Exceptional F(4) Higher-Spin Theory in AdS(6) at One-Loop and other Tests of Duality

We study the higher-spin gauge theory in six-dimensional anti-de Sitter space $AdS_6$ that is based on the exceptional Lie superalgebra $F(4)$. The relevant higher-spin algebra was constructed in arXiv:1409.2185 [hep-th]. We determine the spectrum of the theory and show that it contains the physical fields of the Romans $F(4)$ gauged supergravity. The full spectrum consists of an infinite tower of unitary supermultiplets of $F(4)$ which extend the Romans multiplet to higher spins plus a single short supermultiplet. Motivated by applications to this novel supersymmetric higher-spin theory as well as to other theories, we extend the known one-loop tests of $AdS/CFT$ duality in various directions. The spectral zeta-function is derived for the most general case of fermionic and mixed-symmetry fields, which allows one to test the Type-A and B theories and supersymmetric extensions thereof in any dimension. We also study higher-spin doubletons and partially-massless fields. While most of the tests are successfully passed, the Type-B theory in all even dimensional anti-de Sitter spacetimes presents an interesting puzzle: the free energy as computed from the bulk is not equal to that of the free fermion on the CFT side, though there is some systematics to the discrepancy.


Contents
Higher-spin theories differ from M/Superstring theory in one fundamental way, namely they involve massless fields of arbitrarily high spins and furthermore they favour AdS backgrounds for their consistent formulations [4]. Higher-spin theories are models of AdS/CFT correspondence that should be considerably simpler than the full-fledged strings on AdS 5 × S 5 vs. maximally supersymmetric gauge theory in four dimensions while sharing some of the main features with string theory -dynamical graviton and fields of arbitrarily high spin.
The basic properties of higher-spin (HS) AdS/CFT dualities include: (i) higher-spin theories are in most cases duals of CFT's with matter in fundamental representation rather than in adjoint [5], which simplifies the spectrum of single-trace operators and reduces the field content of HS theories as compared to string theory; (ii) unbroken higher-spin theories are expected to be dual to free CFT's [5][6][7][8]; (iii) models of HS AdS/CFT dualities exist in any spacetime dimension [8]; (iv) interacting CFT's, like the Wilson-Fisher O(N) model, can be duals of the same higher-spin theories for a different choice of boundary conditions [5,9,10]; (v) the duals of CFT's with matter in the adjoint representation, e.g. N = 4 SYM at zero coupling, should also be certain HS theories coupled to matter, [6,[11][12][13].
The simplest free conformal fields provide the basic examples of HS AdS/CFT dualities: free scalar field is dual to Type-A HS theory with spectrum made of totally-symmetric HS fields and free fermion is dual to Type-B whose spectrum contains specific mixed-symmetry fields that include totally-symmetric HS fields too. The results of [8,[24][25][26] establishing a one-to-one correspondence between higher-spin theories and supersymmetric extensions thereof and massless conformal fields and conformal supermultiplets imply that this duality extends to all unbroken higher-spin theories and their supersymmetric extensions.
Symmetries of gauged supergravities in AdS 3,4,5,/ 6,7 are well covered by the classical Lie superalgebras of type OSp(M|N) or SU(N|M) [27][28][29]. The gap in AdS 6 gauged supergravities was bridged by Romans in [30] where the relevant superalgebra turned out to be the exceptional superalgebra F (4). 1 Later it was shown that Romans gauged supergravity arises in a warped S 4 compactification of the massive IIA supergravity [31] as well as from type IIB supergravity [32]. In general much less is known about AdS 6 in the context of AdS/CFT dualities 2 and HS theories than in other dimensions.
One of the original motivations [4] for higher-spin theories had been to overcome the N ≤ 8 restriction on the number of super-symmetries in d = 4 supergravities. In AdS 3,4,5,/ 6,7 one does find infinite families of anti-de Sitter superalgebras with any number of supersymmetries. We find it remarkable that there exists [26] an AdS 6 higher-spin algebra whose maximal finite-dimensional subalgebra is the exceptional Lie superalgebra F (4), which is a unique supersymmetric extension of the 5d conformal algebra SO (5,2). Even subalgebra of F (4) is SO(5, 2) ⊕ SU (2) and the corresponding HS algebra can be realized as the universal enveloping algebra of the minimal unitary supermultiplet 1 We should note that the simple exceptional Lie algebra of rank 4 is denoted as F 4 and does not contain the even subalgebra SO(7) ⊕ SU (2) of the exceptional Lie superalgebra F (4). 2 See e.g. [33][34][35][36][37] and references therein.
of F (4) (super-singleton) obtained via the quasiconformal approach [38][39][40] that consists of two complex scalars in a doublet of R-symmetry group SU(2) R and a symplectic Majorana spinor field. The HS dual contains a tower of totally-symmetric bosonic and fermionic HS fields that are dual to HS conserved currents and super-currents as well as a tower of mixed-symmetry fields. The lowest F (4) supermultiplet in this infinite tower, as we will show, is exactly the Romans' supergravity multiplet.
Relying upon AdS/CFT, higher-spin theories should be consistent quantum field theories, which requires a proof. At present, the full action of any of the higher-spin theories is not yet known. 3 The part of the action that is known does not allow to compute the full one-loop self-energy 4 or betafunction that would provide an access to the quantum properties of HS theories and also a link to the anomalous dimensions of the higher-spin currents in the Wilson-Fisher vector-model. Fortunately, knowledge of kinetic terms is sufficient to perform many nontrivial consistency checks on both sides of the duality by matching various quantities that can be extracted from one-loop partition functions.
Another important ingredient of one-loop computations is the knowledge of the spectrum, which can be inferred from the list of higher-spin algebras [8,22,23,26,51,52]. A simpler way to calculate the spectrum is to enumerate single-trace operators in various free CFT's, which increases considerably the number of examples.
Many one-loop tests have already been performed in a series of papers [12,13,[53][54][55][56][57][58][59][60][61], see also [62,63] for the 3d case. The main lessons are as follows. Each of the fields in the spectrum of HS theories contributes a certain amount to one of the computable quantities: sphere free energy, Casimir Energy, a-and c-anomaly coefficients. The sum over all spins is formally divergent and requires a regularization. Refined in this way the sum over spins becomes finite and matches the corresponding quantity on the CFT side, which in many cases leads to nontrivial tests rather than 0 = 0 equalities.
Motivated by our study of the exceptional F (4) higher-spin theory in AdS 6 we extend the oneloop tests to a number of cases: (i) we derive the spectral zeta-function for arbitrary mixed-symmetry bosonic and fermionic fields; (ii) we compute one-loop determinants for Type-A and Type-B theories; (iii) we study the contributions of fermionic HS fields in diverse dimensions, which is crucial for the consistency of SUSY HS theories; (iv) in AdS 5 we study Type-D,E,... HS theories that are supposed to be dual to higher-spin doubletons with spin greater than one and find that they do not pass the one-loop test; (v) partially-massless fields are also briefly discussed; (vi) a simple expression for the a-anomaly of an arbitrary-spin free field is found; (vii) with the help of the heat kernel technique it is argued that a part of the tadpole diagram of the Type-A theory should vanish; (viii) the spectrum of the F (4) HS theory is worked out and is shown to contain the Romans supermultiplet. 3 The cubic action in de-Donder gauge was recently reconstructed [41] in any dimension for the Type-A theory by the AdS/CFT matching with some partial results in [11,42], see also [43] for 3d. A part of the on-shell quartic action is known in d = 4 thanks to [44]. There are also alternative approaches to the action problem: a generalized Hamiltonian sigma-model action [45], where the Fronsdal kinetic terms are absent, but the theory still can be quantized. See also [46] and [47]. The spectrum of HS theories is determined by HS algebras and for that reason is consistent with linearized Vasiliev equations whenever they are available [48,49]. 4 See [50] for the promising partial results that indicate that the quartic vertex has good chances to cancel all the infinities coming from the bubble made of two cubic vertices.
Extending the findings of [53] we discover that the Type-B theories in all even dimensions lead to puzzling results that call for a better understanding of the duality, the bulk result, however, still can be represented as a change of the F -energy.
The outline is as follows. In Section 2 we review the basic facts about Higher-Spin AdS/CFT correspondence and recall what can be extracted at one-loop order given the fact that the full action is not known. In Section 3 one-loop tests are performed for a number of cases: fermionic HS fields that are necessary present in SUSY HS theories and for mixed-symmetry fields that are omnipresent in AdS 5 and higher. In Section 4 the properties of the exceptional F (4) Higher-Spin Theory are studied. Technicalities are collected in numerous Appendices. The summary of the results and discussion can be found in Section 5.

Higher-Spin Theories at One-Loop
As discussed below, one-loop computations in higher-spin (HS) theories require one simple ingredient as an input data: a CFT with infinitely many conserved higher-rank tensors -higher-spin currents.
Such CFT's are very special -they are free or N → ∞ limits of certain interacting ones, which again behave like free theories in the strict N = ∞ limit. The algebra of HS currents determines the field content of the dual HS theory and allows one to perform many one-loop tests. We briefly review basic facts about higher-spin theories and the scheme of one-loop tests.

Higher-Spin Theories
The intrinsic definition of higher-spin (HS) theories is that they are field theories with infinitely many massless higher-spin fields. A systematic approach is via the Noether procedure, i.e. one starts with the free fields and then tries to add interaction vertices and deform gauge transformations as to maintain gauge invariance of the action.
The AdS/CFT correspondence provides an easier approach to HS theories -HS theories can be thought of as duals to free CFT's [5][6][7][8]. Indeed, HS gauge fields are dual to conserved tensors of rank greater than two, i.e HS conserved tensors 5 The presence of an at least one conserved HS tensor in addition to the stress-tensor in a CF T d in d ≥ 3 makes this CFT a free one in disguise [8,[64][65][66][67][68]. In particular, it implies that conserved tensors of arbitrarily high rank are present. Conserved tensors generate charges and for that reason such CFT's have infinite-dimensional algebras of symmetries, higher-spin algebras, see [69] for the first occurrence of the HS algebra concept in the literature. On the CFT side HS algebra is the algebra of global symmetries and contains the conformal algebra as a subalgebra.
HS currents together with the stress-tensor and few other matter fields that are produced by acting with HS charges form a representation of the HS algebra they generate. On the AdS side the global symmetry should become a gauge symmetry, i.e. a given HS algebra needs to be gauged.
The spectrum of the dual HS theory is induced by the single-trace CFT operators: HS currents and certain other operators as will become clear below.
Let us consider two simplest examples of free CFT's: free scalar and free fermion, whose duals are usually called Type-A and Type-B, respectively, and then extend them to more general cases including the super-symmetric ones.
Type-A. A free scalar field φ = 0 as a representation of the conformal algebra is usually called Rac. With one complex scalar one can construct conserved higher-spin currents, which are totallysymmetric tensors: where we also add the 'spin-zero current'φφ. If the scalar is real then the currents of odd ranks vanish.
For a free theory doing operator product expansion (OPE) is practically equivalent to computing the tensor product of the conformal algebra representations, which in the case ofφφ OPE leads to [17,52,70]: More generally, one can take the scalar field with values in some representation V of some Lie group G and impose the singlet constraint, i.e. project onto G-invariants. In the representation theory language the fundamental fields belong to S = Rac ⊗ V and the spectrum of bilinear operators corresponds to the G-invariant part of the tensor product S ⊗ S. Technically what matters is the symmetry of J s with respect to exchange of two fields and the symmetry of the G-invariant tensors.
For example, if φ i are SO(N)-vectors and N is large, then the relevant invariant tensor is δ ij , which is symmetric. Noting that P (J s ) = (−) s J s , where P exchanges the two scalar fields, we observe that all HS currents with odd spins are projected out and the SO(N)-invariant single-trace operators belong to (Rac ⊗ Rac) S , i.e. have even spins. Therefore, the SO(N)-singlet constraint distinguishes between (anti)-symmetric parts of Rac ⊗ Rac, [17,52,70]: In accordance with (2.4) the spectrum of the Type-A theory is made of bosonic totally-symmetric HS fields that are duals of J s , known as Fronsdal fields [71], and an additional scalar field Φ 0 that is dual to φ 2 . At the free level Fronsdal fields s = 0, 1, 2, 3, ... obey 6 where ξ a(s−1) represents gauge modes. The value of the mass-like term follows directly from the conformal weight of the conserved HS current it is dual to, as usual.
HS theory of totally-symmetric HS fields, s = 0, 1, 2, 3, 4, ... is called the non-minimal Type-A, which is the U(N)-singlet projection, and the one with even spins only, s = 0, 2, 4, ... is the minimal Type-A, which is the O(N)-singlet projection. One can also define the usp(N)-singlet theory whose spectrum is made of three copies of odd spins and one copy of even spins [53].
Type-B. Analogously, one can take a free fermion / ∂ψ = 0, which is called Di. The spectrum of single-trace operators is more complicated [17,52,70,72,73]. The height, (p + 1), of the hook Young diagrams cannot exceed dimension d. Moreover, with the help of the ǫ-tensor the hooks with p + 1 > d/2 can be dualized back to p + 1 ≤ d/2. In the case of d even one may also decompose the hooks with p + 1 = d/2 into two irreducible components. We will give a more detailed description of the Type-B spectrum in Section 3.
Accordingly, the spectrum of the Type-B theory is made of bosonic mixed-symmetry gauge fields with spin Y(s, 1 p ), s > 1, ∀p or s = 1, p = 0: 9 We will refer to such fields simply as hooks, having in mind the shape of Young diagrams Y(s, 1 p ).
The general formula for the mass-like term was found in [74,75]. The anti-symmetric tensors (2.9) are dual to massive 10 anti-symmetric fields, including the scalar Φ: This is the spectrum of the non-minimal Type-B and one can extend the discussion to the duals of (symplectic)(Majorana)-Weyl fermions. It is worth stressing that Type-A theory is in no sense a sub-theory of Type-B. In particular, the cubic couplings are different [11,42], the only exception being the d = 3 case where there are no mixed-symmetry fields and the HS algebras generated by free boson and free fermion are the same.
SUSY HS Theories. The simplest super-symmetric HS theories result from CFT's made of a number of free scalars and fermions. The single-trace operators contain those of Type-A and Type-B combined. Also, there are super-currents: 11 (2.14) 9 Young symmetry requires to add ∇ξ-terms with different permutations, which are hidden in .... 10 There are different definitions of masslessness in anti-de Sitter space. As far as s > 1 2 fields are concerned, the most natural definition seems to be the one where massless fields are those that have gauge symmetries which reduce the number of physical degrees of freedom. The same fields can also be found in the tensor product of two singletons/doubletons, which is the definition of masslessness adopted in [14][15][16]. As for matter fields with s = 0, 1 2 one can either adopt the latter definition or refer to conformally coupled fields as massless instead. Massive h-forms of Type-B theories do not have any gauge symmetries. 11 As primaries the currents must be traceless in a(s) and γ-traceless in a(s); α, the former being a consequence of the latter.
The super-currents, as representations of the conformal algebra, belong to Di ⊗ Rac [17,52,70]: (2.15) The super-currents are dual to totally-symmetric fermionic HS fields, Fang-Fronsdal fields [76,77]: For the purpose of computing the determinants we need to know the square of the HS Dirac operators where the mass-like terms were found in [78] for fermionic fields of any symmetry type.
Therefore, in the simplest super-symmetric HS theory the spectrum is Type-A plus Type-B plus fermionic HS fields, which can be packed symbolically into super-matrices of the form Again, one can take a number of φ's and ψ's and impose the singlet constraint with respect to some global symmetry group. Note that the d = 3 case is special in that there are no mixed-symmetry fields, i.e. p = 0: both Type-A and Type-B have totally-symmetric HS fields only, but φ 2 has weight ∆ = 1 whileψψ has ∆ = 2, which corresponds to the same mass-like term M 2 = −2.
More general HS theories. The general scheme is the following. Given some d there is a list L of free conformal fields that can exist in CF T d . Generically, L always contains free scalar and free fermion. Also, one can add free conformal fields 12 φ S with any spin S obeying k φ S + ... = 0, k = 1, 2, ... equations of motion. However, these are usually non-unitary, which may not be an obstruction to HS AdS/CFT. In even dimension d = 2n doubletons S j with spin-j are also available 13 [8,14,16,80,81], where j = 0, 1 2 are the usual Rac and Di. The j = 1 case corresponds to d 2 -forms, e.g. the Maxwell field-strength F ab in d = 4. It can be further projected onto (anti)-selfdual components, S ± 1 . Therefore, there is some list of free conformal fields of interest in dimension d: In order to build a more general free CFT one can select a number of distinct free fields L i . For every field one can pick some group H i and let it take values in some representation of H i . Also, one should choose some group F ∈ H i that will be used to impose the singlet constraint, i.e. by projecting onto the invariants of F . The higher-spin symmetry or the spectrum of the AdS-dual 12 For a comprehensive list of conformally-invariant equations we refer to [79]. 13 Formally, the so(d = 2n)-spin of doubletons is Y(j, ..., j) = Y(j n ), but we abbreviate it simply as spin-j. theory is then generated by all bilinear quasi-primary operators that are F -singlets. In the case of the adjoint duality one has to consider long trace operators that are dual to certain matter-like massive (HS) fields that couple to the massless sector. Therefore, the duals of free CFT's with matter in adjoint representations look like the duals of CFT's with fundamental matter coupled to certain matter multiplets [6,[11][12][13].
For example, one can take F = u(N), L 1 = Rac, and fields L 1,2 to take values in the Nn and Nm dimensional representations. The resulting F -singlet spectrum has HS fields of Rac ⊗ Rac with values in u(n), fields of Di ⊗ Di with values in u(m) and 2nm fermionic HS fields, see [82] for d = 3.
As another example, one can define Type-C [58] as the dual of the spin-j = 1 doubleton S 1 for d = 2n, i.e. AdS 5 /CF T 4 , AdS 7 /CF T 6 , etc. The spectrum of Type-C contains more complicated mixed-symmetry fields. It is also possible to cook up extended multiplets n b Rac+n f Di+n v S 1 [56]. In AdS 7 /CF T 6 one can take [59] the (2, 0) tensor supermultiplet that contains Rac, Di and a self-dual rank-three tensor T = S 1 , which is spin-one doubleton [14].
The HS algebra based on i n i L i contains diagonal elements L i ⊗ L i and off-diagonal ones L i ⊗ L j . Fermionic HS fields, if any, are always placed into off-diagonal blocks since they result from fermion ⊗ boson products. There are some other fields that can only arise in off-diagonal blocks, e.g.
In dimensions d = 3, 4 and d = 6 there exist conformal superalgebras, namely OSp(N|4, R), SU(2, 2|N) and OSp(8 * |2N), with arbitrary number of supersymmetry generators. Since there is no constraint on the spins of the particles in higher-spin theories there is no constraint on the number of supersymmetry generators of HS superalgebras in these dimensions [19,22,23,52,82]. More general HS superalgebras in higher than six dimensions can also be defined [52,82]. However the supersymmetry in these theories do not obey the usual spin and statistics connection. Only in dimensions d ≤ 6 the HS superalgebras contain the usual spacetime conformal superalgebras as finite-dimensional subalgebras.
Also, as was noted in [84] in the case of AdS 4 /CF T 3 and in [85] for AdS 3 /CF T 2 the AdS/CFT truncates the number of super-symmetries back to the usual one by the boundary conditions. The same is expected to be true in any other dimension where the super-symmetric CFT duals can exist. 14 In other words, usual AdS/CFT restricts the number of super-symmetries in HS theories not to exceed that of supergravities. Dualities between HS theories with any number of SUSY's and free CFT's made of a number of scalars and fermions may still work in any d, though not having standard superalgebras behind.
More recent work has shown that Rac's, which are singletons of SO(d, 2) for odd d and scalar doubletons of SO(d, 2) for even d, are simply the minimal unitary representations of SO(d, 2). For odd d they admit a single deformation (spinor singleton), Di, and for even d they admit an infinite family of deformations (doubletons) [8,[24][25][26]. Furthermore there exists a one-to-one correspondence between the minimal unitary representations of SO(d, 2) and their deformations and massless conformal fields in d dimensional Minkowskian spacetimes [8]. These results were obtained by quantization of the geometric realization of SO(d, 2) as quasiconformal groups [38][39][40]. The geometric realizations of noncompact groups as quasiconformal groups that leave invariant a quartic light-cone was discovered in [38]. The quantization of the geometric quasiconformal realization of a noncompact group leads directly to its minimal unitary representation [40,86].
Quadratic action. Combining the ingredients together the quadratic gauge fixed action of the simplest SUSY HS theory that is cooked up from Rac's and Di's should have the form 20) where the multiplicities N A , N B , N F depend on the multiplet chosen and also, for specific multiplets, can depend on whether spin is even or odd, but HS fermions enter all together s = 1 2 , 3 2 , 5 2 , .... On general grounds the bulk coupling constant G should be related to the fraction N of the fields removed by the singlet constraint as G −1 ∼ N, at least in the large N limit. It was observed [53,54] that in some of the cases this relation should be G −1 = a(N + integer). It can happen that there are higher-order N −1 -corrections as well.

One-Loop Tests
The idea of the one-loop tests of HS AdS/CFT was explained in [53,54]. The AdS partition function as a function of the bulk coupling G should lead to the following expansion of the free energy F AdS : where the first term is the classical action evaluated at an extremum. F 1 stands for one-loop corrections, etc. The large-N counting suggests that G −1 ∼ N. Moreover, N is expected to be quantized [64], which is not yet seen in the bulk. 15 On the dual CFT side there should be a similar expansion 15 It is an interesting question whether the quantization of the bulk HS coupling can be understood as a consequence of invariance under large higher-spin transformations as in Chern-Simons theory.
for the CFT free energy F CF T : A nice property of free CFT's is that all but the first term are zero, which should match F 0 AdS . However, since the classical action is not known, one cannot compute F 0 AdS and compare it to F 0 CF T . Still, one can check that the second term, F 1 AdS vanishes identically or produces a contribution proportional to F 0 CF T , which can be compensated by modifying the simplest relation G −1 = N to G −1 = a(N + integer) [53,54].
This basic idea allows to perform several non-trivial tests thanks to the fact F can be computed on different backgrounds. The simplest ones include S d , R × S d−1 and S 1 × S d−1 that are the boundaries of Euclidean AdS d+1 = H d+1 , global AdS d+1 and thermal AdS d+1 , respectively. 16 In addition, due to the appearance of log-divergences on both sides of AdS/CFT more numbers should agree.
CFT Side. The free energy computed on d-sphere S d of radius R is a well-defined number in odd d provided the power divergences are regularized away and is a d log R in even d, where a is the Weyl anomaly coefficient, see e.g. [89] for conformal scalar.
The free energy on S 1 β × S d−1 with the radius of the circle playing the role of inverse temperature β should have the form where a d is the anomaly and it vanishes for odd d and also for Rac and Di on R ×S d−1 and S 1 ×S d−1 .
The last term F β goes to zero when β → ∞, i.e. for R × S d−1 , and can be easily computed in a free CFT: Here Z 0 (β) is one-particle partition function where d n and ω n are degeneracies and eigen values of the free CFT Hamiltonian. The second term, which is proportional to β, is the Casimir Energy. It is given by a formally divergent sum which is usually regularized via ζ-function. For free fields it vanishes for odd d. The Mellin transform maps Z 0 into ζ 0 . See Appendix B for many explicit values. 16 Note that on more complicated backgrounds one encounters the problem of light states [87,88].
It is crucial to impose the singlet constraint on the CFT side. In a free CFT, e.g. Rac, F β is constructed from the character Z 0 of Rac. After the singlet constraint is imposed, one finds, see e.g. [55], that F β is built from the character Z of the singlet sector instead of the Rac-character Z 0 , i.e.
from the character of Rac⊗Rac if the CFT is just Rac. Also, the Casimir Energy is E sing where N f N is the total number of free fields with the factor of N removed by the singlet constraint.
AdS Side. The one-loop free energy for a number of (massless) fields in AdS d+1 is given by determinant of the bulk kinetic terms where the sum is over all fields Φ s with the ghost contribution 17 subtracted by the second term if Φ s is a gauge field. There is an additional minus for fermions. It can be computed by the standard zeta-function regularization [91,92] of one-loop determinants and leads to where l is the AdS radius, Λ is a UV cutoff.
In Euclidean AdS d+1 the ζ-function is proportional to the regularized volume of AdS d+1 space, which is a well-defined number for AdS d=2n+2 and contains log R for AdS d=2n+1 . Another log-term, which is log lΛ is present in AdS d=2n+2 and is related to the conformal anomaly. The one-loop free energy on the thermal AdS d+1 with boundary S 1 β × S d−1 is expected to be where F β vanishes in the high temperature β → 0 limit. In thermal AdS 2n+1 the a d+1 -anomaly is zero, while in AdS 2n+2 it should be the same as in Euclidean AdS [55]. Therefore, it can be computed from the free energy in Euclidean AdS d+1 with boundary S d , i.e. H d+1 . In the latter case only the total anomaly coefficient can vanish, as was shown in [53,54]. Therefore, once a d+1 = 0 one can scrutinize the rest of the one-loop contribution, which is now well-defined.
The N 0 part of the free energy, F β , counts the spectrum of states and should be automatically the same on both sides of the duality. Indeed, the spectrum of HS theories is determined by the representation theory of HS algebra. In its turn the HS algebras are constructed from free fields.
The spectrum of single-trace operators is the same as the spectrum of HS fields and is given by the tensor product of appropriate (multiplets of) singletons/doubletons. Therefore, the F β part can be ignored on both sides for a moment: it can be attributed to generalized Flato-Fronsdal theorems, see e.g. [55] for some checks. While the representation theory guarantees that the spectra should match, a direct path-integral proof is needed.
17 See [90] for an earlier discussion of quantization of higher-spin fields in AdS 4 .
It is important that the leftover order-N 0 correction, i.e. Casimir Energy E c , does not vanish sometimes (for the minimal theories or for the Type-C [58]), which requires to modify G −1 = N.
What needs to be checked depends heavily on whether d is even or odd.
Tests in AdS 2n+2 /CF T 2n+1 . The CFT partition function on a sphere is a number, while F 1

AdS
in Euclidean AdS contains log lΛ-divergences for individual fields, which have to cancel for the right multiplet, otherwise the finite part is ill-defined. Then the finite part, − 1 2 ζ ′ (0), should be compared to F 1 CF T , which is zero in free CFT's. If F 1 AdS is found to be non-zero, then one can try to adjust the relation between N and bulk coupling G as to make the two sides agree, assuming that F 0 AdS = F 0 CF T and F 1 AdS = integer multiple of F 0 CF T , the latter requirement is due to the quantization of the bulk coupling. It was found [53] that this is the case for the minimal models with even spins and F 1 AdS is equal to F 0 CF T for a free field that is behind the duality [93]. Another test is for Casimir Energy E c . It vanishes on the CFT side, while every field contributes a finite amount on the AdS-side. Therefore, only appropriately regularized sum over spins can vanish.
Tests in AdS 2n+1 /CF T 2n . The regularized volume of AdS-space contains log R, while the sphere free energy F CF T = a d log R is given by the a-coefficient of the Weyl anomaly. Here there is no log lΛterm since it vanishes for every field individually. Again, F 1 AdS either vanishes or should be equal to an integer multiple of the a-anomaly of the dual free CFT, F 0 CF T , and can be compensated by modifying G −1 = N. The same computation then gives the anomaly for the conformal HS fields -Fradkin-Tseytlin fields, −2a HS = a CHS , [54,[94][95][96].
The Casimir Energy test is more non-trivial since it does not have to vanish on the CFT side either. F 1 AdS corresponds to the order-N 0 corrections in CFT, which are absent for free CFT's. It is also important to note that all the tests must be mutually consistent. In particular, if a modification of G −1 = N is needed, it must be the same for all the tests in a given theory.

One-Loop Tests
In this section we perform the one-loop tests reviewed in Section 2. The main emphasis is on the cases that have not yet been widely studied: even dimensions, spectral zeta-function for fermionic and mixed-symmetry HS fields. Less conventional cases of partially-massless fields and higher-spin doubletons are discussed in Appendix C.
The spectrum of SUSY HS theories is made of bosonic and fermionic HS fields. In the simplest case one takes free CFT made of n scalars and m fermions, S = nRac ⊕ mDi. By imposing different singlet constraints the spectrum of bosonic HS fields can be truncated, for example, to even spins only, resulting in minimal theories. The spin of fermionic HS fields, if any, runs over all half-integer values s = 1 2 , 3 2 , 5 2 , .... In the minimal theories the order N 0 one-loop corrections usually do not vanish and it is important for the consistency of SUSY HS theories that the modifications of G −1 = N required for consistency of Type-A and Type-B are the same, which was observed for a, c, E c in AdS 5,7 [56,59] and for E c in all AdS 2n+1 [55].

Casimir Energy Test
The Casimir Energy tests are the simplest since the computation of E c is not difficult and we refer to Appendix B for technicalities. Each field contributes some finite amount to the Casimir Energy.
It is important to use the same regularization that has been already applied for Type-A and Type-B models.
We will discuss HS fermions only, since pure Type-A and Type-B have been already checked.
Vanishing of the Casimir energy can be seen after summation over spins with the exponential regu- where | fin. means to take the finite ǫ-part of the sum evaluated with the exponential regulator. The same can be seen directly from the character of Di ⊗ Rac in any dimension: Moreover, the second derivative of E c with respect to the conformal weight has a very simple form: . (3.5)

Laplace Equation and Zeta Function
The eigenvalue problem of Laplace operator is closely related to construction of zeta-function. We first discuss how to compute the eigenvalues and degeneracies for the Laplace operator on a sphere and then proceed to zeta-function on Euclidean AdS d+1 , i.e. on hyperbolic space H d+1 , which can be obtained from that on a sphere up to few important details.

Laplace Eigenvalue Problem
We are interested in the spectrum of Laplacian on S N = SO(N + 1)/SO(N): where M 2 is the mass-like term and Φ S is a transverse, traceless field with Lorentz spin S, where S can be any representation which we label by a Young diagram, S = Y(s 1 , ..., s k ). As is well-known, the eigenvalues λ n are given by the difference of two Casimir operators with a trivial shift by M 2 : Here the Young diagrams S n of representations that contribute are obtained from S by adding a row of extra length n as the first row: 18 The degeneracy d n is just the dimension of S n . For example, for the scalar Laplacian with M 2 = 0 we have where d n is the number of components of the totally-symmetric rank-n tensor of so(N + 1). Analogously, for totally-symmetric rank-s tensor fields we find

Spectral Zeta-function
Knowing eigen values λ n and degeneracy d n one can compute the spectral ζ-function on S d+1 : Extension to hyperbolic space H d+1 requires some work, see e.g. [62,[97][98][99][100][101][102][103][104][105]. The cases of H 2n+1 and H 2n are very different. Here ζ(z) is the spectral ζ-function, which is the Mellin transform of the traced heat kernel at coincident points: (3.14) In homogeneous spaces the heat kernel at coincident points K(x, x; t) does not depend on coordinates and the volume of the space factorizes out. The volume factor is a source of additional divergences.
The eigenvalues can be computed in a rather simple way for any irreducible representation of weight ∆. The rule established on many examples, see e.g. [99,100] is to replace s 1 + n, which is the length of the first row, by iλ − d 2 where λ is non-negative and real: where we took the standard normalization of the mass-like term, see e.g. [74]: for ∆ corresponding to gauge fields, both unitary [74] and non-unitary [74,106], we have m 2 = 0.
The heat kernel contains only a contribution of the principal series in the odd dimensional case H 2k+1 . In the even dimensional case H 2k a discrete series can contribute [100] too, depending on the type of representation. Effectively, the appearance of the discrete series contribution results in a shift by a constant -the formal degree of the discrete series. A contribution from discrete series arises for higher-spin doubletons -fields in AdS 2k that can be lifted [80] to representations of the conformal algebra so(2k, 2). For such fields the Young diagram S has n rows of non-zero length.
The case of n-forms was studied in [100]. In what follows we will ignore the contribution of discrete series, but it would be interesting to understand if they play any role in HS AdS/CFT in d > 2.
Zeta-function naturally has several different factors and the general expression is usually written in the following form: where µ(λ) is the spectral density that is normalized to its flat-space value: is the number of components of the irreducible transverse traceless tensor that corresponds to the spin of the field. The volume factors are self-evident. There is an extra factor, which is a leftover: Odd dimensions. In the case of odd dimensions, H 2k+1 , d = 2k, the ζ-function is obtained by a simple replacement where the boldface µ(λ) contains all the factors from (3.17) except for the ratio of volumes. We then extract g(s), v d and w d factors. For example, for any even d we find for totally-symmetric spin-s bosonic fields, spin s = m + 1 2 fermionic fields and for bosonic fields with the shape of Y(s, 1 p )-hook: bosons : hooks : where the spin factors are: The s = 1 case of hooks corresponds to (p + 1)-forms studied in [100]; spin-s bosons were investigated in [101]. The most general case in AdS 5 and AdS 7 was studied in [56,59].
Even dimensions. In the case of even dimensions, H 2k+2 , d = 2k + 1, there are two complications: there can be additional discrete modes and the Plancherel measure is not a polynomial. In the cases we are interested in the discrete modes should not contribute and the spectral density is a product of a formally continued dimension d n and a hyperbolic function coth πλ , fermions . (3.28) For example, for any even d we find for totally-symmetric spin-s bosonic fields, spin s = m + 1 2 fermionic fields and for bosonic fields with the shape of Y(s, 1 p )-hook: bosons : where the spin factors are the same. Degenerate hooks with s = 1 again correspond to (p + 1)-forms studied in [100]. For symmetric bosonic fields we refer to [101].
Mixed-Symmetry Fields. As one more example of interest let us take a mixed-symmetry field of shape Y(s 1 , s 2 ): The expression for the most general mixed-symmetry field with spin defined by so(d) Young diagram Y(s 1 , s 2 , ..., s k ) with k rows follows the same pattern: For fermionic mixed-symmetry fields one has to correct f E/O factors only: Let us collect the relevant formulae with all factors now added to µ(λ), which we callμ(λ). The complete spectral zeta-function is It is worth stressing that these are the zeta-functions for transverse, traceless tensors and the ghost contribution is not yet subtracted. Ghosts for massless fields always come with ∆ + 1, s − 1 as compared to ∆, s of the fields themselves.
Four Dimensions. In four-dimensions there are no mixed-symmetry fields and bosons and fermions are described by almost the same formulae [99] bosons/fermions :μ(λ) = coth πλ , fermions . Five Dimensions. The explicit formulae in five dimensions, i.e. AdS 5 , are, see also [56]: Six Dimensions. For application to HS theory based on F (4) we are also interested in sixdimensional anti-de Sitter space: Note that for fermions we use spin s, rather than integer m = s − 1 2 . The only hooks in AdS 6 are of shape Y(s, 1). Also, the bosonic cases are all mutually consistent and follow from the two-row one.
Note that fermions cannot be obtained as s → s + 1/2 from bosons in this case, contrary to d = 3.

Zeta Function Tests: Odd Dimensions
Odd dimensions are easier since evaluation of ζ(0) and ζ ′ (0) is of no technical difficulty. In particular, ζ(0) = 0 for each field individually. The new results are on mixed-symmetry fields that belong to Type-B theories and fermionic HS fields, where all the tests are successfully passed. Also, we found a general formula for the a-anomaly. The zeta-function for the whole multiplet of some HS theory is denoted as ζ HS .

Fermionic HS Fields
Firstly, ζ s (0) = 0 for any s and therefore the bulk result is well-defined. It is proportional to log R due to the regularized volume of AdS 2k+1 . On the boundary it should be equal to the Weyl anomaly coefficient, a log R, but this has been already accounted for by the contribution of bosonic HS fields. Therefore, we should check that ζ ′ HS (0) = 0. To give few examples, in AdS 5 , see also [56], we find that Using the same exponential cut-off exp[−ǫ(s + d−3 2 )] we find the total a-coefficient to vanish In AdS 7 we have a more complicated formulae, but fortunately with the same result that ζ ′ HS (0) = 0, see also [59]: In general dimension the computation can be simplified by introducing P d (λ) = P d (−λ): Then, with the help of the simple integration formula where b = ∆ − d/2, one finds that ζ(0) = 0 and ζ ′ (0) can be obtained from (only even k matters) .
Then, it can be effortlessly checked up to any given dimension that the total ζ ′ HS (0) vanishes identically. In fact, it also vanishes when restricted to 'even half-integer' spins s = 1 2 + 2n.

Symmetric HS Fields
The case of Type-A was studied in [53,54,56,58,59]. Let us quote the results. As always in odd dimensions ζ s (0) = 0, while ζ ′ s (0) can be computed the same way as we did for fermions. The final output is where a d φ is the Weyl-anomaly coefficient of the free scalar field in CF T d , for which one finds, see e.g. [89], . (3.50)

Mixed-Symmetry HS Fields
We will discuss various versions of the Type-B theory that contains mixed-symmetry fields with Young diagrams of hook shape (2.8). The contribution of certain mixed-symmetry fields has been already studied in lower-dimensional cases of AdS 5,7 in [56,58,59]. With the help of the general formula for the zeta-function we can extend these results for the Type-B theory to any dimension.
Here we should find that F 1 AdS is either zero or is a multiple of the free fermion Weyl anomaly a d ψ , see e.g. [107]: First of all, the spectrum of the non-minimal theory is given by the tensor product of Dirac free fermion Di that decomposes into a direct sum Wi ⊕Wi of two Weyl fermions. With the help of Appendix A one finds for AdS 2k+1 : where we indicate the spin of the fields only as the conformal weight/AdS energy is obvious.
For example, in seven dimensions the contribution of the scalar field and the total contributions of hooks of height p = 0, 1, 2 are: 19 , (3.53) while in nine dimensions the contribution of the scalar field and the total contributions of hooks of height p = 0, 1, 2, 3 are: the total sum being zero, as is expected.
As for the minimal theories, there are several surprises. First of all, one can take just U(N)-singlet 19 The zeta-function for hooks with p + 1 > d/2 is the same as for the dual fields with p + 1 < d/2.
sector of Wi. With the help of Appendix A the spectrum is We see that for d = 4k, i.e. AdS 4k+1 , the spectrum does not contain symmetric higher-spin fields at all. In particular, there is no graviton. Nevertheless, the total ζ ′ HS (0) can be found to vanish. For example, consider AdS 9 , for which the results on the row-by-row basis were quoted in (3.54). The For d = 4k + 2, i.e. AdS 4k+3 , the U(N) Weyl fermion does include totally-symmetric HS fields, so the theory looks healthy. The spectrum of the two parts is As for the minimal Type-B theory there are several options. Firstly, one can take the antisymmetric part of Di ⊗ Di, which would be the minimal Type-B. Secondly, one can take the antisymmetric part of only Wi ⊗ Wi, which would be the minimalistic option. The spectrum of the minimalistic Type-B theory is even more peculiar. We refer to Appendix A for more detail, while giving two examples here-below. In AdS 7 we find, see also [59], The total ζ ′ HS (0) is −(1/378) + 211/7560 = 191/7560, which is in accordance with the a-anomaly of one Weyl fermion on S 6 , see also Appendix B. In AdS 9 the spectrum of the minimalistic Type-B is (3.62) and the contribution to ζ ′ HS (0) is 23/5400 − (3463/226800) = 2497/226800, which is again in accordance with the a-anomaly of the free fermion. The contribution of the symmetric part of the tensor which would be relevant for the usp(N)-singlet theory comes with the opposite sign, −2497/226800.
The latter is obvious, of course, without any computation since the total anomaly was found to vanish.
The same pattern can be observed in other dimensions. According to the quite general law [95,108,109], the a-anomaly of conformal HS fields on the boundary can be computed from the AdS side according to a CHS = −2a HS , which is related to more general results on the ratio of determinants [110]. Therefore, vanishing of total a HS for the mixed-symmetry fields of Type-B implies the one-loop consistency of the conformal higher-spin theory with spectrum of conformal HS fields given by the sources to the single-trace operators built out of free fermion. As in the case of Type-A conformal HS theory [94,111], the action is given by the log Λ-part of the generating function of correlators of where ϕ s,p are the sources for J s,p .

Simplifying a-anomaly
The examples above reveal that ζ ′ (0), which is related to the boundary a-anomaly, is a quite complicated expression. However, it comes from a very simple formula. Following earlier results [54,56,59,96], consider the formula for any ∆ and any irreducible representation S defined by some Young diagram Y (s 1 , ..., s n ) with n rows. Then we find that .
a does not have a nice factorized form, but it is always proportional to ∆ − d/2, i.e. it vanishes at ∆ = d/2, which is a boundary condition for the integral that allows to reconstruct a from a ′ :

Zeta Function Tests: Even Dimensions
Even dimensional AdS 2n+2 spaces are much harder due to the complexity of spectral density that is not a simple polynomial, but contains the functions tanh or coth. Moreover, ζ(0) is generally non-zero for each field (which is due to the conformal anomaly for the case of conformally-invariant fields). Below we present the main results with the technicalities devoted to Appendices. The most interesting case is that of mixed-symmetry fields from the Type-B theory.

Fermionic HS Fields
Let us start with few examples. Computation of ζ(0) is not too difficult thanks to a handful of papers [53,99,112]. For example, in AdS 4 and AdS 6 the sum over all fermions is zero The same can be checked for any dimension, see Appendices for the details. As different from odd dimensions, the sum over all 'even half-integer' spins does not vanish.
The computation of ζ ′ (0) is trickier, see Appendices, but it can be shown on a dimension by dimension basis that for AdS 4,6,8,... one finds ζ ′ fermions (0) = 0. Therefore, adding fermionic HS fields is consistent to a given order, which is a necessary condition for the existence of SUSY HS theories.

Symmetric HS Fields
The case of symmetric HS fields was already studied in [53,54]. The summary is that ζ HS (0) = 0 both for minimal and non-minimal Type-A theories while ζ ′ (0) does not vanish for the minimal Type-A and is equal to the sphere free energy of one free scalar: As before, the minimal Type-A requires G −1 = N − 1.

Mixed-Symmetry HS Fields
This is the most interesting case. The Type-B theory in AdS 4 does not differ much from the Type-A -the spectrum consists of totally-symmetric HS fields. This is not the case in d > 3 where the spectrum of Type-B contains mixed-symmetry fields with Young diagrams of hook shape (2.8) in accordance with the singlet spectrum of free fermion Di. Much less is known about these theories 20 except that they should exist in any dimension since Di and Rac do.
Zeta. First of all we check that ζ(0) = 0 and thus the bulk contribution is well-defined. It is convenient to present a contribution of theψψ operator and of the hooks for each height p separately.
Here p can run over 0, ..., d − 2 with p = 0 corresponding to totally-symmetric HS fields. However, one can (and should) take into account only half of the hooks since the rest can be dualized back to p + 1 ≤ d/2 and the zeta function is the same. The latter is in accordance with the generalized Flato-Fronsdal theorem, which we now write for AdS 2k+2 : where there is one scalar and half of the hooks. For example, in AdS 6 we find Here one can see the contribution of the Type-A fields with s ≥ 1, which is −1/1512. In Type-A this is canceled by the ∆ = 3 scalar. Now, the contribution ofψψ is different, but there is the p = 1 sector and ζ HS (0) = 0. In AdS 8 we find It can be checked for higher dimensions that the total ζ HS (0) = 0. Now let us have a look at the minimal theories. The O(N)-singlet version of the Flato-Fronsdal theorem tells that where the scalar is present whenever (k − 1) mod 4 = 0 or (k − 2) mod 4 = 0. Analogously to odd dimensions, simply taking anti-symmetric part of Di ⊗ Di can result in somewhat strange spectra, which may not contain graviton. Nevertheless, such spectra yield vanishing contribution to ζ HS (0). 20 Some cubic interaction vertices for mixed-symmetry fields in AdS were constructed in [113][114][115]. A part of the Type-B cubic action that contains 0 − 0 − s vertices was found in [11].
For example, in AdS 6 we find (3.75) and the contribution of all odd spin fields is zero, while hooks of even spins give exactly 37 7560 to cancel that of the scalar. Similar pattern is true in higher dimensions and both minimal and non-minimal Zeta Prime. The main computational problem is to find ζ ′ HS (0). Below we give the summary of our results in several dimensions, with technicalities devoted to the Appendices. Let us note that despite some analytic regularization, which is needed to make sums over spins well-defined, there are non-trivial self-consistency checks for the computations: certain integrals cannot be evaluated but they cancel each other, also all complicated factors disappear from the final result. For non-minimal theories the total contribution to − 1 2 ζ ′ HS (0) is: 21

79)
(3.80) The case of AdS 4 was studied in [53]. The discrepancy with the sphere free energy of free fermion, F d ψ , is systematic, see Appendix B for some explicit values. However, these numbers are not random. They can be reproduced as a difference in the free energy via RG-flow induced by a double-trace The values of the free scalar F -energy can also be computed as F -difference: (3.82) 21 We list here only those results that fit one line. See also a closely related paper [116]. 22 Here we pass to generalized sphere free energyF that is defined as − sin( πd 2 )F , see e.g. [117].
The numbers that resulted from the tedious computations in AdS 2n+2 arrange themselves into the following sequence: However, the dual of Type-B is supposed to be a fermionic theory, for which a generalization of [93] to fermionic O ∆ in any d gives [117]: Again the free fermion F -energy can be computed as F -difference: We observe that for ∆ = d−2 2 it will give − 1 2 ζ ′ HS (0) up to a factor of ±1/4: For the minimal theories the computations are even more involved, but the unwanted constants do cancel and we find 23 for the total contribution to − 1 2 ζ ′ HS (0): (2) 256 , Again, these numbers do not look random. Curiously enough the AdS 6 result equals 6F φ .

Tadpole
In principle, higher-spin theories should be consistent as quantum theories to all loops as they are duals of well-defined CFT's that are, in general, either free fields or interacting vector-models. It is hard to say anything about higher loops or Feynman-Witten diagrams with legs due to the lack of the complete action. Also, any analog of the non-renormalization theorems for HS theories is not known at present. Moreover, it seems that vectorial super-symmetry cannot help too much and one should better stick to HS extensions of the usual SUSY. Still everything should boil down to the consistency of a simple bosonic HS theory, i.e. HS SUSY should improve the quantum properties, but the need for nontrivial summation over all spins appears unavoidable.
We can see that at least a part of the tadpole diagram vanishes for the reasons similar to the tests performed above. In [44] the quartic scalar vertex 0 − 0 − 0 − 0 was reconstructed from the free scalar CFT at d = 3. Though, the base of structures used there is over-determined and the coefficients are not known in explicit form it seems that the following should be true in any d. The quartic vertex is a double sum where we just meant to indicate that it is a doubly-infinite sum over all independent structures allowed by kinematics. The order of derivatives is unbounded, but the growth of the coefficients is suppressed by locality. The sum is doubly-infinite due to the four-point function it contributes to being the function of two conformally-invariant cross-ratios.
Let us consider the tadpole Feynman graph, ✐ . There is an infinite factor of various derivatives of the Green function at coincident points If we are in the simple Φ 4 theory then the tadpole Φ 2 G(0) contributes to the mass of the field. Now, due to the fact that higher derivatives are present in V 4 we can have a contribution of the kinetic term Φ∇ 2 Φ, which would imply wave-function renormalization. Also, there are infinitely many of unwanted terms Φ∇ n Φ, n > 2 with more than two derivatives, which are absent in the action.
The part of the tadpole that does not have derivatives on G, but can have arbitrarily many derivatives on Φ's, can be related to heat kernel G(x, x) = K(x, x; t). Indeed, Using the general relation between M 2 and conformal weight ∆ we find Formulae of this type have just been shown to facilitate the computation of a-anomaly as an integral of ∂ ∆ ζ ′ (0) over ∆.
Assuming that all terms enter with the same coefficient and with the standard regularization we find that G HS (x, x) vanishes in all even dimensions d. Basically, we just computed ζ HS (1). The contribution of ∆ = d − 2 scalar is always zero, but the sum over HS fields is non-trivial (ghosts need to be subtracted as usual). For example in AdS 5 , evaluation of ζ HS (0) and ζ HS (1) leads to

F(4) Higher-Spin Theory and Romans Supergravity
Exceptional algebraic structures seldom occur in the higher-spin context, see [118] for the discussion of D(2, 1; α) in application to HS AdS/CFT. Hence it is remarkable that there exists an exceptional AdS 6 /CF T 5 HS algebra [26] that is based on the super-singleton of exceptional Lie superalgebra F (4). More specifically it is realized as the enveloping algebra of the minimal unitary realization (super-singleton) of F (4) obtained via the quasiconformal method [26]. The super-singleton multiplet of F (4) consists of an SU(2) R doublet of Rac's and a singlet Di.
As in other cases, one can take the F (4) super-singleton as a free 5d CFT and consider the higherspin theory dual to its singlet sector. The spectrum of fields can be computed as a tensor product of two F (4) super-singletons. As we will show, such HS theory is closely related to the Romans F (4) gauged supergravity in AdS 6 [30].
The original motivation for this work stems from the goal to study this exceptional F (4) HS theory and the known one-loop tests were further developed so as to apply them to it. Below we review the construction of the F (4) HS algebra and work out the full spectrum of HS fields. In particular, we shall prove that the Romans graviton supermultiplet belongs to the spectrum of F (4) HS theory.

Exceptional Lie Superalgebra F (4)
The exceptional Lie superalgebra F (4) has 24 even and 16 odd generators [27]. The real form of F (4) we are interested in has 24 SO(5, 2) ⊕ SU(2) as its even subalgebra with the odd generators transforming in the (8,2) representation. It is the unique simple superconformal algebra in five dimensions. It can be realized as a superconformal symmetry group of an exceptional superspace coordinatized by the exceptional Jordan superalgebra which has no realization in terms of associative super-matrices [119,120]. The minimal unitary realization of F (4) was obtained via quasiconformal methods relatively recently [26], which we shall review below.
where ǫ rs is the two dimensional Levi-Civita tensor and C 7 is the symmetric charge conjugation matrix (C 7 ) αβ = (C 7 ) βα in seven dimensions.  (4):

Minimal Unitary
where Ω IJ is the symplectic invariant metric andB IJ = B † IJ . USp(4) generators satisfy the Hermiticity property The above generators of SO(5, 2) satisfy the commutation relations:  We should note that the Dynkin labels (n 1 , n 2 ) D of USp(4) are related to the Dynkin labels of Spin (5) by interchange of n 1 and n 2 : (n 1 , n 2 ) D of USp(4) ⇐⇒ (n 2 , n 1 ) D of Spin (5) . where where H = H + T 3 is the SO(2) generator that determines the compact 3-grading of F (4). The generators of SU(2) R are denoted as T + , T − and T 3 which satisfy The subsuperalgebra OSp(2|4) has the even subalgebra of SO(2) ⊕ USp(4) whose generators are Z = H + T 3 and U IJ . The odd generators of OSp(2|4) that transform as complex spinors of USp (4) are denoted as R I and R I and satisfy The generators of OSp(2|4) satisfy: (4.17) The odd generators that belong to grade −1 and grade +1 subspaces are denoted as Q I and Q I , respectively, and are related by Hermitian conjugation The remaining commutation relations of the superalgebra F (4) are given below:

Minimal Unitary Supermultiplet of F (4)
In the minimal unitary realization of F (4), as obtained via the quasiconformal method, the generators are expressed in terms of 3 bosonic oscillators, a singular oscillator and two fermionic oscillators [26].
One finds that the supersymmetry generators of the minrep satisfy certain special relations:  generators. Therefore the minimal unitary supermultiplet consists of two complex scalar fields in the doublet of R-symmetry group SU(2) R and a symplectic Majorana spinor. The lowest weight vector corresponding to the second scalar field will be denoted as |Φ + 0 and the lowest vector of the deformed minrep that describes the massless conformal spinor field is denoted as |Ψ 0 I :

Romans F (4) Graviton Supermultiplet in Compact 3-grading
Since there is no invariant concept of mass in AdS spacetimes and hence no universal definition of masslesness it was proposed in [14][15][16]  The singleton and doubleton supermultiplets of these AdS superalgebras do not have a Poincare limit and, as pointed out in these references, their field theories live on the boundaries of AdS spacetimes as superconformal field theories. Taking higher tensor products results in massive supermultiplets of the corresponding superalgebras. In the twistorial oscillator construction of the unitary representations of AdS superalgebras of [14][15][16] tensoring is very straightforward and corresponds to simply increasing the number of colors of the (super)-oscillators since the generators are realized as bilinears of these oscillators. However tensoring of the supersingleton of F (4) is more subtle since its realization, as obtained via the quasiconformal method [26], is nonlinear.
We shall adopt the same definition of massless supermultiplets in AdS 6 and construct the massless graviton supermultiplet that underlies the N = 2 AdS 6 gauged supergravity of Romans [30] by tensoring two singleton supermultiplets of F (4). The fields of the Romans gauged N = 2 AdS 6 supergravity are [30]: graviton e a m , four gravitini ψ m;α satisfying the symplectic Majorana-Weyl condition, an anti-symmetric gauge two-form B mn , an auxiliary abelian gauge vector a m , three SU (2) gauge vectors A a m , four spin-half fields χ α and a scalar σ. Starting with the above fields Romans constructed the AdS 6 gauged supergravity with gauged R-symmetry group SU(2) R . We should note that i) the auxiliary vector a m can be combined with B mn as B mn +∂ m a n −∂ n a m and is a Stueckelberg field that can be gauged away and serves to make B mn into a massive two-form field a-la Higgs; (ii) the mass of the two-form is a free parameter in the Lagrangian. However there is a unique vacuum that enjoys full F (4) symmetry, which corresponds to setting m = g/3, where g is the gauge coupling constant.
At the Lie superalgebra level tensoring is equivalent to taking a sum of two copies of the generators of F (4). Let us denote the generators of F (4) in compact three grading symbolically as follows: where the upper (lower) index A inC A (in C A ) runs over all the generators in grade +1 (-1) space.
In taking direct sum of two copies of the generators we shall label the corresponding generators as follows: (2)) .

(4.24)
In contrast to the maximal supergravity multiplets in AdS 4,5,7 the tensor product of the lowest weight vectors |Φ − 0 (1) and |Φ − 0 (2) of two singleton supermultiplets does not lead to a supermultiplet that includes the graviton. One finds that that the following set of tensor product states subspaces corresponding to Poincare and special conformal supersymmetry generators [26]. The compact decomposition is as follows: The generators in the above 5-grading satisfy the super-commutation relations  where |ψ 0 I is the lowest energy irrep of the spinor singleton, Di,

Romans F (4) Graviton Supermultiplet in Compact 5-grading
The minrep, i.e. Rac, of SO(5, 2) inside the minimal unitary supermultiplet of F (4) occurs with multiplicity two and transforms as a doublet of SU(2) R . Tensoring two copies of the supersingletons corresponds to taking direct sum of two copies of the generators of F (4) and tensoring of the corresponding Hilbert spaces. In the tensor product space there is a unique lowest weight vector that is a singlet of SU(2) R , is annihilated by all the negative grade generators of F (4) in the compact 5-grading and leads to the graviton supermultiplet of Romans' theory. This singlet state is The additional lowest weight vectors of SO(5, 2) inside the resulting unitary representation of F (4) are [5,7,9]  where r = 1, 2 and a = 1, 2, 3 are the spinor and adjoint SU(2) R symmetry indices.
On expanding the Romans supergravity Lagrangian [30] around the unique F (4) supersymmetric vacuum it turns out that the mass of the scalar field is (−6), which is that of a conformally-coupled scalar and the AdS energy is 3; the AdS energy of B mn is 4; the AdS energy of χ is 7/2; the AdS energies of graviton, gravitini and SU(2) gauge field are fixed by gauge symmetry to be 5, 9/2 and 4, respectively. Therefore, the supermultiplet (4.40) thus obtained by tensoring of two F (4) supersingletons with the lowest weight vector |Ω 0 in the compact 5-grading is precisely the Romans supermultiplet, agreeing with the result obtained in compact 3 grading.

F(4) HS Theory Spectrum and One-loop tests
In addition to the Romans supermultiplet, the tensor product of two F (4) super-singletons contains an infinitely many massless F (4) supermultiplets that have higher-spin fields. In fact they include an infinite tower of massless higher spin supermultiplets that extend the graviton supermultiplet which we list below: We should perhaps note that despite the fact that the fields of various supergravity multiplets can occur in the spectrum of HS theories, their appearance is somewhat different. For example the massive two-form that shows up in the product of two F (4) singletons is represented as a matter-like anti-symmetric rank-two tensor in the higher-spin theory. In the Romans F (4) gravity it is realized as a gauge two-form field B mn that is Higgsed via an additional SO(2) gauge field a m .
Remarkably the supermultiplet L(8|8) corresponds simply to the linear multiplet which plays a crucial role in the off-shell formulations of 5d conformal supergravity and their matter couplings [121][122][123]. It is also related to the off-shell (improved) vector multiplet in 5d. Therefore we conclude that the consistent formulation of F (4) HS theory must be based on the reducible multiplet extending the Romans supergravity multiplet by the supermultiplet L(8|8), which plays the role of compensating supermultiplet in 5d conformal supergravity, coupled to the infinite set of higher-spin fields belonging to the Romans tower. The resulting F (4) HS theory passes the one-loop tests by Casimir Energy and its Type-A and the fermionic parts are in agreement with the free energy on five-sphere. The Type-B part reveals a puzzle, which is a general feature of type-B theories that we discuss in the Conclusions.

Discussion and Conclusions
Our results are as follows: • the spectral zeta-function is derived for arbitrary mixed-symmetry fields; • to the list of known one-loop tests we added those that are based on zeta-function for fermions and specific mixed-symmetry fields that arise in Type-B theories; • fermionic HS fields were shown to pass both the Casimir Energy and the zeta-function tests quite easily since they are not expected to generate any one-loop corrections at all, which is what we observed. However, vanishing of the fermions contribution is still nontrivial and involves the summation over all spins; • knowing the zeta-function for a generic mixed-symmetry field allowed us to derive a very simple formula for the derivative ∂ ∆ a(∆) of the a-anomaly that allows one to integrate it to full a(∆).
A similar feature was observed for the second derivative of the Casimir Energy ∂ 2 ∆ E c ; • we showed that ζ HS (1) = 0 at least in some of the cases, which is a different type of equality relying on the spectrum of HS theories. This fact should be related to vanishing of the tadpole diagram, which can be problematic in HS theory; • the spectrum of the Type-B theories, which should be generically dual to a free fermion and • partially-massless fields arising in the duals of the non-unitary higher-order singletons k φ = 0, both minimal and non-minimal, were shown to pass the Casimir Energy tests, see also [124].
They also pass the zeta-function tests in AdS 2n+1 , where for the minimal models the result equals the a-anomaly of higher-order singletons. Such theories provide examples of HS theories with massive HS fields. In addition this series of theories has relation to the A k series of Lie algebra, see Appendix C.2; • higher-spin doubletons with j > 1, which are unitary as representations of conformal algebra but pathological from the CFT point of view in not having a local stress tensor, were shown not to pass the Casimir Energy test in AdS 5 /CF T 4 , see Appendix C.1.
While the tests successfully passed require no further comments, let us discuss the cases where we discovered a mismatch between AdS and CFT sides.
As it was already mentioned, for AdS 2n+1 /CF T 2n the list of unitary conformal fields includes higher-spin doubletons, in addition to the omnipresent Rac and Di. It was shown in [58,59] that the spin-one, j = 1, doubleton in AdS 5 , i.e. the dual of the Maxwell field, and in AdS 7 , i.e. the dual of the self-dual tensor, are consistent with the duality. We observed that for j > 1 the excess of the Casimir energy in the bulk cannot be compensated by a simple modification of G −1 ∼ N relation.
However, higher-spin doubletons are pathological as CFT's so we should not worry that they do pass the test.
We observed that there is a general puzzle about Type-B HS theories that are dual to free fermion.
It has been already noted in [53] that there is a discrepancy in AdS 4 /CF T 3 Type-B duality. At least in AdS 4 /CF T 3 it can be explained almost without computations. The free spectrum of single-trace operators built out of free fermion is identical to that of the 3d critical boson at N = ∞, which was noted in [10]. Therefore, unless a miracle happens the two theories -Type-A with ∆ = 2 boundary condition for the scalar field and Type-B -cannot pass the one-loop test simultaneously.
Our computations extend this puzzle to any AdS 2n+2 /CF T 2n+1 . The fact that the discrepancy is for fermions and it is in odd dimensions makes one think that the problem is due to parity anomaly [53]. The issue could have been easily resolved by allowing fractional coupling constant in the bulk HS theory, i.e. by having a more complicated G −1 (N)-relation. Indeed, the bulk constant has to be quantized [64], but the precise mechanism of how this happens in the bulk is unclear. In particular, it is uncertain if G −1 has to be of the form a(N + integer) or not. However, the need for fractional shift of N would spoil the whole logic of one-loop tests. Moreover, it would render SUSY HS theories inconsistent since the G −1 ∼ N relation for the Type-A subsector of any SUSY HS theory is canonical.
Also, it is not obvious what is the field-realization of the singlet constraint in higher dimensions.
At least in d = 3 there is a natural candidate -Chern-Simons matter theories -that imposes the singlet constraint when coupling is small and provides a family of models that interpolate between free/critical boson/fermion [125]. Therefore, there is no 'sharp difference' between bosons and fermions in 3d. However, the spectrum of single-trace operators of Type-A and Type-B is cleary different in d > 3. In addition, there does not seem to be any natural candidate to impose the singlet constraint.
Lastly, as the sum over spins requires regularization one cannot exclude the possibility that a different kind of regularization is needed for Type-B theories. The latter is unlikely since the same regularization works for Type-B in odd dimensions and all Type-A theories and fermionic HS fields.
Therefore, it seems to be crucial to understand the nature of the singlet constraint and explain the discrepancy for the Type-B. Note added: after the completion of our work we learned that Giombi, Klebanov and Tan have independently obtained some of the results on the one-loop tests of higher-spin theories presented in this paper, see [116].

A Characters, Dimensions and all that
We collect below some useful formulas for the dimensions of various irreducible representations. The classical general formulae for the dimensions of irreducible representations were found by Weyl and for the case of so(2k) and so(2k + 1) read: Y so(2k) (s 1 , ..., s k ) : where the representation is defined by Young diagram Y(s 1 , ..., s k ) with the i-th row having length s i or s i − 1 2 if all s i are half-integer. For some of the particular cases of use we find for so(d): , , , where we use Y 1 2 (m 1 , ...) to denote spinorial representations. For example, Y 1 2 (m) is a symmetric rank-m spin-tensor T a(s);α , i.e. it has spin s = m + 1 2 . Similar formula for symplectic algebra sp(N) yields: which allows to compute the dimension of any representation of so(5) ∼ sp (4): where a, b can be half-integers. Analogously, for special linear algebra sl(d): The isomorphism su(4) ∼ so (6) gives for so (6): Note that the dimension (A.1) in the even case so(2k) is the dimension of irreducible representation, while (A.2) formulas pack (anti)-selfdual representations together, so that (A.2) sometimes gives twice that of (A.1).
Characters. We will discuss only one-particle partition-functions without extra chemical potentials. Character of a generic representation with spin S is obtained by counting ∂ k -descendants assuming there are no relations among them: The characters of more complicated representations are obtained from the resolvent thereof. The simplest representations given by a short exact sequence correspond to partially-massless HS fields: where V (...) denotes generalized Verma module, which can be reducible, and D is the irreducible module. Here, ∆ = d + s i − 1 − i and S ′ is the spin of the gauge parameter in AdS d+1 or, equivalently, the symmetry type of the conservation law for a higher-spin current. 26 An additional parameter t is the depth of partially-masslessness [126] and t = 1 for massless fields. The Casimir Energy of a massless field is simply the difference between that of the two Verma modules -field and its gauge symmetries. Generalization for long exact sequences is straightforward.
In the case of free scalar, Rac, and free fermion, Di, the sequence is short but different. The singular vectors are associated with φ and / ∂ψ: Below we collect some of the blind characters of so(d, 2). The dimensions of irreducible so(d) representations can be found above conserved tensor , , fermion of dimension ∆ , The simplest instance of the Flato-Fronsdal theorem then follows from Given a character Z(q = e −β ), the (anti)-symmetric parts of the tensor product can be extracted in a standard way: symmetric : anti-symmetric : (A.14) 26 In the case of massless totally-symmetric fields we have The character of the weight-∆ spin-(s, 1 h ) operator and the associated conserved current are: Fermionic spin-tensor conformal quasi-primary operator O α;a(s) obeys γ mβ α O β;ma(s−1) = 0, which allows to compute its character and the character of the conserved higher-spin super-current: , .
Tensor Products of Spinors. To derive the decomposition of Di ⊗ Di together with its (anti)symmetric projections we need to know how to take tensor product of two so(d) spinors. For d odd we have Dirac spinors, which we denote D. For d even there are two Weyl spinors, which we denote W andW. 27 There are three distinct cases: so(2k + 1), so(4k) and so(4k + 2). Consulting math literature we can find out that: where the sums are from i = 0 to the maximal value it can take in each of the cases. Defining in even dimensions D = W ⊕W we observe: The decomposition of Di ⊗ Di is known and is quoted in the main text. Let us work out the spectrum of the O(N)-singlet free fermion. In the case of even d we introduce Wi as free Weyl fermion. It 27 Various other possibilities like symplectic Majorana-Weyl spinors in some dimensions will be ignored.
should be taken into account that higher-spin currents dress the tensor productψ(x 1 )ψ(x 2 ) with a Gegenbauer polynomial in derivatives that is (anti)-symmetric for (odd)even number of derivatives in the current. Combing the symmetry of the product of two spinorial representation with the symmetry of the derivative-dressing we find 28 so(2k + 1) : where we indicated the so(d)-spin of the singlet quasi-primary operators, the conformal weight being obvious from Di⊗Di. The above formulae generalize the Flato-Fronsdal theorem to the O(N)-singlet sector of free fermion theory in any dimension. Other versions of the singlet constraint follow from the above results.

B Amusing Numbers
We collect below various numbers associated to the fields discussed in the main text: Casimir Energy, sphere free energy, Weyl a-anomaly coefficients.
Casimir Energy. Casimir Energy, E c , is given by a formally divergent sum for which the standard regularization is to use the exp[−ǫω n ] as a cut-off and then remove all poles in ǫ. All the data can be extracted from the characters. We see that the spin degrees of freedom factor out for massive fields and the Casimir energy is given by Casimir Energy for a massive scalar field of weight ∆: allows one to get the Casimir Energy for any massive representation by multiplying it by dim S.

483840
Note that d = 3 and s = 0 case is special in that the fake ghost contribution does not vanish automatically and the right value is E c = 1 480 . Casimir Energies for higher-spin fermionic fields in lower dimensions are:  29 There is a typo in one of the expressions in the latter paper. 30 The fermion is always a Dirac one. E c for the Weyl fermion is half of the value in the table. 31 When self-duality applies it is the Casimir energy of the two fields. Sphere Free Energy. Also, we will need the free energy on a sphere for free scalar and fermion, see e.g. [93], Weyl Anomaly. The general formula for Weyl anomaly a for real conformal scalar [89] and fermion [107]  Volumes. The volume of d-sphere and the regularized volume of the hyperbolic space, which is Euclidean anti-de Sitter space, are [109]:

C More HS Theories
In this Section we discuss higher-spin doubletons that result in more general mixed-symmetry fields and higher-order singletons that lead to partially-massless fields and mixed-symmetry fields.
spin J, with J = 0, 1 2 being the usual Rac and Di. 33 The J = 1 is free massless spin-one field, i.e. Maxwell. For J > 1 the HS doubletons are unusual CFT's in not having a local stress-tensor, while they still are unitary representations of the conformal algebra.
In [56,58] it was conjectured that there should exist an AdS HS theory that is dual to N free Maxwell fields, called Type-C in analogy with Type-A, J = 0, and Type-B, J = 1 2 . It was found that one-loop tests are successfully passed, but already the non-minimal theory requires the bulk coupling to be G −1 = 2N − 2, i.e. modified. Similar conclusions were arrived at in [59] for the J = 1 doubleton in AdS 7 /CF T 6 [14].
Let us show that all Type-D,E,... theories, i.e. those with J > 1, do not pass the one-loop test.
The Casimir Energy of the spin-J doubleton is easy to find: 34 The spectrum of Type-X theory can be found by evaluating the tensor product of two spin-J doubletons [58,70,114]: where in the first line we see massive and massless mixed-symmetry tensors and massless symmetric HS fields in the second line. The absence of the stress-tensor reveals itself in that the spectrum of massless HS fields is bounded from below by 2J. In particular, there is no dynamical graviton for J > 1.
The Casimir Energies for the three parts of the spectrum: massive, mixed-symmetry massless, and symmetric massless, can be computed with the net result: We see that the total Casimir energy vanishes for J = 0, 1 2 in accordance with [55]. It does not vanish for J = 1 [56,58], rather it equals that of the two Maxwell fields, which still can be compensated by shifting the bulk coupling. However, for J > 1 there does not seem to be any natural way of compensating the excess of the Casimir energy.
The same problem can be understood at the level of characters, which is a simpler approach. The 33 The Young diagram of so(2n) that determines the spin of the field has a form of a rectangular block of length J and height n, i.e. the labels are (J, ..., J). One can also consider higher-spin representations of more complicated symmetry type, however they may be non-unitary. 34 For J = 0 it gives the Casimir Energy of two real scalars. For lower spins J = 0, 1 2 , 1 we therefore find E c = 1 240 , 17 960 , 11 120 .
blind character of the spin-j doubleton is, see e.g. [58]: The singlet partition function is [Z j ] 2 . It is symmetric in β, q = e β , for j = 0, 1 2 . For j = 1 it is not symmetric but the anti-symmetric part can be expressed as a multiple of Z 1 , which can be compensated by modifying G −1 = N [58]. However, for j > 1 the anti-symmetric part cannot be compensated this way, but can be expanded in terms of Z i≤j . Therefore, we see that the duals of HS doubletons J > 1 should have pathologies as quantum theories. Classically, it should be possible to manufacture some interaction vertices in AdS such that they reproduce the correlation functions of conserved HS currents The generating function of three-point correlators was constructed in [129]. That such reconstruction is possible for three-point function follows from counting the number of independent structures that can contribute to j s 1 j s 2 j s 3 [130] and to the cubic vertex V s 1 ,s 2 ,s 3 of three massless HS fields [131,132].
This number is the same n = min(s 1 , s 2 , s 3 ) + 1 and is given by the minimal spin, which is related to the fact that the currents that one can construct from a spin-J doubleton must have s ≥ 2J, see [133] for the explicit form in 4d. Indeed, only those doubletons can give a contribution to j s 1 j s 2 j s 3 that have 2J ≤ min(s 1 , s 2 , s 3 ).
The above considerations pose a puzzle: we see that most of the cubic vertices that exist in principle cannot be a part of any consistent unitary HS theory. 35 In [114] it was shown that deformations of HS spin algebras in any d that are consistent with unitarity in the sense that gauging of such algebras leads to unitary (mixed-symmetry) fields can depend on at most one continuous parameter.
In references [8,22,23]  is the deformation parameter continuous [23,24,134,135] corresponding to helicity [23], while in d > 4 deformations are discrete. The HS algebras resulting from HS doubletons belong to this family as well. We see that restriction to HS doubletons with spin 0, 1 2 , 1 eliminates a considerable part of the mixed-symmetry fields. Therefore, only very restricted Young shapes can arise in HS theories with massless mixed-symmetry fields -no more than two columns of height greater than one. Still a large fraction of massless mixed-symmetry fields is not embedded in any kind of AdS/CFT duality.
Perhaps, they can be brought to existence as duals of non-unitary spinning conformal fields φ S that obey φ S + ... = 0. Massive mixed-symmetry fields of any admissible Young shape are present in string theory, so it should be important to be able to incorporate massless limits thereof into HS 35 HS doubletons exist for even boundary dimension only. However, the number of independent correlators j s1 j s2 j s3 seems to be indifferent to this fact, as if one could formally define HS doubletons in odd dimensions as well.

C.2 Partially-Massless Fields
As it was already noted, the list of free CFT's becomes infinitely richer if the unitarity is abandoned.
The simplest one-parameter family corresponds to higher-order singletons: The spectrum of single-trace operators contains partially-conserved currents [136] The spectrum is encoded in the tensor product of two Rac k [137]: The fields that are dual to partially-conserved currents are partially-massless fields [126,138]: where t is the depth of partially-masslessness. Massless fields occur at t = 1. Therefore, the spectrum of a theory that is dual to Rac k is a nested tower of (partially)-massless fields with the Rac k−1 tower contained in the Rac k one. In particular, usual massless HS fields are present. Note that only odd depths t are found in Rac k ⊗ Rac k .
We can call the dual of Rac k as Type-A k , which is not meaningless for the following reason [83]. One can define HS algebra for the generalized free field of weight-∆. This algebra is naturally described as a centralizer of hs(λ), 36 where ∆ is related to λ. The HS algebras defined by Rac k can be understood as quotients of this algebra that arise at exactly the same values where the dual algebra hs(λ) acquires and ideal and reduces to sl(k). Therefore, the duals of Rac k are related to the A-series of Lie algebras. The (anti)-symmetric parts of Rac k ⊗ Rac k should then be related to the B, C, D series of algebras.
It is important that the operators with s < i are not conserved tensors and are dual to massive fields, which for k > 2 also contain massive HS fields. Therefore, duals of Rac k provide an example of HS theories that contain HS gauges fields and HS massive fields with a spin bounded from above. As a simple test of the AdS/CFT duality we can check the vanishing of Casimir Energy in the non-minimal Type-A k theory, see also [124]. On general grounds the Casimir Energy of Rac k vanishes in odd dimensions. For example, for the simplest case of Rac t we find in d = 3, 4, ...: The Casimir Energy of a depth-t partially-massless spin-s field can be computed in a standard way.
For example, in the d = 3 case we find (g = 2s + 1): Consider the simplest case of Rac 2 . The spectrum contains that of Type-A and massive fields Φ, Φ a , Φ aa plus depth-3 partially-massless fields s = 3, 4, .... The sum over the Type-A spectrum was already found to vanish [124]. At least for odd d we have to ensure that the sum over the rest vanishes as well.
Using the standard exponential cut-off exp[−ǫ(s + x)] we find that this is the case for x = (d − 5)/2.
Therefore, different parts of the spectrum should be summed with different regulators.
The dual of Rac 3 contains the spectrum of Type-A=Type-A 1 , the fields we have just studied plus massive fields Φ a(k) , k = 0, 1, 2, 3, 4 and depth-5 partially-massless fields. The sum of the Casimir Energies of this last part gives zero for x = (d − 7)/2.
Let us turn to the minimal Type-A k theory. It is useful to recall that the Casimir Energy can also be computed as As it was already noted [55], the non-zero contribution to E c comes from the β −1 pole, which is absent if Z(β) is an even function of β. This is typically the case for the tensor product of two singletons, but is not for the (anti)-symmetric projections, which results in where the first term is an even function of β in most cases. Then the contribution to the Casimir Energy is equal to that of the free field due to the last term. A slight generalization of [124,137] implies that the minimal type-A 2 contains fields of even spins only. The excess of the Casimir Energy can be reduced to a linear combination of Rac k by expressing the β-odd part of (Rac k ⊗ Rac k ) S : where Z k is the character of Rac k : This identity directly implies that the Casimir energy of the minimal type-A k theory is equal to that of one Rac k , E k c . If instead we sum over spins with exp[−ǫ(s + x)] cut-off we will have to use x = (d − 3)/2 for depth-1 fields, x = (d − 5)/2 for depth-2 fields etc. In particular, for type-A 2 the sum over its type-A sub-sector gives E c of Rac 1 , while the sum over the depth-2 fields gives E 2 c − E 1 c with the total result E 2 c , as before. Also, it can be checked that the tensor product Rac n ⊗ Rac m with m = n gives zero contribution to the Casimir Energy. Such products should arise in a theory built of several different higher-order singletons.
With the help of the zeta-function we can also check that −2 −1 ζ ′ (0) matches the a-anomaly of k φ = 0 free field. The latter can be extracted from the same zeta-function according to a CHS = −2a HS where the conformal field dual to the order-k singleton has weight (d+2k)/2. The summation over spins can be done as before and we should not forget that the depth-t partially-massless field of spin-s has AdS energy ∆ = d + s − t − 1 and the ghost has spin (s − t) and weight d + s − 1.
Lastly, the contribution of the massive (possibly HS fields) that appear in the tensor product of two higher-order singletons need to be separated. For example, let us consider AdS 5 and set k = 2 as above. We find: so that the total contribution is zero. For the minimal Type-A 2 model, i.e. the one above truncated to even spins only, we have: Therefore, despite non-unitarity, higher-order singletons that lead to partially-massless fields seem to be consistent at one-loop.

D On the Computations in Even Dimensions
In this Section we discuss the computations of ζ and ζ ′ in even dimensions. We presume that the full zeta-function is given in the form where ν = ∆ − d/2 and h(u) is either tanh πu or coth πu. The computation of ζ(0) can be done by which leads to 3) The first integral can be done for large enough z and then continued to z = 0. The second one is perfectly convergent and we can set z = 0 and use To compute ζ ′ (0) we first differentiate ζ(z) with respect to z. This can be directly done for the first part I, with two contributions produced: where p 1,2 are polynomials. In the second part II we find no problem with convergence, but a quite complicated integral Using log[u 2 + ν 2 ] = log u 2 + ν 0 dx 2x(x 2 + u 2 ) −1 we can split it into two parts: .
Now we introduce two types of auxiliary integrals . (D.10) The first one we will not attempt to evaluate since all c n will cancel in the final expressions. The second one can be done iteratively by first finding where in [3.415, Table of integral], Together with a useful formula in [112], J + n (2π) = J − n (2π) − 2J − n (4π), one can get Consider the following equation (D.14) Taking the derivative at a = 1 on both sides, we obtain Therefore, J ± n will contain two types of contributions: The second terms in each equation can be easily integrated over x: Importantly, all log ν now cancel because p 2 (ν) is the same as the one at ∂ z I z=0 . The purely polynomial leftovers p 1 and p 3 from J ± n and ∂ z I z=0 can be added up. We also need to add II.1 to them. Then ν is replaced with ∆ − d/2 and we can sum over all spins as usual. This contribution we call P = P ν,s − P ν+1,s−1 . Importantly, all coefficients c n will be gone and we do not need to deal with their real form, both for Type-A and Type-B.

(E.19)
The second integral is just II = II. (E.21) Repeating the same algorithm as in the case of bosonic theory, we get where, we have returned the degeneracy into the calculation.
, (E. 24) and Q is just It is easy to see that the zeta function for hook fields is not zero, which is not a problem since they make only a part of the Type-B spectrum.
(E. 35) Having these results at hand, we are now able to compute the ζ ′ B for the non-minimal and minimal Type-B theories.

F.2 HS Fermions
Above, we showed that ζ 1 2 and ζ ′ 2 )], we see that the total zeta-functions in d = 3, 5 vanished. As a simple check, one can confirm that for higher dimensions this statement is also true.
Next, to calculate the ζ ′ -function, we again split it into P ν,m and Q ν,m . One can see that for fermions P is non-zero which is different from Type-A theories. For Q ν,m we get d Q ν,m Q 3 − 2(m+1) It is easy to see that P and Q always cancel each other. A further check confirms that ζ ′ (0) is zero in higher dimensions.

F.3 Hook fields
The hook fields only appear in dimensions higher than four. For the computation of the spectral density function µ(u) of hooks with different p, the reader can refer to Section 3.2.2.

F.3.1 Zeta
In We will list the result of ζ-function in both the non-minimal and minimal theory for hook fields below since it is important for our computation of Type-B theory 40 It is interesting that the zeta function for hook fields alone is not zero as in bosonic and fermionic theory. However, when one considers the whole spectrum of Type-B theory, the zeta function will again vanish.

F.3.2 Zeta-prime
Below are the tables for P ν,s and Q ν,s of hook fields. 39 Due to the length of the final results, we only list the zeta function for d = 5, 7 here. 40 The hook fields of minimal theory in d = 5 come with even spins while the hook fields with p = 1 in d = 7 come with odd spins and p = 2 come with even spins. Non-minimal Type-B. Following the method in appendix D, we list the results of Q in d = 5, 7. Minimal Type-B. In the minimal theory, the computations are much longer since there are more derivatives involved when one calculates the Hurwitz-Lersch functions.

F.4 Type-B
We can now combine the results above to get the results for Type-B models. The spectrum of such models is given in Section 3.4.3.

F.4.1 Non-minimal
Scalar Field. The scalar in Type-B has ∆ φ B = ∆ φ A + 1, where ∆ φ A is the conformal weight of the scalar in Type-A theory. One can use this to compute ζ, P, Q using all the formulas in Type-A:  In the main text, our results were generated up to AdS 12 or d = 11, but we checked up to AdS 18 that they agree with the change of F -energy.