∆; j 2 , j 1 ) states are equal so the mixed-symmetry term doubles. For c + this is equivalent to the first relation in (4.12) (where c = −2c + ).
In the minimal type B theory we find where the r.h.s. is the same as the contribution of a single 4d Majorana fermion (again equivalent to (4.12) in the case of c + ). 46 Repeating the same computations for the spectrum of the non-minimal type C theory in Table 5 we find (cf. (A.27) and the discussion of Casimir energy in Appendix A) 2 K + (4; 0, 0) + K + (4; 1, 0) c + 2 Here the sum of all AdS 5 one-loop contributions is no longer zero but is twice K(3; 1 2 , 1 2 ) = ( 11 120 , 31 180 , 1 10 ), i.e. is the same as the contribution of one complex 4d Maxwell field. This suggests that already in the non-minimal type C theory case one needs to assume that the coefficient in front of the corresponding HS classical action in AdS 5 is not N but N − 1. 47 In the minimal type C theory we get a relation similar to (6.20) Since here the boundary vector field is real, this non-vanishing result could be accommodated by the shift N → N − 2 in the coefficient of the classical HS action. This is analogous to what happened in the type A and B theories where one required an extra -1 shift of 46 Here K( 5 2 ; 1 2 , 0)c = 2K( 5 2 ; 1 2 , 0) = ( 17 960 , 11 720 , 1 40 ), see Table 2. 47 An alternative possibility may be to add 4 real massless 5d vectors to the bulk theory, i.e. to put the r.h.s. term in (6.20) to the l.h.s. as in (A.27), but it is unclear why that would lead to a consistent HS theory (and also which should be the corresponding conserved spin 1 currents in the boundary theory). the coefficient of the HS action when going from non-minimal to minimal case. The reason for the N → N − 1 shift required already in the non-minimal type C case remains to be understood.
Let us mention also that as discussed in Appendix A, the one-particle partition functions on S 1 ×S 3 in the non-minimal and minimal type C theories satisfy the relations (A. 22 19) in the type A and type B theories [6]. It is straightforward to derive these relations from the large N limit of the singlet-sector partition function for the boundary spin 1 theory just like that was done in the spin 0 and spin 1 2 cases in [116,6]. 48 Finally, let us note that while supersymmetry is not a necessary ingredient in vectorial AdS/CFT duality, it is possible to consider also supersymmetric AdS 5 /CFT 4 dual pairs. 49 An example of N = 1 supersymmetric higher spin theory in AdS 5 was constructed in [100]. The 4d boundary theory should be represented by N free spin (0, 1 2 ) N = 1 supermultiplets having bosonic integer spin and fermionic half integer spin conserved currents. Equivalently, in addition to the bosonic HS 5d fields there will be the fermionic ones coming from the product of spin 0 and spin 1 2 doubleton representations (cf. (A.29) for n 1 = 0, n 0 = n 1/2 ). The analog of (6.16) for the non-minimal theory should then be given by the sum of the bosonic and fermionic 5d field contributions. The bosonic part vanishes separately due to (6.16) while the fermionic part can be verified to satisfy the required identity (here we use s = s − 1 2 as in (2.16) which takes integer values for the fermions) There is also a minimal-theory analog of this relation ∞ s=1,3,5,...
It should be possible also to consider the case of supersymmetric boundary theory containing spin 1 fields. This will generalize the type A, B and C theory examples considered above. The most supersymmetric case of the free unitary boundary CFT will be a collection of N free N = 4 Maxwell supermultiplets. The spectrum of the dual AdS 5 HS theory will then be given by the product of two N = 4 superdoubletons [45,61,120,97,121] with the low-spin ≤ 2 part [59] being the same as the set of fields of type IIB supergravity compactified on S 5 given in Table 6. This HS theory with AdS 5 vacuum should correspond to the "leading Regge trajectory" part of the zero tension limit of AdS 5 ×S 5 superstring (cf. [66,122]). This may suggest a way to consider a particular maximally supersymmetric case of the vectorial AdS/CFT duality as a truncation of zero gauge coupling limit of the adjoint AdS/CFT. As we have seen in sections 5.1 and 6.1, when 5d fields are combined into supermultiplets many cancellations happen, and this should especially be true in the maximally supersymmetric case. 48 We shall present details of this derivation elsewhere. 49 Supersymmetric AdS4/CFT3 cases were discussed, e.g., in [78,79,119]. KK states of type IIB supergravity on S 5 are contained in tensor products of more than two N = 4 superdoubletons [45]. Their contribution (6.7) was computed in section 6.1 above. We leave the discussion of the contributions of their partner higher spin states for the future.

Conformal higher spin theories
The relation (6.16) written in terms of K = −2K + = (E c , a, c) has also another interpretation: it expresses the vanishing of the total Casimir energy and the total conformal anomaly coefficients in the 4d conformal higher spin (CHS) theory of all symmetric bosonic 50 Here we again use the regularization (6.15) with s = j1 + j2. It turns out that in N = 4 supersymmetric case the total result has no poles in → 0. This is due to supersymmetry and can be understood as follows.
Here we are summing the contributions of bosonic and fermionic fields, where in the fermionic case s = s − 1 2 is an integer. Ignoring regularization and separating finite number of low-spin terms, the remaining sum can be rewritten as s K b (s) + K f (s − 1 2 ) and happens to vanish, implying finiteness of the total result. gauge fields. The vanishing of the total a-coefficient was first observed in the 5d context [1] and then understood also directly from the 4d perspective [3]. The cancellation of E c was demonstrated in [7]. The vanishing of the total c-coefficient requires the use of our proposed expression (4.9) leading to (4.4) with the specific choice of the parameter r b = −1 [3] in (4.11). Similar conclusion applies also to the fermionic CHS theory with the individual field contributions given in (2.34),(3.6),(4.5), (4.11), generalizing earlier demonstration of the vanishing of its total a-coefficient [3].
The consistency of the vectorial AdS/CFT is thus tightly related with the consistency (cancellation of anomalies) of the associated CHS theories. This is not completely surprising in view of the direct connection of the CHS theory (viewed as induced by the boundary CFT [123][124][125]1]) to the CFT conserved currents (CHS fields are shadow fields for the CFT currents) and then, via AdS/CFT, to the 5d higher spins (CHS fields are effectively boundary values for the 5d fields).
While it still remains to prove our conjecture for the c coefficient in (4.9) (implying the values in (4.4),(4.5),(4.11)) this is a strong indication that in addition to the N = 4 supersymmetric theory of conformal supergravity coupled to 4 Maxwell multiplets containing finite number of fields, the theory of an infinite collection of conformal higher spins is also a consistent quantum conformal theory with no Weyl anomalies (both theories are of course perturbatively non-unitary). The same should be true also for the SU(2, 2|N ) supersymmetric conformal higher spin theories like the one constructed in [126,127] and its truncations [100].

Concluding remarks
There are many open questions. One interesting question is to understand better the vectorial AdS/CFT duality in the spin 1 boundary theory case, clarifying the structure of the dual type C theory in Table 5 and providing the interpretation for the equation (6.20).
It remains to explore further the relation between vectorial AdS/CFT duality setup for N = 4 superdoubleton as boundary theory and a tensionless limit of the AdS 5 × S 5 string theory, computing, in particular, the quantities (E c , a, c) and also the twisted and thermodynamic one-particle partition functions for the string spectrum of 5d fields.
Another direction is to attempt to build an example of vectorial AdS/CFT duality by starting with spin > 1 conformal fields at the boundary and considering the set of (in general, non-unitary) 5d higher spin fields corresponding to their conserved currents.
A.1 Characters of products of doubletons Above relations have immediate counterparts in terms of ("blind") characters for the basic representations in (A.1) 52 The character (A.8) has the interpretation of one-particle partition function Z + in (2.7) corresponding to a massive 5d field while the one in (A.9) is the one-particle partition function Z + in (2.7) corresponding to a massless 5d field (2.8) with ∆ 0 = 2 + j 1 + j 2 . For the doubleton partition function we shall also use the notation Z({j, 0}), i.e.
The relations for one-particle partition functions of non-minimal type A and type B theories in Table 5 are direct character counterparts of (A.5) and (A.6): We also get the following character identities that express the relations between one-particle partition functions in minimal type A and type B theories [ For spin 1 doubleton characters we find the following identities Since (A.8) implies that Z(4; 1, 0) = Z(4; 0, 1) = 3Z(4; 0, 0) we get the relation which is the counterpart of (A.7) at the character level: It has the direct interpretation as the relation of one-particle partition functions in nonminimal type C theory in Table 5 It would be interesting to know a group theoretic interpretation of this relation. It is possible to show, just like this was done in the scalar case in [6], that the l.h.s. of (A.23) corresponds to the leading large N term in the singlet-sector partition function of N real 4d Maxwell vectors. Let us now comment on the corresponding (2.28),(2.29) Casimir energy. Note that the expressions (A.24) 53 Here the notation Z({0, 0}) q→q 2 stands for q 2 (1−q 2 ) 3 1 + q 2 , etc.
are invariant under q → q −1 . This implies that the total Casimir energy of the 5d fields appearing in the r.h.s. of (A.5),(A.6) or (A. 16), (A.17). The presence of the additional Z q→q 2 terms in the r.h.s. of (A.18),(A.19) which change sign under q → q −1 implies that the Casimir energy for the representations in the r.h.s. is no longer vanishing in minimal type A and type B theories suggesting the N → N − 1 shift in the 5d classical action of the dual HS theory for a consistent AdS/CFT interpretation [6] (see section 6.2). In contrast, for spin 1 doubleton product (A.7) we get q → q −1 non-invariant expression already in the r.h.s (A. 22). This implies that the Casimir energy in type C theory does not vanish even in the non-minimal case. Observing that one can form a q → q −1 invariant combination as 26) we conclude that one can formally make the Casimir energy vanish by adding four (3; 1 2 , 1 2 ) to the representations in (A.7), getting a theory with field content 4 (3; 1 2 , 1 2 ) + 2 (4; 0, 0) + (4; In the case of the minimal type C theory the l.h.s. of (A.23) contains half of the same term plus an extra q → q −1 non-invariant term Z({1, 0}) q→q 2 , and the two combined together give the same Casimir energy as in the non-minimal theory (see section 6.2).
Let us note that, as follows from (A.14), the partition functions of superdoubletons are the same as of the corresponding super Maxwell theories. Here we shall explicitly compute the one-loop partition functions for low-spin fields that appear in N ≤ 4 conformal supergravities (see Tables 2 and 3). The resulting expressions for the one-particle partition functions will be the same that follow from the operator counting method. The cases of the standard scalar, vector and Weyl graviton were already discussed 55 The term −2 n 2 1 (3; in [7]. For example, for the Maxwell vector (cf. (2.5),(2.6),(2.9),(2.14),(A.25)) Let us start with the familiar case of spin 1 2 Majorana fermion, i.e. L 1 2 = ψ e µ a γ a D µ ψ, . The corresponding partition function is where R = 6 is the scalar curvature of the unit-radius S 3 and ∂ 0 is derivative along the Euclidean time with period β = − ln q. In general, the spectrum of the square of the Dirac operator on unit-radius Then by the standard arguments the corresponding one-particle partition function in (2.2) is given by (see, e.g., [7]) This has direct operator-counting interpretation in R d : counting components of ψ (and their derivative descendants) minus equations of motion / ∂ψ = 0. For d = 4 this gives as in [47] (B.5) Next, let us consider the conformal gravitino [133] with the following quadratic Lagrangian in curved background (we omit DRψψ terms) Considering a Bach (e.g., an Einstein) space background we may fix the gauge symmetries by γ µ ψ µ = 0 and D µ ψ µ = 0, i.e. restrict to transverse γ-traceless field. Then we get This operator factorizes [134,135,29] on an Einstein space background (R µν = 1 4 Rg µν ) as Specializing to the S 1 × S 3 case, we have R µν → R ij = R 3 g ij = 2g ij i.e.
Taking into account that { / D, γ 0 } = 0 the determinant of this operator can written as The spectrum of D D D 2 for a general spin s field on S 3 is (see, e.g., [40]) − D D D 2 → (n + s)(n + s + 2) − s, d n = 2(n + 1)(n + 2s + 1) , (B.13) so that for s = 3 2 we get / D 2 → − n + 5 2 2 , d n = 2(n + 1)(n + 4) . (B.14) Thus the contribution of the spatially transverse and traceless gravitino ψ i to the oneparticle partition function is 2 (n + 1)(n + 4)(q n+ 3 2 + q n+ 5 2 + q n+ 7 2 ) = 4 q To get the full partition function we still need to add the contribution of one Majorana spinor degree of freedom. 56 On R × S 3 , we may further split TT ψ µ into TT ψ i and a spinor. This gives Z = det O TT is defined on transverse γµ-traceless field ψµ (see, e.g., [3]). This correctly accounts for −8 degrees dynamical degrees of freedom: transverse traceless (TT) field ψµ contributes 2 × 4 (with extra factor of 3 being due to the degree of the kinetic operator) and the fermion contributes 2 × 4. 57 This mode must be dropped for the same reason as discussed in Appendix D of [7].
This expression admits the following operator counting interpretation in flat space. The natural gravitino analog of the covariant Weyl tensor field strength for the conformal graviton is its superpartner [136] (tilde denotes the dual field) obeying Φ µν = γ 5Φµν , γ µ Φ µν = 0. These conditions imply that Φ µν has 4 × 2 components (the γ 5 self-duality reduces 6 to 3 and the γ-tracelessness adds one additional constraint). This explains the first 4 × 2q 3 2 term in the numerator of (B.17). The next term 4 × 2q 5 2 is associated with / ∂Φ µν . The equations of motion and the Bianchi identities remove the term 4 × (3 + 3)q 7 2 ; the term 4 × 2q 9 2 compensates for overcounting in this subtraction (cf. [7]). The Lagrangian of the conformal fermion Ψ with / ∂ 3 kinetic term is [51,27] Rg µν γ µ D ν . 2(n + 1)(n + 2)(q n+ 1 2 + q n+ 3 2 + q n+ 5 2 ) = 4 q  where D 2 = ∂ 2 0 + D 2 , and we used that D 2 has the spectrum −n(n + 2) with multiplicity (n + 1) 2 . As a result, Z Φ (q) = ∞ n=0 (n + 1) 2 (q n + q n+2 ) = 1 − q 4 (1 − q) 4 . (B.23) Similar computation can be done in the case of the non-gauge conformal antisymmetric tensor field T µν [51,27,44] with the Lagrangian (corresponding to the Weyl-invariant action) Here we shall just quote the result for the corresponding partition function which is much easier to find by the counting method in flat space. T µν has 6 components with dimension 1. The equations of motion C Spectral ζ-function for 2nd-order operator on (∆; j 1 , j 2 ) fields in AdS 5 The computation of a-coefficient requires consideration of (in general, massive) higher spin field partition function in Euclidean AdS 5 with boundary S 4 . The relevant kinetic operator O given in (1.14) is defined on transverse fields. In general, for the operator O on a space M one can express the corresponding ζfunction in terms of heat kernel as ζ(z) = 1 Γ(z) Here tr is the trace over the Lorentz indices of the operator and ζ(z; x) does not actually depend on x.
while (4.2) is reproduced if r b = 1 2 as in (4.6). Our choice (D.4) ensures, in particular, that the consistency conditions for vectorial AdS/CFT discussed in section 6.2 that hold for a-coefficient and E c are valid also for the c-coefficient. E AdS 5 field content of type IIB 10d supergravity compactified on S 5 In Table 6 we summarize the field content of S 5 compactification of IIB supergravity [57,67]. For each KK level p we list the corresponding SO(2, 4) and SU (4) representations.