AdS one-loop partition functions from bulk and edge characters

We show that the one-loop partition function of any higher spin field in $(d+1)$-dimensional Anti-de Sitter spacetime can be expressed as an integral transform of an $\text{SO}(2,d)$ bulk character and an $\text{SO}(2,d-2)$ edge character. We apply this character integral formula to various higher-spin Vasiliev gravities and find miraculous (almost) cancellations between bulk and edge characters that lead to agreement with the predictions of HS/CFT holography. We also discuss about the relation between the character integral representation and Rindler-AdS thermal partition function.

The first test on one-loop free energy, for (non)minimal type-A higher spin theory in AdS 4 , was carried out by Giombi and Klebanov in [6] where the authors used the zeta function regularization for (i) computing the free energy of a single field and (ii) summing over free energies of all field content. Then the same method was applied to higher dimensional AdS [7], type-B higher spin gravity [8] and partially massless higher spin fields [10]. However, one has to work dimension by dimension using this method, in particular for even dimensional AdS, due to the technical difficulty in summing over field content. This difficulty was bypassed in [12,13] where the authors, inspired by earlier works [11,14], expressed the higher spin spectral zeta functions, cf. (2.4), as an integral transformation of the corresponding characters of the AdS isometry group and apply it to a large class of higher spin theories.
On the other hand, a character integral representation of one-loop free energies on de Sitter spacetime is found in [15] recently by using the heat kernel regularization. For example, for a massive vector field in dS d+1 of mass m = (d−2) 2 4 + ν 2 , the one-loop free energy can be expressed as (suppressing the regularization) log Z = log Z bulk − log Z edge = The bulk character χ dS bulk (u) is the Harish-Chandra character of the spin-1, ∆ = d 2 +iν representation of SO(1, d + 1) and the edge character χ dS edge (u) is the Harish-Chandra character of the spin-0, ∆ = d− 2 2 +iν representation of SO (1, d−1). Apart from the standard meaning as the one-loop path integral on S d+1 , which is the Euclidean Wick rotation of dS d+1 , it is also pointed out in the same paper that the character integral representation (1.1) can alternatively be interpreted as the bulk thermal free energy in the static patch of dS d+1 subject to possible corrections from edge modes living on the cosmological horizon. The appearance of edge modes here should not be surprising.
In fact, it was already noticed in [16][17][18] that edge modes can be used to explain the long-standing discrepancy between the Euclidean path integral and bulk canonical definitions of the entanglement entropy for Maxwell field in Rindler space [19] .
In this paper, we extend this dS character integral representation to AdS spacetime. Essentially this extension amounts to replacing the bulk SO(1, d+1) characters by the corresponding SO (2, d) characters and the replacing the edge SO(1, d − 1) characters by the corresponding SO (2, d − 2) characters. For example, for a massive vector field in AdS d+1 , the unregularized free energy (the properly regularized version will be derived in section 5) can be expressed as log Z = log Z bulk − log Z edge = (1 − e −u ) d (1.5) and the AdS edge character, corresponding to spin-0, ∆ = d−2 2 + ν representation of SO(2, d − 2) is Compared to the character integral representation found in [12,13], eq. (1.3) or (1.4) seems quite different and considerably simpler because there is no angular integrals associated to the SO(d) Cartan subgroup. However, in all cases we have checked, explicit evaluation of the angular integrals in [12,13] reproduces (the UV-finite part of) the character integral representations derived in this paper. In particular, when applied to Vasiliev higher spin theories, the two methods completely agree, though in our setup the (almost) vanishing of total one-loop free energy is a result of the miraculous (almost) cancellations between the total bulk characters and the total edge characters.
For nonmiminal type-A theory, the cancellation between the total bulk character and the total edge character is exact. For example, in AdS 4 , the total bulk character is For minimal type-A theory, the cancellation between bulk and edge characters yields an SO (1, d) Harish-Chandra character corresponding a conformally couple scalar, cf. (8.15) log Z The dS character integral representation of sphere partition functions obtained in [15] directly identifies Z Apart from these technical advantages, our character integral representation admits a thermal interpretation (at least for even dimensional AdS). More explicitly, we argue that the bulk part of the one-loop path integral on EAdS d+1 can be interpreted as the quasi-canonical partition function in the Rindler patch of Lorentzian AdS [20] (consult the appendix E for details about the Rindler patch which is also called Rindler-AdS) and the edge part is associated to edge modes localized on the horizon of the Rindler patch.
The organization of the paper is as follows. In section 2, we review the one-loop partition function and heat kernels in AdS d+1 . In section 3 and 4, we show a very simple but not rigorous derivation of the unregularized character integral representation in the case of even dimensional AdS. In section 5, we give a rigorous version of the regularized character integral formula that works for both even and odd dimensional AdS by adding a UV-regulator and specifying the appropriate integral contour. An explicit evaluation of the regularized character integral is done in section 6. In section 7, we derive the effect of (higher spin) double trace deformation on one-loop free energy. In section 8, we review various Flato-Fronsdal theorems which fix the free spectrum of higher spin theory group theoretically and apply the character integral formula to (non)minimal type-A/B Vasiliev theories. In section 9, we comment on the thermal interpretation of the character integral representation. Finally, appendices contain various technical results, some of which are of independent interest. For example, in appendix B, we give a physical derivation of the SO(1, d + 1) Plancherel measure supported on the scalar principal series. In appendix F, we compute the SO(2, 1) Harish-Chandra character corresponding to unitary highest-weight representations and in appendix G, we explain some of the physics it encodes, including the quasinormal mode spectrum and normal mode density of Rindler-AdS.

One-loop partition functions and heat kernels on AdS
The one-loop partition function Z of a (real bosonic) quantum field theory is given by a functional determinant log Z = − 1 2 log det(D) (2.1) where D is the "Laplacian" in the quadratic Lagrangian L 2 = 1 2 φDφ. (There might be some nontrivial factors that are not captured by the functional determinant due to subtleties like zero modes when the quantum field theory is defined on a compact manifold. We ignore these subtleties in this general discussion as they will not appear in this paper). When the theory has a gauge symmetry, we should also subtract the functional determinant of the corresponding ghost field. In general, the functional determinant is UV-divergent and needs to be regularized. The regularization scheme for log Z we'll use in this paper is where K D (t) ≡ Tr e −tD is the heat kernel of D. In terms of the spectrum of D, the heat kernel is formally 1 defined as where λ n is the eigenvalue of D of degeneracy d n . Performing a Mellin transformation for the heat kernel K D (t) yields the spectral zeta function which is also a common tool to regularize partition function.
Given a free field in AdS d+1 carrying a generic unitary irreducible representation (UIR) [∆, s] of SO(2, d) with ∆ = d 2 + ν, the corresponding heat kernel K s,ν (t) is constructed explicitly in [21] by a group theoretical method. Alternatively, we can infer heat kernel from the associated spectral zeta function [12] by an inverse Mellin transformation: Explanations of the various notations appearing in eq. (2.5) are given as follows: • Vol(S d ): the volume of a d-dimensional sphere.
. For example, when s = (s, 0, · · · , 0), it gives the dimension of spin-s representation and when s = (n, s, 0, · · · , 0), it gives When we interpret D d n,s as the dimension of an so(d) representation, n cannot be smaller than s. On the other hand, D d n,s as a function of n, s defined by (2.9), admits a meromorphic continuation to C 2 . Using the second point of view, we can find some interesting and useful relations like D d n−1,s = −D d s−1,n which holds for all n, s ∈ C.
• µ (d) s (λ): the spins spectral density up to normalization [26] (but we still call it the "spectral density" for simplicity) where j ≡ s j + d 2 − j, and = ±1 for bosonic/fermionic fields. The so(d) spin label s will be dropped when we deal with a scalar field. We focus on bosonic fields in the main text of this paper and an example about Dirac spinors can be found in the appendix A. The spectral density µ (d) s (λ) with a Wick rotation λ → iλ, is also the Plancherel measure on the principal series of SO(1, d+1) [27]. The simplest way to derive it [21] is based on an analytical continuation of the SO(d + 2) Plancherel measure, which is proportional to the dimension of SO(d + 2) representation (we also provide a physical derivation/interpretation of the spectral density in the appendix B). As a result of this analytical continuation, the polynomial part P s (λ) ≡ λ r j=1 (λ 2 + 2 j ) of the spectral density also appears in the following formula that relates D d+2 (s 0 , s) and D d s [12] D d+2 with s 0 replaced by − d 2 +iλ. This equation is extremely useful when we compare the character integral representations for AdS and dS in appendix C.
The various Γ-functions in (2.5) can be greatly simplified for integer dimension d and we're left 3 Warm-up example: scalar fields in even dimensional AdS

Scalar fields in AdS 2
As a warm-up, let's consider a scalar field ϕ of mass m 2 = ν 2 − 1 4 in AdS 2 which corresponds to the scaling dimension ∆ + = 1 2 + ν representation of the isometry group SO(2, 1). Applying (2.12) to this field yields the partition function where we've extended the integration domain of λ to the whole real line. To perform the integral over λ, we use the Hubbard-Stratonovich trick: The naive Fourier transformation (3.3) is ill-defined. However for our practical purpose i.e. to get the character integral formula as soon as possible, we pretend that it's well-defined and the contour can be closed at infinity. A more rigorous treatment is postponed until section 5 and 6 where we give a fully regularized character integral and justify our naive result obtained here is indeed reasonable and sufficient for most applications, in particular the total partition function of Vasiliev theories. From now on, keeping the above comments in mind, we're free to close the λ-contour in W (u) in either upper or lower half-plane depending on the sign of u and we also write the partition function log Z ν in the unregularized form by putting = 0 formally. When u > 0, we close the contour in the upper half-plane picking up simple poles at λ = i(n + 1 2 ), n ∈ N and when u < 0, we close the contour in the lower half-plane picking up simple poles at λ = −i(n + 1 2 ), n ∈ N. By summing over residues in both cases, we find Thus the unregularized partition function is given by where χ AdS 2 ∆ + (u) is the character of scaling dimension ∆ + representation of SO (2,1). Before moving to the higher dimensional examples, let's notice that when we evaluate the t-integral in eq. (3.2), we implicitly assume ν > 0, so ∆ + = 1 2 + ν corresponds to the "standard quantization" in bulk. On the other hand, we can safely send ν to −ν in eq. (3.5) by analytic continuation, as long as 0 < ν < 1 2 . Altogether we conclude that, with the standard boundary condition, the partition function of ϕ is and with the alternate boundary condition, the partition function is where ∆ − = 1 2 − ν.

Scalar fields in AdS 2r+2
The computation above in AdS 2 can be generalized straightforwardly to scalar field in any AdS d+1 with d = 2r + 1 odd. Assuming that the scalar field has scaling dimension ∆ + = d 2 + ν, its partition function is given by where the scalar spectral density is Following the same steps as in section 3.1, we obtain The "Fourier transform" W (d) (u) can be evaluated according to the comments below eq. (3.3) Plugging (3.11) into (3.10) yields a character integral expression for the unregularized log Z ν where χ AdS d+1 ∆ + (u) is the character of scaling dimension ∆ + representation of SO (2,d). Before turning to the higher spin case, let's comment on the relation between the character integral and the original heat kernel integral. The heat kernel in AdS is defined through the spectral density µ(λ) whose explicit construction is given in [26]. Briefly speaking, the authors of [26] found a complete set of δ-function normalizable eigenfunctions of the Laplacian operator −∇ 2 in EAdS d+1 : where σ is a discrete label for distinguishing eigenfunctions of the same λ. (Note that the inner product for h (λσ) involves an integration over the whole EAdS rather than a spatial slice as in the standard Klein-Gordon inner product). The spectral density is defined via these eigenfunctions: . Therefore the original heat kernel method involves an integral over the whole continuous spectrum labeled by (λ, σ). On the other hand, the character can be expanded into a discrete sum This expansion encodes a whole tower of solutions to the equation of motion (−∇ 2 +∆ + (∆ + −d))φ = 0 in global AdS that furnish a representation of SO (2, d). More explicitly, φ 0 = e −i∆ + t (1+r 2 ) ∆ + /2 is the primary mode, i.e. ground state, in the global coordinate: ds 2 = −(1 + r 2 )dt 2 + dr 2 1+r 2 + r 2 dΩ 2 . It solves the equation of motion, falls like r −∆ + at the boundary but its Wick rotation under t → −iτ is not normalizable in the sense of (3.14). By acting the conformal algebra so(2, d) on φ 0 repeatedly, we get a collection of modes that also solve the equation of motion and have the same boundary condition. At each frequency ω n = ∆ + + n, the degeneracy of these modes are exactly d+n−1 d−1 . Therefore while switching from the heat kernel integral to the character integral, we effectively turn a continuous spectrum into a discrete spectrum and curiously both of them encode the information of partition function. This observation is the main point of [28]. Actually the character integral representation we found is equivalent to the "zero mode method" used in that paper. For example, using the unregularized expression (3.12), we obtain formally which recovers the result in [28] up to some holomorphic function denoted by Pol(∆ + ) there.
Pol(∆ + ) is a polynomial in ∆ + and depends on the UV-cutoff. In section 6, we'll show that it can also be recovered if we use the fully regularized character integral.

Higher spin fields in AdS 2r+2
In this section, we turn to the character integral representation of higher spin fields in AdS d+1 with d = 2r + 1 2 . Unlike scalar fields, a spin-s field ϕ µ 1 ···µs in AdS can carry either massive or massless irreducible representation [29] depending on the scaling dimension. When ∆ = ∆ s,t ≡ d+t−1 with t ∈ {0, 1, · · · , s − 1}, ϕ µ 1 ···µs is called a partially massless (PM) field of depth t and it has a gauge symmetry δϕ µ 1 ···µs = ∇ (µ t+1 ···µs ξ µ 1 ···µt) + · · · [10]. In this case, we should include the contribution of the ghost field, which has spin-t and scaling dimension ∆ t,s = d + s − 1, in the one-loop partition function. When ∆ = d 2 + ν is not in the discrete set {∆ s,t }, the field ϕ µ 1 ···µs falls into the massive representations and doesn't have gauge symmetry. Due to the emergence of gauge symmetry, the characters corresponding to massive and massless representations take very different forms Let's start from computing the partition function of a spin-s field in the massive representations where the spin-s spectrum density is Following the same steps as in the scalar case, we obtain s (λ) e iλu is an even function in u and when u > 0, it is Plugging (4.5) into (4.4) yields a new character integral The second term in the bracket corresponds to subtracting the partition function of D d+2 s−1 scalars on EAdS d−1 with scaling dimension d−2 2 + ν since it involves an SO(2, d − 2) character of scalar representation. As in the de Sitter case [15], we tend to identify these scalar degrees of freedom as edge modes living on the horizon of Rindler-AdS [20] which is a Lorentzian Wick rotation of EAdS and has a EAdS d−1 shaped horizon. More discussions about Rindler-AdS to will be left to section 9 and appendix E. We can also write log Z s,ν in terms of an SO(2, d) character and an SO(2, d + 2) This form of character integral representation doesn't have the physically meaningful edge character structure but it turns out to be much more convenient than (4.6) computationally when we sum over all field content in Vasiliev higher spin gravities.
Finally, let's move to our main interest: (partially) massless fields. Due to gauge symmetry, the partition function of a PM field with spin-s and depth t ∈ {0, 1, · · · , s − 1} is given by where the first term corresponds to the spin-s gauge field and the second term arises from the spin-t ghost field. By using the explicit expression of W [d+t,s−1] (u) (4.9) where the characters of PM fields are is a massless representation of SO(2, d + 2) with spin-(s−1) and depth-(t−1).

Regularization, contour prescription and odd dimensional AdS
In the previous sections, we've derived a formal character integral formula for the unregularized one-loop partition functions of both scalar fields and higher spin fields in even dimensional AdS.
However, to make sense of the character integral mathematically and apply it to actual computation of renormalized partition functions, we have to use a well-defined and efficient regularization scheme.
In this section, we'll sort out this issue. Surprisingly the resolution turns out to have a very important byproduct: a character integral representation that works in odd dimensional AdS.

Regularization and contour prescription
We use a real scalar field to illustrate the regularization scheme. But it will be clear in the end that the same regularization also works for higher spin fields.

d = 2r + 1
It's mentioned in section 3.1 that W (d) (u), Fourier transformation of the scalar spectral density µ (d) (λ), is not well-defined. As a manifestation of this point, W (d) (u) is singular at u = 0. This singularity may lead to two inequivalent definitions of inverse Fourier transformation of W (d) (u) since the contour can either go above or below u = 0: where δ is a small positive number. (At this stage, the size of δ is not important as long as it's smaller than 2π. We'll later impose a more stringent constraint on it). It's clear that the two definitions of inverse Fourier transformation differ by the residue of the integrand at u = 0. In We use the function H d,iλ (u) here because its residue at u = 0 is given by eq. (D.8) and the residues at other poles u = 2πin, n ∈ Z can be easily inferred by using its quasi-periodicity H d,iλ (u+2πin) = (−e 2πλ ) n H d,iλ (u). With the information of residues known, we close the contour at infinity and get Therefore in order to recover the spectral density µ (d) , cf. (3.9), the contour prescription should be the average of R±iδ : Plugging this equation into the scalar partition function (3.8) , we obtain where δ has to be smaller than , otherwise the contours would cross the branch cut of √ u 2 + 2 , which has two disconnected pieces with one piece going upwards from i to i∞ and the other going downwards from −i to −i∞. Due to the manifest u → −u symmetry of the integrand for odd d, the two contours R ± iδ are actually equivalent and it suffices to use one of them The computation above also works for higher spin fields. It suffices to show that the higher spin generalization of (5.4) holds. Notice that W (d) s (u) corresponding to a spin-s field is related to the scalar version W (d) (u) by Applying the integral (5.4) to this relation indeed yields Therefore the regularized character integral for a spin-s field in massive representations is Starting from the eq. (5.9), we'll derive a complete expression for the regularized partition function Z s,ν in section 6.

d = 2r
The odd d case above tells us the correct strategy to get a regularized character integral. First, we (1−e −u ) d and compute its inverse Fourier transformations defined by two different contour choices. Then one proper linear combination of these choices can give the correct spectral density. Plug this integral expression of spectral density into the original partition function (2.12) and we finally obtain the regularized character integral. Now let's apply this strategy to the d = 2r case where we choose W (d) (u) to be The two possible inverse Fourier transformations defined as in eq. (5.1) arẽ Substituting this equation into (2.12) and performing the t, λ integrals, we end up with Since there is no poles in the strip bounded by R ± iδ, we can further deform the contour to be a small circle C 0 around u = 0. Then the integral is equivalent to evaluating the residue at u = 0: Similarly for higher spin fields, by using appropriate square root regularization, we can also get

Conclusion
Altogether, we can conclude that the regularized one-loop partition function (with the UV regu- For a field in massless representations, we need to include the corresponding ghost contribution

Evaluation of the regularized character integrals
In this section, we give an efficient and general recipe to compute the regularized character integral formula following the appendix C of [15]. For the simplicity of notation, we'll use scalar fields as an illustration of this recipe. But our reasoning and final result can be easily generalized to fields of arbitrary spin. In addition, the result can be used to justify that the unregularized character integral is sufficient for the application to Vasiliev gravities. First, let's briefly review the standard heat kernel method in computing one-loop partition functions where I've plugged in the explicit form of Vol(S d ) compared to eq. (2.5). By using the small t we can separate it into UV and IR parts without ambiguity The UV-part of the heat kernel expansion in AdS is fairly simple. When d is even, K ν (t) can be evaluated exactly because it's a Gaussian integral in λ. When d is odd, using tanh πλ = 1 − 2 1+e 2πλ we can split the heat kernel into two parts. The first part is a simple Gaussian integral as in the even d case. The second part is of O(t 0 ) and hence we can put t = 0 which yields an exactly solvable integral. In addition, by direct computation, one can show that α k is nonvanishing only for even k. For example, in d = 3, we obtain the nonzero heat kernel coefficients: where the UV regulator has been dropped in the IR integral because it's by construction UV finite. The IR regulator e −κ 2 t is inserted in the UV integral because the integrand has a 1 t term when α d+1 = 0, i.e. when d is odd. In the end, the log κ terms in log Z uv ν and log Z ir ν will cancel out and we're left with is the spectral zeta function and α d+1 = ζ ν (0). Next, we'll apply this UV-IR separation idea to the evaluation of partition function in the regularized character integral formalism.

Even dimensional AdS 2r+2
In section 5.1.1, we've found that the square-root regularized partition function in AdS 2r+2 is given where d = 2r + 1. Putting = 0 we recover the formal UV-divergent character formula: The unregularized integrand 1 2u H d,ν (u) admits a Laurent expansion around u = 0 with coefficients being polynomials in ν: The . Therefore we separate them from the remaining UV-finite terms Similarly using an infinitesimal IR regulator κ → 0 + , we obtain a UV-IR separation for the regu- where r = √ u 2 + 2 /u. In the IR part of the partition function, we've deformed the contour and safely put = 0. The log κ terms will drop out at the end when summing up log Z uv ν and log Z ir ν .

Evaluation of UV part
Using the expansion (6.8), the UV part of log Z ν ( ) can be written as where the IR regulator κ is not necessary for 0 ≤ k ≤ d. Then it suffices to evaluate the following two u-integrals When is odd in I (k, ), we can close the contour in the upper half plane ( fig. 6.1a) and the integral vanishes because (1 + u 2 ) −1 2 has no pole in this case. Thus I (k, ) is nonvanishing only when is even. On the other hand, the coefficient of I (k, ), i.e. b k , vanishes for k + odd. Therefore, only terms with even k, survive in log Z uv ν ( ). When is even in I (k, ), the previous "no poles" argument doesn't hold any more due to the presence of a branch cut from i to i∞ in the upper half plane. However, we can deform the contour to integrate along the branch cut (fig. 6.1b) which yields . For the integral J ( ), we split it into an IR-divergent part and an IR-finite part. In the IR-finite part, we can drop the IR regulator e −κu . . When is odd, the contour is closed at infinity. When is even, the red contour in deformed to the blue one running along the branch cut of √ u 2 + 1.
Since the coefficient of J ( ) is b d+1, , it suffices to consider even which implies that (1 + u −2 ) 2 can be expanded into a polynomial of u −2 and for each term in the polynomial we can use the same contour trick as in the I case to evaluate the integral. The IR-divergent part can be analytically evaluated in mathematica and it has a very simple small behavior. Altogether, the final result for J ( ) is where H is the harmonic number of order . Plugging (6.12) and (6.14) into (6.10) yields the regularized UV-part of the partition function (6.16) which reproduces the Pol(∆) part of log Z ν in [28] 3 without using the heat kernel coefficients α k . In fact, by comparing (6.16) and (6.4), we can express the nonzero heat kernel coefficients α k in terms of b kl

Evaluation of IR part
The IR part can be evaluated through certain zeta function method as in [15] log Notice that the "character zeta function"ζ ν (z) is originally defined by the integral above for z sufficiently large and then analytically continued to small z. b d+1 (ν) is related to the character zeta function as b d+1 (ν) = 1 2ζ ν (0). Combing the UV part (6.15) and IR part (6.18) leads to Compared to the standard heat kernel results, we find the discrepancy between the character zeta functionζ ν (z) and the spectral zeta function ζ ν (z) = 1 This difference is the so-called multiplicative anomaly, arising as a UV correction to the formal factorization log det(AB) = log det A + log det B, as reviewed in [36]. Multiplicative anomaly is computed specifically for fields in AdS in [12], where it's called "secondary contribution". Though the information about multiplicative anomaly is lost , the formal factorization makes the evaluation ofζ ν (z) much simpler than ζ ν (z). For example, we can expand H d,ν (u) with respect to e −u as n e −(∆+n)u and for each fixed n, the u-integral yields (n + ∆) −z . Then we can immediately expressζ ν (z) as a finite sum of Hurwitz zeta functions where δ is an operator defined as and thus Altogether, the full one-loop partition function of a real scalar field with scaling dimension ∆ = 3 2 +ν in AdS 4 is consistent with [28].

Odd dimensional AdS 2r+1
The UV-regularized partition function in AdS 2r+1 can be written in terms of the following character where d = 2r. As in the even dimensional AdS case, we can separate H d,ν (u) into a UV-part and an IR-part. But the resulting IR partition function log Z ir ν = Res u→0 t have a pole at u = 0 by our prescription for the UV-IR separation. Therefore, it suffices to compute the UV-part of the partition function where b k is nonvanishing only when k + is even. By evaluating the residue explicitly, we find where we rewrite b d+1 (ν) as a residue. As expected, there is no log divergence or multiplicative anomaly and hence we can define a renormalized partition function by subtracting all the divergent terms unambiguously where the residue is evaluated in appendix D: For example, according to this expression, the renormalized scalar partition function in AdS 3 is − ν 3 6 , consistent with [37].

Summary
The computations in this section provide a well-defined and efficient rule to obtain a regularized partition function using only the unregularized character integral formula derived in section 3 and section 4. Here we summarize this rule for both even and odd dimensional AdS.

Odd dimensional AdS: d = 2r
Given the total character , χ e (u) are bulk character and edge character respectively, the renormalized partition function (with all negative powers of dropped) is given by For example, a massive spin-s field with ∆ = 2 + ν on AdS 5 has bulk character and edge character χ e (u) = D 6 This result agrees with the computation of [7] based on direct spectral zeta function regularization.
Another interesting example is linearized gravity in AdS 3 . In this case, the bulk character is where the overall factor 2 is spin degeneracy for any field of nonzero spin, and the edge character is χ e (u) = 4 e −u − e −2u . Therefore, according to eq. (6.29) the renormalized partition function of 3D gravity is 13 3 log R, consistent with [37].

Even dimensional AdS: d = 2r + 1
For odd d, we need to consider massive and massless representations separately because in massless representations both gauge field and ghost field can contribute to the total multiplicative anomaly.
Given a massive representation d 2 + ν, s , the unregularized partition function is and similarly forζ e s,ν (z). In AdS 4 , for example, the bulk and edge character zeta functions arē For a massless representation of spin-s and depth-t, the unregularized partition consists of four The renormalized partition function can be expressed as For example, the renormalized partition function of linearized gravity in AdS 4 is where A is the Glaisher-Kinkelin constant. (6.38) reproduces the s = 2 result in [6].

Comment:
Apart from the multiplicative anomaly, the remaining part of the partition function for both massive and massless representations is completely captured by 1 , where χ tot (u) consists of bulk and edge characters. Thus, by inserting a UV regulator u z in the unregularized partition function, we recover the correct log-divergence and finite part up to multiplicative anomaly, which was proved to vanish when summing up the whole spectrum of Vasiliev theories [12]. (We'll also show in section 8 the vanishing of multiplicative anomaly for type-A Vasiliev gravity using the character integral formalism). Keeping this comment in mind, we are free to use the unregularized character integral formula to compute the full partition function of Vasiliev theories in section 8.  [22,25,[32][33][34][35]. In particular, in [25] the authors thoroughly computed the effect of any higher spin currents. Let O µ 1 ···µs be a spin-s current and they found the change of free energy induced by O µ 1 ···µs O µ 1 ···µs has log-divergence when ∆ O = d + s − 2, i.e. O is a conserved current, and the change is of order 1 when ∆ O takes other values. In this section, we'll reproduce the main results of [25] on bulk side by using character integral formula (5.9). As we've just mentioned that the double trace deformation induces the dual boundary condition, it suffices to compute log Z s,ν − log Z s,−ν . We'll focus on the odd d case (the even d case can be analyzed similarly) and see that it's extremely convenient to use the character integral representation to do this computation because flipping the boundary condition is equivalent to switching to the character of the dual representation, i.e. χ

Double trace deformation
We'll start from considering a scalar field with complex scaling dimension ∆ = d 2 + iν and then Wick rotate ∆ to a real number. Before performing any actual computation, we want to mention the following observation which is based on the explicit evaluation in last section, that the UVdivergent part including multiplicative anomaly of log Z ν is an even function in ν when d is odd (see equation (6.16) as an explicit example). This observation implies that log Z iν − log Z −iν is UV finite. In addition, in the difference the integrand is indeed a single-valued function because when u goes around one of the branch picks an extra minus sign. Then we are free to shift the u-contour upwards such that δ > . With this new contour and using the UV-finiteness of log Z iν − log Z −iν , we can safely put → 0 which amounts to sending √ u 2 + 2 to u: To proceed further, we introduce a new integral that can eliminate u in the denominator of (7.2) and then switch the order of integrals After these manipulations, the u-integral is essentially the definition ofμ where we've used that the even part ofμ is the spectral density µ (d) (λ). It's straightforward to generalize this method to higher spin fields by including the edge-mode contribution and using eq. (5.8) Surprisingly, the change of free energy triggered by the higher spin double trace deformation at large N is completely encoded in the higher spin spectral density. Given the eq. (7.5), we make a Wick rotation ν → iν to obtain result for real scaling dimension ∆ = d For example, at d = 3 we get where we've changed variable λ = x − 3 2 . This equation agrees with the result in [25]. When ν reaches some half integer number, say ν s−1 = d 2 + s − 2, which corresponds to a double trace deformation triggered by a spin-s conserved current in free U (N ) vector model, the integral (7.6) is divergent no matter what contour we use because the singularity ν = ν s−1 is at the end point of the integration contour. To extract the leading divergence, we need the singular behavior of µ which is a consequence of eq. (2.9) and D d+2 p−1,s = −D d+2 s−1,p . Notice that D d+2 s−1,s−1 ≡ n KT s−1 is the number of spin-(s−1) Killing tensors on S d+1 and also the number of spin-(s − 1) conformal Killing tensors on S d [25]. Therefore if we truncate the integral (7.6) at ν = ν s − , the change of log Z induced by a spin-s conserved current has a log-divergence part 1 2 n KT s−1 log( ).

Application to Vasiliev theories
With the character integral method developed in the previous sections, we're finally able to compute partition function of Vasiliev theories in all even dimensional AdS (The odd dimensional AdS case can be analyzed similarly and is indeed much simpler). We'll use (non)minimal type-A theory and type-B Vasiliev theory, which are reviewed below, to illustrate the application of the character integral method. Before that we want to stress again, due to the comment at the end of the section 6, the unregularized version of character integral formula is sufficient.

A brief review of Vasiliev theories and Flato-Fronsdal theorems
The simplest and best understood higher spin theory is the nonminimal type-A Vasiliev theory in AdS d+1 , which contains a ∆ = d−2 real scalar and a tower of massless higher spin gauge fields. This theory is believed to be dual to a free U (N ) vector model on boundary described by Lagrangian One direct result of the duality is a one-to-one correspondence between the field content in bulk and the single-trace operators in U (N ) vector model. In representation theory, this is confirmed by Flato-Fronsdal theorem [38]: which can be proved by using the following identity of characters χ so(2,d) There are a lot of variants of the original type-A Vasiliev theory. For example, if we relax the requirement of unitarity, we can take the boundary CFT to be L = 1 For more details about the theory, we refer readers to [10,39]. The bulk dual of this nonunitary CFT is called the type-A higher spin gravity with field content given by a generalized Flato-Fronsdal theorem [39][40][41] Rac ⊗ Rac = where [d + s − p − 1, s] corresponds to a PM field of spin-s and depth-(s − p) for p ≤ s. At the level of characters, this tensor product decomposition is equivalent to In type-A theory, we can further replace the complex scalars by real scalars that are in the fundamental representation of O(N ). The resulting AdS dual is called the minimal type-A theory and its field content can be extracted from the symmetrized tensor product of two Rac : where (s, 1 m ) is a shorthand notation for an so (

Type-A higher spin gravity
Nonminimal theory: Field content of the nonmonimal type-A higher spin gravity is given by eq. where r = √ u 2 + 2 /u. The multiplicative anomaly, if exists, should appear as the coefficient of 0 in the small expansion of (8.10), which can be realized by a change of variable u → u and expanding the integrand around small : ) d (8.11) Notice that the integrand of (8.11) is an odd function of and hence cannot have any 0 term in small expansion. This observation leads to the vanishing of the total multiplicative anomaly in nonminimal type-A theory.
Minimal theory: Since minimal type-A theory contains only fields of even spins, its total partition function can be written as [d+2+s−2,s] (u) (8.12) where the spin in edge characters is shifted by 1. The sum over all bulk characters lead to χ where we've used eq. (8.2) and (8.6). The sum of edge characters, since only odd spin fields are involved, yields the difference between the nonminimal character and minimal character in AdS d+3 : Plugging eq. (8.13) and (8.14) into (8.12), the type-A characters cancel out as in the nonminimal theory and thus the remaining term is log Z (8.15) where in the second line u has been rescaled u → u 2 . Notice that e − d is the Harish-Chandra character of the ∆ = d−2 2 representation of SO (1, d). Then according to the character integral representation of the sphere partition functions found in [15], the partition function of minimal type-A theory on AdS d+1 is the same as the partition function of a conformally coupled scalar on S d . This result agrees with [6], where the appearance of this scalar partition function is interpreted as an N → N − 1 shift in the identification of Newton's constant G N ∼ 1 N . Again, starting from the square root regularized character and following the same argument as in the nonminimal case, one can also show the vanishing of total multiplicative anomaly for minimal type-A theory. Let's also mention that when d = 2r is even, the analogue of (8.15) implies the coefficient of log R of minimal type-A theory in AdS 2r+1 matches the Weyl anomaly of a conformally couple scalar on the boundary, which is a d-dimensional sphere of radius R.

Type-A higher spin gravities
Nonminimal theory: Given the spectrum of nonminimal type-A theory (8.3), the total partition function can be written as where the spin label s is shifted by 1 in the sum of edge characters. Using the generalized Flato- Therefore the total free energy of nonminimal type-A also vanishes.
Minimal theory: Following the same steps as in the minimal type-A case, we can directly write down the result for minimal type-A theory log Z where u is rescaled in the second line. Naively speaking, eq.  [15]. Defining a collection of scalar can be rewritten as where we are allowed to put the Laplacian into a product form because there is no multiplicative anomaly on an odd dimensional manifold. Notice that S d , the Weyl-covariant generalization of , is a GJMS operator on S d [42][43][44][45] and when = 1 it is reduced to the conformal Laplacian on S d . Therefore, the one-loop partition function of minimal type-A theory on AdS d+1 is the same as the one-loop partition function of the -theory on S d . This is again consistent with the N → N − 1 interpretation.

Type-B higher spin gravities
which represents the change of partition function induced by a double-trace deformation. To evaluate this integral, we can either regularize it by inserting u z and express it in terms of Hurwitz zeta function or directly use eq. (7.7) Unlike in AdS 4 , the m = 0 partition function log Z AdS 6 m=0 itself doesn't have the double tracedeformation interpretation because ∆ = 3 and ∆ = 4 are not conjugate scaling dimensions. We'll see that the double-trace deformation pattern can be restored with the m = 1 sector taken into account. The m = 1 sector is more involving since it consists of fields with mixed symmetry. Following the same steps as in section 4, we derive the character integral formula for fields in massive representation 5 2 + ν, (s, 1) sharpen the understanding about edge modes, we'll not try to provide a precise interpretation for it. Summing over all the fields in the m = 1 sector, including the ghosts associated with the s ≥ 2 ones, we end up with a very simple expression Combing (8.24) and (8.26) leads to the total partition function of type-B theory in AdS 6 which apparently has the interpretation of double-trace deformation of a conformally coupled scalar field on S 5 . In higher dimensions, the partition function of m = 0 sector is still trivial. For m ≥ 1, the partition function restricted to the m sector is given by which we've checked up to AdS 16 by mathematica. Summing over all log Z AdS d+1 m , we recover the structure of double-trace deformation of a conformally coupled scalar on S d up to a sign [8] log Z consistent with [7]. For completeness, let's also give the explicit result for any even dimensional (2n)! (2k)! a n (r)ζ(2n + 1) (8.32) where {a n (r)} are defined as r−1 j=0 (x−j 2 ) = r n=1 a n (r) x n . This result contracts with the proposed boundary duality which predicts vanishing one-loop free energy and meanwhile it is too complicated to be accommodated by a shift of N .

Comments on thermal interpretations
In [15], it's argued by using the character integral representations like eq. (1.1) that the oneloop partition function Z (1) PI of a field ϕ on S d+1 is related to the bulk quasi-canonical partition function of ϕ in the static patch of dS d+1 , subject to possible edge corrections localized on the dS cosmological horizon. In this section, we will first briefly review this argument and then explore the generalization to the path integral on EAdS d+1 .

Review the thermal picture in dS
For an inertial observer in dS d+1 , the perceived universe is the (southern) static patch (the de Sitter radius is taken to be 1) which has a cosmological horizon at r = 1 of temperature T = 1 2π . Given certain field content in bulk, the field quanta are in thermal equilibrium with the horizon and we can compute the bulk quasi-canonical partition function as Tr S e −2πH S , where H S is the Hamiltonian which generates times translation in the southern static patch and Tr S denotes trace over the southern multi-particle Hilbert space. At a very formal level, Tr S e −2πH S is supposed to be "equal to" 5 the Euclidean path integral on S d+1 , because operationally Tr S e −2πH S means the path integral on a manifold obtained from the static patch by Wick rotation t = −iτ and identification τ ∼ τ + 2π, which is nothing but the unit sphere S d+1 . We can make sense of this formal argument by using the SO(1, d + 1) Harish-Chandra characters at least at one-loop level. On the path integral side, the (unregularized) one-loop sphere partition function of a (bosonic) field ϕ is given by [15] log Z where χ dS ϕ (u) is the SO(1, d + 1) Harish-Chandra character tr G e −iuH corresponding to the UIR carried by ϕ. Here tr G means tracing over the global single-particle Hilbert space of ϕ. In the thermal picture, we use the ideal gas approximation for thermal partition function at the one-loop level log Tr S e −2πH S = − where ρ S (ω) = tr S δ(ω − H S ) is the density of single-particle states of ϕ in the southern Hilbert space. Define χ S (u) ≡ ∞ 0 dωρ S (ω)(e iωu + e −iωu ) and then log Tr S e −2πH S can be expressed as The density ρ S (ω) is badly divergent because H S has a continuous spectrum due to the infinite redshift near horizon. To make sense of ρ S (ω), we need two steps: • Step 1: Identify ρ S (ω) with the global density of states ρ G (ω) = tr G δ(ω − H) for positive ω.
This identification holds because there exists a one-to-one map between southern and global single-particle states of the same H = ω > 0 induced by the Bogoliubov transformations [46].
• Step 2: ρ G (ω) can be extracted from the Harish-Chandrea character χ dS ϕ (u) (with suitable UV-regularizations like Pauli-Villas, cf. appendix G.2, or dimensional regularization which are suppressed here) Altogether, we have χ S (u) = χ dS ϕ (u) and Tr S e −2πH S is the same as the bulk part of log Z (1) PI . This is the thermal interpretation of the one-loop sphere partition functions.

Thermal picture in AdS
To explore the thermal interpretation of the character integral representation of partition functions AdS, we first need to find coordinate systems of AdS that have a horizon structure. Such coordinates are summarized in the appendix E.

AdS 2
In the 2D Lorentzian AdS, there exist a black hole solution [28] with coordinates, cf. (E.5) and metric ds 2 = −(ρ 2 − 1)dt 2 S + dρ 2 ρ 2 −1 , which shows a point-like horizon at ρ = 1 of temperature T = 1 2π . Wick rotation t S → −iτ and identification τ ∼ τ + 2π yield the 2D Euclidean AdS. Compared to the conformal global coordinate of AdS 2 X 0 = cos t G cos θ , X 1 = tan θ, X 2 = sin t G cos θ (9.7) the black hole solution (9.6) covers the region: θ ∈ (0, π 2 ) and sin θ > | sin t G |, cf. fig (9.1). This scenario is very similar to its dS counter part and hence we're allowed to use the dS argument to claim that the thermal partition function of a field ϕ in the black hole patch of AdS 2 is given by where the noncompact Lorentz generator L 21 ∈ so(2, 1) generates time translation t S → t S + const and χ S (u) is the "character" defined with respect to the single-particle Hilbert space in the black hole patch. Using the Bogoliubov transformations [46], χ S (u) can be replaced by the SO(2, 1) Harish-Chandra character χ HC ϕ (u) = tr G e −iuL 21 which is traced over the global single-particle Hilbert space: At this stage, we want to emphasize that the Harish-Chandra character χ HC ϕ (u), by definition, is completely different from the characters we've used in the previous sections, like χ AdS 2 ∆ (u) = e −∆ 1−e −u . The latter are defined as tr G e −uH for positive u, where H is the global Hamiltonian generating global time translation t G → t G + const. Indeed, these characters are not group characters. So it seems that we cannot naively identify the thermal partition function Tr S e −2πL 21 as the one-loop path integral on Euclidean AdS 2 . However, in the appendix F, we explicitly compute the Harish-Chandra character χ HC ϕ (u) when ϕ is a scalar field of scaling dimension ∆ and we find perhaps  (9.11) in agreement with the path integral result cf. (3.5). Therefore the one-loop path integral on EAdS 2 can be interpreted as the quasi-canonical partition function Tr S e −2πL 21 in the black hole patch.
To further understand why the Harish-Chandra character χ HC ϕ (u) appears in the quasi-canonical partition function Tr s e −2πL 21 , we explore the underlying physical meanings of χ HC ϕ (u) in appendix G. In section G.1, we show that χ HC ϕ (u) encodes the quasinormal spectrum of ϕ in the black hole patch of AdS 2 and in section G.2, we extract a well-defined single-particle density of states in the black patch from χ HC ϕ (u) and show numerically that it can be realized as the continuous limit of the density of states in some simple model with a finite dimensional Hilbert space.

Higher dimensions
In higher dimensional AdS d+1 , the universe perceived by an accelerating observer is called (southern) Rindler-AdS due to the presence of a Rindler horizon [20]. The Rindler-AdS admits a dS d+1 foliation, cf. appendix E: (9.12) and hence has temperature T = 1 2π . In the Rindler-AdS patch, we can use the dS type argument to show that the quasi-canonical partition function of ϕ with spin-s and scaling dimension ∆ is where the noncompact Lorentz generator L d+1,d ∈ so(2, d) generates time translation t S → t S + const and χ HC ϕ (u) is the Harish-Chandra character tr G e −iuL d+1,d . In the appendix F, we argue that x, x). Another approach to this difference is dimensional regularization which works for both the UV and IR divergences, along the line of [11,22]. But the physical picture is not clear if we implement this formal regularization scheme.
We will leave this to future work. I also thank Frederik Denef for reading the paper and providing precious comments. ZS was supported in part by the U.S. Department of Energy grant de-sc0011941.

A Partition function of Dirac spinors
As an example of applying the character integral method to fermions, let's consider a complex Dirac spinor of scaling dimension ∆ = d 2 + ν in AdS d+1 with d = 2r + 1. It carries a highest weight representation s = ( 1 2 , · · · , 1 2 ) ≡ 1 2 of so(d) which has real dimension 2 r+1 . The one-loop partition function of this field is given by It's very likely to have a R 0 piece even when d is even if we implement the IR regulator properly. Of course, the where the spinor spectral function µ 1 2 (λ) is After using the standard Hubbard-Stratonovich trick, the partition function is completely encoded To perform the λ-integral in W 1 2 (u) , we can close the contour in the upper half plane and pick up the poles at λ = in, n ≥ r + 1, where λ tanh(πλ) has residue in π .

Plugging (A.4) into (A.3) yields the unregularized partition function
We can also easily write down the regularized version following the derivation in section 5 Compared to the bosonic case, the only difference is that the representation-independent factor

B Physical interpretation of spectral density/Plancherel measure
In an ordinary quantum mechanical system, given a Hamiltonian H, the associated density of state (DOS) is defined as ρ(E) = Tr δ(H − E) where we trace over the whole Hilbert space. Using the well-known distributional identity 1 x±i = P( 1 x ) ∓ iπδ(x), the DOS can also be formally expressed where R(E) ≡ Tr 1 H−E is the so-called resolvent and the limit → 0 + is understood. In this appendix, we will show that the (scalar) Plancherel measure given by eq. (2.10) can be interpreted as a DOS in the sense of (B.1).
For a real scalar field in EAdS d+1 , we choose Hamiltonian H to be the Laplace-Beltrami operator −∇ 2 which has a continuous spectrum E λ ≡ d 2 4 + λ 2 for all λ ∈ R ≥0 [26]. In this case, the operator 1 H−E is nothing but a scalar Green function G ∆ (X, X ) with mass m 2 = −E ≡ ∆(∆ − d) [47][48][49]: where X, X are points in the embedding space representation of EAdS d+1 . Plugging in E = E λ ±i , the corresponding resolvent R(E λ ± i ) is given by Therefore, combining eq. (B .1) and (B.3), we find that the DOS per volume of −∇ 2 in EAdS d+1 is simply where P → −1 − means that P approaches −1 from the left. (Technically the direction of limit is important, and physically the direction is also fixed because X · X ≤ −1 for any two points on and Notice that what we've obtained here is the number of states per unit "energy" E λ rather than spectral density because the latter is the number of states per unit λ. However, they can be easily mapped to each other by a change of integral measure dE λ = 2λ dλ. This observation suggests us to define the spectral density as where the λ-dependent factor is exactly what we call µ (d) (λ) in eq. (3.9). As a final consistency check, let's reconstruct the scalar heat kernel associated to (−∇ 2 + ν 2 − d 2 4 ) from its canonical definition, i.e. "summing" over all energy eigenfunctions Altogether, the computations in this appendix help us to identify the spectral density or the Plancherel measure of SO(1, d + 1) which has a rigorous mathematical definition in the pure group theory setup [27,51], as the density of states associated to the Hamiltonian H = −∇ 2 in a unit volume of EAdS d+1 up to some representation-independent normalization factors.

C Comparison with dS character integral
In sections 3 and 4, we derived character integral formulae for one-loop partition functions of both scalars and spin-s fields in even dimensional AdS. These formulae are very similar with their dS counterpart derived in [15] , where the unregularized one-loop partition function of a massive spin-s field with scaling dimension ∆ = d 2 + iν is given by (the following formula works for both even and odd d in dS) where extending the sum to n = −1 is a result of locality. Summing over n yields a (bulk+edge) type contribution as in W In this appendix, we'll show that the origin of such similarity between AdS and dS can be traced back to the eq. (2.11).
On the AdS side, we know that the unregularized partition function of a field carrying the where W where in the second line we've shifted n by r. Now let's focus on the spin-s representation, i.e. s = (s, 0, · · · , 0). In this case D d+2 n,s vanishes for n ∈ {−r, −(r − 1), · · · , −2} (and also n = s − 1 but this is irrelevant to our discussion) and hence the sum in eq. (C.5) effectively starts from n = −1 Compared to (C.1), it's clear that the only difference is the absence of e −∆u because in AdS only one boundary mode is dynamical and the other one is identified as a source.

D Evaluation of various residues
This appendix is a collection of technical proofs and results about residues of certain functions appearing in the character integrals. The ultimate goal here is to compute the residue of which itself is also very interesting because we need it to verify the contour prescription proposed in section 5.

First we show by induction that
It's straightforward to check that eq. To use the induction condition, we should lower the power in the denominator which can be realized by integration by part: Applying the induction condition (D.3) to eq. (D.5) yields This confirms that (D.3) holds for all d.
To bridge the gap between Res u→0 G d,ν (u) and Res u→0 F d,ν (u), we need to define another func- It's direct to write down the residue of H d,ν (u) at u = 0 by using its relation with the G-functions which is a polynomial in ν for positive integer d. In addition, the R.H.S of eq. (D.8) is an even function in ν when d is even and an odd function when d is odd. More explicitly, for d = 2r, and for d = 2r + 1, To achieve our original goal, the residue of F d,ν (u) at u = 0 for even d, we need the following differential relation between F d,ν (u) and H d,ν (u) which yields The unknown constant can be easily fixed for even d without any extra effort. This claim follows from the observation that F 2r,0 (u) is an even function in u. Thus Res u→0 F 2r,ν (u) vanishes when ν = 0 and the integration constant has to be zero: where the numerical coefficients {a n (r)} are defined through the following generating function a n (r) x n (D.14)

E Various coordinate systems in Euclidean/Lorentzian AdS
We begin with the embedding space representation of Lorentzian AdS d+1 of unit radius By Wick rotation X d+1 → −iX d+1 , we obtain Euclidean AdS in embedding space The global coordinate for Euclidean AdS is chosen to be where η ≥ 0 and Ω d denotes a point on S d . In this coordinate, the metric is given by In particular when d = 1, choosing Ω 1 = (cos ϕ, sin ϕ), the metic is ds 2 EAdS 2 = dη 2 + sinh 2 η dϕ 2 . Under the Wick rotation ϕ → it, we transform back to Lorentzian signature. In embedding space, it means we choose the following coordinate systems on two patches that cover different portions of Lorentzian AdS Southern : where I've replaced cosh η by ρ ≥ 1. The metric of southern/northern patch can be expressed as which describes a black hole solution with a point-like horizon at ρ = 1, the intersection of the southern and northern patches. The temperature of this black hole is T = 1 2π . When d ≥ 2, there exists a similar Wick rotation that describes a spacetime of the same temperature. Notice that the global coordinate system (E.3) realizes a S d foliation of EAdS d+1 .
By using the Wick rotation between de Sitter static patch and sphere, we obtain a dS foliation of AdS d+1 . More explicitly, we choose the following coordinate system for Ω d where Ω d−2 denotes the usual spherical coordinates of S d−2 . Upon a Wick rotation ϕ → it, Ω d becomes a point on dS d and (E.4) becomes the Rindler-AdS metric [20]. As before, the Wick-rotated coordinate system describes two patches of Lorentzian AdS Southern : in either of which the metric is The two patches intersect at the horizon r = 1 which has the geometry of EAdS d−1 .
Finally, let's also introduce the global coordinate of AdS Global : At time t G = t S = t N = 0, i.e. X d+1 = 0, the southern and northern patches cover the X d ≥ 0 and X d ≤ 0 parts of the global spatial slice respectively. where η M N = diag(−, +, · · · , +, −). The physical interpretation of this algebra will be clear using the following Cartan-Weyl type basis where the trivial commutation relations are omitted. While acting on the AdS d+1 quantum Hilbert space, H can be identified with the Hamiltonian which generates time translation in global coordinates, and M ij can be identified with angular momentum operators. The L ± i can then be viewed as raising/lowering operators for energy eigenstates. We're mainly interested in single-particle Hilbert space H ∆ built from a primary state |∆ (also known as the lowest energy state), i.e. H|∆ = ∆|∆ , L − i |∆ = 0. By construction the Hilbert space H ∆ furnishes a representation of so(2, d) 7 .
In most of the physics literature [29,[52][53][54], the SO(2, d) character of representation H ∆ is computed with respect to a compact Cartan algebra, in particular Hamiltonian and rotations.
Here, for our purpose of thermal interpretations in section 9, we illustrate in (unitary) scalar primary representations how to compute SO(2, d) character associated to a noncompact generator, i.e. generator of boost in AdS d+1 .
On these holomorphic function, the generators of so(2, 1) act as The normalizable function f (z) = z k is an eigenfunction of H with eigenvalue ∆ + k and hence the character associated to H is Upper half-plane realization: Using a fractional linear transformation z → w = 1−iz z−i , we can map the disc D to the upper half-plane H = {x+iy ∈ C : y > 0} and the new Hilbert space consists of normalizable holomorphic functions f (w) on H with inner product [51,55] When acting on a holomorphic function f (w) on H, the algebra so(2, 1) is realized as R + realization [55] and evaluation of character: Given a holomorphic function f (w) on the upper half-plane, we can define a new function F on R + : 10) and the inverse transformation is given by Under the integral transformation (F.10), the inner product (F.8) is mapped to and the so(2, 1) action (F.9) is mapped to To find all eigenfunctions of H in (F.13), let's start from the primary state φ 0 (w) = (w + i) −2∆ in the upper half plane realization. φ 0 can be expressed as a Schwinger parameterization: Comparing (F.14) with (F.11), we immediately get that the dual function of φ 0 (w) in R + is G 0 (ξ) = ξ 2∆−1 e −ξ (dropping unimportant normalization constants). For the dual function of φ k (w), we use the ansatz G k (ξ) = G 0 (ξ)P k (ξ), where P k (ξ) is a polynomial in ξ. Then the eigenequation HG k = (∆ + k)G k yields a second order differential equation of P k (ξ) whose polynomial solution is the generalized Laguerre polynomial P k (ξ) = L (2∆−1) k (2ξ). Thus the spectrum of H is given by By using the recurrence relation of Laguerre polynomial, we can also show that L ± indeed behaves like lower/raise operator As another self-consistency check of this representation, we show the Fourier/Laplace transformation (F.11) of G k (ξ) is φ k (w) up to normalization factors. To do this, we need the series expansion of a generalized Laguerre polynomial L (α) Finally, we are at a stage of actually evaluating the character associated with L 12 by using the new basis G k (ξ) of H ∆ : where (e itL 12 G k )(ξ) = e (1−∆)t G k (e t ξ) is obtained by exponentiating the action of L 12 in (F.13) and (G k , G k ) = Γ(2∆+k) 2 2∆ k! is a result of the orthogonality of generalized Laguerre polynomial: Thus the character Tr e itL 12 can be expressed as If we switch the order of summation and integration mindlessly, the sum is not convergent. To makes sense of this procedure, we introduce a factor (1 − δ) k , δ > 0 This equation is called "Hardy-Hille formula" [56]. With the summation regularized and evaluated, the remaining integral can be computed by using the result on page 91 of [50] Tr e itL 12 = (1 − δ)

G Physics of SO(2, d) character
We will try to build up some physical intuitions about the character χ(t) ≡ Tr e itL d,d+1 that is computed by brutal force in the last appendix. In section G.1, we construct quansinormal modes in Rinder-AdS and show that they are counted by the character χ(t). In section G.2, we compute the density of L 21 eigenstates numerically in AdS 2 by imposing an upper bound on the eigenvalues of the global Hamiltonian H and compare it with the density of states defined as the Fourier transformation of χ(t).

G.1 Quasinormal modes in Rindler-AdS
It is clear that the character Tr e itH , Im t > 0 counts normal modes in AdS d+1 because H is the Hamiltonian in global coordinate. However, the same interpretation does not hold for Tr e itL d,d+1 because L d,d+1 is not a positive definite opeator. Instead, as we will show in the following, it counts resonances/quasinormal modes in Rindler-AdS.
To construct quasinormal modes in an efficient algebraic way [57], it's convenient to define the following dS-type conformal generators: subject to commutation relations (we only show the nontrivial ones that will be used in the deriva- where 0 ≤ µ ≤ d − 1. In terms of embedding space coordinates, the differential operator realization of D, P µ , K µ is Consider a scalar field of scaling dimension ∆. Then its "primary mode" i.e. eigenfunction of D with eigenvalue ∆ which is annihilated by K µ , is given by In the southern Rindler coordinate of AdS 8 , this "primary mode" ψ ∆ (X) descends to linearly independent quasinormal modes whose quasinormal frequency ω n is related to their scaling dimension under D by iω n = ∆ + n.
Notice that and thus the character Tr e −tD counts quasinormal modes. The same construction can be easily generalized to higher spin fields, either massive of massless.

G.2 Numerical computation of density of state
For the southern Rindler-AdS Hamiltonian L d+1,d , the associated density of (single-particle) states can be formally defined as which is also a Fourier transformation of the character χ(t) = Tr e itL d,d+1 The Fourier transformation above is UV-divergent but it can be easily regularized by using a hard cutoff 1 Λ for the lower bound of the t integral. For example, for a scalar field of scaling dimension ∆ in AdS 2 , this regularization yields where γ E is the Euler constant and ψ(x) = Γ (x)/Γ(x) is the digamma function.
On the other hand, approximating ρ Λ (ω) by a model of finite dimensional Hilbert space would provide a more physical interpretation for it. Such an approximation can be easily implemented by imposing a UV cutoff on the spectrum of H. For example in the AdS 2 case, consider a truncated Hilbert space H K generated by G k (ξ) with 0 ≤ k ≤ K and thus the highest energy of H is ∆ + K.
Normalizing G k (ξ) and using the recurrence relations (F.17), L ± are realized as finite dimensional matrices in H K L + k+1,k = − (k + 1)(k + 2∆), L − k,k+1 = − (k + 1)(k + 2∆) (G.10) 9 There are two different definitions of descendants depending on the choice of Hamiltonian: either L 0,d+1 or L d+1,d . Since the Hamiltonian in Rindler-AdS is L d+1,d , the descendants are obtained by acting Pµ on ψ ∆ . Altogether, in this truncated model, the noncompact "Hamiltonian" L 21 = i 2 (L − − L + ) is a sparse (K + 1) × (K + 1) matrix, which admits an efficient numerical diagonalization. With the eigenspectrum {ω k } 0≤k≤K (which is ordered such that ω k+1 ≥ ω k ) obtained from diagonalization, a coarse-grained density of eigenstates can be defined as To compare the character induced density ρ Λ and the discretized densityρ K for a fixed K, we adjust the UV cut-off Λ such that they coincide around ω ≈ 0, i.e. more precisely at the lowest non-negative eigenvalue of L 21 . Such a comparison for ∆ = 3 and K = 2999 is shown in fig. G.1. They agree fairly well in the IR region and hence the UV truncated model is a pretty good approximation for computing density of states.