The ABC of Higher-Spin AdS/CFT

In recent literature one-loop tests of the higher-spin AdS$_{d+1}$/CFT$_d$ correspondences were carried out. Here we extend these results to a more general set of theories in $d>2$. First, we consider the Type B higher spin theories, which have been conjectured to be dual to CFTs consisting of the singlet sector of $N$ free fermion fields. In addition to the case of $N$ Dirac fermions, we carefully study the projections to Weyl, Majorana, symplectic, and Majorana-Weyl fermions in the dimensions where they exist. Second, we explore theories involving elements of both Type A and Type B theories, which we call Type AB. Their spectrum includes fields of every half-integer spin, and they are expected to be related to the $U(N)/O(N)$ singlet sector of the CFT of $N$ free complex/real scalar and fermionic fields. Finally, we explore the Type C theories, which have been conjectured to be dual to the CFTs of $p$-form gauge fields, where $p=\frac d 2 -1$. In most cases we find that the free energies at $O(N^0)$ either vanish or give contributions proportional to the free-energy of a single free field in the conjectured dual CFT. Interpreting these non-vanishing values as shifts of the bulk coupling constant $G_N\sim 1/(N-k)$, we find the values $k=-1, -1/2, 0, 1/2, 1, 2$. Exceptions to this rule are the Type B and AB theories in odd $d$; for them we find a mismatch between the bulk and boundary free energies that has a simple structure, but does not follow from a simple shift of the bulk coupling constant.


Introduction
Extensions of the original AdS/CFT correspondence [1][2][3] to relations between the "vectorial" ddimensional CFTs and the Vasiliev higher-spin theories in (d + 1)-dimensional AdS space [4][5][6][7][8] have attracted considerable attention (for recent reviews of the higher-spin AdS d+1 /CFT d correspondence see [9,10]). The CFTs in question are quite well understood; their examples include the singlet sector of the free U (N )/O(N ) symmetric theories where the dynamical fields are in the vectorial representation (rather than in the adjoint representation), or of the vectorial interacting CFTs such as the d = 3 Wilson-Fisher and Gross-Neveu models [11][12][13]. Some years ago the singlet sectors of U (N )/O(N ) symmetric d-dimensional CFTs of scalar fields were conjectured to be dual to the Type A Vasiliev theory in AdS d+1 [11], while the CFTs of fermionic fields -to the type B Vasiliev theory [12,13]. In d = 3 the U (N )/O(N ) singlet constraint is naturally imposed by coupling the massless matter fields to the Chern-Simons gauge field [14,15]. While the latter is in the adjoint representation, it carries no local degrees of freedom so that the CFT remains vectorial. More recently, a new similar set of dualities was proposed in even d and called Type C [16][17][18]; it involves the CFTs consisting of some number N of d 2 − 1 -form gauge fields projected onto the U (N )/O(N ) singlet sector.
The higher-spin AdS/CFT conjectures were tested through matching of three-point correlation functions of operators at order N , corresponding to tree level in the bulk [9,19]; further work on the correlation functions includes [20][21][22][23][24][25]. Another class of tests, which involves calculations at order N 0 , corresponding to the one-loop effects in the bulk, was carried out in [16][17][18][26][27][28][29]. It concerned the calculation of one-loop vacuum energy in Euclidean AdS d+1 , corresponding to the sphere free energy F = − log Z S d in CFT d ; in even/odd d this quantity enters the a/F theorems [30][31][32][33][34][35][36]. Similar tests using the thermal AdS d+1 , where the Vasiliev theory is dual to the vectorial CFT on S d−1 ×S 1 , have also been conducted [16-18, 28, 37]. Such calculations serve as a compact way of checking the agreement of the spectra in the two dual theories. The quantities of interest are the formula for the thermal free energies at arbitrary temperature β, as well as the temperature-independent Casimir energy E c .
In this paper we continue and extend the earlier work [16][17][18][26][27][28][29] on the one-loop tests of higher-spin AdS/CFT. In particular, we will compare the Type B theories in various dimensions d and their dual CFTs consisting of the Dirac fermionic fields (we also consider the theories with Majorana, symplectic, Weyl, or Majorana-Weyl fermions in the dimensions where they are admissible). Let us also comment on the Sachdev-Ye-Kitaev (SYK) model [38,39], which is a quantum mechanical theory of a large number N of Majorana fermions with random interactions; it has been attracting a great deal of attention recently [40][41][42][43][44]. After the use of replica trick, this model has manifest O(N ) symmetry [40], and it is tempting to look for its gravity dual using some variant of type B higher spin theory. Following [45] one may speculate that the SYK model provides an effective IR description of a background of a type B Vasiliev theory asymptotic to AdS 4 which is dual to a theory of Majorana fermions; this background should describe RG flow from AdS 4 to AdS 2 (one could also search for RG flow from HS theory in AdS d+1 to AdS 2 with d = 2, 4, . . .).
Two other types of theories with no explicitly constructed Vasiliev equations are also explored.
First, we consider the theories whose CFT duals are expected to consist of both scalar and fermionic fields, with a subsequent projection onto the singlet sector. These theories, which we call of Type AB, are then expected to have half-integral spin gauge fields in addition to the integral spin gauge fields of Type A and Type B theories. Depending on the precise scalar and fermion field content, the Type AB theories may be supersymmetric in some specific dimension d. For example, the U (N ) singlet sector of one fundamental Dirac fermion and one fundamental complex scalar is supersymmetric in d = 3, and a similar theory with one fundamental Dirac fermion and two fundamental complex scalars is supersymmetric in d = 5. 1 Second, we study the Type C theories, where the CFT dual consists of some number of p-form gauge fields, with p = d 2 − 1; the self-duality condition on the field strength may also be imposed. Such theories were studied in [16][17][18] for d = 4 and 6, and we extend them to more general dimensions.
The organization of the paper is as follows. In Section 2, we review how the comparison of the partition functions of the higher-spin theory and the corresponding CFT allows us to draw useful conclusions about their duality. We will also go through the various HS theories that will be examined in this paper. This will allow us to summarize our results in Tables 1, 2, and 6. In Section 3, we present our results for the free energy of Vasiliev theory in Euclidean AdS d+1 space asymptotic to the round sphere S d . In addition, in Appendix A, we detail the calculations for the free energy of Vasiliev theory in the thermal AdS d+1 space, which is asymptotic to S d−1 × S 1 .
Note Added: Shortly before completion of this paper we became aware of independent forthcoming work on related topics by M. Gunaydin, E. Skvortsov and T. Tran [46].

Higher spin partition functions in Euclidean AdS spaces
According to the AdS/CFT dictionary, the CFT partition function Z CFT on the round sphere S d has to match the partition function of the bulk theory on the Euclidean AdS d+1 asymptotic to S d . This is the hyperbolic space H d+1 with the metric, ds 2 = dρ 2 + sinh 2 ρ dΩ d , where dΩ d is the metric of a unit d-sphere. After defining the free energy F = − log Z, the AdS/CFT correspondence implies F CFT = F bulk . 1 This theory may be coupled to the U (N ) 5d Chern-Simons gauge theory to impose the singlet constraint.
For a vectorial CFT with U (N ), O(N ) or U Sp(N ) symmetry, the large N expansion is For a CFT consisting of N free fields, one obviously has f (n) = 0 for all n ≥ 1.
For the bulk gravitational theory with Newton constant G N the perturbative expansion of the free energy assumes the form The leading contribution is the on-shell classical action of the theory; it should match the leading term in the CFT answer which is of order N . Such a matching seems impossible at present due to the lack of a conventional action for the higher spin theories. 2 However, as first noted in [26], the one-loop correction F (1) requires the knowledge of only the free quadratic actions for the higherspin fields in AdS d+1 ; it can be obtained by summing the logarithms of functional determinants of the relevant kinetic operators. The latter were calculated by Camporesi and Higuchi [47][48][49][50], who derived the spectral zeta function for fields of arbitrary spin in (A)dS. What remains is to carry out the appropriately regularized sum over all spins present in a particular version of the higher spin theory.
The corresponding sphere free energy in a free CFT is given by F CFT = N F , where F may be extracted from the determinant for a single conformal field (see, for example, [35]); the examples of the latter are conformally coupled scalars, massless fermions, or p-form gauge fields. For vectorial theories with double-trace interactions, such as the Wilson-Fisher and Gross-Neveu models, the CFT itself has a non-trivial 1 N expansion, and so F CFT = N F + O(N 0 ). To match the large N scaling, the Newton constant of the bulk theory must behave as in the large N limit. If one assumes that 1 G N F (0) = F CFT , then all the higher-loop corrections to F bulk must vanish for F CFT = F bulk to hold. In [26,27], it was found that for the Vasiliev Type A theories in all dimensions d, the non-minimal theories containing each integer spin indeed have a vanishing one-loop correction to F . However, the minimal theories with even spins only were found to have a non-vanishing one-loop contribution that matched exactly the value of the sphere freeenergy of a single conformal real scalar. This surprising result was then interpreted as a one-loop where the one-loop contribution cancels exactly the shift in the coupling constant. Such an integer shift is consistent with the quantization condition for 1 G N established in [20,21]. The rule N → N −1 does not apply to all the variants of the HS theory. In [16,17] it was shown that the one-loop calculations in Type C higher spin theories dual to free U (N )/O(N ) Maxwell fields in d = 4 required that 1 G N ∼ N − 1 or N − 2 respectively. If the Maxwell fields are taken to be self-dual then 1 G N ∼ N − 1/2; in view of this half-integer shift it is not clear if such a theory is consistent.

Variants of Higher Spin Theories and Key Results
The simplest and best understood HS theory is the type A Vasiliev theory in AdS d+1 , which is known at non-linear level for any d [7]. The spectrum consists of a scalar with m 2 = −2(d − 2) and a tower of totally symmetric HS gauge fields (in the minimal theory, only the even spins are present). This is in one to one correspondence with the spectrum of O(N )/U (N ) invariant "single trace" operators on the CFT side, which consists of the ∆ = d − 2 scalar and the tower of conserved currents This spectrum can be confirmed for instance by computing the tensor product of two free scalar representatios, which yields the result [8,51,52] where the notation (∆; m 1 , m 2 , . . .) indicates a representation of the conformal algebra with conformal dimension ∆ and SO(d) representation labelled by [m 1 , m 2 , . . .] (on the left-hand side, 0)). Equivalently, one may obtain the same result by computing the "thermal" partition function of the free CFT on S 1 × S d−1 , using a flat connection to impose the U (N ) singlet constraint [28,37]. Similarly one can consider real scalars and O(N ) singlet constraint, where one obtains the same spectrum but with odd spins removed (this corresponds to symmetrizing the product in (2.7)).
Another version of the HS theory is the so-called "type B" theory, which is defined to be the HS gauge theory in AdS d+1 dual to the free fermionic CFT d restricted to its singlet sector. The field content of such theories can be deduced from CFT considerations, by deriving the spectrum of singlet operators which are bilinears in the fermionic fields. In the case of Dirac fermions, one has the following results for the tensor product of two free fermion representations [8,52]: in even d Note that in the case d = 3, the spectra of the type A and type B theory are the same, except for the fact that the m 2 = −2 scalar is parity even in the former and parity odd in the latter (and also quantized with conjugate boundary conditions, ∆ = 1 versus ∆ = 2). In this special case, the fully non-linear equations for the type B HS gauge theory in AdS 4 are known and closely related to those of the type A theory [6]. For all d > 3, however, the spectra of Type B theories differ considerably from Type A theories, since they contain towers of spins with various mixed symmetries, see (2.8)-(2.9), and the corresponding non-linear equations are not known. As an example, and to clarify the meaning of (2.8)-(2.9), let us consider d = 4 [28,[53][54][55]. In this case, on the CFT side one can construct the two scalar operators as well as (schematically) the totally symmetric and traceless bilinear currents and a tower of mixed higher symmetry bilinear current, which has an equivalent form for integer d For example, for d = 3, 5, 7 one finds (2. 16) and similarly for higher d. Obviously, these complicated shifts cannot be accommodated by an integer shift of N . While the reason for this is not fully clear to us, it may be related to the fact that the imposition of the singlet constraint requires introduction of other terms in F . For example, in d = 3 the theory also contains a Chern-Simons sector, whose leading contribution to F is of order N 2 . Perhaps a detailed understanding of these additional terms holds the key to resolving the puzzle for the fermionic theories in odd d.
We note that (2.14) always produces only linear combinations of ζ(2k + 1)/π 2 with rational coefficients. Interestingly, these formulas are related to the change in F due to certain double-trace deformations [56]. In particular, the first formula gives (up to sign) the change in free energy due , and the second formula is proportional to the change in free energy due to the deformation ∼ The reason for this formal relation to the double-trace flows is unclear to us.
We also consider bulk Type B theories where various truncations have been imposed on the non-minimal Type B theory and we provide evidence that they are dual to the singlet sectors of various free fermionic CFTs. In d = 2, 3, 4, 8, 9 mod 8 we study the CFT of N Majorana fermions with the O(N ) singlet constraint, while in d = 5, 6, 7 mod 8 we study the theory of N symplectic Majorana fermions with the U Sp(N ) singlet constraint. We also study the CFT of Weyl fermions in even d, and of Majorana-Weyl fermions when d = 2 mod 8. We will discuss these truncations in more detail in section 3.2.1. For even d, we find that under the Weyl truncation, the Type B theories have vanishing F at the one-loop level. Under the Majorana/symplectic Majorana condition, the free energy of the truncated Type B theory gives (up to sign) the free energy of one free conformal fermionic field on S d . This is logarithmically divergent due to the CFT a-anomaly, where the anomaly coefficient a f is given by [56]:  Table 6.
We did not find a simple analytic formula that reproduces these numbers, but we note that, as in the non-minimal type B result (2.14), these values are always linear combinations of ζ(2k + 1)/π 2 with rational coefficients. It would be very interesting to understand the origin of these "anomalous" results in the type B theories.
One may also consider free CFTs which involve both the conformal scalars and fermions in the When we impose the U (N ) singlet constraint, the spectrum of single trace operators contains not only the bilinears in φ and ψ, which are the same as discussed above, but also fermionic operators of the form The dual HS theory in AdS should then include, in addition to the bosonic fields that appear in type A and type B theories, a tower of massless half integer spin particles with s = 3/2, 5/2, . . ., plus a s = 1/2 matter field. We will call the resulting HS theory the "type AB" theory. Note that in d = 3 this leads to a supersymmetric theory, but in general d the action (2.19) is not supersymmetric. One The partition function for the Type AB theory is, with F b being for the contributions from bosonic higher-spin fields, which arise from purely Type A and purely Type B contributions, and F (1) f is the contribution of the HS fermions dual to (2.20).
Up to one-loop level, the bosonic and fermionic contributions are decoupled, as indicated in (2.21).
A similar decoupling of the Casimir energy occurs at the one-loop level, i.e. E (1) c,b . Our calculations for the Euclidean-AdS higher spin theory shows that F E c,f = 0. In even d, from our results on the Type B theories and the earlier results on Vasiliev Type A theories, we see that F = 0 for the non-minimal Type AB theory, and this suggests that Type AB theories at one-loop have vanishing F (1) . For odd d, F (1) is non-vanishing with the non-zero contribution coming from the Type B theory's free energy, as discussed above.
Finally, we consider the Type C higher-spin theories, which are conjectured to be dual to the singlet sector of massless p-forms, where p = ( d 2 − 1). 3 The first two examples of these theories are the d = 4 case discussed in [16,17], where the dynamical fields are the N Maxwell fields, and the d = 6 case [18] where the dynamical fields are N 2-form gauge fields with field strength H µνρ . In these theories, there are also an infinite number of totally symmetric conserved higher-spin currents, in addition to various fields of mixed symmetry. We will extend these calculations to even d > 6.
As for type B theories in d > 3, there are no known equations of motion for type C theories, but we can still infer their free field spectrum from CFT considerations, using the results of [52]. The non-minimal theory is obtained by taking N complex (d/2 − 1)-form gauge fields A, and imposing a U (N ) singlet constraint. One may further truncate these models by taking real fields and O(N ) singlet constraint, which results in the "minimal type C" theory. In addition, one can further impose a self-duality condition on the d/2-form field strength F = dA. Since * 2 = +1 in d = 4m + 2 and * 2 = −1 in d = 4m, where * is the Hodge-dual operator, one can impose the self-duality condition F = * F only in d = 4m + 2 (for m integer); this can be done both for real (O(N )) and complex (U (N )) fields. In d = 4m, and only in the non-minimal case with N complex fields, one can impose the self-duality condition F = i * F . Decomposing F = F 1 + iF 2 into its real and imaginary parts, this condition implies F 1 = − * F 2 , and selfdual and anti-selfdual parts of F are complex conjugate of each other.
As an example, let us consider d = 4 and take N complex Maxwell fields with a U (N ) singlet constraint. The spectrum of the the single trace operators arising from the tensor productF i µν ⊗F ρσ i can be found to be [17,52] (2; 1, 1) c ⊗ (2; 1, 1) c = 2(4; 0, 0) + (4; 1, 1) c + (4; 2, 2) c where we use the notation (2; 1, 1) c = (2; 1, 1) + (2; 1, −1), corresponding to the sum of the selfdual and anti selfdual 2-form field strength with ∆ = 2, and similarly for the representations appearing on the right-hand side. Note that we use SO(4) notations [m 1 , m 2 ] to specify the representation. Similarly, one may obtain the spectrum in all higher dimensions d = 4m and d = 4m + 2, as will be explained in detail in section 3.2.3. As an example, in the d = 8 type C theory we find the Our results for the one-loop calculations in type C theories are summarized in Table 1. We find that the non-minimal U (N ) theories have non-zero one-loop contributions, unlike the type A   3 Matching the Sphere Free Energy

The AdS spectral zeta function
Let us first review the calculation of the one-loop partition function on the hyperbolic space in the case of the totally symmetric HS fields [26,27]. After gauge fixing of the linearized gauge invariance, the contribution of a spin s (s ≥ 1) totally symmetric gauge field to the bulk partition function is obtained as [57][58][59] where the label ST T stands for symmetric traceless transverse tensors, and the numerator corresponds to the contributions of the spin s − 1 ghosts. The mass-like terms in the above kinetic operators are related to the conformal dimension of the dual fields. For a totally symmetric field with kinetic operator −∇ 2 + κ 2 , the dual conformal dimension is given by  For the values of κ in (3.1), one finds for the physical spin s field in the denominator 4 which corresponds to the scaling dimension of the dual conserved current in the CFT. Similarly, the conformal dimension obtained from the ghost kinetic operator in (3.1) is From CFT point of view, this is the dimension of the divergence ∂ · J s , which is a null state that one has to subtract to obtain the short representation of the conformal algebra corresponding to a conserved current.
The determinants in (3.1) can be computed using the heat kernel, or equivalently spectral zeta functions techniques. 5 The spectral zeta function for a differential operator on a compact space with discrete eigenvalues λ n and degeneracy d n is defined as In our case, the differential operators in hyperbolic space have continuous spectrum, and the sum over eigenvalues is replaced by an integral. Let us consider a field labelled by the representation 6 , where we have denoted by m 1 = s the length of the first row in the corresponding Young diagram, which we may call the spin of the particle (for example, for a totally symmetric field, we have α s = [s, 0, 0, . . . , 0]). For a given representation α s , the spectral zeta function takes the form where µ αs (u) is the spectral density of the eigenvalues, which will be given shortly, and g αs is the dimension of the representation α s (see eq. (3.18) and (3.19) below). The denominator corresponds to the eigenvalues of the kinetic operator, and ∆ is the dimension of the dual CFT operator. 7 The regularized volume of AdS is given explicitly by [61][62][63] vol where R is the radius of the boundary sphere. The logarithmic dependence on R in even d is related to the presence of the Weyl anomaly in even dimensional CFTs. Finally, the volume of the round sphere of unit radius is Once the spectral zeta function is known, the contribution of the field labelled by (∆; α s ) to the bulk free energy is obtained as where σ = +1 or −1 depending on whether the field is bosonic or fermionic. Here is the AdS curvature, which we will set to 1 henceforth, and Λ is a UV cut-off. In general, the coefficient of the logarithmic divergence ζ (∆;αs) (0) vanishes for each α s in even dimension d, but it is non-zero for odd d.
When the dimension ∆ = s+d−2, the field labelled by α s is a gauge field and one has to subtract the contribution of the corresponding ghosts in the α s−1 representation. 8 We find it convenient to introduce the notation to indicate the spectral zeta function of the HS gauge fields in the α s representation, with ghost contribution subtracted. The full one-loop free energy may be then obtained by summing over all representations α s appearing in the spectrum. For instance, in the case of the non-minimal type A theory, we may define the "total" spectral zeta function from which we can obtain the full one-loop free energy Similarly, one can obtain ζ HS total (z) and the one-loop free energy in the other higher spin theories we discuss. As these calculations requires summing over infinite towers of fields, one has of course to suitably regularized the sums, as discussed in [26,27] and reviewed in the explicit calculations below.

The spectral density for arbitrary representation
A general formula for the spectral density for a field labelled by the representation α = [m 1 , m 2 , . . .] was given in [50], and we summarize their result below.
In AdS d+1 , arranging the weights for the irreps of SO(d) as m 1 ≥ m 2 ≥ · · · ≥ |m p |, where p = d−1 2 for odd d and p = d 2 for even d, we may define In terms of these, the spectral density takes the form of The pre-factor of π (2 d−1 Γ( d+1 2 )) 2 arises as a normalization constant found by imposing the condition that as we approach flat space from hyperbolic space, the spectral density should approach that of flat space.
The number of degrees of freedom g α is equal to the dimension of the corresponding representation of SO(d), and is given by [64] and As an example, in the Type A case in AdS d+1 , the only representation we need to consider is m 1 = s, and for all j = 1, m j = 0. This gives us (3.20) and The results agree with the formulas derived in [47] and used in [27].
In type AB theories, we need the spectral density for fermion fields in the α = [s, 1/2, 1/2, . . . , 1/2] representation. We find that the above general formulas for even and odd d can be expressed in the compact form valid for all d and (3.23) The spectral densities for the mixed symmetry fields appearing in type B and C theories can be obtained in a straightforward way from the above general formulas, and we present the explicit results in the next sections.

Type B Theories
Spectrum The non-minimal Type B higher spin theory, which is conjectured to be dual to the U (N ) singlet sector of the free Dirac fermion theory, contains towers of mixed symmetry gauge fields of all integer spins. From the spectrum given in (2.8), we obtain the total spectral zeta function In the third line of (3.24), the representations [s, 1, 1, . . . , 1, 1] and [s, 1, 1, . . . , 1, −1] give the selfdual and anti-selfdual parts of the corresponding fields. At the level of the spectral ζ functions, they yield equal contributions. 9 Using the spectral zeta function formulas listed in Section 3.1.1 and summing over all representations given above, we find that for all even d Weyl projection The projection from the non-minimal Type B theory described above is slightly different when the theory is in d = 4m or d = 4m + 2. Using the results of [52] for the product of chiral fermion representations, we find 10 Note that under this projection, there are no scalars in the spectrum. The case d = 4 (AdS 5 ) was already discussed in [28]. Summing over all representations, we find that for all even d and so Minimal Theory (Majorana projection) The Majorana conditionψ = ψ T C, where C is the charge conjugation matrix, can be imposed in d = 2, 3, 4, 8, 9 (mod 8), see for instance [65]. In these dimensions, we can consider the theory of N free Majorana fermions and impose an O(N ) singlet constraint. In d = 6 (mod 8), provided one has an even number N of fermions, one can impose instead a symplectic Majorana conditionψ i = ψ T j CΩ ij , where C is the charge conjugation matrix and Ω ij the antisymmetric symplectic metric. In this case, we consider the theory of N free symplectic Majoranas with a U Sp(N ) singlet constraint.
The operator spectrum in the minimal theory can be deduced by working out which operators of the non-minimal theory are projected out by the Majorana constraint. The bilinear operators in the non-minimal theory are of the schematic form where n = 0, . . . , d 2 −1, and Γ (n) is the antisymmetrized product of n gamma matrices. For Majorana fermions, we haveψ = ψ T C, and so the operators are projected out or kept depending on whether CΓ (n) is symmetric or antisymmetric. If CΓ (n) is symmetric, then the operators with an even number of derivatives (i.e. odd spin) are projected out; if it is antisymmetric, then the operators with an odd number of derivatives (i.e. even spin) are projected out. In addition to (3.30), the non-minimal For instance, in d = 4, the non-minimal theory contains the operators given in (2.10), (2.11) and (2.12). In d = 4, one has that both C and Cγ 5 are antisymmetric, so both scalars in (2.10) are retained. Then, one has that Cγ µ is symmetric while Cγ µ γ 5 antisymmetric, and so we keep Higher dimensions can be analyzed similarly, using the symmetry/antisymmetry properties of CΓ (n) in various d [65]. The results are summarized in Table 3 after which this cycle repeats. The number of scalars with ∆ = d − 1 to be included also changes in a cycle of 4. In AdS 5 , we have 2 scalars; in AdS 7 , we have 1 (this case, though, should be discussed separately, see below); in AdS 9 , we have 0; in AdS 11 , we have 1, and the cycle repeats. In a more compact notation, the total spectral zeta function in the minimal type B theories dual to the O(N ) 11 As an example, consider the bilinear ψ T M ψ. If M is symmetric, this operator clearly vanishes. On the other hand, consider ψ T M ∂µψ. In this case, if M is an antisymmetric matrix, then this is equal to +∂µψ T M ψ. In turn, this means that ψ T M ∂µψ = 1 2 ∂µ(ψ T M ψ), and so this operator is a total derivative and is not included in the spectrum of primaries.
where χ(d) = 1, 2, 0 when d = 0, 2, 4 ( mod 8) respectively. Explicit illustrations of this formula are given in Table 3. Using these spectra we find, in all even d where the Majorana condition is possible where R is the radius of the boundary sphere, and a f is the a-anomaly coefficient of a single Majorana fermion in dimension d, given in (2.18). As explained earlier, this is consistent with the duality, provided G type B Maj.
N ∼ 1/(N − 1). As mentioned above, in d = 6 (mod 8), i.e. AdS 7(mod 8) , we should impose a symplectic Majorana condition and consider the U Sp(N ) invariant operators. In terms of the operators (3.30), sincē ψ = ψ T CΩ with Ω antisymmetric, all this means is that now odd spins are projected out when CΓ (n) is antisymmetric, and even spins are projected out when CΓ (n) is symmetric. Similarly, the scalar operatorsψ i ψ i and ψ i γ * ψ i are now projected out when C and Cγ * are antisymmetric, respectively. In d = 6 (mod 8), one has that C is symmetric and Cγ * is antisymmetric, so we retain a single scalar field. On the other hand, Cγ µ and Cγ µ γ * are both antisymmetric, and so we have two towers of totally symmetric representations of all even s. 12 The projection of the mixed symmetry representations can be deduced similarly. The total spectral zeta function is given by the formula An illustration of the formula is given in Table 3 for the AdS 7 and AdS 15 cases. Using these spectra, we find that the one loop free energy of the minimal type B theory corresponding to the symplectic Majorana projection is given by i.e. the opposite sign compared to (3.32). This is consistent with the duality, provided G type B sympl.Maj.
An illustration of this can be seen in Table 4, where we list the spectra of AdS 11 and AdS 19 .
Summing up over these spectra, we find the result which is the a-anomaly coefficient of a single Majorana-Weyl fermion at the boundary.
In d = 6 (mod 8) one may impose a symplectic Majorana-Weyl projection. The resulting spectra are the overlap between the symplectic Majorana and Weyl projections. For instance, in d = 6 we     Table 4). In this case (and similarly for higher

Sample Calculations
AdS 5 Following (3.24) for the non-minimal Type B theory, In the above, we made use of the formula, , (3.40) to go from the first to the second line.
In our regularization scheme, we sum over the physical modes separately from the ghost modes.
Similarly, for the totally symmetric representation [s, 0], we have Finally, for the massive scalar with ∆ = 3, we have To illustrate the zeta-regularization, let us consider the last term, where on the second line we used the substitution s → 2s, followed by rewriting 2s + 1 = 2(s + 1 2 ). 13 The partial results coming from summing each tower are given in Table 9. Putting everything together, we obtain F AdS 11 We skip the d = 7, 9 case, whose spectrum for the various theories follow from the discussion in Section 3.3.3. For reference, the calculated free energy of each weight F (1) is given in Tables 10 and 11.

Fermionic Higher Spins in Type AB Theories
Spectrum We described earlier that there is only one irrep of SO(d) of interest here that describes the tower of spins corresponding to the fermionic bilinears in type AB theories, namely α s = [s, 1 2 , 1 2 , . . . , 1 2 ]. Therefore, in the non-minimal theories dual to complex scalars and fermions in the U (N ) singlet sector, the purely fermionic contribution to the total zeta function is  with ∆ ph = 2 + s. For the massive fermion contribution, Then, This gives us, The technicalities of the shift to the Hurwitz Zeta function in the sum above is similar to the case for the minimal Type B theory in AdS 5 which we worked out earlier. More details can be found in type AB ferm = 0, consistently with the duality. For reference, we also report the expected expression of ζ HS type AB ferm (z) for AdS 7 and AdS 9 , expanded in z up to the second order, in Appendix C.2.

Type C Theories
Calculations for Type C theories are similar to those described above and we will not go through all details explicitly. In the following sections, we list the spectrum of fields in these theories, including Spectrum The spectrum of the non-minimal type C theories, dual to the free theory of N complex d/2-form gauge fields with U (N ) singlet constraint, can be obtained from the character formulas derived in [52]. While the resulting spectra may look complicated, they follow a clear pattern that can be rather easily identified if one refers to the tables given in Appendix C.3. The results are split into the cases d = 4m and d = 4m + 2. For d = 4m, the total spectral zeta function is given by 16 17 ζ HS type C (z) = 2 and for d = 4m + 2 ζ (4m+2;[k 1 ,k 1 ,...,km,km,0]) (z) 16 (3.58) and (3.59) correspond to equations (4.20)-(4.21) and (4.22)-(4.23) of [52] respectively, and the tensorial decomposition in these quoted equations can be further simplified by the formulas on p. 104 of [66]. 17 For all Type C theories, the field of spin s = 2 in the towers of spins of representation [s, 2, . . .] are not gauge fields, but we will still use the symbol Z for conciseness. See footnote 9 for similar remarks.
Using these spectra and (3.15) to compute the zeta functions, we find the results

F
(1) where a d/2−form is the a-anomaly coefficient of a single real (d/2 − 1)-form gauge field in dimension d. Thus, we see that (3.60) is consistent with the duality provided G type C Minimal Type C O(N ) The "minimal type C" theory corresponds to the O(N ) singlet sector of the free theory of N (d/2 − 1)-form gauge fields. Its spectrum can be in principle obtained by appropriately "symmetrizing" the character formulas given in [52] and used above to obtain the non-minimal spectrum. The spectra in d = 4 and d = 6 were obtained in [16][17][18]. Generalizing those results for all d, we arrive at the following total spectral zeta functions. In d = 4m, ζ HS min. type C (z) = 2     These correspond to the shifts given in Table 1. Interestingly, in the minimal type C theory in d = 6, 10, . . . the bulk one-loop free energy vanishes and no shift of the coupling constant is required.
Self-dual U (N ) In d = 4m, we can impose a self-duality constraint F i = i * F i in the theory of N complex p-forms. The resulting spectrum of U (N ) invariant bilinears leads to the following total zeta function in the bulk 18 In d = 4m + 2, we can impose the self-duality condition F i = * F i , and the resulting truncated spectrum gives the following total zeta function 19 Using these spectra, we find the results which correspond to the shifts given in Table 1.

Self-dual O(N )
In d = 4m + 2, we can impose a self-duality condition on the theory of N real forms with O(N ) singlt constraint. The spectrum is given by the "overlap" of the minimal type C and self-dual U (N ) spectra given above. The resulting total zeta function is given by ζ HS min. type C SD (z) = s=2,4,6,...

Preliminaries
Alternate Regulators In the calculations for even d discussed above, we chose to sum over the spins before sending the spectral parameter z → 0. This analytic continuation in z is a natural way to regulate the sums. In practice, this is possible in the even d case because the spectral density is polynomial in the integrating variable u. In the case of odd d, summing before sending z → 0 is not easy to do, and we will instead first send z → 0 and then evaluate the regularized sums over spins.
There are two equivalent ways to do this. The first involves using exponential factors to suppress the spins all spins in αs where we recall that ∆ ph = s+d−2 and ∆ gh = s+d−1.
This method, which is closely related to the one previously used in [27], 20 will be described in the next sections in greater detail.
Note that, while in even d ζ (∆s;αs) (0) vanishes identically for any representation, this is not true in odd d. Vanishing of the logarithmic divergence in the one-loop free energy requires in this case summing over all the bulk fields, as reviewed below.
Integrals In all odd d calculations, we encounter the integrals of the type We define There exists a recursive relation between the various A k 's and B k 's for any odd integer 2k + 1 (see Appendix B.2 for a proof): As a consequence of this relation, we only need the explicit analytic expressions of the integrals A ± 1 , 21 which is given by [47] A + 1 (x) = where ψ(x) is the digamma function ψ(x) = Γ (x)/Γ(x).

Calculational method and Type A example
To illustrate the method of calculation, we first review the calculation in the non-minimal Type A theory [26,27]. The calculations for the various Type B theories are similar and we will not give 20 In that paper, an "averaged" regulator of ( ∆ ph +∆ gh 2 − d 2 ) − was preferred for the Type A theory calculations, and it can be shown to give the same result as the regulators (3.72)-(3.73) that we will use in our calculations. In type AB theories, however, it appears that "averaged" regulator does not work, and we will use the shifts defined in (3.72)-(3.73) in all theories consistently. 21 While not needed, the integral results for B ± k , can be identified with the Hurwitz-Lerch Phi function Φ(z, s, v), all details. Calculations for the Type AB theory are similar with slight differences that will be discussed below.
Unlike the even d case, the spectral function µ α (u) is no longer polynomial in u, but a polynomial in u multiplied by a hyperbolic function, for bosons, Then, for a particular spectral weight α, the partition function can be written as (3.81) We will use the example of the Type A theory in AdS 4 to walk us through the calculations. In the non-minimal Type A theory in AdS 4 , the only representations are the totally symmetric ones α = [s], s ≥ 0, and the spectral zeta function for a given spin s is To calculate the one-loop free energy, we will need to evaluate ζ (∆;α) (0) and ζ (∆;α) (0).
It is remarkable that when we sum over the entire spectrum of bulk fields, we get ζ HS Total (0) = 0, (3.87) which indicates that the one-loop free energies have no logarithmic divergences. We find that this result holds not only in type A theories [26,27], but also in all of the type B and type AB theories we discuss below.
Computing ζ (∆;αs) (0): The evaluation of ζ (0) in odd d is considerably more complicated. One may begin by splitting the f ± (u) term as f ± (u) = 1 ∓ 2 e 2πiu ±1 so that And, by differentiating (3.88), The integral in ζ poly (∆;α) (z) may be evaluated at arbitrary z, and after taking the derivative and summing over spins, one finds a zero contribution to the free energy. The evaluation of ζ exp (∆;α) (z) is more involved, and we refer the reader to Appendix B.3 and [26,27] for more details. The final result is that, in the non-minimal theory [26] which implies that the one loop free energy vanishes. In the non-minimal theory, one finds instead which is the free energy of a single real conformal scalar on S 3 . An analogous result is found for the type A theory in AdS d+1 for all d [27].

Type B Theories
Non-minimal Theory The full spectral zeta function for the non-minimal type B theory in odd d follows from eq. (2.9), and reads Note that instead of two towers, there is only one tower for each representation, due to the lack of the chirality matrix. Using this spectrum and the procedure outlined above to regulate the sums, we find that the logarithmic divergence correctly cancels ζ HS type B (0) = 0 . However, as summarized in section 2.2, the evaluation of (ζ HS type B ) (0) leads to a surprising result. The one-loop free energy of the non-minimal type B theories in all odd d does not vanish, but is given by (2.14), or equivalently by (2.15). This apparent mismatch with the expected result F (1) = 0 remains to be understood.   (1) can be found in Table 6. spectra of the minimal theories can be again deduced from the symmetry/antisymmetry properties of the CΓ (n) matrices. In the Majorana case, if CΓ (n) is symmetric the operators of the form (3.30) are retained for even spins and projected out for odd spins, and vice-versa if CΓ (n) is antisymmetric.

Minimal theories
The scalar operatorψ i ψ i is projected out if C is symmetric. For instance, in d = 3 the C matrix is antisymmetric and Cγ µ is symmetric, and so the spectrum of the minimal theory includes the ∆ = 2 (pseudo)-scalar and the tower of totally symmetric fields of even spin. Higher dimensional cases can be worked out similarly, and the first few examples are listed in Table 5. In a compact notation, the total spectral zeta function of the minimal theories dual to the Majorana projected fermion model reads where χ(d) = 0, 1 when d = 5, 7 (mod 8) respectively. In both versions of the minimal truncation, we find that the coefficient of the logarithmic divergence still vanishes after summing up over the full spectrum. However, similarly to the non-minimal case, the minimal Type B theories in odd d appear to have a non-zero one-loop free energy, which we report in Table 6. We did not find an analytic formula for these results similar to (2.14). However, we note that all these "anomalous" values only involve the Riemann zeta functions ζ(2k + 1) divided by π 2 , and interestingly all other

Type AB Theories
Spectrum and Results As in the even d case, the only irrep of SO(d) describing the tower of half-integer spins is α s = [s, 1 2 , 1 2 , . . . , 1 2 ]. Thus, the total spectral zeta function is given by the same equation as in (3.51).
The calculation is rather similar to the one we outlined for the Type A theory. The only difference is that the spectral density µ α (u) includes coth(πu) instead of tanh(πu). For example, in the Type AB theory in AdS 4 , the higher-spin zeta-function is given by (3.98) The calculations for ζ (∆;αs) (0) are essentially identical to that of Type A theories, and in particular we find that the contribution to the logarithmic divergence due to the fermionic fields vanishes after summing over the whole tower. Heading straight to the calculation of ζ (∆;αs) (0), if we follow the procedure outlined for the Type A case, we have Rewriting the exponential terms using (3.74), we should use In any case, the terms involving 1 2 √ x , which we can call ζ exp−sqrt (∆;[s]) (0), will not contribute to the value of ζ exp (∆;[s]) (0). Only the contributions from the terms involving ψ( √ x), namely the third term inside the bracket of (3.100) will contribute. After putting all together, the end result is i.e., the tower of fermionic fields in type AB theories yields a vanishing contribution to the bulk one-loop free energy. This result extends to all higher d.

Acknowledgments
The with S d boundary. The results below follow and generalize [28], which considered type A theories in all d and type B theories in d = 2, 3, 4, and [16][17][18], where type B theories in d = 6 and type C theories in d = 4, 6 were discussed.
The free energy on S 1 × S d−1 takes the form where F β depends non-trivially on the temperature and goes to zero at large β, and E c is the Casimir energy. The latter is related to the "one-particle" partition function on S 1 × S d−1 by (see e.g. [28] for a review) where σ = +1 for bosonic fields, and σ = −1 for fermionic ones, and Z(β) denotes the one-particle partition function. This also determines F β by Note that E c vanishes for a CFT d in odd d, but it is non-zero in even d.
In the vector models restricted to the singlet sector, one finds that F β = O(N 0 ), due to the integration over the flat connection which enforces the gauge singlet constraint [28,37]. This term should then match the temperature dependent part of the bulk one-loop thermal free energy, ob- to the Casimir energy should precisely be consistent with such a shift. We will see below that this is the case in all higher spin theories we considered in this paper.
On the CFT side, the one-particle partition functions of a conformal scalar and Majorana (or Weyl) fermion are given by Using (A.2) and the identity where ∆ ph = s + d − 2 and g αs is the dimension of the representation α s (the number of propagating degrees of freedom in the bulk is g αs − g α s−1 ). For the massive fields, the ghost contribution is not present, and one has (A.11) One may obtain a "total" one-particle partition function Z(β) in the bulk by summing over all representations in the spectrum, and from it one may then find the bulk one-loop Casimir energy by (A.2) and F β by (A.3). In the following we summarize the result of these calculations in the various higher spin theories considered in this paper.
The bulk Casimir energy can be obtained by inserting the right-hand side of (A.12) and (A. 13) into (A.2) (alternatively, one may compute the Casimir contributions spin by spin, and sum up at the end). One finds that [Z 0 (q)] 2 gives zero contribution to the Casimir energy, 24 while Z 0 (q 2 ) gives a contribution equal to 2E c,0 . Then, E c,type A = 0 and E c,min. type A = E c,0 , consistently with the expected shift of G N deduced from the S d calculations.
Type AB Theories In the purely fermionic sector of the type AB theories, the only representations are given by the weights [s, 1 2 , . . . , 1 2 ], which lead to a simple computation that gives for a generic d, (A.26) A quick calculation gives us E c = 0 for the contribution of the fermionic tower in the Type AB theories, which is nicely consistent with what we obtained in the S d calculations, namely that there are no shifts due to the purely fermionic fields.
Type C Theories In type C theories, summing up over the relevant bulk spectra given in Section 3.

2.3, we find
Non-Minimal Type C: where Z d 2 -form (q) is the one-particle partition function (A.7) of a single real (d/2 − 1)-form gauge field. The results on the right-hand side have the correct structure expected from the CFT thermal free energy in the U (N )/O(N ) singlet sector of the theory of N differential form gauge fields. This calculation was carried out explicitly in the S 1 × S 3 case in [17], and we expect it to generalize to all d.  Table 1. A few explicit values are reported in Table 8. 25 See Appendix D of [18] for a discussion of this.  To implement ζ-function regularization, we identify the conventionally divergent term ∞ s=1 1/(s + ν) k as ∞ s=0 1/(s + ν + 1) k , and treating it as the Hurwitz zeta function, where we then analytically extend to the full complex plane. This allows us to regulate systematically the sums to obtain their finite contributions.
Suppose we want to start summing all integer spins s ≥ ≥ 0, then, This is the convention we applied in this paper, and avoids potential inconsistencies that can occur with the Hurwitz zeta function. We might also consider sums that only incorporate a particular subset of spins, such as either all odd integer spins or all even integer spins. To do so, we can transform the summing variable of the original Hurwitz zeta function appropriately. We give two examples: To sum over all even spins, consider ∞ s=2,4,6,...

(s +
which allows us to obtain the regularization over both the odd or the even integer spins by just doing one of the two calculation.

B.2 Identity for odd d free energy calculations
The relationship described in (3.76) can be derived by Here we collect some details on the evaluation of the term ∂ z ζ exp (∆;αs) (z)| z=0 in (3.90), in the explicit example of the type A theory in AdS 4 . The calculations in the other theories studied in this paper go through in a similar way. After some integral identities and algebraic manipulations, we may The only overall non-zero contribution will come from the fourth term, ζ exp−ψ (∆;αs) (0), and the contributions of the first three will cancel out, after taking into account the ghost modes and all other particles in the entire spectra of the theory.
To understand what these three terms are, let's return to the Type A non-minimal theory, the l.h.s. of (B.7) is now, Using (3.74), we can rewrite the above term into: where ∆ ph − 3 2 = s − 1 2 . The second term can then be explicitly integrated using the recursive relation for , where B ± k := ∞ 0 du u k e 2πu ±1 , and ψ(x) is the digamma function ψ(x) = Γ (x)/Γ(x). We concentrate on the last term including the digamma function, since it is the only term that contributes to the final partition function. To integrate the digamma function, we make use of its integral representation so that we get 1 − e −t − 1 6 x(2s + 1) + 1 24 (2s + 1) 3 The terms in the integrand above split into those that include a prefactor of e −st , and those that do not. For the terms with the prefactor, we can sum over the spins easily and without a regulator, For those terms without the prefactor, we sum using the same regulator as in the previous segment, (B.14) Combining (B.13) and (B.14) under the integrand, we obtain the expression for ζ exp−ψ ∆ ph ;[s] (0). Then, repeating the calculations for the ghost calculations, we obtain 2e t e t + 1 After combining these above with the integral representation for the scalar term, we then make use of the integral representation of the Hurwitz-Lerch transcendental function, to transform the expressions into sums of derivatives of Hurwitz-Lerch transcendental functions 26 .
Finally, the Type A non-minimal theory will give us an expression of     In this case, Z HS total (z) = (1) f = 0, as we set z → 0.

C.3 Free Energy Values for Type C Theories in AdS 9
AdS9 Type C Towers of Spins Contribution to F from one tower summed over:

C.3.1 Spectra of Spins for Type C Theories
In these following results, α = [t