Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions

Motivated by the prospect of constraining microscopic models, we calculate the exact one-loop corrected de Sitter entropy (the logarithm of the sphere partition function) for every effective field theory of quantum gravity, with particles in arbitrary spin representations. In doing so, we universally relate the sphere partition function to the quotient of a quasi-canonical bulk and a Euclidean edge partition function, given by integrals of characters encoding the bulk and edge spectrum of the observable universe. Expanding the bulk character splits the bulk (entanglement) entropy into quasinormal mode (quasiqubit) contributions. For 3D higher-spin gravity formulated as an sl($n$) Chern-Simons theory, we obtain all-loop exact results. Further to this, we show that the theory has an exponentially large landscape of de Sitter vacua with quantum entropy given by the absolute value squared of a topological string partition function. For generic higher-spin gravity, the formalism succinctly relates dS, AdS$^\pm$ and conformal results. Holography is exhibited in quasi-exact bulk-edge cancelation.

The horizon consumes everything once seen, growing until it reaches its de Sitter equilibrium area A.

Introduction
As seen by local inhabitants [1][2][3][4][5][6] of a cosmology accelerated by a cosmological constant, the observable universe is evolving towards a semiclassical equilibrium state asymptotically indistinguishable from a de Sitter static patch, enclosed by a horizon of area A = Ω d−1 d−1 , ∝ 1/ √ Λ, with the de Sitter universe globally in its Euclidean vacuum state. A picture is shown in fig. 1.1b, and the metric in (2.1)/(D.10)S. The semiclassical equilibrium state locally maximizes the observable entropy at a value S semiclassically given by [2] S = log Z , (1.1) where Z = e −S E [g,··· ] is the effective field theory Euclidean path integral, expanded about the round sphere saddle related by Wick-rotation (D.12) to the de Sitter universe of interest.
At tree level in Einstein gravity, the familiar area law is recovered: The interpretation of S as a (metastable) equilibrium entropy begs for a microscopic understanding of its origin. By aspirational analogy with the Euclidean AdS partition function for effective field theories with a CFT dual (see [7] for a pertinent discussion), a natural question is: are there effective field theories for which the semiclassical expansion of S corresponds to a large-N expansion of a microscopic entropy? Given a proposal, how can it be tested?
In contrast to EAdS, without making any assumptions about the UV completion of the effective field theory, there is no evident extrinsic data constraining the problem. The sphere has no boundary, all symmetries are gauged, and physically meaningful quantities must be gauge and field-redefinition invariant, leaving little. In particular there is no invariant information contained in the tree-level S (0) other than its value, which in the low-energy effective field theory merely represents a renormalized coupling constant; an input parameter. However, in the spirit of [7][8][9][10], nonlocal quantum corrections to S do offer unambiguous, intrinsic data, directly constraining models. To give a simple example, discussed in more detail under (8.17), say someone posits that for pure 3D gravity, the sought-after microscopic entropy is S micro = log d(N ), where d(N ) is the number of partitions of N . This is readily ruled out. Both macroscopic and microscopic entropy expansions can uniquely be brought to a form characterized by absence of odd (=local) powers of 1/S (0) . The microscopic theory predicts (a, b) = 2, log(π 2 /6 √ 3) , refuted by the macroscopic one-loop result (a, b) = 3, 5 log(2π) . Some of the models in [11] are sufficiently detailed to be tested along these lines.
In this work, we focus exclusively on collecting macroscopic data, more specifically the exact one-loop (in some cases all-loop) corrected S = log Z. The problem is old, and computations for s ≤ 1 are relatively straightforward, but for higher spin s ≥ 2, sphere-specific complications crop up. Even for pure gravity [12][13][14][15][16][17][18][19][20][21], virtually no complete, exact results have been obtained at a level brining tests of the above kind to their full potential.
Building on results and ideas from [22][23][24][25][26][27][28][29][30][31][32], we obtain a universal formula solving this problem in general, for all d ≥ 2 parity-invariant effective field theories, with matter in arbitrary representations, and general gauge symmetries including higher-spin: q ≡ e −t/ . Below we explain the ingredients in sufficient detail to allow application in practice. A sample of explicit results is listed in (1.12). We then summarize the content of the paper by section, with more emphasis on the physics and other results of independent interest.
G 0 is the subgroup of (possibly higher-spin) gravitational gauge transformations acting trivially on the S d+1 saddle. This includes rotations of the sphere. vol G 0 is the volume for the invariant metric normalized such that the standard rotation generators have unit norm, implying in particular vol SO(d+2) = (D.7). The other G i , i = 1, . . . , K are Yang-Mills group factors, with vol G i the volume in the metric defined by the trace in the action, as in (D.8).
The γ a are proportional to the (algebraically defined) gauge couplings:
The functions χ tot (t) are determined by the bosonic/fermionic physical particle spectrum of the theory. They take the form of a "bulk" minus an "edge" character: The bulk character χ bulk (t) is defined as follows. Single-particle states on global dS d+1 furnish a representation R of the isometry group SO(1, d + 1). The content of R is encoded in its Harish-Chandra characterχ(g) ≡ tr R(g) (appendix A). Restricted to SO(1, 1) isometries g = e −itH acting as time translations on the static patch,χ(g) becomes χ bulk (t) ≡ tr e −itH . For example for a massive integer spin-s particle it is given by (A.14): where D d s is the spin degeneracy (A. 15), e.g. D 3 s = 2s + 1, and ν is related to the mass: For arbitrary massive matter χ bulk is given by (A. 16). Massless spin-s characters are more intricate, but can be obtained by applying a simple "flipping" recipe (5.7) to (5.5), or from the general formulae (G. 2) The q-expansion of χ bulk gives the static patch quasinormal mode degeneracies, its Fourier transform gives the normal mode spectral density, and the bulk part of (1.4) is the quasicanonical ideal gas partition function at β = 2π , as we explain below (1.15).
The edge character χ edge (t) is inferred from path integral considerations in sections 3-5. It vanishes for spin s < 1. For integer s ≥ 1 we get (4.8): 10) e.g. N 1 = 1, N 2 = d + 2. Note this is the bulk character of N s scalars in two lower dimensions.
Thus the edge correction effectively subtracts the degrees of freedom of N s scalars living on S d−1 , the horizon "edge" of static time slices (yellow dot in fig. 1.1). (4.14) yields analogous results for more general matter; e.g. bulk field → edge field, bulk → (d + 2) × edge.
The general closed-form evaluation of the integral in (1.4) is given by (C. 19) in heat kernel regularization. In even d, the finite part is more easily obtained by summing residues.
The term ∝ s 2 is the edge contribution. It is negative for all m and dominates the bulk contribution ( fig. 1.2). It diverges at the unitarity/Higuchi bound m = s − 1.
In the 4D gravity example, L is a minimal subtraction scale canceling out of S (0) + S (1) . In this case, constant terms in S (1) cannot be distinguished from constants in S (0) and are as such physically ambiguous. 2 The term α 4 log( /L) with α 4 = − 571 45 arises from the log-divergent term α 4 log( / ) of the regularized character integral.

Overview
We summarize the content of sections 2-9, highlighting other results of interest, beyond (1.4).
From (1.17), covariantly regularized quasicanonical bulk thermodynamic quantities can be analytically computed for general particle content, as illustrated in section 2.3. Substituting the expansion (1.14) expresses these quantities as a sum of quasinormal mode contributions, generalizing and refining [50]. In particular the contribution to the entropy and heat capacity from each physical quasinormal mode is finite and positive ( fig. 2.4).
S bulk can alternatively be viewed as a covariantly regularized entanglement entropy between two hemispheres in the global dS Euclidean vacuum (red and blue lines in figs. 2.1, D.1).
In the spirit of [47], the quasinormal modes can then be viewed as entangled quasiqubits.

Sphere partition functions (sections 3,4,5)
In sections 3-5 we obtain character integral formulae computing exact heat-kernel regularized one-loop sphere partition functions Z (1) PI for general field content, leading to (1.4). For scalars and spinors (section 3), this is easy. For massive spin s (section 4), the presence of conformal Killing tensors on the sphere imply naive reduction to a spin-s Laplacian determinant is inconsistent with locality [26]. The correct answer can in principle be obtained by path integrating the full off-shell action [51], but this involves an intricate tower of spin s < s Stueckelberg fields. Guided by intuition from section 2, we combine locality and unitary constraints with path integral considerations to find the terms in log Z missed by naive reduction. They turn out to be obtained simply by extending the spin-s Laplacian eigenvalue sum to include its "subterranean" levels with formally negative degeneracies, (4.6).
The extra terms capture contributions from unmatched spin s < s conformal Killing tensor ghost modes in the gauge-fixed Stueckelberg path integral. The resulting sum yields the bulk−edge character integral formula (4.7). Locality and unitarity uniquely determine the generalization to arbitrary parity-symmetric matter representations, (4.14).
In the massless case (section 5), new subtleties arise: negative modes requiring contour rotations (which translate into the massless character "flipping" recipe mentioned above (1.9)), and ghost zeromodes which must be omitted and compensated by a carefully normalized group volume division. Non-universal factors cancel out, yielding (5.19) modulo renormalization. 3D de Sitter HS n quantum gravity and the topological string (section 6) The sl(2) Chern-Simons formulation of 3D gravity [52,53] can be extended to an sl(n) Chern-Simons formulation of s ≤ n higher-spin (HS n ) gravity [54]. The action for positive cosmo- The number of vacua grows as N vac ∼ e 2π √ n/6 . The maximal entropy vacuum is m = {n}.
We obtain the all-loop exact quantum entropy S m = log Z m by analytic continuation k ± → l ± iκ of the SU (n) k + × SU (n) k − Chern-Simons partition function on S 3 , (6.7). In the weak-coupling limit κ → ∞, this reproduces S (1) as computed by (1.4) in the metric-like formulation of the theory, given in (1.12) for the maximal-entropy vacuum m = {n}.
When n grows large and reaches a value n ∼ κ, the 3D higher-spin gravity theory becomes strongly coupled. (In the vacuum m = {n} this means n 4 ∼ /G N .) In this regime, Gopakumar-Vafa duality [55,56] can be used to express the quantum de Sitter entropy S in terms of a weakly-coupled topological string partition function on the resolved conifold, (6.8): S m = log Z top (g s , t) e −πT m ·2πi/gs 2 (1. 19) where g s = 2π n+l+iκ , and the conifold Kähler modulus t ≡ S 2 J + iB = ig s n = 2πin n+l+iκ .
Euclidean thermodynamics of the static patch (section 7) In section 7 we consider the Euclidean thermodynamics of a QFT on a fixed static patch/sphere background. The partition function Z PI is the Euclidean path integral on the sphere of radius , the Euclidean energy density is ρ PI = −∂ V log Z PI , where V = Ω d+1 d+1 is the volume of the sphere, and the entropy is S PI = log Z PI + 2π U PI = log Z PI + V ρ PI = (1 − V ∂ V ) log Z PI , or Using the exact one-loop sphere partition functions obtained in sections 3-5, this allows general exact computation of the one-loop Euclidean entropy S (1) PI , illustrated in section 7.2. Euclidean Rindler results are recovered in the limit m → ∞. The sphere computation avoids introducing the usual conical deficit angle, varying the curvature radius instead.
For minimally coupled scalars, S (1) PI = S bulk , but more generally this is false, due to edge (and other) corrections. Our results thus provide a precise and general version of observations made in the work reviewed in appendix E.5. Of note, these "corrections" actually dominate the one-loop entropy, rendering it negative, increasingly so as s grows large.

Quantum gravitational thermodynamics (section 8)
In section 8 (with details in appendix I), we specialize to theories with dynamical gravity.
Denoting Z PI , ρ PI and S PI by Z, and S in this case, (1.20) trivially implies = 0, S = log Z, reproducing (1.1). All UV-divergences can be absorbed into renormalized coupling constants, rendering the Euclidean thermodynamics well-defined in an effective field theory sense.
Integrating over the geometry is similar in spirit to integrating over the temperature in statistical mechanics, as one does to extract the microcanonical entropy S(U ) from the canonical partition function. 3 The analog of this in the case of interest is S(ρ) ≡ log Dg · · · e −S E [g,...] + ρ √ g , (1.21) for some suitable metric path integration contour. In particular S(0) = S. The analog of the microcanonical β ≡ ∂ U S is V ≡ ∂ ρ S, and the analog of the microcanonical free energy is the Legendre transform log Z ≡ S − V ρ, satisfying ρ = −∂ V log Z. If we furthermore define by Ω d+1 d+1 ≡ V , the relation between log Z, ρ and S is by construction identical to (1.20).
Equivalently, the free energy Γ ≡ − log Z can be thought of as a quantum effective action for the volume. At tree level, Γ equals the classical action S E evaluated on the round sphere of radius . For example for 3D Einstein gravity, The tree-level on-shell radius 0 maximizes log Z (0) , i.e. ρ (0) ( 0 ) = 0.
We compare the situation to analogous one-loop expressions [32,58] for Euclidean AdS with standard (AdS + d+1 ) [9] and alternate (AdS − d+1 ) [25] gauge field boundary conditions, and to the associated conformal higher-spin theory on the boundary S d (CHS d ) [26,59]. For AdS + the above problems are absent, as g is not gauged and ∆ s > s. Like a summed KK tower, the spin-summed bulk character has increased UV dimensionality d bulk eff = 2d − 2. However, the edge character almost completely cancels this, leading to a reduced d eff = d − 1 in (9.10)- (9.12). This realizes a version of a stringy picture painted in [33] repainted in fig. E.4. A HS "swampland" is identified: lacking a holographic dual, characterized by d eff > d − 1.
For AdS − and CHS, the problems listed for dS all reappear. g is gauged, and the character spin sum divergences are identical to dS, as implied by the relations (9.16): The spin sum divergences are not UV. Their origin lies in low-energy features: an infinite number of quasinormal modes decaying as slowly as e −2T / for d ≥ 4 (cf. discussion below (G.5)). We see no justification for zeta-regularizing such divergences away. However, in certain supersymmetric extensions, the spin sum divergences cancel in a rather nontrivial way, leaving a finite residual as in (9.23). This eliminates problem 1, but leaves problem 2.
Problem 2 might be analogous to vol G = ∞ for the bosonic string or vol G = 0 for supergroup Chern-Simons: removed by appropriate insertions. This, and more, is left to future work.

Quasicanonical bulk thermodynamics 2.1 Problem and results
From the point of view of an inertial observer, such as Pelagibacter ubique in fig. 2.1c, the global de Sitter vacuum appears thermal [1,2,60]: P. ubique perceives its universe, the southern static patch (S in fig. 2.1a), as a static ball of finite volume, whose boundary r = is a horizon at temperature T = 1/2π , and whose bulk is populated by field quanta in thermal equilibrium with the horizon. P. ubique wishes to understand its universe, and figures the easiest thing to understand should be the thermodynamics of its thermal environment in the ideal gas approximation. The partition function of an ideal gas is Tr e −βH = exp ∞ 0 dω −ρ(ω) bos log e βω/2 − e −βω/2 + ρ(ω) fer log e βω/2 + e −βω/2 , (2.2) where ρ(ω) = ρ(ω) bos + ρ(ω) fer is the density of bosonic and fermionic single-particle states at energy ω. However to its dismay, it immediately runs into trouble: the dS static patch mode spectrum is continuous and infinitely degenerate, leading to a pathologically divergent density ρ(ω) = δ(0) m··· . It soon realizes the unbounded redshift is to blame, so it imagines a brick wall excising the horizon, or some variant thereof (appendix E. 3). Although this allows some progress, it is aware this alters what it is computing and depends on choices. To check to what extent this matters, it tries to work out nontrivial examples. This turns out to be painful. It feels there should be a better way, but its efforts come to an untimely end [61].
Here we will make sense of the density of states and the static patch bulk thermal partition function in a different way, manifestly preserving the underlying symmetries, allowing general exact results for arbitrary particle content. The main ingredient is the Harish-Chandra group character (reviewed in appendix A) of the SO(1, d + 1) representation R furnished by the physical single-particle Hilbert space of the free QFT quantized on global dS d+1 . Letting H be the global SO(1, 1) generator acting as time translations in the southern static patch and globally as in fig. 2.1a, the character restricted to group elements e −itH is (2.3) Here tr G traces over the global dS single-particle Hilbert space furnishing R. (More generally we denote tr ≡ single-particle trace, Tr ≡ multi-particle trace, G ≡ global, S ≡ static patch. Our default units set the dS radius ≡ 1 .) For example for a scalar field of mass m 2 = ( d 2 ) 2 + ν 2 , as computed in (A. 13), For a massive spin-s field this simply gets an additional spin degeneracy factor D d s , (A.14). Massless spin-s characters take a similar but somewhat more intricate form, (G.2)-(G.4).
As mentioned in the introduction, (1.14), the character has a series expansion (2.5) encoding the degeneracy N r of quasinormal modes ∝ e −rT of the dS static patch background.
For example expanding the scalar character yields two towers of quasnormal modes with r n± = d 2 ± iν + n and degeneracy N n± = n+d−1 n .
Our main result, shown in 2.2 below, is the observation that 1 + e −2πt/β 1 − e −2πt/β χ(t) bos − 2 e −πt/β 1 − e −2πt/β χ(t) fer , (2.6) suitably regularized, provides a physically sensible, manifestly covariant regularization of the static patch bulk thermal partition. The basic idea is that ρ(ω) can be obtained as a well-defined Fourier transform of the covariantly UV-regularized character χ(t), which upon substitution in the ideal gas formula (2.2) yields the above character integral formula. Arbitrary thermodynamic quantities at the horizon equilibrium β = 2π can be extracted from this in the usual way, for example S bulk = (1 − β∂ β ) log Z bulk | β=2π , which can alternatively be interpreted as the "bulk" entanglement entropy between the northern and southern S d hemispheres (red and blue lines fig. 2.1a). 4 We work out various examples of such thermodynamic quantities in section 2.3. General exact solution are easily obtained. The expansion (2.5) also allows interpreting the results as a sum over quasinormal modes along the lines of [50].
We conclude this part with some comments on the relation with the Euclidean partition In sections 3-5 we will obtain the exact edge corrections by direct computation, logically independent of these considerations, but guided by the physical expectation (2.7) and more generally the intuition developed in this section.

Derivation
We first give a formal derivation and then refine this by showing the objects of interest become rigorously well-defined in a manifestly covariant UV regularization of the QFT.

Formal derivation
Our starting point is the observation that the thermal partition function Tr e −βH of a bosonic resp. fermionic oscillator of frequency ω has the integral representation (E.14): − log e βω/2 − e −βω/2 = + (2. 8) with the pole in the factor f (t) = c t −2 + O(t 0 ) multiplying e −iωt + e iωt resolved by Now consider a free QFT on some space of finite volume, viewed as a system S of bosonic and/or fermionic oscillator modes of frequencies ω with mode (or single-particle) density ρ S (ω) = ρ S (ω) bos + ρ S (ω) fer . The system is in thermal equilibrium at inverse temperature β.
Using the above integral representation, we can write its thermal partition function (2.2) as log Tr S e −βH S = ∞ 0 dt 2t 1 + e −2πt/β 1 − e −2πt/β χ S (t) bos − 2 e −πt/β 1 − e −2πt/β χ S (t) fer , (2.10) where we exchanged the order of integration, and we defined χ S (t) ≡ ∞ 0 dω ρ S (ω) e −iωt + e iωt (2.11) We want to apply (2.10) to a free QFT on the southern static patch at inverse temperature β, with the goal of finding a better way to make sense of it than P. ubique's approach. To this end, we note that the global dS d+1 Harish-Chandra character χ(t) defined in (2.3) can formally be written in a similar form by using the general property (A.4), χ(t) = χ(−t):

12)
This looks like (2.11), except ρ G (ω) = tr G δ(ω − H) is the density of single-particle excitations of the global Euclidean vacuum, while ρ S (ω) is the density of single-particle excitations of the southern vacuum. The global and southern vacua are very different. Nevertheless, there is a simple kinematic relation between their single-particle creation and annihilation operators: the Bogoliubov transformation (E.10) (suitably generalized to d > 0 [60]). This provides an explicit one-to-one, inner-product-preserving map between southern and global single-particle states with H = ω > 0. Hence, formally, While formal in the continuum, this relation becomes precise whenever ρ is rendered effectively finite, e.g. by a brick-wall cutoff or by considering finite resolution projections (say if we restrict to states emitted/absorbed by some apparatus built by P. ubique).
At first sight this buys us nothing though, as computing ρ G (ω) = tr G δ(ω − H) for say a scalar in dS 4 in a basis |ω m G immediately leads to ρ G (ω) = δ(0) m , in reassuring but discouraging agreement with P. ubique's result for ρ S (ω). On second thought however, substituting this into (2.12) leads to a nonsensical χ(t) = 2πδ(t)δ(0) m , not remotely resembling the correct expression (2.4). How could this happen? As explained under (A.17), the root cause is the seemingly natural but actually ill-advised idea of computing χ(t) = tr G e −iHt by diagonalizing H: despite its lure of seeming simplicity, |ω m G is in fact the worst possible choice of basis to compute the character trace. Its wave functions on the global future boundary S d of dS d+1 are singular at the north and south pole, exactly the fixed points of H at which the correct computation of χ(t) in appendix A.2 localizes. Although |ω m is a perfectly fine basis on the cylinder obtained by a conformal map from sphere, the information needed to compute χ is irrecoverably lost by this map.
However we can turn things around, and use the properly computed χ(t) to extract ρ G (ω) as its Fourier transform, inverting (2.12). As it stands, this is not really possible, for (2.4) implies χ(t) ∼ |t| −d as t → 0, so its Fourier transform does not exist. Happily, this problem is automatically resolved by standard UV-regularization of the QFT, as we will show explicitly below. For now let us proceed formally, as at this level we have arrived at our desired result: combining (2.13) with (2.12) and (2.11) implies χ S (t) = χ(t), which by (2.10) yields with Z bulk (β) as defined in (2.6). The above equation formally gives it its claimed thermal interpretation. In what follows we will make this a bit more precise, and spell out the UV regularization explicitly.

Covariant UV regularization of ρ and Z bulk
We begin by showing that ρ G (ω) in (2.12) becomes well-defined in a suitable standard UVregularization of the QFT. As in [63], it is convenient to consider Pauli-Villars regularization, which is manifestly covariant and has a conceptually transparent implementation on both the path integral and canonical sides. For e.g. a scalar of mass m 2 = ( d 2 ) 2 + ν 2 , a possible implementation is adding k n , n = 1, . . . , k ≥ d 2 fictitious particles of mass m 2 = ( d 2 ) 2 +ν 2 +nΛ 2 and positive/negative norm for even/odd n, 5 turning the character χ ν 2 (t) of (2.4) into (−1) n k n χ ν 2 +nΛ 2 (t). (2.15) This effectively replaces falls off exponentially at large t, which is always the case for unitary representations [64], χ Λ (t) has a well-defined Fourier transform, analytic in ω: (2.16) The above character regularization can immediately be transported to arbitrary massive SO(1, d + 1) representations, as their characters χ s,ν 2 (A.16) only differ from the scalar one by an overall spin degeneracy factor. 6 Although we won't need to in practice for computations of thermodynamic quantities (which are most easily extracted directly as character integrals), ρ G,Λ (ω) can be computed explicitly. For the dS d+1 scalar, using (2.4) regularized with k = 1, we get for ω Λ where ψ(x) = Γ (x)/Γ(x). Denoting the Λ-independent parts of the above ω Λ expansions byρ ν 2 (ω), the exact ρ G,Λ (ω) for general ω and k is ρ G,Λ (ω) = k n=0 (−1) n k n ρ ν 2 +nΛ 2 (ω), illustrated in fig. 2.3 for k = 1, 2. The ω Λ result is independent of k up to rescaling of Λ. The result for massive higher-spin fields is the same up to an overall degeneracy factor D d s from (A.14).
To make sense of the southern static patch density ρ S (ω) directly in the continuum, we define its regularized version by mirroring the formal relation (2.13), thus ensuring all of the well-defined features and physics this relation encapsulates are preserved: This definition of the regularized static patch density evidently inherits all of the desirable properties of ρ G (ω): manifest general covariance, independence of arbitrary choices such as brick wall boundary conditions, and exact analytic computability. The physical sensibility of this identification is also supported by the fact that the quasinormal mode expansion (2.5) 5 This is equivalent to inserting a heat kernel regulator f (τ Λ 2 ) = 1 − e −τ Λ 2 k in (3.2), with k ≥ d 2 + 1. 6 For massless spin-s, the PV-regulating characters to add to the physical character (e.g. (5.9),(G.3)) arê  of χ(t) produces the physically expected static patch quasinormal resonance pole structure ρ S (ω) = 1 2π r,± Nr r±iω , cf. appendix B.3. Putting things together in the way we obtained the formal relation (2.14), the correspondingly regularized version of the static patch thermal partition function (2.10) is then Note that if we take k ≥ d 2 + 1, then χ Λ (t) ∼ t 2k−d with 2k − d ≥ 2 and we can drop the i prescription (2.9). Z bulk (or equivalently χ) can be regularized in other ways, including by cutting off the integral at t = Λ −1 , or as in (A.21), or by dimensional regularization. For most of the paper we will use yet another variant, defined in section 3, equivalent, like Pauli-Villars, to a manifestly covariant heat-kernel regularization of the path integral.
In view of the above observations, Z bulk,Λ (β) is naturally interpreted as a well-defined, covariantly regularized and ambiguity-free definition of the static patch ideal gas thermal partition in the continuum. However we refrain from denoting Z bulk (β) as Tr S,Λ e −βH S , because it is not constructed as an actual sum over states of some definite regularized static patch Hilbert space H S,Λ . This (together with the role of quasinormal modes) is also why we referred to Z bulk (β) as a "quasi"-canonical partition function in the introduction.

Example computations
In this section we illustrate the use and usefulness of the character formalism by computing some examples of bulk thermodynamic quantities at the equilibrium inverse temperature β = 2π of the static patch. The precise relation of these quantities with their Euclidean counterparts will be determined in 3-5 and 7.
Character formulae for bulk thermodynamic quantities at β = 2π At β = 2π, the bulk free energy, energy, entropy and heat capacity are obtained by taking the appropriate derivatives of (2.6) and putting β = 2π, using the standard thermodynamic

21)
and similarly for S bulk and C bulk . The characters χ for general massive representation are given by (A.16), for massless spin-s representations by (G.2)-(G.4), and for partially massless (s, s ) representations by (G.32). Regularization is implicit here.

Leading divergent term
The leading t → 0 divergence of the scalar character (2.4) is χ(t) ∼ 2/t d . For more general representations this becomes χ(t) ∼ 2n/t d with n the number of on-shell internal (spin) degrees of freedom. The generic leading divergent term of the bulk (free) energy is then given by F bulk , U bulk ∼ − 1 π (n bos −n fer ) dt t d+2 ∼ ±Λ d+1 d , while for the bulk heat capacity and entropy we get where we reinstated the dS radius . In particular S bulk ∼ +Λ d−1 × horizon area, consistent with an entanglement entropy area law.
The energy diverges more strongly because we included the QFT zero point energy term in its definition, which drops out of S and C.

Coefficient of log-divergent term
The coefficient of the logarithmically divergent part of these thermodynamic quantities is universal. A pleasant feature of the character formalism is that this coefficient can be read off trivially as the coefficient of the 1/t term in the small-t expansion of the integrand, easily computed for any representation. In odd d + 1, the integrand is even in t, so log-divergences are absent. In even d + 1, the integrand is odd in t, so generically we do get a log-divergence = a log Λ. For example the log Z integrand for a dS 2 scalar is expanded as For a ∆ = d 2 + iν spin-s particle in even d + 1, the log Λ coefficient for U bulk is similarly read off as where Li n is the polylogarithm, Li n (x) ≡ ∞ k=1 x k /k n . For future reference, note that • Entropy: Combined with our earlier result for the bulk energy U bulk , the above also gives the finite part of the bulk entropy S bulk = log Z bulk + 2πU bulk . In the Pauli-Villars regularization (2.15), the UV-divergent part is obtained from the finite part by mirrorring (2.15). For where we used (2.26). S bulk decreases monotonically with m 2 = 1 + ν 2 . In the massless limit m → 0, it diverges logarithmically: S bulk = − log m + · · · . For ν 1, S bulk = π 6 (Λ − ν) up to exponentially small corrections. Thus S bulk > 0 within the regime of validity of the low-energy field theory, consistent with its quasi-canonical/entanglement entropy interpretation. For a conformally coupled scalar ν = i 2 , this gives S fin bulk = 3ζ(3) 16π 2 − log(2) 8 .
s bos (r)/ log 2 s fer (r)/ log 2 c fer (r) c bos (r) Figure 2.4: Contribution to β = 2π bulk entropy and heat capacity of a quasinormal mode ∝ e −rT , r ∈ C, Re r > 0. Only the real part is shown here because complex r come in conjugate pairs r n,± = d 2 + n ± iν. The harmonic oscillator case corresponds to the imaginary axis.

Quasinormal mode expansion
Substituting the quasinormal mode expansion (2.5), (2.28) in the PV-regularized log Z bulk (β) (2.19), rescaling t → β 2π t, and using (C.31) gives Truncating the integral to the IR part (C.31) is justified because the Pauli-Villars sum (2.15) cancels out the UV part. The dependence on µ likewise cancels out, as do some other terms, but it is useful to keep the above form. At the equilibrium β = 2π, log Z bulk is given by (2.29) with b = 1. This provides a PV-regularized version of the quasinormal mode expansion of [50]. Since it is covariantly regularized, it does not require matching to a local heat kernel expansion. Moreover it applies to general particle content, including spin s ≥ 1. 7 QNM expansions of other bulk thermodynamic quantities are readily derived from (2.29) by taking derivatives where the entropy s(r) carried by a single QNM ∝ e −rT at β = 2π is given by The expansion of [50] pertains to Z PI for s ≤ 1 2 . In the following sections we show Z PI = Z bulk for s ≤ 1 2 but not for s ≥ 1. Hence in general the QNM expansion of [50] computes Z bulk , not Z PI .
Note the µ-dependence has dropped out, reflecting the fact that the contribution of each individual QNM to the entropy is UV-finite, not requiring any regularization. For massive representations, r can be complex, but will always appear in a conjugate pair r n± = d 2 +n±iν. Taking this into account, all contributions to the entropy are real and positive for the physical part of the PV-extended spectrum. The small and large r asymptotics are The QNM entropies at general β are obtained simply by replacing The entropy of a normal bosonic mode of frequency ω,s(ω) = − log 1 − e −βω + βω e βω −1 , is recovered for complex conjugate pairs r ± in the scaling limit β → 0, βν = ω fixed, and likewise for fermions. At any finite β, the n → ∞ UV tail of QNM contributions is markedly different however. Instead of falling off exponentially, if falls off as s ∼ 1/n. PV or any other regularization effectively cuts off the sum at n ∼ Λ , so since The bulk heat capacity C bulk = −β∂ β S bulk , so the heat capacity of a QNM at β = 2π is c(r) = −r∂ r s(r) . (2.34) The real part of s(r) and c(r) on the complex r-plane are shown in fig. 2.4.

An application of the quasinormal expansion
The above QNM expansions are less useful for exact computations of thermodynamic quantities than the direct integral evaluations discussed earlier, but can be very useful in computations of certain UV-finite quantities. A simple example is the following. In thermal equilibrium with a 4D dS static patch horizon, which set of particle species has the largest bulk heat capacity: (A) six conformally coupled scalars + graviton, (B) four photons? The answer is not obvious, as both have an equal number of local degrees of freedom: 6 + 2 = 4 × 2 = 8.
One could compute each in full, but the above QNM expansions offers a much easier way to get the answer. From (2.4) and (G.3) we read off the scalar and massless spin-s characters: where q = e −|t| . We see χ A − χ B = χ 2 + 6 χ 0 − 4 χ 1 = 6 q, so χ A and χ B are almost exactly equal: A has just 6 more quasinormal modes than B, all with r = 1. Thus, using (2.34), (2.36) Pretty close, but π > 3 [65], so A wins. The difference is ∆C ≈ 0.87. Along similar lines, Another UV-finite example: relative entropies of graviton, photon, neutrino Less trivial to compute but more real-world in flavor is the following UV-finite linear combination of the 4D graviton, photon, and (assumed massless) neutrino bulk entropies: S graviton + 60 7 S neutrino − 37 7 S photon = 48 7 ζ (−1) − 60 7 ζ (−3) + 6γ + 149 56 − 33 14 log(2π) ≈ 0.61 (2.37) Finiteness can be checked from the small-t expansion of the total integrand computing this, and the integral can then be evaluated along the lines of (C.15)-(C.16). We omit the details.

Vasiliev higher-spin example
Non-minimal Vasiliev higher-spin gravity on dS 4 has a single conformally coupled scalar and a tower of massless spin-s particles of all spins s = 1, 2, 3, . . .. The prospect of having to compute bulk thermodynamics for this theory by brick wall or other approaches mentioned in appendix E.3 would be terrifying. Let us compare this to the character approach. The total character obtained by summing the characters of (2.35) takes a remarkably simple form: (2.38) The first expression is two times the square of the character of a 3D conformally coupled scalar, plus the character of 3D conformal higher-spin gravity (9.17). 8 The second expression equals the character of one ν = i and three ν = 0 scalars on dS 5 . Treating the character integral as such, we immediately get, in k = 3 Pauli-Villars regularization (2.15), π ≈ 0.032. The tower of higher-spin particles alters the bulk UV dimensionality much like a tower of KK modes would. (We will later see edge "corrections" rather dramatically alter this.) 3 Sphere partition function for scalars and spinors

Problem and result
In this section we consider the one-loop Gaussian Euclidean path integral Z (1) PI of scalar and spinor field fluctuations on the round sphere. For a free scalar of mass m 2 on S d+1 , 8 For AdS 4 , the analogous χ tot equals one copy of the 3D scalar character squared, reflecting the single-trace spectrum of its holographic dual U (N ) model 9.2 . The dS counterpart thus encodes the single-trace spectrum of two copies of this 3D CFT + 3D CHS gravity, reminiscent of [66]. This is generalized by (9.16).
A convenient UV-regularized version is defined using standard heat kernel methods [67]: The insertion e − 2 /4τ implements a UV cutoff at length scale ∼ . We picked this regulator for convenience in the derivation below. We could alternatively insert the PV regulator of footnote 5, which would reproduce the PV regularization (2.15). However, being uniformly applicable to all dimensions, the above regulator is more useful for the purpose of deriving general evaluation formulae, as in appendix C.
In view of (2.7) we wish to compare Z PI to the corresponding Wick-rotated dS static patch bulk thermal partition function Z bulk (β) (2.6), at the equilibrium inverse temperature β = 2π.
Here and henceforth, Z bulk by default means Z bulk (2π): Below we show that for free scalars and spinors, with the specific regularization (3.2) for Z PI mapping to a specific regularization (3.9) for Z bulk . The relation is exact, for any . This makes the physical expectation (2.7) precise, and shows that for scalars and spinors, there are in fact no edge corrections.
In appendix C we provide a simple recipe for extracting both the UV and IR parts in the → 0 limit in the above regularization, directly from the unregularized form of the character

Derivation
The derivation is straightforward:

Scalars:
The eigenvalues of −∇ 2 on a sphere of radius ≡ 1 are λ n = n(n+d), n ∈ N, with degeneracies D d+2 n given by (A.15), that is D d+2 2) can be written as To perform the sum over n, we use the Hubbard-Stratonovich trick, i.e. we write . with integration contour A = R + iδ, δ > 0, as shown in fig. 3.1. The sum evaluates to

7)
We first consider the case m > d 2 , so ν is real and positive. Then, keeping Im u = δ < , we can perform the τ -integral first in (3.5) to get Deforming the contour by folding it up along the two sides of the branch cut to contour B in fig. 3.1, changing variables u = it and using that the square root takes opposite signs on both sides of the cut, we transform this to an integral over C in fig. 3.1:

9)
The result for 0 < m ≤ d 2 , i.e. ν = iµ with 0 ≤ µ < d 2 can be obtained from this by analytic continuation. Putting = 0, this formally becomes Following the same steps as for the scalar case, this can be rewritten as Putting = 0, this formally becomes which we recognize as the fermionic (3.3) with χ(t) the character of the ∆ = 3 2 + im unitary SO (1,4) representation carried by the single-particle Hilbert space of a Dirac spinor quantized on dS 4 , given by twice the character (A.16) of the irreducible representation (∆, S) with S = ( 1 2 ). Thus we conclude Z PI = Z bulk . The comment below (4.16) generalizes this to all d.

Massive higher spins
We first formulate the problem, explaining why it is not nearly as simple as one might have hoped, and then state the result, which turns out to be much simpler than one might have feared. The derivation of the result is detailed in appendix F.1.

Problem
Consider a massive spin-s ≥ 1 field, more specifically a totally symmetric tensor field φ µ 1 ···µs on dS d+1 satisfying the Fierz-Pauli equations of motion: Upon quantization, the global single-particle Hilbert space furnishes a massive spin-s representation of SO(1, d + 1) with ∆ = d 2 + iν, related to the effective mass m s appearing above (see e.g. [68]), and to the more commonly used definition of mass m (see e.g. [69]) as Then m = 0 for the photon, the graviton and their higher-spin generalizations, and for s = 1, m is the familiar spin-1 Proca mass. The massive spin-s bulk thermal partition function is immediately obtained by substituting the massive spin-s character (A.14) into the character formula (3.3) for Z bulk . For d ≥ 3, 9 with spin degeneracy factor read off from (A.15) or (D.3).
The corresponding free massive spin-s Euclidean path integral on S d+1 takes the form where Φ includes at least φ. However it turns out that in order to write down a local, manifestly covariant action for massive fields of general spin s, one also needs to include a tower of auxiliary Stueckelberg fields of all spins s < s [51], generalizing the familiar Stueckelberg action (F.20) for massive vector fields. These come with gauge symmetries, which in turn require the introduction of a gauge fixing sector, with ghosts of all spins s < s.
The explicit form of the action and gauge symmetries is known, but intricate [51].
Classically, variation of the action with respect the Stueckelberg fields merely enforces the transverse-traceless (TT) constraints in (4.1), after which the gauge symmetries can be used to put the Stueckelberg fields equal to zero. One might therefore hope the intimidating off-shell Z PI (4.4) likewise collapses to just the path integral Z TT over the TT modes of φ with kinetic term given by the equations of motion (4.1). This is easy to evaluate. The TT eigenvalue spectrum on the sphere follows from SO(d + 2) representation theory. As detailed in eqs. (F.1)-(F.3), we can then follow the same steps as in section 3, ending up with 10 Here D d+2 n,s is the dimension of the SO(d + 2) representation labeled by the two-row Young diagram (n, s), given explicitly by the dimension formulae in appendix D.1.
Unfortunately, Z TT is not equal to Z PI on the sphere. The easiest way to see this is to consider an example in odd spacetime dimensions, such as (F.4), and observe the result has a logarithmic divergence. A manifestly covariant local QFT path integral on an odd-dimensional sphere cannot possibly have logarithmic divergences. Therefore Z PI = Z TT . The appearance of such nonlocal divergences in Z TT can be traced to the existence of (normalizable) zeromodes in tensor decompositions on the sphere [15,26]. For example the decomposition φ µ = φ T µ + ∇ µ ϕ has the constant ϕ mode as a zeromode, φ µν = φ TT µν + ∇ (µ ϕ ν) + g µν ϕ has conformal Killing vector zeromodes, and φ µ 1 ···µs = φ TT µ 1 ···µs + ∇ (µ 1 ϕ µ 2 ···µs) + g (µ 1 µ 2 ϕ µ 3 ···µs) has rank s − 1 conformal Killing tensor zeromodes. As shown in [15,26], this implies log Z TT contains a nonlocal UV-divergent term c s log Λ, where c s is the number of rank s − 1 conformal Killing tensors. This divergence cannot be canceled by a local counterterm. Instead it must be canceled by contributions from the non-TT part. Thus, in principle, the full off-shell path integral must be carefully evaluated to obtain the correct result. Computing Z PI for general s on the sphere is not as easy as one might have hoped.

Result
Rather than follow a brute-force approach, we obtain Z PI in appendix F.1 by a series of relatively simple observations. In fact, upon evaluating the sum in (4.5), writing it in a way that brings out a term log Z bulk as in (4.3), and observing a conspicuous finite sum of terms bears full responsibility for the inconsistency with locality, the answer suggests itself right away: the non-TT part restores locality simply by canceling this finite sum. This turns out to be equivalent to the non-TT part effectively extending the sum n ≥ s in (4.5) to n ≥ −1: 11 where D d+2 n,s is given by the explicit formulae in appendix D.1, in particular (D.4). For n < s, this is no longer the dimension of an SO(d + 2) representation, but it can be rewritten as minus the dimension of such a representation, as D d+2 n,s = −D d+2 s−1,n+1 . This extension also turns out to be exactly what is needed for consistency with the unitarity bound (F. 19) and more refined unitarity considerations. A limited amount of explicit path integral considerations combined with the observation that the coefficients D d+2 s−1,n+1 count conformal Killing tensor mode mismatches between ghosts and longitudinal modes then suffice to establish this is indeed the correct answer. We refer to appendix F.1 for details.
Using the identity (F.9), we can write this in a rather suggestive form: where χ bulk and χ edge are explicitly given by 12 The log Z bulk term is the character integral for the bulk partition function (4.3). Strikingly, the correction log Z edge also takes the form a character integral, but with an "edge" character χ edge in two lower dimensions. By our results of section 3 for scalars, Z edge effectively equals the Euclidean path integral of D d+2 s−1 scalars of massm 2 = d−2 2 2 + ν 2 on S d−1 : in (F.15)), and that they become massless at the unitarity bound ν = ±i( d 2 − 1), eq. (F. 19), where a partially massless field emerges with a scalar gauge parameter.
Independent of any interpretation, we can summarize the result (4.7)-(4.8) as Examples That is, Z PI = Z bulk /Z edge , with the finite part of log Z bulk explicitly given by twice (2.25), and with Z edge equal to the Euclidean path integral of D 4 s−1 = s 2 harmonic oscillators of frequency ν on S 1 , naturally identified with the S 1 horizon of the dS 3 static patch, with finite part (4.12) The heat-kernel regularized Z PI is then, restoring and recalling ν = m 2 2 − (s − 1) 2 , The d = 3 spin-s case is worked out as another example in (C.27).
General massive representations (4.6) has a natural generalization, presented in appendix F.2, to arbitrary parity-invariant massive SO(1, d + 1) representations R = ⊕ a (∆ a , S a ), ∆ a = d 2 + iν a , S a = (s a1 , . . . , s ar ): (4.14) where Θ(x) is the Heaviside step step function with Θ(0) ≡ 1 2 , and F a = 0, 1 for bosons resp. fermions. This is the unique TT eigenvalue sum extension consistent with locality and unitarity constraints. As in the S = (s) case, this can be rewritten as a bulk-edge decomposition log Z PI = log Z bulk − log Z edge . For example, using (D.5) and the notation explained above it, the analog of (4.10) for an S = (s, 1 m ) field becomes so here Z edge is the path integral of D d+2 s−1 massive m-form fields living on the S d−1 edge. In particular this implies the recursion relation log Z d+1 . Similarly for a spin s = k + 1 2 Dirac fermion, in the notation explained under table (D.3) where Z bulk now takes the form of the fermionic part of (3.3), with χ bulk as in (A. 16) with , the factor 2 due to the field being Dirac. The edge fields are Dirac spinors.
Note that because D d+2 . We do not have a systematic group-theoretic or physical way of identifying the edge field content. For evaluation of Z PI using (C.19), this identification is not needed however. Actually the original expansions (4.6), (4.14) are more useful for this, as illustrated in (C.23)-(C.27).

Bulk thermal partition function Z bulk
Massless spin-s fields on dS d+1 are in many ways quite a bit more subtle than their massive spin-s counterparts. This manifests itself already at the level of the characters χ bulk,s needed to compute the bulk ideal gas thermodynamics along the lines of section 2. The SO(1, d + 1) unitary representations furnished by their single-particle Hilbert space belong to the discrete series for d = 3 and to the exceptional series for d ≥ 4 [31]. The corresponding characters, discussed in appendix G.1, are more intricate than their massive (principal and complementary series) counterparts. A brief look at the general formula (G.2) or the table of examples (G.3) suffices to make clear they are far from intuitively obvious -as is, for that matter, the identification of the representation itself. Moreover, [31] reported their computation of the exceptional series characters disagrees with the original results in [64].
As noted in section 2 and appendix B.3, the expansion χ bulk (q) = q N k q k can be interpreted as counting the number N k of static patch quasinormal modes decaying as e −kT / .
This gives some useful physics intuition for the peculiar form of these characters, explained in appendix G.1. The characters χ bulk,s (q) can in principle be computed by explicitly constructing and counting physical quasinormal modes of a massless spin-s field. This is a rather nontrivial problem, however.
Thus we see that for massless fields, complications appear already in the computation of Z bulk . Computing Z PI adds even more complications, due to the presence of negative and zero modes in the path integral. Happily, as we will see, the complications of the latter turn out to be the key to resolving the complications of the former. Our final result for Z PI confirms the identification of the representation made in [31] and the original results for the corresponding characters in [64]. This is explicitly verified by counting quasinormal modes in [48].

Euclidean path integral Z PI
We consider massless spin-s fields in the metric-like formalism, that is to say totally symmetric double-traceless fields φ µ 1 ···µs , with linearized gauge transformation with ξ is traceless symmetric in its s = s − 1 indices, and α s picked by convention. 13 We use the notation s ≡ s − 1 as it makes certain formulae more transparent and readily generalizable to the partially massless (0 ≤ s < s) case. The dimensions of φ s and ξ s are Note that this value of ν φ assign a mass m = 0 to φ according to (4.2). The Euclidean path integral of a collection of (interacting) gauge fields φ on S d+1 is formally given by where G is the group of local gauge transformations. At the one-loop (Gaussian) level S[φ] is the quadratic Fronsdal action [70]. Several complications arise compared to the massive case: 1. For s ≥ 2, the Euclidean path integral has negative ("wrong sign" Gaussian) modes, generalizing the well-known issue arising for the conformal factor in Einstein gravity [71]. These can be dealt with by rotating field integration contours. A complication on the sphere is that rotations at the local field level ensuring positivity of short-wavelength modes causes a finite subset of low-lying modes to go negative, requiring these modes to be rotated back [18].
2. The linearized gauge transformations (5.1) have zeromodes: symmetric traceless tensors ξ µ 1 ...µ s−1 satisfying ∇ (µ 1ξ µ 2 ···µs) = 0, the Killing tensors of S d+1 . This requires omitting associated modes from the BRST gauge fixing sector of the Gaussian path integral. As a result, locality is lost, and with it the flexibility to freely absorb various normalization constants into local counterterms without having to keep track of nonlocal residuals. 13 As explained in appendix G.4, for compatibility with certain other conventions we adopt, we will pick α s ≡ √ s with symmetrization conventions such that φ (µ1···µs) = φ µ1···µs . 3. At the nonlinear level, the Killing tensors generate a subalgebra of the gauge algebra.
The structure constants of this algebra are determined by the TT cubic couplings of the interacting theory [27]. At least when it is finite-dimensional, as is the case for Yang-Mills, Einstein gravity and the 3D higher-spin gravity theories of section 6, the Killing tensor algebra exponentiates to a group G. For example for Einstein gravity, To compensate for the zeromode omissions in the path integral, one has to divide by the volume of G. The appropriate measure determining this volume is inherited from the path integral measure, and depends on the UV cutoff and the coupling constants of the theory. Precisely relating the path integral volume vol(G) PI to the "canonical" vol(G) c defined by a theory-independent invariant metric on G requires considerable care in defining and keeping track of normalization factors.
Note that these complications do not arise for massless spin-s fields on AdS with standard boundary conditions. In particular the algebra generated by the (non-normalizable) Killing tensors in this case is a global symmetry algebra, acting nontrivially on the Hilbert space.
These problems are not insuperable, but they do require some effort. A brute-force path integral computation correctly dealing with all of them for general higher-spin theories is comparable to pulling a molar with a plastic fork: not impossible, but necessitating the sort of stamina some might see as savage and few would wish to witness. The character formalism simplifies the task, and the transparency of the result will make generalization obvious.

Ingredients and outline of derivation
We derive an exact formula for Z PI in appendix G.2-G. 4. In what follows we merely give a rough outline, just to give an idea what the origin is of various ingredients appearing in the final result. To avoid the d = 2 footnotes of section 4 we assume d ≥ 3 in what follows.

Naive characters
Naively applying the reasoning of section 4 to the massless case, one gets a character formula of the form (4.7), with "naive" bulk and edge charactersχ given bŷ where χ φ , χ ξ are the massive bulk/edge characters for the spin-s, ∆ = s +d−1 field φ and the spin-s , ∆ = s + d − 1 gauge parameter (or ghost) field ξ, recalling s ≡ s − 1. The subtraction −χ ξ arises from the BRST ghost path integral. More explicitly, from (4.8), (5.5) For example for s = 2 in d = 3, Because of the presence of non-positive powers of q,χ bulk is manifestly not the character of any unitary representation of SO(1, d + 1). Indeed, the character integral (4.7) using these naiveχ is badly IR-divergent, due to the presence of non-positive powers of q.

Contributions to Z PI
To be more precise, after implementing the appropriate contour rotations and zeromode subtractions, we get the following expression for the path integral: where n s is the number of massless spin-s fields in the theory, and the different factors appearing here are defined as follows: 1. Z char,s is defined by the character integral where × 0 means ∞ 0 with the IR divergence due to the constant term removed: The flipped χ bulk,s + turns out to be precisely the massless spin-s exceptional series character χ bulk,s : (G.32) = (G.2). Thus the χ bulk contribution = ideal gas partition function Z bulk , pleasingly consistent with the physics picture. The second term is an edge correction as in the massive case. The third term has no massive counterpart, tied to the presence of gauge zeromodes: D d+2 s−1,s−1 counts rank s − 1 Killing tensors on S d+1 .

2.
A s is due to the zeromode omissions. Denoting M = 2e −γ / as in (C.30), This term looks ugly. Happily, it will drop out of the final result.

i −Ps is the spin-s generalization of Polchinski's phase of the one-loop path integral of
Einstein gravity on the sphere [18]. It arises because every negative mode contour rotation adds a phase factor −i to the path integral. Explicitly, (5.14) In particular P 1 = 0, P 2 = D d+2 1 + D d+2 0 = d + 3 in agreement with [18]. For d + 1 = 4, vol(G) PI is the volume of G according to the QFT path integral measure. We wish to relate it to a "canonical" vol(G) c . We use the word "canonical" in the sense of defined in a theory-independent way. We determine vol(G) PI /vol(G) c given our normalization conventions in appendix G.4. Below we summarize the most pertinent definitions and results.
For Einstein gravity, the Killing vector Lie algebra is g = so(d + 2). For higher-spin gravity, the Killing tensor Lie algebra g contains so(d + 2) as a subalgebra with generators M IJ . We define ·|· c on g to be the unique g-invariant bilinear form [27,28] normalized by (5.15). vol(G) c is defined using the corresponding metric ds 2 c on G. The Killing tensor commutators are determined by the local gauge algebra [δ ξ , δ ξ ] = δ [ξ,ξ ] as in [27]. For Einstein or HS gravity, in our conventions (canonical φ + footnote 13), this gives for the so(d + 2) Killing vector (sub)algebra of g where [·, ·] Lie is the standard vector field Lie bracket. In Einstein gravity, G N is the Newton constant. In Einstein + higher-order curvature corrections (section 8) or in higher-spin gravity we take it to define the Newton constant. It is related to a "central charge" C in (G.62).
Building on [27,30], we find the bilinear ·|· c determining vol(G) c can then be written as This turns out to cancel the awkward eigenvalue product of (5.13), up to a universal factor: for all theories covered by [27], i.e. all parity-invariant HS theories consistent at cubic level.
For Yang-Mills, vol(G) c is computed using the metric ds 2 c on g defined by the canonically normalized YM action S =:

Result and examples
Thus we arrive at the following universal formula for the one-loop Euclidean path integral for parity-symmetric (higher-spin) gravity and Yang-Mills gauge theories on S d+1 , d ≥ 3: • G = G 0 × G 1 × · · · G K is the subgroup of (higher-spin) gravitational and Yang-Mills gauge transformations acting trivially on the background, where A n ≡ Ω n n , Ω n = (D.6), the gravitational and YM coupling constants G N and g 1 , . . . , g K are defined by the canonically normalized so(d+2) and YM gauge algebras as explained around (5.16), and vol G a is the canonically normalized volume of G a , defined in the same part.
• For a theory with n s massless spin-s fields where χ s = [χ s ] + are the flipped versions (5.7) of the naive characters (5.5), with examples in (5.9) and general formulae in (G.32) and (G.34), and P s = (5.14) is the spin-s generalization of the s = 2 phase P 2 = d + 3 found in [18].
• The heat-kernel regularized integral can be evaluated using (C. 19), as spelled out in appendix C.3. For odd d + 1, the finite part can alternatively be obtained by summing residues.
× 0 means integration with the IR log-divergence from the constant −2 dim G term removed as in (5.12). The constant term contribution is then dim G · c −1 + log(2π) , so when keeping track of linearly divergent terms is not needed, one can replace (5.19) by Example: coefficient α d+1 of log-divergent term The heat kernel coefficient α d+1 , i.e. the coefficient of the log-divergent term of log Z, can be read off simply as the coefficient of the 1/t term in the small-t expansion of the integrand. As explained in C.3, we can just use the original, naive integrandF (t) = 1 2t (χ bulk −χ edge ) for this purpose, obtained from (5.5). For e.g. a massless spin-s field on S 4 this immediately gives   Checks: We rederive the S 3 result in the Chern-Simons formulation of 3D gravity [52] in appendix H.2, and find precise agreement, with the phase matching for odd framing of the Chern-Simons partition function (it vanishes for even framing). The coefficient − 571 45 of the log-divergent term of the S 4 result agrees with [14]. The phases agree with [18]. The powers of G N agree with zeromode counting arguments of [22,25]. The full one-loop partition function on S 4 was calculated using zeta-function regularization in [21]. Upon correcting an error in the second number of their equation (A.36) we find agreement. As far as we know, the zeta-function regularized Z (1) PI has not been explicitly computed before for S d+1 , d ≥ 4.

Higher-spin theories
Generic Vasiliev higher-spin gravity theories have infinite spin range and dim G = ∞, evidently posing problems for (5.19). We postpone discussion of this case to section 9. Below we consider a 3D higher-spin gravity theory with finite spin range s = 2, . . . , n.

3D HS n gravity and the topological string
As reviewed in appendix H.2, 3D Einstein gravity with positive cosmological constant in Lorentzian or Euclidean signature can be formulated as an SL(2, C) resp. SU (2) × SU (2) Chern-Simons theory [52]. 15 This has a natural extension to an SL(n, C) resp. SU (n)×SU (n) Chern-Simons theory, discussed in appendix H.3, which can be viewed as an s ≤ n dS 3 higher-spin gravity theory, analogous to the AdS 3 theories studied e.g. in [54,73,74]. The Lorentzian/Euclidean actions S L /S E are Tr n A ∧ dA + 2 3 A ∧ A ∧ A and A ± are sl(n)-valued connections with reality condition A * ± = A ∓ for the Lorentzian theory and A † ± = A ± for the Euclidean theory. The Chern-Simons formulation allows all-loop exact results, providing a useful check of our result (5.19) for Z (1) PI obtained in the metric-like formulation. Besides this, we observed a number of other interesting features, collected in appendix H.3, and summarized below.

Landscape of vacua (H.3.1)
The theory has a set of dS 3 vacua (or round S 3 solutions in the Euclidean theory), corresponding to different embeddings of sl(2) into sl(n), labeled by n-dimensional representations of su(2), i.e. by partitions of n = a m a . The radius in Planck units /G N and Z (0) = e −S E depend on the vacuum R as Note that S (0) = log Z (0) takes the standard Einstein gravity horizon entropy form. The entropy is maximized for the principal embedding, i.e. R = n, for which T n = 1 6 n(n 2 − 1). The number of vacua equals the number of partitions of n: For, say, n ∼ 2 × 10 5 , we get N vac ∼ 10 500 , with maximal entropy S (0) | R=n ∼ 10 15 κ.
Higher-spin algebra and metric-like field content (H.

One-loop and all-loop partition function (H.3.3-H.3.4)
In view of the above higher-spin interpretation, we can compute the one-loop Euclidean path integral on S 3 for l = 0 from our general formula ( where vol(G) Trn = √ n n s=2 (2π) s /Γ(s) 2 as in (D.8).
This can be compared to the weak-coupling limit of the all-loop expression (H.45)-(H.47), obtained from the known exact partition function of SU (n) k + ×SU (n) k − Chern-Simons theory on S 3 by analytic continuation k ± → l ± iκ, (6.7) Here φ = π 4 ± c(l ± iκ) with c(k) ≡ (n 2 − 1) 1 − n n+k , and r ∈ Z labels the choice of framing needed to define the Chern-Simons theory as a QFT, discussed in more detail below (H.25). Canonical framing corresponds to r = 0. Z(R) is interpreted as the all-loop quantumcorrected Euclidean partition function of the dS 3 static patch in the vacuum R.
The weak-coupling limit κ → ∞ of (6.7) precisely reproduces (6.6), with the phase matching for odd framing r. Alternatively this can be seen more directly by a slight variation of the computation leading to (H.11). This provides a check of (5.19), in particular its normalization in the metric-like formalism, and of the interpretation of (6.1) as a higher-spin gravity theory.

Large-n limit and topological string dual (H.3.5)
Vasliev-type hs(so(d + 2)) higher-spin theories (section 9) have infinite spin range but finite d−1 /G N . To mimic this case, consider the n → ∞ limit of the theory at l = 0. The semiclassical expansion is reliable only if n κ. Using /G N ∼ κT R , this translates to n T R /G N , which becomes n 4 /G N for the principal vacuum R = n, and n /G N at the other extreme for R = 2 ⊕ 1 ⊕ · · · ⊕ 1. Either way, the Vasiliev-like limit n → ∞ at fixed S (0) = 2π /4G N is strongly coupled. However (6.7) continues to make sense in any regime, and in particular does have a weak coupling expansion in the n → ∞ 't Hooft limit. Using the large-n duality between U (n) k Chern-Simons on S 3 and closed topological string theory on the resolved conifold [55,56], the partition function (6.7) of de Sitter higher-spin quantum gravity in the vacuum R can be expressed in terms of the weakly-coupled topological string partition functionZ top , (H.50): where (in the notation of [56]) the string coupling constant g s and the resolved conifold Kähler modulus t ≡ S 2 J + iB are given by Note that e −πT R ·2πi/gs 2 = e 2πκT R = e S (0) , and that κ > 0 implies S 2 J > 0 and Im g s = 0. The dependence on n at fixed S (0) is illustrated in fig. 1.4. We leave further exploration of the dS quantum gravity -topological string duality suggested by these observations to future work.

Euclidean thermodynamics
In section 2.3 we defined and computed the bulk partition function, energy and entropy of the static patch ideal gas. In this section we define and compute their Euclidean counterparts, building on the results of the previous sections.

Generalities
Consider a QFT on a dS d+1 background with curvature radius . Wick-rotated to the round sphere metric g µν of radius D.3 , we get the Euclidean partition function: where Φ collectively denotes all fields. The quantum field theory is to be thought of here as a (weakly) interacting low-energy effective field theory with a UV cutoff .
Recalling the path integral definition (E.18) of the Euclidean vacuum |O paired with its dual O| as Z PI = O|O , the Euclidean expectation value of the stress tensor is The last equality, in which ρ PI is a constant, follows from SO(d + 2) invariance of the round sphere background. Denoting the volume of the sphere by Reinstating the radius , the sphere metric in the S coordinates of (D.12) takes the form where τ τ + 2π . Wick rotating τ → iT yields the static patch metric. Its horizon at r = has inverse temperature β = 2π . On a constant-T slice, the vacuum expectation value of the Killing energy density corresponding to translations of T equals ρ PI at the location r = 0 of the inertial observer. Away from r = 0, it is redshifted by a factor 1 − r 2 / 2 . The Euclidean vacuum expectation value U PI of the total static patch energy then equals ρ PI 1 − r 2 / 2 integrated over a constant-T slice: Note that v is the volume of a d-dimensional ball of radius in flat space, so effectively we can think of U PI as the energy of an ordinary ball of volume v with energy density ρ PI . Combining (7.3) and (7.5), the Euclidean energy on this background is obtained as and the corresponding Euclidean entropy S PI ≡ log Z PI + β U PI is S PI can thus be viewed as the Legendre transform of log Z PI trading V for ρ PI : The above differential relations express the first law of (Euclidean) thermodynamics for the system under consideration: using V = βv and ρ PI = U PI /v, they can be rewritten as Viewing v as the effective thermodynamic volume as under (7.5), these take the familiar form of the first law, with pressure p = −ρ, the familiar cosmological vacuum equation of state.
The expression (7.7) for the Euclidean entropy and (7.8) naturally generalize to Euclidean partition functions Z PI ( ) for arbitrary background geometries g µν ( ) ≡ 2g µν with volume V ( ) = d+1Ṽ . In contrast, the expression (7.6) for the Euclidean energy is specific to the sphere. A generic geometry has no isometries, so there is no notion of Killing energy to begin with. On the other hand, the density ρ PI appearing in (7.8) does generalize to arbitrary backgrounds. The last equality in (7.2) and the physical interpretation of ρ PI as a Killing energy density no longer apply, but (7.3) remains valid.

Examples
Free d = 0 scalar To connect to the familiar and to demystify the ubiquitous Li n (e −2πν ) = k e −2πkν /k n terms encountered later, consider a scalar of mass m on an S 1 of radius , a.k.a. a harmonic oscillator of frequency m at β = 2π = V . Using (C.32) and applying (7.6)-(7.7) with ν( ) ≡ m , Mod ∆E 0 ∝ − −1 , these are the textbook canonical formulae turned into polylogs by (2.26).

Free scalar in general d
The Euclidean action of a free scalar on S d+1 is Neither Z PI nor the bulk thermodynamic quantities of section 2 distinguish between the m 2 and ξR contributions to m 2 eff , but U PI and S PI do, due to the ∂ derivatives in (7.6)-(7.7). This results in an additional explicit dependence on ξ, as For the minimally coupled case ξ = 0, the Euclidean and bulk thermodynamic quantities agree, but in general not if ξ = 0. To illustrate this we consider the d = 2 example. Using 14) The corresponding Euclidean energy U PI = ρ PI π 2 (7.6) is given by For minimal coupling ξ = 0 (i.e. η = 1), U fin PI equals U fin bulk (2.24), but not for ξ = 0. For general d, ξ, U fin PI is given by (2.23) with the overall factor m 2 the mass m 2 appearing in the action rather than m 2 eff , in agreement with [75] or (6.178)-(6.180) of [76]. The entropy S PI = log Z PI + 2π U PI (7.7) is where we used coth(πν) = 1 + 2 Li 0 (e −2πν ) (2.26). Since Z PI = Z bulk in general for scalars and U PI = U bulk for minimally coupled scalars, S PI = S bulk for minimally coupled scalars. Indeed, after conversion to Pauli-Villars regularization, (7.16) equals (2.27) if η = 1. As a check on the results, the first law dS PI = V dρ PI (7.8) can be verified explicitly.
In the m → ∞ limit, S PI → π 6 η( −1 − m) , reproducing the well-known scalar one-loop Rindler entropy correction computed by a Euclidean path integral on a conical geometry [33-36, 40, 77]. Note that S PI < 0 when η < 0. Indeed as reviewed in the Rindler context in appendix E.5, S PI does not have a statistical mechanical interpretation on its own. Instead it must be interpreted as a correction to the large positive classical gravitational horizon entropy. We discuss this in the de Sitter context in section 8.
A pleasant feature of the sphere computation is that it avoids replicated or conical geometries: instead of varying a deficit angle, we vary the sphere radius , preserving manifest SO(d + 2) symmetry, and allowing straightforward exact computation of the Euclidean entropy directly from Z PI ( ), for arbitrary field content.

Free 3D massive spin s
Recall from (4.13) that for a d = 2 massive spin-s ≥ 1 field of mass m, the bulk part of log Z PI is twice that of a d = 2 scalar (7.14) with ν = (m ) 2 − η, η = (s − 1) 2 , while the edge part is −s 2 times that of a d = 0 scalar, as in (7.10), with the important difference however that As a result, for the bulk contribution, we can just copy the scalar formulae (7.15) and (7.16) for U PI and S PI setting η = (s − 1) 2 , while for the edge contribution we get something rather different from the harmonic oscillator energy and entropy (7.10): The edge contribution renders S PI negative for all . In particular, in the m → ∞ limit, d+1 = 0 for even d and given by (5.23) for odd d. L is an arbitrary length scale canceling out of the sum of finite and divergent parts, and K d+1 an exactly computable numerical constant. Explicitly for d = 2, 3, 4, from (5.27): The one-loop energy and entropy (7.6)-(7.7) are split accordingly. The finite parts are where as always 2π U = V ρ with V = Ω d+1 d+1 . For d = 2, 3, 4: Like their quasicanonical bulk counterparts, the Euclidean quantities obtained here are UVdivergent, and therefore ill-defined from a low-energy effective field theory point of view.
However if the metric itself, i.e. gravity, is dynamical, these the UV-sensitive terms can be absorbed into standard renormalizations of the gravitational coupling constants, rendering the Euclidean thermodynamics finite and physically meaningful. We turn to this next.

Quantum gravitational thermodynamics
In section 7 we considered the Euclidean thermodynamics of effective field theories on a fixed background geometry. In general the Euclidean partition function and entropy depend on the choice of background metric; more specifically on the background sphere radius . Here we specialize to field theories which include the metric itself as a dynamical field, i.e. we consider gravitational effective field theories. We denote Z PI , ρ PI and S PI by Z, and S in this case: The geometry itself being dynamical, we have ∂ Z = 0, so (7.6)-(7.7) reproduce (1.1):

2)
We will assume d ≥ 2, but it is instructive to first consider d = 0, i.e. 1D quantum gravity coupled to quantum mechanics on a circle. Then Z = dβ 2β Tr e −βH , where β is the circle size and H is the Hamiltonian of the quantum mechanical system shifted by the 1D cosmological constant. To implement the conformal factor contour rotation of [71] implicit in (8.2), we pick an integration contour β = 2π + iy with y ∈ R and > 0 the background circle radius. Then Z = πi N (0) where N (E) is the number of states with H < E. This being -independent implies = 0. A general definition of microcanonical entropy is S mic (E) = log N (E). Thus, modulo the content-independent πi factor in Z, S = log Z is the microcanonical entropy at zero energy in this case.
Of course d = 0 is very different from the general-d case, as there is no classical saddle of the gravitational action, and no horizon. For d ≥ 2 and Λ → 0, the path integral has a semiclassical expansion about a round sphere saddle or radius 0 ∝ 1/ √ Λ, and S is dominated by the leading tree-level horizon entropy (1.2). As in the AdS-Schwarzschild case reviewed in E.5.1, the microscopic degrees of freedom accounting for the horizon entropy, assuming they exist, are invisible in the effective field theory. A natural analog of the dual large-N CFT partition function on S 1 × S d−1 microscopically computing the AdS-Schwarzschild free energy may be some dual large-N quantum mechanics coupled to 1D gravity on S 1 microscopically computing the dS static patch entropy. These considerations suggest interpreting S = log Z as a macroscopic approximation to a microscopic microcanonical entropy, with the semiclassical/low-energy expansion mapping to some large-N expansion.
The one-loop corrected Z is obtained by expanding the the action to quadratic order about its sphere saddle. The Gaussian Z (1) PI was computed in previous sections. Locality and dimensional analysis imply that one-loop divergences are ∝ R n with 2n ≤ d + 1. Picking counterterms canceling all (divergent and finite) local contributions of this type in the limit  In the examples given earlier, these two look rather different. Can these considerations be reconciled? 2. Besides local UV contributions absorbed into renormalized coupling constants determining the tree-level radius 0 , there will be nonlocal IR vacuum energy contributions (pictorially Hawking radiation in equilibrium with the horizon), shifting the radius from To address these and other questions, we follow a slghtly less direct path, summarized below, and explained in more detail including examples in appendix I.

Free energy/quantum effective action for volume
We define an off-shell free energy/quantum effective action Γ(V ) = − log Z(V ) for the volume, the Legendre transform of the off-shell entropy/moment-generating function S(ρ): 16 At large V , the geometry semiclassically fluctuates about a round sphere. Parametrizing the mean volume V by a corresponding mean radius as V ( ) ≡ Ω d+1 d+1 , we have where tree dρ means saddle point evaluation, i.e. extremization. The Legendre transform (8.3) is the same as (7.8), so we get thermodynamic relations of the same form as (7.6)-(7.8): On-shell quantities are obtained at ρ = 0, i.e. at the minimum¯ of the free energy − log Z( ):

Tree level
At tree level (8.4) evaluates to readily evaluated for any action using R µνρσ = (g µρ g νσ − g µσ g νρ )/ 2 , taking the general form The z n are R n+1 coupling constants and l s is the length scale of UV-completing physics.
The off-shell entropy and energy density are obtained from log Z (0) as in (8.5).

One loop
The one-loop order, (8.4) is a by construction tadpole-free Gaussian path integral, (I.31): PI as computed in sections 4-5 and log Z ct ( ) = −S E,ct [g ] a polynomial counterterm. We define renormalized coupling constants as the coefficients of the d+1−2n terms in the → ∞ expansion of log Z, and fix log Z ct by equating tree-level and renormalized coefficients of the polynomial part, which amounts to the renormalization condition Example: 3D Einstein gravity + minimally coupled scalar ( The last two terms are counterterms. The first two are nonlocal graviton terms. The scalar Defining ρ (1) and S (1) from log Z (1) as in (8.5), and the quantum on-shell¯ = 0 + O(G) as in (8.6), the quantum entropy can be expressed in two equivalent ways, (I.38)-(I.39): where the dots denote terms neglected in the one-loop approximation. This simultaneously answers questions 1 and 2 on our list, reconciling intuitive (A) and (8.2)-based (B) expectations. To make this physically obvious, consider the quantum static patch as two subsystems, geometry (horizon) + quantum fluctuations (radiation), with total energy ∝ ρ = ρ (0) +ρ (1) = 0.
If ρ (0) = 0, the horizon entropy is S (0) ( 0 ). But here we have ρ = 0, so the horizon entropy Adding the radiation entropy S (1) and recalling log Thus A = B is just the usual small+large = system+reservoir approximation, the horizon being the reservoir, and the Boltzmann factor e −V ρ (1) = e −βU (1) in Z (1) accounting for the reservoir's entropy change due to energy transfer to the system.
Viewing the quantum contributions as (Hawking) radiation has its picturesque merits and correctly conveys their nonlocal/thermal character, e.g. Li(e −2πν ) ∼ e −βm for m 1 in (8.13), but might incorrectly convey a presumption of positivity of ρ (1) and S (1)

Computational recipe and examples
For practical purposes, (B) is the more useful expression in (8.14). Together with (8.10) computing S (0) , the exact results for Z (1) PI obtained in previous sections (with γ 0 = 2π/S (0) , see (8.16) below), and the renormalization prescription outlined above, it immediately gives in terms of the renormalized coupling constants, for general effective field theories of gravity coupled to arbitrary matter and gauge fields. in the low-energy effective action. The former is field-redefinition invariant; the latter is not.
In Einstein frame (z n = 0) the two definitions coincide, hence in a general frame

Constraints on microscopic models
For pure 3D gravity 0 + · · · , and to one-loop order we have (I.62): Granting 17 (H.24) with l = 0 gives the all-loop expansion of pure 3D gravity, taking into there, to all-loop order, where S 0 ≡ S (0) to declutter notation. Note all quantum corrections are strictly nonlocal, i.e. no odd powers of 0 appear, reflected in the absence of odd powers of 1/S 0 .
Though outside the scope of this paper, let us illustrate how such results may be used to constrain microscopic models identifying large-0 and large-N expansions in some way.

Say a modeler posits a model consisting of
There is a unique identification of S 0 bringing this in a form with the same analytic/locality structure as (8.18), to wit, S 0 = log 4 · N + n c n N 1−2n , resulting in . ., fully failing to match (8.18), starting at one loop. The model is ruled out.
A slightly more sophisticated modeler might posit S mic = log d(N ), where d(N ) is the N -th level degeneracy of a chiral boson on S 1 . To leading order S mic ≈ 2π N/6 ≡ K. Beyond, √ 3) and c n given by [79]. Identifying S 0 = K + n c 2n−1 K −(2n−1) brings this to the form (8.18), yielding . . -ruled out. We actually did not need the higher-loop corrections at all to rule out the above models. In higher dimensions, or coupled to more fields, one-loop constraints moreover become increasingly nontrivial, evident in (1.12). For pure 5D gravity (I.62), It would be quite a miracle if a microscopic model managed to match this. 9 dS, AdS±, and conformal higher-spin gravity Vasiliev higher-spin gravity theories [57] have infinite spin range and an infinite-dimensional higher-spin algebra, g = hs(so(d + 2)), leading to divergences in the one-loop sphere partition function formula (5.19) untempered by the UV cutoff. In this section we take a closer look at these divergences. We contrast the situation to AdS with standard boundary conditions (AdS+), where the issue is entirely absent, and we point out that, on the other hand, for AdS with alternate HS boundary conditions (AdS−) as well as conformal higher-spin (CHS) theories, similar issues arise. We end with a discussion of their significance.

dS higher-spin gravity
Nonminimal type A Vasiliev gravity on dS d+1 has a tower of massless spin-s fields for all s ≥ 1 and a ∆ = d − 2 scalar. We first consider d = 3. The total bulk and edge characters are obtained by summing (5.9) and adding the scalar, as we did for the bulk part in (2.38): Quite remarkably, the bulk and edge contributions almost exactly cancel: For d = 4 however, we see from (5.9) that due to the absence of overall q s suppression factors, the total bulk and edge characters each diverge separately by an overall multiplicative factor: This pattern persists for all d ≥ 4, as can be seen from the explicit form of bulk and edge characters in (G.3), (G.32), (G.34). For any d, there is moreover an infinite-dimensional group volume factor in (5.19) to make sense of, involving a divergent factor and the volume of an object of unclear mathematical existence [80].
Before we continue the discussion of what, if anything, to make of this, we consider AdS± and CHS theories within the same formalism. Besides independent interest, this will make clear the issue is neither intrinsic to the formalism, nor to de Sitter.

AdS± higher-spin gravity AdS characters for standard and alternate HS boundary conditions
Standard boundary conditions on massless higher spin fields ϕ in AdS d+1 lead to quantization such that spin-s single-particle states transform in a UIR of so(2, d) with primary dimension Higher-spin Euclidean AdS one-loop partition functions with these boundary conditions were computed in [9]. In [25], the Euclidean one-loop partition function for alternate boundary conditions (∆ ϕ = ∆ − = 2 − s) was considered. In the EAdS+ case, the complications listed under (5.3) are absent, but for EAdS− close analogs do appear.
EAdS path integrals can be expressed as character integrals [32,58], in a form exactly paralleling the formulae and bulk/edge picture of the present work [58]. 18 The AdS analog of the dS bulk and edge characters (4.8) for a massive spin-s field ϕ with Thus, as functions of q, The AdS analog of (5.4) for a massless spin-s field φ s with gauge parameter field ξ s iŝ More explicitly, analogous to (5.5), The presence of non-positive powers of q in χ AdS− has a similar path integral interpretation as in the dS case summarized in section 5.2. The necessary negative mode contour rotation and zeromode subtractions are again implemented at the character level by flipping characters.
In particular the proper χ s to be used in the character formulae for EAdS± are The group G they generate acts nontrivially on the Hilbert space as a global symmetry group. 18 In this picture, EAdS is viewed as the Wick-rotated AdS-Rindler wedge, with dS d static patch boundary metric, as in [81]. The bulk character is χ ≡ tr G q iH , with H the Rindler Hamiltonian, not the global AdS Hamiltonian. Its q-expansion counts quasinormal modes of the Rindler wedge. The one-loop results are interpreted as corrections to the gravitational thermodynamics of the AdS-Rindler horizon [58,81].

AdS+
For standard boundary conditions, the character formalism reproduces the original results of [9] by two-line computations [58]. We consider some examples: For nonmimimal type A Vasiliev with ∆ 0 = d − 2 scalar boundary conditions, dual to the free U (N ) model, using (9.7) and the scalar χ 0 = q d−2 /(1 − q) d , the following total bulk and edge characters are readily obtained: The total bulk character takes the singleton-squared form expected from the Flato-Fronsdal theorem [82]. More interestingly, the edge characters sum up to exactly the same. Thus the generally negative nature of edge "corrections" takes on a rather dramatic form here: As Z AdS+ bulk has an Rindler bulk ideal gas interpretation analogous to the static patch ideal gas of section 2 [58], the exact bulk-edge cancelation on display here is reminiscent of analogous one-loop bulk-edge cancelations expected in string theory according to the qualitative picture reviewed in appendix E.5.2.
For minimal type A, dual to the free O(N ) model, the sum yields an expression which after rescaling of integration variables t → t/2 is effectively equivalent to the so(2, d) singleton character, which is also the so(1, d) character of a conformally coupled (ν = i/2) scalar on S d . Using (3.10), this means Z AdS+ PI equals the sphere partition function on S d , immediately implying the N → N − 1 interpretation of [9].
For nonminimal type A with ∆ 0 = 2 scalar boundary conditions, dual to an interacting U(N) CFT, the cancelation is almost exact but not quite: (9.12)

AdS+ higher-spin swampland
In the above examples it is apparent that although the spin-summed χ bulk has increased effective UV-dimensionality d bulk eff = 2d − 2, as if we summed KK modes of a compactification manifold of dimension d−2, the edge subtraction collapses this back down to a net d eff = d−1, decreasing the original d. Correspondingly, the UV-divergences of Z (1) PI are not those of a d + 1 dimensional bulk-local theory, but rather of a d-dimensional boundary-local theory. In fact this peculiar property appears necessary for quantum consistency, in view of the non-existence of a nontrivially interacting local bulk action [83]. It appears to be true for all AdS+ higher spin theories with a known holographic dual [58], but not for all classically consistent higher-spin theories. Thus it appears to be some kind of AdS higher-spin "swampland" criterion: Higher-spin theories violating this criterion do exist. Theories with a tower of massless spins s ≥ 2 and an a priori undetermined number n of real scalars can be constructed in AdS 3 [84,85]. Assuming all integer spins s ≥ 2 are present, the total character sums up to (9.14) For To satisfy (9.13), the number of scalars must be n = 2. This is inconsistent with the n = 4 AdS 3 theory originally conjectured in [85] to be dual to a minimal model CFT 2 , but consistent with the amended conjecture of [86,87].

AdS−
For alternate boundary conditions, one ends up with a massless higher-spin character formula [25]. (9.5) implies the massless AdS± and dS bulk and edge characters are related as

Conformal higher-spin gravity Conformal HS characters
Conformal (higher-spin) gravity theories [88] have (higher-spin extensions of) diffeomorphisms and local Weyl rescalings as gauge symmetries. If one does not insist on a local action, a general way to construct such theories is to view them as induced theories, obtained by integrating out the degrees of freedom of a conformal field theory coupled to a general background metric and other background fields. In particular one can consider a free U (N ) CFT d in a general metric and higher-spin source background. For even d, this results in a local action, which at least at the free level can be rewritten as a theory of towers of partially massless fields with standard kinetic terms [26,59]. Starting from this formulation of CHS theory on S d (or equivalently dS d ), using our general explicit formulae for partially massless higher-spin field characters (G.32) and (G. 34), and summing up the results, we find where χ CdS d s are the CHS bulk and edge characters and the second equality uses (9.15). Since we already know the explicit dS and AdS HS bulk and edge characters, this relation also provides the explicit CHS bulk and edge characters. For example bulk,s computed from (9.16) agree with the so(2, d) q-characters obtained in [89]. Edge characters were not derived in [89], as they have no role in the thermal 19 The one-loop Euclidean path integral of the CHS theory on S d is given by (5.19) using the bulk and edge CHS characters χ CdS d s and with G the CHS symmetry group generated by the conformal Killing tensors on S d (counted by D d+3 s−1,s−1 ). The coefficient of the log-divergent term, the Weyl anomaly of the CHS theory, is extracted as usual, by reading off the coefficient of the 1/t term in the small-t expansion of the integrand in (5.19), or more directly from the "naive" integrand 1 2t 1+q 1−qχ . For example for conformal s = 2 gravity on S 2 coupled to D massless scalars, also known as bosonic string theory in D spacetime dimensions, we have dim G = ± D 4 1,±1 = 6, generating G = SO(1, 3), and from the above table (9.17), The small-t expansion of the integrand in (5.19) for this case is 1 2t reassuringly informing us the critical dimension for the bosonic string is D = 26. Adding a massless s = 3 2 field, we get 2D conformal supergravity. For half-integer conformal spin s, Furthermore adding D massless Dirac spinors, the total fermionic character is The symmetry algebra has ± D 4 from which we read off supersymmetry + conformal symmetry requires D = D = 10.

Physics pictures
Cartoonishly speaking, the character relation (9.16) translates to one-loop partition function The first relation can then be understood as a consequence of the holographic duality between AdS d+1 higher-spin theories and free CFT d vector models [25,26,59], while the second relation can be understood as an expression at the Gaussian/one-loop level of where ψ HH (σ) = ψ HH (0) e − 1 2 σKσ+··· is the late-time dS Hartle-Hawking wave function, related by analytic continuation to the EAdS partition function with boundary conditions σ [90]. The factor Z EAdS d+1 + 2 can then be identified with the bulk one-loop contribution to |ψ HH (0)| 2 , and Z cnf S d with Dσ e −σKσ , along the lines of [25]. Along the lines of footnote 8, perhaps another interpretation of the spin-summed relation (9.16) exists within the picture of [66].

Comments on infinite spin range divergences
Let us return now to the discussion of section 9.1. Above we have seen that for EAdS+, summing spin characters leads to clean and effortless computation of the one-loop partition function. The group volume factor is absent because the global higher-spin symmetry algebra g generated by the Killing tensors is not gauged. The character spin sum converges, and no additional regularization is required beyond the UV cutoff at t ∼ we already had in place.
The underlying reason for this is that in AdS+, the minimal energy of a particle is bounded below by its spin, hence a UV cutoff is effectively also a spin cutoff. In contrast, for dS, AdS− and CHS theories alike, g is gauged, leading to the group volume division factor, and moreover, for d ≥ 4, the quasinormal mode levels (or energy levels for CHS on R × S d−1 ) are infinitely degenerate, not bounded below by spin, leading to character spin sum divergences untempered by the UV cutoff. The geometric origin of quasinormal modes decaying as slowly as e −2T / for every spin s in d ≥ 4 was explained below (G.5).
One might be tempted to use some form of zeta function regularization to deal with divergent sums s χ s such as (9.3), which amounts to inserting a convergence factor ∝ e −δs and discarding the divergent terms in the limit δ → 0. This might be justified if the discarded divergences were UV, absorbable into local counterterms, but that is not the case here. The divergence is due to low-energy features, the infinite multiplicity of slow-decaying quasinormal modes, analogous to the divergent thermodynamics of an ideal gas in a box with an infinite number of different massless particle species. Zeta function regularization would give a finite result, but the result would be meaningless.
As discussed at the end of section 6, the Vasiliev-like 20 limit of the 3D HS n higher-spin gravity theory, n → ∞ with l = 0 and S (0) fixed, is strongly coupled as a 3D QFT. Unsurprisingly, the one-loop entropy "correction" S (1) = log Z (1) diverges in this limit: writing the explicit expression for the maximal-entropy vacuum R = n in (1.12) as a function of The higher-spin decomposition (H.41) might inspire an ill-advised zeta function regularization along the lines . This gives S (1) = 2 3 log S (0) + c with c a computable constant -a finite but meaningless answer. In fact, using (6.7), the all-loop quantum correction to the entropy can be seen to vanish in the limit under consideration, as illustrated in fig. 1.4. As discussed around (6.8), there are more interesting n → ∞ limits one can consider, taking S (0) → ∞ together with n. In these cases, the weakly-coupled description is not a 3D QFT, but a topological string theory.
Although these and other considerations suggest massless higher-spin theories with infinite spin range cannot be viewed as weakly-coupled field theories on the sphere, one might wonder whether certain quantities might nonetheless be computable in certain (twisted) supersymmetric versions. We did observe some hints in that direction. One example, with details omitted, is the following. First consider the supersymmetric AdS 5 higher-spin theory dual to the 4D N = 2 supersymmetric free U (N ) model, i.e. the U (N ) singlet sector of N massless hypermultiplets, each consisting of two complex scalars and a Dirac spinor. The AdS 5 bulk field content is obtained from this following [91]. In their notation, the hypermultiplet corresponds to the so(2, 4) representation Di + 2 Rac. Decomposing (Di + 2 Rac) ⊗ (Di + 2 Rac) into irreducible so(2, 4) representations gives the AdS 5 free field content: four ∆ = 2 and two ∆ = 3 scalars, one ∆ = 3, S = (1, ±1) 2-form field, six towers of massless spin-s fields for all s ≥ 1, one tower of massless S = (s, ±1) fields for all s ≥ 2, one ∆ = 5 2 Dirac spinor, and four towers of massless spin s = k + 1 2 fermionic gauge fields for all k ≥ 1. Consider now the same field content on S 5 . The bulk and edge characters are obtained paralleling the steps summarized in section 5.2, generalized to the present field content using (4.15) and (4.16).
Each individual spin tower gives rise to a badly divergent spin sum similar to (9.3). However, a remarkable conspiracy of cancelations between various bosonic and fermionic bulk and edge 20 "Vasiliev-like" is meant only in a superficial sense here. The higher-spin algebras are rather different [28].
contributions in the end leads to a finite, unambiguous net integrand: 21 dt 2t Note that the effective UV dimensionality is reduced by two in this case.
An analogous construction for S 4 starting from the 3D N = 2 U (N ) model, gives two ∆ ± = 1, 2 scalars, a ∆ = 3 2 Dirac spinor and two massless spin-1, 3 2 , 2, 5 2 , . . . towers, as in [92,93]. The fermionic bulk and edge characters cancel and the bosonic part is twice (9.2). In this case we moreover get a finite and unambiguous dim G = lim δ→0  [94]. In this case the indeterminacy is resolved by a construction replacing the Wilson loop by an auxiliary worldline quantum mechanics [94]. Perhaps in this spirit, getting a meaningful path integral on the sphere in the present context may require inserting an auxiliary "observer" worldline quantum mechanics, with a natural action of the higher-spin algebra on its phase space, allowing to soak up the residual gauge symmetries.
One could consider other options, such as breaking the background isometries, models with a finite-dimensional higher-spin algebra [95], models with an α -like parameter breaking the higher-spin symmetries, or models of a different nature, perhaps along the lines of [96], or bootstrapped bottom-up. We leave this, and more, to future work.

Acknowledgments
We are grateful to Simone Giombi 21 The spin sums are performed by inserting a convergence factor such as e −δs , but the end result is finite and unambiguous when taking δ → 0, along the lines of lim δ→0 s∈ 1 2 N (−1) 2s (2s + 1) e −δs = 1 4 .

A Harish-Chandra characters
A.1 Definition of χ A central ingredient in this work is the Harish-Chandra group character for unitary represen- More rigorously this should be viewed as a distribution to be integrated against smooth test are trace-class operators, and is always a locally integrable function on G, analytic away from its poles [97]. The group of interest to us is SO(1, d + 1), the isometry group of global dS d+1 , generated by M IJ as defined under (D.9). The representations of SO(1, d + 1) were classified and their characters explicitly computed in [64]. For a recent review and an extensive dictionary between fields and representations, see [31].
For example for a spin-0 UIR corresponding to a scalar field of mass m 2 = ∆(d − ∆), as we will explicitly compute below, this takes the form for all d. The latter property holds for any SO(1, d + 1) representation: This follows from the fact that the SO(1, 1) boost generator H = M 0,d+1 can be conjugated to a boost −H in the opposite direction by a 180-degree rotation: −H = uHu −1 for e.g.

A.2 Computation of χ
Here we show how characters χ(t) = tr e −itH can be computed by elementary means. The full characters χ(t, φ) = tr e −itH+iφ·J can be computed similarly, but we will focus on the former.
Simplest example: d = 1, s = 0 We first consider a d = 1, spin-0 principal series representation with ∆ = 1 2 + iν, ν ∈ R. This corresponds to a massive scalar field on dS 2 with mass m 2 = 1 4 + ν 2 . This unitary irreducible representation of SO(1, 2) can be realized on the Hilbert space of square-integrable wave functions ψ(ϕ) on S 1 , with standard inner product. The circle can be thought of as the future conformal boundary of global dS 2 in global coordinates (cf. (D.10)), which for dS 2 becomes ds 2 = (cos ϑ) −2 (dϑ 2 + dϕ 2 ). Kets |ϕ can be thought of as states produced by a boundary conformal field 22 O(ϕ) of dimension ∆ = 1 2 + iν acting on an SO(1, 2)-invariant global vacuum state |vac such as the global Euclidean vacuum: In conventions in which H, P and K are hermitian, the Lie algebra of so (1, 2) is the action of these generators on kets |ϕ in the above representation is Note that his implies that the action of for example H on wave functions ψ(ϕ) is given by One gets simpler expressions after conformally mapping this to planar boundary coordinates Then H, P, K take the familiar planar dilatation, translation and special conformal form: In particular this makes exponentiation of H easy: However one has to keep in mind that planar coordinates miss a point of the global boundary, here the point ϕ = π. This will actually turn out to be important in the computation of the character. Let us first ignore this though and compute We see that the computation localizes at the point x = 0, singled out because it is a fixed point of H. Actually there is another fixed point, which we missed here because it is exactly the point at infinity in planar coordinates. This is clear from the global version (A.8): one fixed point of H is at ϕ = 0, which maps to x = 0 and was picked up in the above computation, while the other fixed point is at ϕ = π, which maps to x = ∞ and so was missed.
This is easily remedied though. The most straightforward way is to repeat the computation in the global boundary basis |ϕ , which is sure not to miss any fixed points. It suffices to consider an infinitesimally small neighborhood of the fixed points. For ϕ = y → 0, we get H ≈ i(y∂ y + ∆), which coincides with the planar expression, while for ϕ = π + y with y → 0, we get H ≈ −i(y∂ y + ∆), which is the same except with the opposite sign. Thus we obtain However we can now define a "shadow" bra (ϕ| ∝ dϕ 1 − cos(ϕ − ϕ ) −∆ ϕ | satisfying (ϕ|ϕ = δ(ϕ − ϕ ) and compute the trace as χ(t) = dϕ (ϕ|e −itH |ϕ . The computation then proceeds in exactly the same way, with the same result (A.12).

General dimension and spin
The generalization to d > 1 is straightforward. Again the trace only picks up contributions from fixed points of H. The fixed point at the origin in planar coordinates contributes , while the fixed point at the other pole of the global boundary sphere gives a contribution of the same form but with t → −t. Together we get For massive spin-s representations, the basis merely gets some additional SO(d) spin labels, and the trace picks up a corresponding degeneracy factor, so 23 where∆ = d − ∆ as before, and D d s is the spin degeneracy, for example D 3 s = 2s + 1. More generally for d > 2 it is the number of totally symmetric traceless tensors of rank s: (For d = 2 we get spin ±s irreps of SO(2) with D 2 ±s = 1. However both of these appear when quantizing a Fierz-Pauli spin-s field.) Explicit low-d spin-s degeneracies are listed in (D.3).
The most general massive unitary representation of SO(1, d + 1) is labeled by an irrep [31,64]. The spin-s case discussed above corresponds to S = (s 1 , 0, . . . , 0). The character for general S is where the generalized spin degeneracy factor D d S is the dimension of the SO(d) irrep S, explicitly given for general S in appendix D.1.

Massless and partially massless representations
(Partially) massless representations correspond to higher-spin gauge fields and are in the exceptional or discrete series. These representations and their characters χ(t) are considerably more intricate. We give the general expression and examples in appendix G.1 for the massless case. Guided by our path integral results of section 5, we are led to a simple recipe for constructing these characters from their much simpler "naive" counterparts, spelled out in (5.7). This generalizes straightforwardly to the partially massless case, leading to the explicit general-d formula (G.32).

A.3 Importance of picking a globally regular basis
Naive evaluation of the character trace χ(t) = tr e −itH by diagonalization of H results in nonsense. In this section we explain why: emphasizing the importance of using a basis on which finite SO(1, d + 1) transformations act in a globally regular way. 23 Here ∆ = d 2 + iν with either ν ∈ R (principal series) or ν = iµ with |µ| < d 2 for s = 0 and |µ| < d 2 − 1 for s ≥ 1 (complementary series). For s = 0 the mass is

Failure of computation by diagonalization of H
Naively, one might have thought the easiest way to compute χ(t) = tr e −itH would be to diagonalize H and sum over its eigenstates. The latter are given by |ωσ , where H = ω ∈ R, σ labels SO(d) angular momentum quantum numbers, and ωσ|ω σ = δ(ω−ω ) δ σσ . However this produces a nonsensical result, not even remotely resembling the correct χ(t) as computed earlier in A.2.
Our method of computation there also illuminates why this naive computation fails. To make this concrete, let us go back to the d = 1 scalar example with ∆ = 1 2 + iν, ν ∈ R. Recalling the action of H on wave functions ψ(ϕ) mentioned below (A.8), it is straightforward to find the wave functions ψ ωσ (ϕ) of the eigenkets |ωσ = π −π dϕ ψ ωσ (ϕ) |ϕ of H: where Θ is the step function. Alternatively we can first conformally map S 1 to the "cylinder" 24 Explicitly T = log | tan ϕ 2 |, Ω = sign ϕ, which analogously to the global → planar map (A.9) yields

Globally regular bases
More generally, to ensure correct computation of the full Harish-Chandra group character χ R (g) = tr R(g), g ∈ SO(1, d + 1), we must use a basis on which finite SO(1, d + 1) transformations g act in a globally nonsingular way. This is the case for a global dS d+1 boundary basis |Ω S d ,Ω ∈ S d , generalizing the d = 1 global S 1 basis |ϕ S 1 , but not for a planar basis |x R d or a cylinder basis |T Ω R×S d−1 . Indeed generic SO(d + 1) rotations of the global S d move the poles of the sphere, thus mapping finite points to infinity in planar or cylindrical coordinates. This singular behavior is inherited by the corresponding Fourier dual bases |p ∝ d d x e ipx |x and |ωσ ∝ dT dΩ e iωT Y σ (Ω) |T Ω . From a bulk point of view these are the states obtained by standard mode quantization in the planar patch resp. future wedge.
The singular behavior is evident here from the fact that these patches have horizons that are moved around by global SO(d + 1) rotations. Naively computing χ(g) in these frames will in general give incorrect results. More precisely the result will be wrong unless the fixed points where tr traces over the global single-particle Hilbert space and we recall H = M 0,d+1 is a global SO(1, 1) boost generator, which becomes a spatial momentum operator in the future wedge and the energy operator in the southern static patch (cf. fig. D.1c). This is to be contrasted with the familiar character of the unitary lowest-weight representation of a scalar of mass m 2 = −( d 2 ) 2 + µ 2 on global AdS d+1 with standard boundary conditions: Here the so(2) generator H is the energy operator in global AdS d+1 . Besides the occurrence of both ∆ ± in (B.1), another notable difference is the absence of factors of i in the exponents.
The physics content of χ AdS is clear: χ AdS (−iβ) = tr e −βH is the single-particle partition function at inverse temperature β for a scalar particle trapped in the global AdS gravitational potential well. Equivalently for Im t < 0, the expansion counts normalizable single-particle states of energy H = λ, or equivalently global normal modes of frequency λ. The corresponding density of single-particle states is For dS, we can likewise expand the character as in (B.3). For t > 0, However λ is now complex, so evidently N λ does not count physical eigenstates of the hermitian operator H. Rather, as further discussed in section B.3, it counts resonances, or quasinormal modes. The density of physical states with H = ω ∈ R is formally given by where ω can be interpreted as the momentum along the T -direction of the future wedge (F in fig. D.1   AdS potential well. Note that although the above ρ dS 3 ,Λ (ω) formally goes negative in the large-ω limit, it is positive within its regime of validity, that is to say for ω, ν Λ.  .7) is The algebraic way goes as follows. A normalizable basis |n of the global dS 2 scalar singleparticle Hilbert space can be constructed from the SO(1, 2) conformal algebra (A.7), using a basis of generators L 0 , L ± related to H, K and P as L 0 = 1 2 (P + K), L ± = 1 2 (P − K) ± iH. Then L 0 is the global angular momentum generator i∂ φ along the future boundary S 1 and L ± Plot on the right shows wider range of t. Plot in the middle smaller range of t, but larger χ.
are its raising and lowering operators. In some suitable normalization of the L 0 eigenstates |n , we have L 0 |n = n|n , L ± |n = (n ± ∆)|n ± 1 . Cutting off the single-particle Hilbert space at −N < n ≤ N , 25 the operator H = i 2 (L − − L + ) acts as a sparse 2N × 2N matrix on the truncated basis |n .
A minimally coarse-grained density of states can then be defined as the inverse spacing of its eigenvalues ω i , i = 1, . . . , 2N , obtained by numerical diagonalization: (B.10) The continuum limit corresponds to N → ∞ in the discretized model, and to Λ → ∞ in (B.8).
To We can make a similar comparison directly at the (UV-finite) character level. The discrete character is i e −iω i t , which is a wildly oscillating function. At first sight this seems very different from the character χ(t) = tr e −iHt in (B.6). However to properly compare the two, we should coarse grain this at a small but finite resolution δ. We do this by convolution with a Gaussian kernel, that is to say we consider  Note that there is no reason to expect any discretization scheme will converge to χ(t) or ρ(ω). For example it is not clear a brick wall discretization along the lines described in section E.3 would. On the other hand, the convergence of the above global angular momentum cutoff scheme to the continuum χ(t) was perhaps to be expected, given (A.20) and the discussion preceding it.

B.3 Resonances and quasinormal mode expansion
Substituting the expansion (B.5) of the dS character, , we can formally express the density of states as From this we read off that ρ(ω) analytically continued to the complex plane has poles at ω = ±λ which for massive representations means ω = ∓i(∆ ± + n). This can also be checked there to the quasinormal mode expansion of [50] for scalar and spinor path integrals.

"S-matrix" formulation
The appearance of resonance poles in the analytically continued density of states is wellknown in quantum mechanical scattering off a fixed potential V . They are directly related to the poles/zeros in the S-matrix S(ω) at energy ω through the relation [100] ρ where ρ 0 (ω) is the density of states at V = 0.
In [101], a general (flat space) S-matrix formulation of statistical mechanics for interacting QFTs was developed. In this formulation, the canonical partition function is expressed as where the subscript c indicates restriction to connected diagrams (where "connected" is defined with the rule that particle permutations are interpreted as interactions [101]). Combined with the above observations, this hints at a possible generalization of our free QFT results to interacting theories.

C Evaluation of character integrals
The most straightforward way of UV-regularizing character integrals is to simply cut off the t-integral at some small t = . However to compare to the standard heat kernel (or spectral zeta function) regularization for Gaussian Euclidean path integrals [67], it is useful to have explicit results in the latter scheme. In this appendix we give an efficient and general recipe to compute the exact heat kernel-regularized one-loop Euclidean path integral, with regulator e − 2 /4τ as in (3.2), requiring only the unregulated character formula as input. For concreteness we consider the scalar case in the derivation, but because the scalar character χ 0 (t) provides the basic building block for all other characters χ S (t), the final result will be applicable in general. We spell out the derivation is some detail, and summarize the final result together with some examples in section C.2. Application to the massless higher-spin case is discussed in section C.3, where we work out the exact one-loop Euclidean path integral for Einstein gravity on S 4 as an example. In section C.4 we consider different regularizations, such as the simple t > cutoff.

C.1 Derivation
As shown in section 3, the scalar Euclidean path integral regularized as where D = −∇ 2 + d 2 4 + ν 2 , can be written in character integral form as Putting = 0 we recover the formal (UV-divergent) character formula To evaluate (C.3), we split the integral into UV and IR parts, each of which can be evaluated in closed form in the limit → 0.

Separation into UV and IR parts
The separation of the integral in UV and IR parts is analogous to the usual procedure in heat kernel regularization, where one similarly separates out the UV part of the τ integral by isolating the leading terms in the τ → 0 heat kernel expansion Introducing an infinitesimal IR cutoff µ → 0, we may write log Z = log Z uv + log Z ir where Dropping the UV regulator in the IR integral is allowed because all UV divergences have been removed by the subtraction. The factor e −µ 2 τ serves as an IR regulator needed for the separate integrals when F uv has a term α d+1 2τ = 0, that is to say when d + 1 is even. The resulting log µ terms cancel out of the sum at the end. Evaluating this using the specific UV regulator of (C.1) gives dτ τ τ z Tr e −τ D is the zeta function of D and α d+1 = ζ D (0). We can apply the same idea to the square-root regulated character formula (C.3) for Z .
The latter is obtained from the simpler integrand of the formal character formula (C.4) for Z =0 by dividing it by r( , t) ≡ √ t 2 − 2 /t and replacing ν by νr( , t): Note that 0 < r < 1 for all t > , r ∼ O(1) for t ∼ and r → 1 for t . Therefore, given the t → 0 behavior of the integrand in the formal character formula for Z =0 , we get the t ∼ → 0 behavior of the integrand for the exact Z : Thus we can separate log Z = logZ uv + logZ ir , with (C.11) Again the limit µ → 0 is understood. We were allowed to put = 0 in the IR part because it is UV finite.

Evaluation of UV part
Using the expansion (C.10), the UV part can be evaluated explicitly as where B(x, y) = Γ(x)Γ(y) Γ(x+y) is the Euler beta function and H x = γ + Γ (1+x) Γ(1+x) which for integer x is the x-th harmonic number H x = 1 + 1 2 + · · · + 1 x . For example for d = 3, we get This gives an explicit expression for the part of log Z denoted Pol(∆) in [50], without having to invoke an independent computation of the heat kernel coefficients. Indeed, turning this around, by comparing (C.12) to (C.7), we can express the heat kernel coefficients α k explicitly in terms of the character coefficients b k, . In particular the Weyl anomaly coefficient is simply given by the coefficient b d+1 = b d+1, ν of the 1/t term in the integrand of the formal character formula (C.4). More generally, For example for d = 3, this becomes α 0 = 1 12 b 00 , α 2 = 1 2 b 20 + ν

Evaluation of IR part
As we explain momentarily, the IR part can be evaluated as where like for the spectral zeta function ζ D (z), the "character zeta function" ζ ν (z) is defined by the above integral for z sufficiently large and by analytic continuation for z → 0. This zeta function representation of log Z ir follows from the following observations. If we define then since the integral remains finite for z → 0, while Γ(z) ∼ 1/z and ∂ z (1/Γ(z)) → 1, we trivially have 1 2 ∂ z ζ ir ν (z)| z=0 = logZ ir . Moreover for z sufficiently large we have in the limit µ → 0 that 1 2 upon analytic continuation we have 1/2∂ z ζ uv (z)| z=0 = −b d+1 log µ, and (C.15) follows.

C.2 Result and examples
Result Altogether we conclude that given a formal character integral formula for a field corresponding to a dS d+1 irrep of dimension d 2 + iν, with IR and UV expansions where b k (ν) = b k ν , we obtain the exact Z PI with heat kernel regulator e − 2 /4τ as , which for integer x is the x-th harmonic number H x = 1 + 1 2 + · · · + 1 x , andδ is the unit shift operator acting on the first argument of the Hurwitz zeta function ζ(z, ∆): the polynomial P ∆ (δ − ∆) is to be expanded in powers ofδ, settingδ n ζ (0, ∆) ≡ ζ (−n, ∆). Finally the heat kernel coefficients are If we are only interested in the finite part of log Z, only the first three terms in (C. 19) matter. Note that the third and the second term M ν ≡ b d+1, H − 1 2 H /2 is in general nonvanishing for even d + 1. By comparing (C. 19) to (C.7), say in the scalar case discussed earlier, we see that ζ D (0) = ζ ν (0)+2M ν . Thus 2M ν can be thought of as correcting the formal factorization n log(n + ∆ + )(n + ∆ − ) = n log(n + ∆ + ) + n log(n + ∆ − ) in zeta function regularization. For this reason M ν is called the multiplicative "anomaly", as reviewed in [102]. The above thus generalizes the explicit formulae in [102] for M ν to fields of arbitrary representation content.

For a scalar on
The heat kernel coefficients are In particular for a conformally coupled scalar, i.e. ∆ = 1 2 , 3 2 or equivalently ν = i/2, we get for the finite part the familiar result log Z PI = 3ζ (3) 16π 2 − log(2) 8 . For ∆ = 1, i.e. ν = 0, we get log Z PI = − ζ(3) 4π 2 . Notice that the finite part looks quite different from (2.25) obtained by contour integration. Nevertheless they are in fact the same function.

Single-mode contributions
Contributions from single path integral modes and contributions of single quasinormal modes are of use in some of our derivations and applications. These are essentially special cases of the above general results, but for convenience we collect some explicit formulae here: • Path integral single-mode contributions: For our choice of heat-kernel regulator e − 2 /4τ , the contribution to log Z PI, from a single bosonic eigenmode with eigenvalue λ is Different regulator insertions lead to a similar result in the limit → 0, with M = c/ for some regulator-dependent constant c. A closely related formula is obtained for the contribution from an individual term in the sum (C.2) or equivalently in the IR expansion of (C.18), which amounts to computing (C.17) with F ν (t) ≡ e −ρt , ρ = a ± iν. The small-t expansion is 1 2t F ν (t) = 1 2t + O(t 0 ), so the UV part is given by the log term in (C.19) with coefficient 1 2 , and the IR part is 1 2 ζ ν (0) = − 1 2 log ρ as in (C.15). Thus where the integral is understood to be regularized as in (C.2), left implicit here. The similarities between (C.29) and (C.30) are of course no accident, since in our setup, the former splits into the sum of two integrals of the latter type: writing λ = a 2 + ν 2 = (a + iν)(a − iν), we have I λ = I a+iν + I a−iν .

C.3 Massless case
Here we give a few more details on how to use (C.19) to explicitly evaluate Z PI in the massless case, and work out the exact Z PI for Einstein gravity on S 4 as an example.
Our final result for the massless one-loop Z PI = Z G · Z char is given by (5.19): and vol(G) c = (D.7). • UV part: As always, the coefficient of the log-divergent term simply equals the coefficient of the 1/t term in the small-t expansion of the integrand in (C.34). For the other UV terms in (C.19) (including the "multiplicative anomaly"), a problem might seem to be that we need a continuously variable dimension parameter ∆ = d 2 + iν, whereas massless fields, and our explicit formulae forχ → [χ] + , require fixed integer dimensions. This problem is easily solved, as the UV part can actually be computed from the original naive character formula (G.8): Indeed sinceF → F = {F } + in (G.10) affects just a finite number of terms c k q k → c k q −k , it does not alter the small-t (UV) part of the integral. Moreoverχ s =χ s,ν φ −χ s,ν ξ , whereχ s,ν is a massive spin-s character. Thus the UV part may be obtained simply by combining the results of (C.19) for general ν and s, substituting the values ν φ , ν ξ set by (5.2).
• IR part: The IR part is the ζ part of (C. 19), obtained from the q-expansion of F (q) in (C.34). This can be found in general by using Example: Einstein gravity on S 4 As a simple application, let us compute the exact one-loop Euclidean path integral for pure gravity on S 4 . In this case G = SO(5), dim G = 10, d = 3 and s = 2. From (5.2) we read off iν φ = 3 2 , iν ξ = 5 2 , and from (5.9) we get The small-t expansion of the integrand in (C.34) in agreement with [14]. The complete heat-kernel regularized UV part of (C.19) can be read off directly from our earlier results for massive spin-s in d = 3 as Here M = 715 48 is the "multiplicative anomaly" term. The integrated heat kernel coefficients are similarly obtained from (C. where P b (n) = D 5 n = 1 6 (n + 1)(n + 2)(2n + 3), P e (n) = D 3 n = 2n + 1. According to (C.19) this gives a contribution to log Z char | IR equal to where the polynomials are to be expanded in powers ofδ, puttingδ n ζ (0, ∆) ≡ ζ (−n, ∆).
Working this out and adding the contribution (C.37), we find Combining this with the UV part and reinstating , we get 26 log Z char = 8 3 where L is an arbitrary length scale introduced to split off a finite part: To compute the group volume factor Z G in (C.34), we use (D.7) for G = SO(5) to get vol(G) c = 2 3 (2π) 6 , and γ = 8πG N /4π 2 . Finally, i −P = i −(d+3) = −1. Thus we conclude that the one-loop Euclidean path integral for Einstein gravity on S 4 is where Z char is given by (C.44).

C.4 Different regularization schemes
If we simply cut off the character integral at t = , we get the following instead of (C.19): . Unsurprisingly, this differs from (C.19) only in its UV part, more specifically in the terms polynomial in ν, including the "multiplicative anomaly" term discussed below (C.20). The transcendental (ζ ) part and the log coefficient remain unchanged. This remains true in any other regularization.
If we stick with heat-kernel regularization but pick a different regulator f (τ / 2 ) instead of e − 2 /4τ (e.g. the f = (1 − e −τ Λ 2 ) k PV regularization of section 2) or use zeta function regularization, more is true: the same finite part is obtained for any choice of f provided logarithmically divergent terms (arising in even d + 1) are expressed in terms of M defined as in (C.29) with e − 2 /4τ → f . The relation M ( ) will depend on f , but nothing else.
In dimensional regularization, some polynomial terms in ν will be different, including the "multiplicative anomaly" term. Of course no physical quantity will be affected by this, as long as self-consistency is maintained. In fact any regularization scheme (even (C.48)) will lead to the same physically unambiguous part of the one-loop corrected dS entropy/sphere partition function of section 8. However to go beyond this, e.g. to extract more physically unambiguous data by comparing different saddles along the lines of (??) and (I.70), a portable covariant regularization scheme, like heat-kernel regularization, must be applied consistently to each saddle. A sphere-specific ad-hoc regularization as in (C.48) is not suitable for such purposes.

D Some useful dimensions, volumes and metrics D.1 Dimensions of representations of SO(K)
General irreducible representations of SO(K) with K = 2r or K = 2r + 1 are labeled by r-row Young diagrams or more precisely a set S = (s 1 , . . . , s r ) of highest weights ordered from large to small, which are either all integer (bosons) or all half-integer (fermions). When K = 2r, s r can be either positive of negative, distinguishing the chirality of the representation. For various applications in this paper we need the dimensions D K S of these SO(K) representations S. The Weyl dimension formula gives a general expression for the dimensions of irreducible representations of simple Lie groups. For the SO(K) this is • K = 2r: where N K is fixed as above: For convenience we list here some low-dimensional explicit expressions:

D.2 Volumes
The volume of the unit sphere S n is The volume of SO(d + 2) with respect to the invariant group metric normalized such that minimal SO(2) orbits have length 2π is .

D.3 de Sitter and its Wick rotations to the sphere
Global dS d+1 has a convenient description as a hyperboloid embedded in R 1,d+1 , Below we set ≡ 1 . The isometry group is SO(1, d+1), with generators M IJ = X I ∂ J −X J ∂ I .
Various coordinate patches are shown in fig. D.1a,c, with coordinates and metric given by co embedding (X 0 , . . . , X d+1 ) coordinate range metric 10) illustrated in fig. D.1a,c. N is obtained from S by X d+1 → −X d+1 , and P from F by

E Euclidean vs canonical: formal & physics expectations
Given a QFT on a static spacetime R × M with metric ds 2 = −dt 2 + ds 2 M , Wick rotating t → −iτ yields a Euclidean QFT on a space with metric ds 2 = dτ 2 + ds 2 M . The Euclidean path integral Z PI (β) = DΦ e −S[Φ] on S 1 β × M obtained by identifying τ τ + β equals the thermal partition function: Z PI (β) = Tr e −βH , as follows from cutting the path integral along constant-τ slices and viewing e −τ H as the Euclidean time evolution operator.
At least for noninteracting theories, it is in practice much more straightforward to compute the partition function as the state sum Tr e −βH of an ideal gas in a box M than as a one-loop path integral Z PI = DΦ e −S[Φ] on S 1 β ×M , in particular for higher-spin fields. In view of this, it is reasonable to wonder if a free QFT path integral on the sphere could perhaps similarly be computed as a simple state sum, by viewing the sphere as the Wick-rotated static patch ( fig. D.1d), with inverse temperature β = 2π given by the period of the angular coordinate τ : (E.1) Below we review the formal path integral slicing argument suggesting the above relation and = Z bulk , with Z bulk defined as a character integral as in section 2. This improves the situation, but is still incorrect for spin s ≥ 1. In more detail, the content is as follows: In E.1 we consider the d = 0 case: a scalar of mass ω on dS 1 in its Euclidean vacuum state, i.e. an entangled pair of harmonic oscillators. Though surely superfluous to most readers, we use the occasion to provide a pedagogical introduction to some standard constructions.
In E.2 we formally apply the same template to general d, ignoring yellow-dot issues, leading to the standard formal "thermofield double" description of the static patch of de Sitter [60], and more specifically to Z PI Tr e −2πH Z bulk . We review the pathological divergences that ensue when one attempts to evaluate the trace, and some of its proposed fixes such as the "brick-wall" cutoff [49] and refinements thereof. We contrast these to Z bulk defined as a character integral.
In E.5, we turn to the edge corrections missed by such formal arguments, explaining from various points of view why they are to be expected.
The dS 1  are parametrized by T , and each contains one harmonic oscillator, respectively φ S and φ N .
The Euclidean path integral Z PI on S 1 is most easily computed by reverting to the canonical formalism with e −τ H S = e −τ H as the Euclidean time evolution operator, which maps it to the harmonic oscillator thermal partition function at inverse temperature β = 2π: (E.5) We can alternatively consider the S 1 to be obtained as the Wick rotation of global dS 1 as in fig. D.1b/E.1b, G in (D.12), parametrizing the S 1 by (τ ,Ω), − π 2 ≤τ ≤ π 2 ,Ω ∈ S 0 = {S, N }, identifying (± π 2 , S) = (± π 2 , N ). The global action (E.2) then Wick rotates to where φ 0 = (φ S,0 , φ N,0 ). The notation τ ≤0 Dφ| φ 0 means the path integral of φ = (φ S , φ N ) is G of "single-particle" excitations of the global Euclidean vacuum is two-dimensional, spanned by The character χ(t) of the SO(1, 1) representation furnished by H G is The above constructions are straightforwardly generalized to fermionic oscillators. The character of a collection of bosonic and fermionic oscillators of frequencies ω i and ω j is (E.14) 27 The t −2 pole of the integrand is resolved by the i -prescription t −2 → 1 2 (t − i ) −2 + (t + i ) −2 , left implicit here and in the formulae below. The integral formula can be checked by observing the integrand is even in t, extending the integration contour to the real line, closing the contour, and summing residues.
Combining this with (E.13) expresses the thermal partition function of a collection of bosonic and harmonic oscillators as an integral transform of its SO(1, 1) character: log Tr e −βH = ∞ 0 dt 2t The Euclidean path integral on an S 1 of radius = 1 for a collection of free bosons and fermions (the latter with thermal, i.e. antiperiodic, boundary conditions) is then given by putting β = 2π in the above: The arguments in this section will be formal, following the template of section E.1 while glossing over some important subtleties, the consequence of which we discuss in section E.5.
Wick-rotating a QFT on dS d+1 to S d+1 , we get the Euclidean path integral where Φ collects all fields in the theory. Just like in the d = 0 case, the two different paths where τ ≤0 Dφ| φ 0 means the path integral is performed on the lower hemisphereτ ≤ 0 of The static patch Wick rotation of fig. D.1d on the other hand leads to an interpretation of Z PI as a thermal partition function at inverse temperature β = 2π, analogous to (E.5): slicing 28 For kind enough theories, such as a scalar field theory, this pairing can be identified with the Hilbert space inner product. However not all theories are kind enough, as is evident from the negative-mode rotation phase i −(d+3) in the one-loop graviton contribution to Z PI = O|O according to (5.19) and [18]. Indeed for gravity this pairing is not in an obvious way related to the semiclassical inner product of [105]. On the other hand, in the CS formulation of 3D gravity it appears to be framing-dependent, vanishing in particular for canonical framing (cf. (H.28) and discussion below it). The phase also drops out of A ≡ O|A|O / O|O . the path integral along constant-τ slices as in fig. D we formally get the analog of (E.9): |O " n "e −βEn/2 |E n , E n ,ˆ S e −βH S (β = 2π) , (E.20) where we have put the sum in quotation marks because the spectrum is actually continuous, as we will describe more precisely for free QFTs below. Granting this, we conclude that an inertial observer in de Sitter space sees the global Euclidean vacuum as a thermal state at inverse temperature β = 2π, the Hawking temperature of the observer's horizon [1,2,60].
Applying (E. 19) to a free QFT on dS d+1 , we can write the corresponding Gaussian Z PI on S d+1 as the thermal partition function of an ideal gas in the southern static patch: where ρ S (ω) ≡ tr S δ(ω − H) is the density of single-particle states at energy ω > 0 above the vacuum energy in the static patch, split into bosonic and fermionic parts as ρ S = ρ S+ + ρ S− .

E.3 Brick wall regularization
Here we review how attempts at evaluating the ideal gas partition function (E.21) directly hit a brick wall. Consider for example a scalar field of mass m 2 on dS d+1 . Denoting ∆ ± = where ω > 0, and Y σ (Ω) is a basis of spherical harmonics on S d−1 labeled by σ, which includes the total SO(d) angular momentum quantum number . A basis of energy and SO(d) angular momentum eigenkets is therefore given by |ωσ) satisfying (ωσ|ω σ ) = δ(ω − ω ) δ σσ .
Naive evaluation of the density of states in this basis gives a pathologically divergent result ρ S (ω) = dω σ (ω σ|δ(ω − ω )|ω σ) = σ δ(0), and commensurate nonsense in (E.21). Pathological divergences of this type are generic in the presence of a horizon. Physically they can be thought of as arising from the fact that the infinite horizon redshift enables the existence of field modes with arbitrary angular momentum and energy localized in the vicinity of the horizon. One way one therefore tries to deal with this is to replace the horizon by a "brick wall" at a distance δ away from the horizon [49], with some choice of boundary conditions, say φ(T, Ω, 1 − 1 2 δ 2 ) = 0 in the example above. This discretizes the energy spectrum and lifts the infinite angular momentum degeneracy, allowing in principle to control the divergences as δ → 0. However, inserting a brick wall alters what one is actually computing, introduces ambiguities (e.g. Dirichlet/Neumann), potentially leads to new pathologies (e.g. Dirichlet boundary conditions for the graviton are not elliptic [106]), and breaks most of the symmetries in the problem.
A more refined version of the idea considers the QFT in Pauli-Villars regularization [63].
This eliminates the dependence on δ in the limit δ → 0 at fixed PV-regulator scale Λ. It was shown in [63] that for scalar fields the remaining divergences for Λ → ∞ agree with those of the PV-regulated path integral. 30 A somewhat different approach, reviewed in [40,107], first maps the equations of motion in the metric ds 2 = g µν dx µ dx ν by a (singular) Weyl transformation to formally equivalent equations of motion in the "optical" metric ds 2 = |g 00 | −1 ds 2 . In the case at hand this would be ds 2 = −dT 2 + (1 − r 2 ) −2 dr 2 + (1 − r 2 ) −1 r 2 dΩ 2 , corresponding to R × hyperbolic d-ball. The thermal trace is then mapped to a path integral on the Euclidean optical geometry with an S 1 of constant radius β and a Weyl-transformed action. (This is not a standard covariant path integral. In the case at hand, unless the theory happens to be conformal, non-metric r-dependent terms break the SO(1, d) symmetry of the hyperbolic ball to SO(d).) This path integral can be expressed in terms of a heat kernel trace x x|e −τD |x .
The divergences encountered earlier now arise from the fact that the optical metric ds 2 has infinite volume near r = 1. This is regularized by cutting the x integral off at r = 1 − δ, analogous to the brick wall cutoff, though computationally more convenient. For scalars and spinors, Pauli-Villars or dimensional regularization again allows trading the δ → 0 divergences for the standard UV divergences [107].
Unfortunately, certainly for general field content and in the absence of conformal invariance, none of these variants offers any simplification compared to conventional Euclidean path integral methods. In the case of interest to us, the large underlying SO(1, d + 1) symmetry is broken, and with it one's hope for easy access to exact results. Generalization to higher-spin 30 This work directly inspired the use of Pauli-Villars regularization in section 2. fields, or even just the graviton, appears challenging at best.

E.4 Character regularization
The character formula (E.23) is formally equivalent to the ideal gas partition function (E. 21), and indeed at first sight, naive evaluation in a global single-particle basis |ωσ = a G † ωσ |0 diagonalizing H = ω ∈ R, obtained e.g. by quantization of the natural cylindrical mode functions of the future wedge (F in fig. D.1 and table D.10), gives a similarly pathological ; hardly a surprise in view of the Bogoliubov relation ρ G (ω) ρ S (ω) and our earlier result ρ S (ω) = σ δ(0). Thus the conclusion would appear to be that the situation is as bad, if not worse, than it was before.
However this is very much the wrong conclusion. As reviewed in appendix A, χ(t), properly defined as a Harish-Chandra character, is in fact rigorously well-defined, analytic in t for t = 0, and moreover easily computed. For example for a scalar of mass m 2 on dS d+1 , we get (A.3): as explicitly computed in appendix A.2. The reason why naive computation by diagonalization of H fails so badly is explained in detail in appendix A.3: it is not the trace itself that is sick, but rather the basis |ωσ used in the naive computation.  . D.1b). The angular momentum cutoff means we only have a finite number of spherical harmonics available to build our field modes. This makes it impossible in particular to build field modes sharply localized in the southern or northern hemisphere: the harmonic expansion of a localized mode always has infinitely many terms. Cutting off this expansion will necessarily leave some support on the other hemisphere.
Quite similar in this way again to the Euclidean path integral, this offers some intuition on why the UV-regularized character integral avoids the pathological divergences induced by sharply cutting space.

E.5 Edge corrections
In view of all this and (E.26), one might be tempted at this point to jump to the conclusion that the arguments of section E.2, while formal and glossing over some subtle points, are apparently good enough to give the right answer provided we use the character formulation, and that likewise Z PI on the sphere for a field of arbitrary spin s, despite its off-shell baroqueness, is just the ideal gas partition function Z bulk on the dS static patch, calculable with on-shell ease: mission accomplished. As further evidence in favor of declaring footnote 29 overly cautious, one might point to the fact that in the context of theories of quantum gravity, identifying Z grav PI = Tr H e −βH elegantly reproduces the thermodynamics of horizons inferred by other means [2], and that such identifications are moreover known to be valid in a quantitatively precise way in many well-understood cases in string theory and AdS-CFT.
If the formal argument is good enough for quantum gravity, then surely it is good enough for field theory, one might think.
These naive considerations are wrong: the formal relation Z PI Z bulk for fields of spin s ≥ 1 receives "edge" corrections. In sections 4 and 5, we determine these for massive resp. massless spin-s fields on S d+1 by direct computation. The results are eqs. Similar edge corrections, to the entropy S PI S bulk in the conceptually analogous case of Rindler space, were anticipated long ago in [33] and explicitly computed shortly thereafter for massless spin-1 fields in [34]. The result of [34] was more recently revisited in several works including [24,44], relating it to the local factorization problem of constrained QFT Hilbert spaces [43] and given an interpretation in terms of the edge modes arising in this context.
We leave the precise physical interpretation of the explicit edge corrections we obtain in this paper to future work. Below we will review why they were to be expected, and how

E.5.2 Strings on Rindler considerations
To gain some insight from a bulk point of view, we consider the simplest example of a spacetime with a horizon: the Rindler wedge ds 2 = −ρ 2 dt 2 + dρ 2 + dx 2 ⊥ of Minkowski space. While not quite at the level of AdS-CFT, we do have a perturbative theory of quantum gravity in Minkowski space: string theory. In fact, that Z (1) PI on a Euclidean geometry with a contractible thermal circle cannot be interpreted as a statistical mechanical partition function in general, even if the full Z grav PI has such an interpretation, was anticipated long ago in [33], in an influential attempt at developing a string theoretic understanding of the thermodynamics of the Rindler horizon. Rindler space Wick rotates to with the conical defect = 0 on-shell. The argument given in [33] is This picture is qualitative, as the individual contributions corresponding to a sharp split of the worldsheet path integrals along these lines are likely ill-defined/divergent [112]. Moreover, even without any splitting, an actual string theory calculation of S PI = (1 − β∂ β ) log Z PI | β=2π is problematic, as Euclidean Rindler with a generic conical defect = 2π − β is off-shell.
Shortly after [33], [113] proposed to compute Z PI on the orbifold R 2 /Z N for general integer N and then analytically continue the result to N → 1 + . Unfortunately such orbifolds have closed string tachyons leading to befuddling IR-divergences [113,114]. Recently, progress was made in resolving some of these issues: in an open string version of the idea, arranged in type II string theory by adding a sufficiently low-dimensional D-brane, it was shown in [115] that upon careful analytic continuation, the tachyon appears to disappear at N = 1 + .

E.5.3 QFT considerations
The problem of interest to us is really just a problem involving Gaussian path integrals in free quantum field theory, so there should be no need to invoke quantum gravity to gain some insight in what kind of corrections we should expect to the naive Z PI Z bulk . Indeed the above stringy Rindler considerations have much more straightforwardly computable lowenergy counterparts in QFT.
Motivated by [33], [34] computed Z space. For scalars and spinors, this was found to coincide with the ideal gas partition function, whereas for Maxwell an additional contact term was found, expressible in terms of a "edge" worldline path integral with coincident start and end points at ρ = 0, fig. E.5c. This term contributes negatively to S (1) PI = (1 − β∂ β ) log Z PI | β=2π and thus has no thermal interpretation on its own. In fact it causes the total S (1) PI to be negative in less than 8 dimensions. The results of [34] and more generally the picture of [33] were further clarified by low-energy effective field theory analogs in [35], emphasizing in particular that whereas S PI remains invariant under Wilsonian RG, the division between contributions with or without a low-energy statistical interpretation does not, the former gradually turning into the latter as the UV-cutoff Λ is The result of [34] was more recently revisited in [24], relating the appearance of edge corrections to the local factorization problem of QFT Hilbert spaces with gauge constraints [43] like Gauss' law ∇ · E = 0 in Maxwell theory. This problem arises more generally when contemplating the definition of entanglement entropy S R = −Tr R log R of a spatial subregion R in gauge theories. In principle R is obtained by factoring the global Hilbert space H G = H R ⊗ H R c and tracing out H R c . As mentioned at the end of E.4, local factorization is impossible in the continuum limit of any QFT, including scalar field theories, but the issue raised there can be dealt with by a suitable regularization. However for a gauge theory such as free Maxwell theory, there is an additional obstruction to local factorization, which persists after regularization, and indeed is present already at the classical phase space level: the Gauss law constraint ∇ · E = 0 prevents us from picking independent initial conditions in both R and R c ( fig. E.6), unless the boundary is a physical object that can accommodate compensating surface charges -but this is not the case here. One way to resolve this is to decompose the global phase space into sectors labeled by "center" variables located at the boundary surface [43], for example the normal component E ⊥ of the electric field. The center variables Poisson-commute with all local observables inside R and R c . In any given sector, factorization then becomes possible.
Building on this framework it was shown in [24] that in a suitable brick wall-like regularization scheme and for some choice of measure DE ⊥ , the edge correction of [34] arises as a classical contribution DE ⊥ e −S E [E ⊥ ] to the thermal statistical partition function. Here is the on-shell action for static electromagnetic field modes in Euclidean Rindler space with prescribed E ⊥ , localized vanishingly close to ρ = 0 when the brick-wall cutoff is taken to zero, and thus interpreted as edge modes. They also find a more precise form for the result of [34] for Rindler with its transverse dimensions compactified on a torus, which is identical Similar results for massive vector fields were obtained in [44]. (The Stueckelberg action for a massive vector has a U (1) gauge symmetry, so from that point of view it may fit into the above considerations.) An open string realization of the above ideas was proposed in [116].
It has been suggested that edge modes and "soft hair" might be related [117].

F Derivations for massive higher spins F.1 Massive spin-s fields
Here we derive (4.7) and (4.6). The starting point is the path integral (4.4). To get a result guaranteed to be consistent with QFT locality and general covariance, we should in principle start with the full off-shell system [51] involving auxiliary Stueckelberg fields of all spin s < s.

Transverse-traceless part Z TT
One's initial hope might be that Z PI ends up being equal to the path integral Z TT restricted to the propagating degrees of freedom, the transverse traceless modes of φ, with kinetic operator given by the second-order equation of motion in (4.1). Regularized as in (3.2), this is The index TT indicates the object is defined on the restricted space of transverse traceless modes. This turns out to be correct for Euclidean AdS with standard boundary conditions [25]. However, this is not quite true for the sphere, related to the presence of normalizable tensor decomposition zeromodes.
The easiest way to convince oneself that Z PI = Z TT on the sphere is to just compute Z TT and observe it is inconsistent with locality, in a sense made clear below. To evaluate Z TT , all we need is the spectrum of −∇ 2 TT +m 2 s [118,119]. The eigenvalues are λ n = n + d 2 2 +ν 2 , n ≥ s with degeneracy given by the dimension D d+2 n,s of the so(d + 2) representation corresponding to the two-row Young diagram (n, s), for example for d = 3, (n, s) = (7, 3), Explicit dimension formulae and tables can be found in appendix D.1.
Following the same steps as for the scalar case in section 3, we end up with Now let us evaluate this explicitly for the example of a massive vector on S 5 , i.e. d = 4, s = 1.
From (D.3) we read off D 6 n,1 = 1 3 n(n + 2) 2 (n + 4). Performing the sum we end up with The first term inside the brackets can be recognized as the d = 4 massive spin-1 bulk character.
The small-t expansion of the integrand in (F.3) contains a term 1/t. This term arises from the term +q in the above expression, as the other parts give contributions to the integrand that are manifestly even under t → −t. The presence of this 1/t term in the small-t expansion implies log Z TT has a logarithmic UV divergence log Z TT | log div = log M where M is the UV cutoff scale. More precisely in the heat-kernel regularization under consideration, the contribution of the term +q to log Z TT is, according to (C.30), Note that m = √ 1 + ν 2 is the Proca mass (4.2) of the vector field. The presence of a logarithmic divergence means Z PI = Z TT , for log Z PI itself is defined as a manifestly covariant, local QFT path integral on S 5 , which cannot have any logarithmic UV divergences, as there are no local curvature invariants of mass dimension 5.
For s = 2 and d = 4 we get similarly The terms 6 q + 15 q 2 produce a nonlocal logarithmic divergence log Z TT | log div = c log M , where c = 6 + 15 = 21, so again Z PI = Z TT . Note that 21 = 7×6 2 = dim so (1,6), the number of conformal Killing vectors on S 5 . That this is no coincidence can be ascertained by repeating the same exercise for general d ≥ 3 and s = 2: The q, q 2 terms generate a log-divergence c 2 log M , c 2 = D d+2 , the number of conformal Killing vectors on S d+1 . The identity N CKV = D d+3 1,1 = D d+2 1,1 + D d+2 1,0 and its generalization to the spin-s case will be a crucial ingredient in establishing our claims. It has a simple group theoretic origin. As a complex Lie algebra, the conformal algebra so(1, d + 2) generated by the conformal Killing vectors is the same as so(d + 3), which is generated by antisymmetric matrices and therefore forms the irreducible representation with Young diagram of so(d + 3). This decomposes into irreps of so(d + 2) by the branching rule (F.10) To rewrite the finite sum we used D d+2 n,s = −D d+2 s−1,n+1 and D d+2 s−1,s = 0, both of which follow from (F.9). Substituting this into the integral (F.3), we get where log Z bulk and log Z edge are the character integrals defined in (4.7)-(4.8), and, evaluating the integral of the remaining finite sum as in (F.5), This can be checked for any given d from the Weyl dimension formula of appendix D.1, or from the general d formula given in e.g. [118,119], or proven directly (with some effort) by an SO(d+2) → SO(2)×SO(d) reduction.
It has a stronger version as an so(d + 2) character relation: χ The term log Z res has a logarithmic UV-divergence: where N CKT = D d+3 s−1,s−1 is the number of rank s − 1 conformal Killing tensors on S d+2 [120]. This identity has a group theoretic origin as an so(d+3) → so(d+2) branching rule generalizing (F.8). For example for s = 4: Geometrically this reflects the fact that the rank s − 1 = 3 Killing tensor modes split up into 4 types: Schematically where the ϕ ir µ 1 ···µr span the eigenspace of the TT spin-r Laplacian labeled by the above Young diagrams. As pointed out in examples above and discussed in more detail below, the log-divergence of log Z res is inconsistent with locality, hence Z PI = Z TT : locality must be restored by the non-TT part of the path integral. Below we argue this part in fact exactly cancels the log Z res term, thus ending up with log Z PI = log Z bulk − log Z edge , i.e. the character formula (4.7).

Full path integral Z PI : locality constraints
The full, manifestly covariant, local path integral takes the form (a simple example is (F.22)): All UV-divergences of log Z PI are local, in the sense they can be canceled by local counterterms, more specifically local curvature invariants of the background metric. In particular for odd d+1, this implies there cannot be any logarithmic divergences at all, as there are no curvature invariants of odd mass dimension. Recall from (F.14) that the term log Z res is logarithmically divergent. For odd d + 1, this is clearly the only log-divergent contribution to log Z TT , as the integrands of both log Z bulk and log Z edge are even in t in this case. More generally, for even or odd d + 1, log Z res is the only nonlocal log-divergent contribution to log Z TT , as follows from the result of [15,26] mentioned below (4.4), combined with the observation in (F.14) that c s = N CKT . Therefore the log-divergence of log Z res must be canceled by an equal log-divergence in log Z non-TT of the opposite sign.
The simplest way this could come about is if Z non-TT exactly cancels Z res , that is if Note furthermore that from (F.10), or from (F.13) and D d+2 s−1,n+1 = −D d+2 n,s , it follows this identification is equivalent to the following simple prescription: The full Z PI is obtained from Z TT by extending the TT eigenvalue sum n≥s in (F.3) down to n≥−1 : i.e. (4.6). In what follows we establish this is indeed the correct identification. We start by showing it precisely leads to the correct spin-s unitarity bound, and that it moreover exactly reproduces the critical mass ("partially massless") thresholds at which a new set of terms in the action defining the path integral Z non-TT fails to be positive definite. Assisted by those insights, it will then be rather clear how (F.17) arises from explicit path integral computations.

Unitarity constraints
A significant additional piece of evidence beyond consistency with locality is consistency with unitarity. It is clear that both the above integral (F.18) for log Z PI and the integral (F.3) for log Z TT are real provided ν is either real or imaginary. Real ν corresponds to the principal series ∆ = d 2 + iν, while imaginary ν = iµ corresponds to the complementary series ∆ = d 2 − µ ∈ R. In the latter case there is in addition a bound on |µ| beyond which the integrals cease to make sense, due to the appearance of negative powers of q = e −t and the integrand blowing up at t → ∞. The bound can be read off from the term with the smallest value of n in the sum. In the Z TT integral (F.3) this is the n = s term ∝ q d 2 +s±µ , yielding a bound |µ| < d 2 + s. In the Z PI integral (F.18), assuming s ≥ 1, this is the n = −1 term ∝ q d 2 −1±µ , so the bound becomes much tighter: This is exactly the correct unitarity bound for the spin-s ≥ 1 complementary series representations of SO(1, d + 1) [31,64]. In terms of the mass m 2 = ( d 2 + s − 2) 2 − µ 2 in (4.2), this becomes m 2 > (s − 1)(d − 3 + s), also known as the Higuchi bound [121] (a convenient concise summary is given in [122] s.a. [123]). From a path integral perspective, this bound can be understood as the requirement that the full off-shell action is positive definite [51], so indeed log Z PI should diverge exactly when the bound is violated. Moreover, we get new divergences in the integral formula for log Z non-TT , according to the above identifications, each time |µ| crosses a critical value µ * n = d 2 + n, where n = −1, 0, 1, 2, . . . , s − 2. These correspond to critical masses m 2 * n = ( d 2 + s − 2) 2 − ( d 2 + n) 2 = (s − 2 − n)(d + s − 2 + n), which on the path integral side precisely correspond to the points where a new set of terms in the action fails to be positive definite. [51].
This establishes the terms in the integrand of (F.13), or equivalently the extra terms n = −1, . . . , s − 2 in (F.18), have exactly the correct powers of q to match with log Z non-TT . It does not yet confirm the precise values of the coefficients D d+2 n,s -except for their sum (F.14), which was fixed earlier by the locality constraint. To complete the argument, we determine the origin of these coefficients from the path integral point of view in what follows.

Explicit path integral considerations
Complementary to but guided by the above general considerations, we now turn to more concrete path integral calculations to confirm the expression (F.17) for Z non−TT , focusing in particular on the origin of the coefficients D d+2 n,s .

Spin 1:
We first consider the familar s = 1 case, a vector field of mass m, related to ν by (4.2) as The local field content in the Stueckelberg description consists of a vector φ µ and a scalar χ, with action and gauge symmetry given by Gauge fixing the path integral by putting χ ≡ 0, we get the gauge-fixed action with BRST ghosts c,c. Decomposing φ µ into a transversal and longitudinal part, φ µ = φ T µ +φ µ , we can decompose the path integral as Z PI = Z TT · Z non-TT with  34 A priori there might be a relative numerical factor κ between ghost and longitudinal factors, depending on the so far unspecified normalization of the measure Dc. But because c is local, unconstrained, rescaling Dc = i dc i → i (λdc i ) merely amounts to a trivial constant shift of the bare cc. So we are free to take κ = 1.

Spin 2:
For s = 2, the analogous Stueckelberg action involves a symmetric tensor φ µν , a vector χ µ , and a scalar χ, subject to the gauge transformations [51] where a 0 ≡ m and a −1 ≡ m 2 − (d − 1). Equivalently, recalling (4.2), a n = ( d 2 + n) 2 + ν 2 . Gauge fixing by putting χ = 0, χ µ = 0, we get a ghost action We can decompose φ µν into a TT part and a non-TT part orthogonal to it as φ µν = φ TT µν +φ µν , where φ µν can be decomposed into vector and scalar modes as φ µν = ∇ (µ ϕ ν) +g µν ϕ. Analogous to the s = 1 example, we should expect that integrating out φ cancels against integrating out the ghosts, up to unmatched modes of the latter. The unmatched modes correspond to mixed vector-scalar modes solving ∇ (µ ϕ ν) + g µν ϕ = 0. This is equivalent to the conformal Killing where the dots indicate terms we won't technically need -which is to say, as transpired from in agreement with (F.17) for general s. This establishes our claims.
The above computation was somewhat schematic of course, and one could perhaps still worry about missed purely numerical factors independent of ν, perhaps leading to an additional finite constant term being added to our final formulae (4.7) -(4.6) for log Z PI . However at fixed UV-regulator scale, the limit ν → ∞ of these final expressions manifestly approaches zero, as should be the case for particles much heavier than the UV cutoff scale. This would not be true if there was an additional constant term. Finally, we carefully checked the analogous result in the massless case (which has a more compact off-shell formulation [70]), discussed in section 5, by direct path integral computations in complete gory detail [124], for all s.
Also, the result is pretty.

F.2 General massive representations
Here we give a generalization of (4.6) for arbitrary massive representations of the dS d+1 isometry group SO(1, d + 1).
Massive irreducible representations of SO(1, d + 1) are labeled by a dimension ∆ = d 2 + iν and an so(d) highest weight S = (s 1 , . . . , s r ) [64]. The massive spin-s case considered in (4.6) corresponds to S = (s, 0, . . . , 0), a totally symmetric tensor field. More general irreps correspond to more general mixed-symmetry fields. The analog of (F.1) in this generalized setup is where for bosons the sum runs over integer n with an overall + sign and for fermions the sum runs over half-integer n with an overall − sign. The dimensions of the so(d + 2) irreps for odd d, and adding an overall factor 1 2 to correct for double counting, suggesting where σ d (x) ≡ 1 for even d and σ d (x) ≡ sign(x) for odd d. Equivalently, in view of (F.30) where n ∈ Z for bosons and n ∈ 1 2 + Z for fermions, and At first sight this seems to be different from the extension to n ≥ −1 in (4.6) for the spin-s case S = (s, 0, . . . , 0). However it is actually the same, as (D.1)-(D.2) imply that D n,S vanishes for 2 − d ≤ n ≤ −2 when S = (s, 0, . . . , 0).
The obvious conjecture is then that (F.32) is true for general massive representations.
Here are some consistency checks, which are satisfied precisely for the sum range in (F.32): • Locality: For even d, the summand in (F.31) is analytic in n. Applying the Euler-Maclaurin formula to extract the τ → 0 asymptotic expansion of the sum gives in this case The symmetry (F.30) tells us that the integrand on the right hand side is even in x ≡ n + d 2 .
Since dx x 2k e −τ x 2 ∝ τ −k−1/2 , this implies the absence of 1/τ terms in the τ → 0 expansion of the integrand in (F.31), and therefore, in contrast to (F.29), the absence of nonlocal logdivergences, as required by locality of Z PI in odd spacetime dimension d + 1.
• Bulk − edge structure: By following the usual steps, we can rewrite (F.32) as where n * (S) is the lowest value of n in the sum for which D d+2 n,S is nonvanishing. This coincides again with the unitarity bound on µ for massive representations of SO(1, d + 1) [31,64].
Recalling the discussion below (F. 19), this can be viewed as a generalization of the Higuchi bound to arbitrary representations.
Combining (F.32) with (C.19), we thus arrive at an exact closed-form solution for the Euclidean path integral on the sphere for arbitrary massive field content.

G Derivations for massless higher spins
In this appendix we derive (5.19) and provide details of various other points summarized in section 5.

G.1 Bulk partition function: Z bulk
The bulk partition function Z bulk as defined in (3.3) for a massless spin-s field is given by where q = e −t , and χ bulk,s (q) = tr q iH in the case at hand is the (restricted) q-character of the massless spin-s SO(1, d + 1) UIR. For generic d, this UIR is part of the exceptional series [31].
With the exception of the d = 3 case, the above so(1, d + 1) q-characters encoding the Hspectrum of massless spin-s fields in dS d+1 are very different from the so(2, d) characters encoding the energy spectrum of massless spin-s fields in AdS d+1 with standard boundary conditions, the latter being χ In particular for d ≥ 4, the lowest power q ∆ appearing in the q-expansion of the character is ∆ = 2, and is associated with the so(d) representation S = (s, s), i.e. , , , . . . for s = 1, 2, 3, . . ., whereas for the so(2, d) character this is ∆ = s + d − 2 and S = (s). An explanation for this was given in [31]: in dS, S should be thought of as associated with the higher-spin Weyl curvature tensor of the gauge field rather than the gauge field itself.
This fits well with the interpretation of the expansion as counting the number N r of physical static patch quasinormal modes decaying as e −rT (cf. is special is that the Weyl tensor vanishes identically in this case. To get a nonvanishing gauge-invariant tensor, one has to act with at least 2s − 1 derivatives (spin-s Cotton tensor), yielding ∆ = (2s − 1) + (2 − s) = s + 1. An extensive analysis is given in [48].
Note on a literature disagreements: The characters (G.2) agree with the characters listed in the original work [64], computed by undisclosed methods. They do not agree with those listed in the more recent work [31], computed by Bernstein-Gelfand-Gelfand resolutions. Indeed [31] emphasized they disagreed with [64] for even d. More precisely, in their eq. (2.14) applied to p = 2, Y p = (s, s, 0, . . . , 0), x = 0, they find a factor 2 = (1 + (−1) d ) instead of the factor . It is stated in [31] that on the other hand their results do agree with [64] for odd d. Actually we find this is not quite true either, as in that case eq. (2.13) in [31] applied to p = 2, Y p = (s, s, 0, . . . , 0), x = 0) has a factor 1 instead of the factor 37 More precisely, invariant under linearized gauge transformations acting on the conformal boundary.
Further support is provided in [48] by direct construction of higher-spin quasinormal modes.

G.2 Euclidean path integral:
The Euclidean path integral of a collection of gauge fields φ on S d+1 is formally given by where G is the local gauge group generated by the local field ξ appearing in (5.1). This ill-defined formal expression is turned into something well-defined by BRST gauge fixing. A convenient gauge for higher-spin fields is the de Donder gauge. At the Gaussian level, the resulting analog of (F.1) is 38 where s sums over the spin-s gauge fields in the theory (possibly with multiplicities) and m 2 φ,s and m 2 ξ,s are obtained from the relations below (4.1) using (5.2). The first term arises from the path integral over the TT modes of φ, while the second arises from the TT part of the gauge fixing sector in de Donder gauge -a combination of integrating out the TT part of the spin-(s − 1) ghost fields and the corresponding longitudinal degrees of freedom of the spin-s gauge fields. The above (G.7) is the difference of two expressions of the form (F.1).
Naively applying the formula (F.35) or (4.6) for the corresponding full Z PI , we get However this is clearly problematic. One problem is that for s ≥ 2, the aboveF s (q) contains negative powers of q = e −t , making (G.8) exponentially divergent at t → ∞. The appearance of such "wrong-sign" powers of q is directly related to the appearance of "wrong-sign" Gaussian integrals in the path integral, as can be seen for instance from the relation between (F.35) and the heat kernel integral (F.32). In the path integral framework, one deals with this problem by analytic continuation, generalizing the familiar contour rotation prescription for negative modes in the gravitational Euclidean path integral [12]. Thus one defines 38 A detailed discussion of normalization conventions left implicit here is given above and below (G.40).
dx e −λx 2 /2 for λ < 0 by rotating x → ix, or equivalently by rotating τ → −τ in the heat kernel integral. Essentially this just boils down to flipping any λ < 0 to −λ > 0. Since the Laplacian eigenvalues are equal to the products of the exponents appearing in the pairs q ∆+n + q d−∆+n in (G.9), the implementation of this prescription in our setup is to flip the negative powers q k inF s (q) = k c k q k to positive powers q −k , that is to say replacê (G.10) In addition, each negative mode path integral contour rotation produces a phase ±i, resulting in a definite, finite overall phase in Z PI [18]. The analysis of [18] translates to each corresponding flip in (G.10) contributing with the same sign, 39 hence to an overall phase i −Ps with P s the total degeneracy of negative modes in (G.9). Using D d+2 n,s = −D d+2 s−1,n+1 : In particular this implies P 1 = 0 and P 2 = d + 3 in agreement with [18].
After having taken care of the negative powers of q, the resulting amended formula Z PI = dt 2t F s (q) is still problematic, however, as F s (q) still contains terms proportional to q 0 , causing the integral to diverges (logarithmically) for t → ∞. These correspond to zeromodes in the original path integral. Indeed such zermodes were to be expected: they are due to the existence of normalizable rank s − 1 traceless Killing tensorsξ (s−1) , which by definition satisfy ∇ (µ 1ξ µ 2 ···µs) = 0, and therefore correspond to vanishing gauge transformations (5.1), leading in particular to ghost zeromodes. Zeromodes of this kind must be omitted from the Gaussian path integral. They are easily identified in (G.9) as the values of n for which a term proportional to q 0 appears. Since we are assuming d > 2, this is n = s − 2 in the first sum and n = s − 1 in the second. Thus we should refine (G.10) tô  We cannot just drop the zeromodes and move on, however. The original formal path integral expression (G.6) is local by construction, as both numerator and denominator are defined with a local measure on local fields. In principle BRST gauge fixing is designed to maintain manifest locality, but if we remove any finite subset of modes by hand, including in particular zeromodes, locality is lost. Indeed the −F (0) s subtraction results in nonlocal log-divergences in the character integral, i.e. divergences which cannot be canceled by local counterterms. From the point of view of (G.6), the loss of locality is due the fact that we are no longer dividing by the volume of the local gauge group G, since we are effectively omitting the subgroup G generated by the Killing tensors. To restore locality, and to correctly implement the idea embodied in (G.6), we must divide by the volume of G by hand. This volume must be computed using the same local measure defining vol(G), i.e. the invariant measure on G normalized such that integrating the gauge fixing insertion in the path integral over the gauge orbits results in a factor 1. Hence the appropriate measure defining the volume of G in this context is inherited from the BRST path integral measure. As such we will denote it by vol(G) PI . A detailed general discussion of the importance of these specifications for consistency with locality and unitarity in the case of Maxwell theory can be found in [23].
Relating vol(G) PI to a "canonical", theory-independent definition of the group volume vol(G) c (such as for example vol(U (1)) c ≡ 2π) is not trivial, requiring considerable care in keeping track of various normalization factors and conventions. Moreover vol(G) PI depends on the nonlinear interaction structure of the theory, as this determines the Lie algebra of G. We postpone further analysis of vol(G) PI to section G.4.

Conclusion
To summarize, instead of the naive (G.8), we get the following formula for the 1-loop Euclidean path integral on S d+1 for a collection of massless spin-s gauge fields: where F s = {F s } + and F 0 s were defined in (G.9), (G.10) and (G.13); G is the subgroup of gauge transformations generated by the Killing tensorsξ (s−1) , i.e. the zeromodes of (5.1); and i −Ps is the phase (G.11). We can split up the integrals by introducing an IR regulator: where the notation × means we IR regulate by introducing a factor e −µt , take µ → 0, and subtract the log µ divergent term. For a function f (t) such that lim t→∞ f (t) = c, this means For example for f (t) = t t+1 this gives × 0 dt t t t+1 = log µ − log(e γ µ) = −γ, and for f (t) = 1 with the integral UV-regularized as in (C.30) we get × dt t = log(2e −γ / ). In section G.3 we recast Z char as a character integral formula. In section G.4 we express Z G in terms of the canonical group volume vol(G) c and the coupling constant of the theory.
G.3 Character formula: Z char = Z bulk /Z edge Z KT In this section we derive a character formula for Z char in (G.15). If we start from the naivê F s given by (G.9) and follow the same steps as those bringing (4.6) to the form (4.7), we get Note that these take the form of "field − ghost" characters obtained respectively by substituting the values of ν φ and ν ξ given by (5.2) into the massive spin s and spin s − 1 characters where the "flipped" character [χ] + is obtained fromχ = k c k q k by flipping c k q k → −c k q −k for k < 0 and dropping the k = 0 terms: Thus this flipping prescription can be thought of as the character analog of contour rotations for "wrong-sign" Gaussians in the path integral. Notice the slight differences in the map χ → [χ] + and the related but different mapF → {F } + defined in (G.10). Substituting (G.20) into (G.15), we conclude 40 To check (G.20) starting from (G.10), observe that 1+q 1−q q k + q −k − 2 + = 0 for any integer k, so 1+q 1−q q k + = 1+q 1−q −q −k + 2 for k < 0, while of course 1+q 1−q q k + = 1+q 1−q q k for k ≥ 0, . This accounts for the k < 0 and k > 0 terms in the expansion k c k q k of (G.20). The coefficient 2N KT s−1 of the q 0 term is most easily checked by comparing the q 0 terms on the left and right hand sides of (G.20), taking into account that, by definition, [χ] + has no q 0 term, and that the q 0 terms of the left hand side are given by (G.13).

Consistency with ideal gas bulk character formula
Consistency with the physical ideal gas picture used to derive Z bulk in section G.1 requires the bulk part of log Z char as given in (G.22) agrees with (G.1), that is to say it requires where χ bulk,s is one of the intricate exceptional series characters (G.2), while [χ bulk,s ] + is obtained from the simple naive bulk character (G.18) just by flipping some polar terms as in (G.21). At first sight this might seems rather unlikely. Nevertheless, quite remarkably, it turns out to be true. Let us first check this in some simple examples: • s = 2 in d = 3: The naive character (G.18) iŝ The polar and q 0 terms are obtained by expandingχ bulk = − 3 q − 4 + O(q). Thus correctly reproducing the d = 3, s = 2 character in (G.3).
Using Mathematica, it is straightforward to check an arbitrary large number of examples in this way. Below we will derive a general explicit formula for the character flip mapχ → [χ] + .
This will provide a general proof of (G.23), and more generally will enable efficient closedform computation of the proper bulk and edge characters for general s and d. Generalizations are implemented with equal ease: as an illustration thereof we compute the bulk and edge characters for partially massless fields.

Flipping formula
We wish to derive a general explicit formula for q ∆ Here D d 1−∆,1 m is the dimension of the irrep of so(d) corresponding to the Young diagram S = (1 − ∆, 1, . . . , 1, 0, . . . , 0), with 1 repeated m times, explicitly given in (D.1)-(D.2). We obtained this formula using Mathematica and also obtained a proof of (G.29) by expressing (D.1)-(D.2) in terms of gamma functions and comparing to (G.28). This is somewhat tedious and not especially illuminating, so we omit it here.

Bulk characters for (partially) massless fields
Now let us apply this to a slight generalization of the masslessχ bulk,s given in (G.18), This is the naive bulk character for a partially massless spin-s field φ µ 1 ···µs with a spin-s (0 ≤ s ≤ s − 1) gauge parameter field ξ µ 1 ···µ s [125]. The massless case (G.18) corresponds to We consider the more general partially massless case here to illustrate the versatility of (G.29), and because in a sense the resulting formulae are more elegant than in the massless case, due to the neat s ↔ s symmetry already evident in (G.30). Applying (G.29), still with r = d 2 , Remarkably, (G.32) precisely reproduces the massless exceptional series characters (G.2) for s = s − 1, further strengthening our physical picture, adding evidence for (F.35), proving (G.23), and generalizing it moreover to partially massless gauge fields. Comparing to [64], the partially massless gauge field characters we find here coincide with those of the unitary exceptional series D j S,p with p = 0, S = (s, s + 1), and j = (d − 3)/2 for odd d, j = (d − 4)/2 for even d. In the notation of [31] this is the exceptional series with ∆ = p = 2, S = (s, s + 1), which was indeed identified in [31] as the so(1, d + 1) UIR for partially massless fields.

Edge characters for (partially) massless fields
For the edge correction we proceed analogously. The naive PM edge character iŝ reducing to the massless case (G.19) for s = s − 1. Applying (G.29) gives, still with r = d 2 , ,1 m . Note that in the massless spin-1 case In the notation of [64],

Conclusion
We conclude that (G.22) can be written as where the bulk and edge contributions are explicitly given by (G.32)-(G.34) with s = s − 1, 41 The finite (IR) part of Z KT is given by (C.37):

G.4 Group volume factor: Z G
The remaining task is to compute the factor Z G defined in (G.15), that is We imagine the spin range to be finite, or cut off in some way. (The infinite spin range case is discussed in section 9.) In the heat kernel regularization scheme of appendix C, we can then evaluate the integral using (C.30): On general grounds, the nonlocal UV-divergent factors M appearing here in Z G should cancel against factors of M in vol(G) PI , as we will explicitly confirm below.

Generalities
Recall that G is the group of gauge transformations generated by the Killing tensors. Equiva- constants multiplying kinetic operators, as these can no longer be automatically discarded by adjusting local counterterms. Consequently, exact, direct path integral computationz of Z G for general higher-spin theories requires great care and considerable persistence, although it can be done [124]. Below we obtain an exact expression for Z G in terms of vol(G) c and the Newton constant in a comparatively painless way, by combining results and ideas from [22,23,25,[27][28][29][30], together with the observation that the form of (G.7) actually determines all the normalization factors we need. Although the expressions at intermediate stages are still a bit unpleasant, the end result takes a strikingly simple and universal form.
If G is finite-dimensional, as is the case for example for Yang-Mills, Einstein gravity and certain (topological) higher-spin theories [54,73,95] including the dS 3 higher-spin theory analyzed in section 6, we can then proceed to compute vol(G) c , and we are done. If G is infinite-dimensional, as is the case in generic higher-spin theories, one faces the remaining problem of making sense of vol(G) c itself. Glossing over the already nontrivial problem of exponentiating the higher-spin algebra to an actual group [80], the obvious issue is that vol(G) c is going to be divergent. We discuss and interpret this and other infinite spin range issues in section 9. In what follows we will continue to assume the spin range is finite or cut off in some way as before, so G is finite-dimensional.
We begin by determining the path integral measure to be used to compute vol(G) PI in (G.39). Then we compute Z G in terms of vol(G) c and the coupling constant of the theory, first for Yang-Mills, then for Einstein gravity, and finally for general higher-spin theories.

Path integral measure
To determine vol(G) PI we have to take a quick look at the path integral measure. This is fixed by locality and consistency with the regularized heat kernel definition of Gaussian path integrals we have been using throughout. For example for a scalar field as in (3.2), We work with canonically normalized fields. For a spin-s field φ this means the quadratic part of the action evaluated on its transverse-traceless part φ TT takes the form Consistency with (F.1) or (G.7) then requires the measure for φ to be derived again from the metric ds 2 φ = M 2 2π (δφ) 2 . If φ has a gauge symmetry, the formal division by the volume of the gauge group G is conveniently implemented by BRST gauge fixing. For example for a spin-1 field with gauge symmetry δφ µ = ∂ µ ξ, we can gauge fix in Lorenz gauge by adding the BRST-exact action S BRST = iB ∇ µ φ µ −c∇ 2 c. This requires specifying a measure for the Lagrange multiplier field B and the ghosts c,c. It is straightforward to check that a ghost measure derived from ds 2 cc = M 2 δc δc (which translates to a mode measure i 1 M 2 dc i dc i ) combined with a B-measure derived from ds 2 B = 1 2π (δB) 2 , reproduces precisely the second term in (G.7) upon integrating out B, c,c and the longitudinal modes of φ. It is likewise straightforward to check that BRST gauge fixing is then formally equivalent to dividing by the volume of the local gauge group G with respect to the measure derived from the following metric on the algebra of local gauge transformations: Note that all of these metrics take the same form, with the powers of M fixed by dimensional analysis. An important constraint in the above was that the second term in (G.7) is exactly reproduced, without some extra factor multiplying the Laplacian. This matters when we omit zeromodes. For this to be the case with the above measure prescriptions, it was important that the gauge transformation took the form δφ µ = α 1 ∂ µ ξ with α 1 = 1 as opposed to some different value of α 1 , as we a priori allowed in (5.1). For a general α 1 , we would have obtained an additional factor α 1 in the ghost action, and a corresponding factor α 2 1 in the kinetic term in the second term of (G.7). To avoid having to keep track of this, we picked α 1 ≡ 1. For Yang-Mills theories, everything remains the same, with internal index contractions understood, e.g.
(δξ a ) 2 . For higher-spin fields, we gauge fix in the de Donder gauge. All metrics remain unchanged, except for the obvious additional spacetime index contractions. The second term of (G.7) is exactly reproduced upon integrating out the TT sector of the BRST fields together with the corresponding longitudinal modes of φ, provided we pick  [27].

Yang-Mills
Consider a Yang-Mills theory with with a simple Lie algebra with the L a some standard basis of anti-hermitian matrices and f abc real and totally antisymmetric. For example for su(2) Yang-Mills, L a = − 1 2 iσ a and [L a , L b ] = abc L c . Consistent with our general conventions, we take the gauge fields φ µ = φ a µ L a to be canonically normalized: the curvature takes the form F a µν L a = F µν = ∂ µ φ ν − ∂ ν φ µ + g[φ µ , φ ν ], and the action is The quadratic part of S is invariant under the linearized gauge transformations δ Evidently this is isomorphic to the original YM Lie algebra. Being a simple Lie algebra, g has an up to normalization unique invariant bilinear form/metric. The path integral metric ds 2 PI of (G.41) corresponds to such an invariant bilinear form with a specific normalization: We define the theory-independent "canonical" invariant bilinear form ·|· c on g as follows.
First pick a "standard" basis M , where we used vol(S d+1 ) = 2π d vol(S d−1 ). (Recall we have been assuming d > 2. The case d = 2 is discussed in appendix H.1.) The quantity γ may look familiar: the Coulomb potential energy for two unit charges at a distance r in flat space is V (r) = γ 2 /r d−2 .
Practically speaking, the upshot is that Z G is given by (G.51), with vol(G) c the volume of the Yang-Mills gauge group with respect to the metric defined by the Yang-Mills action (G.44). For example for G = SU (2) with f abc = abc as before, vol(G) c = 16π 2 , because SU (2) in this metric is the round S 3 with circumference 4π, hence radius 2.
The relation (G.46) can be viewed as defining the coupling constant g given our normalization conventions for the kinetic terms and linearized gauge transformations. Of course the final result is independent of these conventions. Conventions without explicit factors of g in the curvature and gauge transformations are obtained by rescaling φ → φ/g, ξ → ξ/g. Then there won't be a factor g in (G.46), but instead g is read off from the action S = 1 4g 2 (F a ) 2 .
We could also write this without explicit reference to a basis as S = 1 4g 2 Tr F 2 , where the trace "Tr" is normalized such that Tr(L a L b ) ≡ δ ab . Then we can say the canonical bilinear/metric/volume is defined by the trace norm appearing in the YM action. We could choose a differently normalized trace Tr = λ 2 Tr. The physics remains unchanged provided g = λg. Then vol(G) c = λ dim G vol(G) c , hence, consistently, Z G = Z G .
As a final example, for SU (N ) Yang-Mills with su(N ) viewed as anti-hermitian N × N matrices, S = − 1 4g 2 Tr N F 2 in conventions without a factor g in the gauge algebra, and Tr N the ordinary N × N matrix trace, vol(SU (N )) c = (D.8).

Einstein gravity
The Einstein gravity case proceeds analogously. Now we have single massless spin-2 field φ µν . With respect to the corresponding metric ds 2 c , orbits g(ϕ) = e ϕM 12 with ϕ ranging from 0 to 2π have length 2π. The canonical volume is then given by (D.7).
To identify the standard generators M IJ more precisely in our normalization conventions forξ, we need to look at the field theory realization in more detail. The so(d + 2) algebra generated by the Killing vectorsξ is realized in the interacting Einstein gravity theory as a subalgebra of the gauge (diffeomorphism) algebra. As in the Yang-Mills case (G.46), the bracket [[·, ·]] of this subalgebra is inherited from the gauge algebra. Writing the Killing vectors asξ =ξ µ ∂ µ , the standard Lie bracket is [ξ,ξ ] L = ξ µ ∂ µξ ν −ξ µ ∂ µξ ν ∂ ν . If we had normalized φ µν as φ µν ≡ g µν − g 0 µν with g 0 µν the background sphere metric, and if we had normalized ξ µ by putting α 2 ≡ 1 in (5.1), the bracket [[·, ·]] would have coincided with the Lie bracket [·, ·] L . However, we are working in different normalization conventions, in which φ µν is canonically normalized and α 2 = √ 2 according to (G.42). In these conventions we have instead where G N is the Newton constant. This can be checked by starting from the Einstein-Hilbert action, expanding to quadratic order (see e.g. [126] for convenient and reliable explicit expressions in dS d+1 ), and making the appropriate convention rescalings. This is the Ein- ξ ·ξ , so Thus we obtain the following relation between PI and canonical metrics and volumes for G = SO(d + 2):

Higher-spin gravity
We follow the same template for the higher-spin case. In the interacting higher-spin theory, fixed by the transverse-traceless cubic couplings of the theory [27]. The Killing tensor algebra includes in particular an so(d + 2) subalgebra, that is to say an algebra of the same general form (G.53) as in Einstein gravity, with some constant appearing on the right-hand side determined by the spin-2 cubic coupling in the TT action. We define the "Newton constant" G N of the higher-spin theory to be this constant, that is to say we read off G N from the so(d + 2) Killing vector subalgebra by writing it as The standard Killing vector basis is then again given by where B s are certain constants fixed in principle by the algebra. The arguments given in [27] moreover imply that up to overall normalization, the coefficients B s are independent of the coupling constants in the theory. More specifically, adapted (with some work, as described below) to our setting and conventions, and correcting for what we believe is a typo in [27], the coefficients are B s ∝ (2s + d − 4)(2s + d − 2). We confirmed this by comparison to [30], where the invariant bilinear form for minimal Vasiliev gravity in AdS d+1 , dual to the free O(N ) model, was spelled out in detail, building on [27][28][29]. Analytically continuing to positive cosmological constant, implementing their ambient space X-contractions by a Gaussian integral, and reducing this integral to the sphere by switching to spherical coordinates, the expression in [30] can be brought to the form (G.58). This transformation almost completely cancels the factorials in the analogous coefficients b s in [30], reducing to the simple . (The alternating signs of [30] are absent here due to the analytic continuation to positive cc.) Taking into account our normalization prescription (G.57) (which is different from the normalization chosen in [30]), we thus get with A d−1 = vol(S d−1 ) as before. In view of the independence of the coefficients B s of the couplings within the class of theories considered in [27], i.e. all parity-invariant massless higher-spin gravity theories consistent to cubic order, this result is universal, valid for this entire class.
As before for Einstein gravity and Yang Mills, from (G.59) we get the ratio (G.60) Combining this with (G.39) we see that, rather delightfully, all the unpleasant-looking factors cancel, leaving us with This takes exactly the same form as the Einstein gravity result (G.55) except G is now the higher-spin symmetry group rather than the SO(d + 2) spin-2 symmetry group.
The cancelation of the UV divergent factors M is as expected from consistency with locality. The cancelation of the s-dependent factors on the other hand seems surprising, in view of the different origin of the numerator (spectrum of quadratic action) and the denominator (invariant bilinear form on higher spin algebra of interactions). Apparently the former somehow knows about the latter. We do not see an obvious reason why this is the case, although the simplicity and universality of the result suggests we should, and that this entire section should be replaceable by a one-line argument. Perhaps it is obvious in a frame-like formalism.

Newton constant from central charge
Recall that the Newton constant G N appearing in (G.61) was defined by the so(d + 2) algebra (G.56) in our normalization conventions. An analogous definition can be given in dS d+1 or AdS d+1 where the algebra becomes so(1, d + 1) resp. so (2, d). Starting from this definition, G N can also be formally related to the Cardy central charge C of a putative 42 boundary CFT for AdS or dS, defined as the coefficient of the CFT 2-point function of the putative energymomentum tensor. With our definition of G N , the computation of [127] remains unchanged, so we can just copy the result obtained there: where as before . 42 There is no assumption whatsoever this CFT actually exists. One just imagines it exists and uses the formal holographic dictionary to infer the two-point function of this imaginary CFT's stress tensor. In dS, this "dual CFT" can be thought of as computing the Hartle-Hawking wave function of the universe [90].
The dimensions D 2 s = D 2 −s = 1 and D 4 n,s = D 4 n,−s = (1 + n − s)(1 + n + s) of the SO(2) and SO(4) irreps are correctly reproduced by the Weyl dimension formula (D.1), rather than (A.15). It should however be kept in mind that the single-particle Hilbert space of for instance a massive spin-s ≥ 1 Pauli-Fierz field on dS 3 carries both helicity versions (∆, ±s) of the massive spin-s SO (1,3) irrep, hence the character χ to be used in expressions for Z PI in this case is χ = χ +s + χ −s = 2χ +s = 2(q ∆ + q 2−∆ )/(1 − q) 2 . On the other hand for a real scalar field, we just have Relatedly we get an extra q 0 term in q 2s+d−4 + q 2s+d−2 + 2 = q 2 + 3 in (G.38), and we end up with 2π instead of (G.51). Everything else remains the same. Finally, the phase i −Ps (G.11) is somewhat modified. For s ≥ 2, from (H.2), Note that P 2 = 5, in agreement with [18]. P 1 = 0 as before, since there are no negative modes.

Conclusion
The final result for Z where as before γ = 8πG N /A 1 = 4G N / , P = s P s , and vol(G) c is the volume with respect to the metric for which the standard so (4) For example for Einstein gravity with G = SO(4), we get • For Yang-Mills theories with gauge group G and coupling constant g, we get . (H.8) Using (C. 19), (C.37), the finite part evaluates to As in (5.19), vol(G) c is the volume of G with respect to the metric defined by the trace appearing in the Yang-Mills action. As a check, for G = U (1) we have vol(G) c = 2π, so Z = g e −ζ(3)/4π 2 √ in agreement with [128] eq. (3.25).
• We could also consider the Chern-Simons partition function on S 3 , with k > 0 suitably quantized (k ∈ Z for G = SU (N ) with Tr the trace in the Ndimensional representation). Because in this case the action is first order in the derivatives and not parity-invariant, it falls outside the class of theories we have focused on in this paper. It is not too hard though to generalize the analysis to this case. The main difference with Yang-Mills is that χ bulk = 0 = χ edge : like in the s ≥ 2 case, the s = 1 Chern-Simons theory has no particles. The functionF 1 is no longer given by the Maxwell version (H.3), but rather by (H.2), except without the factors of 2, related to the fact that the CS action is first order in the derivatives. This immediately gives The computation of the volume factor is analogous to our earlier discussions. The result (in canonical framing [129]) is where vol(G) Tr is the volume with respect to the metric defined by the trace appearing in the Chern-Simons action (H.10). This agrees with the standard results in the literature, nicely reviewed in section 4 of [130].

H.2 Chern-Simons formulation of Einstein gravity
3D Einstein gravity can be reformulated as a Chern-Simons theory [52,131]. Although wellknown, we briefly review some of the basic ingredients and conceptual points here to facilitate the discussion of the higher-spin generalization in section H.3. A more detailed review of certain aspects, including more explicit solutions, can be found in section 4 of [132]. Explicit computations using the Chern-Simons formulation of Λ > 0 Euclidean quantum gravity with emphasis on topologies more sophisticated than the sphere can be found in [133][134][135].

Lorentzian gravity
For the Lorentzian theory with positive cosmological constant, amplitudes are computed by path integrals DA e iS L with real Lorentzian SL(2, C) Chern-Simons action [53] where S CS is as in (H.10) with A ± an sl(2, C)-valued connection and Tr = Tr 2 . The vielbein e and spin connection ω are the real and imaginary parts of the connection: For the last equality we decomposed e = e i L i in a basis L i of sl(2, R), say −1, 1) . (H.14) Note that [L i , L j ] = − ijk L k with L k ≡ η kk L k . When l = 0, the action reduces to the firsorder form of the Einstein action with Newton constant G N = /4κ and cosmological constant Λ = 1/ 2 . The equations of motion stipulate A ± must be flat connections: equivalent with the Einstein gravity torsion constraint (with ω i j ≡ η il ljk ω k ) and the Einstein equations of motion [52]. Turning on l deforms the action by parity-odd terms of gravitational Chern-Simons type. This does not affect the equations of motion (H.15). We can take l ≥ 0 without loss of generality. The part of the action multiplied by l has a discrete ambiguity forcing l to be integrally quantized, like k in (H.10). Summarizing, A flat connection corresponding to the de Sitter metric can be obtained as follows. (We will be brief because the analog for the sphere below will be simpler and make this more clear.) Define Q(X) ≡ 2 (X 4 L 4 + iX i L i ) with L 4 ≡ 1 2 1 and note that det Q = X 2 4 + η ij X i X j =: η IJ X I X J , so M ≡ {X| det Q(X) = 1} is the dS 3 hyperboloid, and Q is a map from M into SL(2, C).
which is the de Sitter metric of radius on M.

Euclidean gravity
Like the Einstein-Hilbert action -or any other action for that matter -(H.12) may have complex saddle points, that is to say flat connections A ± which do not satisfy the reality constraint (H.12), or equivalently solutions for which some components of the vielbein and spin connection are not real. Of particular interest for our purposes is the solution corresponding to the round metric on S 3 . This can be obtained from the dS 3 solution as usual by a Wick rotation of the time coordinate. Given our choice of sl(2, R) basis (H.14), this means At the level of the vielbein e = e i L i such a Wick rotation is implemented as Similarly, recalling ω ij = ijk ω k , the spin connection ω = ω i L i rotates as ω 1 → iω 1 , Thus the Lorentzian metric η ij gets replaced by the Euclidean metric δ ij , the Lorentzian sl(2, C) = so(1, 3) reality condition gets replaced by the Euclidean su(2) ⊕ su(2) = so(4) reality condition, and the Lorentzian path integral DA e iS L becomes a Euclidean path integral DA e −S E , where S E ≡ −iS L is the Euclidean action: , and in addition here it also exchanges the ± parts of the decomposition so(4) = su(2) + ⊕su(2) − into self-dual and anti-self-dual parts, that is to say it exchanges A + ↔ A − . Thus orientation

Round sphere solutions
Parametrizing S 3 by g ∈ SU (2) S 3 , it is easy to write down a flat su(2) ⊕ su(2) connection yielding the round metric of radius : The radius can be checked by observing that along an orbit g(ϕ) = e ϕS 3 , we get g −1 dg = dϕ S 3 so ds = 1 2 dϕ and the orbit length is where we used (H. 16). This reproduces the standard Gibbons-Hawking result [2] for dS 3 .
More generally we can consider flat connections of the form (A + , A − ) = (h −1 where n ± ∈ Z if we take the gauge group to be G = SU (2)×SU (2) and n ± ∈ Z or n ± ∈ 1 2 + Z if we take G = SU (2) × SU (2) /Z 2 SO(4). These are all related to the trivial connection (0, 0) by a large gauge transformation g ∈ S 3 → (h + , h − ) ∈ G. All other flat connections on S 3 are obtained from these by gauge transformations continuously connected to the identity, which are equivalent to diffeomorphisms and vielbein rotations continuously connected to the identity in the metric description [52]. Large gauge transformations on the other hand are in general not equivalent to large diffeomorphisms. Indeed, so evidently different values of n are physically inequivalent. Conversely, for a fixed value of n but different values ofñ, we get the same metric, so these solutions are geometrically equivalent. In particular the n = 1 solutions all produce the same round metric (H. 19). For n = 0, the metric vanishes. For n < 0, we get a vielbein with negative determinant. Only vielbeins with positive determinant reproduce the Einstein-Hilbert action with the correct sign, so from the point of view of gravity we should discard the n < 0 solutions. Finally the cases n > 1 correspond to a metric describing a chain of n spheres connected by throats of zero size, presumably more appropriately thought of as n disconnected spheres. The Wick rotation X 2 → −iX 2 of our earlier constructed Lorentzian dS 3 equals the (n+, n − ) = ( 1 2 , − 1 2 ) solution constructed here.

Euclidean path integral
The object of interest to us is the Euclidean path integral Z = DA e −S E [A] , defined perturbatively around an n = n + − n − = 1 round sphere solution (Ā + ,Ā − ) = (g −n + dg n + , g −n − dg n − ), such as the (1, 0) solution (H. 19). Physically, this can be interpreted as the all-loop quantumcorrected Euclidean partition function of the dS 3 static patch. For simplicity we take G = SU (2) × SU (2), so n ± ∈ Z and we can formally factorize Z as an SU (2) k + × SU (2) k − CS partition function where k ± = l ± iκ, with l ∈ Z + and κ ∈ R + : Here the complex-k CS partition function Z CS (SU (2) k |Ā m ) ≡ m DA e ikS CS [A] is defined perturbatively around the critical pointĀ = g −m dg m . It is possible, though quite nontrivial in general, to define Chern-Simons theories at complex level k on general 3-manifolds M 3 [136,137]. Our goal is less ambitious, since we only require a perturbative expansion of Z around a given saddle, and moreover we restrict to M 3 = S 3 . In contrast to generic M 3 , at least for integer k, the CS action on S 3 has a unique critical point modulo gauge transformations, and its associated perturbative large-k expansion is not just asymptotic, but actually converges to a simple, explicitly known function: in canonical framing [129], The dependence on the choice of critical pointĀ = g −m dg m actually drops out for integer k, as S CS [Ā] = −2πm ∈ 2πZ. We have kept it in the above expression to because this is no longer the case for complex k. Analytic continuation to k ± = l ± iκ with l ∈ Z + and κ ∈ R + in (H.22) then gives: where I(ω) = 1 4π Tr 3 (ω ∧ dω + 2 3ω ∧ω ∧ω) is the gravitational Chern-Simons action. The action I(ω) can be defined more precisely as explained under (2.22) of [138], by picking a 4-manifold M with boundary ∂M = S 3 and putting Taking  where, using c(k) = 3 1 − 2 2+k , the phase is given by In the weak-coupling limit κ → ∞, (H.30) Using (H.16) and taking into account that we took G = SU (2) × SU (2) here, the absolute value agrees with our general one-loop result (H.6) in the metric formulation, with the phase (−i) r matching Polchinski's phase i −P = i −5 = −i in (5.19) for odd framing r. 43 We do not have any useful insights into why (or whether) CS framing and the phase i −P might have anything to do with each other, let alone why odd but not even framing should reproduce the phase of [18]. Perhaps different contour rotation prescriptions as those assumed in [18] might reproduce the canonically framed (r = 0) result in the metric formulation of Euclidean gravity. We leave these questions open.

H.3 Chern-Simons formulation of higher-spin gravity
The SL(2, C) Chern-Simons formulation of Einstein gravity (H.12) has a natural extension to an SL(n, C) Chern-Simons formulation of higher-spin gravity -the positive cosmological constant analog of the theories studied e.g. in [54,73,74]. The Lorentzian action is Tr n A ∧ dA + 2 3 A ∧ A ∧ A , an A is an sl(n, C)-valued connection, κ ∈ R + and l ∈ Z + . The corresponding Euclidean action S E = −iS L extending (H.17) is given by where A ± are now independent su(n)-valued connections.

H.3.1 Landscape of dS 3 vacua
The solutions A of the original (n = 2) Einstein gravity theory can be lifted to solutions A = R(A) of the extended (n > 2) theory by choosing an embedding R of sl(2) into sl(n).
More concretely, such lifts are specified by picking an n-dimensional representation R of su (2), Here J irrep is j(j + 1) = 1 4 (m 2 − 1), so   Note that S (0) ≡ −S E = 2π 4G N is the usual dS 3 Gibbons-Hawking horizon entropy [2]. Its value S (0) = 2πκ T R depends on the vacuum R = ⊕ a m a through T R as given by (H.34). The vacuum R maximizing e −S E corresponds to the partition of n = a m a maximizing T R . Clearly the maximum is achieved for R = n: The corresponding embedding of su(2) into su(n) is called the "principal embedding". Thus the "principal vacuum" maximizes the entropy at S GH,n = 1 6 n(n 2 − 1) 2πκ, exponentially dominating the Euclidean path integral in the semiclassical (large-κ) regime. In the remainder we focus on the Euclidean version of the theory.

H.3.2 Higher-spin field spectrum and algebra
Of course for n > 2, there are more degrees of freedom in the 2(n 2 −1) independent components of A ± than just the 3+3 vielbein and spin connection degrees of freedom A i ± S i . The full set of fluctuations around the vacuum solution can be interpreted in a metric-like formalism as higher-spin field degrees of freedom. The precise spectrum depends on the vacuum R. For the principal vacuum R = n, we get the higher-spin vielbein and spin connections of a set of massless spin-s fields of s = 2, 3, . . . , n, as was worked out in detail for the AdS analog in [73].

H.3.3 One-loop Euclidean path integral from metric-like formulation
In view of the above higher-spin interpretation of the theory, we can apply our general massless HS formula (H.6) with G = SU (n) × SU (n) to obtain the one-loop contribution to the Euclidean path integral (for l = 0). In combination with (H.36) this takes the form Recall that vol(G) c is the volume of G with respect to the metric normalized such that where M is one of the standard so(4) = su(2) ⊕ su(2) generators, which we can for instance take to be the rotation generator M = S 3 ⊕ S 3 . In the context of Chern-Simons , the T R -dependent factors cancel out. Finally, using (H.4), we get P = n s=2 P s = 1 3 (2s − 3)(2s − 1)(2s + 1) = 2 3 n 2 (n − 1)(n + 1) − (n 2 − 1). Because (n − 1) · n · (n + 1) is divisible by 3, the first term is an integer, and moreover a multiple of 8 because either n 2 or (n + 1)(n − 1) is a multiple of 4.
Hence i −P = i (n 2 −1) , which equals −i for even n and +1 for odd n. Thus we get  [56,139] is

H.3.4 Euclidean path integral from CS formulation
(H.44) The corresponding higher-spin generalization of (H.24) is therefore hence the generalization of (H.28) for arbitrary framing I M = 6πr, r ∈ Z, is In the limit κ → ∞, Recognizing n 2 − 1 = dim SU (n) and √ n n s=2 (2π) s /Γ(s) = vol(SU (n)) Trn (D.8), we see this precisely reproduces the one-loop result (H.43). Like in the original n = 2 case, the phase again matches for odd framing r. (The agreement at one loop can also be seen more directly by a slight variation of the computation leading to (H.11).) This provides a nontrivial check of our higher-spin gravity formula (H.6) and more generally (5.19).

I Quantum dS entropy: computations and examples
Here we provide the details for section 8. I.1 Classical gravitational dS thermodynamics I.1.1 3D Einstein gravity example For concreteness we start with pure 3D Einstein gravity as a guiding example, but we will phrase the discussion so generalization will be clear. The Euclidean action in this case is with Λ > 0. The tree-level contribution to the entropy (8.2) is The dominant saddle of (I.2) is a round S 3 metric g of radius = 0 minimizing S E ( ) ≡ S E [g ]: where tree means evaluation at the saddle point, here at the on-shell radius = 0 : reproducing the familiar area law S (0) = A/4G for the horizon entropy.
We now recast the above in a way that will allow us to make contact with the formulae of section 7.1 and will naturally generalize beyond tree level in a diffeomorphism-invariant way.
To this end we define an "off-shell" tree-level partition function at fixed (off-shell) volume V : Evaluating the integral is equivalent to a constrained extremization problem with Lagrange multiplier σ enforcing the constraint √ g = V . The dominant saddle is the round sphere g = g of radius (V ) fixed by the volume constraint: Paralleling (7.6) and (7.7), we define from this an off-shell energy density and entropy, 4G .
(I.7) ρ (0) is the sum of the positive cosmological constant and negative curvature energy densities.
It is the Legendre transform of log Z (0) : Note that evaluating tree dσ in (I.5) sets σ = −∂ V log Z (0) = ρ (0) (V ). On shell, Paralleling (7.8), the differential relations in (I.8) can be viewed as the first law of tree-level de Sitter thermodynamics. We can also consider variations of coupling constants such as Λ.
As an example, consider the general action up to order R 2 written as where E µν ≡ R µν − 1 d+1 R g µν , C µνρσ is the Weyl tensor, l s is a length scale and the λ i are dimensionless. The Weyl tensor vanishes on the round sphere and R µν = d g µν / 2 , hence 44 By virtue of its SO(d + 2) symmetry, the round sphere metric g with Ω d+1 d+1 = V is a saddle of (I.5).
Spheres of dimension ≥ 5 admit a plentitude of Einstein metrics that are not round [141][142][143][144], but as explained e.g. in [145], by Bishop's theorem [146], these saddles are subdominant in Einstein gravity. In the large-size limit, higher-order curvature corrections are small, hence the round sphere dominates in this regime.

I.1.3 Effective field theory expansion and field redefinitions
Curvature corrections such as those considered above naturally appear as terms in the derivative expansion of low-energy effective field theories of quantum gravity, with l s the characteristic length scale of UV-completing physics and higher-order curvature corrections terms suppressed by higher powers of l 2 s / 2 1. The action (I.12) is then viewed as a truncation at order l 2 s , and (I.15) can be solved perturbatively to obtain 0 and S (0) as a function of Λ. Suppose someone came up with some fundamental theory of de Sitter quantum gravity, producing both a precise microscopic computation of the entropy and a precise low-energy effective action, with the large-0 /l s expansion reproduced as some large-N expansion. At least superficially, the higher-order curvature-corrected entropy obtained above looks like a Wald entropy [78]. In the spirit of for instance the nontrivial matching of R 2 corrections to the macroscopic BPS black hole entropy computed in [147] and the microscopic entropy computed from M-theory in [148], it might seem then that matching microscopic 1/N -corrections and macroscopic l 2 s / 2 0 -corrections to the entropy such as those in (I.15) could offer a nontrivial way of testing such a hypothetical theory.
However, this is not the case. Unlike the Wald entropy, there are no charges Q (such as energy, angular momentum or gauge charges) available here to give these corrections physical meaning as corrections in the large-Q expansion of a function S(Q). Indeed, the detailed structure of the l s / 0 expansion of S (0) = S (0) ( 0 ) has no intrinsic physical meaning at all, because all of it can be wiped out by a local metric field redefinition, order by order in l s / 0 , bringing the entropy to pure Einstein area law form, and leaving only the value of S (0) itself as a physically meaningful, field-redefinition invariant, dimensionless quantity. This is essentially a trivial consequence of the fact that in perturbation theory about the round sphere, the round sphere itself is the unique solution to the equations of motion. Let us however recall in more detail how this works at the level of local field redefinitions, and show how this is expressed at the level of log Z (0) ( ), as this will be useful later in interpreting quantum corrections. For concreteness, consider again (I.12) viewed as a gravitational effective field theory action expanded to order l 2 s R 2 . Under a local metric field redefinition g µν → g µν + δg µν + O(l 4 s ) , δg µν ≡ l 2 s u 0 Λ g µν + u 1 R g µν + u 2 R µν , (I. 16) where the u i are dimensionless constants, the action transforms as S E → S E + 1 16πG √ g R µν − 1 2 Rg µν + Λg µν δg µν + O(l 4 s ) , (I. 17) shifting λ R 2 , λ E 2 and rescaling G, Λ in (I.12). A suitable choice of u i brings S E to the form Equivalently, this is obtained by using the O(l 0 s ) equations of motion R µν = 2 d−1 Λg µν in the O(l 2 s ) part of the action. Since λ R 2 = 0, the entropy computed from this equivalent action takes a pure Einstein area law form S (0) = Ω d+1 d−1 0 /4G, with 0 = d(d − 1)/2Λ . The on-shell value S (0) itself remains unchanged of course under this change of variables.
In the above we picked a field redefinition keeping G = G. Further redefining g µν → α g µν leads to another equivalent set of couplings G , Λ , . . . rescaled with powers of α according to their mass dimension. We could then pick α such that instead Λ = Λ, or such that 0 = 0 , now with G = G. If we keep 0 = 0 , we get The point of considering quantum corrections to the entropy S is that these include nonlocal contributions, not removable by local redefinitions, and thus, unlike the tree-level entropy S (0) , offering actual data quantitatively constraining candidate microscopic models.

I.2 Quantum gravitational thermodynamics
The quantum off-shell partition function Z(V ) generalizing the tree-level Z (0) (V ) (I.5) is defined by replacing tree Dg → Dg in that expression: 45 Z(V ) ≡ tree dσ Dg e −S E [g]+σ( √ g−V ) . (I.22) The quantum off-shell energy density and entropy generalizing (I.7) are hence S(ρ) is the generating function for moments of the volume. In particular is the expectation value of the volume in the presence of a source ρ shifting the cosmological constant Λ 8πG → Λ 8πG − ρ. Γ(V ) can be viewed as a quantum effective action for the volume, in the spirit of the QFT 1PI effective action [150,151]  The quantum on-shell value of is denoted by¯ and satisfies ρ(¯ ) = 0.
The magnitude of quantum fluctuations of the volume about its mean value is given by 45 Z(V ) is reminiscent of but different from the fixed-volume partition function considered in the 2D quantum gravity literature, e.g. (2.20) in [149]. The latter would be defined as above but with 1 2πi iR dσ instead of tree dσ, constraining the volume to V , whereas Z(V ) constrains the expectation value of the volume to V . To one-loop order, we may replace ρ by its tree-level approximation ρ (0) = −∂ V log Z (0) . By construction this ensures the round sphere metric g = g of radius (V ) given by (I.28) is a saddle. Expanding the action to quadratic order in fluctuations about this saddle then gives a massless spin-2 Gaussian path integral of the type solved in general by (5.19), or more explicitly in (7.19)- (7.20). For 3D Einstein gravity, using (7.20), Here c 0 and c 2 arise from O(G 0 ) local counterterms S E,ct = √ g c 0 − c 2 2 R , (I. 33) split off from the bare action (I.1) to keep the tree-level couplngs Λ and G equal to their "physical" (renormalized) values to this order. We define these physical values as the coefficients of the local terms ∝ 3 , in the V → ∞ asymptotic expansion of the quantum log Z(V ).
That is to say, we fix c 0 and c 2 by imposing the renormalization condition log Z(V ) = Ω 3 8πG −Λ 3 + 3 + · · · (V → ∞) . where log Z ct = −S E,ct [g ] with g the round sphere metric of volume V . On S 3 we have log Z ct = c 0 3 + c 2 , and the → ∞ condition fixes c 0 and c 2 . Recalling (7.6), we can physically interpret this as requiring the renormalized one-loop Euclidean energy U (1) of the static patch vanishes in the → ∞ limit.
For general d, the UV-divergent terms in log Z PI come with non-negative powers ∝ d+1−2n , canceled by counterterms consisting of n-th order curvature invariants. For example on S 5 , log Z ct = c 0 5 + c 2 3 + c 4 . In odd d + 1, the renormalization prescription (I.36) then fixes the c 2n . In even d + 1, log Z ct has a constant term c d+1 , which is not fixed by (I.36). As The equivalence of (I.38) and (I.40) can be checked directly here noting¯ = 0 − 2 π G + O(G 2 ), so 2π¯ 4G = 2π 0 4G − 1 + O(G). The −1 cancels the +1 in (I.39), reproducing (I.41). More generally and more physically, the relation between these two expressions can be understood as follows. At tree level, the entropy equals the geometric horizon entropy S (0) ( 0 ), with radius 0 such that the geometric energy density ρ (0) vanishes. At one loop, we get additional contributions from quantum field fluctuations. The UV contributions are absorbed into the gravitational coupling constants. The remaining IR contributions shift the entropy by S (1) and the energy density by ρ (1) . The added energy backreacts on the fluctuating geometry: its mean radius changes from 0 to¯ such that the geometric energy density

I.4.4 4D massive spin s
4D massive spin-s fields can be treated similarly, starting from (C.27). In particular the edge contribution log Z edge equals minus the log Z (1) of D 5 s−1 = 1 6 s(s + 1)(2s + 1) scalars on S 2 , computed earlier in (I.55), with the same ν = m 2 2 − η s as the bulk spin-s field, which according to (4.2) means η s = (s − 1 2 ) 2 . The corresponding contribution to the renormalized energy density is ρ caution is in order then if we wish to apply (5.19) or (7.19)-(7.20) to compute log Z (1) . The formula (5.19) for Z (1) PI depends on γ = 8πG N /A d−1 , gauge-algebraically defined by (5.16) and various normalization conventions. We picked these such that in pure Einstein gravity, γ = 8πG/A( 0 ), 0 = d(d − 1)/2Λ, with G and Λ read off from the gravitational Lagrangian. However this expression of γ in terms of Lagrangian parameters will in general be modified in the presence of higher-order curvature terms. This is clear from the discussion in I.1.3, and (I. 19) in particular. Since γ 0 is field-redefinition invariant, and since after transforming to a pure Einstein frame we have γ 0 = 2π/S (0) , with the right hand side also invariant, we have in general (for Einstein + perturbative higher-order curvature corrections) γ = 2π/S (0) .
(I.60) From (7.19) we thus get (ignoring the phase) The constant K 4 could be absorbed into λ R 2 at this level. Below, in (I.70), we will give it relative meaning however, by considering saddles different from the round S 4 .

I.5 Classical and quantum observables
Here we address question 3 in our list below (8.2). To answer this, we need "observables" of the Λ > 0 Euclidean low-energy effective field theory probing independent gravitational couplings (for simplicity we restrict ourselves to purely gravitational theories here), i.e. diffeomorphism and field-redefinition invariant quantities, analogous to scattering amplitudes in asymptotically flat space. For this to be similarly useful, an infinite amount of unambiguous data should be extractable, at least in principle, from these observables.
As discussed above, S (0) = log Z (0) = −S E [g 0 ] invariantly probes the dimensionless coupling given by d−1 0 /G ∝ 1/GΛ (d−1)/2 in Einstein frame. The obvious tree-level invariants probing different couplings in the gravitational low-energy effective field theory are then the analogous S

3D
In d = 2, the Weyl tensor vanishes identically, so higher-order curvature invariants involve R µν only and can be removed from the action by a field redefinition in large-0 perturbation theory, reducing it to pure Einstein form in general. As a result, S (0) is the only independent invariant in pure 3D gravity, all g M are Einstein, and the S The corrections terms in the expansion are all nonlocal (no odd powers of 0 ), and the coefficients provide an unambiguous, infinite data set.

odd D ≥ 5
In 5D gravity, there are infinitely many independent coupling constants. There are also infinitely many different Λ > 0 Einstein metrics on S 5 , including a discrete infinity of Böhm metrics with SO(3) × SO(3) symmetry [142] amenable to detailed numerical analysis [143], and 68 Sasaki-Einstein families with moduli spaces up to real dimension 10 [144]. Unlike the round S 5 , these are not conformally flat, and thus, unlike S (0) , the corresponding S (0) M will pick up couplings such as the Weyl-squared coupling λ C 2 in (I.12). It is plausible that this set of known Einstein metrics (perturbed by small higher-order corrections to the Einstein equations of motion at finite 0 ) more than suffices to invariantly probe all independent couplings of the gravitational action, delivering moreover infinitely many quantum observables S M , providing an infinity of unambiguous low-energy effective field theory data to any order in perturbation theory, without ever leaving the sphere -at least in principle.
The landscape of known Λ > 0 Einstein metrics on odd-dimensional spheres becomes increasingly vast as the dimension grows, with double-exponentially growing numbers [144].
For example there are at least 8610 families of Sasaki-Einstein manifolds on S 7 , spanning all 28 diffeomorphism classes, with the standard class admitting a 82-dimensional family, and there are at least 10 828 distinct families of Einstein metrics on S 25 , featuring moduli spaces of dimension greater than 10 833 .

4D
4D gravity likewise has infinitely many independent coupling constants. It is not known if S 4 has another Einstein metric besides the round sphere. In fact the list of 4D topologies known to admit Λ > 0 Einstein metrics is rather limited [153]: S 4 , S 2 × S 2 , CP 2 , and the connected sums CP 2 #kCP 2 , 1 ≤ k ≤ 8. However for k ≥ 5 these have a moduli space of nonzero dimension [154,155], which might suffice to probe all couplings. (The moduli space would presumably be lifted at sufficiently high order in the l s expansion upon turning on higher-order curvature perturbations.) Below we illustrate in explicit detail how the Weyl-squared coupling can be extracted from suitable linear combinations of pairs of S The Weyl-squared coupling λ C 2 in S E [g] = (I.12) + · · · is invisible to S (0) (I.63) but it is picked up by S where D M is the number of Killing vectors of M : D S 4 = 10, D S 2 ×S 2 = 6, D CP 2 = 8, and α M can be obtained from the local expressions in [14]: α S 4 = − 571 45 , α S 2 ×S 2 = − 98 45 , α CP 2 = − 359 60 . Computing the constants K M generalizing K S 4 given in (I.62) would require more work.
Moreover, computing them for one or two saddles would provide no unambiguous information because they may be absorbed into λ C 2 and λ R 2 . However, since there only two undetermined coupling constants at this order, computing them for all three does provide unambiguous information, extracted by the quantum counterpart of (I.68): S cub ≡ 16 S CP 2 − 9 S S 2 ×S 2 − 6 S S 4 = −7 log S (0) + 16 K CP 2 − 9 K S 2 ×S 2 − 6 K S 4 + · · · (I.70) The log( 0 /L) terms had to cancel in this linear combination because the tree-level parts at this order cancel by design and L∂ L S cub = 0.