Normal modes in thermal AdS via the Selberg zeta function

The heat kernel and quasinormal mode methods of computing 1-loop partition functions of spin $s$ fields on hyperbolic quotient spacetimes $\mathbb{H}^{3}/\mathbb{Z}$ are related via the Selberg zeta function. We extend that analysis to thermal $\text{AdS}_{2n+1}$ backgrounds, with quotient structure $\mathbb{H}^{2n+1}/\mathbb{Z}$. Specifically, we demonstrate the zeros of the Selberg function encode the normal mode frequencies of spin fields upon removal of non-square-integrable modes. With this information we construct the 1-loop partition functions for symmetric transverse traceless tensors in terms of the Selberg zeta function and find exact agreement with the heat kernel method.


Introduction
In Euclidean quantum gravity, the main object of interest is the partition function Z = DgDφ e −S E (g,φ)/ , (1.1) where g is the dynamical metric and φ represents all other matter fields. The leading order quantum effects are captured by the 1-loop partition function Z (1) φ . For a free field φ on a gravitational background M, computing Z (1) φ involves calculating functional determinants of kinetic operators ∇ 2 φ,M . This perturbative approach is useful when finding quantum corrections to black hole entropy [1][2][3] and holographic entanglement entropy [4].
There are several methods for computing functional determinants of kinetic operators. Two such methods we consider are known as the heat kernel method (c.f. [5]) and the quasinormal mode method [6]. We outline the heat kernel method in Appendix B and give a more extensive review of the quasinormal mode method in Section 2.2.

(1.2)
Here ∆ is a function of the mass m of the field in question, and d 0 and d p are the degeneracies of the zeros ∆ 0 and poles ∆ p , respectively. For examples we consider, ∆ is the conformal dimension of the field theory operator dual to the bulk field in question. It was shown in [6,7] that Z (1) (∆) for a massive scalar field φ living on a thermal AdS d+1 background may be expressed in terms of quasinormal modes 1 ω * (∆): Here * stands for a collection of additional quantum numbers, such as angular momentum, and ω n (T ) are the Matusubara frequencies of the thermal background at temperature T arising from the condition that φ is regular in the Euclidean time coordinate. For stationary spacetimes, the Matsubara frequencies will generalize from ω n (T ) = 2πinT to a function that depends on the angular momentum quantum number [8].
Recently the authors of [9] showed how to connect the heat kernel and quasinormal mode methods on the Bañados, Teitelboim and Zanelli (BTZ) black hole [10]. In particular, the two methods were formally related via the Selberg zeta function Z Z (z) [11], a zeta function that is built entirely from the quotient structure of H 3 /Z: 1 − e 2ibk 1 e −2ibk 2 e −2a(k 1 +k 2 +z) .
(1. 4) In (1.4), a and b are geometric quantities 2 . Specifically, the authors of [9,12] showed that when the zeros of the Selberg zeta function z * are identified with ∆ s ± isb a , and when the Selberg integers k 1 and k 2 are appropriately recast via a relabeling inspired by scattering theory [13], the quasinormal mode frequencies ω * are equal to the Matsubara frequencies ω n (T ): ∆ s ± isb a = z * ⇐⇒ ω * (∆) = ω n (T ) . (1.5) Condition (1.5) leads to the observation: if any two of (i) the Selberg zeta function, (ii) the Matsubara frequencies, or (iii) the quasinormal mode frequencies of a spacetime and field are known, then one can reconstruct the third. This observation provides a means of predicting quasinormal mode (or Matsubara) frequencies of fields on locally thermal AdS 3 spacetimes. In this article we extend the results of [9,12] in two ways. First we confine ourselves to thermal AdS 3 , and build the connection between heat kernel and quasinormal mode methods presented in [12] using instead the normal mode frequencies of general spin s fields. We find that the relabelings of the Selberg integers k 1 and k 2 are the same as for the BTZ black hole 1 Quasinormal modes are eigenmodes of dissipative systems, such as those modes obeying infalling boundary conditions at a black hole horizon. The quasinormal mode method can be applied to spacetimes without horizons, such as thermal AdSN . However, in these cases the quasinormal modes are replaced by normal modes as damping does not occur in such backgrounds.
2 For the BTZ black hole a = πr+ and b = π|r−|, where r− and r+ are the inner and outer horizon radii.
For thermal AdS3, a = 1/(2T ) and b = θ/2, where T is the temperature and θ is an angular potential.
[13], except with the thermal quantum number n and angular quantum number switched. This connection is not surprising, in that the BTZ black hole and thermal AdS 3 are related via a modular transformation. However, this extension provides a testbed in which to study the ideas presented in [9,12]. Indeed, using the Selberg formalism we are able to "predict" the known normal modes of spin s fields on thermal AdS 3 . Second, we extend [9,12] to higher dimensional thermal AdS 2n+1 . To do this, we employ the higher dimensional generalization of (1.4), the Selberg zeta function on H 2n+1 /Z [11,14]: (1.6) We conjecture an augmented relabeling for the integers k i , generalizing [13]. We do this first for a complex scalar field. We then move to higher spin s fields, and write an explicit formula for the 1-loop partition function for symmetric transverse traceless tensor fields on thermal AdS 5 in terms of the Selberg zeta function. We then discuss a generalization to AdS 2n+1 . Our note is organized as follows. We begin Section 2 with a brief review of the geometry and quotient structure of thermal AdS 3 (with non-zero angular potential), and demonstrate how the quasinormal mode method is used to calculate the 1-loop partition function for scalar fields on this spacetime. We then relate the zeros of the Selberg zeta function and normal mode frequencies of arbitrary spin fields on thermal AdS 3 . In Section 3 we extend our analysis to thermal AdS 2n+1 , both without and with non-zero angular potentials. Concluding remarks are given in Section 4. To keep the article self-contained, Appendix A gives an overview of the geometry of Euclidean AdS 2n+1 , and Appendix B reviews basic elements of the group theoretic construction of the heat kernel on hyperbolic spaces [15,16].

Geometry of Thermal AdS 3
Anti-de Sitter space in three dimensions in global coordinates takes the form where L is the AdS radius. These coordinates have ranges −∞ < t < ∞, 0 < ρ < ∞, and 0 < φ < 2π. Written in global coordinates, it is clear AdS 3 is static and axially symmetric, symmetries generated by the Killing vectors H ≡ i∂ t and J ≡ i∂ φ . These vectors can be used to define a notion of energy and angular momentum and define a pair of conserved charges. We are interested in studying free quantum field theory on a fixed AdS 3 background. Upon quantization the vectors H and J become operators on the field theory Hilbert space such that the Hilbert space is organized into states of fixed energy and angular momentum. To properly define quantum field theory on an AdS 3 background we analytically continue t → −it E , to obtain the Euclidean AdS 3 metric Euclidean AdS 3 is equivalent to the hyperbolic space H 3 . To obtain a thermal spacetime, we periodically identify the Euclidean time coordinate t E and the angular coordinate φ via where β is defined as the inverse temperature and θ is an angular potential. The identifications (2.3) allow us to recast the path integral (1.1) as a thermal partition function 3) is known as thermal AdS 3 . The identifications (2.3) generate the group Z, and so thermal AdS 3 is topologically equivalent to the hyperbolic quotient H 3 /Z. We can view H 3 /Z as a solid torus with a T 2 S 1 θ × S 1 β boundary and modular parameter 2πτ = θ + iβ [5]. We can see from the ρ = 0 behavior of (2.2) that the Euclidean time circle t E ∼ t E + β is non-contractible, so S 1 θ fills in the solid torus.

Normal modes and the 1-loop partition function
Here we review the derivation of the 1-loop partition function of a spin-s field living on thermal AdS 3 via the quasinormal mode method [6]. Our discussion is slightly more general than the one provided in [6] as we consider arbitrary spin-s fields and θ = 0. For concreteness we first consider a massive complex scalar field ϕ of mass m. The chief idea of the quasinormal mode method is to assume Z (1) is a meromorphic function of some mass parameter ∆, and then analytically continue this mass parameter to the complex plane. For a scalar field the mass parameter is the conformal dimension ∆ = 1 + 1 + (mL) 2 (2.5) of the conformal field theory operator dual to ϕ. If Z (1) (∆) is meromorphic in ∆, we may use Weierstrass's factorization theorem (1.2) and express the 1-loop partition function as a product over its zeros and poles up to a entire function 3 Poly(∆). Since Z (1) scalar ∝ (det∇ 2 ) −1 , it has no zeros but will have a pole whenever ∇ 2 has a zero mode. Zero modes occur when ∆ is tuned such that the Klein-Gordon equation has a smooth, single-valued solution ϕ = ϕ * ,n in Euclidean signature which obeys the asymptotic boundary conditions, (2.8). Here n labels the mode number in the Euclidean time 3 As described in [6], Poly(∆) hides UV divergences. For example, a scalar field in AdS will contribute its zero point energy, ∞ κ=0 (κ + 1) κ+∆ 2T , to Poly(∆). To have a complete accounting of the 1-loop partition function we must determine Poly(∆). We are not interested in this divergent term and will often drop it from our calculations. Poly(∆) is fixed by imposing the correct large ∆ behavior and using the heat kernel coefficients of the Laplacian ∇ 2 as in [17]. The term Poly(∆) is proportional to the volume of H 3 /Z. direction and * represents all other quantum numbers. The associated ∆ for which ϕ * ,n solve the Klein-Gordon equation are denoted ∆ * ,n . Thus, poles in Z (1) (∆) occur when ∆ = ∆ * ,n .
It was first realized in [6] that for scalar fields on AdS black hole backgrounds, the (anti)quasinormal modes are Lorentzian modes that are purely (out)ingoing at the horizon, and satisfy the asymptotic boundary conditions. Both conditions can only be satisfied at a set of discrete frequencies ω * (∆), and setting ∆ = ∆ * ,n is equivalent to setting the black hole quasinormal frequencies to the Matsubara frequencies ω n (T ) arising from the Euclidean periodicity condition, ω * (∆ * ,n ) = ω n . The 1-loop partition function is therefore computed using (1.3). Despite AdS not having a horizon, the quasinormal method of computing 1-loop determinants can nonetheless be applied in this context. Because there is no horizon, the key change is that there is no dissipation, and so the quasi normal frequencies ω * become normal frequencies, and that there is no sensible difference between ingoing and outgoing modes.
The exact normal frequencies 4 for scalar fields are known [6,19] and can be written as: where p = 0, 1, 2, ... is the radial quantum number and is the angular momentum quantum number, ∈ Z. The mode frequencies (2.7) are found by solving the Klein-Gordon equation and imposing Dirichlet boundary conditions. From here on we will set L = 1. The Matsubara frequencies are found via the large ρ limit of the solution to the Klein-Gordon equation after imposing that ϕ is periodic under identifications (2.3). The large ρ limit of ϕ is [19] ϕ Upon Wick rotation t → −it E and demanding that ϕ be periodic under (2.3) sets the frequency ω to a particular form ω n (T ), Here the thermal integer n ranges over all integers. The frequencies ω n (T ) in (2.9) 5 are known as the Matsubara frequencies with a shift by a chemical potential µ = −θT . We can now construct the 1-loop determinant for a scalar field on thermal AdS 3 with θ = 0. Substituting the normal mode frequencies (2.7) and Matsubara frequencies (2.9) into 4 The AdS3 normal frequencies can be obtained from the BTZ quasinormal frequencies via the identifications, r− → 0, and r+ → ±iL where L is the AdS length scale, and r± are the outer and horizon radii. Correspondingly, TBTZ → ∓2πiL. These are the same changes which transform the (Lorentzian) rotating BTZ metric in Boyer-Lindquist form with ADM mass and angular momentum M = −1 and J = 0, respectively, into (Lorentzian) AdS3 in global coordinates [18]. 5 Using the modular transformation τ AdS 3 = −1/τBTZ it is straightforward to show the Matsubara frequencies of thermal AdS3 with θ = 0 transform into the Matsubara frequencies for the Euclidean BTZ black hole with rotation (e.g., take Eqn. (2.8) of [8], rewrite the left and right temperatures TL and TR in terms of τBTZ and then perform the modular transformation).
the expression for the 1-loop partition function over its poles (1.3), we find: (2.10) To get to the second line we cast the product over n ∈ Z into a single product over n > 0, absorbing any UV divergent pieces into the Poly(∆) contribution, and moving to the third line regulated the product over n using the identity n>0 1 + x 2 n 2 = e πx πx (1 − e −2πx ), again absorbing any UV divergent contribution into the entire function Poly(∆).
We recast the product over ∈ Z into one over ≥ 0, introducing a degeneracy factor D (1) : .

(2.11)
Here D . When we set θ = 0 we recover the expression of the 1-loop partition function found in [6].
Taking the logarithm and evaluating the sums over p and , we arrive at (2.12) Up to the entire function Poly(∆), our expression (2.12) matches the 1-loop determinant − log det(−∇ 2 +m 2 ) of a scalar field on thermal AdS 3 found using the heat kernel method [15] (set s = 0 and n = 1 in (B.11)). The expression (2.12) is also the 1-loop determinant for the BTZ black hole because it is invariant under the modular transformation τ AdS 3 → −1/τ BTZ .

Normal Modes from Selberg Zeros and Higher Spin
In [9,12] the authors formally connected the heat kernel and quasinormal mode methods using the Selberg zeta function. The Selberg zeta function Z Γ is a zeta function that is built entirely from the geometry of the hyperbolic quotient The parameters a and b are related to the identifications (2.3) of thermal AdS 3 ; specifically, 2a = β and 2b = θ. The zeros z * of the Selberg zeta function (2.13) are 14) where N ∈ Z.
One-loop determinants can be recast in terms of the Selberg zeta function [9,13,20], e.g., for a scalar, log det∇ 2 s=0 = 2 log Z Γ (∆). It is interesting to see what happens when we set the argument of the Selberg zeta function, ∆, to the zeros (2.14): Notice that when we suggestively relabel the integers k 1 , k 2 and N such that we find that (2.15) becomes That is, tuning the conformal dimension ∆ to the zeros z * of the Selberg zeta function gives us the condition that ω n (T ) = ω * (∆). The relabeling of integers k 1 , k 2 is not ad hoc. Rather, (2.16) comes from spectral theory on the hyperbolic quotient H 3 /Z [13], where k 1 , k 2 are repackaged into new integers p ≥ 0 and ∈ Z such that the zeros of the Selberg zeta function coincide with the poles of the so-called scattering operator ∆ Γ , i.e., the positive Laplacian acting on the Hilbert space.
The observation that ∆ = z * leads to ω n (T ) = ω * was noted in [9], and generalized to include higher spin fields in [12] in the context of a rotating BTZ background. Our above analysis for thermal AdS 3 is largely the same as the one performed for the BTZ black hole [9], however, there are some key differences, specifically the physical interpretation of the integers appearing in the redefinition (2.16). For the BTZ black hole, the relabeling of k 1 , k 2 is Comparing to the relabeling for thermal AdS 3 (2.16), we observe that the roles of the thermal integer n and angular momentum quantum are swapped. This is reminiscent of the topology of each of these spacetimes: the thermal time circle for the BTZ black hole is contractible, while it is non-contractible for thermal AdS 3 . Therefore, we see that the interpretation of k 1 − k 2 and N are linked to spacetime topology, and signals the fact that the BTZ black hole and thermal AdS 3 are related via a modular transformation. It was also emphasized in [9] that given knowledge of any two of (i) Matsubara frequencies, (ii) (quasi)normal mode frequencies, and (iii) the zeros of the Selberg zeta function, the third can be constructed. This provides us a means of predicting the normal modes of a field on thermal AdS 3 . Let us use this predictive power to uncover the normal mode frequencies for an arbitrary spin-s field in a thermal AdS 3 background.
We begin by considering spin-s bosonic fields of mass m s . The 1-loop determinant can be cast as a product of Selberg zeta functions [9,12,20] log det(−∇ 2 (s) + m 2 where ∆ s = 1 + s + 1 + m 2 s is the conformal dimension of the dual CFT 2 operator [21]. At this point the arguments ∆ s ± isθ β are perhaps unmotivated, but we will shed light on them shortly. Setting the arguments ∆ s + σ ∆ isθ β , where we use σ ∆ to denote the ± sign, to the zeros (2.14) leads to: Then, motivated by the relabeling (2.16) of [13], and using the form of the Matsubara frequencies (2.9), we may write down where σ s = ±1. For example, when σ ∆ = −1, i.e., considering the argument ∆ s − isθ β , and selecting σ s = −1, we have This allows us to read off the normal mode frequency ω * (∆ s ) = 2p + | + s| + ∆ s . Collectively, the relabeling (2.21) substituted into (2.20) gives the condition ω * (∆ s ) = ω n (T ) where we identify the normal mode frequencies of a spin-s boson in thermal AdS 3 (2.24) Our relabeling (2.21) can be understood as a generalization of the redefinitions of integers k 1 and k 2 from [13]. A similar generalized relabeling was found for spin-s bosons on the rotating BTZ background in [12], where the relabeling led to ω n = ω * (∆ s ) for only square-integrable zero modes. More specifically, the Euclidean zero modes of higher spin fields will be non-square integrable for low lying values of p. The removal of these non-square integrable modes leads to the conditions ω n = ω * (∆ s ) (see, e.g., Appendix B of [8]). Consequently, it was further shown in [8] that the conditions ω n = ω * (∆ s ) are equivalent to ω n = ω scalar * ∆ s + σ ∆ isθ BTZ β BTZ , where σ ∆ depends on the sign of m s , i.e., σ ∆ = −1 corresponds to m s > 0.
Likewise, higher spin fields in thermal AdS 3 will have unphysical non-square integrable Euclidean zero modes for particularly low lying values of p. The removal of these modes leads to the our conditions ω n = ω * (∆ s ) summarized by (2.20) with the relabeling (2.21). As such, we also have that arguments of the Selberg zeta functions in (2.19) come from removing the non-square integrable zeros modes and reexpressing ω n = ω * (∆ s ) for scalar fields with ∆ → ∆ s + σ ∆ isθ β . Indeed, the 1-loop partition function for a spin-s boson can be written as where Z (1) (∆) is the 1-loop partition function for a scalar field (2.11) without Poly(∆). Lastly, we note that our method works equally well for spin-s fermions. The 1-loop determinant for spin-s fermions with anti-periodic boundary conditions is The anti-periodic boundary conditions along the Euclidean circle force n → n+1/2. As in the case of spin-s fermions on a BTZ background [12], we again find that anti-periodic boundary conditions along the φ cycle are imposed upon us, such that → + 1/2. Our spin-s fermion result is the same as the spin-s boson one (2.21), except with n → n + 1/2 and → + 1/2. In summary, the zeros of the Selberg zeta function encode the normal mode frequencies of arbitrary spin-s fields which propagate on thermal AdS 3 . Moreover, setting the arguments ∆ s ± isθ β (for bosons) or ∆ s ± isθ β + iπ β (for fermions) equal to the zeros of the Selberg zeta function leads to the condition the Matsubara frequencies are aligned with the normal mode frequencies:

Extending to Thermal AdS 2n+1
We now extend the analysis presented in Section 2 to thermal AdS 2n+1 . The geometry of AdS 2n+1 is reviewed in Appendix A. Generally, thermal AdS 2n+1 can be viewed as the quotient space H 2n+1 /Z. The quotient structure arises from the periodic identification of the Euclidean time coordinate t E and the remaining angular coordinates {φ 1 , ..., φ n+1 } being identified via (t E , φ 1 , ..., φ n ) ∼ (t E + β, φ 1 + θ 1 , ..., φ n + θ n ) , where β is the inverse temperature and θ i are angular potentials. Written in global coordinates (A.4), we see AdS 2n+1 has symmetries generated by the Killing vectors H ≡ i∂ t and J i ≡ i∂ φ i , defining phase space charges associated with energy H and angular momenta J i . Upon quantization, the vectors H and J i organize the field theory Hilbert space into states of fixed energy and angular momenta. Field theory quantities are computed using the canonical ensemble partition function Z(β, θ i ), generalizing the thermal partition function in AdS 3 , (2.4). Evaluating the partition function (3.2) is equivalent to calculating a Euclidean path integral on H 2n+1 /Z. The leading order quantum effects are quantified by the 1-loop partition function Z (1) . The 1-loop partition function for complex scalar fields on thermal AdS d+1 without angular potentials was computed using the quasinormal mode method in [6]. The 1loop determinant for STT tensor fields on thermal AdS 2n+1 with non-zero angular potentials was constructed using the heat kernel method [16,22] (also reviewed in Appendix B).
Here we explore the connection between the 1-loop partition function, the Selberg zeta function on H 2n+1 /Z, and the normal modes of STT tensor fields on thermal AdS 2n+1 , both with and without angular potentials. We begin by considering STT tensor fields on thermal AdS 2n+1 with θ i = 0. The remaining discussion will mirror the presentation in Section 2, where we begin with a scalar field on thermal AdS 2n+1 when θ i = 0. We will then consider higher spin field partition functions, where we explicitly write the 1-loop partition function in AdS 5 in terms of a higher dimensional Selberg zeta function.

Thermal AdS
It is straightforward to generalize the complex scalar field 1-loop partition function in [6] to include symmetric, transverse, traceless (STT) tensor fields of spin-s 6 where ∆ s is the conformal dimension ∆ s = d 2 + s + d 2 4 + m 2 s . Here d s is the dimension (B.6) when d + 1 is odd, and D (d−1) is the degeneracy of the th angular momentum eigenvalue Some degenerate cases of note include when d = 1 for which D where we ignore the Poly(∆ s ) term. Performing the sums of p and we find, (check steps here) matching the heat kernel result (B.9), with θ i = 0. Interestingly, one can build the 1-loop partition function (3.6) in odd dimensions using multiple copies of the AdS 3 result, (2.12) (when θ = 0). For example, consider thermal AdS 5 .
6 By spin-s we mean unitary irreducible representations of SO(2n + 1) under which the fields transform. We further restrict ourselves to symmetric transverse traceless representations of SO(2n + 1), the highest weight representations, which greatly simplifies our study. Such fields include bosons of spin-s.
which matches the scalar result (3.6). We can likewise build the full 1-loop determinant in AdS 2n+1 , where for each additional odd dimension, introduce another p i and i . For example, in the AdS 7 case let p → p 1 + p 2 + p 3 and → 1 + 2 + 3 , and so forth 7 . We will show that introducing additional integers p i and i is motivated from the higher dimensional Selberg zeta function.

Scalar fields
To compute the 1-loop partition function, we must have knowledge of the Matsubara frequencies ω n and the normal mode frequencies for a scalar field on AdS 2n+1 . The periodic identification (3.1) imposed on a scalar field ϕ leads to a generalization of the Matsubara frequencies in thermal AdS 3 (2.9), whereñ ∈ Z is the thermal integer and i ∈ Z is the angular momentum quantum number with respect to each φ i in the geometry (A.5). The normal mode frequencies for scalar fields on AdS 2n+1 were calculated explicitly in [19] in terms of a single radial quantum number p and angular momentum quantum number . Let us instead use the Selberg zeta function to "predict" these normal mode frequencies, extending our algorithm developed for thermal AdS 3 to higher-dimensional thermal AdS. The Selberg zeta function of H 2n+1 /Z is given by [11,14] where 2a is the length of the primitive closed geodesic and e 2ib i are the eigenvalues of a rotation matrix A describing the rotation of nearby closed geodesics under the Poincaré recurrence map. In the context of thermal AdS 2n+1 , we identify 2a = β and 2b i = θ i . The zeros of Z Z (z) occur at the special value of z * , (3.10) where N ∈ Z.
Taking the logarithm and evaluating the resulting sums over integers k 1 , ..., k 2n leads to where we have introduced a 'modular' parameter for each angular potential, such that q 1 ≡ e 2πiτ = e i(θ 1 +iβ) , q 2 ≡ e 2πiτ = e i(θ 2 +iβ) , ... Setting ∆ = z * gives us (3.14) Motivated by the relabeling (2.16) of k 1 and k 2 from thermal AdS 3 , we consider the following relabeling where p i is a non-negative integer, i ∈ Z, and σ is the sign of i . For example, in thermal AdS 5 , we have that (3.15) The sign σ is written such that σ = +1 when k 1 − k 2 = + 1 and k 3 − k 4 = + 2 , and σ = −1 when k 1 − k 2 = − 1 and k 3 − k 4 = − 2 . We do not consider any mixture of signs, e.g., k 1 − k 2 = + 1 and k 3 − k 4 = − 2 . This can also be accomplished by setting the sign of 1 , and then relabeling each subsequent k 2i−1 − k 2i to have the same sign as 1 . Using our conjectured 8 relabeling (3.15) we arrive at where p ≡ p 1 + p 2 + ... + p n . Recognizing the right hand side as the Matsubara frequencies (3.8), we 'predict' the normal mode frequencies for a scalar field in AdS 2n+1 to be where we again find that tuning the conformal dimension ∆ to the zeros of the Selberg zeta function, we uncover the condition ω * (∆) = ω n (T ). As a consistency check, we can substitute the Matsubara (3.8) and normal mode frequencies (3.18) into our expression for the 1-loop partition function (1.3) and show that we reproduce the 1-loop partition function calculated using the heat kernel method (B.12). For concreteness, consider AdS 5 . Following similar steps as in the AdS 3 case, (2.11), we obtain log Z Or, using (q 1q1 ) = (q 2q2 ) = e −2β we may write log Z (3.20) Note that when we turn off the angular potentials, define p 1 + p 2 ≡ p, | 1 | + | 2 | ≡ and make the replacement D we recover the logarithm of (3.3) for scalars. Taking the logarithm and evaluating the sums over integers p 1 , p 2 , 1 , and 2 , we arrive to the 1-loop partition function computed using the heat kernel method (B.12) at s = 0. The above procedure holds for the scalar field AdS 2n+1 . Each higher dimension includes an additional p i and i , from which it is straightforward to show log Z where ... implies the permutations q 1 q 2 ...q n and their complex conjugates corresponding to rewriting the products over i to range from all integers to all non-negative integers. Evaluting the sum over i and p i yields the s = 0 1-loop partition function computed using the heat kernel method (B.11).

Higher Spin
We now turn to building the 1-loop partition functions of STT spin-s fields on thermal AdS 2n+1 . This was accomplished using the heat kernel method in [16]. To use the quasinormal mode method, we need the Matsubara frequencies (3.8) and the normal mode frequencies for STT tensor fields. We will follow our approach in Section 2 and uncover the normal mode frequencies using the zeros of the Selberg zeta function. In the AdS 3 case, recall we first recast the 1-loop determinant for arbitrary spin-s fieldsfound via the heat kernel method -in terms of a product of Selberg zeta functions. We then tuned the arguments of the Selberg zeta functions to their zeros and extended the relabeling of integers k 1 and k 2 from [13]. Our first task then is to rewrite the 1-loop partition function (B.11) in terms of Selberg zeta functions. For concreteness, we will consider first the AdS 5 case and then comment on general AdS 2n+1 .
The Selberg zeta function for H 5 /Z is (3.11) leading to the 1-loop partition function for STT tensors in thermal AdS 5 in terms of Selberg zeta functions For example, for spin-2 fields we have log Z (3. 26) We see that, just as in the AdS 3 case (2.25), the spin-s 1-loop partition function on AdS 5 breaks into a product of scalar 1-loop partition functions, where Z (1) (∆) is given by (3.20). The arguments of the Selberg zeta function (3.25) arise from an analysis similar to the AdS 3 set-up where we must remove the non-square integrable zero modes for particularly low lying values of p i andñ. Setting the arguments of the Selberg zeta functions (3.25) to the zeros (3.10) for n = 2, we obtain (3.28) with m, m = 0, 1, ..., s. We now extend the relabeling presented in (3.16): Since there are (s + 1) 2 combinations of m and m , we have (s + 1) 2 different relabelings of the pairs k 1 , k 2 and k 3 , k 4 . Collectively, the relabeling (3.29) gives where p ≡ p 1 + p 2 . We are inclined to interpret the left hand side as the set of normal mode frequencies ω * (∆ s ), such that (3.30) becomes ω * (∆ s ) = ω n (T ). Unsurprisingly, it is straightforward to confirm that substituting the (s+1) 2 collection of normal mode frequencies into the 1-loop partition function (1.3) leads to the spin-s result found via the heat kernel method. We expect that the normal mode frequencies displayed in (3.30) will arise from a spin-s quasinormal mode analysis on thermal AdS 2n+1 , whereupon we remove non-square integrable Euclidean zero modes. We leave this confirmation for future work.
Our study of 1-loop partition functions of higher spin on AdS 5 informs us of how to extend the result to AdS 2n+1 . Specifically, the spin-s 1-loop partition function Z  s (∆ s ) is a product of two scalar field partition functions, such that ∆ is replaced by ∆ s ± isθ 1 /β, and the magnitude of the integer coefficient to iθ 1 /β matches the spin of the field (2.25), corresponding to d (1) s = 2 for any s > 0. Moreover, for AdS 5 , the 1-loop partition function is the product of (s + 1) 2 scalar field partition functions, such that the magnitude of the integer coefficients in front of each iθ i /β sum to the total spin of the field. For example, for s = 2, we replace ∆ with each argument appearing in the Selberg zeta functions in (3.26).
Likewise, we may build Z (1) s (∆ s ) on AdS 2n+1 from a product of scalar field partition functions on AdS 2n+1 , where the number of terms in the product correspond to the number of replacements to ∆, where the magnitude of the integer coefficients in front of each iθ i /β must sum to the spin of the field s. For example, for a spin-1 field on AdS 7 , Z 1 (∆ 1 ) is given by a product of six scalar field 1-loop partition functions corresponding to the replacements of ∆ by ∆ → ∆ 1 ± iθ i β , for i = 1, 2, 3 (using d (5) s = (s + 3)(s + 2) 2 (s + 1)/12). For each higher dimension, the number of relabelings (3.29) will also go as d

Discussion
We have extended the relationship between the heat kernel and quasinormal mode methods for computing 1-loop determinants, as explored in [9,12], to H 2n+1 /Z, i.e., thermal AdS 2n+1 . First we considered arbitrary spin-s fields propagating on a thermal AdS 3 background and showed that by tuning the zeros of the Selberg zeta function to conformal dimensions ∆ s , we arrive at the condition that the normal modes must be identified with the Matsubara frequencies of thermal AdS 3 . Comparing to our previous work with the BTZ black hole, we observed the relabeling of the integers k 1 , k 2 depends on which circle of the solid torus H 3 /Z is filled in. We then extended our analysis on H 3 /Z to higher dimensional thermal AdS 2n+1 for arbitrary STT tensor fields. These generalizations to [9,12] allowed us to derive the normal modes for STT tensors on thermal AdS 2n+1 with angular potentials, using the zeros of the Selberg zeta function. With the normal modes we verified for consistency that the sum over radial and angular momentum quantum numbers p and i build up the global characters of SO(2n + 1, 1), and the sum over images arises from taking the logarithm of the functional determinant, matching the heat kernel results found in [16]. Finally, we developed an algorithm for recasting the higher spin field 1-loop partition functions as a product of Selberg zeta functions on H 2n+1 /Z.
An extension to this work would be to consider non-hyperbolic spacetimes that possess sufficient symmetry, e.g., the sphere S N and the quotients S N /Γ. In fact, it is straightforward to extend our analysis from AdS N to S N via a formal Wick rotation, where the normal modes of thermal AdS become the quasinormal modes of Euclidean de Sitter space. Understanding the relationship between the heat kernel and quasinormal mode methods of computing 1-loop determinants on S N and S N /Γ may lead to deeper insights into de Sitter quantum gravity. Currently this work is underway.

A Geometry of AdS 2n+1
One way to obtain the metric of global Euclidean AdS 2n+1 is to perform a double Wick rotation of the sphere S 2n+1 metric. Specifically, define the coordinates of the S 2n+1 sphere in terms complex numbers (z 1 , z 2 , ..., z n+1 ) such that each with a phase φ i . For example, for S 3 there are two complex numbers z i with phases φ 1 , φ 2 ; for S 5 there are three complex numbers z i with φ 1 , φ 2 , φ 3 , and so forth. It is often useful to decompose the complex numbers into real coordinates {x i } that embed the sphere S 2n+1 into R 2n+2 . For S 5 the three complex numbers are decomposed into six real x 1 = cos θ cos φ 1 , x 2 = cos θ sin φ 1 , x 3 = sin θ cos ψ cos φ 2 , The corresponding line element for S 5 is where dΩ 2 3 is the 3-sphere line element written in Hopf coordinates, dΩ 2 3 = dψ 2 + sin 2 ψdφ 2 3 + cos 2 ψdφ 2 2 . Generalizing this procedure to 2n + 1 dimensions, the metric for S 2n+1 is Euclidean AdS 2n+1 is now obtained by Wick rotating θ → −iρ and φ 1 → it E , where ρ, t E ∈ R, and ds → ids, leading to In what follows we will relabel the phases φ i appearing in dΩ 2 2n−1 so that they run from i = 1, ..., n − 1. For example, replace dΩ 2 3 → dψ 2 + sin 2 ψdφ 2 1 + cos 2 ψdφ 2 2 in the line element for AdS 5 .

B Group Theoretic Construction of the Heat Kernel
The heat kernel method [15,16] is used to compute 1-loop functional determinants by constructing the heat kernel K (s) (x, y; t) with respect to the normalized eigenfunctions ψ The subscripts a and b to denote the local Lorentz indices of the spin-s field [16]. The 1-loop partition function Z (1) s is given in terms of the trace of the coincident heat kernel K (s) (t) When M is a highly symmetric homogeneous space G/H, group theoretic techniques can be used to write down the eigenfunctions ψ (s) n,a of the spin-s Laplacian ∇ 2 (s) in terms of matrix elements of representations of the symmetry group. These techniques were developed by Camporesi and Higuchi [23][24][25][26][27] and adapted for thermal AdS by [15,16]. Here we summarize the findings of [15,16].
Euclidean AdS N is the homogeneous space H N SO(N, 1)/SO(N ). Building the heat kernel on H N requires that we construct the eigenfunctions associated with a section 10 in SO (N, 1). Specifically, the eigenfunctions are determined by the matrix elements of unitary representations of SO(N, 1) containing unitary representations of SO(N ). When N = 2n + 1 these are the principal series representations 11 of SO(2n + 1, 1) labeled by the array R = (iλ, m) for m = (m 2 , ..., m n+1 ) with each m i being a non-negative half integer, λ ∈ R. For STT tensor fields, the array simplifies such that m = (s, 0, ..., 0). The eigenvalues E (s) for STT tensors is given by the quadratic Casimir Notice that for any n the scalar s = 0 yields d 0 = 1, for n = 1 we have d s>0 = 2, and when n = 2, d s = (s + 1) 2 . The trace of the coincident heat kernel for massless spin-s fields on thermal AdS 2n+1 with angular potentials θ i is built using the method of images [16] 10 Recall a section σ(x) in the principal bundle G of a homogeneous space G/H is a map σ : G/H → G such that π • σ = e, where π is the projection map from G to G/H, π(g) = gH for all g ∈ G, e is the identity element in G, and x are coordinates on G/H. 11 For even-dimensional AdS2n (SO(2n, 1)/SO(2n)) the principal series of SO(2n, 1) carries an additional discrete series of representations not present in the odd-dimensional case [16]. It turns out, however, that for STT tensor fields, this additional discrete series does not contribute for n > 1 [26]. The odd-dimensional analysis is then readily extended to even-dimensional thermal AdS2n for n > 1, when all of the angular potentials are switched off. The only change to the functional determinant in (B.9) is that, the dimension ds (B.6) must be replaced with its even-dimensional counterpart.