Black hole one-loop determinants in the large dimension limit

We calculate the contributions to the one-loop determinant for transverse traceless gravitons in an n + 3-dimensional Schwarzschild black hole background in the large dimension limit, due to the SO(n + 2)-type tensor and vector fluctuations, using the quasinormal mode method. Accordingly we find the quasinormal modes for these fluctuations as a function of a fiducial mass parameter ∆. We show that the behavior of the one-loop determinant at large ∆ accords with a heat kernel curvature expansion in one lower dimension, lending further evidence towards a membrane picture for black holes in the large dimension limit.


Introduction
Quantum effects in nontrivial gravitational backgrounds are of great interest to the theoretical physics community. Even the leading one-loop effects can contain important physical results, such as quantum corrections to the entropy of black holes, which a series of papers [1][2][3][4][5][6][7][8][9][10] found via calculations of one-loop determinants.
Since the computation of one-loop determinants, and thus one-loop partition functions, is technically difficult in generic curved spacetimes, several methods have been developed to handle the computations. There are three primary strategies: heat kernel methods, group JHEP06(2020)099 theoretic approaches, and the quasinormal mode method. Heat kernel methods may be exact, as in the eigenfunction expansion (e.g. [11][12][13][14]) or the method of images (e.g. [15]); alternatively, the heat kernel curvature approximation is only appropriate for fluctuations of massive fields (see [16] for a review of the heat kernel approach). For spacetimes with a simple symmetry structure, physicists have applied two group theoretic approaches. First, the authors of [17,18] use characters of representations to build the explicit expression for the one loop determinant of a fields with arbitrary spin in odd-dimensional AdS spaces. In the same spirit, the authors of [19][20][21][22] use the group theoretical structure of black hole solutions to predict their one loop determinants.
Around a decade ago, in two papers [23,24], Denef, Hartnoll and Sachdev developed a method for the computation of one-loop determinant not directly based on the heat kernel approach. The key insight of [23] is to build the one-loop determinant as a function of mass parameter ∆ using quasinormal frequencies of the field fluctuations in the bulk. These frequencies turn out to be poles (zeros) of the one-loop determinant for bosonic (fermionic) field fluctuations. A series of papers have employed this approach for one-loop calculations [25][26][27][28][29]. We will use this method to compute the one-loop determinant for gravitational perturbations around the Schwarzschild black hole in the large dimension limit.
In a finite number of dimensions, the Einstein equation lacks a small parameter; in the limit where the number of dimensions D is taken large, then 1/D can provide a perturbative parameter [30]. For the Schwarzschild black hole in a small number of dimensions, there is no separation of scales between the horizon dynamics and the asymptotic behavior. Conversely, in the large D limit, the gravitational field of a black hole becomes strongly localized near the horizon, effectively decoupling the black hole dynamics from the asymptotic structure. As suggested in [31,32], we can think of the dynamics of a such a large dimension black hole as equivalent to those of a membrane propagating in flat space. Since these developmental works, there has been significant interest in black hole spacetimes in the large dimension limit .
Of particular relevance for our work, [34] computes the spectrum of massless quasinormal modes in a 1/D expansion. Using the gauge-invariant formalism of [51,52], the authors of [34] find two distinct sets of quasinormal modes. First, they consider modes whose frequencies are of order D, finding that the dynamics of these modes depends mostly on the flat asymptotics of the spacetime. The second sequence of modes has frequencies of order D 0 and lives entirely in the near-horizon region. Hence, the dynamics of these decoupled modes reflects the fluctuations of the near-horizon membrane region. As we will show, it is this second decoupled sector of modes which are of greatest importance in the one-loop determinant.
We compute the one loop determinant for gravitational fluctuations in the membrane region of the large dimension Schwarzschild black hole, via the quasinormal mode method. Like [34], we adopt the gauge-invariant formalism of Kodama+Ishibashi [51,52] to compute the quasinormal frequencies, now as a function of a mass parameter ∆. We consider only transverse traceless fluctuating modes, as they have been successful previously for 3 dimensional gravitons in AdS 3 [25], combined with appropriate gauge fixing, in obtaining JHEP06(2020)099 the one loop determinant for graviton fluctuations in Einstein gravity. We emphasize that these transverse traceless gravitons, often referred to as 'tensor' fluctuations due to their behavior under the local Lorentz symmetry, further break down into tensor, vector and scalar components with respect to the spherical symmetry of the spacetime. Accordingly we use the terminology 'scalar', 'vector', and 'tensor' to refer to this spherical symmetry, and not to the full local Lorentz symmetry (as is otherwise common, e.g. in [53]). At ∆ of order D 0 , we find that only the decoupled, membrane-region quasinormal modes provide mass-plane poles. There are no contributing modes in the tensor sector, while in the vector sector the poles are simply defined by the quantum numbers, at least at leading order in D. Using these poles, we build the one-loop determinant for vector modes and express it in terms of Hurwitz zeta function. We find that the leading behavior of the oneloop determinant is proportional to ∆ D−1 . This dimensionality reduction in the exponent corresponds with the membrane paradigm picture of black holes in the large dimension limit. We also compare our results with heat kernel curvature calculations.
The paper is outlined as follows: in section 2 we review the large dimension limit of the Schwarzschild black hole and the quasinormal mode method. In section 3 we compute the quasinormal modes for the graviton field with an added fiducial mass. In section 4 we write the expression for the one-loop determinant in terms of elementary zeta functions, and compare this expression with the heat kernel curvature calculation. In section 5 we conclude with some comments and open questions.

Review
In this section we review the large dimension limit (section 2.1) and the quasinormal mode method (section 2.2) and set our notation. The reader familiar with both topics may wish to skip to section 3.

The large dimension limit
We review the large dimension limit of a Schwarzschild black hole, as exhibited in [34] and [31]. Since our goal will be to calculate quasinormal mode frequencies for massive modes, we will largely follow the method and notation of [34], although we will also draw upon the conceptual approach in [31]. The metric for a Schwarzschild black hole of radius r 0 in D total dimensions is where n = D − 3. As we can see from the metric, in the large D (or n) limit there are two important regions outside the horizon, with markedly different behavior. First, for any r > r 0 , with r 0 held fixed as D → ∞, f (r) → 1, so the metric reduces to flat space. We refer to this region as the 'far' region.
On the other hand, if we examine a very near-horizon region by setting r = r 0 1+ λ D−3 and instead keeping λ fixed as D → ∞, then f (r) → 1 − e −λ . Thus the nontrivial gravitational field, where f (r) is substantially different from one, is strongly localized in JHEP06(2020)099 Overlap region Near-horizon region a thin near-horizon region of thickness ∼ r 0 /D. This near-horizon 'membrane' region encodes the most important black hole physics [31,32].
In the near-horizon zone, we will use the radial coordinate ρ = (r/r 0 ) n where the metric becomes The near and far regions actually overlap, as depicted in figure 1: Or, equivalently in terms of ρ: near region : ln ρ D; far region : ln ρ 1 (2.4) The overlap region is thus (2.5) [34] studied the linearized gravitational perturbations around the Schwarzschild black hole in the large dimension limit, and computed the quasinormal spectrum of its oscillations in analytic form in the 1/D expansion. They found two distinct sets of modes: • Non-decoupled modes (heavy), with frequencies of order D/r 0 , lying between the near-horizon and asymptotic region. Most quasinormal modes fall in this class but they carry very little information about the black hole and so are of least physical interest.
• Decoupled modes (light), with frequencies of order 1/r 0 , are decoupled from the asymptotic region and are localized entirely inside the membrane region. There are only three such modes (two scalars and one vector). 1

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The decoupled modes capture the interesting physics specific to each black hole, such as stability properties. Accordingly, we will choose a limit that focuses on these decoupled modes. In [34], the authors recover both sets of modes by studying the linearized gravitational perturbations h µν = e −iωt h µν (r, Ω) of the metric (2.1) under the vacuum GR equations. The modes can additionally be separated according to their angular dependence; that is, via their transformation properties under the SO(n + 2) symmetry of the S n+1 sphere. There are thus three types of linearized graviton modes: scalar-type (S), vector-type (V ) and tensor-type (T ). 2 Following [51], we can study each of these modes in terms of a single gauge invariant master variable ψ(r) that satisfies master equations of the form where s stands for either Tensor(T), Vector(V) or Scalar(S). The effective tensor potential is given by where l is the conserved angular momentum from the SO(n + 2) symmetry. For vectors the effective potential is The scalar potential can be found in the appendix in equation (C.2).
When finding the quasinormal modes, as done in [34], the key idea is to first solve the linearized field equation (2.6) perturbatively in the 1/n expansion in the near region, with ingoing boundary conditions imposed at the horizon. Next the equation should be solved in the far region, with outgoing or normalizable boundary conditions at infinity. Last, the near and far region solutions should be matched in the overlap region.
We can understand the distinction between the near-horizon, decoupled modes and the far, coupled modes by examining the form of these potentials. In figure 2, we present an example of one such potential, for a fixed but large value of the dimension, D − 3 = n = 1000, and a representative value of l = 5 for the angular momentum. The lightest modes of O(n 0 ) frequency will live in the near-horizon dip of the potential. These modes are the decoupled modes. At large frequencies, another set of modes lives exclusively in the far region, at larger radius than the maximum of the potential.
Again, our primary interest will be in the physics of the black hole itself, so we will concentrate on the light, decoupled, near-horizon modes. As we will see in the next section, we need to extend the calculation of [34] to include a formal mass for the graviton.

Quasinormal mode method
In this section we review the quasinormal mode method for computing one-loop determinants as developed in [23]. For simplicity, we present the method here for a complex scalar, although we will use it for gravitons in our work below. For bosonic degrees of freedom, the one-loop determinant for a complex scalar field ψ of mass m is where ∇ 2 is the kinetic operator on the given spacetime background. 3 If the background is non-compact, then we examine the boundary conditions we want to impose on fields at infinity, and choose a mass parameter ∆, a function of the mass m 2 , accordingly. For the case of AdS backgrounds, this parameter simply becomes the conformal dimension, hence the notation ∆(m 2 ); more generally it is chosen so the boundary conditions depend meromorphically on ∆. Next, we assume that Z (1) (∆(m 2 )), analytically continued to the ∆ complex plane, is a meromorphic function. 4 This assumption allows us to use the Weierstrass factorization theorem to find the determinant. As a consequence of this theorem, a meromorphic function is determined (up to a single entire function) by the locations and degeneracies of its zeros JHEP06(2020)099 and poles. As we can see from the right hand side of (2.9) that the determinant for bosonic degrees of freedom will have no zeros, so we can write where the ∆ i are the poles (of Z (1) (∆)) each with degeneracy d i . Pol(∆) is a polynomial in ∆, since e Pol(∆) is entire (that is, it is meromorphic and has no zeros or poles itself). Poles occur when det = 0, which happens when the mass parameter ∆(m 2 ) is set so there is a ψ that solves the Klein-Gordon equation (−∇ 2 + m 2 )ψ = 0 while simultaneously being smooth and regular everywhere in the Euclidean background space. [23] shows that for static Euclidean thermal spacetimes, the allowed zero modes Wick rotate to (anti)quasinormal modes of the corresponding Lorentzian black holes with frequencies satisfying ω(∆ i ) = 2πipT, (2.11) where p ∈ Z. For p ≥ 0, the Euclidean modes match onto ingoing quasinormal modes, while for p ≤ 0, they instead match onto outgoing quasinormal modes (or anti-quasinormal modes). Consequently, if we know the (anti)quasinormal mode frequencies as a function of mass for a complex scalar field in a black hole background, then we can immediately write down the one-loop determinant, up to an entire function, as an infinite product. Once all poles are determined, we take the logarithm of both sides and use zeta functions to perform the infinite sum. If we can characterize the large ∆ behavior of the infinite sum, then we can determine Pol(∆) by matching the large ∆ behavior of the right hand side of (2.10) to the local heat kernel curvature expansion in 1/m for Z(∆) (see e.g. [16]).
For our calculation, we will introduce a fictitious mass for the graviton (e.g. following [25,59,60]) in order to facilitate this mathematical trick.

Calculating the quasinormal modes
In this section we extend the quasinormal mode calculations done in [34] by adding a fictitious mass to the graviton field. 5 Here we consider only transverse traceless fluctuations, as they have been successful previously for 3 dimensional gravitons in AdS 3 [25]. Our goal is to obtain the quasinormal mode spectrum as a function of this fictitious mass; then we will use the relation (2.11) to find the corresponding poles in the one loop determinant, in section 3.4.

Setting up the equations
We want to study the transverse traceless metric perturbation around our background (2.1) using the quasinormal mode method. Accordingly we add a fiducial mass, resulting in the action

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where m 2 is the added mass. Additionally, g µν = g 0 µν + h µν , where g 0 µν is the fixed background metric and h µν is the metric perturbation. We consider these perturbations to be transverse and traceless. Varying the action with respect to h µν gives the linearized field equation where δR µν and δR are taken from appendix B of [52]. Following [51], we study (3.2) for three different decompositions of the transverse traceless gravitational perturbation h µν , depending on the transformation properties under the SO(n + 2) symmetry of the S n+1 sphere: scalar-type (S), vector-type (V ) and tensor type (T ).
In this paper we will focus on calculations for vector and tensor type perturbations; we partially address the scalar perturbations in appendix C. For tensor type perturbations, the equation of motion (3.2), in terms of the master variable Ψ(t, r) = ψ(r)e −iωt as defined above equation (A.9), becomes For vector type perturbations, under a parameterization choice we define in appendix A, (3.2) similarly reduces to The V s are defined in (2.7) and (2.8). For both the tensor and vector modes, we follow the basic idea of [34], solving these field equations perturbatively order by order in the 1/n expansion in order to compute the quasinormal modes. We choose the convenient mass parameter because our boundary conditions on the field will be analytic in ∆, as we discuss below. We additionally introduce the parameter µ = m 2 /n 2 + 1/4, which satisfies the relations Following the quasinormal mode method, we wish to study the quasinormal mode frequencies in the complex ∆ plane.
Since we are interested in only the near-horizon contribution to the one-loop determinant, we expect that the poles of interest should satisfy ∆ = O(n 0 ). Our argument can be understood from figures 3a and 3b. Figure 3a shows a cartoon of the poles for finite n in the complex ∆ plane. The poles inside the circle are O(1) and the ones outside are O(n) and higher. As we increase n, the circle will get bigger and O(n) poles are pushed further out. In the n → ∞ limit, as JHEP06(2020)099 shown in 3b, there will only be O(1) poles left. Accordingly, in the large n limit we will only be interested in the O(1) poles; any poles that scale faster in n have been pushed to the edge of the complex plane. Since we only assume our function Z(∆) is meromorphic on the complex delta plane and not on the full Riemann sphere, we recognize that these poles should not be counted as poles on the ∆ plane in the large n limit. Rather their effect should be accounted for by the entire function exp(Poly), which we fix by other means. 6 Although we have not provided a rigorous mathematical proof, our intuition here accords with the physical picture of the large dimension limit, n → ∞, as taking a nearhorizon limit. These O(1) poles, as discussed in later sections, correspond to modes that live entirely in the near-horizon region and thus capture the relevant physics of interest in the near-horizon limit.
We now wish to study the equations (3.3) and (3.4) to solve for quasinormal modes that indicate poles at ∆ = O(1). Since ∆ = n(µ − 1/2), we can rewrite the requirement for ∆ to be O(1) as the requirement that µ satisfy In other words, we want to find quasinormal modes whose frequencies satisfy the regularity condition equation (2.11) when the mass parameter µ = 1/2 + O(1/n). As we show below, these modes will primarily live in the near-horizon region, so we now rewrite our equations in terms of the ρ = (r/r 0 ) n coordinate from the near-horizon metric (2.2). Note that these equations are still exact, in that we have not yet taken either a near-horizon nor a large-dimension limit.

3) and (3.4) become
where s is set to either T or V as appropriate. Here we have introduced In order to find the quasinormal modes, we also need to know their boundary conditions; ψ should satisfy the ingoing horizon boundary condition near the horizon, ρ = 1. It is easiest to express the ingoing condition in the radial coordinate r * = 1 n ln(ρ − 1). In this coordinate, the horizon is at r * = −∞. For an ingoing wave, we should expect (3.10) The ∝ here indicates that the function can have a regular component as it approaches the horizon; since we do not care about the overall normalization of ψ we are free to insist this component approach 1 at the horizon. Rewriting the horizon boundary condition again in our near-horizon coordinate ρ, we have Anti-quasinormal modes have the same condition except for a change of sign in the exponent: Quasinormal modes must also satisfy the no-incoming-flux condition at large radius. Here we closely follow the analysis done in [34]. In the far zone (2.4), any term of the form 1/ρ is exponentially small in n. Accordingly, we can set f = 1, and drop other terms with negative powers of ρ, so our wave equations (3.8) simplify; for both the tensor and vector modes the solution becomes just a Hankel function: Here, we have chosen to use H (1) because it is outgoing near r → ∞, provided that ω 2 > m 2 . Following the results of [27], we will take the boundary condition appropriate to the physical value of m 2 , in this case m = 0, and simply analytically continue it to all m 2 . Consequently, the boundary condition at r = ∞ is really that we should have none of the H (2) function present there.
In the sections below, we will proceed by a matched expansion. That is, we will solve the wave equation in the near-horizon region first, apply the horizon boundary condition, and then match this solution to the solution ψ ∞ from the far zone by using the overlap region. The requirement to have no H (2) behavior by the time we reach the far zone will result in a discrete set of allowed values for ω.

SO(n + 2) vector modes
Since we want to concentrate on the near-horizon physics, we will first consider the quasinormal modes that [34] found to have nontrivial dependence on the near-horizon region: these are their decoupled modes. For vectors, these modes have both frequency and angular momentum satisfying l, ω = O(1) in a 1/n expansion. As we will see in section 3.2.1, these modes can indeed only be obtained by setting µ 0 = 1/2, or ∆ = O(1), as we discussed in section 3.1.
After finding the ω for these decoupled modes, we then proceed to argue that indeed the choice ∆ = O(1) eliminates all of the modes with ω or l at size n or larger; that is, it eliminates all of the modes which have support in the far zone, the ones which are not decoupled from the asymptotics in which the black hole is placed. Consequently, in these two sections, we show that our choice of ∆ combined with the restriction ∆ = O(1), effectively studies only the physics of the near-horizon, decoupled modes.

Decoupled vector modes
The decoupled modes correspond to l, ω = O(1). As found in [34], these modes are necessarily normalizable within the near-horizon region. They evade having any H (2) behavior at r = ∞ by being exponentially suppressed there. In other words, decoupled modes have their primary support within the near-horizon region, and are thus normalizable in the asymptotic range of the near-horizon region.
Following [34], we study the wave equation (3.8) perturbatively, order by order in the 1/n expansion. In order to perform the expansion, we expand all quantities in powers of 1/n: (3.14) We additionally expand ∆, using the definition of µ from (3.7): Note we have left µ 0 arbitrary rather than setting it to 1/2; we do so in order show that the only quasinormal modes with l, , this result will support our choice of ∆ as the meromorphic parameter.
To leading order in 1/n, (3.8) reduces to with and

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Using ω = O(1), the horizon boundary condition (3.11) can also be written order by order: where we have fixed the overall amplitude of ψ via (3.19) for convenience. As highlighted above, at the boundary of the near-horizon region ρ 1, we will insist on normalizability. The solution to the lowest order equation (3.16) and the boundary condition (3.19) is To get the behavior in the overlap region, we expand the remaining piece around ρ → ∞: From this form we can easily see that for µ > 0, the second line will be non-normalizable. We also find that is, any ρ µ 0 behavior should be absent. We use the analytic continuation of this boundary condition when we consider negative or imaginary values for µ 0 . Since the values of µ 0 that can possibly solve our boundary condition turn out to be half-integers, we actually need to revaluate the expansion in (3.23). When the first two parameters in a hypergeometric function differ by an integer, in this case 2µ 0 , then the expansion takes a more complicated form. We first rewrite the function around ρ = 1 in terms of k: ψ For both k = 0 and k = 1, this solution becomes exactly However, if we rewrite this solution in terms of µ 0 , we find

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Now we can see that only the first option, k = 0, satisfies the normalizability condition (3.24). If we can also rule out k ≥ 2, then we will have µ 0 = 1/2, or ∆ = O(1) in 1/n, as we wanted. Before we analyze the k = 0 case in more detail, let us examine these k ≥ 2 solutions. Although the expansion (3.23) is no longer valid, we can see from the rewriting (3.26) that for all integers k ≥ 2, the solution (3.22) takes the exact form Since the series always terminates with a 0 , no ρ µ 0 behavior is present, so these solutions do satisy the boundary condition at infinity (3.24). However, as we will show in appendix B, continuing the perturbation procedure to one higher order results in quasinormal modes ω which do not satisfy the Euclidean regularity condition (2.11) for any value of µ 1 . Accordingly, we will not consider k ≥ 2 further in the main body of the paper.
Since the only mode remaining is k = 0 (corresponding to µ 0 = 1/2 or ∆ = O(1) in 1/n), we now find its quasinormal mode frequencies. Using the O(1) solution (3.27) and expanding to next order in 1/n, the wave equation (3.8) becomes The general solution for ψ (1) for the physical region ρ > 1 becomes where C 1/2 and C −1/2 are arbitrary constants. Since the C 1/2 √ ρ term violates the normalizability condition (3.24), we set C 1/2 = 0. Expanding the solution around ρ = 1 in order to apply the horizon boundary condition (3.20), we obtain Matching to (3.19) requires C −1/2 = 0, while matching to (3.20) gives the leading order quasinormal frequency We have now found the leading-order term in the quasinormal mode frequencies under the assumption that ω, l = O(1) in 1/n, and that the modes we are interested in should be normalizable in the near zone. Normalizability in the near zone ensures that these modes capture information about the dynamics of the near-horizon, membrane region, and additionally justifies our choice to concentrate on ∆ = O(1).

Non-decoupled vector modes
As we will now show, if we allow either (or both) of ω and l to be of order O(n) or larger, the resulting modes are non-normalizable in the near-horizon geometry.

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Although we have justified the choice of the mass parameter ∆ and the condition ∆ = O(1), we will begin by allowing a ∆ to grow at order n 2 : ∆ = n(µ − 1/2) = µ −1 n 2 + n(µ − 1/2) + µ 1 + µ 2 /n + O(1/n 2 ). (3.35) Next, we assume both l, ω are no larger than O(n 2 ): This assumption is just to simplify the argument; as we will show, both l −2 and ω −2 (as well as µ −1 are forced to be zero when solving the vector version of (3.8), and any larger terms would be as well. With these assumptions, the vector wave equation (3.8) at order n 2 is A nonzero solution is only possible if ω −2 = 0 and l 2 −2 + µ 2 −1 = 0. Next, we examine the equation (3.8) at order n, imposing ω −2 = 0; we find Since the log ρ term is independent of the others, its coefficient must vanish; thus, we find l −2 = 0 which implies µ −1 = 0 from the order n 2 condition. Consequently, we have µ −1 = l −2 = ω −2 = 0 from the O(n 2 ) and O(n) equations combined. Now, using our intuition from section 3.2.1 above, we expect µ 0 = 1/2, so that the first nonzero term in ∆ is µ 1 at order n 0 . Thus in the near-horizon coordinates at leading order O(n 0 ), (3.8) becomes where ω c 0 is the O(n 0 ) component of ω c from the far region solution (3.13), so ω c 0 = 1/2 + l −1 . The solution to the above equation satisfying the horizon boundary condition (3.11) is where In the overlap zone we have large ρ, so the solution (3.41) becomes (3.43) As we can see, these modes have both normalizable and non-normalizable behaviors in the asymptotic regime of the near-horizon region, that is, at large ρ. In order to remove the JHEP06(2020)099 nonnormalizable behavior (which could be either term depending on the parameters), we would need either q + or q − to solve Rewriting as a condition on ω −1 , we have In the last line, we have added the requirement to additionally satisfy the regularity condition (2.11), for a black hole with r 0 = 1 or 2πT = 1. The only possible solution with integers is However, when ω −1 = 0, then ω = O(1) and we have Thus the expansion (3.43) fails to be valid as (q + − q − ) is an integer, but (3.41) has a terminating expansion of the form We see that these modes have only non-normalizable behavior at large ρ, and thus should not be included. Consequently, we find that there are no poles with ∆ = O(1) due to vector modes with either ω = O(n) or larger, or with l = O(n) or larger. Non-decoupled modes do not contribute poles in the ∆ = O(1) region. We will return to finding the poles associated with the decoupled vector modes in section 3.4 below, after providing a brief argument that no tensor modes contribute.

SO(n + 2) tensor modes
In the tensor case [34] found only non-decoupled modes, that is only modes with large profile far from the horizon. Given our vector result (that no non-decoupled modes contribute relevant poles), we expect a similar result for tensors. The argument is quite similar to the vector case. Since the only difference between the vector potential (2.8) and the tensor potential (2.7) first shows up at O(1) in 1/n, the arguments around (3.38) leading to the result ω −2 = l −2 = 0 apply to tensors as well as vectors, again assuming µ 0 = 1/2, by our definition.
Consequently, the tensor equation (3.8) becomes, at O(1) in 1/n, where again ω −1 is the O(n) piece in ω, and ω c 0 = 1/2 + l −1 with l −1 the O(n) piece in the angular momentum l, as in the vector case.

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The solution to the above equation is where q ± are as in (3.42). In the overlap zone at large ρ, (3.50) expands as For generic values of the parameters, we see these modes have both normalizable and nonnormalizable behaviors in the asymptotic regime. As before, we might guess that q ± = −k for a nonnegative integer k might allow us to remove the non-normalizable behavior. Quite similarly to (3.45), upon adding the requirement to satisfy the regularity condition (2.11), we find Again, there is only one possible solution with integers: As before, showing ω −1 = 0 is equivalent to requiring ω = O(1) in 1/n; the expansion in (3.51) is not valid here because q + − q − is an integer, but fortunately the solution (3.50) has a terminating expansion, of the form and its general solution is This mode is also not allowed, because regardless of the subleading values of ∆, there is no way to satisfy the normalization (3.24) and horizon (3.20) boundary conditions. Accordingly, as per our expectation, no tensor modes contribute poles in the region ∆ = O(1).

Quasinormal mode results
In sections 3.2.2 and 3.3 we showed that there are no allowed tensor and non-decoupled vector modes, so the only poles we are interested in are the ones corresponding to decoupled vector modes. In order to find the poles, we will need the Euclidean regularity condition (2.11). Since we have set r 0 = 1, the black hole temperature becomes 2πT = 1, and we have

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(3.57) Requiring that the decoupled vector mode frequencies (3.34) satisfy (3.57) gives Thus the corresponding poles occur at where we have used µ 0 = 1/2. For anti-quasinormal modes, the poles can be obtained in a similar way. The boundary condition at the horizon has a sign change, as in (3.12), which means the first order boundary condition (3.20) should become instead while the regularity condition (2.11) becomesω (0) = ip, where p ∈ Z <0 . We find that the poles for anti-quasinormal modes instead occur at We note that the p = 0 case is not allowed here as it is already accounted for in (3.59). We summarize our results in table 1. For both the QNM and anti-QNM series, additional care should be taken for mode numbers less than the spin of the particle being considered; since we are considering gravitons, we should reanalyze the solutions with l = 0, 1 and p = 0, 1, to ensure the modes are actually integrable at the origin of the Euclidean thermal space. However, since any necessary adjustment would only result in removing a finite number of poles, and we will concern ourselves below mainly with the behavior of the one-loop determinant in the ∆ → ∞ limit, we postpone this analysis to future work. For more details on this issue, the interested reader may consult Append B.3 of [25].

Writing the one-loop determinant
As we will show, it is possible to express the SO(n + 2) vector and tensor portions of the graviton one-loop determinant in the large-dimension Schwarzschild background directly JHEP06(2020)099 in terms of the Hurwitz zeta function and its derivatives. Since the SO(n + 2) tensor does not contribute any poles in the ∆ = O(1) regime, our work in this section primarily relies on the poles due to vector quasinormal modes (3.59) and anti-quasinormal modes (3.61). We defer treating the SO(n + 2)-scalar graviton contribution for future work, but make some commentary in appendix C.

Expressing Z V in terms of Hurwitz ζ
Using Weierstrass's factorization theorem as reviewed in section 2.2 the one-loop determinant in the large n limit becomes where the degeneracy D l of each frequency equals the degeneracy of the lth angular momentum eigenvalue on S n+1 Taking the logarithm and rearranging, we find whereD(j) is given byD (j) = n + (2j + n) 2 n(n + 1) We now follow [23] to rewrite this sum in terms of the Hurwitz zeta function ζ(s, x), defined as

Matching with the heat kernel expression
In this section we compute the large ∆ limit of log Z V . We will then compare these results to the heat kernel curvature expansion, at least for the largest terms. In a successful comparison, this procedure fixes the polynomial piece Pol(∆ V ), as done in [23]. In our case, we do find a match between the non-polynomial terms in the heat kernel expansion appropriate to D − 1 dimensions, and the non-polynomial terms in the quasinormal mode method, indicating that a higher order calculation should be able to fix the polynomial piece. We discuss our interpretation of this result, as well as some caveats, in this section and in the discussion in section 5.

Large ∆ limit using QNM method
To get the leading large ∆ behavior of (4.8), we begin by finding the highest power of j in the degeneracy polynomial (4.4): where # refers to a coefficient we will not need below. We will also use the expansion of the Hurwitz zeta derivative at large argument: where B k are the Bernoulli numbers. Using these expressions, we expand the final term in (4.8) at large ∆ V , keeping only the leading terms. We find In order to find the highest power of ∆ V in this expression, we need to find the largest

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We then find the leading terms at large ∆ V arẽ (4.13) Following (4.8), these terms are also the largest in log Z V − Pol(∆ V ) as well. In particular the logarithmic terms cannot be cancelled by the polynomial piece, so the leading behavior of the determinant satisfies (4.14)

Heat kernel calculation
We will now find the large ∆ behavior for the full log Z, via the heat kernel curvature expansion (for a review, see e.g. [16]). In this expansion, we expand the determinant at large mass as whereD denotes the effective total dimension of the spacetime, a q denotes the heat kernel curvature coefficients and is a regulator which we will take to zero below. The heat kernel curvature coefficients are zero for odd q s for spacetimes without a boundary; as we will show below, the a 0 term gives the leading large ∆ behavior, and is present regardless of the dimension. We find the leading large m behavior for the qth term in (4.15) by evaluating the integral and expanding for small regulator . For odd dimensions, the leading term goes like mD −q , or in terms of ∆ it goes like ∆D −q : (4.16) For evenD, we find instead where c 1 and c 2 are constants dependent on the a n . In both cases, the largest power of m, and thus of ∆, corresponds to the term with q = 0. For q = 0, (4.17) explicitly becomes where H(x) is the xth harmonic number. Expanding
In terms of ∆, the leading behavior of this volume-dependent term becomes ∆D for odd D, and ∆D log ∆ for evenD. We also note that no log ∆ terms are present in the oddD expansion, regardless of how many terms we consider. Now, we want to compare Z V from the quasinormal mode method (4.8) to the heat kernel result. Expanding (4.8) at large ∆ (using (4.13)), we find (4.20) Comparing this quasinormal mode expression with the heat kernel results (4.16) and (4.17), we see that matching will only be possible whenD is even.
Comparing further with the large ∆ expansion for evenD as in (4.19), we find that the logarithmic behaviors in (4.17) and (4.20) match for evenD whenD = D − 1. Although this result may at first be surprising, it shows that the heat kernel analysis should really be using an effective total dimension that is reduced by one, which reflects a clear picture of the membrane paradigm as mentioned in [31]. This interpretation is quite strong since the log Z in (4.3) is actually built out of poles corresponding to modes living in the membrane region, which is itself D − 1 dimensional. Now, by matching the coefficient of log(∆ V )∆D in (4.20) and (4.19), usingD = D − 1, we can calculate 7 a 0 : Once we have obtained a 0 , by equating the polynomial terms in (4.19) and (4.20) at large ∆ we can compute the leading behavior at large ∆ V of Pol(∆ V ): Thus, we claim the contribution to the partition function from the SO(n + 2) vector and tensor modes of the graviton in the large D limit of a Schwarzschild background, for D odd, is given by (4.8), where the largest contribution to the polynomial is set by (4.22).
There are a few caveats to this interpretation. First, for even D or oddD, the heat kernel analysis and the quasinormal mode analysis disagree, since the quasinormal mode method indicates a logarithmic term whereas the heat kernel one only produces polynomials in ∆. We believe this behavior occurs because the large D limit is not analytic when including both parities; since many behaviors depend on whether D is odd or even, we should specify if we are taking a large D limit for the odd case or the even case. The presence 7 Since we do not have a precise definition of the membrane region whose regulated volume a0 should calculate, we cannot find it from first principles and must instead do a comparison. Since a0 should be proportional to the (regulated) volume, we find this form encouraging; it is the ratio of sphere volumes for dimensions D − 1 and (D − 1)/2. We do not have an interpretation of why this ratio should be the regulated volume appropriate to the membrane region.

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of a logarithmic term in the quasinormal mode method indicates that the effective total heat kernel dimensionD should be thought of as even, or the total spacetime dimension D should be odd.
In addition to the concern about analyticity of the large D limit, we should also mention an order of limits concern between large ∆ and large D. Recall that the quasinormal mode result in (4.3) is calculated by taking the large D limit; we then take the large ∆ limit to compare with the heat kernel result. Contrastingly, the heat kernel calculation first takes the large ∆ limit for fixedD.
Another possible concern is that the heat kernel analysis may be expecting a fully gauge-fixed result, whereas we have only been able to analyze the gauge-independent tensor and vector variables in calculating (4.14). Accordingly, we have only claimed to calculate the polynomial piece due to the vector (and tensor) perturbations, and thus call our result in (4.22) Pol V . We expect a more complete calculation may have further contributions, particularly at subleading order in ∆, to the polynomial piece due to scalar and gauge fixing perturbations.
Even given these caveats, we have shown that a consistent interpretation of our quasinormal mode calculation in the large dimension limit points to an effective total dimension of D − 1 in the heat kernel analysis, which corresponds with a membrane-paradigm picture for the large dimension limit we analyze.

Conclusion
We have studied the SO(n + 2) vector and tensor modes of the one-loop determinant, for fluctuations of the n + 3-dimensional transverse traceless graviton, in a Schwarzschild black hole background in the large dimension limit. We found that no tensor modes contribute poles to the one-loop determinant, while the vector modes result in poles at locations set by integers, specifically at Using the locations of these poles, we constructed the expression (4.8) for the one loop determinant due to the vector modes, in terms of Hurwitz zeta functions. Importantly, we found that the large ∆ behavior of this determinant is ∆ D−1 log ∆, as in (4.14). We compared this result to that obtained from a preliminary heat kernel curvature expansion, presented in (4.16) for odd dimensions and (4.17) for even dimensions; we find that the results match when the heat kernel effective dimensionD is even, and when D = D − 1 from the quasinormal mode analysis. As discussed in section 4.2, this result reflects the dimensionality reduction in the leading power of ∆ in log Z in agreement with the membrane paradigm picture of the large dimension black hole as presented in [31].
We do believe the quasinormal mode result (4.14) as compared to the heat kernel result (4.17) indicates a picture of effectively reduced dimensions in the large dimension limit. However, there are a few caveats on our result. First, there are two possible issues with the large dimension limit. As evidenced by the dependence on dimensional parity, the large dimension limit may not be analytic, or rather may not preserve the meromorphicity JHEP06(2020)099 of the one-loop determinant. Additionally, there is a possible order of limits issue regarding taking the large D vs. large ∆ limits. The quasinormal mode analysis relies on first taking a large D limit, and then large ∆; the heat kernel curvature expansion is itself already a large ∆ limit, and we impose the large D limit afterwards.
Next, we do not perform the gauge fixing to obtain the full one-loop partition function for quasinormal modes. The primary obstruction is that the generic gauge-fixing analysis of the partition function for the gauge-independent variables of [51,52] has not been done. Since their (and our) decomposition into scalar, vector, and tensor modes refers to the SO(n + 2) spherical symmetry of the spacetime, and not to the local Lorentz symmetry, we cannot use the [54] formula. Accordingly, we leave this gauge-fixing problem to future work.
Next, our choice of mass parameter ∆ = n(µ−1/2) = −n/2+ m 2 + n 2 /4 may appear somewhat arbitrary. However, as we can see from the explicit behavior of the near-horizon modes in (3.23), the appropriate boundary condition on ψ in the asymptotia of the nearhorizon region is indeed analytic in µ and thus in ∆. In fact this analyticity looks somewhat similar to that found for AdS in previous quasinormal mode studies (e.g [23,[25][26][27]; we picked the name ∆ for this parameter in deference to this analogy. We should also justify the specific linear combination of the square root, that is, why we chose ∆ = n(µ − 1/2) and not some other dependence on n. As we showed in section 3.2.1, this choice captures the physics of the near-horizon region at order ∆ ∼ O(n 0 ). If we instead chose ∆/n as our mass parameter, then the order O(n 0 ) poles would also include non-decoupled modes. In other words, this choice of mass parameter would include the effect of modes which have nontrivial support in the asymptotic flat region. Contrastingly, if we chose n∆, then all of the vector poles we found would also be pushed to infinite mass parameter; we could not resolve their effect even though these modes live entirely in the near-horizon region. As we point out in appendix C, the scalar modes actually arise at n 2 (µ − 1/2) ∼ O(n 0 ). Accordingly, the choice of n∆ would capture the scalar modes; however it is not an acceptable mass parameter because it leaves out the near-horizon vector modes. The choice ∆ = n(µ − 1/2) thus captures exactly the near-horizon physics for the tensor and vector modes that are our main focus, at order O(n 0 ). We leave the study of scalar modes, which appear at ∆ ∼ O(1/n), to future work.
Better control over the large D limit of the heat kernel curvature expansion (or perhaps of the heat kernel curvature expansion of the large D limit) would elucidate the order of limits issues highlighted previously, and also aid in understanding correction terms present for finite D. Our results are only appropriate in the strict D → ∞ limit, but most physics of interest is at finite D. For a finite D, the dynamics of the asymptotic region may be important for the one loop determinant. In the infinite D limit, only the near-horizon dynamics matter. Since our analysis is only of the near-horizon dynamics, we lack a characterization of the correction terms for finite D. We leave such a characterization to future work.
We should also note that the similarity to the AdS analysis appears deeper than the choice of ∆; the functional forms for ψ at the pole locations ∆ = ∆ * as found in section 3.2.1 also appear quite similar to the AdS equivalents. Additionally, as pointed out in [61], the near-horizon region of the large D limit is described by a two-dimensional string theory JHEP06(2020)099 black hole geometry which has conformal symmetry. We hope to further explore this emergent symmetry in the future.

A.2 SO(n + 2) vector modes
For vector modes, the metric perturbation is given by where in the massless case f a and H T are gauge dependent variables which can be combined in terms of a single gauge invariant variable F a as described in [51,52]: Here, V i are vector harmonics defined by with the propertiesD i V i = 0, (A.14) where k 2 = l(l + n) − 1, l ∈ Z, l > 0, and the index sums are taken only over the n + 1 spherical coordinates i, j. Writing in terms of F a and H T the ij component of (A.1) becomes:

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Because we have added a mass, H T is no longer pure gauge (no gauge freedom remains). However, we are not interested in the behavior of the H T mode, since it is not present when we return to m = 0. Accordingly, we set (A.20) Under this choice, the modes we are interested in lie in Ω, whereas h a contains the 'gauge' modes. These modes do interact, but as we will see the coupling is a suppressed term in the 1/n expansion. Of course a proper gauge fixing procedure at m = 0 should consider the effect of the ghosts which would arise when choosing the gauge H T = 0, but this procedure should be done separately. We discuss this consideration further in the conclusion, and for now focus on the Ω modes.
The ai component of (A.1) is which under the definition (A. 19) becomes we can rewrite the antisymmetric term inside the curly brackets on the left hand side of (A.22). We find We now move all of the h terms to the right hand side. Since we can commute the s through the covariant derivatives, and since ed dc = δ e c , we find

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