The ABC of Higher-Spin AdS/CFT

Abstract

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 $U\left(N\right)/O\left(N\right)$ 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 $O\left(N^0\right)$ 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.

1. Introduction

Extensions of the original AdS/CFT correspondence [1,2,3] to relations between the “vectorial” d-dimensional 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 $U\left(N\right)/O\left(N\right)$ 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 $U\left(N\right)/O\left(N\right)$ 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 $U\left(N\right)/O\left(N\right)$ 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 $\left(\frac{d}{2}-1\right)$-form gauge fields projected onto the $U\left(N\right)/O\left(N\right)$ 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 $S^{d-1}\times S^1$, have also been conducted [16,17,18,28,37]. 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 $\beta$, 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 the replica trick, this model has manifest $O\left(N\right)$ 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 IRdescription 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 RGflow from AdS${}_4$ to $Ad{S}_2$ (one could also search for RG flow from HStheory in AdS${}_{d+1}$ to AdS${}_2$ with $d=2,4,\dots$).

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 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 $U\left(N\right)$ 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 $d=5$ [46].1 Second, we study the Type C theories, where the CFT dual consists of some number of p-form gauge fields, with $p=\frac{d}{2}-1$; 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 Table 1, Table 2 and Table 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.1, we detail the calculations for the free energy of Vasiliev theory in the thermal AdS${}_{d+1}$ space, which is asymptotic to $S^{d-1}\times S^1$.

Note: Shortly before completion of this paper, we became aware of independent forthcoming work on related topics by M.Gunaydin, E. Skvortsov and T. Tran [47]. After the original submission of this paper, we also noticed a related paper by Y. Pang, E. Sezgin and Y. Zhu [48].

2. Review and Summary of Results

2.1. Higher Spin Partition Functions in Euclidean AdS Spaces

According to the AdS/CFT dictionary, the CFT partition function $Z_{\mathrm{CFT}}$ 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 ${\mathbb{H}}_{d+1}$ with the metric, $d{s}^2=d{\rho }^2+{sinh}^2\rho d{\Omega }_d$, where $d{\Omega }_d$ is the metric of a unit d-sphere. After defining the free energy $F=-logZ$, the AdS/CFT correspondence implies $F_{\mathrm{CFT}}=F_{\mathrm{bulk}}$.

For a vectorial CFT with $U\left(N\right)$, $O\left(N\right)$ or $USp\left(N\right)$ symmetry, the large N expansion is:

$$\begin{array}{c}\hfill F_{\mathrm{CFT}}=N{f}^{\left(0\right)}+f^{\left(1\right)}+\frac{1}{N}{f}^{\left(2\right)}+\dots .\end{array}$$

(1)

For a CFT consisting of N free fields, one obviously has $f^{\left(n\right)}=0$ for all $n\ge 1$.

For the bulk gravitational theory with Newton constant $G_N$ the perturbative expansion of the free energy assumes the form:

$$\begin{array}{c}\hfill F_{\mathrm{bulk}}=\frac{1}{G_N}{F}^{\left(0\right)}+F^{\left(1\right)}+G_N{F}^{\left(2\right)}+\dots \end{array}$$

(2)

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 $F^{\left(1\right)}$ requires the knowledge of only the free quadratic actions for the higher-spin 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 [49,50,51,52], 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 $F_{\mathrm{CFT}}=NF$, 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 $\frac{1}{N}$ expansion, and so $F_{\mathrm{CFT}}=NF+\mathcal{O}\left(N^0\right)$. To match the large N scaling, the Newton constant of the bulk theory must behave as:

$$\begin{array}{c}\hfill \frac{1}{G_N}\propto N,\end{array}$$

(3)

in the large N limit. If one assumes that $\frac{1}{G_N}{F}^{\left(0\right)}=F_{\mathrm{CFT}}$, then all the higher-loop corrections to $F_{\mathrm{bulk}}$ must vanish for $F_{\mathrm{CFT}}=F_{\mathrm{bulk}}$ 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 free-energy of a single conformal real scalar. This surprising result was then interpreted as a one-loop shift:

$$\begin{array}{c}\hfill \frac{1}{G_N}\sim N-1,\end{array}$$

(4)

where the one-loop contribution cancels exactly the shift in the coupling constant. Such an integer shift is consistent with the quantization condition for $\frac{1}{G_N}$ established in [20,21]. The rule $N\to N-1$ 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 $U\left(N\right)/O\left(N\right)$ Maxwell fields in $d=4$ required that $\frac{1}{G_N}\sim N-1$ or $N-2$ respectively. If the Maxwell fields are taken to be self-dual then $\frac{1}{G_N}\sim N-1/2$; in view of this half-integer shift it is not clear if such a theory is consistent.

2.2. 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 $O\left(N\right)$/$U\left(N\right)$ invariant “single trace” operators on the CFT side, which consists of the $\Delta =d-2$ scalar:

$$J_0={\varphi }_i^{*}{\varphi }^i$$

(5)

and the tower of conserved currents:

$$J_{{\mu }_1\cdots {\mu }_s}={\varphi }_i^{*}{\partial }_{({\mu }_1}\cdots {\partial }_{{\mu }_s)}{\varphi }^i+\cdots ,s\ge 1.$$

(6)

This spectrum can be confirmed for instance by computing the tensor product of two free scalar representations, which yields the result [8,53,54]

$$\left(\frac{d}{2}-1;0\right)\otimes \left(\frac{d}{2}-1;0\right)=(d-2;0,\dots ,0)+\sum _{s=1}^{\infty }(d-2+s;s,0,\dots ,0)$$

(7)

where the notation $(\Delta ;m_1,m_2,\dots )$ indicates a representation of the conformal algebra with conformal dimension $\Delta$ and $SO\left(d\right)$ representation labeled by $[m_1,m_2,\dots ]$ (on the left-hand side, $(d/2-1;0)$ is a shorthand for $(d/2-1;0,\dots ,0)$). Equivalently, one may obtain the same result by computing the “thermal” partition function of the free CFT on $S^1\times S^{d-1}$, using a flat connection to impose the $U\left(N\right)$ singlet constraint [28,37]. Similarly one can consider real scalars and $O\left(N\right)$ singlet constraint, where one obtains the same spectrum but with odd spins removed (this corresponds to symmetrizing the product in (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 [8,54]: in even d:

$$\begin{array}{c}\left(\frac{d-1}{2};\frac{1}{2}\right)\otimes \left(\frac{d-1}{2};\frac{1}{2}\right)=2(d-1;0,\dots ,0)+2{\sum }_{s=1}^{\infty }\left[(d-2+s;s,0,\dots ,0)+(d-2+s;s,1,0,\dots ,0)\right.\hfill \\ +(d-2+s;s,1,1,0,\dots ,0)+\dots +(d-2+s;s,1,1,1,\dots ,1,0)\hfill \\ \left.+(d-2+s;s,1,1,\dots ,1,1)+(d-2+s;s,1,1,\dots ,1,-1)\right]\hfill \end{array}$$

(8)

and in odd d:

$$\begin{array}{c}\left(\frac{d-1}{2};\frac{1}{2}\right)\otimes \left(\frac{d-1}{2};\frac{1}{2}\right)=(d-1;0,\dots ,0)+\sum _{s=1}^{\infty }\left[(d-2+s;s,0,\dots ,0)+\right.\hfill \\ \left.+(d-2+s;s,1,0,\dots ,0)+\dots +(d-2+s;s,1,1,\dots ,1,0)+(d-2+s;s,1,1,\dots ,1,1)\right].\hfill \end{array}$$

(9)

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, $\Delta =1$ versus $\Delta =2$). 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 (8) and (9), and the corresponding non-linear equations are not known. As an example, and to clarify the meaning of (8) and (9), let us consider $d=4$ [28,55,56,57]. In this case, on the CFT side one can construct the two scalar operators:

$$\begin{array}{c}\hfill J_0={\overline{\psi }}_i{\psi }^i,{\tilde{J}}_0={\overline{\psi }}_i{\gamma }_5{\psi }^i,\end{array}$$

(10)

as well as (schematically) the totally symmetric and traceless bilinear currents:

$$\begin{array}{c}\hfill J_{{\mu }_1\cdots {\mu }_s}={\overline{\psi }}_i{\gamma }_{({\mu }_1}{\partial }_{{\mu }_2}\cdots {\partial }_{{\mu }_s)}{\psi }^i+\cdots ,{\tilde{J}}_{{\mu }_1\cdots {\mu }_s}={\overline{\psi }}_i{\gamma }_5{\gamma }_{({\mu }_1}{\partial }_{{\mu }_2}\cdots {\partial }_{{\mu }_s)}{\psi }^i+\cdots ,s\ge 1,\end{array}$$

(11)

and a tower of mixed higher symmetry bilinear current,

$$\begin{array}{c}\hfill M_{{\mu }_1\cdots {\mu }_s,\nu }={\overline{\psi }}_i{\gamma }_{\nu ({\mu }_1}{\partial }_{{\mu }_2}\cdots {\partial }_{{\mu }_s)}{\psi }^i+\cdots ,s\ge 1,\end{array}$$

(12)

where ${\gamma }_{\nu {\mu }_1}={\gamma }_{[\nu }{\gamma }_{{\mu }_1]}$ is the antisymmetrized product of the gamma matrices. These operators are dual to corresponding HS fields in AdS${}_5$. In particular, in addition to two towers of Fronsdal fields and a tower of mixed symmetry gauge fields, there are two bulk scalar fields and a massive antisymmetric tensor dual to ${\overline{\psi }}_i{\gamma }_{\mu \nu }{\psi }^i$. Similarly, in higher dimensions one can construct the tower of mixed symmetry operators appearing in (8) and (9) by using the antisymmetrized product of several gamma matrices. In the Young tableaux notation, these operators correspond to the hook type diagrams:

(13)

where $1<j\le p$, with $p=d/2$ for even d and $p=(d-1)/2$ for odd d. For $s>1$, these operators are conserved currents and are dual to massless gauge fields in the bulk, while for $s=1$ they are dual to massive antisymmetric fields.

For even d, we find evidence that the non-minimal Type B theory is exactly dual to the singlet sector of the $U\left(N\right)$ free fermionic CFT. The one-loop free energy of the Vasiliev theory vanishes exactly. This generalizes the result given in [16] for the non-minimal Type B theory in AdS${}_5$; namely, there is no shift to the coupling constant in the non-minimal Type B theory dual to the singlet sector of Dirac fermions.

However, for all odd d, the one-loop free energy does not vanish. Instead, it follows a surprising formula:

$$\begin{array}{cc}\hfill F_{\mathrm{type}\mathrm{B}}^{\left(1\right)}=& -\frac{1}{\Gamma (d+1)}{\int }_0^{1/2}duusin\left(\pi u\right)\Gamma \left(\frac{d}{2}+u\right)\Gamma \left(\frac{d}{2}-u\right),\hfill \end{array}$$

(14)

which has an equivalent form for integer d:

$$\begin{array}{cc}\hfill F_{\mathrm{type}\mathrm{B}}^{\left(1\right)}=& \frac{1}{2\Gamma (d+1)}{\int }_0^{1}ducos\left(\pi u\right)\Gamma \left(\frac{d+1}{2}+u\right)\Gamma \left(\frac{d+1}{2}-u\right),\hfill \end{array}$$

(15)

For example, for $d=3,5,7$, one finds:

$$\begin{array}{cc}{F}_{\mathrm{type}\mathrm{B}}^{\left(1\right)}=-\frac{\zeta \left(3\right)}{8{\pi }^2},\hfill & \hfill d=3,\\ F_{\mathrm{type}\mathrm{B}}^{\left(1\right)}=-\frac{\zeta \left(3\right)}{96{\pi }^2}-\frac{\zeta \left(5\right)}{32{\pi }^2},\hfill & \hfill d=5,\\ F_{\mathrm{type}\mathrm{B}}^{\left(1\right)}=-\frac{\zeta \left(3\right)}{720{\pi }^2}-\frac{\zeta \left(5\right)}{192{\pi }^2}-\frac{\zeta \left(7\right)}{128{\pi }^2},\hfill & \hfill d=7.\end{array}$$

(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 (14) always produces only linear combinations of $\zeta (2k+1)/{\pi }^2$ with rational coefficients. Interestingly, these formulas are related to the change in F due to certain double-trace deformations [58]. In particular, the first formula gives (up to sign) the change in free energy due to the double-trace deformation $\sim \int d^{d}x{O}_{\Delta }^2$, where $O_{\Delta }$ is a scalar operator of dimension $\Delta =\frac{d-1}{2}$, and the second formula is proportional to the change in free energy due to the deformation $\sim \int d^{d}x{\overline{\Psi }}_{\Delta }{\Psi }_{\Delta }$, where ${\Psi }_{\Delta }$ is a fermionic operator of dimension $\Delta =\frac{d-2}{2}$. 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 $O\left(N\right)$ singlet constraint, while in $d=5,6,7$ mod 8 we study the theory of N symplectic Majorana fermions with the $USp\left(N\right)$ singlet constraint.3 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, $F_f^{S^d}=a_{f}log\left(\mu R\right)$, where the anomaly coefficient $a_f$ is given by [58]:

$$\begin{array}{cc}\hfill a_f& =2^{\frac{d}{2}}\frac{{(-1)}^{\frac{d}{2}}}{\pi \Gamma (1+d)}{\int }_0^{1}ducos\left(\frac{\pi u}{2}\right)\Gamma \left(\frac{1+d+u}{2}\right)\Gamma \left(\frac{1+d-u}{2}\right)\hfill \end{array}$$

(17)

$$\begin{array}{c}=\left\{-\frac{1}{6},\frac{11}{180},-\frac{191}{7560},\frac{2497}{\mathrm{226,800}},-\frac{\mathrm{14,797}}{\mathrm{2,993,760}},\frac{\mathrm{92,427,157}}{\mathrm{40,864,824,000}},-\frac{\mathrm{36,740,617}}{\mathrm{35,026,992,000}},\cdots \right\}\hfill \end{array}$$

(18)

for $d=\{2,4,6,8,\dots \}$. Finally, under the Majorana–Weyl condition, the free energy of the corresponding truncated Type B theory reproduces half of the anomaly coefficients given in (18), corresponding to a single Majorana–Weyl fermion.

For the odd d case, the minimal Type B theories dual to the Majorana (or symplectic Majorana) projections again have unexpected values of their one-loop free energies. They are listed in 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 (14), these values are always linear combinations of $\zeta (2k+1)/{\pi }^2$ 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 fundamental of $U\left(N\right)$ (or $O\left(N\right)$), with action:

$$\begin{array}{c}\hfill S=\int d^{d}x\sum _{i=1}^N\left[\left({\partial }_{\mu }{\varphi }_i^{*}\right)\left({\partial }^{\mu }{\varphi }^i\right)+{\overline{\psi }}_i(\cancel{\partial }){\psi }^i\right].\end{array}$$

(19)

When we impose the $U\left(N\right)$ singlet constraint, the spectrum of single trace operators contains not only the bilinears in $\varphi$ and $\psi$, which are the same as discussed above, but also fermionic operators of the form:

$$\begin{array}{c}\hfill {\Psi }_{{\mu }_1\cdots {\mu }_s}={\overline{\psi }}^i{\partial }_{({\mu }_1}\cdots {\partial }_{{\mu }_{s-\frac{1}{2}})}{\varphi }^i+\dots ,\mathrm{where}s=\frac{1}{2},\frac{3}{2},\cdots ,\end{array}$$

(20)

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 $s=3/2,5/2,\dots$, 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 (19) is not supersymmetric. One may also truncate the model to the $O\left(N\right)/USp\left(N\right)$ by imposing suitable reality conditions. There is no qualitative difference in the spectrum of the half-integer operators in the truncated models, with the only quantitative difference being a doubling of the degrees of freedom of each half-integer spin particle when going from $O\left(N\right)/USp\left(N\right)$ to $U\left(N\right)$ in the dual CFT.

The partition function for the Type AB theory is,

$$\begin{array}{c}\hfill Z=e^{-F}=e^{-\frac{1}{G_N}{F}^{\left(0\right)}+F^{\left(1\right)}+G_N{F}^{\left(2\right)}+\cdots },\mathrm{where}{F}^{\left(1\right)}=F_f^{\left(1\right)}+F_b^{\left(1\right)},\end{array}$$

(21)

with $F_b$ being for the contributions from bosonic higher-spin fields, which arise from purely Type A and purely Type B contributions, and $F_f^{\left(1\right)}$ is the contribution of the HS fermions dual to (20). Up to one-loop level, the bosonic and fermionic contributions are decoupled, as indicated in (21). A similar decoupling of the Casimir energy occurs at the one-loop level, i.e., $E_c^{\left(1\right)}=E_{c,f}^{\left(1\right)}+E_{c,b}^{\left(1\right)}$.

Our calculations for the Euclidean-AdS higher spin theory shows that $F_f^{\left(1\right)}=0$ at the one-loop level for both $U\left(N\right)/O\left(N\right)$ theories for all d. Similarly, the Casimir energies are found to vanish: $E_{c,f}=0$. In even d, from our results on the Type B theories and the earlier results on Vasiliev Type A theories, we see that $F_b^{\left(1\right)}=0$ for the non-minimal Type AB theory, and this suggests that Type AB theories at one-loop have vanishing $F^{\left(1\right)}$. For odd d, $F^{\left(1\right)}$ 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 $p=(\frac{d}{2}-1)$.4 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 $H_{\mu \nu \rho }$. 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 [54]. The non-minimal theory is obtained by taking N complex $(d/2-1)$-form gauge fields A, and imposing a $U\left(N\right)$ singlet constraint. One may further truncate these models by taking real fields and $O\left(N\right)$ 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 ${\ast }^2=+1$ in $d=4m+2$ and ${\ast }^2=-1$ in $d=4m$, where ∗ is the Hodge-dual operator, one can impose the self-duality condition $F=\ast F$ only in $d=4m+2$ (for m integer); this can be done both for real ($O\left(N\right)$) and complex ($U\left(N\right)$) fields. In $d=4m$, and only in the non-minimal case with N complex fields, one can impose the self-duality condition $F=i\ast F$. Decomposing $F=F_1+i{F}_2$ into its real and imaginary parts, this condition implies $F_1=-\ast F_2$, and self-dual and anti-self-dual 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 $U\left(N\right)$ singlet constraint. The spectrum of the the single trace operators arising from the tensor product ${\overline{F}}_{\mu \nu }^i\otimes F_i^{\rho \sigma }$ can be found to be [17,54]:

$$\begin{array}{c}{(2;1,1)}_c\otimes {(2;1,1)}_c=2(4;0,0)+{(4;1,1)}_c+{(4;2,2)}_c\hfill \\ +2\sum _{s=2}^{\infty }(s+2;s,0)+\sum _{s=3}^{\infty }{(s+2;s,2)}_c\hfill \end{array}$$

(22)

where we use the notation ${(2;1,1)}_c=(2;1,1)+(2;1,-1)$, corresponding to the sum of the self-dual and anti self-dual 2-form field strength with $\Delta =2$, and similarly for the representations appearing on the right-hand side. Note that we use $SO\left(4\right)$ notations $[m_1,m_2]$ to specify the representation. The operators in the first line are dual to matter fields in AdS${}_5$ in the corresponding representations, while the second line corresponds to massless HS gauge fields. Note that a novel feature compared to Type A and Type B is the presence of mixed symmetry representations with two boxes in the second row:

(23)

Imposing a reality condition and $O\left(N\right)$ singlet constraint, one obtains the minimal spectrum [17]:

$$\begin{array}{c}{\left[{(2;1,1)}_c\otimes {(2;1,1)}_c\right]}_{\mathrm{symm}}=2(4;0,0)+{(4;2,2)}_c\hfill \\ +{\sum }_{s=2}^{\infty }(s+2;s,0)+{\sum }_{s=4,6,\dots }^{\infty }{(s+2;s,2)}_c.\hfill \end{array}$$

(24)

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 representations:

(25)

Our results for the one-loop calculations in type C theories are summarized in Table 1. We find that the non-minimal $U\left(N\right)$ theories have non-zero one-loop contributions, unlike the Type A and type B theories (in even d). The results can be grouped into two subclasses depending on the spacetime dimension, namely those in $d=4m$ or in $d=4m+2$, where m is an integer. In the minimal type C theories with $O\left(N\right)$ singlet constraint, we find that for all $d=4m$ the identification of the bulk coupling constant is $1/G_N\sim N-2$, while in $d=4m+2$, the bulk one-loop free energy vanishes, and therefore no shift is required. In the self-dual $U\left(N\right)/O\left(N\right)$ theories, the one-loop free energy does not vanish, but can be accounted for by half-integer shifts $1/G_N\sim N\pm 1/2$, as mentioned earlier. We find that all of these results are consistent with calculations of Casimir energies in thermal AdS space, which are collected in the Appendix.

3. Matching the Sphere Free Energy

3.1. 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 ($s\ge 1$) totally symmetric gauge field to the bulk partition function is obtained as [59,60,61]:

$$\begin{array}{c}\hfill Z_s=\frac{{\left[{\det }_{s-1}^{STT}\left(-{\nabla }^2+(s+d-2)(s-1)\right)\right]}^{\frac{1}{2}}}{{\left[{\det }_s^{STT}\left(-{\nabla }^2+(s+d-2)(s-2)-s\right)\right]}^{\frac{1}{2}}}\end{array}$$

(26)

where the label $STT$ 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 $-{\nabla }^2+{\kappa }^2$, the dual conformal dimension is given by:

$$\Delta (\Delta -d)-s={\kappa }^2.$$

(27)

For the values of $\kappa$ in (26), one finds for the physical spin s field in the denominator5

$${\Delta }^{\mathrm{ph}}=s+d-2$$

(28)

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 (26) is:

$${\Delta }^{\mathrm{ph}}=s+d-1.$$

(29)

From CFT point of view, this is the dimension of the divergence $\partial ·J_s$, 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 (26) can be computed using the heat kernel, or equivalently spectral zeta functions techniques.6 The spectral zeta function for a differential operator on a compact space with discrete eigenvalues ${\lambda }_n$ and degeneracy $d_n$ is defined as

$$\begin{array}{c}\hfill \zeta \left(z\right)=\sum _n{d}_n{\lambda }_n^{-z}.\end{array}$$

(30)

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 labeled by the representation ${\alpha }_s=[s,m_2,m_3,\dots ]$ of $SO\left(d\right)$7, 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 ${\alpha }_s=[s,0,0,\dots ,0]$). For a given representation ${\alpha }_s$, the spectral zeta function takes the form:

$$\begin{array}{cc}\hfill {\zeta }_{(\Delta ;{\alpha }_s)}\left(z\right)& =\frac{\mathrm{vol}\left({\mathrm{AdS}}_{d+1}\right)}{\mathrm{vol}\left(S^d\right)}\frac{2^{d-1}}{\pi }{g}_{{\alpha }_s}{\int }_0^{\infty }du\frac{{\mu }_{{\alpha }_s}\left(u\right)}{{\left[u^2+{\left(\Delta -\frac{d}{2}\right)}^2\right]}^z},\hfill \end{array}$$

(31)

where ${\mu }_{{\alpha }_s}\left(u\right)$ is the spectral density of the eigenvalues, which will be given shortly, and $g_{{\alpha }_s}$ is the dimension of the representation ${\alpha }_s$ (see Equations (43) and (44) below). The denominator corresponds to the eigenvalues of the kinetic operator, and $\Delta$ is the dimension of the dual CFT operator.8 The regularized volume of AdS is given explicitly by [63,64,65]:

$$\begin{array}{c}\hfill \mathrm{vol}(\mathrm{AdS}{S}_{d+1})=\left\{\begin{array}{cc}{\pi }^{d/2}\Gamma (-\frac{d}{2}),\hfill & d\mathrm{odd},\hfill \\ \frac{2{(-\pi )}^{d/2}}{\Gamma (1+\frac{d}{2})}logR,\hfill & d\mathrm{even},\hfill \end{array}\right.\end{array}$$

(32)

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:

$$\begin{array}{c}\hfill \mathrm{vol}\left(S^d\right)=\frac{2{\pi }^{(d+1)/2}}{\Gamma \left[\right(d+1)/2]}.\end{array}$$

(33)

Once the spectral zeta function is known, the contribution of the field labeled by $(\Delta ;{\alpha }_s)$ to the bulk free energy is obtained as:

$$F_{(\Delta ;{\alpha }_s)}^{\left(1\right)}=\sigma \left[-\frac{1}{2}{\zeta }_{(\Delta ;{\alpha }_s)}^{\prime }\left(0\right)-{\zeta }_{(\Delta ;{\alpha }_s)}\left(0\right)log(\ell \Lambda )\right],$$

(34)

where $\sigma =+1$ or $-1$ depending on whether the field is bosonic or fermionic. Here ℓ is the AdS curvature, which we will set to one henceforth, and $\Lambda$ is a UVcut-off. In general, the coefficient of the logarithmic divergence ${\zeta }_{(\Delta ;{\alpha }_s)}\left(0\right)$ vanishes for each ${\alpha }_s$ in even dimension d, but it is non-zero for odd d.

When the dimension $\Delta =s+d-2$, the field labeled by ${\alpha }_s$ is a gauge field and one has to subtract the contribution of the corresponding ghosts in the ${\alpha }_{s-1}$ representation.9 We find it convenient to introduce the notation:

$${\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}}=s+d-2;{\alpha }_s)}(z)\equiv {\zeta }_{({\Delta }^{\mathrm{ph}};{\alpha }_s)}(z)-{\zeta }_{({\Delta }^{\mathrm{ph}}+1;{\alpha }_{s-1})}\left(z\right)$$

(35)

to indicate the spectral zeta function of the HS gauge fields in the ${\alpha }_s$ representation, with ghost contribution subtracted. The full one-loop free energy may be then obtained by summing over all representations ${\alpha }_s$ appearing in the spectrum. For instance, in the case of the non-minimal Type A theory, we may define the “total” spectral zeta function:

$${\zeta }_{\mathrm{type}\mathrm{A}}^{\mathrm{HS}}\left(z\right)={\zeta }_{(d-2;[0,\dots ,0\left]\right)}\left(z\right)+\sum _{s=1}^{\infty }{\mathfrak{Z}}_{(s+d-2;[s,0,\dots ,0\left]\right)}\left(z\right)$$

(36)

from which we can obtain the full one-loop free energy:

$$F_{\mathrm{type}\mathrm{A}}^{\left(1\right)}=\left[-\frac{1}{2}{({\zeta }_{\mathrm{type}\mathrm{A}}^{\mathrm{HS}})}^{\prime }\left(0\right)-\frac{1}{2}{\zeta }_{\mathrm{type}\mathrm{A}}^{\mathrm{HS}}\left(0\right)log({\ell }^2{\Lambda }^2)\right].$$

(37)

Similarly, one can obtain ${\zeta }_{\mathrm{total}}^{\mathrm{HS}}\left(z\right)$ 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.

3.1.1. The Spectral Density for Arbitrary Representation

A general formula for the spectral density for a field labeled by the representation $\alpha =[m_1,m_2,\dots ]$ was given in [52], and we summarize their result below.

In AdS${}_{d+1}$, arranging the weights for the irreps of $SO\left(d\right)$ as $m_1\ge m_2\ge \cdots \ge \left|m_p\right|$, where $p=\frac{d-1}{2}$ for odd d and $p=\frac{d}{2}$ for even d, we may define:

$$\begin{array}{cc}\hfill {\ell }_j& =m_{p-j+1}+j-1,\mathrm{for}d=\mathrm{even},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\ell }_j& =m_{p-j+1}+j-\frac{1}{2},\mathrm{for}d=\mathrm{odd}.\hfill \end{array}$$

(39)

In terms of these, the spectral density takes the form of:

$$\begin{array}{cc}\hfill {\mu }_{\alpha }\left(u\right)& =\frac{\pi }{{\left(2^{d-1}\Gamma \left(\frac{d+1}{2}\right)\right)}^2}\prod _{j=1}^p(u^2+{\ell }_j^2),\mathrm{for}d=\mathrm{even},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\mu }_{\alpha }\left(u\right)& =\frac{\pi }{{\left(2^{d-1}\Gamma \left(\frac{d+1}{2}\right)\right)}^2}f\left(u\right)u\prod _{j=1}^p(u^2+{\ell }_j^2),\mathrm{for}d=\mathrm{odd},\hfill \end{array}$$

(41)

where:

$$\begin{array}{c}\hfill f\left(u\right)=\left\{\begin{array}{c}tanh\left(\pi u\right),{\ell }_j=\text{half-integer},\hfill \\ coth\left(\pi u\right),{\ell }_j=\mathrm{integer}.\hfill \end{array}\right.\end{array}$$

(42)

The pre-factor of $\frac{\pi }{{\left(2^{d-1}\Gamma \left(\frac{d+1}{2}\right)\right)}^2}$ 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 $g_{\alpha }$ is equal to the dimension of the corresponding representation of $SO\left(d\right)$, and is given by [66]:

$$\begin{array}{ccc}\hfill g_{{\alpha }_s}& ={\prod }_{1\le i<j\le p}\frac{m_i-m_j+j-i}{j-i}{\prod }_{1\le i<j\le p}\frac{m_i+m_j+2p-i-j}{2p-i-j},\hfill & \mathrm{for}d=2p,\hfill \end{array}$$

(43)

and:

$$\begin{array}{cc}\hfill g_{{\alpha }_s}& ={\prod }_{1\le i\le p}\frac{2m_i+2p-2i+1}{2p-2i+1}{\prod }_{1\le i<j\le p}\frac{m_i-m_j+j-i}{j-i}\hfill \\ & \times {\prod }_{1\le i<j\le p}\frac{m_i+m_j+2p-i-j+1}{2p-i-j+1},\hfill & \mathrm{for}d=2p+1,\hfill \end{array}$$

(44)

where $\alpha =[m_1,\dots ,m_p]$. 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 $j\ne 1$, $m_j=0$. This gives us:

$$\begin{array}{cc}\hfill {\mu }_{[s,0,\dots ,0]}\left(u\right)& =\frac{\pi }{{\left(2^{d-1}\Gamma \left(\frac{d+1}{2}\right)\right)}^2}\left[u^2+{\left(s+\frac{d-2}{2}\right)}^2\right]{\left|\frac{\Gamma \left(iu+\frac{d-2}{2}\right)}{\Gamma \left(iu\right)}\right|}^2\hfill \\ & =\left\{\begin{array}{cc}{\displaystyle \frac{\pi }{{\left(2^{d-1}\Gamma \left(\frac{d+1}{2}\right)\right)}^2}\left[u^2+{\left(s+\frac{d-2}{2}\right)}^2\right]\prod _{j=0}^{(d-4)/2}(u^2+j^2),}\hfill & d=\mathrm{even},\hfill \\ {\displaystyle \frac{\pi }{{\left(2^{d-1}\Gamma \left(\frac{d+1}{2}\right)\right)}^2}utanh\left(\pi u\right)\left[u^2+{\left(s+\frac{d-2}{2}\right)}^2\right]\prod _{j=0}^{(d-5)/2}\left[u^2+{(j+\frac{1}{2})}^2\right],}\hfill & d=\mathrm{odd}.\hfill \end{array}\right.\hfill \end{array}$$

(45)

and:

$$\begin{array}{c}\hfill g_{[s,0,\dots ,0]}=\frac{(2s+d-2)(s+d-3)!}{(d-2)!s!},d\ge 3.\end{array}$$

(46)

The results agree with the formulas derived in [49] and used in [27].

In Type AB theories, we need the spectral density for fermion fields in the $\alpha =[s,1/2,1/2,\dots ,1/2]$ representation. We find that the above general formulas for even and odd d can be expressed in the compact form valid for all d:

$$\begin{array}{c}\hfill {\mu }_{[s,\frac{1}{2},\dots ,\frac{1}{2}]}\left(u\right)=\frac{\pi }{{\left(2^{d-1}\Gamma \left(\frac{d+1}{2}\right)\right)}^2}\left[u^2+{\left(s+\frac{d-2}{2}\right)}^2\right]{\left|\frac{\Gamma \left(iu+\frac{d-1}{2}\right)}{\Gamma (iu+\frac{1}{2})}\right|}^2,\end{array}$$

(47)

and:

$$\begin{array}{c}\hfill g_{[s,\frac{1}{2},\dots ,\frac{1}{2}]}=\frac{(s-\frac{5}{2}+d)!}{(s-\frac{1}{2})!(d-2)!}{n}_F\left(d\right),n_F\left(d\right)=\left\{\begin{array}{cc}{2}^{\frac{d-2}{2}},\hfill & \mathrm{if}d=\mathrm{even},\hfill \\ 2^{\frac{d-1}{2}},\hfill & \mathrm{if}d=\mathrm{odd}.\hfill \end{array}\right.\end{array}$$

(48)

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.

3.2. Calculations in Even d

3.2.1. Type B Theories

Spectrum

The non-minimal Type B higher spin theory, which is conjectured to be dual to the $U\left(N\right)$ singlet sector of the free Dirac fermion theory, contains towers of mixed symmetry gauge fields of all integer spins. From the spectrum given in (8), we obtain the total spectral zeta function:

$$\begin{array}{cc}\hfill {\zeta }_{\mathrm{type}\mathrm{B}}^{\mathrm{HS}}\left(z\right)& =2{\zeta }_{(\Delta =d-1;[0,0,\dots ,0\left]\right)}\left(z\right)\hfill \\ & +2{\sum }_{s=1}^{\infty }[{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1,1,\dots ,1,0])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1,1,\dots ,1,0,0])}\left(z\right)+\dots +{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1,0,\dots ,0])}\left(z\right)\hfill \\ & +{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,0,0,\dots ,0])}\left(z\right)]\hfill \\ & +{\sum }_{s=1}^{\infty }\left[{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1,1,\dots ,1,1])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1,1,\dots ,1,1,-1])}\left(z\right)\right].\hfill \end{array}$$

(49)

In the third line of (49), the representations $[s,1,1,\dots ,1,1]$ and $[s,1,1,\dots ,1,-1]$ give the self-dual and anti-self-dual parts of the corresponding fields. At the level of the spectral $\zeta$ functions, they yield equal contributions.10 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:

$${\zeta }_{\mathrm{type}\mathrm{B}}^{\mathrm{HS}}\left(z\right)=\mathcal{O}\left(z^2\right),$$

(50)

and consequently the one-loop free energy in the non-minimal Type B theory in even d exactly vanishes:

$$F_{\mathrm{type}\mathrm{B}}^{\left(1\right)}=0.$$

(51)

There are various truncations to the non-minimal Type B theory that results in the Weyl, Majorana and Majorana–Weyl projections on the free fermionic CFT. While the Weyl projection can be applied in all even dimensions d, the Majorana projection can be applied in dimensions $d=2,3,4,8,9$ (mod 8), and the Majorana–Weyl projection only in dimensions $d=2$ (mod 8). An interesting example is $d=10$ (AdS${}_{11}$), where we can consider all four types of Type B theories.

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 [54] for the product of chiral fermion representations, we find11

$$\begin{array}{cc}\hfill {\zeta }_{\mathrm{type}\mathrm{B}\mathrm{Weyl}}^{\mathrm{HS}}\left(z\right)& =\left\{\begin{array}{cc}{\displaystyle \sum _{s=1}^{\infty }\left[{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,0,0,\dots ,0])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1,1,0,\dots ,0])}\left(z\right)+\dots +{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1,1,\dots ,1])}\left(z\right)\right],}\hfill & \mathrm{for}d=4m+2,\hfill \\ {\displaystyle \sum _{s=1}^{\infty }\left[{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,0,0,\dots ,0])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1,1,0,\dots ,0])}\left(z\right)+\dots +{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1,1,\dots ,1,0])}\left(z\right)\right],}\hfill & \mathrm{for}d=4m,\hfill \end{array}\right.\hfill \\ & =\left\{\begin{array}{cc}{\displaystyle \sum _{s=1}^{\infty }\sum _{\begin{array}{l}{t}_i\ge 0\\ t_i\ge t_{i+1}\end{array}}^1{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_1,\dots ,t_m,t_m])}\left(z\right),}\hfill & \mathrm{for}d=4m+2,\hfill \\ {\displaystyle \sum _{s=1}^{\infty }\sum _{\begin{array}{l}{t}_i\ge 0\\ t_i\ge t_{i+1}\end{array}}^1{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_1,\dots ,t_{m-1},t_{m-1},0])}\left(z\right),}\hfill & \mathrm{for}d=4m,\hfill \end{array}\right.\hfill \end{array}$$

(52)

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:

$${\zeta }_{\mathrm{type}\mathrm{B}\mathrm{Weyl}}^{\mathrm{HS}}\left(z\right)=\mathcal{O}\left(z^2\right),$$

(53)

and so

$$F_{\mathrm{type}\mathrm{B}\mathrm{Weyl}}^{\left(1\right)}=0.$$

(54)

Minimal Theory (Majorana projection) The Majorana condition $\overline{\psi }={\psi }^T\mathcal{C}$, where $\mathcal{C}$ is the charge conjugation matrix, can be imposed in $d=2,3,4,8,9$ (mod 8), see for instance [67]. In these dimensions, we can consider the theory of N free Majorana fermions and impose an $O\left(N\right)$ singlet constraint. In $d=6$ (mod 8), provided one has an even number N of fermions, one can impose instead a symplectic Majorana condition ${\overline{\psi }}^i={\psi }_j^T\mathcal{C}{\Omega }^{ij}$, where $\mathcal{C}$ is the charge conjugation matrix and ${\Omega }^{ij}$ the antisymmetric symplectic metric. In this case, we consider the theory of N free symplectic Majoranas with a $USp\left(N\right)$ 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:

$$\begin{array}{c}\hfill J_{{\mu }_1\cdots {\mu }_s,{\nu }_1\cdots {\nu }_{n-1}}\sim {\overline{\psi }}_i{({\Gamma }^{\left(n\right)})}_{{\nu }_1\cdots {\nu }_{n-1}({\mu }_1}{\partial }_{{\mu }_2}{\partial }_{{\mu }_3}\cdots {\partial }_{{\mu }_s)}{\psi }^i+\dots \end{array}$$

(55)

where $n=0,\dots ,\frac{d}{2}-1$, and ${\Gamma }^{\left(n\right)}$ is the antisymmetrized product of n gamma matrices. For Majorana fermions, we have $\overline{\psi }={\psi }^T\mathcal{C}$, and so the operators are projected out or kept depending on whether $\mathcal{C}{\Gamma }^{\left(n\right)}$ is symmetric or antisymmetric. If $\mathcal{C}{\Gamma }^{\left(n\right)}$ 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 (55), the non-minimal Type B theories in even d include two scalars $J_0={\overline{\psi }}_i{\psi }^i$ and ${\tilde{J}}_0={\overline{\psi }}_i{\gamma }_{*}{\psi }^i$, where ${\gamma }_{*}\sim {\Gamma }^{\left(d\right)}$ is the chirality matrix. When $\mathcal{C}$ is symmetric, $J_0$ is projected out, and when $\mathcal{C}{\gamma }_{*}$ is symmetric, ${\tilde{J}}_0$ is projected out.12

For instance, in $d=4$, the non-minimal theory contains the operators given in (10)–(12). In $d=4$, one has that both $\mathcal{C}$ and $\mathcal{C}{\gamma }_5$ are antisymmetric, so both scalars in (10) are retained. Then, one has that $\mathcal{C}{\gamma }^{\mu }$ is symmetric while $\mathcal{C}{\gamma }^{\mu }{\gamma }^5$ antisymmetric, and so we keep the first tower in (11) for even s and the other tower for odd s: together, they make up a single tower in the $[s,0]$ representation with all integer spins. Finally, $\mathcal{C}{\Gamma }_{\mu \nu }$ is symmetric, so we keep the mixed symmetry fields (12) with an odd number of derivatives, i.e., the spectrum contains the representations ${[s,1]}_c=[s,1]+[s,-1]$ for all even s.

Higher dimensions can be analyzed similarly, using the symmetry/antisymmetry properties of $\mathcal{C}{\Gamma }^{\left(n\right)}$ in various d [67]. The results are summarized in Table 3. One finds that under the Majorana projections the operators with the “heaviest” weight ${[s,1,1,\dots ,1]}_c$ always form a tower containing all even s. The next representation $[s,1,\dots ,1,0]$ form a tower of all integer s. Then, $[s,1,\dots ,1,0,0]$ appears in two towers of all odd s. Finally, $[s,1,\dots ,1,0,0,0]$ form a tower of all integer spins, after which this cycle repeats. The number of scalars with $\Delta =d-1$ 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 $O\left(N\right)$ Majorana theories is:

$$\begin{array}{c}{\zeta }_{\mathrm{type}\mathrm{B}\mathrm{Maj}.}^{\mathrm{HS}}\left(z\right)=\chi \left(d\right){\zeta }_{(d-1;[0,0,\dots ,0\left]\right)}\left(z\right)\hfill \\ +\sum _{s=2,4,6,\dots }^{\infty }\sum _{\begin{array}{c}{t}_i\ge t_{i+1}\\ t_i\ge 0\\ {\sum }_w{t}_w=w\left(\mathrm{mod}4\right)\end{array}}^1\left({\mathfrak{Z}}_{(\Delta ;[s,t_1,t_2,\dots ,t_{w-1},t_w])}\left(z\right)+{\mathfrak{Z}}_{(\Delta ;[s,t_1,t_2,\dots ,t_{w-1},-t_w])}\left(z\right)\right)\hfill \\ +\sum _{s=1,2,3,\dots }\sum _{\begin{array}{c}{t}_i\ge t_{i+1}\\ t_i\ge 0\\ {\sum }_w{t}_w=(w-1)\left(\mathrm{mod}4\right)\end{array}}^1{\mathfrak{Z}}_{(\Delta ;[s,t_1,t_2,\dots ,t_{w-1},t_w])}\left(z\right)\hfill \\ +\sum _{s=1,2,3,\dots }\sum _{\begin{array}{c}{t}_i\ge t_{i+1}\\ t_i\ge 0\\ {\sum }_w{t}_w=(w-3)\left(\mathrm{mod}4\right)\end{array}}^1{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_2,\dots ,t_{w-1},t_w])}\left(z\right)\hfill \\ +\sum _{s=1,3,5,\dots }\sum _{\begin{array}{c}{t}_i\ge t_{i+1}\\ t_i\ge 0\\ {\sum }_w{t}_w=(w-2)\left(\mathrm{mod}4\right)\end{array}}^1\left({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_2,\dots ,t_{w-1},t_w])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_2,\dots ,t_{w-1},-t_w])}\left(z\right)\right)\hfill \end{array}$$

(56)

where $\chi \left(d\right)=1,2,0$ when $d=0,2,4(mod8)$ 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:

$$F_{\mathrm{type}\mathrm{B}\mathrm{Maj}.}^{\left(1\right)}=a_{f}logR$$

(57)

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 (18). As explained earlier, this is consistent with the duality, provided $G_N^{\mathrm{type}\mathrm{B}\mathrm{Maj}.}\sim 1/(N-1)$. As mentioned above, in $d=6$ (mod 8), i.e., AdS${}_{7\left(\mathrm{mod}8\right)}$, we should impose a symplectic Majorana condition and consider the $USp\left(N\right)$ invariant operators. In terms of the operators (55), since $\overline{\psi }={\psi }^T\mathcal{C}\Omega$ with $\Omega$ antisymmetric, all this means is that now odd spins are projected out when $\mathcal{C}{\Gamma }^{\left(n\right)}$ is antisymmetric, and even spins are projected out when $\mathcal{C}{\Gamma }^{\left(n\right)}$ is symmetric. Similarly, the scalar operators ${\overline{\psi }}_i{\psi }^i$ and ${\psi }_i{\gamma }_{*}{\psi }^i$ are now projected out when $\mathcal{C}$ and $\mathcal{C}{\gamma }_{*}$ are antisymmetric, respectively. In $d=6$ (mod 8), one has that $\mathcal{C}$ is symmetric and $\mathcal{C}{\gamma }_{*}$ is antisymmetric, so we retain a single scalar field. On the other hand, $\mathcal{C}{\gamma }_{\mu }$ and $\mathcal{C}{\gamma }_{\mu }{\gamma }_{*}$ are both antisymmetric, and so we have two towers of totally symmetric representations of all even s.13 The projection of the mixed symmetry representations can be deduced similarly. The total spectral zeta function is given by the formula:

$$\begin{array}{c}{\zeta }_{\mathrm{type}\mathrm{B}\mathrm{Symp}.\mathrm{Maj}.}^{\mathrm{HS}}\left(z\right)={\zeta }_{(d-1;[0,0,\dots ,0\left]\right)}\left(z\right)\hfill \\ +\sum _{s=1,3,5,\dots }^{\infty }\sum _{\begin{array}{c}{t}_i\ge t_{i+1}\\ t_i\ge 0\\ {\sum }_w{t}_w=w\left(\mathrm{mod}4\right)\end{array}}^1\left({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_2,\dots ,t_{w-1},t_w])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_2,\dots ,t_{w-1},-t_w])}\left(z\right)\right)\hfill \\ +\sum _{s=1,2,3,\dots }\sum _{\begin{array}{c}{t}_i\ge t_{i+1}\\ t_i\ge 0\\ {\sum }_w{t}_w=(w-1)\left(\mathrm{mod}4\right)\end{array}}^1{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_2,\dots ,t_{w-1},t_w])}\left(z\right)\hfill \\ +\sum _{s=1,2,3,\dots }\sum _{\begin{array}{c}{t}_i\ge t_{i+1}\\ t_i\ge 0\\ {\sum }_w{t}_w=(w-3)\left(\mathrm{mod}4\right)\end{array}}^1{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_2,\dots ,t_{w-1},t_w])}\left(z\right)\hfill \\ +\sum _{s=2,4,6,\dots }\sum _{\begin{array}{c}{t}_i\ge t_{i+1}\\ t_i\ge 0\\ {\sum }_w{t}_w=(w-2)\left(\mathrm{mod}4\right)\end{array}}^1\left({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_2,\dots ,t_{w-1},t_w])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_2,\dots ,t_{w-1},-t_w])}\left(z\right)\right)\hfill \end{array}$$

(58)

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:

$$F_{\mathrm{type}\mathrm{B}\mathrm{sympl}.\mathrm{Maj}.}^{\left(1\right)}=-a_{f}logR,$$

(59)

i.e., the opposite sign compared to (57). This is consistent with the duality, provided $G_N^{\mathrm{type}\mathrm{B}\mathrm{sympl}.\mathrm{Maj}.}\sim 1/(N+1)$.

Majorana–Weyl Projection

Finally the spectra arising from the Majorana–Weyl projection, which can be imposed in dimensions $d=2(mod8)$, is the overlap of the individual Majorana and Weyl projection. The resulting spectrum yields the total zeta function:

$$\begin{array}{cc}\hfill {\zeta }_{\mathrm{Type}\mathrm{B}\mathrm{MW}}^{\mathrm{HS}}\left(z\right)& =\sum _{s=2,4,6,\dots }^{\infty }\sum _{\begin{array}{c}{t}_i\ge t_{i+1}\\ t_i\ge 0\\ {\sum }_w{t}_w=w\left(\mathrm{mod}4\right)\end{array}}^1{\mathfrak{Z}}_{(\Delta ;[s,t_1,t_2,\dots ,t_{w-1},t_w])}\left(z\right)\hfill \\ & +\sum _{s=1,3,5,\dots }\sum _{\begin{array}{c}{t}_i\ge t_{i+1}\\ t_i\ge 0\\ {\sum }_w{t}_w=(w-2)\left(\mathrm{mod}4\right)\end{array}}^1{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_2,\dots ,t_{w-1},t_w])}\left(z\right).\hfill \end{array}$$

(60)

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:

$$F_{\mathrm{type}\mathrm{B}\mathrm{MW}}^{\left(1\right)}=\frac{1}{2}{a}_{f}logR$$

(61)

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 find a minimal theory with a totally symmetric tower $[s,0,0]$ of all even spins, and a tower of the mixed symmetry fields $[s,1,1]$ of all odd spins (see Table 4). In this case (and similarly for higher $d=14,22,\dots$), we find:

$$F_{\mathrm{type}\mathrm{B}\mathrm{SMW}}^{\left(1\right)}=-\frac{1}{2}{a}_{f}logR.$$

(62)

Since the a-anomaly of the boundary free theory of N symplectic Majorana–Weyl fermions is $a_{N\mathrm{SMW}}=\frac{N}{2}{a}_f$, this result is consistent with a shift $G_N^{\mathrm{type}\mathrm{B}\mathrm{SMW}}\sim 1/(N+1)$.

Sample Calculations

AdS${}_{\mathbf{5}}$ Following (49) for the non-minimal Type B theory,

$$\begin{array}{c}\hfill {\zeta }_{\mathrm{type}\mathrm{B}}^{\mathrm{HS}}\left(z\right)=2{\zeta }_{(3;[0,0\left]\right)}\left(z\right)+\sum _{s=1}^{\infty }\left({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,-1])}\left(z\right)\right)+2\sum _{s=1}^{\infty }{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,0])}\left(z\right).\end{array}$$

(63)

We see that there are two weights to consider in AdS${}_5$, corresponding to $[s,0]$ and $[s,\pm 1]$ representation. Using (31) and (40), we have:

$$\begin{array}{cc}\hfill \frac{{\zeta }_{(\Delta ;[s,1\left]\right)}\left(z\right)}{logR}& ={\pi }^2{\int }_0^{\infty }du\frac{\left(u^2+1\right)\left[{(s+1)}^2+u^2\right]}{12{\pi }^3}\frac{s(2+s)}{{\left[u^2+{(\Delta -2)}^2\right]}^z}\hfill \\ & =\frac{s(s+2)}{8\sqrt{\pi }\Gamma \left(z\right)}[\frac{{(s+1)}^2{(\Delta -2)}^{1-2z}\Gamma \left(z-\frac{1}{2}\right)}{3}+\frac{\left({(s+1)}^2+1\right){(\Delta -2)}^{3-2z}\Gamma \left(z-\frac{3}{2}\right)}{6}\hfill \\ & +\frac{{(\Delta -2)}^{5-2z}\Gamma \left(z-\frac{5}{2}\right)}{4}]\hfill \end{array}$$

(64)

In the above, we made use of the formula,

$$\begin{array}{c}\hfill {\int }_0^{\infty }du\frac{u^{2p}}{{\left[u^2+{\nu }^2\right]}^z}={\nu }^{2p+1-2z}{\int }_0^{\infty }du\frac{u^{2p}}{{\left[u^2+1\right]}^z}={\nu }^{2p+1-2z}\frac{\Gamma (p+\frac{1}{2})}{2}\frac{\Gamma (z-p-\frac{1}{2})}{\Gamma \left(z\right)},\end{array}$$

(65)

to go from the first to the second line.

In our regularization scheme, we sum over the physical modes separately from the ghost modes. We introduce $\zeta (k,\nu )$, the Hurwitz zeta function (analytically extended to the entire complex plane), which is given by:

$$\begin{array}{c}\hfill \zeta (k,\nu )=\sum _{s=0}^{\infty }\frac{1}{{(s+\nu )}^k}.\end{array}$$

(66)

Then, using ${\Delta }^{\mathrm{gh}}=s+3$,

$$\begin{array}{c}\frac{1}{logR}{\sum }_{s=1}^{\infty }{\zeta }_{({\Delta }^{\mathrm{gh}};[s-1,1])}\left(z\right)\hfill \\ =\frac{1}{96\sqrt{\pi }\Gamma \left(z\right)}[2\zeta (2z-7)\Gamma \left(z-\frac{3}{2}\right)+3\zeta (2z-7)\Gamma \left(z-\frac{5}{2}\right)+8\zeta (2z-6)\Gamma \left(z-\frac{3}{2}\right)\hfill \\ +6\zeta (2z-6)\Gamma \left(z-\frac{5}{2}\right)+4\zeta (2z-5)\Gamma \left(z-\frac{1}{2}\right)+12\zeta (2z-5)\Gamma \left(z-\frac{3}{2}\right)\hfill \\ +16\zeta (2z-4)\Gamma \left(z-\frac{1}{2}\right)+8\zeta (2z-4)\Gamma \left(z-\frac{3}{2}\right)+20\zeta (2z-3)\Gamma \left(z-\frac{1}{2}\right)\hfill \\ +8\zeta (2z-2)\Gamma \left(z-\frac{1}{2}\right)].\hfill \end{array}$$

(67)

Similarly, using ${\Delta }^{\mathrm{ph}}=s+2$,

$$\begin{array}{c}\frac{1}{logR}{\sum }_{s=1}^{\infty }{\zeta }_{({\Delta }^{\mathrm{ph}};[s,1])}\left(z\right)\hfill \\ =\frac{1}{96\sqrt{\pi }\Gamma \left(z\right)}\{\Gamma \left(z-\frac{5}{2},1\right)\left[6\zeta (2z-6,1)+3\zeta (2z-7,1)\right]\hfill \\ +\Gamma \left(z-\frac{3}{2}\right)\left[12\zeta (2z-5,1)+2\zeta (2z-7,1)+8\zeta (2z-6,1)\Gamma +8\zeta (2z-4,1)\right]\hfill \\ +\left(z-\frac{1}{2}\right)\left[16\zeta (2z-4,1)\Gamma +8\zeta (2z-2,1)+20\zeta (2z-3,1)+4\zeta (2z-5,1)\right]\}\hfill \end{array}$$

(68)

Putting (67) and (68) together,

$$\begin{array}{c}\frac{1}{logR}{\sum }_{s=1}^{\infty }\left[{\zeta }_{({\Delta }^{\mathrm{ph}};[s,1])}\left(z\right)-{\zeta }_{({\Delta }^{\mathrm{gh}};[s-1,1])}\left(z\right)\right]\hfill \\ =-\frac{1}{90}z+\frac{1}{180}\left[56{\zeta }^{\prime }(-6)-160{\zeta }^{\prime }(-4)-120{\zeta }^{\prime }(-2)-2\gamma +3\psi \left(-\frac{5}{2}\right)-5\psi \left(-\frac{3}{2}\right)\right]z^2+\mathcal{O}\left(z^3\right)\hfill \end{array}$$

(69)

where $\psi \left(x\right)$ is the digamma function, $\gamma$ the Euler–Mascheroni constant, and ${\zeta }^{\prime }$ the derivative of the Riemann Zeta function $\zeta \left(z\right)$ (which is related to the Hurwitz Zeta function $\zeta \left(z\right)=\zeta (z,1)$). Similarly, for the totally symmetric representation $[s,0]$, we have:

$$\begin{array}{c}\frac{1}{logR}{\sum }_{s=1}^{\infty }{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,0])}\left(z\right)\hfill \\ =\frac{1}{logR}{\sum }_{s=1}^{\infty }\left[{\zeta }_{({\Delta }^{\mathrm{ph}};[s,0])}\left(z\right)-{\zeta }_{({\Delta }^{\mathrm{gh}};[s-1,0])}\left(z\right)\right]\hfill \\ =\frac{1}{24\sqrt{\pi }\Gamma \left(z\right)}\left[3\zeta \left(2(z-3)\right)\Gamma \left(z-\frac{5}{2}\right)+4(\zeta \left(2(z-3)\right)+\zeta \left(2(z-2)\right))\Gamma \left(z-\frac{3}{2}\right)\right]\hfill \\ =\left(\frac{14}{45}{\zeta }^{\prime }(-6)+\frac{4}{9}{\zeta }^{\prime }(-4)\right)z^2+\mathcal{O}\left(z^3\right)\hfill \end{array}$$

(70)

Finally, for the massive scalar with $\Delta =3$, we have:

$$\begin{array}{cc}\hfill \frac{{\zeta }_{(3;[0,0\left]\right)}\left(z\right)}{logR}& ={\left.{\int }_0^{\infty }du\frac{{(s+1)}^2{u}^2\left[{(s+1)}^2+u^2\right]}{12\pi {\left[{(\Delta -2)}^2+u^2\right]}^z}\right|}_{\begin{array}{c}\Delta =3,\\ s=0\end{array}}\hfill \\ & ={\left[\frac{{(s+1)}^4{(\Delta -2)}^{3-2z}\Gamma \left(z-\frac{3}{2}\right)}{48\sqrt{\pi }\Gamma \left(z\right)}+\frac{{(s+1)}^2{(\Delta -2)}^{5-2z}\Gamma \left(z-\frac{5}{2}\right)}{32\sqrt{\pi }\Gamma \left(z\right)}\right]}_{\begin{array}{c}\Delta =3,\\ s=0\end{array}}\hfill \\ & =\frac{\Gamma \left(z-\frac{3}{2}\right)}{48\sqrt{\pi }\Gamma \left(z\right)}+\frac{\Gamma \left(z-\frac{5}{2}\right)}{32\sqrt{\pi }\Gamma \left(z\right)}\hfill \\ & =\frac{1}{90}z+\frac{1}{180}\left[2\gamma -3\psi \left(-\frac{5}{2}\right)+5\psi \left(-\frac{3}{2}\right)\right]z^2+\mathcal{O}\left(z^3\right).\hfill \end{array}$$

(71)

When summing (69)–(71) together, there are no terms of order $\mathcal{O}\left(z^0\right)$ or $\mathcal{O}\left(z^1\right)$ in the sum, and hence, taking $z\to 0$, we obtain $F^{\left(1\right)}=0$ for the non-minimal Type B theory.

For the Type B minimal theory, we should evaluate, according to (56), the following sum:

$$\begin{array}{c}\hfill {\zeta }_{\mathrm{Total}-\mathrm{Type}\mathrm{B}}^{\mathrm{HS}}\left(z\right)=2{\zeta }_{(3;[0,0\left]\right)}\left(z\right)+\sum _{s=2,4,6,\dots }({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,-1])}\left(z\right))+\sum _{s=1}^{\infty }{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,0])}\left(z\right).\end{array}$$

(72)

The first and third term of the sum have already been evaluated for in the non-minimal theory in (70) and (71) respectively. For the second term,

$$\begin{array}{c}{\sum }_{s=2,4,6,\dots }({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,1])}+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,-1])})\hfill \\ =2{\sum }_{s=2,4,6,\dots }[\frac{{(s+1)}^2(s+2)s^{2-2z}\Gamma \left(z-\frac{1}{2}\right)}{24\sqrt{\pi }\Gamma \left(z\right)}+\frac{(s+2)\left({(s+1)}^2+1\right)s^{4-2z}\Gamma \left(z-\frac{3}{2}\right)}{48\sqrt{\pi }\Gamma \left(z\right)}\hfill \\ +\frac{(s+2)s^{6-2z}\Gamma \left(z-\frac{5}{2}\right)}{32\sqrt{\pi }\Gamma \left(z\right)}]logR\hfill \\ -2{\sum }_{s=2,4,6,\dots }[\frac{(s-1)s^2{(s+1)}^{2-2z}\Gamma \left(z-\frac{1}{2}\right)}{24\sqrt{\pi }\Gamma \left(z\right)}+\frac{(s-1)\left(s^2+1\right){(s+1)}^{4-2z}\Gamma \left(z-\frac{3}{2}\right)}{48\sqrt{\pi }\Gamma \left(z\right)}\hfill \\ +\frac{(s-1){(s+1)}^{6-2z}\Gamma \left(z-\frac{5}{2}\right)}{32\sqrt{\pi }\Gamma \left(z\right)}]logR\hfill \end{array}$$

(73)

To illustrate the zeta-regularization, let us consider the last term,

$$\begin{array}{c}{\sum }_{s=2,4,6,\dots }\frac{(s-1){(s+1)}^{6-2z}\Gamma \left(z-\frac{5}{2}\right)}{32\sqrt{\pi }\Gamma \left(z\right)}\hfill \\ ={\sum }_{s=2,4,6,\dots }\left[\frac{{(s+1)}^{7-2z}\Gamma \left(z-\frac{5}{2}\right)}{32\sqrt{\pi }\Gamma \left(z\right)}-2\frac{{(s+1)}^{6-2z}\Gamma \left(z-\frac{5}{2}\right)}{32\sqrt{\pi }\Gamma \left(z\right)}\right]\hfill \\ ={\sum }_{s=1,2,3,\dots }\left[\frac{2^{7-2z}{(s+\frac{1}{2})}^{7-2z}\Gamma \left(z-\frac{5}{2}\right)}{32\sqrt{\pi }\Gamma \left(z\right)}-2\frac{2^{6-2z}{(s+\frac{1}{2})}^{6-2z}\Gamma \left(z-\frac{5}{2}\right)}{32\sqrt{\pi }\Gamma \left(z\right)}\right]\hfill \\ =\frac{2^{2-2z}\zeta \left(2z-7,\frac{3}{2}\right)\Gamma \left(z-\frac{5}{2}\right)}{\sqrt{\pi }\Gamma \left(z\right)}-\frac{2^{2-2z}\zeta \left(2z-6,\frac{3}{2}\right)\Gamma \left(z-\frac{5}{2}\right)}{\sqrt{\pi }\Gamma \left(z\right)}\hfill \end{array}$$

(74)

where on the second line we used the substitution $s\to 2s$, followed by rewriting $2s+1=2(s+\frac{1}{2})$.14 The partial results coming from summing each tower are given in Table A3. Putting everything together, we obtain $F_{\mathrm{type}\mathrm{B}\mathrm{Maj}.}^{\left(1\right)}=\frac{11}{180}logR=a_f^{d=4}logR$, which agrees with the results of [16].

Finally, for the Weyl truncated theory,

$$\begin{array}{c}\hfill {\zeta }_{\mathrm{type}\mathrm{B}\mathrm{Weyl}}^{\mathrm{HS}}\left(z\right)=\sum _{s=1}^{\infty }{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,0])}\left(z\right)=\mathcal{O}\left(z^2\right),\end{array}$$

(75)

which gives us $F^{\left(1\right)}=0$.

AdS${}_{\mathbf{11}}$ 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 $F^{\left(1\right)}$ is given in Table A4 and Table A5.

Instead, let us consider the $d=11$ case, where we can compare the four different types of fermions: non-minimal ($U\left(N\right)$), Weyl, minimal ($O\left(N\right)$), and Majorana–Weyl. The calculations of $F^{\left(1\right)}$ for each the various weights and their spectra are given in Table A6. In the non-minimal and Weyl projected theories, the bulk $F^{\left(1\right)}$ contributions sum to zero, whereas in the minimal and Majorana–Weyl theories, the bulk $F^{\left(1\right)}$ contributions are $-\mathrm{14,797}/\mathrm{2,993,760}logR$ and $-\mathrm{14,797}/\mathrm{5,987,520}logR$ respectively. The numerical parts of these free energies correspond exactly to the values of the free energy of one real fermion, $-\mathrm{14,797}/\mathrm{2,993,760}$ and one real Weyl fermion on $S^{10}$, $-\mathrm{14,797}/\mathrm{5,987,520}$.

3.2.2. Fermionic Higher Spins in Type AB Theories

Spectrum

We described earlier that there is only one irrep of $SO\left(d\right)$ of interest here that describes the tower of spins corresponding to the fermionic bilinears in Type AB theories, namely ${\alpha }_s=[s,\frac{1}{2},\frac{1}{2},\dots ,\frac{1}{2}]$. Therefore, in the non-minimal theories dual to complex scalars and fermions in the $U\left(N\right)$ singlet sector, the purely fermionic contribution to the total zeta function is:

$$\begin{array}{c}\hfill {\zeta }_{\mathrm{type}\mathrm{AB}\mathrm{ferm}}^{\mathrm{HS}}\left(z\right)=2{\zeta }_{({\Delta }_{1/2};[\frac{1}{2},\frac{1}{2},\frac{1}{2},\dots ,\frac{1}{2}])}^{\mathrm{HS}}\left(z\right)+2\sum _{s=\frac{3}{2},\frac{5}{2},\dots }^{\infty }{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,\frac{1}{2},\frac{1}{2},\dots ,\frac{1}{2}])}\left(z\right),\end{array}$$

(76)

where ${\Delta }_{1/2}=\frac{1}{2}+d-2=d-\frac{3}{2}$. Thus, the spectrum of spins gives us a massive Dirac fermion15, and a tower of complex massless higher-spin fermionic fields.16

Sample Calculation: AdS${}_5$

After collecting our equations following (76), we have,

$$\begin{array}{c}\hfill {\zeta }_{\mathrm{type}\mathrm{AB}\mathrm{ferm}}^{\mathrm{HS}}\left(z\right)=2{\zeta }_{({\Delta }_{1/2};[\frac{1}{2},\frac{1}{2}])}^{\mathrm{HS}}\left(z\right)+2\sum _{s=\frac{3}{2},\frac{5}{2},\dots }^{\infty }{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,\frac{1}{2}])}\left(z\right),\end{array}$$

(77)

with ${\Delta }^{\mathrm{ph}}=2+s$. For the massive fermion contribution,

$$\begin{array}{cc}\hfill {\zeta }_{({\Delta }_{1/2};[s,\frac{1}{2}])}\left(z\right)& =\frac{2^{2z-10}\left(36\Gamma \left(z-\frac{1}{2}\right)+20\Gamma \left(z-\frac{3}{2}\right)+3\Gamma \left(z-\frac{5}{2}\right)\right)}{3\sqrt{\pi }\Gamma \left(z\right)}.\hfill \end{array}$$

(78)

Then,

$$\begin{array}{cc}\hfill \frac{{\zeta }_{(\Delta ;[s,\frac{1}{2}])}\left(z\right)}{logR}& =\frac{(s+\frac{1}{2})(s+\frac{3}{2})}{48\pi {(\Delta -2)}^{2z-1}\Gamma \left(z\right)}\hfill \\ & \times \left[{(\Delta -2)}^4\Gamma \left(\frac{5}{2}\right)\Gamma \left(z-\frac{5}{2}\right)+\Gamma \left(\frac{3}{2}\right)\Gamma \left(z-\frac{3}{2}\right){(\Delta -2)}^2\left(\frac{1}{4}+{(s+1)}^2\right)\right.\hfill \\ & \left.+\Gamma \left(\frac{1}{2}\right)\Gamma \left(z-\frac{1}{2}\right)\frac{{(s+1)}^2}{4}\right].\hfill \end{array}$$

(79)

This gives us,

$$\begin{array}{c}{\sum }_{s=\frac{3}{2},\frac{5}{2},\dots }{\zeta }_{({\Delta }^{\mathrm{ph}};[s,\frac{1}{2}])}\left(z\right)\hfill \\ =\frac{1}{1536\sqrt{\pi }\Gamma \left(z\right)}\left\{6\left[4\zeta \left(2z-7,\frac{3}{2}\right)+8\zeta \left(2z-6,\frac{3}{2}\right)+3\zeta \left(2z-5,\frac{3}{2}\right)\right]\Gamma \left(z-\frac{5}{2}\right)\right.\hfill \\ +\left[16\zeta \left(2z-7,\frac{3}{2}\right)+64\zeta \left(2z-6,\frac{3}{2}\right)+96\zeta \left(2z-5,\frac{3}{2}\right)+64\zeta \left(2z-4,\frac{3}{2}\right)\right.\hfill \\ \left.+15\zeta \left(2z-3,\frac{3}{2}\right)\right]\Gamma \left(z-\frac{3}{2}\right)+2\left[4\zeta \left(2z-5,\frac{3}{2}\right)+16\zeta \left(2z-4,\frac{3}{2}\right)\right.\hfill \\ \left.\left.+23\zeta \left(2z-3,\frac{3}{2}\right)+14\zeta \left(2z-2,\frac{3}{2}\right)+3\zeta \left(2z-1,\frac{3}{2}\right)\right]\Gamma \left(z-\frac{1}{2}\right)\right\}.\hfill \end{array}$$

(80)

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 Appendix B.1. Similarly,

$$\begin{array}{c}{\sum }_{s=\frac{3}{2},\frac{5}{2},\dots }{\zeta }_{({\Delta }^{\mathrm{gh}};[s-1,\frac{1}{2}])}\left(z\right)\hfill \\ =\frac{1}{1536\sqrt{\pi }\Gamma \left(z\right)}\{6\left[4\zeta \left(2z-7,\frac{5}{2}\right)-8\zeta \left(2z-6,\frac{5}{2}\right)+3\zeta \left(2z-5,\frac{5}{2}\right)\right]\Gamma \left(z-\frac{5}{2}\right)\hfill \\ +\left[16\zeta \left(2z-7,\frac{5}{2}\right)-64\zeta \left(2z-6,\frac{5}{2}\right)+96\zeta \left(2z-5,\frac{5}{2}\right)-64\zeta \left(2z-4,\frac{5}{2}\right)\right.\hfill \\ \left.+15\zeta \left(2z-3,\frac{5}{2}\right)\right]\Gamma \left(z-\frac{3}{2}\right)+2\left[4\zeta \left(2z-5,\frac{5}{2}\right)-16\zeta \left(2z-4,\frac{5}{2}\right)\right.\hfill \\ \left.+23\zeta \left(2z-3,\frac{5}{2}\right)-14\zeta \left(2z-2,\frac{5}{2}\right)+3\zeta \left(2z-1,\frac{5}{2}\right)\right]\Gamma \left(z-\frac{1}{2}\right)\}.\hfill \end{array}$$

(81)

Quite clearly, the Hurwitz-zeta function shifts differently for the physical and ghost modes. Adding all three contributions and expanding near $z=0$,

$$\begin{array}{c}\hfill {\zeta }_{\mathrm{type}\mathrm{AB}\mathrm{ferm}}^{\mathrm{HS}}\left(z\right)=\mathcal{O}\left(z^2\right)\end{array}$$

(82)

which implies that $F_{\mathrm{type}\mathrm{AB}\mathrm{ferm}}^{\left(1\right)}=0$, consistently with the duality.

For reference, we also report the expected expression of ${\zeta }_{\mathrm{type}\mathrm{AB}\mathrm{ferm}}^{\mathrm{HS}}\left(z\right)$ for AdS${}_7$ and AdS${}_9$, expanded in z up to the second order, in Appendix C.2.

3.2.3. 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 their various possible truncations. The free energy contributions in a few explicit examples are collected for reference in Appendix C.3.

Spectrum

The spectrum of the non-minimal type C theories, dual to the free theory of N complex $d/2$-form gauge fields with $U\left(N\right)$ singlet constraint, can be obtained from the character formulas derived in [54]. 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 by17,18

$$\begin{array}{cc}\hfill {\zeta }_{\mathrm{type}\mathrm{C}}^{\mathrm{HS}}& \left(z\right)=2\sum _{\begin{array}{c}{k}_i\ge 0\\ k_i\ge k_{i+1}\end{array}}^1{\zeta }_{(4m;[k_1,k_1,\dots ,k_m,k_m])}\left(z\right)\hfill \\ & +\sum _{s=2}^{\infty }[\sum _{\begin{array}{c}{t}_i\ge 0\\ t_i\ge t_{i+1}\end{array}}^{2}2{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_1,t_2,t_2,\dots ,t_{m-1},t_{m-1},0])}\left(z\right)\hfill \\ & +\sum _{\begin{array}{c}{j}_i\ge 0\\ j_i\ge j_{i+1}\end{array}}^2\left({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,2,j_1,j_1,j_2,j_2,\dots ,j_{m-1},+j_{m-1}])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,2,j_1,j_1,j_2,j_2,\dots ,j_{m-1},-j_{m-1}])}\left(z\right)\right)]\hfill \end{array}$$

(83)

and for $d=4m+2$:

$$\begin{array}{cc}\hfill {\zeta }_{\mathrm{type}\mathrm{C}}^{\mathrm{HS}}& \left(z\right)=2\sum _{\begin{array}{c}{k}_i\ge 0\\ k_i\ge k_{i+1}\end{array}}^1{\zeta }_{(4m+2;[k_1,k_1,\dots ,k_m,k_m,0])}\left(z\right)\hfill \\ & +\sum _{s=2}^{\infty }[\sum _{\begin{array}{c}{t}_i\ge 0\\ t_i\ge t_{i+1}\end{array}}^{2}2{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,2,t_1,t_1,t_2,t_2,\dots ,t_{m-1},t_{m-1},0])}\left(z\right)\hfill \\ & +\sum _{\begin{array}{c}{j}_i\ge 0\\ j_i\ge j_{i+1}\end{array}}^2\left({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,j_1,j_1,j_2,j_2,\dots ,j_{m-1},+j_{m-1}])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,j_1,j_1,j_2,j_2,\dots ,j_{m-1},-j_{m-1}])}\left(z\right)\right)]\hfill \end{array}$$

(84)

Using these spectra and (40) to compute the zeta functions, we find the results:

$$\begin{array}{c}{F}_{\mathrm{type}\mathrm{C}}^{\left(1\right)}=2a_{d/2-\mathrm{form}}logR,d=4m\hfill \\ F_{\mathrm{type}\mathrm{C}}^{\left(1\right)}=-2a_{d/2-\mathrm{form}}logR,d=4m+2\hfill \end{array}$$

(85)

where $a_{d/2-\mathrm{form}}$ is the a-anomaly coefficient of a single real $(d/2-1)$-form gauge field in dimension d.

The first few values in $d=4,6,8,\dots$ read [69]:

$$a_{d/2-\mathrm{form}}=\left\{\frac{62}{90},-\frac{221}{210},\frac{8051}{5670},-\frac{\mathrm{1,339,661}}{\mathrm{748,440}},\frac{\mathrm{525,793,111}}{\mathrm{243,243,000}},-\frac{\mathrm{3,698,905,481}}{\mathrm{1,459,458,000}},\dots \right\}.$$

(86)

Thus, we see that (85) is consistent with the duality provided $G_N^{\mathrm{type}\mathrm{C}}\sim 1/(N-1)$ in $d=4m$, and $G_N^{\mathrm{type}\mathrm{C}}\sim 1/(N+1)$ in $d=4m+2$.

Minimal Type C $\mathit{O}\left(\mathit{N}\right)$ The “minimal type C” theory corresponds to the $O\left(N\right)$ 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 [54] 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 $d=4m$,

$$\begin{array}{c}{\zeta }_{\min .\mathrm{type}\mathrm{C}}^{\mathrm{HS}}\left(z\right)=2\sum _{\begin{array}{c}{k}_i\ge 0\\ k_i\ge k_{i+1}\end{array}}^1{\zeta }_{(4m;[k_1,k_1,k_1,k_1,\dots ,k_{\lfloor \frac{m}{2}\rfloor },k_{\lfloor \frac{m}{2}\rfloor },k_{\lfloor \frac{m}{2}\rfloor },k_{\lfloor \frac{m}{2}\rfloor },0])}\left(z\right)\hfill \\ +\sum _{s=2}^{\infty }\sum _{\begin{array}{c}{t}_i\ge 0\\ t_i\ge t_{i+1}\end{array}}^2{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_1,t_2,t_2,\dots ,t_{m-1},t_{m-1},0])}\left(z\right)\hfill \\ +\sum _{s=2,4,6,\dots }\sum _{\begin{array}{c}{j}_i\ge 0\\ j_i\ge j_{i+1}\\ {\sum }_i{j}_i=0\left(\mathrm{mod}2\right)\end{array}}^2\left({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,2,j_1,j_1,j_2,j_2,\dots ,j_{m-1},+j_{m-1}])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}},s;[s,2,j_1,j_1,j_2,j_2,\dots ,j_{m-1},-j_{m-1}])}\left(z\right)\right)\hfill \\ +\sum _{s=3,5,7,\dots }\sum _{\begin{array}{c}{j}_i\ge 0\\ j_i\ge j_{i+1}\\ {\sum }_i{j}_i=1\left(\mathrm{mod}2\right)\end{array}}^2\left({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,2,j_1,j_1,j_2,j_2,\dots ,j_{m-1},+j_{m-1}])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,2,j_1,j_1,j_2,j_2,\dots ,j_{m-1},-j_{m-1}])}\left(z\right)\right)\hfill \end{array}$$

(87)

and in $d=4m+2$,

$$\begin{array}{c}{\zeta }_{\min .\mathrm{type}\mathrm{C}}^{\mathrm{HS}}\left(z\right)=\sum _{\begin{array}{c}{k}_i\ge 0\\ k_i\ge k_{i+1}\end{array}}^1{\zeta }_{(4m+2;[k_1,k_1,\dots ,k_m,k_m,0])}\left(z\right)\hfill \\ +\sum _{s=2}^{\infty }\sum _{\begin{array}{c}{t}_i\ge 0\\ t_i\ge t_{i+1}\end{array}}^2{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,2,t_1,t_1,t_2,t_2,\dots ,t_{m-1},t_{m-1},0])}\left(z\right)\hfill \\ +\sum _{s=2,4,6,\dots }\sum _{\begin{array}{c}{j}_i\ge 0\\ j_i\ge j_{i+1}\\ {\sum }_i{j}_i=0\left(\mathrm{mod}2\right)\end{array}}^2\left({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,j_1,j_1,j_2,j_2,\dots ,j_m,+j_m])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,j_1,j_1,j_2,j_2,\dots ,j_m,-j_m])}\left(z\right)\right)\hfill \\ +\sum _{s=3,5,7,\dots }\sum _{\begin{array}{c}{j}_i\ge 0\\ j_i\ge j_{i+1}\\ {\sum }_i{j}_i=1\left(\mathrm{mod}2\right)\end{array}}^2\left({\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,j_1,j_1,j_2,j_2,\dots ,j_m,+j_m])}\left(z\right)+{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,j_1,j_1,j_2,j_2,\dots ,j_m,-j_m])}\left(z\right)\right),\hfill \end{array}$$

(88)

where $\lfloor \frac{m}{2}\rfloor$ denotes the integer part of $\frac{m}{2}$.

As a consistency check of these spectra, in Appendix A.1 we computed the corresponding partition functions in thermal AdS with $S^1\times S^{d-1}$ boundary. After summing up over all representations appearing in the zeta functions above, the result matches the (symmetrized) square of the one-particle partition function of a $(d/2-1)$-form gauge field, see Equation (A29).

Evaluating the spectral zeta functions with the help of the formulas in Section 3.1.1, we obtain the results:

$$\begin{array}{cccc}\hfill F_{\min .\mathrm{type}\mathrm{C}\mathrm{SD}}^{\left(1\right)}& =2a_{d/2-\mathrm{form}}logR,\hfill & \hfill d& =4m\hfill \\ \hfill F_{\min .\mathrm{type}\mathrm{C}\mathrm{SD}}^{\left(1\right)}& =0,\hfill & \hfill d& =4m+2\hfill \end{array}$$

(89)

These correspond to the shifts given in Table 1. Interestingly, in the minimal type C theory in $d=6,10,\dots$ the bulk one-loop free energy vanishes and no shift of the coupling constant is required.

Self-dual $\mathit{U}\left(\mathit{N}\right)$ In $d=4m$, we can impose a self-duality constraint $F^i=i\ast F^i$ in the theory of N complex p-forms. The resulting spectrum of $U\left(N\right)$ invariant bilinears leads to the following total zeta function in the bulk19

$$\begin{array}{cc}\hfill {\zeta }_{\mathrm{type}\mathrm{C}\mathrm{SD}}^{\mathrm{HS}}\left(z\right)& =\sum _{s=2}^{\infty }\sum _{\begin{array}{c}{t}_i\ge 0\\ t_i\ge t_{i+1}\end{array}}^2{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,t_1,t_1,t_2,t_2,\dots ,t_{m-1},t_{m-1},0])}\left(z\right).\hfill \end{array}$$

(90)

In $d=4m+2$, we can impose the self-duality condition $F^i=\ast F^i$, and the resulting truncated spectrum gives the following total zeta function20

$$\begin{array}{cc}\hfill {\zeta }_{\mathrm{type}\mathrm{C}\mathrm{SD}}^{\mathrm{HS}}\left(z\right)& =\sum _{s=2}^{\infty }\sum _{\begin{array}{c}{j}_i\ge 0\\ j_i\ge j_{i+1}\end{array}}^2{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};[s,j_1,j_1,j_2,j_2,\dots ,j_{m-1},+j_{m-1}])}\left(z\right).\hfill \end{array}$$

(91)

Using these spectra, we find the results:

$$\begin{array}{cccc}\hfill F_{\mathrm{type}\mathrm{C}\mathrm{SD}}^{\left(1\right)}=& \frac{1}{2}{a}_{d/2-\mathrm{form}}logR,\hfill & \hfill & d=4m\hfill \\ \hfill F_{\mathrm{type}\mathrm{C}\mathrm{SD}}^{\left(1\right)}=& -\frac{1}{2}{a}_{d/2-\mathrm{form}}logR,\hfill & \hfill & d=4m+2\hfill \end{array}$$

(92)

which correspond to the shifts given in Table 1.

Self-dual $\mathit{O}\left(\mathit{N}\right)$ In $d=4m+2$, we can impose a self-duality condition on the theory of N real forms with $O\left(N\right)$ singlet constraint. The spectrum is given by the “overlap” of the minimal type C and self-dual $U\left(N\right)$ spectra given above. The resulting total zeta function is given by:

$$\begin{array}{cc}\hfill {\zeta }_{\min .\mathrm{type}\mathrm{C}\mathrm{SD}}^{\mathrm{HS}}\left(z\right)& =\sum _{s=2,4,6,\dots }\sum _{\begin{array}{c}{j}_i\ge 0\\ j_i\ge j_{i+1}\\ {\sum }_i{j}_i=0\left(\mathrm{mod}2\right)\end{array}}^2{\mathfrak{Z}}_{(\Delta ;[s,j_1,j_1,j_2,j_2,\dots ,j_m,+j_m])}\left(z\right)\hfill \\ & +\sum _{s=3,5,7,\dots }\sum _{\begin{array}{c}{j}_i\ge 0\\ j_i\ge j_{i+1}\\ {\sum }_i{j}_i=1\left(\mathrm{mod}2\right)\end{array}}^2{\mathfrak{Z}}_{(\Delta ;[s,j_1,j_1,j_2,j_2,\dots ,j_m,+j_m])}\left(z\right)\hfill \end{array}$$

(93)

from which we find the result:

$$\begin{array}{c}\hfill F_{\min .\mathrm{type}\mathrm{C}\mathrm{SD}}^{\left(1\right)}=\frac{1}{4}{a}_{d/2-\mathrm{form}}logR,d=4m+2.\end{array}$$

(94)

3.3. Calculations in Odd d

3.3.1. Preliminaries

Alternate Regulators

In the calculations for even d discussed above, we chose to sum over the spins before sending the spectral parameter $z\to 0$. 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 $z\to 0$ is not easy to do, and we will instead first send $z\to 0$ 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:

$$\begin{array}{cc}\hfill \sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}& \left[{\left.{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};{\alpha }_s)}\left(z\right)\right|}_{z=0}\right]\hfill \\ & ={lim}_{ϵ\to 0}\sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}{e}^{-ϵ({\Delta }^{\mathrm{ph}}-\frac{d}{2})}({\zeta }_{({\Delta }^{\mathrm{ph}};{\alpha }_s)})\left(0\right)-{lim}_{ϵ\to 0}\sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}{e}^{-ϵ({\Delta }^{\mathrm{gh}}-\frac{d}{2})}({\zeta }_{({\Delta }^{\mathrm{gh}};{\alpha }_{s-1})})\left(0\right),\hfill \end{array}$$

(95)

and similarly:

$$\begin{array}{cc}\hfill \sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}& \left[{\left.\frac{\partial }{\partial z}{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};{\alpha }_s)}\left(z\right)\right|}_{z=0}\right]\hfill \\ & ={lim}_{ϵ\to 0}\sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}{e}^{-ϵ({\Delta }^{\mathrm{ph}}-\frac{d}{2})}{({\zeta }_{({\Delta }^{\mathrm{ph}};{\alpha }_s)})}^{\prime }\left(0\right)-\underset{ϵ\to 0}{lim}\sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}{e}^{-ϵ({\Delta }^{\mathrm{gh}}-\frac{d}{2})}{({\zeta }_{({\Delta }^{\mathrm{gh}};{\alpha }_{s-1})})}^{\prime }\left(0\right),\hfill \end{array}$$

(96)

where we recall that ${\Delta }^{\mathrm{ph}}=s+d-2$ and ${\Delta }^{\mathrm{gh}}=s+d-1$. In even d, one can show that this procedure, with the shifted exponentials as above, gives the same result as first summing over all representations and then sending the spectral parameter $z\to 0$. Equivalently, instead of the exponential regulators, one can use the analytic continuation of the Hurwitz zeta function by evaluating:

$$\begin{array}{cc}\hfill \sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}& \left[{\left.{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};{\alpha }_s)}\left(z\right)\right|}_{z=0}\right]\hfill \\ & ={lim}_{ϵ\to 0}\sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}{\left({\Delta }^{\mathrm{ph}}-\frac{d}{2}\right)}^{-ϵ}({\zeta }_{({\Delta }^{\mathrm{ph}},s;{\alpha }_s)}^{\mathrm{HS}})\left(0\right)-\underset{ϵ\to 0}{lim}\sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}{\left({\Delta }^{\mathrm{gh}}-\frac{d}{2}\right)}^{-ϵ}({\zeta }_{({\Delta }^{\mathrm{gh}};{\alpha }_{s-1})}^{\mathrm{HS}})\left(0\right),\hfill \end{array}$$

(97)

and:

$$\begin{array}{cc}\hfill \sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}& \left[{\left.\frac{\partial }{\partial z}{\mathfrak{Z}}_{({\Delta }^{\mathrm{ph}};{\alpha }_s)}\left(z\right)\right|}_{z=0}\right]\hfill \\ & ={lim}_{ϵ\to 0}\sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}\alpha \end{array}}{\left({\Delta }^{\mathrm{ph}}-\frac{d}{2}\right)}^{-ϵ}{({\zeta }_{({\Delta }^{\mathrm{ph}};{\alpha }_s)})}^{\prime }\left(0\right)-\underset{ϵ\to 0}{lim}\sum _{\begin{array}{c}\mathrm{all}\mathrm{spins}\\ \mathrm{in}{\alpha }_s\end{array}}{\left({\Delta }^{\mathrm{gh}}-\frac{d}{2}\right)}^{-ϵ}{({\zeta }_{({\Delta }^{\mathrm{gh}};{\alpha }_{s-1})}^{\mathrm{HS}})}^{\prime }\left(0\right),\hfill \end{array}$$

(98)

This method, which is closely related to the one previously used in [27],21 will be described in the next sections in greater detail.

Note that, while in even d ${\zeta }_{({\Delta }_s;{\alpha }_s)}\left(0\right)$ 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:

$$\begin{array}{c}\hfill {\int }_0^{\infty }\frac{u^k}{e^{2\pi u}\pm 1}log[u^2+b^2]={\int }_0^{\infty }\frac{u^{k}du}{e^{2\pi u}\pm 1}\left[log\left(u^2\right)+{\int }_0^{b^2}\frac{1}{u^2+x}dx\right].\end{array}$$

(99)

We define:

$$\begin{array}{c}\hfill A_k^{\pm }\left(x\right)\equiv {\int }_0^{\infty }\frac{u^k}{e^{2\pi u}\pm 1}\frac{du}{u^2+x},B_k^{\pm }\equiv {\int }_0^{\infty }\frac{u^k}{e^{2\pi u}\pm 1}.\end{array}$$

(100)

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):

$$\begin{array}{cc}\hfill A_{2k+1}^{\pm }\left(x\right)& ={(-x)}^k{A}_1^{\pm }\left(x\right)+\sum _{j=1}^k{(-x)}^{k-j}{B}_{2j-1}^{\pm }.\hfill \end{array}$$

(101)

As a consequence of this relation, we only need the explicit analytic expressions of the integrals $A_1^{\pm }$,22 which is given by [49]:

$$\begin{array}{cc}\hfill A_1^{+}\left(x\right)& =\frac{1}{2}\left[-log\left(\sqrt{x}\right)+\psi \left(\sqrt{x}+\frac{1}{2}\right)\right]\hfill \end{array}$$

(103)

$$\begin{array}{cc}\hfill A_1^{-}\left(x\right)& =\frac{1}{2}\left[log\left(\sqrt{x}\right)-\frac{1}{2\sqrt{x}}-\psi \left(\sqrt{x}\right)\right],\hfill \end{array}$$

(104)

where $\psi \left(x\right)$ is the digamma function $\psi \left(x\right)={\Gamma }^{\prime }\left(x\right)/\Gamma \left(x\right)$.

3.3.2. 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 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 ${\mu }_{\alpha }\left(u\right)$ is no longer polynomial in u, but a polynomial in u multiplied by a hyperbolic function,

$$\begin{array}{c}\hfill {\mu }_{\alpha }\left(u\right)={\mu }_{\alpha }^{\mathrm{poly}}\left(u\right)\times f_{\pm }\left(u\right),\mathrm{where}{f}_{\pm }\left(u\right)=\left\{\begin{array}{c}{f}_{+}\left(u\right)=tanh\left(\pi u\right),\mathrm{for}\mathrm{bosons},\hfill \\ f_{-}\left(u\right)=coth\left(\pi u\right),\mathrm{for}\mathrm{fermions}.\hfill \end{array}\right.\end{array}$$

(105)

Then, for a particular spectral weight $\alpha$, the partition function can be written as:

$$\begin{array}{c}\hfill {\zeta }_{(\Delta ;\alpha )}\left(z\right)=\frac{\mathrm{vol}\left({\mathrm{AdS}}_{d+1}\right)}{\mathrm{vol}\left(S^d\right)}\frac{2^{d-1}}{\pi }{\int }_0^{\infty }du\frac{g_{\alpha }{\mu }_{\alpha }^{\mathrm{poly}}\left(u\right)}{{\left[u^2+{\left(\Delta -\frac{d}{2}\right)}^2\right]}^z}{f}_{\pm }\left(u\right).\end{array}$$

(106)

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 $\alpha =\left[s\right]$, $s\ge 0$, and the spectral zeta function for a given spin s is:

$$\begin{array}{cc}\hfill {\zeta }_{(\Delta ;{\alpha }_s)}\left(z\right)& =\frac{\mathrm{vol}\left({\mathrm{AdS}}_4\right)}{\mathrm{vol}\left(S^3\right)}\frac{4}{\pi }{\int }_0^{\infty }du\frac{g_{\left[s\right]}\left(s\right){\mu }_{\left[s\right]}^{\mathrm{poly}}(}{}\hfill \end{array}$$