PI}=\frac{\pi\ell^{3}}{2\epsilon^{3}}-\frac{\pi\nu^{2}\ell}{4\epsilon}\,\,+\frac{\pi\nu^{3}}{6}-\sum_{k=0}^{2}\frac{\nu^{k}}{k!}\,\frac{{\rm
Li}_{3-k}(e^{-2\pi\nu})}{(2\pi)^{2-k}}\,.$ (143)

The corresponding Euclidean energy $U_{\rm PI}=\rho_{\rm PI}\,\pi\ell^{2}$
(135) is given by

$\displaystyle 2\pi\ell\,U_{\rm PI}=V\rho_{\rm
PI}=-\frac{\pi\ell^{3}}{2\epsilon^{3}}+\frac{\pi(\nu^{2}+\tfrac{2}{3}\eta)\ell}{4\epsilon}-\frac{\pi}{6}(\nu^{2}+\eta)\nu\coth(\pi\nu)$
(144)

where $V={\rm vol}(S^{3}_{\ell})=2\pi^{2}\ell^{3}$. For minimal coupling
$\xi=0$ (i.e. $\eta=1$), $U^{\rm fin}_{\rm PI}$ equals $U^{\rm fin}_{\rm
bulk}$ (49), but not for $\xi\neq 0$. For general $d,\xi$, $U_{\rm PI}^{\rm
fin}$ is given by (48) with the overall factor $m^{2}$ the mass $m^{2}$
appearing in the action rather than $m_{\rm eff}^{2}$, in agreement with
Dowker:1975tf ; Candelas:1975du  or (6.178)-(6.180) of birrell_davies_1982 .
The entropy $S_{\rm PI}=\log Z_{\rm PI}+2\pi\ell\,U_{\rm PI}$ (136) is

$\displaystyle S_{\rm
PI}=\frac{\pi\eta}{6}\Bigl{(}\frac{\ell}{\epsilon}-\nu\coth(\pi\nu)\Bigr{)}-\sum_{k=0}^{3}\frac{\nu^{k}}{k!}\,\frac{{\rm
Li}_{3-k}(e^{-2\pi\nu})}{(2\pi)^{2-k}}\,\,,$ (145)

where we used $\coth(\pi\nu)=1+2\,{\rm Li}_{0}(e^{-2\pi\nu})$ (51). Since
$Z_{\rm PI}=Z_{\rm bulk}$ in general for scalars and $U_{\rm PI}=U_{\rm bulk}$
for minimally coupled scalars, $S_{\rm PI}=S_{\rm bulk}$ for minimally coupled
scalars. Indeed, after conversion to Pauli-Villars regularization, (145)
equals (52) if $\eta=1$. As a check on the results, the first law $dS_{\rm
PI}=Vd\rho_{\rm PI}$ (137) can be verified explicitly.

In the $m\ell\to\infty$ limit, $S_{\rm
PI}\to\frac{\pi}{6}\eta(\epsilon^{-1}-m)\ell$, reproducing the well-known
scalar one-loop Rindler entropy correction computed by a Euclidean path
integral on a conical geometry Susskind:1994sm ; Callan:1994py ; Kabat:1995eq
; Kabat:1995jq ; Larsen:1995ax ; Solodukhin:2011gn . Note that $S_{\rm PI}<0$
when $\eta<0$. Indeed as reviewed in the Rindler context in appendix E.5,
$S_{\rm PI}$ 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 $\ell$, preserving manifest $SO(d+2)$ symmetry, and allowing
straightforward exact computation of the Euclidean entropy directly from
$Z_{\rm PI}(\ell)$, for arbitrary field content.

#### Free 3D massive spin $s$

Recall from (90) that for a $d=2$ massive spin-$s\geq 1$ field of mass $m$,
the bulk part of $\log Z_{\rm PI}$ is twice that of a $d=2$ scalar (143) with
$\nu=\sqrt{(m\ell)^{2}-\eta}$, $\eta=(s-1)^{2}$, while the edge part is
$-s^{2}$ times that of a $d=0$ scalar, as in (139), with the important
difference however that $\nu=\sqrt{(m\ell)^{2}-\eta}$ instead of $\nu=m\ell$.
Another important difference with (139) is that in the case at hand, (135)
stipulates $V\rho_{\rm PI}=2\pi\ell\,U_{\rm
PI}=-\frac{1}{d+1}\ell\partial_{\ell}\log Z_{\rm PI}$ with $d=2$ instead of
$d=0$. As a result, for the bulk contribution, we can just copy the scalar
formulae (144) and (145) for $U_{\rm PI}$ and $S_{\rm PI}$ setting
$\eta=(s-1)^{2}$, while for the edge contribution we get something rather
different from the harmonic oscillator energy and entropy (139):

$\displaystyle V\rho_{\rm PI}$
$\displaystyle=2\times(\ref{UPIscalar})-s^{2}\bigl{(}-\tfrac{\pi}{3}\tfrac{1}{\epsilon}\ell+\tfrac{\pi}{3}\bigl{(}\nu^{2}+\eta\bigr{)}\nu^{-1}\coth(\pi\nu)\bigr{)}$
(146) $\displaystyle S_{\rm PI}$
$\displaystyle=2\times(\ref{SPIscalar})-s^{2}\bigl{(}\tfrac{2\pi}{3}(\tfrac{1}{\epsilon}\ell-\nu)+\tfrac{\pi}{3}\eta\nu^{-1}\coth(\pi\nu)+{\rm
Li_{1}}(e^{-2\pi\nu})+\tfrac{2\pi}{3}\nu\,{\rm Li}_{0}(e^{-2\pi\nu})\bigr{)}$
(147)

The edge contribution renders $S_{\rm PI}$ negative for all $\ell$. In
particular, in the $m\ell\to\infty$ limit, $S_{\rm
PI}\to\tfrac{\pi}{3}\bigl{(}(s-1)^{2}-2s^{2}\bigr{)}\,\bigl{(}\epsilon^{-1}-m\bigr{)}\ell\to-\infty$:
although the bulk part gives a large positive contribution for $s\geq 2$, the
edge part gives an even larger negative contribution. Going in the opposite
direction, to smaller $m\ell$, we hit the $d=2$, $s\geq 1$ unitarity bound at
$\nu=0$, i.e. at $m\ell=\sqrt{\eta}=s-1$. Approaching this bound, the bulk
contribution remains finite, while the edge part diverges, again negatively.
For $s=1$, $S_{\rm PI}\to\log(m\ell)$, due to the ${\rm Li}_{1}(e^{-2\pi\nu})$
term, while for $s\geq 2$, more dramatically, we get a pole $S_{\rm
PI}\to-\frac{s^{2}(s-1)}{6}\bigl{(}m\ell-(s-1)\bigr{)}^{-1}$, due to the
$\eta\nu^{-1}\coth(\pi\nu)$ term. Below the unitarity bound, i.e. when
$\ell<(s-1)/m$, $S_{\rm PI}$ becomes complex. To be consistent as a
perturbative low-energy effective field theory valid down to some length scale
$l_{s}$, massive spin-$s\geq 2$ particles on dS3 must satisfy
$m^{2}>(s-1)^{2}/l_{s}^{2}$.

#### Massless spin 2

From the results and examples in section 5.3, $\log Z_{\rm PI}^{(1)}=\log
Z_{\rm PI,div}^{(1)}+\log Z_{\rm PI,fin}^{(1)}-\frac{(d+3)\pi}{2}\,i$,

$\displaystyle\log Z_{\rm
PI,fin}^{(1)}(\ell)=-\frac{D_{d}}{2}\log\frac{A(\ell)}{4G_{\rm
N}}+\alpha^{(2)}_{d+1}\log\frac{\ell}{L}+K_{d+1}$ (148)

$D_{d}=\dim{\rm so}(d+2)=\frac{(d+2)(d+1)}{2}$,
$A(\ell)=\Omega_{d-1}\ell^{{d-1}}$, $\alpha^{(2)}_{d+1}=0$ for even $d$ and
given by (116) 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 (120):

$\begin{array}[]{l|l|l}d&\log Z_{\rm PI,div}^{(1)}&\log Z_{\rm
PI,fin}^{(1)}\\\ \hline\cr
2&0-\frac{9\pi}{2}\frac{1}{\epsilon}\ell&-3\log(\frac{\pi}{2G_{\rm
N}}\ell)+5\log(2\pi)\\\
3&\frac{8}{3}\frac{1}{\epsilon^{4}}\ell^{4}-\frac{32}{3}\frac{1}{\epsilon^{2}}\ell^{2}-\frac{571}{45}\log(\frac{2e^{-\gamma}}{\epsilon}L)&-5\log(\frac{\pi}{G_{\rm
N}}\ell^{2})-\frac{571}{45}\log(\frac{1}{L}\ell)-\log(\frac{8\pi}{3})+\frac{715}{48}-\frac{47\,\zeta^{\prime}(-1)}{3}+\frac{2\,\zeta^{\prime}(-3)}{3}\\\
4&\frac{15\pi}{8}\frac{1}{\epsilon^{5}}\ell^{5}-\frac{65\pi}{24}\frac{1}{\epsilon^{3}}\ell^{3}-\frac{105\pi}{16}\frac{1}{\epsilon}\ell&-\frac{15}{2}\log(\frac{\pi^{2}}{2G_{\rm
N}}\ell^{3})+\log(12)+\frac{27}{2}\log(2\pi)+\frac{65\,\zeta(3)}{48\,\pi^{2}}+\frac{5\,\zeta(5)}{16\,\pi^{4}}\end{array}$
(149)

The one-loop energy and entropy (135)-(136) are split accordingly. The finite
parts are

$\displaystyle S_{\rm PI,fin}^{(1)}=\log Z_{\rm PI,fin}^{(1)}+V\rho_{\rm
fin}^{(1)}\,,\qquad V\rho_{\rm
fin}^{(1)}=\tfrac{1}{2}\tfrac{d-1}{d+1}\,D_{d}-\tfrac{1}{d+1}\alpha_{d+1}^{(2)}\,,$
(150)

where as always $2\pi\ell\,U=V\rho$ with $V=\Omega_{d+1}\ell^{d+1}$. For
$d=2,3,4$:

$\begin{array}[]{l|l|l|l}d&V\rho_{\rm div}^{(1)}&V\rho_{\rm fin}^{(1)}&S_{\rm
PI,div}^{(1)}\\\ \hline\cr
2&0+\frac{3\pi}{2}\frac{1}{\epsilon}\ell&1&-3\pi\frac{1}{\epsilon}\ell\\\
3&-\frac{8}{3}\frac{1}{\epsilon^{4}}\ell^{4}+\frac{16}{3}\frac{1}{\epsilon^{2}}\ell^{2}&\frac{5}{2}+\frac{571}{180}&-\frac{16}{3}\frac{1}{\epsilon^{2}}\ell^{2}-\frac{571}{45}\log(\frac{2e^{-\gamma}}{\epsilon}L)\\\
4&-\frac{15\pi}{8}\frac{1}{\epsilon^{5}}\ell^{5}+\frac{13\pi}{8}\frac{1}{\epsilon^{3}}\ell^{3}+\frac{21\pi}{16}\frac{1}{\epsilon}\ell&\frac{9}{2}&-\frac{13\pi}{12}\frac{1}{\epsilon^{3}}\ell^{3}-\frac{21\pi}{4}\frac{1}{\epsilon}\ell\end{array}$
(151)

Like their quasicanonical bulk counterparts, the Euclidean quantities obtained
here are UV-divergent, 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.

##  8 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 $\ell$. 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 $Z_{\rm PI}$,
$\rho_{\rm PI}$ and $S_{\rm PI}$ by ${\cal Z}$, $\varrho$ and ${\cal S}$ in
this case:

$\displaystyle{\cal Z}=\mbox{\Large$\int$}{\cal
D}g\,\cdots\,e^{-S_{E}[g,\ldots]}\,,\qquad S_{E}[g,\ldots]=\frac{1}{8\pi
G}\int\\!\\!\sqrt{g}\,\bigl{(}\Lambda-\tfrac{1}{2}R+\cdots\bigr{)}\,.$ (152)

The geometry itself being dynamical, we have $\partial_{\ell}{\cal Z}=0$, so
(135)-(136) reproduce (1):

$\displaystyle\varrho=0\,,\quad{\cal S}=\log{\cal Z}\,,$ (153)

We will assume $d\geq 2$, but it is instructive to first consider $d=0$, i.e.
1D quantum gravity coupled to quantum mechanics on a circle. Then ${\cal
Z}=\int\frac{d\beta}{2\beta}\,{\rm Tr}\,e^{-\beta H}$, where $\beta$ 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 Gibbons:1978ac  implicit in (153), we pick an integration
contour $\beta=2\pi\ell+iy$ with $y\in{\mathbb{R}}$ and $\ell>0$ the
background circle radius. Then ${\cal Z}=\pi i\,{\cal N}(0)$ where ${\cal
N}(E)$ is the number of states with $H<E$. This being $\ell$-independent
implies $\varrho=0$. A general definition of microcanonical entropy is $S_{\rm
mic}(E)=\log{\cal N}(E)$. Thus, modulo the content-independent $\pi i$ factor
in ${\cal Z}$, ${\cal S}=\log{\cal Z}$ 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 $d\geq 2$
and $\Lambda\to 0$, the path integral has a semiclassical expansion about a
round sphere saddle or radius $\ell_{0}\propto 1/\sqrt{\Lambda}$, and ${\cal
S}$ is dominated by the leading tree-level horizon entropy (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 $S^{1}\times S^{d-1}$ 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 ${\cal S}=\log{\cal Z}$ 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 ${\cal Z}$ is obtained by expanding the action to
quadratic order about its sphere saddle. The Gaussian $Z_{\rm PI}^{(1)}$ was
computed in previous sections. Locality and dimensional analysis imply that
one-loop divergences are $\propto\int R^{n}$ with $2n\leq d+1$. Picking
counterterms canceling all (divergent and finite) local contributions of this
type in the limit $\ell_{0}\propto 1/\sqrt{\Lambda}\to\infty$, we get a well-
renormalized ${\cal S}=\log{\cal Z}$ to this order. Proceeding along these
lines would be the most straightforward path to the computational objectives
of this section. However, when pondering comparisons to microscopic models,
one is naturally led to wondering what the actual physics content is of what
has been computed. This in turn leads to small puzzles and bigger questions,
such as:

  1. 1.

A natural guess would have been that the one-loop correction to the entropy
${\cal S}$ is given by a renormalized version of the Euclidean entropy $S_{\rm
PI}^{(1)}$ (136). However (153) says it is given by a renormalized version of
the free energy $\log Z_{\rm PI}^{(1)}$. In the examples given earlier, these
two look rather different. Can these considerations be reconciled?

  2. 2.

Besides local UV contributions absorbed into renormalized coupling constants
determining the tree-level radius $\ell_{0}$, there will be nonlocal IR vacuum
energy contributions (pictorially Hawking radiation in equilibrium with the
horizon), shifting the radius from $\ell_{0}$ to $\bar{\ell}$ by gravitational
backreaction. The effect would be small, $\bar{\ell}=\ell_{0}+O(G)$, but since
the leading-order horizon entropy is $S(\ell)\propto\ell^{d-1}/G$, we have
$S(\bar{\ell})=S(\ell_{0})+O(1)$, a shift at the one-loop order of interest.
The horizon entropy term in (153) is ${\cal S}^{(0)}=S(\ell_{0})$, apparently
not taking this shift into account. Can these considerations be reconciled?

  3. 3.

At any order in the large-$\ell_{0}$ perturbative expansion, UV-divergences
can be absorbed into a renormalization of a finite number of renormalized
coupling constants, but for the result to be physically meaningful, these must
be defined in terms of low-energy physical “observables”, invariant under
diffeomorphisms and local field redefinitions. In asymptotically flat space,
one can use scattering amplitudes for this purpose. These are unavailable in
the case at hand. What replaces them?

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 $\Gamma(V)=-\log Z(V)$ for the volume, the
Legendre transform of the off-shell entropy/moment-generating function
$S(\rho)$:161616Non-metric fields in the path integral are left implicit. Note
“off-shell” = on-shell for c.c. $\Lambda^{\prime}=\Lambda-8\pi G\,\rho$.

$\displaystyle S(\rho)\equiv\log\mbox{\Large$\int$}{\cal
D}g\,e^{-S_{E}[g]+\rho\int\\!\sqrt{g}}\,,\qquad\log Z(V)\equiv
S-V\rho\,,\qquad
V=\partial_{\rho}S=\bigl{\langle}\,\mbox{$\int\\!\sqrt{g}$}\,\bigr{\rangle}_{\rho}\,.$
(154)

At large $V$, the geometry semiclassically fluctuates about a round sphere.
Parametrizing the mean volume $V$ by a corresponding mean radius $\ell$ as
$V(\ell)\equiv\Omega_{d+1}\ell^{d+1}$, we have

$\displaystyle
Z(\ell)=\mbox{\Large$\int$}_{\\!\\!\\!\text{tree}}\,d\rho\,\mbox{\Large$\int$}{\cal
D}g\,e^{-S_{E}[g]+\rho(\int\\!\sqrt{g}-V(\ell))}\,,$ (155)

where $\int_{\text{tree}}d\rho$ means saddle point evaluation, i.e.
extremization. The Legendre transform (154) is the same as (137), so we get
thermodynamic relations of the same form as (135)-(137):

$\displaystyle dS=Vd\rho\,,\quad d\log
Z=-\rho\,dV\,,\qquad\rho=-\tfrac{1}{d+1}\ell\partial_{\ell}\log
Z\,/\,V\,,\quad S=\bigl{(}1-\tfrac{1}{d+1}\ell\partial_{\ell}\bigr{)}\log
Z\,.$ (156)

On-shell quantities are obtained at $\rho=0$, i.e. at the minimum $\bar{\ell}$
of the free energy $-\log Z(\ell)$:

$\varrho=\rho(\bar{\ell})=0\,,\qquad{\cal S}=S(\bar{\ell})=\log
Z(\bar{\ell})\,,\qquad\bigl{\langle}\,\mbox{$\int\\!\sqrt{g}$}\,\bigr{\rangle}=\Omega_{d+1}\bar{\ell}^{d+1}\,.$
(157)

Tree level At tree level (155) evaluates to

$\displaystyle\log Z^{(0)}(\ell)=-S_{E}[g_{\ell}]\,,\qquad\mbox{$g_{\ell}$ =
round $S^{d+1}$ metric of radius $\ell$}\,,$ (158)

readily evaluated for any action using
$R_{\mu\nu\rho\sigma}=(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho})/\ell^{2}$,
taking the general form

$\displaystyle\log Z^{(0)}=\frac{\Omega_{d+1}\ell^{d+1}}{8\pi
G}\bigl{(}-\Lambda+\tfrac{d(d+1)}{2}\,\ell^{-2}+z_{1}\,l_{s}^{2}\,\ell^{-4}+z_{2}\,l_{s}^{4}\,\ell^{-6}+\cdots\bigr{)}\,.$
(159)

The $z_{n}$ are $R^{n+1}$ coupling constants and $l_{s}\ll\ell$ is the length
scale of UV-completing physics. The off-shell entropy and energy density are
obtained from $\log Z^{(0)}$ as in (156).

$S^{(0)}=\frac{\Omega_{d-1}\ell^{d-1}}{4G}\bigl{(}1+s_{1}\,l_{s}^{2}\,\ell^{-2}+\cdots\bigr{)},\qquad\rho^{(0)}=\frac{1}{8\pi
G}\bigl{(}\Lambda-\tfrac{d(d-1)}{2}\,\ell^{-2}+\rho_{1}\,l_{s}^{2}\,\ell^{-4}+\cdots\bigr{)}$
(160)

where $s_{n},\rho_{n}\propto z_{n}$ and we used
$\Omega_{d+1}=\frac{2\pi}{d}\Omega_{d-1}$. The on-shell entropy and radius are
given by

$\displaystyle{\cal
S}^{(0)}=S^{(0)}(\ell_{0})\,,\qquad\rho^{(0)}(\ell_{0})=0\,,$ (161)

either solved perturbatively for $\ell_{0}(\Lambda)$ or, more conveniently,
viewed as parametrizing $\Lambda(\ell_{0})$.

One loop  The one-loop order, (155) is a by construction tadpole-free Gaussian
path integral, (500):

$\displaystyle\log Z=\log Z^{(0)}+\log Z^{(1)}\,,\qquad\log Z^{(1)}=\log
Z^{(1)}_{\rm PI}+\log Z_{\rm ct}\,,$ (162)

with $Z_{\rm PI}^{(1)}$ as computed in sections 4-5 and $\log Z_{\rm
ct}(\ell)=-S_{E,\rm ct}[g_{\ell}]$ a polynomial counterterm. We define
renormalized coupling constants as the coefficients of the $\ell^{d+1-2n}$
terms in the $\ell\to\infty$ expansion of $\log Z$, and fix $\log Z_{\rm ct}$
by equating tree-level and renormalized coefficients of the polynomial part,
which amounts to the renormalization condition

$\displaystyle\lim_{\ell\to\infty}\partial_{\ell}\log Z^{(1)}=0\,,$ (163)

in even $d+1$ supplemented by $\log Z_{\rm
ct}(0)\equiv-\alpha_{d+1}\log(2e^{-\gamma}L/\epsilon)$, implying
$L\partial_{L}\log Z^{(0)}=\alpha_{d+1}$.

Example: 3D Einstein gravity + minimally coupled scalar (I.4.1), putting
$\nu\equiv\sqrt{m^{2}\ell^{2}-1}$,

$\displaystyle\log
Z^{(1)}=-3\log\frac{2\pi\ell}{4G}+5\log(2\pi)\,\,-\,\,\sum_{k=0}^{2}\frac{\nu^{k}}{k!}\,\frac{{\rm
Li}_{3-k}(e^{-2\pi\nu})}{(2\pi)^{2-k}}+\frac{\pi\nu^{3}}{6}\,\,-\,\,\frac{\pi
m^{3}\ell^{3}}{6}+\frac{\pi m\ell}{4}\,.$ (164)

The last two terms are counterterms. The first two are nonlocal graviton
terms. The scalar part is $O(1/m\ell)$ for $m\ell\gg 1$ but goes nonlocal at
$m\ell\sim 1$, approaching $-\log(m\ell)$ for $m\ell\ll 1$.

Defining $\rho^{(1)}$ and $S^{(1)}$ from $\log Z^{(1)}$ as in (156), and the
quantum on-shell $\bar{\ell}=\ell_{0}+O(G)$ as in (157), the quantum entropy
can be expressed in two equivalent ways, (507)-(508):

$\displaystyle{A}:\,\,{\cal
S}=S^{(0)}(\bar{\ell})+S^{(1)}(\bar{\ell})+\cdots\,,\qquad{B}:\,\,{\cal
S}=S^{(0)}(\ell_{0})+\log Z^{(1)}(\ell_{0})+\cdots$ (165)

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 (153)-based (${B}$) expectations. To make this physically obvious,
consider the quantum static patch as two subsystems, geometry (horizon) +
quantum fluctuations (radiation), with total energy
$\propto\rho=\rho^{(0)}+\rho^{(1)}=0$. If $\rho^{(0)}=0$, the horizon entropy
is $S^{(0)}(\ell_{0})$. But here we have $\rho=0$, so the horizon entropy is
actually $S^{(0)}(\bar{\ell})=S^{(0)}(\ell_{0})+\delta S^{(0)}$, where by the
first law (156), $\delta S^{(0)}=V\delta\rho^{(0)}=-V\rho^{(1)}$. Adding the
radiation entropy $S^{(1)}$ and recalling $\log Z^{(1)}=S^{(1)}-V\rho^{(1)}$
yields ${\cal S}={A}={B}$. Thus ${A}={B}$ is just the usual small+large =
system+reservoir approximation, the horizon being the reservoir, and the
Boltzmann factor $e^{-V\rho^{(1)}}=e^{-\beta U^{(1)}}$ 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. ${\rm
Li}(e^{-2\pi\nu})\sim e^{-\beta m}$ for $m\ell\gg 1$ in (164), but might
incorrectly convey a presumption of positivity of $\rho^{(1)}$ and $S^{(1)}$.
Though positive for minimally coupled scalars (fig. 21), they are in fact
negative for higher spins (figs. 22, 23), due to edge and group volume
contributions. Moreover, although the negative-energy backreaction causes the
horizon to grow, partially compensating the negative $S^{(1)}$ by a positive
$\delta S^{(0)}=-V\rho^{(1)}$, the former still wins: ${\cal
S}^{(1)}\equiv{\cal S}-{\cal S}^{(0)}=S^{(1)}-V\rho^{(1)}=\log Z^{(1)}<0$.

Computational recipe and examples For practical purposes, (B) is the more
useful expression in (165). Together with (161) computing ${\cal S}^{(0)}$,
the exact results for $Z^{(1)}_{\rm PI}$ obtained in previous sections (with
$\gamma_{0}=\sqrt{2\pi/{\cal S}^{(0)}}$, see (167) below), and the
renormalization prescription outlined above, it immediately gives

$\displaystyle{\cal S}={\cal S}^{(0)}+{\cal S}^{(1)}+\cdots\,,\qquad{\cal
S}^{(0)}=S^{(0)}(\ell_{0})\,,\qquad{\cal S}^{(1)}=\log Z^{(1)}(\ell_{0})$
(166)

in terms of the renormalized coupling constants, for general effective field
theories of gravity coupled to arbitrary matter and gauge fields.

For 3D gravity, this gives ${\cal S}={\cal S}^{(0)}-3\log{\cal
S}^{(0)}+5\log(2\pi)+O(1/{\cal S}^{(0)})$. We work out and plot several other
concrete examples in appendix I.4: 3D Einstein gravity + scalar (I.4.1, fig.
21), 3D massive spin $s$ (I.4.2, fig. 22), 2D scalar (I.4.3), 4D massive spin
$s$ (I.4.4, fig. 23), and 3D,4D,5D gravity (including higher-order curvature
corrections) (I.4.5). Table 12 in the introduction lists a few more sample
results.

Local field redefinitions, invariant coupling constants and physical
observables

Although the higher-order curvature corrections to the tree-level dS entropy
${\cal S}^{(0)}=S^{(0)}(\ell_{0})$ (160) seem superficially similar to
curvature corrections to the entropy of black holes in asymptotically flat
space Wald:1993nt ; Iyer:1994ys , there are no charges or other asymptotic
observables available here to endow them with physical meaning. Indeed, they
have no intrinsic low-energy physical meaning at all, as they can be removed
order by order in the $l_{s}/\ell$ expansion by a metric field redefinition,
bringing the entropy to pure Einstein form (2). In $Z^{(0)}(\ell)$ (159), this
amounts to setting all $z_{n}\equiv 0$ by a redefinition
$\ell\to\ell\sum_{n}c_{n}\ell^{-2n}$ (490). The value of ${\cal
S}^{(0)}=\max_{\ell\gg l_{s}}\log Z^{(0)}(\ell)$ remains of course unchanged,
providing the unique field-redefinition invariant combination of the coupling
constants $G,\Lambda\mbox{(or $\ell_{0}$)},z_{1},z_{2},\ldots$.

Related to this, as discussed in I.4.5, caution must be exercised when porting
the one-loop graviton contribution in (112) or (148): $G_{\rm N}$ appearing in
$\gamma_{0}=\sqrt{8\pi G_{\rm N}/A}$ is the algebraically defined Newton
constant (109), as opposed to $G$ defined by the Ricci scalar coefficient
$\frac{1}{8\pi G}$ 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

$\displaystyle\gamma_{0}=\sqrt{2\pi/{\cal S}^{(0)}}\,.$ (167)

Since $\log{\cal S}^{(0)}=\log\frac{A}{4G}+\log(1+O(l_{s}^{2}/\ell_{0}^{2}))$,
this distinction matters only at $O(l_{s}^{2}/\ell_{0}^{2})$, however.

In $d=2$, ${\cal S}^{(0)}$ is in fact the only invariant gravitational
coupling: because the Weyl tensor vanishes identically, any 3D parity-
invariant effective gravitational action can be brought to Einstein form by a
field redefinition. In the Chern-Simons formulation of H.2, ${\cal
S}^{(0)}=2\pi\kappa$. In $d\geq 3$, the Weyl tensor vanishes on the sphere,
but not identically. As a result, there are coupling constants not picked up
by the sphere’s ${\cal S}^{(0)}=-S_{E}[g_{\ell_{0}}]$. Analogous ${\cal
S}^{(0)}_{M}\equiv-S_{E}[g_{M}]$ for different saddle geometries $g_{M}$,
approaching Einstein metrics in the limit $\Lambda\propto\ell_{0}^{-2}\to 0$,
can be used instead to probe them, and analogous ${\cal S}_{M}\equiv\log{\cal
Z}_{M}$ expanded about $g_{M}$ provide quantum observables. Section I.5
provides a few more details, and illustrates extraction of unambiguous linear
combinations of the 4D one-loop correction for 3 different $M$.

This provides the general picture we have in mind as the answer, in principle,
to question 3 on our list below (153): the tree-level ${\cal S}^{(0)}_{M}$ are
the analog of tree-level scattering amplitudes, and the analog of quantum
scattering amplitudes are the quantum ${\cal S}_{M}$.

Constraints on microscopic models For pure 3D gravity ${\cal
S}^{(0)}=\frac{2\pi}{4G}\bigl{(}\ell_{0}+s_{1}\ell_{0}^{-1}+s_{2}\,\ell_{0}^{-3}+\cdots\bigr{)}$,
and to one-loop order we have (531):

$\displaystyle{\cal S}={\cal S}^{(0)}-3\log{\cal
S}^{(0)}+5\log(2\pi)+\cdots\,.$ (168)

Granting171717This does not affect the 1-loop based conclusions below, but
does affect the $c_{n}$. One could leave $l$ general. (442) with $l=0$ gives
the all-loop expansion of pure 3D gravity, taking into account $G\equiv SO(4)$
here while $G\equiv SU(2)\times SU(2)$ there, to all-loop order,

${\cal S}={\cal S}_{0}+\log\left|\sqrt{\tfrac{4}{2+i\,{\cal
S}_{0}/{2\pi}}}\,\sin\bigl{(}\tfrac{\pi}{2+i\,{{\cal
S}_{0}}/{2\pi}}\bigr{)}\right|^{2}={\cal S}_{0}-3\log{\cal
S}_{0}+5\log(2\pi)+\mbox{$\sum_{n}$}\,c_{n}\,{\cal S}_{0}^{-2n}$ (169)

where ${\cal S}_{0}\equiv{\cal S}^{(0)}$ to declutter notation. Note all
quantum corrections are strictly nonlocal, i.e. no odd powers of $\ell_{0}$
appear, reflected in the absence of odd powers of $1/{\cal S}_{0}$.

Though outside the scope of this paper, let us illustrate how such results may
be used to constrain microscopic models identifying large-$\ell_{0}$ and
large-$N$ expansions in some way. Say a modeler posits a model consisting of
$2N$ spins $\sigma_{i}=\pm 1$ with $H\equiv\sum_{i}\sigma_{i}=0$. The
microscopic entropy is $S_{\rm mic}=\log{2N\choose N}=2\log 2\cdot
N-\frac{1}{2}\log(\pi N)+\sum_{n}c^{\prime}_{n}N^{1-2n}$. There is a unique
identification of ${\cal S}_{0}$ bringing this in a form with the same
analytic/locality structure as (169), to wit, ${\cal S}_{0}=\log 4\cdot
N+\sum_{n}c^{\prime}_{n}N^{1-2n}$, resulting in

$S_{\rm mic}\,=\,{\cal S}_{0}-\tfrac{1}{2}\log{\cal
S}_{0}+\log(\tfrac{\pi}{2\log
2})+\mbox{$\sum_{n}$}\,c_{n}^{\prime\prime}\,{\cal S}_{0}^{-2n}\,,$ (170)

where $c_{1}^{\prime\prime}=-\frac{1}{8}\log
2,c_{2}^{\prime\prime}=\frac{3}{64}(\log 2)^{2}+\frac{1}{48}(\log
2)^{3},\ldots$, fully failing to match (169), starting at one loop. The model
is ruled out.

A slightly more sophisticated modeler might posit $S_{\rm mic}=\log d(N)$,
where $d(N)$ is the $N$-th level degeneracy of a chiral boson on $S^{1}$. To
leading order $S_{\rm mic}\approx 2\pi\sqrt{N/6}\equiv K$. Beyond, $S_{\rm
mic}=K-a^{\prime}\log K+b^{\prime}+\sum_{n}c_{n}^{\prime}K^{-n}+O(e^{-K/2})$,
where $a^{\prime}=2$, $b^{\prime}=\log(\pi^{2}/6\sqrt{3})$ and
$c_{n}^{\prime}$ given by 10.1112/plms/s2-17.1.75 . Identifying ${\cal
S}_{0}=K+\sum_{n}c^{\prime}_{2n-1}K^{-(2n-1)}$ brings this to the form (169),
yielding $S_{\rm mic}={\cal S}_{0}-a^{\prime}\log{\cal
S}_{0}+b^{\prime}+\sum_{n}c_{n}^{\prime\prime}{\cal S}_{0}^{-2n}+O(e^{-{\cal
S}_{0}/2})$, with $c_{1}^{\prime\prime}=-\frac{5}{2}$,
$c_{2}^{\prime\prime}=\frac{37}{12}$, $\ldots$ — 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 (12). For pure
5D gravity (531),

${\cal S}={\cal S}^{(0)}-\frac{15}{2}\log{\cal
S}^{(0)}+\log(12)+\frac{27}{2}\log(2\pi)+\frac{65\,\zeta(3)}{48\,\pi^{2}}+\frac{5\,\zeta(5)}{16\,\pi^{4}}\,.$
(171)

It would be quite a miracle if a microscopic model managed to match this.

##  9 dS, AdS$\pm$, and conformal higher-spin gravity

Vasiliev higher-spin gravity theories Vasiliev:1990en ; Vasiliev:2003ev ;
Bekaert:2005vh  have infinite spin range and an infinite-dimensional higher-
spin algebra, ${\mathfrak{g}}={\rm hs}({\rm so}(d+2))$, leading to divergences
in the one-loop sphere partition function formula (112) 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.

###  9.1 dS higher-spin gravity

Nonminimal type A Vasiliev gravity on dSd+1 has a tower of massless spin-$s$
fields for all $s\geq 1$ and a $\Delta=d-2$ scalar. We first consider $d=3$.
The total bulk and edge characters are obtained by summing (102) and adding
the scalar, as we did for the bulk part in (63):

$\displaystyle\chi_{\rm
bulk}\,=\,2\cdot\biggl{(}\frac{q^{1/2}+q^{3/2}}{(1-q)^{2}}\biggr{)}^{2}-\frac{q}{(1-q)^{2}}\,,\qquad\chi_{\rm
edge}=2\cdot\biggl{(}\frac{q^{1/2}+q^{3/2}}{(1-q)^{2}}\biggr{)}^{2}\,.$ (172)

Quite remarkably, the bulk and edge contributions almost exactly cancel:

$\displaystyle\chi_{\rm bulk}-\chi_{\rm edge}=-\frac{q}{(1-q)^{2}}\,.$ (173)

For $d=4$ however, we see from (102) 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:

$\displaystyle\chi_{\rm
bulk}=\sum_{s}(2s+1)\cdot\frac{2\,q^{2}}{(1-q)^{4}}\,,\qquad\chi_{\rm
edge}=\sum_{s}\tfrac{1}{6}s(s+1)(2s+1)\cdot\frac{2\,q}{(1-q)^{2}}\,.$ (174)

This pattern persists for all $d\geq 4$, as can be seen from the explicit form
of bulk and edge characters in (359), (388), (390). For any $d$, there is
moreover an infinite-dimensional group volume factor in (112) to make sense
of, involving a divergent factor $(\ell^{d-1}/G_{\rm N})^{\dim G/2}$ and the
volume of an object of unclear mathematical existence Monnier:2014tfa .

Before we continue the discussion of what, if anything, to make of this, we
consider AdS$\pm$ 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.

###  9.2 AdS$\pm$ higher-spin gravity

#### AdS characters for standard and alternate HS boundary conditions

Standard boundary conditions on massless higher spin fields $\varphi$ in
AdSd+1 lead to quantization such that spin-$s$ single-particle states
transform in a UIR of ${\rm so}(2,d)$ with primary dimension
$\Delta_{\varphi}=\Delta_{+}=s+d-2$. Higher-spin Euclidean AdS one-loop
partition functions with these boundary conditions were computed in
Giombi:2013fka ; Giombi:2014iua ; Gunaydin:2016amv ; Giombi:2016pvg ;
Skvortsov:2017ldz . In Giombi:2013yva , the Euclidean one-loop partition
function for alternate boundary conditions ($\Delta_{\varphi}=\Delta_{-}=2-s$)
was considered. In the EAdS$+$ case, the complications listed under (96) are
absent, but for EAdS$-$ close analogs do appear.

EAdS path integrals can be expressed as character integrals Sun:2020ame ;
Basile:2018zoy ; Basile:2018acb , in a form exactly paralleling the formulae
and bulk/edge picture of the present work Sun:2020ame .181818In this picture,
EAdS is viewed as the Wick-rotated AdS-Rindler wedge, with dSd static patch
boundary metric, as in Keeler:2014hba ; Keeler:2016wko . The bulk character is
$\chi\equiv{\rm tr}_{G}\,q^{iH}$, 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 Sun:2020ame ;
Keeler:2014hba ; Keeler:2016wko . The AdS analog of the dS bulk and edge
characters (85) for a massive spin-$s$ field $\varphi$ with
$\Delta_{\varphi}=\Delta_{\pm}$ is Sun:2020ame

$\displaystyle\chi^{\rm AdS\pm}_{\rm bulk,\varphi}\equiv
D_{s}^{d}\,\frac{q^{\Delta_{\pm}}}{(1-q)^{d}}\,,\qquad\chi^{\rm AdS\pm}_{\rm
edge,\varphi}\equiv D_{s-1}^{d+2}\,\frac{q^{\Delta_{\pm}-1}}{(1-q)^{d-2}}\,,$
(175)

where $\Delta_{-}=d-\Delta_{+}$. Thus, as functions of $q$,

$\displaystyle\chi_{\varphi}^{\rm dS}=\chi^{\rm AdS+}_{\varphi}+\chi^{\rm
AdS-}_{\varphi}\,.$ (176)

The AdS analog of (97) for a massless spin-$s$ field $\phi_{s}$ with gauge
parameter field $\xi_{s^{\prime}}$ is

$\displaystyle\hat{\chi}_{s}^{{\rm AdS}\pm}\equiv\chi^{\rm
AdS\pm}_{\phi}-\chi^{\rm AdS\pm}_{\xi}\,,$ (177)

where $\Delta_{\phi,+}=s^{\prime}+d-1$, $\Delta_{\xi,+}=s+d-1$,
$s^{\prime}\equiv s-1$. More explicitly, analogous to (98),

$\displaystyle\hat{\chi}^{\rm AdS+}_{{\rm bulk},s}$
$\displaystyle=\frac{D_{s}^{d}\,q^{s^{\prime}+d-1}-D_{s^{\prime}}^{d}\,q^{s+d-1}}{(1-q)^{d}}\,,$
$\displaystyle\hat{\chi}^{\rm AdS+}_{{\rm edge},s}$
$\displaystyle=\frac{D^{d+2}_{s-1}\,q^{s^{\prime}+d-2}-D^{d+2}_{s^{\prime}-1}\,q^{s+d-2}}{(1-q)^{d-2}}$
(178) $\displaystyle\hat{\chi}^{\rm AdS-}_{{\rm bulk},s}$
$\displaystyle=\frac{D_{s}^{d}\,q^{1-s^{\prime}}-D_{s^{\prime}}^{d}\,q^{1-s}}{(1-q)^{d}}\,,$
$\displaystyle\hat{\chi}^{\rm AdS-}_{{\rm edge},s}$
$\displaystyle=\frac{D_{s-1}^{d+2}\,q^{-s^{\prime}}-D_{s^{\prime}-1}^{d+2}\,q^{-s}}{(1-q)^{d-2}}\,.$
(179)

The presence of non-positive powers of $q$ in $\chi^{\rm AdS-}$ 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 $\chi_{s}$ to be used in the character formulae for EAdS$\pm$ are

$\displaystyle\chi_{s}^{\rm AdS-}=\bigl{[}\hat{\chi}_{s}^{\rm
AdS-}\bigr{]}_{+}\,,\qquad\chi_{s}^{\rm AdS+}=\bigl{[}\hat{\chi}_{s}^{\rm
AdS+}\bigr{]}_{+}=\hat{\chi}_{s}^{\rm AdS+}\,,$ (180)

with $[\hat{\chi}]_{+}$ defined as in (100). The omission of Killing tensor
zeromodes for alternate boundary conditions must be compensated by a a
division by the volume of the residual gauge group $G$ generated by the
Killing tensors. Standard boundary conditions on the other hand kill these
Killing tensor zeromodes: they are not part of the dynamical, fluctuating
degrees of freedom. The group $G$ they generate acts nontrivially on the
Hilbert space as a global symmetry group.

#### AdS$+$

For standard boundary conditions, the character formalism reproduces the
original results of Giombi:2013fka ; Giombi:2014iua ; Gunaydin:2016amv ;
Giombi:2016pvg ; Skvortsov:2017ldz  by two-line computations Sun:2020ame . We
consider some examples:

For nonmimimal type A Vasiliev with $\Delta_{0}=d-2$ scalar boundary
conditions, dual to the free $U(N)$ model, using (178) and the scalar
$\chi_{0}=q^{d-2}/(1-q)^{d}$, the following total bulk and edge characters are
readily obtained:

$\displaystyle\chi_{\rm bulk}^{{\rm AdS}+}=\sum_{s=0}^{\infty}\chi_{{\rm
bulk},s}^{{\rm
AdS}+}=\biggl{(}\frac{q^{\frac{d}{2}-1}+q^{\frac{d}{2}}}{(1-q)^{d-1}}\biggr{)}^{2}\,,\quad\chi_{\rm
edge}^{{\rm AdS}+}=\sum_{s=0}^{\infty}\chi_{{\rm edge},s}^{{\rm
AdS}+}=\biggl{(}\frac{q^{\frac{d}{2}-1}+q^{\frac{d}{2}}}{(1-q)^{d-1}}\biggr{)}^{2}\,.$
(181)

The total bulk character takes the singleton-squared form expected from the
Flato-Fronsdal theorem Flato:1978qz . 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:

$\displaystyle\chi_{\rm tot}^{{\rm AdS}+}=\chi_{\rm bulk}^{{\rm
AdS}+}-\chi_{\rm edge}^{{\rm AdS}+}=0\qquad\Rightarrow\qquad\log Z_{\rm
PI}^{{\rm AdS}+}=0\,.$ (182)

As $Z^{{\rm AdS}+}_{\rm bulk}$ has an Rindler bulk ideal gas interpretation
analogous to the static patch ideal gas of section 2 Sun:2020ame , 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 $t\to t/2$ is
effectively equivalent to the ${\rm so}(2,d)$ singleton character, which is
also the ${\rm so}(1,d)$ character of a conformally coupled ($\nu=i/2$) scalar
on $S^{d}$. Using (74), this means $Z_{\rm PI}^{{\rm AdS}+}$ equals the sphere
partition function on $S^{d}$, immediately implying the $N\to N-1$
interpretation of Giombi:2013fka ; Giombi:2014iua ; Gunaydin:2016amv ;
Giombi:2016pvg ; Skvortsov:2017ldz .

For nonminimal type A with $\Delta_{0}=2$ scalar boundary conditions, dual to
an interacting U(N) CFT, the cancelation is almost exact but not quite:

$\displaystyle\chi_{\rm tot}^{{\rm
AdS}+}=\frac{\sum_{k=2}^{d-3}\,q^{k}}{(1-q)^{d-1}}\,.$ (183)

#### AdS$+$ higher-spin swampland

In the above examples it is apparent that although the spin-summed $\chi_{\rm
bulk}$ has increased effective UV-dimensionality $d^{\rm bulk}_{\rm
eff}=2d-2$, as if we summed KK modes of a compactification manifold of
dimension $d-2$, the edge subtraction collapses this back down to a net
$d_{\rm eff}=d-1$, decreasing the original $d$. Correspondingly, the UV-
divergences of $Z^{(1)}_{\rm PI}$ 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
Sleight:2017pcz . It appears to be true for all AdS$+$ higher spin theories
with a known holographic dual Sun:2020ame , but not for all classically
consistent higher-spin theories. Thus it appears to be some kind of AdS
higher-spin “swampland” criterion:

$\displaystyle\mbox{AdS${}_{d+1}$ HS theory has holographic
dual}\qquad\Rightarrow\qquad d_{\rm eff}=d-1\,.$ (184)

Higher-spin theories violating this criterion do exist. Theories with a tower
of massless spins $s\geq 2$ and an a priori undetermined number $n$ of real
scalars can be constructed in AdS3 Vasiliev:1999ba ; Gaberdiel:2010pz .
Assuming all integer spins $s\geq 2$ are present, the total character sums up
to

$\displaystyle\chi_{\rm
tot}=\frac{2\,q^{2}}{(1-q)^{2}}-\frac{4\,q}{(1-q)^{2}}+\sum_{i=1}^{n}\frac{q^{\Delta_{i}}}{(1-q)^{3}}\,.$
(185)

For $t\to 0$ diverges as $\chi_{\rm HS}\sim(n-2)/t^{2}+O(1/t)$. To satisfy
(184), the number of scalars must be $n=2$. This is inconsistent with the
$n=4$ AdS3 theory originally conjectured in Gaberdiel:2010pz  to be dual to a
minimal model CFT2, but consistent with the amended conjecture of Chang:2011mz
; Gaberdiel:2012ku ; Gaberdiel:2012uj .

#### AdS$-$

For alternate boundary conditions, one ends up with a massless higher-spin
character formula similar to (112). The factor $\gamma^{\dim G}$ in (112) is
consistent with $\log Z^{{\rm AdS}-}_{\rm PI}\propto(G_{\rm
N})^{\frac{1}{2}\sum_{s}N^{\rm KT}_{s-1}}$ found in Giombi:2013yva . (176)
implies the massless AdS$\pm$ and dS bulk and edge characters are related as

$\displaystyle\boxed{\chi_{s}^{{\rm AdS}-}=\chi_{s}^{\rm dS}-\chi_{s}^{{\rm
AdS}+}}$ (186)

hence we can read off the appropriate flipped $\chi_{s}^{{\rm
AdS}-}=[\hat{\chi}_{s}^{{\rm AdS}-}]_{+}$ characters from our earlier explicit
results (388) and (390) for $\chi_{s}^{\rm dS}$. Just like in the dS case, the
final result involves divergent spin sums when the spin range is infinite.

###  9.3 Conformal higher-spin gravity

#### Conformal HS characters

Conformal (higher-spin) gravity theories FRADKIN1985233  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
$U(N)$ CFTd 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 Tseytlin:2013jya ; Tseytlin:2013fca . Starting from this
formulation of CHS theory on $S^{d}$ (or equivalently dSd), using our general
explicit formulae for partially massless higher-spin field characters (388)
and (390), and summing up the results, we find

$\displaystyle\boxed{\chi_{s}^{\rm CdS_{d}}=\chi_{s}^{{\rm
AdS_{d+1}}-}-\chi_{s}^{{\rm AdS_{d+1}}+}=\chi_{s}^{{\rm
dS_{d+1}}}-2\,\chi_{s}^{{\rm AdS_{d+1}}+}}$ (187)

where $\chi_{s}^{\rm CdS_{d}}$ are the CHS bulk and edge characters and the
second equality uses (186). 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

$\begin{array}[]{l|l|l|l}d&s&\chi^{\rm CdS_{d}}_{{\rm
bulk},s}\cdot(1-q)^{d}&\chi^{\rm CdS_{d}}_{{\rm edge},s}\cdot(1-q)^{d-2}\\\
\hline\cr 2&\geq
2&-4q^{s}(1-q)&-2\bigl{(}s^{2}q^{s-1}-(s-1)^{2}q^{s}\bigr{)}\\\ 3&\geq
1&0&0\\\ 3&0&-q(1-q)&0\\\ 4&\geq
0&2(2s\\!+\\!1)\,q^{2}+2s^{2}q^{s+3}-2(s\\!+\\!1)^{2}q^{s+2}&\frac{s(s+1)(2s+1)}{3}\,q+\frac{(s-1)s^{2}(s+1)}{6}\,q^{s+2}-\frac{s(s+1)^{2}(s+2)}{6}\,q^{s+1}\\\
5&\geq
0&\frac{(s+1)(2s+1)(2s+3)}{3}\,q^{2}(1-q)&\frac{s(s+1)(s+2)(2s+1)(2s+3)}{30}\,q(1-q)\end{array}$
(188)

The bulk $SO(1,d)$ $q$-characters $\chi^{\rm CdS_{d}}_{{\rm bulk},s}$ computed
from (187) agree with the ${\rm so}(2,d)$ $q$-characters obtained in
Beccaria:2014jxa . Edge characters were not derived in Beccaria:2014jxa , as
they have no role in the thermal $S^{1}\times S^{d-1}$ CHS partition functions
studied there.191919A priori the interpretation of the bulk characters in
(188) and those in Beccaria:2014jxa  is different. Their mathematical equality
is a consequence of the enhanced ${\rm so}(2,d)$ symmetry allowing to map
$S^{d}\to{\mathbb{R}}\times S^{d-1}$.

The one-loop Euclidean path integral of the CHS theory on $S^{d}$ is given by
(112) using the bulk and edge CHS characters $\chi_{s}^{\rm CdS_{d}}$ 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 (112), or more directly from the “naive” integrand
$\frac{1}{2t}\frac{1+q}{1-q}\,\hat{\chi}$. 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 $\dim
G=\sum_{\pm}D^{4}_{1,\pm 1}=6$, generating $G=SO(1,3)$, and from the above
table (188),

$\displaystyle\chi_{\rm
tot}=D\cdot\frac{1+q}{1-q}-\frac{4q^{2}}{1-q}+2(4q-q^{2})\,.$ (189)

The small-$t$ expansion of the integrand in (112) for this case is

$\displaystyle\frac{1}{2t}\frac{1+q}{1-q}\bigl{(}\chi_{\rm
tot}-12\bigr{)}\to\frac{2(D-2)}{t^{3}}+\frac{D-26}{3\,t}+\cdots\,,$ (190)

reassuringly informing us the critical dimension for the bosonic string is
$D=26$. Adding a massless $s=\frac{3}{2}$ field, we get 2D conformal
supergravity. For half-integer conformal spin $s$, $\chi_{\rm
bulk}=-4q^{s}/(1-q)$ and $\chi_{\rm
edge}=-2\bigl{(}(s-\frac{1}{2})(s+\frac{1}{2})q^{s-1}-(s-\frac{3}{2})(s-\frac{1}{2})q^{s}\bigr{)}$.
Furthermore adding $D^{\prime}$ massless Dirac spinors, the total fermionic
character is

$\displaystyle\chi_{\rm tot}^{\rm
fer}=D^{\prime}\cdot\frac{2\,q^{1/2}}{1-q}-\frac{4\,q^{3/2}}{1-q}+4\,q^{1/2}\,.$
(191)

The symmetry algebra has $\sum_{\pm}D^{4}_{\frac{1}{2},\pm\frac{1}{2}}=4$
fermionic generators, contributing negatively to $\dim G$ in (112). Putting
everything together,

$\displaystyle\frac{1}{2t}\,\frac{1+q}{1-q}\bigl{(}\chi^{\rm bos}_{\rm
tot}-2(6-4)\bigr{)}-\frac{1}{2t}\,\frac{\sqrt{q}}{1-q}\,\chi^{\rm fer}_{\rm
tot}\to\frac{2(D-D^{\prime})}{t^{3}}+\frac{2D+D^{\prime}-30}{6\,t}+\cdots\,,$
(192)

from which we read off supersymmetry + conformal symmetry requires
$D^{\prime}=D=10$.

More systematically, the Weyl anomaly $\alpha_{d,s}$ can be read off by
expanding $\frac{1}{2t}\frac{1+q}{1-q}\,\hat{\chi}^{\rm CS^{d}}$ with
$\hat{\chi}^{\rm CS^{d}}=\hat{\chi}^{{\rm AdS_{d+1}}-}-\hat{\chi}^{{\rm
AdS_{d+1}}+}$ given by (178)-(179) for integer $s$. For example,

$\begin{array}[]{l|l}d&-\alpha_{d,s}\\\ \hline\cr
2&\frac{2(6s^{2}-6s+1)}{3}\\\ 4&\frac{s^{2}(s+1)^{2}(14s^{2}+14s+3)}{180}\\\
6&\frac{(s+1)^{2}(s+2)^{2}(22s^{6}+198s^{5}+671s^{4}+1056s^{3}+733s^{2}+120s-50)}{151200}\\\
8&\frac{(s+1)(s+2)^{2}(s+3)^{2}(s+4)(150s^{8}+3000s^{7}+24615s^{6}+106725s^{5}+261123s^{4}+351855s^{3}+225042s^{2}+31710s-14560)}{2286144000}\end{array}$
(193)

This reproduces the $d=2,4,6$ results of Tseytlin:2013fca ; Tseytlin:2013jya
and generalizes them to any $d$.

#### Physics pictures

Cartoonishly speaking, the character relation (187) translates to one-loop
partition function relations of the form $Z^{{\rm CS}^{d}}\sim Z^{{\rm
EAdS_{d+1}}-}/Z^{{\rm EAdS_{d+1}}+}$ and $Z^{S^{d+1}}\sim Z^{{\rm
CS}^{d}}\bigl{(}Z^{{\rm EAdS_{d+1}}+}\bigr{)}^{2}$. The first relation can
then be understood as a consequence of the holographic duality between AdSd+1
higher-spin theories and free CFTd vector models Tseytlin:2013fca ;
Giombi:2013yva ; Tseytlin:2013jya , while the second relation can be
understood as an expression at the Gaussian/one-loop level of
$Z^{S^{d+1}}\sim\int{\cal D}\sigma\,\bigl{|}\psi_{\rm
HH}(\sigma)\bigr{|}^{2}$, where $\psi_{\rm HH}(\sigma)=\psi_{\rm
HH}(0)\,e^{-\frac{1}{2}\sigma K\sigma+\cdots}$ is the late-time dS Hartle-
Hawking wave function, related by analytic continuation to the EAdS partition
function with boundary conditions $\sigma$ Maldacena:2002vr . The factor
$\bigl{(}Z^{{\rm EAdS}_{d+1}+}\bigr{)}^{2}$ can then be identified with the
bulk one-loop contribution to $|\psi_{\rm HH}(0)|^{2}$, and $Z^{{\rm
cnf}\,S^{d}}$ with $\int{\cal D}\sigma\,e^{-\sigma K\sigma}$, along the lines
of Giombi:2013yva . Along the lines of footnote 8, perhaps another
interpretation of the spin-summed relation (187) exists within the picture of
Alishahiha:2004md .

###  9.4 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 ${\mathfrak{g}}$ generated by the
Killing tensors is not gauged. The character spin sum converges, and no
additional regularization is required beyond the UV cutoff at $t\sim\epsilon$
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, ${\mathfrak{g}}$ is gauged, leading to the group volume
division factor, and moreover, for $d\geq 4$, the quasinormal mode levels (or
energy levels for CHS on ${\mathbb{R}}\times S^{d-1}$) 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 $e^{-2T/\ell}$ for every spin $s$ in $d\geq 4$ was
explained below (361).

One might be tempted to use some form of zeta function regularization to deal
with divergent sums $\sum_{s}\chi_{s}$ such as (174), which amounts to
inserting a convergence factor $\propto e^{-\delta s}$ and discarding the
divergent terms in the limit $\delta\to 0$. 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-like202020“Vasiliev-like”
is meant only in a superficial sense here. The higher-spin algebras are rather
different Joung:2014qya . limit of the 3D HSn higher-spin gravity theory,
$n\to\infty$ with $l=0$ and ${\cal S}^{(0)}$ fixed, is strongly coupled as a
3D QFT. Unsurprisingly, the one-loop entropy “correction” ${\cal S}^{(1)}=\log
Z^{(1)}$ diverges in this limit: writing the explicit expression for the
maximal-entropy vacuum $R={\bf n}$ in (12) as a function of $\dim
G=2(n^{2}-1)$, one gets ${\cal S}^{(1)}=\dim G\cdot\log\bigl{(}\dim
G/\sqrt{{\cal S}^{(0)}}\bigr{)}+\cdots\to\infty$. The higher-spin
decomposition (459) might inspire an ill-advised zeta function regularization
along the lines of $\dim
G=2\sum_{r=1}^{\infty}2r+1=4\,\zeta(-1)+2\,\zeta(0)=-\frac{4}{3}$. This gives
${\cal S}^{(1)}=\frac{2}{3}\log{\cal S}^{(0)}+c$ with $c$ a computable
constant — a finite but meaningless answer. In fact, using (127), the all-loop
quantum correction to the entropy can be seen to vanish in the limit under
consideration, as illustrated in fig. 4. As discussed around (128), there are
more interesting $n\to\infty$ limits one can consider, taking ${\cal
S}^{(0)}\to\infty$ 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 AdS5 higher-spin theory dual to the 4D
${\cal N}=2$ 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 AdS5 bulk field content is obtained from this following
Basile:2018dzi . In their notation, the hypermultiplet corresponds to the
${\rm so}(2,4)$ representation ${\rm Di}+2\,{\rm Rac}$. Decomposing $({\rm
Di}+2\,{\rm Rac})\otimes({\rm Di}+2\,{\rm Rac})$ into irreducible ${\rm
so}(2,4)$ representations gives the AdS5 free field content: four $\Delta=2$
and two $\Delta=3$ scalars, one $\Delta=3$, $S=(1,\pm 1)$ 2-form field, six
towers of massless spin-$s$ fields for all $s\geq 1$, one tower of massless
$S=(s,\pm 1)$ fields for all $s\geq 2$, one $\Delta=\frac{5}{2}$ Dirac spinor,
and four towers of massless spin $s=k+\frac{1}{2}$ fermionic gauge fields for
all $k\geq 1$. 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 (92) and (93). Each individual
spin tower gives rise to a badly divergent spin sum similar to (174). However,
a remarkable conspiracy of cancelations between various bosonic and fermionic
bulk and edge contributions in the end leads to a finite, unambiguous net
integrand:212121The spin sums are performed by inserting a convergence factor
such as $e^{-\delta s}$, but the end result is finite and unambiguous when
taking $\delta\to 0$, along the lines of $\lim_{\delta\to
0}\sum_{s\in\frac{1}{2}{\mathbb{N}}}\,(-1)^{2s}(2s+1)\,e^{-\delta
s}=\frac{1}{4}$.

$\displaystyle\int\frac{dt}{2t}\biggl{(}\frac{1+q}{1-q}\,\chi_{\rm tot}^{\rm
bos}-\frac{2\sqrt{q}}{1-q}\,\chi_{\rm tot}^{\rm
fer}\biggr{)}=-\frac{3}{4}\int\frac{dt}{2t}\,\frac{1+q}{1-q}\,\frac{q}{(1-q)^{2}}\,.$
(194)

Note that the effective UV dimensionality is reduced by two in this case.

An analogous construction for $S^{4}$ starting from the 3D ${\cal N}=2$ $U(N)$
model, gives two $\Delta_{\pm}=1,2$ scalars, a $\Delta=\frac{3}{2}$ Dirac
spinor and two massless spin-$1,\frac{3}{2},2,\frac{5}{2},\ldots$ towers, as
in Sezgin:2012ag ; Hertog:2017ymy . The fermionic bulk and edge characters
cancel and the bosonic part is twice (173). In this case we moreover get a
finite and unambiguous $\dim G=\lim_{\delta\to
0}\sum_{s\in\frac{1}{2}{\mathbb{N}}}^{\infty}\,(-1)^{2s}\,2\,D^{5}_{s-1,s-1}\,e^{-\delta
s}=\frac{1}{4}$.

The above observations are tantalizing, but leave several problems unresolved,
including what to make of the supergroup volume ${\rm vol}\,G$. Actually
supergroups present an issue of this kind already with a finite number of
generators, as their volume is generically zero. In the context of supergroup
Chern-Simons theory this leads to indeterminate $0/0$ Wilson loop expectation
values Mikhaylov:2014aoa . In this case the indeterminacy is resolved by a
construction replacing the Wilson loop by an auxiliary worldline quantum
mechanics Mikhaylov:2014aoa . 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 Boulanger:2011qt ;
Joung:2015jza ; Manvelyan:2013oua ; Brust:2016gjy , models with an
$\alpha^{\prime}$-like parameter breaking the higher-spin symmetries, or
models of a different nature, perhaps along the lines of Anninos:2017eib , or
bootstrapped bottom-up. We leave this, and more, to future work.

###### Acknowledgements.

We are grateful to Simone Giombi, Sean Hartnoll, Chris Herzog, Austin Joyce,
Igor Klebanov, Marcos Mariño, Ruben Monten, Beatrix Mühlmann, Bob Penna,
Rachel Rosen and Charlotte Sleight for discussions, explanations and past
collaborations inspiring this work. FD and DA are particularly grateful to Eva
Depoorter and Carolina Martes for near-infinite patience. FD, AL and ZS were
supported in part by the U.S. Department of Energy grant de-sc0011941. DA is
funded by the Royal Society under the grant The Atoms of a deSitter Universe.

##  Appendix A Harish-Chandra characters

###  A.1 Definition of $\chi$

A central ingredient in this work is the Harish-Chandra group character for
unitary representations $R$ of Lie groups $G$,

$\displaystyle\tilde{\chi}_{R}(g)\equiv{\rm tr}\,R(g)\,,\qquad g\in G\,.$
(195)

More rigorously this should be viewed as a distribution to be integrated
against smooth test functions $f(g)$ on $G$. The smeared operators
$\int[dg]\,f(g)R(g)$ are trace-class operators, and $\tilde{\chi}_{R}(g)$ is
always a locally integrable function on $G$, analytic away from its poles
bams/1183520006 ; bams/1183525024 .

The group of interest to us is $SO(1,d+1)$, the isometry group of global
dSd+1, generated by $M_{IJ}$ as defined under (289). The representations of
$SO(1,d+1)$ were classified and their characters explicitly computed in
10.3792/pja/1195522333 ; 10.3792/pja/1195523378 ; 10.3792/pja/1195523460 . For
a recent review and an extensive dictionary between fields and
representations, see Basile:2016aen .

For our purposes in this work we only need to consider characters restricted
to group elements of the form $g=e^{-itH}$, where $H=M_{0,d+1}$ generates
global $SO(1,1)$ transformations acting as time translations $T\to T+t$ on the
southern static patch (fig. 5):

$\displaystyle\chi(t)\equiv{\rm tr}\,e^{-itH}\,.$ (196)

For example for a spin-0 UIR corresponding to a scalar field of mass
$m^{2}=\Delta(d-\Delta)$, as we will explicitly compute below, this takes the
form

$\displaystyle\chi(t)={\rm
tr}\,e^{-itH}=\frac{e^{-t\Delta}+e^{-t\bar{\Delta}}}{|1-e^{-t}|^{d}}\,,\qquad\bar{\Delta}\equiv
d-\Delta\,.$ (197)

Putting $\Delta=\frac{d}{2}+i\nu$, we get $m^{2}=(\frac{d}{2})^{2}+\nu^{2}$,
so $m^{2}>0$ if $\nu\in{\mathbb{R}}$ (principal series) or $\nu=i\mu$ with
$|\mu|<\frac{d}{2}$ (complementary series). Since
$\bar{\Delta}=d-\Delta=\frac{d}{2}-i\nu$, this implies $\chi(t)=\chi(t)^{*}$,
as follows more generally from $H^{\dagger}=H$. The absolute value signs
moreover ensure $\chi(t)=\chi(-t)$ for all $d$. The latter property holds for
any $SO(1,d+1)$ representation:

$\displaystyle\chi(-t)=\chi(t)\,.$ (198)

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. $u=e^{i\pi M_{d,d+1}}$, implying
$\chi(-t)={\rm tr}\,e^{iHt}={\rm tr}\,u\,e^{-iHt}u^{-1}={\rm
tr}\,e^{-iHt}=\chi(t)$.

###  A.2 Computation of $\chi$

Here we show how characters $\chi(t)={\rm tr}\,e^{-itH}$ can be computed by
elementary means. The full characters $\chi(t,\phi)={\rm
tr}\,e^{-itH+i\phi\cdot J}$ 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
$\Delta=\frac{1}{2}+i\nu$, $\nu\in{\mathbb{R}}$. This corresponds to a massive
scalar field on dS2 with mass $m^{2}=\frac{1}{4}+\nu^{2}$. This unitary
irreducible representation of $SO(1,2)$ can be realized on the Hilbert space
of square-integrable wave functions $\psi(\varphi)$ on $S^{1}$, with standard
inner product. The circle can be thought of as the future conformal boundary
of global dS2 in global coordinates (cf. (290)), which for dS2 becomes
$ds^{2}=(\cos\vartheta)^{-2}(-d\vartheta^{2}+d\varphi^{2})$. Kets
$|\varphi\rangle$ can be thought of as states produced by a boundary conformal
field222222${\cal O}(\varphi)$ arises from the bulk scalar
$\phi(\vartheta,\varphi)$ as $\phi(\frac{\pi}{2}-\epsilon,\varphi)\sim{\cal
O}(\varphi)\,\epsilon^{\Delta}+\bar{\cal O}(\varphi)\,\epsilon^{\bar{\Delta}}$
in the infinite future $\epsilon\to 0$ ${\cal O}(\varphi)$ of dimension
$\Delta=\frac{1}{2}+i\nu$ acting on an $SO(1,2)$-invariant global vacuum state
$|{\rm vac}\rangle$ such as the global Euclidean vacuum:

$\displaystyle|\varphi\rangle\equiv{\cal O}(\varphi)|{\rm
vac}\rangle\,,\qquad\langle\varphi|\varphi^{\prime}\rangle=\delta(\varphi-\varphi^{\prime})\,.$
(199)

This pairing is $SO(1,2)$ invariant. Normalizable states $|\psi\rangle$ are
then superpositions

$\displaystyle|\psi\rangle=\int_{-\pi}^{\pi}d\varphi\,\psi(\varphi)\,|\varphi\rangle\,,\qquad\langle\psi|\psi\rangle=\int_{-\pi}^{\pi}d\varphi\,|\psi(\varphi)|^{2}<\infty\,.$
(200)

In conventions in which $H$, $P$ and $K$ are hermitian, the Lie algebra of
${\rm so(1,2)}$ is

$\displaystyle[H,P]=iP\,,\qquad[H,K]=-iK\,,\qquad[K,P]=2iH\,,$ (201)

the action of these generators on kets $|\varphi\rangle$ in the above
representation is

$\displaystyle H|\varphi\rangle$
$\displaystyle=i\bigl{(}\sin\varphi\,\partial_{\varphi}+\Delta\cos\varphi\bigr{)}|\varphi\rangle$
(202) $\displaystyle P|\varphi\rangle$
$\displaystyle=i\bigl{(}(1+\cos\varphi)\partial_{\varphi}-\Delta\sin\varphi\bigr{)}|\varphi\rangle$
$\displaystyle K|\varphi\rangle$
$\displaystyle=i\bigl{(}(1-\cos\varphi)\partial_{\varphi}+\Delta\sin\varphi\bigr{)}|\varphi\rangle\,.$

Note that this implies that the action of for example $H$ on wave functions
$\psi(\varphi)$ is given by $H|\psi\rangle=\int d\varphi\,{\cal
H}\psi(\varphi)\,|\varphi\rangle$ where ${\cal
H}\psi(\varphi)=-i\bigl{(}\sin\varphi\,\partial_{\varphi}+\bar{\Delta}\cos\varphi\bigr{)}\psi(\varphi)$,
with $\bar{\Delta}=1-\Delta=\frac{1}{2}-i\nu$. One gets simpler expressions
after conformally mapping this to planar boundary coordinates
$x=\tan\frac{\varphi}{2}$, that is to say changing basis from
$|\varphi\rangle_{S^{1}}$ to $|x\rangle_{\mathbb{R}}$, $x\in{\mathbb{R}}$,
where

$\displaystyle|x\rangle_{{\mathbb{R}}}\equiv\bigl{(}\tfrac{\partial\varphi}{\partial
x}\bigr{)}^{\Delta}\bigl{|}\varphi(x)\bigr{\rangle}_{S^{1}}=\bigl{(}\tfrac{2}{1+x^{2}}\bigr{)}^{\Delta}\,\bigl{|}2\arctan
x\bigr{\rangle}_{S^{1}}\,,\qquad\langle
x|x^{\prime}\rangle=\delta(x-x^{\prime})\,.$ (203)

Then $H,P,K$ take the familiar planar dilatation, translation and special
conformal form:

$\displaystyle H|x\rangle=i(x\partial_{x}+\Delta)|x\rangle\,\qquad
P|x\rangle=i\partial_{x}|x\rangle\,,\qquad
K|x\rangle=i(x^{2}\partial_{x}+2\Delta x)|x\rangle\,.$ (204)

In particular this makes exponentiation of $H$ easy:

$\displaystyle e^{-itH}|x\rangle=e^{t\Delta}|e^{t}x\rangle\,.$ (205)

However one has to keep in mind that planar coordinates miss a point of the
global boundary, here the point $\varphi=\pi$. This will actually turn out to
be important in the computation of the character. Let us first ignore this
though and compute

$\displaystyle\chi(t)|_{\rm planar}=\int dx\,\langle
x|e^{-itH}|x\rangle=e^{t\Delta}\int
dx\,\delta\bigl{(}x-e^{t}x\bigr{)}=e^{t\Delta}\int
dx\,\frac{\delta(x)}{|1-e^{t}|}=\frac{e^{t\Delta}}{|1-e^{t}|}\,.$

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 (202): one fixed point of $H$ is at
$\varphi=0$, which maps to $x=0$ and was picked up in the above computation,
while the other fixed point is at $\varphi=\pi$, which maps to $x=\infty$ and
so was missed.

This is easily remedied though. The most straightforward way is to repeat the
computation in the global boundary basis $|\varphi\rangle$, which is sure not
to miss any fixed points. It suffices to consider an infinitesimally small
neighborhood of the fixed points. For $\varphi=y\to 0$, we get $H\approx
i(y\partial_{y}+\Delta)$, which coincides with the planar expression, while
for $\varphi=\pi+y$ with $y\to 0$, we get $H\approx-i(y\partial_{y}+\Delta)$,
which is the same except with the opposite sign. Thus we obtain

$\displaystyle\chi(t)=\int
d\varphi\,\langle\varphi|e^{-itH}|\varphi\rangle=\frac{e^{t\Delta}}{|1-e^{t}|}+\frac{e^{-t\Delta}}{|1-e^{-t}|}\,=\frac{e^{-t\Delta}+e^{-t\bar{\Delta}}}{|1-e^{-t}|}\,,$
(206)

where $\bar{\Delta}=1-\Delta=\frac{1}{2}-i\nu$, reproducing (197) for $d=1$.

For the complementary series $0<\Delta<1$, we have $\Delta^{*}=\Delta$ instead
of $\Delta^{*}=\bar{\Delta}\equiv 1-\Delta$, so the conjugation properties of
$H$, $D$, $K$ are different. As a result they are no longer hermitian with
respect to the inner product (199), but rather with respect to
$\langle\varphi|\varphi^{\prime}\rangle\propto\bigl{(}1-\cos(\varphi-\varphi^{\prime})\bigr{)}^{-\Delta}$.
However we can now define a “shadow” bra $(\varphi|\propto\int
d\varphi^{\prime}\bigl{(}1-\cos(\varphi-\varphi^{\prime})\bigr{)}^{-\bar{\Delta}}\langle\varphi^{\prime}|$
satisfying $(\varphi|\varphi^{\prime}\rangle=\delta(\varphi-\varphi^{\prime})$
and compute the trace as $\chi(t)=\int
d\varphi\,(\varphi|e^{-itH}|\varphi\rangle$. The computation then proceeds in
exactly the same way, with the same result (206).

#### 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 $e^{t\Delta}\int
d^{d}x\,\delta^{d}(x-e^{t}x)=\frac{e^{t\Delta}}{|1-e^{t}|^{d}}\,,$ while the
fixed point at the other pole of the global boundary sphere gives a
contribution of the same form but with $t\to-t$. Together we get

$\displaystyle\chi_{0,\Delta}(t)=\frac{e^{-t\Delta}+e^{-t\bar{\Delta}}}{|1-e^{-t}|^{d}}\,,$
(207)

where $\bar{\Delta}=d-\Delta$.

For massive spin-$s$ representations, the basis merely gets some additional
$SO(d)$ spin labels, and the trace picks up a corresponding degeneracy factor,
so232323 Here $\Delta=\frac{d}{2}+i\nu$ with either $\nu\in{\mathbb{R}}$
(principal series) or $\nu=i\mu$ with $|\mu|<\frac{d}{2}$ for $s=0$ and
$|\mu|<\frac{d}{2}-1$ for $s\geq 1$ (complementary series). For $s=0$ the mass
is $m^{2}=(\frac{d}{2})^{2}+\nu^{2}=\Delta(d-\Delta)$ while for $s\geq 1$ it
is given by (79):
$m^{2}=(\tfrac{d}{2}+s-2)^{2}+\nu^{2}=(\Delta+s-2)(d-\Delta+s-2)$.

$\displaystyle\chi_{s,\Delta}(t)=D_{s}^{d}\,\frac{e^{-t\Delta}+e^{-t\bar{\Delta}}}{|1-e^{-t}|^{d}}\,,$
(208)

where $\bar{\Delta}=d-\Delta$ as before, and $D_{s}^{d}$ is the spin
degeneracy, for example $D_{s}^{3}=2s+1$. More generally for $d>2$ it is the
number of totally symmetric traceless tensors of rank $s$:

$\displaystyle D^{d}_{s}=\mbox{\large$\binom{s+d-1}{d-1}-\binom{s+d-3}{d-1}$}$
(209)

(For $d=2$ we get spin $\pm s$ irreps of $SO(2)$ with $D^{2}_{\pm s}=1$.
However both of these appear when quantizing a Fierz-Pauli spin-$s$ field.)
Explicit low-$d$ spin-$s$ degeneracies are listed in (283).

The most general massive unitary representation of $SO(1,d+1)$ is labeled by
an irrep $S=(s_{1},\ldots,s_{r})$ of $SO(d)$ (cf. appendix D.1) and
$\Delta=\frac{d}{2}+i\nu$, $\nu\in{\mathbb{R}}$ (principal series) or
$\Delta=\frac{d}{2}+\mu$, $|\mu|<\mu_{\rm max}(S)\leq\frac{d}{2}$
(complementary series) 10.3792/pja/1195522333 ; 10.3792/pja/1195523378 ;
10.3792/pja/1195523460 ; Basile:2016aen . The spin-$s$ case discussed above
corresponds to $S=(s_{1},0,\ldots,0)$. The character for general $S$ is

$\displaystyle\chi_{S,\Delta}(t)=D_{S}^{d}\,\frac{e^{-t\Delta}+e^{-t\bar{\Delta}}}{|1-e^{-t}|^{d}}\,,$
(210)

where the generalized spin degeneracy factor $D_{S}^{d}$ 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 $\chi(t)$ 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 (100). This generalizes straightforwardly to the partially
massless case, leading to the explicit general-$d$ formula (388).

###  A.3 Importance of picking a globally regular basis

Naive evaluation of the character trace $\chi(t)={\rm tr}\,e^{-itH}$ 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.

#### Failure of computation by diagonalization of $H$

Naively, one might have thought the easiest way to compute $\chi(t)={\rm
tr}\,e^{-itH}$ would be to diagonalize $H$ and sum over its eigenstates. The
latter are given by $|\omega\sigma\rangle$, where $H=\omega\in{\mathbb{R}}$,
$\sigma$ labels $SO(d)$ angular momentum quantum numbers, and
$\langle\omega\sigma|\omega^{\prime}\sigma^{\prime}\rangle=\delta(\omega-\omega^{\prime})\,\delta_{\sigma\sigma^{\prime}}$.
However this produces a nonsensical result,

$\displaystyle\chi(t)={\rm
tr}\,e^{-itH}\,\,\stackrel{{\scriptstyle\text{naive}}}{{=}}\,\,\int
d\omega\sum_{\sigma}\langle\omega\sigma|e^{-itH}|\omega\sigma\rangle=2\pi\,\sum_{\sigma}\delta(0)\,\delta(t)\,\qquad\mbox{(naive)}\,,$
(211)

not even remotely resembling the correct $\chi(t)$ 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
$\Delta=\frac{1}{2}+i\nu$, $\nu\in{\mathbb{R}}$. Recalling the action of $H$
on wave functions $\psi(\varphi)$ mentioned below (202), it is straightforward
to find the wave functions $\psi_{\omega\sigma}(\varphi)$ of the eigenkets
$|\omega\sigma\rangle=\int_{-\pi}^{\pi}d\varphi\,\psi_{\omega\sigma}(\varphi)\,|\varphi\rangle$
of $H$:

$\displaystyle\psi_{\omega\sigma}(\varphi)=\frac{\Theta(\sigma\sin\varphi)}{\sqrt{2\pi}}\,|\sin\varphi|^{-\bar{\Delta}}\left|\tan\frac{\varphi}{2}\right|^{i\omega}\,,\qquad\omega\in{\mathbb{R}}\,,\quad\sigma=\pm
1\,.$ (212)

where $\Theta$ is the step function. Alternatively we can first conformally
map $S^{1}$ to the “cylinder” ${\mathbb{R}}\times S^{d-1}={\mathbb{R}}\times
S^{0}$ parametrized by $(T,\Omega)$, $T\in{\mathbb{R}}$,
$\Omega\in\\{-1,+1\\}=S^{0}$, that is to say change basis
$|\varphi\rangle_{S^{1}}\to|T\Omega\rangle_{{\mathbb{R}}\times S^{0}}$.242424
Explicitly $T=\log|\tan\frac{\varphi}{2}|$, $\Omega={\rm sign}\,\varphi$,
which analogously to the global $\to$ planar map (203) yields
$\bigl{|}T\pm\bigr{\rangle}_{{\mathbb{R}}\times S^{0}}=(\cosh
T)^{-\Delta}\bigl{|}\pm 2\arctan e^{T}\bigr{\rangle}_{S^{1}}$, satisfying
$\langle
T\Omega|T^{\prime}\Omega^{\prime}\rangle=\delta(T-T^{\prime})\,\delta_{\Omega\Omega^{\prime}}$
and $H|T\Omega\rangle=i\partial_{T}|T\Omega\rangle$. Then $H$ generates
translations of $T$, so the wave functions of $|\omega\sigma\rangle$ in this
basis are simply

$\displaystyle\psi_{\omega\sigma}(T,\Omega)=\frac{1}{\sqrt{2\pi}}\,\delta_{\Omega,\sigma}\,e^{i\omega
T}\,.$ (213)

The cylinder is the conformal boundary of the future wedge, $F$ in fig. 14
(which actually splits in two wedges at $\Omega=\pm 1$ in the case of dS2),
and the $|\omega\pm\rangle$ are the states obtained by the usual free field
quantization corresponding to the natural modes $\phi_{\omega\pm}(T,r)$ in
this patch.

It is now clear why the naive computation (211) of $\chi(t)$ in the basis
$|\omega\sigma\rangle$ fails to produce the correct result: the wave functions
$\psi_{\omega\sigma}(\varphi)$ are singular precisely at the fixed points
$\varphi=0,\pi$ of $H$ (top corners of Penrose diagram in fig. 5), which are
exactly the points at which the character trace computation of section A.2
localizes. Closely related failure would be met in the basis
$|T\Omega\rangle$: $H$ acts as by translating $T$, seemingly without fixed
points, oblivious to their actual presence at $T=\pm\infty$. In other words,
despite their lure as being the bases in which the action of $H$ is maximally
simple, $|T\Omega\rangle$ or its Fourier dual $|\omega\sigma\rangle$ are in
fact the worst possible choice one could make to compute the trace.

Similar observations hold for in higher dimensions. The wave functions
diagonalizing $H$ take the form $\psi_{\omega\sigma}(T,\Omega)\propto
e^{i\omega T}Y_{\sigma}(\Omega)$ in ${\mathbb{R}}\times S^{d-1}$ cylinder
coordinates. Transformed to global $S^{d}$ coordinates, these are singular
precisely at the fixed points of $H$, excluded from the cylinder, making this
frame particularly ill-suited for computing ${\rm tr}\,e^{-itH}$.

#### Globally regular bases

More generally, to ensure correct computation of the full Harish-Chandra group
character $\chi_{R}(g)={\rm tr}\,R(g)$, $g\in SO(1,d+1)$, 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 dSd+1 boundary basis
$|\bar{\Omega}\rangle_{S^{d}}$, $\bar{\Omega}\in S^{d}$, generalizing the
$d=1$ global $S^{1}$ basis $|\varphi\rangle_{S^{1}}$, but not for a planar
basis $|x\rangle_{{\mathbb{R}}^{d}}$ or a cylinder basis
$|T\Omega\rangle_{{\mathbb{R}}\times S^{d-1}}$. 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
$|p\rangle\propto\int d^{d}x\,e^{ipx}|x\rangle$ and
$|\omega\sigma\rangle\propto\int dT\,d\Omega\,e^{i\omega
T}Y_{\sigma}(\Omega)\,|T\Omega\rangle$. 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 $\chi(g)$ in these frames will in general give incorrect
results. More precisely the result will be wrong unless the fixed points of
$g$ lie at finite position on the corresponding conformal boundary patch.

On the other hand the normalizable dual basis $|\bar{\sigma}\rangle=\int
d\bar{\Omega}\,Y_{\bar{\sigma}}(\bar{\Omega})\,|\bar{\Omega}\rangle$ inherits
the global regularity of $|\bar{\Omega}\rangle_{S^{d}}$. Here
$Y_{\bar{\sigma}}(\Omega)$ is a basis of spherical harmonics on $S^{d}$, with
$\bar{\sigma}$ labeling the global $SO(d+1)$ angular momentum quantum numbers,
and
$\langle\bar{\sigma}|\bar{\sigma}^{\prime}\rangle=\delta_{\bar{\sigma}\bar{\sigma}^{\prime}}$.
(From the bulk point of view this is essentially the basis obtained by
quantizing the natural mode functions of the global dSd+1 metric in table
290.) Although in practice much harder than computing $\chi(t)=\int
d\bar{\Omega}\,\langle\bar{\Omega}|e^{-iHt}|\bar{\Omega}\rangle$ as in section
A.2, computing

$\displaystyle\chi(t)={\rm
tr}\,e^{-itH}=\sum_{\bar{\sigma}}\langle\bar{\sigma}|e^{-itH}|\bar{\sigma}\rangle$
(214)

gives in principle the correct result. Note that this suggests a natural UV
regularization of $\chi(t)$ for $t\to 0$, by cutting off the global $SO(d+1)$
angular momentum. For example for a scalar on dS3 with $SO(3)$ angular
momentum cutoff $L$, this would be

$\displaystyle\chi_{L}(t)\equiv\sum_{\ell=0}^{L}\langle\ell m|e^{-itH}|\ell
m\rangle\,.$ (215)

##  Appendix B Density of states and quasinormal mode resonances

The review in appendix A focuses mostly on mathematical and computational
aspects of the Harish-Chandra character $\chi(t)={\rm tr}\,e^{-itH}$. Here we
focus on its physics interpretation, in particular the density of states
$\rho(\omega)$ obtained as its Fourier transform. We define this in a general
and manifestly covariant way using Pauli-Villars regularization in section 2.
Here we will not be particularly concerned with general definitions or
manifest covariance, taking a more pedestrian approach. At the end we briefly
comment on an “S-matrix” interpretation and a possible generalization of the
formalism including interactions.

In B.1, we contrast the spectral features encoded in the characters of unitary
representations of the ${\rm so}(1,d+1)$ isometry algebra of global dSd+1 with
the perhaps more familiar characters of unitary representations of the ${\rm
so}(2,d)$ isometry algebra of AdSd+1: in a sentence, the latter encodes bound
states, while the former encodes scattering states. In B.2 we explicitly
compare $\rho(\omega)$ obtained as the Fourier transform of $\chi(t)$ for dS2
to the coarse-grained eigenvalue density obtained by numerical diagonalization
of a model discretized by global angular momentum truncation, and confirm the
results match at large $N$. In B.3 we identify the poles of $\rho(\omega)$ in
the complex $\omega$ plane as scattering resonances/quasinormal modes, counted
by the power series expansion of the character. As a corollary this implies
the relation $Z_{\rm PI}=Z_{\rm bulk}$ of (68) can be viewed as a precise
version of the formal quasinormal mode expansion of $\log Z_{\rm PI}$ proposed
in Denef:2009kn .

###  B.1 Characters and the density of states: dS vs AdS

We begin by highlight some important differences in the spectrum encoded in
the characters of unitary ${\rm so}(1,d+1)$ representations furnished by
global dSd+1 single-particle Hilbert spaces and the characters of unitary
${\rm so}(2,d)$ representations furnished by global AdSd+1 single-particle
Hilbert spaces. Although the discussion applies to arbitrary representations,
for concreteness we consider the example of a scalar of mass
$m^{2}=(\frac{d}{2})^{2}+\nu^{2}$ on dSd+1. Its character as computed in (207)
is

$\displaystyle\chi_{\rm dS}(t)\equiv{\rm
tr}\,e^{-itH}=\frac{e^{-\Delta_{+}t}+e^{-\Delta_{-}t}}{|1-e^{-t}|^{d}}\,,\qquad\Delta_{\pm}=\tfrac{d}{2}\pm
i\nu\,,\qquad t\in{\mathbb{R}}.$ (216)

where ${\rm tr}$ 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. 14c). This is to be contrasted with the
familiar character of the unitary lowest-weight representation of a scalar of
mass $m^{2}=-(\frac{d}{2})^{2}+\mu^{2}$ on global AdSd+1 with standard
boundary conditions:

$\displaystyle\chi_{\rm AdS}(t)\equiv{\rm
tr}\,e^{-itH}=\frac{e^{-i\Delta_{+}t}}{(1-e^{-it})^{d}}\,,\qquad\Delta_{+}=\tfrac{d}{2}+\mu\,,\qquad{\rm
Im}\,t<0\,.$ (217)

Here the ${\rm so}(2)$ generator $H$ is the energy operator in global AdSd+1.
Besides the occurrence of both $\Delta_{\pm}$ in (216), another notable
difference is the absence of factors of $i$ in the exponents.

The physics content of $\chi_{\rm AdS}$ is clear: $\chi_{\rm
AdS}(-i\beta)={\rm tr}\,e^{-\beta H}$ is the single-particle partition
function at inverse temperature $\beta$ for a scalar particle trapped in the
global AdS gravitational potential well. Equivalently for ${\rm Im}\,t<0$, the
expansion

$\displaystyle\chi_{\rm
AdS}(t)=\sum_{\lambda}N_{\lambda}\,e^{-it\lambda}\,,\qquad\lambda=\Delta_{+}+n\,,\quad
n\in{\mathbb{N}}\,,$ (218)

counts normalizable single-particle states of energy $H=\lambda$, or
equivalently global normal modes of frequency $\lambda$. The corresponding
density of single-particle states is

$\displaystyle\rho_{\rm
AdS}(\omega)=\int_{-\infty}^{\infty}\frac{dt}{2\pi}\,\chi_{\rm
AdS}(t)\,e^{i\omega t}=\sum_{\lambda}N_{\lambda}\,\delta(\omega-\lambda)\,.$
(219)

Figure 10:  Density of states $\rho_{\Lambda}(\omega)$ for dS3 scalars with
$\Delta=1+2i$, $\Delta=\frac{1}{2}$, $\Delta=\frac{1}{10}$, and UV cutoff
$\Lambda=100$, according to (222). The red dotted line represents the term
$2\Lambda/\pi$. The peak visible at $\Delta=\frac{1}{10}$ is due to a
resonance approaching the real axis, as explained in section B.3.

For dS, we can likewise expand the character as in (218). For $t>0$,

$\displaystyle\chi_{\rm
dS}(t)=\sum_{\lambda}N_{\lambda}\,e^{-it\lambda}\,,\qquad\lambda=-i(\Delta_{\pm}+n)=-i(\tfrac{d}{2}+n)\pm\nu\,,\quad
n\in{\mathbb{N}}\,.$ (220)

However $\lambda$ is now complex, so evidently $N_{\lambda}$ 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 $H=\omega\in{\mathbb{R}}$ is formally given by

$\displaystyle\rho_{\rm
dS}(\omega)=\int_{-\infty}^{\infty}\frac{dt}{2\pi}\,\chi_{\rm
dS}(t)\,e^{i\omega t}=\int_{0}^{\infty}\frac{dt}{2\pi}\,\chi_{\rm
dS}(t)\bigl{(}e^{i\omega t}+e^{-i\omega t}\bigr{)}\,,$ (221)

where $\omega$ can be interpreted as the momentum along the $T$-direction of
the future wedge ($F$ in fig. 14 and table 290). Alternatively for $\omega>0$
it can be interpreted as the energy in the southern static patch, as discussed
in section 2.2. A manifestly covariant Pauli-Villars regularization of the
above integral is given by (41). For our purposes here a simple
$t>\Lambda^{-1}$ cutoff suffices. For example for dS3,

$\displaystyle\rho_{\rm dS_{3},\Lambda}(\omega)$
$\displaystyle\equiv\int_{\Lambda^{-1}}^{\infty}\frac{dt}{2\pi}\,\frac{e^{-(1+i\nu)t}+e^{-(1-i\nu)t}}{(1-e^{-t})^{2}}\bigl{(}e^{i\omega
t}+e^{-i\omega t}\bigr{)}$ (222)
$\displaystyle=\frac{2\Lambda}{\pi}-\frac{1}{2}\sum_{\pm}(\omega\pm\nu)\coth\bigl{(}\pi(\omega\pm\nu)\bigr{)}\,.$

Some examples are illustrated in fig. 10. In contrast to AdS, $\rho_{\rm
dS}(\omega)$ is continuous. Indeed energy eigenkets $|\omega\sigma\rangle$ of
the static patch form a continuum of scattering states, coming out of and
going into the horizon, instead of the discrete set of bound states one gets
in the global AdS potential well. Note that although the above $\rho_{\rm
dS_{3},\Lambda}(\omega)$ formally goes negative in the large-$\omega$ limit,
it is positive within its regime of validity, that is to say for
$\omega,\nu\ll\Lambda$.

###  B.2 Coarse-grained density of states in globally truncated model

Figure 11: Density of states for a $\Delta=\frac{1}{2}+i\nu$ scalar with
$\nu=2$ in dS2. The red dots show the local eigenvalue density
$\bar{\rho}_{N}(\omega)$, (225), of the truncated model with global angular
momentum cutoff $N=2000$, obtained by numerical diagonalization. The blue line
shows $\rho(\omega)$ obtained as the Fourier transform of $\chi(t)$,
explicitly (223) with $e^{-\gamma}\Lambda\approx 4000$. The plot on the right
zooms in on the IR region. The peaks are due to the proximity of quasinormal
mode poles in $\rho(\omega)$, discussed in B.3.

For a $\Delta=\frac{1}{2}+i\nu$ scalar on dS2, the density of states
regularized by as in (222) is

$\displaystyle\rho(\omega)=\frac{2}{\pi}\log(e^{-\gamma}\Lambda)-\frac{1}{2\pi}\sum_{\pm,\pm}\psi\bigl{(}\tfrac{1}{2}\pm
i\nu\pm i\omega)\bigr{)}\,,$ (223)

where $\gamma$ is the Euler constant, $\psi(x)=\Gamma^{\prime}(x)/\Gamma(x)$
is the digamma function, and the sum is over the four different combinations
of signs. To ascertain it makes physical sense to identify this as the density
of states, we would like to compare this to a model with discretized spectrum
of eigenvalues $\omega$.

An efficient discretization — which does not require solving bulk equations of
motion and is quite natural from the point of view of dS-CFT approaches to de
Sitter quantum gravity Maldacena:2002vr ; Strominger:2001pn ; Anninos:2011ui
— is obtained by truncating the global dSd+1 angular momentum $SO(d+1)$ of the
single-particle Hilbert space, considering instead of $H$ a finite-dimensional
matrix

$\displaystyle
h_{\bar{\sigma}\bar{\sigma}^{\prime}}\equiv\langle\bar{\sigma}|H|\bar{\sigma}^{\prime}\rangle\,,$
(224)

where $\bar{\sigma}$ are $SO(d+1)$ quantum numbers, as in (214). For dS2 this
is $SO(2)$ and $\bar{\sigma}=n\in{\mathbb{Z}}$, truncated e.g. by $|n|\leq N$.
The matrix $h$ is sparse and can be computed either directly using
$|n\rangle\propto\int d\varphi\,e^{in\varphi}|\varphi\rangle$ and the explicit
form of $H$ given in (202), or algebraically.

The algebraic way goes as follows. A normalizable basis $|n\rangle$ of the
global dS2 scalar single-particle Hilbert space can be constructed from the
$SO(1,2)$ conformal algebra (201), using a basis of generators $L_{0}$,
$L_{\pm}$ related to $H$, $K$ and $P$ as $L_{0}=\frac{1}{2}(P+K)$,
$L_{\pm}=\frac{1}{2}(P-K)\pm iH$. Then $L_{0}$ is the global angular momentum
generator $i\partial_{\phi}$ along the future boundary $S^{1}$ and $L_{\pm}$
are its raising and lowering operators. In some suitable normalization of the
$L_{0}$ eigenstates $|n\rangle$, we have $L_{0}|n\rangle=n|n\rangle$,
$L_{\pm}|n\rangle=(n\pm\Delta)|n\pm 1\rangle$. Cutting off the single-particle
Hilbert space at $-N<n\leq N$,252525The asymmetric choice here allows us to
use the simple coarse graining prescription (225) and keep this discussion
short. A symmetric choice $|n|\leq N$ would lead to an enhanced
${\mathbb{Z}}_{2}$ and two families of eigenvalues distinguished by their
${\mathbb{Z}}_{2}$ parity, inducing persistent microstructure in the level
spacing. The most efficient way to proceed then is to compute
$\bar{\rho}_{N,\pm}(\omega)$ as the inverse level spacing for these two
families separately and then add the contributions together as interpolated
functions. For dS3 with $SO(3)$ cutoff $\ell\leq N$ one similarly gets $2N+1$
families of eigenvalues, labeled by $SO(2)$ angular momentum $m$, and one can
proceed analogously. Alternatively, one can compute $\bar{\rho}_{N}(\omega)$
directly by binning and counting, but this requires larger $N$. the operator
$H=\frac{i}{2}(L_{-}-L_{+})$ acts as a sparse $2N\times 2N$ matrix on the
truncated basis $|n\rangle$.

A minimally coarse-grained density of states can then be defined as the
inverse spacing of its eigenvalues $\omega_{i}$, $i=1,\ldots,2N$, obtained by
numerical diagonalization:

$\displaystyle\bar{\rho}_{N}(\omega_{i})\equiv\frac{2}{\omega_{i+1}-\omega_{i-1}}.$
(225)

The continuum limit corresponds to $N\to\infty$ in the discretized model, and
to $\Lambda\to\infty$ in (223). To compare to (223), we adjust $\Lambda$, in
the spirit of renormalization, to match the density of states at some scale
$\omega$, say $\omega=0$. The results of this comparison for $\nu=2$, $N=2000$
are shown in fig. 11. Clearly they match remarkably well indeed in the regime
where they should, i.e. well below the UV cutoff scale.

Figure 12: Comparison of $d=1$ character $\chi(t)$ defined in (216) (blue) to
the coarse-grained discretized character $\bar{\chi}_{N,\delta}(t)$ defined in
(226) (red), with $\delta=0.1$ and other parameters as in fig. 11. Plot on the
right shows wider range of $t$. Plot in the middle smaller range of $t$, but
larger $\chi$.

We can make a similar comparison directly at the (UV-finite) character level.
The discrete character is $\sum_{i}e^{-i\omega_{i}t}$, which is a wildly
oscillating function. At first sight this seems very different from the
character $\chi(t)={\rm tr}\,e^{-iHt}$ in (221). However to properly compare
the two, we should coarse grain this at a small but finite resolution
$\delta$. We do this by convolution with a Gaussian kernel, that is to say we
consider

$\displaystyle\bar{\chi}_{N,\delta}(t)\equiv\frac{1}{\sqrt{2\pi}\delta}\int_{-\infty}^{\infty}dt^{\prime}\,e^{-(t-t^{\prime})^{2}/2\delta^{2}}\,\sum_{i}e^{-i\omega_{i}t^{\prime}}=\sum_{i}e^{-it\omega_{i}-\delta^{2}\omega_{i}^{2}/2}\,.$
(226)

A comparison of $\bar{\chi}_{N,\delta}$ to $\chi$ is shown in fig. 12 for
$\delta=0.1$. The match is nearly perfect for $|t|$ not too large and not too
small. For small $t$, the $\bar{\chi}_{N,\delta}(t)$ caps off at a finite
value, the number of eigenvalues $|\omega_{i}|\lesssim 1/\delta$, while
$\chi(t)\sim 1/|t|\to\infty$. The approximation gets better here when $\delta$
is made smaller. For larger values of $t$, $\bar{\chi}_{N,\delta}(t)$ starts
showing some oscillations again. These can be eliminated by increasing
$\delta$, at the cost of accuracy at smaller $t$. In the $N\to\infty$ limit,
the discretized approximation gets increasingly better over increasingly large
intervals of $t$, with $\lim_{\delta\to
0}\lim_{N\to\infty}\bar{\chi}_{N,\delta}(t)=\chi(t)$.

Note that there is no reason to expect any discretization scheme will converge
to $\chi(t)$ or $\rho(\omega)$. 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 $\chi(t)$ was perhaps to be expected, given (214) and the
discussion preceding it.

###  B.3 Resonances and quasinormal mode expansion

Figure 13: Plot of $|\rho(\omega)|$ in complex $\omega$-plane corresponding to
the dS3 examples of fig. 10, that is $\Delta_{\pm}=\\{1+2i,1-2i\\}$,
$\\{\frac{1}{2},\frac{3}{2}\\}$, $\\{0.1,1.9\\}$, and $2\Lambda/\pi\approx
64$. Lighter is larger with plot range $58\text{ (black)}<|\rho|<67\text{
(white)}$. Resonance poles are visible at $\omega=\mp i(\Delta_{\pm}+n)$,
$n\in{\mathbb{N}}$.

Substituting the expansion (220) of the dS character,

$\displaystyle\chi(t)=\sum_{\lambda}N_{\lambda}\,e^{-it\lambda}\qquad(t>0)\,,$
(227)

into (221),
$\rho(\omega)=\frac{1}{2\pi}\int_{0}^{\infty}dt\,\chi(t)\,(e^{i\omega
t}+e^{-i\omega t})$, we can formally express the density of states as

$\displaystyle\rho(\omega)=\frac{1}{2\pi
i}\sum_{\lambda}N_{\lambda}\Bigl{(}\frac{1}{\lambda-\omega}+\frac{1}{\lambda+\omega}\Bigr{)}\,,$
(228)

From this we read off that $\rho(\omega)$ analytically continued to the
complex plane has poles at $\omega=\pm\lambda$ which for massive
representations means $\omega=\mp i(\Delta_{\pm}+n)$. This can also be checked
from explicit expressions such as the dS3 scalar density of states (222),
illustrated in fig. 13. These values of $\omega$ are precisely the frequencies
of the (anti-)quasinormal field modes in the static patch, that is to say
modes with purely ingoing/outgoing boundary conditions at the horizon, regular
in the interior. If we think of the normal modes as scattering states, the
quasinormal modes are to be thought of as scattering resonances. Indeed the
poles of $\rho(\omega)$ are related to the poles/zeros of the static patch
$S$-matrix $S(\omega)$, cf. (229) below. Thus we see the coefficients
$N_{\lambda}$ in (227) count resonances (or quasinormal modes), rather than
states (or normal modes) as in AdS. This expresses at the level of characters
the observations made in Ng:2012xp . It holds for any $SO(1,d+1)$
representation, including massless representations, as explored in more depth
in Sun:2020sgn  (see also appendix G.1). Some corresponding quasinormal mode
expansions of bulk thermodynamic quantities are given in (56) and (59), and
related there to the quasinormal mode expansion of Denef:2009kn  for scalar
and spinor path integrals.

#### “S-matrix” formulation

The appearance of resonance poles in the analytically continued density of
states is well-known in quantum mechanical scattering off a fixed potential
$V$. They are directly related to the poles/zeros in the S-matrix $S(\omega)$
at energy $\omega$ through the relation262626An elementary exposition can be
found e.g. in chapter 39 of chaosbook .

$\displaystyle\rho(\omega)-\rho_{0}(\omega)=\frac{1}{2\pi
i}\,\frac{d}{d\omega}\,{\rm tr}\log S(\omega)\,,$ (229)

where $\rho_{0}(\omega)$ is the density of states at $V=0$.

Using the explicit form of the dS2 dimension-$\Delta$ scalar static patch mode
functions (316) $\phi^{\Delta}_{\omega\ell}(r,T)$, expanding these for
$r=:\tanh X\to 1$ as

$\displaystyle\phi^{\Delta}_{\omega\ell}(r)\to
A^{\Delta}_{\ell}(\omega)\,e^{-i\omega(T+X)}+B^{\Delta}_{\ell}(\omega)\,e^{-i\omega(T-X)}\,,$
(230)

and defining $S^{\Delta}_{\ell}(\omega)\equiv
B^{\Delta}_{\ell}(\omega)/A^{\Delta}_{\ell}(\omega)$, one can check that
$\rho^{\Delta}(\omega)$ as obtained in (223) satisfies

$\displaystyle\rho^{\Delta}(\omega)-\rho_{0}(\omega)=\frac{1}{2\pi
i}\,\frac{d}{d\omega}\sum_{\ell=0,1}\log S^{\Delta}_{\ell}(\omega)\,,$ (231)

where $\rho_{0}(\omega)=\frac{1}{\pi}(\psi(i\omega)+\psi(-i\omega))+{\rm
const.}$ does not depend on $\Delta$. This can be viewed as a rough analog of
(229), although the interpretation of $\rho_{0}(\omega)$ in the present
setting is not clear to us. Similar observations can be made in higher
dimensions.

In PhysRev.187.345 , 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

$\displaystyle\log Z-\log Z_{0}=\frac{1}{2\pi i}\int dE\,e^{-\beta
E}\,\frac{d}{dE}\,\bigl{[}{\rm Tr}\log S(E)\bigr{]}_{c}\,,$ (232)

where the subscript $c$ indicates restriction to connected diagrams (where
“connected” is defined with the rule that particle permutations are
interpreted as interactions PhysRev.187.345 ). Combined with the above
observations, this hints at a possible generalization of our free QFT results
to interacting theories.

##  Appendix 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 $t=\epsilon$. However to compare
to the standard heat kernel (or spectral zeta function) regularization for
Gaussian Euclidean path integrals Vassilevich:2003xt , 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 $e^{-\epsilon^{2}/4\tau}$ as in (66),
requiring only the unregulated character formula as input. For concreteness we
consider the scalar case in the derivation, but because the scalar character
$\chi_{0}(t)$ provides the basic building block for all other characters
$\chi_{S}(t)$, 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 $t>\epsilon$ cutoff.

###  C.1 Derivation

As shown in section 3, the scalar Euclidean path integral regularized as

$\displaystyle\log
Z_{\epsilon}=\int_{0}^{\infty}\frac{d\tau}{2\tau}\,e^{-\frac{\epsilon^{2}}{4\tau}}\,F_{D}(\tau)\,,\qquad
F_{D}(\tau)\equiv{\rm Tr}\,e^{-\tau
D}=\sum_{n}D_{n}^{d+2}\,e^{-\left(n+\frac{d}{2}+i\nu\right)\left(n+\frac{d}{2}-i\nu\right)\tau}\,,$
(233)

where $D=-\nabla^{2}+\frac{d^{2}}{4}+\nu^{2}$, can be written in character
integral form as

$\displaystyle\log Z_{\epsilon}$
$\displaystyle=\int_{\epsilon}^{\infty}\frac{dt}{2\sqrt{t^{2}-\epsilon^{2}}}\sum_{n}D_{n}^{d+2}\Bigl{(}e^{-(n+\frac{d}{2})t-i\nu\sqrt{t^{2}-\epsilon^{2}}}+e^{-(n+\frac{d}{2})t+i\nu\sqrt{t^{2}-\epsilon^{2}}}\Bigr{)}$
(234)
$\displaystyle=\int_{\epsilon}^{\infty}\frac{dt}{2\sqrt{t^{2}-\epsilon^{2}}}\,\,\frac{1+e^{-t}}{1-e^{-t}}\,\frac{e^{-\frac{d}{2}t-i\nu\sqrt{t^{2}-\epsilon^{2}}}+e^{-\frac{d}{2}t+i\nu\sqrt{t^{2}-\epsilon^{2}}}}{(1-e^{-t})^{d}}\,,$
(235)

Putting $\epsilon=0$ we recover the formal (UV-divergent) character formula

$\displaystyle\log Z_{\epsilon=0}$
$\displaystyle=\int_{0}^{\infty}\frac{dt}{2t}\,F_{\nu}(t)\,,$ $\displaystyle
F_{\nu}(t)$
$\displaystyle\equiv\sum_{n}D_{n}^{d+2}\Bigl{(}e^{-(n+\frac{d}{2}+i\nu)t}+e^{-(n+\frac{d}{2}-i\nu
n)t}\Bigr{)}=\frac{1+e^{-t}}{1-e^{-t}}\,\frac{e^{-(\frac{d}{2}+i\nu)t}+e^{-(\frac{d}{2}-i\nu)t}}{(1-e^{-t})^{d}}\,.$
(236)

To evaluate (235), we split the integral into UV and IR parts, each of which
can be evaluated in closed form in the limit $\epsilon\to 0$.

#### 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 $\tau$ integral by isolating the leading terms in the $\tau\to
0$ heat kernel expansion

$\displaystyle F_{D}(\tau):={\rm Tr}\,e^{-\tau
D}\,\to\,\sum_{k=0}^{d+1}\alpha_{k}\,\tau^{-(d+1-k)/2}=:F_{D}^{\rm
uv}(\tau)\,.$ (237)

Introducing an infinitesimal IR cutoff $\mu\to 0$, we may write $\log
Z_{\epsilon}=\log Z^{\rm uv}_{\epsilon}+\log Z^{\rm ir}$ where

$\displaystyle\log Z^{\rm
uv}_{\epsilon}\equiv\int_{0}^{\infty}\frac{d\tau}{2\tau}\,e^{-\frac{\epsilon^{2}}{4\tau}}\,F_{D}^{\rm
uv}(\tau)\,e^{-\mu^{2}\tau}\,,\quad\log Z^{\rm
ir}\equiv\int_{0}^{\infty}\frac{d\tau}{2\tau}\bigl{(}F_{D}(\tau)-F_{D}^{\rm
uv}(\tau)\bigr{)}\,e^{-\mu^{2}\tau}\,.$ (238)

Dropping the UV regulator in the IR integral is allowed because all UV
divergences have been removed by the subtraction. The factor
$e^{-\mu^{2}\tau}$ serves as an IR regulator needed for the separate integrals
when $F^{\rm uv}$ has a term $\frac{\alpha_{d+1}}{2\tau}\neq 0$, that is to
say when $d+1$ is even. The resulting $\log\mu$ terms cancel out of the sum at
the end. Evaluating this using the specific UV regulator of (233) gives

$\displaystyle\log
Z_{\epsilon}=\frac{1}{2}\zeta_{D}^{\prime}(0)+\alpha_{d+1}\log\bigl{(}\tfrac{2}{e^{\gamma}\epsilon}\bigr{)}+\frac{1}{2}\sum_{k=0}^{d}\alpha_{k}\,\Gamma\bigl{(}\tfrac{d+1-k}{2}\bigr{)}\left(\tfrac{2}{\epsilon}\right)^{d+1-k}\,,$
(239)

where $\zeta_{D}(z)={\rm
Tr}\,D^{-z}=\frac{1}{\Gamma(z)}\int\frac{d\tau}{\tau}\,\tau^{z}\,{\rm
Tr}\,e^{-\tau D}$ is the zeta function of $D$ and $\alpha_{d+1}=\zeta_{D}(0)$.

We can apply the same idea to the square-root regulated character formula
(235) for $Z_{\epsilon}$. The latter is obtained from the simpler integrand of
the formal character formula (236) for $Z_{\epsilon=0}$ by dividing it by
$r(\epsilon,t)\equiv\sqrt{t^{2}-\epsilon^{2}}/t$ and replacing $\nu$ by $\nu
r(\epsilon,t)$:

$\displaystyle\log
Z_{\epsilon=0}=\int_{0}^{\infty}\frac{dt}{2t}\,F_{\nu}(t)\qquad\Rightarrow\qquad\log
Z_{\epsilon}=\int_{\epsilon}^{\infty}\frac{dt}{2rt}\,F_{r\nu}(t)\,,\qquad
r\equiv\frac{\sqrt{t^{2}-\epsilon^{2}}}{t}\,.$ (240)

Note that $0<r<1$ for all $t>\epsilon$, $r\sim{\cal O}(1)$ for $t\sim\epsilon$
and $r\to 1$ for $t\gg\epsilon$. Therefore, given the $t\to 0$ behavior of the
integrand in the formal character formula for $Z_{\epsilon=0}$,

$\displaystyle\frac{1}{2t}F_{\nu}(t)\to\frac{1}{t}\sum_{k=0}^{d+1}b_{k}(\nu)\,t^{-(d+1-k)}=:\frac{1}{2t}F_{\nu}^{\rm
uv}(t)\,,\qquad b_{k}(\nu)=\sum_{\ell=0}^{k}b_{k\ell}\,\nu^{\ell}\,,$ (241)

we get the $t\sim\epsilon\to 0$ behavior of the integrand for the exact
$Z_{\epsilon}$:

$\displaystyle\frac{1}{2rt}\,F_{r\nu}(t)\to\frac{1}{2rt}\,F^{\rm
uv}_{r\nu}(t)=\frac{1}{rt}\sum_{k,\ell}b_{k\ell}\,\nu^{\ell}\,r^{\ell}\,t^{-(d+1-k)}\,.$
(242)

Thus we can separate $\log Z_{\epsilon}=\log\tilde{Z}^{\rm
uv}_{\epsilon}+\log\tilde{Z}^{\rm ir}$, with

$\displaystyle\log\tilde{Z}^{\rm
uv}_{\epsilon}\equiv\int_{\epsilon}^{\infty}\frac{dt}{2rt}\,F^{\rm
uv}_{r\nu}(t)\,e^{-\mu t}\,,\qquad\log\tilde{Z}^{\rm
ir}\equiv\int_{0}^{\infty}\frac{dt}{2t}\,\bigl{(}F_{\nu}(t)-F^{\rm
uv}_{\nu}(t)\bigr{)}e^{-\mu t}\,.$ (243)

Again the limit $\mu\to 0$ is understood. We were allowed to put $\epsilon=0$
in the IR part because it is UV finite.

#### Evaluation of UV part

Using the expansion (242), the UV part can be evaluated explicitly as

$\displaystyle\log\tilde{Z}^{\rm uv}_{\epsilon}=\frac{1}{2}\sum_{\ell,k\leq
d}b_{k\ell}\,B\bigl{(}\tfrac{d+1-k}{2},\tfrac{\ell+1}{2}\bigr{)}\,\nu^{\ell}\,\epsilon^{-(d+1-k)}-\sum_{\ell}b_{d+1,\ell}\bigl{(}H_{\ell}-\tfrac{1}{2}H_{\ell/2}+\log(\tfrac{e^{\gamma}\,\epsilon\,\mu}{2})\bigr{)}\nu^{\ell}$
(244)

where $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ is the Euler beta
function and $H_{x}=\gamma+\frac{\Gamma^{\prime}(1+x)}{\Gamma(1+x)}$ which for
integer $x$ is the $x$-th harmonic number
$H_{x}=1+\frac{1}{2}+\cdots+\frac{1}{x}$. For example for $d=3$, we get

$\displaystyle\log\tilde{Z}^{\rm
uv}_{\epsilon}=\tfrac{4}{3}\,\epsilon^{-4}-\tfrac{4\nu^{2}+1}{12}\,\epsilon^{-2}-\bigl{(}\tfrac{\nu^{4}}{9}+\tfrac{\nu^{2}}{24}\bigr{)}-\bigl{(}\tfrac{\nu^{4}}{12}+\tfrac{\nu^{2}}{24}-\tfrac{17}{2880}\bigr{)}\log\bigl{(}\tfrac{e^{\gamma}\epsilon\mu}{2}\bigr{)}\,.$
(245)

This gives an explicit expression for the part of $\log Z$ denoted ${\rm
Pol}(\Delta)$ in Denef:2009kn , without having to invoke an independent
computation of the heat kernel coefficients. Indeed, turning this around, by
comparing (244) to (239), we can express the heat kernel coefficients
$\alpha_{k}$ explicitly in terms of the character coefficients $b_{k,\ell}$.
In particular the Weyl anomaly coefficient is simply given by the coefficient
$b_{d+1}=\sum_{\ell}b_{d+1,\ell}\nu^{\ell}$ of the $1/t$ term in the integrand
of the formal character formula (236). More generally,

$\displaystyle\alpha_{k}=\sum_{\ell}\frac{\Gamma(\frac{\ell+1}{2})}{2^{d+1-k}\Gamma(\frac{d+1-k+\ell+1}{2})}\,b_{k\ell}\,\nu^{\ell}\,.$
(246)

For example for $d=3$, this becomes $\alpha_{0}=\frac{1}{12}b_{00}$,
$\alpha_{2}=\frac{1}{2}b_{20}+\frac{\nu^{2}}{6}b_{22}$ and $\alpha_{4}=b_{4}$.
From the small-$t$ expansion $\frac{1}{2t}F_{\nu}(t)\to\sum_{k}b_{k}t^{3-k}$
in (236) we read off $b_{0}=2$, $b_{2}=-\frac{1}{12}-\nu^{2}$ and
$b_{4}=-\frac{17}{2880}+\frac{1}{24}\nu^{2}+\frac{1}{12}\nu^{4}$. Thus
$\alpha_{0}=\frac{1}{6}$, $\alpha_{2}=-\frac{1}{24}-\frac{1}{6}\nu^{2}$ and
$\alpha_{4}=-\frac{17}{2880}+\frac{1}{24}\nu^{2}+\frac{1}{12}\nu^{4}$.

#### Evaluation of IR part

As we explain momentarily, the IR part can be evaluated as

$\displaystyle\log\tilde{Z}^{\rm
ir}=\frac{1}{2}\zeta_{\nu}^{\prime}(0)+b_{d+1}\log\mu\,,\qquad\zeta_{\nu}(z)\equiv\frac{1}{\Gamma(z)}\int_{0}^{\infty}\frac{dt}{t}\,t^{z}\,F_{\nu}(t)\,,$
(247)

where like for the spectral zeta function $\zeta_{D}(z)$, the “character zeta
function” $\zeta_{\nu}(z)$ is defined by the above integral for $z$
sufficiently large and by analytic continuation for $z\to 0$. This zeta
function representation of $\log Z^{\rm ir}$ follows from the following
observations. If we define $\zeta^{\rm
ir}_{\nu}(z)\equiv\frac{1}{\Gamma(z)}\int_{0}^{\infty}\frac{dt}{t}\,t^{z}\bigl{(}F_{\nu}(t)-F_{\nu}^{\rm
uv}(t)\bigr{)}\,e^{-\mu t}$, then since the integral remains finite for $z\to
0$, while $\Gamma(z)\sim 1/z$ and $\partial_{z}(1/\Gamma(z))\to 1$, we
trivially have $\frac{1}{2}\partial_{z}\zeta^{\rm
ir}_{\nu}(z)|_{z=0}={\log\tilde{Z}^{\rm ir}}$. Moreover for $z$ sufficiently
large we have in the limit $\mu\to 0$ that $\frac{1}{2}\zeta^{\rm
uv}(z)\equiv\frac{1}{\Gamma(z)}\int_{0}^{\infty}\frac{dt}{2t}\,t^{z}F^{\rm
uv}_{\nu}(t)\,e^{-\mu t}=b_{d+1}\mu^{-z}$, so upon analytic continuation we
have ${1/2}\partial_{z}\zeta^{\rm uv}(z)|_{z=0}=-b_{d+1}\log\mu$, and (247)
follows.

In contrast to the spectral zeta function, the character zeta function can
straightforwardly be evaluated in terms of Hurwitz zeta functions. Indeed,
denoting $\Delta_{\pm}=\frac{d}{2}\pm i\nu$, we have
$F_{D}(t)=\sum_{n}Q(n)\,e^{-t(n+\Delta_{+})(n+\Delta_{-})}$ where the spectral
degeneracy $Q(n)$ is some polynomial in $n$, and
$\zeta_{D}(z)=\sum_{n=0}^{\infty}Q(n)\,\bigl{(}(n+\Delta_{+})(n+\Delta_{-})\bigr{)}^{-z}$,
which is quite tricky to evaluate, whereas
$F_{\nu}(t)=\sum_{n}Q(n)\bigl{(}e^{-t(n+\Delta_{+})}+e^{-t(n+\Delta_{-})}\bigr{)}$,
and we can immediately express the associated character zeta function as a
finite sum of Hurwitz zeta functions
$\zeta(z,\Delta)=\sum_{n=0}^{\infty}(n+\Delta)^{-z}$:

$\displaystyle\zeta_{\nu}(z)=\sum_{\pm}\sum_{n=0}^{\infty}Q(n)(n+\Delta_{\pm})^{-z}=\sum_{\pm}Q(\hat{\delta}-\Delta_{\pm})\,\zeta(z,\Delta_{\pm})\,.$
(248)

Here $\hat{\delta}$ is the unit $z$-shift operator acting as
$\hat{\delta}^{n}\zeta(z,\Delta)=\zeta(z-n,\Delta)$; for example if
$Q(n)=n^{2}$ we have
$Q(\hat{\delta}-\Delta)\,\zeta(z,\Delta)=(\hat{\delta}^{2}-2\Delta\hat{\delta}+\Delta^{2})\zeta(z,\Delta)=\zeta(z-2,\Delta)-2\Delta\zeta(z-1,\Delta)+\Delta^{2}\zeta(z,\Delta)$.

###  C.2 Result and examples

#### Result

Altogether we conclude that given a formal character integral formula

$\displaystyle\log Z_{\rm PI}=\int_{0}^{\infty}\frac{dt}{2t}\,F_{\nu}(t)\,,$
(249)

for a field corresponding to a dSd+1 irrep of dimension $\frac{d}{2}+i\nu$,
with IR and UV expansions

$\displaystyle
F_{\nu}(t)=\sum_{\Delta}\sum_{n=0}^{\infty}P_{\Delta}(n)\,e^{-(n+\Delta)t}\,,\qquad\frac{1}{2t}F_{\nu}(t)=\frac{1}{t}\sum_{k=0}^{d+1}b_{k}(\nu)\,t^{-(d+1-k)}\,+{\cal
O}(t^{0})\,,$ (250)

where $b_{k}(\nu)=\sum_{\ell}b_{k\ell}\,\nu^{\ell}$, we obtain the exact
$Z_{\rm PI}$ with heat kernel regulator $e^{-\epsilon^{2}/4\tau}$ as

$\boxed{\begin{aligned} \log Z_{{\rm
PI},\epsilon}=&\frac{1}{2}\sum_{\Delta}P_{\Delta}(\hat{\delta}-\Delta)\,\zeta^{\prime}(0,\Delta)-\sum_{\ell=0}^{d+1}b_{d+1,\ell}\bigl{(}H_{\ell}-\tfrac{1}{2}H_{\ell/2}\bigr{)}\nu^{\ell}+b_{d+1}(\nu)\log(2e^{-\gamma}/\epsilon)\\!\\\
&+\frac{1}{2}\sum_{k=0}^{d}\sum_{\ell=0}^{k}b_{k\ell}\,B\bigl{(}\tfrac{d+1-k}{2},\tfrac{\ell+1}{2}\bigr{)}\,\nu^{\ell}\,\epsilon^{-(d+1-k)}\,.\end{aligned}}$
(251)

Here $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$,
$H_{x}=\gamma+\frac{\Gamma^{\prime}(1+x)}{\Gamma(1+x)}$, which for integer $x$
is the $x$-th harmonic number $H_{x}=1+\frac{1}{2}+\cdots+\frac{1}{x}$, and
$\hat{\delta}$ is the unit shift operator acting on the first argument of the
Hurwitz zeta function $\zeta(z,\Delta)$: the polynomial
$P_{\Delta}(\hat{\delta}-\Delta)$ is to be expanded in powers of
$\hat{\delta}$, setting
$\hat{\delta}^{n}\zeta^{\prime}(0,\Delta)\equiv\zeta^{\prime}(-n,\Delta)$.
Finally the heat kernel coefficients are

$\displaystyle\alpha_{k}=\sum_{\ell}\frac{\Gamma(\frac{\ell+1}{2})}{2^{d+1-k}\Gamma(\frac{d+1-k+\ell+1}{2})}\,b_{k\ell}\,\nu^{\ell}\,.$
(252)

If we are only interested in the finite part of $\log Z$, only the first three
terms in (251) matter. Note that the third and the second term ${\cal
M}_{\nu}\equiv\sum_{\ell}b_{d+1,\ell}\bigl{(}H_{\ell}-\tfrac{1}{2}H_{\ell/2}\bigr{)}$
is in general nonvanishing for even $d+1$. By comparing (251) to (239), say in
the scalar case discussed earlier, we see that
$\zeta_{D}^{\prime}(0)=\zeta_{\nu}^{\prime}(0)+2{\cal M}_{\nu}$. Thus $2{\cal
M}_{\nu}$ can be thought of as correcting the formal factorization
$\sum_{n}\log(n+\Delta_{+})(n+\Delta_{-})=\sum_{n}\log(n+\Delta_{+})+\sum_{n}\log(n+\Delta_{-})$
in zeta function regularization. For this reason ${\cal M}_{\nu}$ is called
the multiplicative “anomaly”, as reviewed in Dowker:2014xca . The above thus
generalizes the explicit formulae in Dowker:2014xca  for ${\cal M}_{\nu}$ to
fields of arbitrary representation content.

#### Examples

1. A scalar on $S^{2}$ ($d=1$) with $\Delta_{\pm}=\frac{1}{2}\pm i\nu$ has $F_{\nu}(t)=\frac{1+e^{-t}}{1-e^{-t}}\frac{e^{-\Delta_{+}t}+e^{-\Delta_{-}t}}{1-e^{-t}}$ so the IR and UV expansions are $F_{\nu}(t)=\sum_{\pm}\sum_{n=0}^{\infty}(2n+1)e^{-(\Delta_{\pm}+n)t}$ and $\frac{1}{2t}\,F_{\nu}(t)=\frac{2}{t^{3}}+\frac{\frac{1}{12}-\nu^{2}}{t}+{\cal O}(t^{0})$. Therefore according to (251)

$\displaystyle\log Z_{\rm PI,\epsilon}=$
$\displaystyle\sum_{\Delta=\frac{1}{2}\pm
i\nu}\Bigl{(}\zeta^{\prime}(-1,\Delta)-(\Delta-\tfrac{1}{2})\zeta^{\prime}(0,\Delta)\Bigr{)}+\nu^{2}+\bigl{(}\tfrac{1}{12}-\nu^{2}\bigr{)}\log\bigl{(}2\,e^{-\gamma}/\epsilon\bigr{)}+\frac{2}{\epsilon^{2}}\,.$
(253)

The heat kernel coefficients are obtained from (252) as $\alpha_{0}=1$ and
$\alpha_{2}=\frac{1}{12}-\nu^{2}$.

2. For a scalar on $S^{3}$, $F_{\nu}(t)=\sum_{\pm}\sum_{n=0}^{\infty}(n+1)^{2}e^{-(\Delta_{\pm}+n)t}$, $\frac{1}{2t}F_{\nu}(t)\to\frac{2}{t^{4}}-\frac{\nu^{2}}{t^{2}}+{\cal O}(t^{0})$, so

$\displaystyle\log Z_{\rm PI,\epsilon}=$
$\displaystyle\sum_{\pm}\Bigl{(}\tfrac{1}{2}\zeta^{\prime}(-2,1\pm i\nu)\mp
i\nu\zeta^{\prime}(-1,1\pm i\nu)-\tfrac{1}{2}\nu^{2}\zeta^{\prime}(0,1\pm
i\nu)\Bigr{)}-\frac{\pi\nu^{2}}{4\epsilon}+\frac{\pi}{2\epsilon^{3}}\,.$ (254)

The heat kernel coefficients are $\alpha_{0}=\frac{\sqrt{\pi}}{4}$,
$\alpha_{2}=-\frac{\sqrt{\pi}}{4}\nu^{2}$. In particular for a conformally
coupled scalar, i.e. $\Delta=\frac{1}{2},\frac{3}{2}$ or equivalently
$\nu=i/2$, we get for the finite part the familiar result $\log Z_{\rm
PI}=\frac{3\zeta(3)}{16\pi^{2}}-\frac{\log(2)}{8}$. For $\Delta=1$, i.e.
$\nu=0$, we get $\log Z_{\rm PI}=-\frac{\zeta(3)}{4\pi^{2}}$. Notice that the
finite part looks quite different from (50) obtained by contour integration.
Nevertheless they are in fact the same function. 3. A more interesting example
is the massive spin-$s$ field on $S^{4}$ with $\Delta_{\pm}=\frac{3}{2}\pm
i\nu$. In this case, (83) combined with (329) or equivalently (84) gives
$F_{\nu}=F_{\rm bulk}-F_{\rm edge}$ with

$\displaystyle F_{\rm bulk}(t)$ $\displaystyle=\sum_{\Delta=\frac{3}{2}\pm
i\nu}\sum_{n=-1}^{\infty}D^{3}_{s}D^{5}_{n}\,e^{-(n+\Delta)t}=D_{s}^{3}\,\frac{1+e^{-t}}{1-e^{-t}}\frac{e^{-(\frac{3}{2}+i\nu)t}+e^{-(\frac{3}{2}-i\nu)t}}{(1-e^{-t})^{3}}\,,\qquad\,$
(255) $\displaystyle F_{\rm edge}(t)$
$\displaystyle=\sum_{\Delta=\frac{1}{2}\pm
i\nu}\sum_{n=-1}^{\infty}D^{5}_{s-1}D^{3}_{n+1}\,e^{-(n+\Delta)t}=D_{s-1}^{5}\,\frac{1+e^{-t}}{1-e^{-t}}\frac{e^{-(\frac{1}{2}+i\nu)t}+e^{-(\frac{1}{2}-i\nu)t}}{(1-e^{-t})}\,,$
(256)

where $D_{p}^{3}=2p+1$, $D^{5}_{p}=\frac{1}{6}(2p+3)(p+2)(p+1)$. In particular
note that with $g_{s}\equiv D_{s}^{3}=2s+1$, we have
$D_{s-1}^{5}=\frac{1}{24}g_{s}(g_{s}^{2}-1)$. The small-$t$ expansions are

$\displaystyle\tfrac{1}{2t}F_{\rm bulk}(t)$ $\displaystyle\to
g_{s}\Bigl{(}2\,t^{-5}-\bigl{(}\nu^{2}+\tfrac{1}{12}\bigr{)}t^{-3}+\bigl{(}\tfrac{\nu^{4}}{12}+\tfrac{\nu^{2}}{24}-\tfrac{17}{2880}\bigr{)}t^{-1}+{\cal
O}(t^{0})\Bigr{)}$ (257) $\displaystyle\tfrac{1}{2t}F_{\rm edge}(t)$
$\displaystyle\to\tfrac{1}{24}g_{s}(g_{s}^{2}-1)\Bigl{(}2\,t^{-3}+\bigl{(}\tfrac{1}{12}-\nu^{2}\bigr{)}t^{-1}+{\cal
O}(t^{0})\Bigr{)}\,.$ (258)

Thus the exact partition function for a massive spin-$s$ field is

$\displaystyle\log Z_{\rm PI,\epsilon}$
$\displaystyle=g_{s}\sum_{\Delta=\frac{3}{2}\pm
i\nu}\Bigl{(}\tfrac{1}{6}\zeta^{\prime}(-3,\Delta)\mp\tfrac{1}{2}i\nu\zeta^{\prime}(-2,\Delta)-\bigl{(}\tfrac{1}{2}\nu^{2}+\tfrac{1}{24}\bigr{)}\zeta^{\prime}(-1,\Delta)\pm
i\bigl{(}\tfrac{1}{24}\nu+\tfrac{1}{6}\nu^{3}\bigr{)}\zeta^{\prime}(0,\Delta)\Bigr{)}$
$\displaystyle\quad-\tfrac{1}{24}g_{s}(g_{s}^{2}-1)\sum_{\Delta=\frac{1}{2}\pm
i\nu}\Bigl{(}\zeta^{\prime}(-1,\Delta)\mp
i\nu\zeta^{\prime}(0,\Delta)\Bigr{)}\,-\tfrac{1}{24}g_{s}^{3}\nu^{2}-\tfrac{1}{9}g_{s}\nu^{4}$
(259)
$\displaystyle\quad+\Bigl{(}g_{s}^{3}\bigl{(}\tfrac{1}{24}\nu^{2}-\tfrac{1}{288}\bigr{)}+g_{s}\bigl{(}\tfrac{1}{12}\nu^{4}-\tfrac{7}{2880}\bigr{)}\Bigr{)}\log(2\,e^{-\gamma}/\epsilon)-\bigl{(}\tfrac{1}{12}g_{s}^{3}+\tfrac{1}{3}g_{s}\nu^{2}\bigr{)}\epsilon^{-2}+\tfrac{4}{3}g_{s}\epsilon^{-4}\,.$

Finally the heat kernel coefficients are

$\displaystyle\alpha_{0}=\tfrac{1}{6}g_{s}\,,\qquad\alpha_{2}=-\tfrac{1}{24}g_{s}^{3}-\tfrac{1}{6}g_{s}\nu^{2}\,,\qquad\alpha_{4}=g_{s}^{3}\bigl{(}\tfrac{1}{24}\nu^{2}-\tfrac{1}{288}\bigr{)}+g_{s}\bigl{(}\tfrac{1}{12}\nu^{4}-\tfrac{7}{2880}\bigr{)}\,.$
(260)

#### 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:
$\bullet$ Path integral single-mode contributions: For our choice of heat-
kernel regulator $e^{-\epsilon^{2}/4\tau}$, the contribution to $\log Z_{{\rm
PI},\epsilon}$ from a single bosonic eigenmode with eigenvalue $\lambda$ is

$\displaystyle
I_{\lambda}=\int_{0}^{\infty}\frac{d\tau}{2\tau}\,e^{-{\epsilon^{2}}/{4\tau}}\,e^{-\tau\lambda}=K_{0}(\epsilon\sqrt{\lambda})\to-\frac{1}{2}\log\frac{\lambda}{M^{2}}\,,\qquad
M\equiv\frac{2e^{-\gamma}}{\epsilon}\,,$ (261)

Different regulator insertions lead to a similar result in the limit
$\epsilon\to 0$, with $M=c/\epsilon$ for some regulator-dependent constant
$c$. A closely related formula is obtained for the contribution from an
individual term in the sum (234) or equivalently in the IR expansion of (250),
which amounts to computing (249) with $F_{\nu}(t)\equiv e^{-\rho t}$,
$\rho=a\pm i\nu$. The small-$t$ expansion is
$\frac{1}{2t}F_{\nu}(t)=\frac{1}{2t}+{\cal O}(t^{0})$, so the UV part is given
by the log term in (251) with coefficient $\frac{1}{2}$, and the IR part is
$\frac{1}{2}\zeta_{\nu}^{\prime}(0)=-\frac{1}{2}\log\rho$ as in (247). Thus

$\displaystyle I^{\prime}_{\rho}=\int_{0}^{\infty}\frac{dt}{2t}\,e^{-\rho
t}\to-\frac{1}{2}\log\frac{\rho}{M}\,,\qquad
M=\frac{2e^{-\gamma}}{\epsilon}\,,$ (262)

where the integral is understood to be regularized as in (234),
$I^{\prime}_{\rho}=\int_{\epsilon}^{\infty}\frac{dt}{2\sqrt{t^{2}-\epsilon^{2}}}\,e^{-ta-i\nu\sqrt{t^{2}-\epsilon^{2}}}$,
left implicit here. The similarities between (261) and (262) are of course no
accident, since in our setup, the former splits into the sum of two integrals
of the latter type: writing $\lambda=a^{2}+\nu^{2}=(a+i\nu)(a-i\nu)$, we have
$I_{\lambda}=I^{\prime}_{a+i\nu}+I^{\prime}_{a-i\nu}$. $\bullet$ Quasinormal
mode contributions: Considering a character quasinormal mode expansion
$\chi(t)=\sum_{r}N_{r}\,e^{-r|t|}$ as in (14), the IR contribution from a
single bosonic/fermionic QNM is

$\displaystyle\int_{0}^{\infty}\frac{dt}{2t}\frac{1+e^{-t}}{1-e^{-t}}\,e^{-r\,t}\biggr{|}_{\rm
IR}=\log\frac{\Gamma(r+1)}{\mu^{r}\sqrt{2\pi
r}}\,,\qquad-\int_{0}^{\infty}\frac{dt}{2t}\frac{2e^{-t/2}}{1-e^{-t}}\,e^{-r\,t}\biggr{|}_{\rm
IR}=-\log\frac{\Gamma(r+\frac{1}{2})}{\mu^{r}\sqrt{2\pi}}$ (263)

$\bullet$ Harmonic oscillator: The character of a $d=0$ scalar of mass $\nu$
is $\chi(t)=e^{-i\nu t}+e^{i\nu t}$, hence

$\displaystyle\log Z_{\rm
PI,\epsilon}=\int_{0}^{\infty}\frac{dt}{2t}\,\frac{1+e^{-t}}{1-e^{-t}}\,\bigl{(}e^{-i\nu
t}+e^{i\nu
t}\bigr{)}=\frac{\pi}{\epsilon}-\log\bigl{(}e^{\pi\nu}-e^{-\pi\nu}\bigr{)}\,.$
(264)

The finite part gives the canonical bosonic harmonic oscillator thermal
partition function ${\rm Tr}\,e^{-\beta
H}=\sum_{n}e^{-\beta\nu(n+\frac{1}{2})}=\bigl{(}e^{\beta\nu/2}-e^{-\beta\nu/2}\bigr{)}^{-1}$
at $\beta=2\pi$. The fermionic version is

$\displaystyle\log Z_{\rm
PI,\epsilon}=-\int_{0}^{\infty}\frac{dt}{2t}\,\frac{2e^{-t/2}}{1-e^{-t}}\,\bigl{(}e^{-i\nu
t}+e^{i\nu
t}\bigr{)}=-\frac{\pi}{\epsilon}+\log\bigl{(}e^{\pi\nu}+e^{-\pi\nu}\bigr{)}\,.$
(265)

###  C.3 Massless case

Here we give a few more details on how to use (251) to explicitly evaluate
$Z_{\rm PI}$ in the massless case, and work out the exact $Z_{\rm PI}$ for
Einstein gravity on $S^{4}$ as an example.

Our final result for the massless one-loop $Z_{\rm PI}=Z_{G}\cdot Z_{\rm
char}$ is given by (112):

$\displaystyle Z_{\rm PI}=i^{-P}\,\frac{\gamma^{\rm dimG}}{{\rm
vol}{(G)}_{{\rm c}}}\cdot\exp\int^{\times}\frac{dt}{2t}\,F\,,\qquad
F=\frac{1+q}{1-q}\Bigl{(}\bigl{[}\hat{\chi}_{\rm
bulk}\bigr{]}_{+}-\bigl{[}\hat{\chi}_{\rm edge}\bigr{]}_{+}-2\dim
G\Bigr{)}\,,$ (266)

where for $s=2$ gravity $\gamma=\sqrt{\frac{8\pi G_{\rm N}}{A_{d-1}}}$,
$P=d+3$, $G=SO(d+2)$ and ${\rm vol}{(G)}_{{\rm c}}=(\ref{vcSO})$.

$\bullet$ 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 (266). For the other UV terms in (251) (including the
“multiplicative anomaly”), a problem might seem to be that we need a
continuously variable dimension parameter $\Delta=\frac{d}{2}+i\nu$, whereas
massless fields, and our explicit formulae for
$\hat{\chi}\to[\hat{\chi}]_{+}$, require fixed integer dimensions. This
problem is easily solved, as the UV part can actually be computed from the
original naive character formula (364):

$\displaystyle\log Z_{\rm PI}\bigr{|}_{\rm
UV}=\int\frac{dt}{2t}\,\hat{F}\Bigr{|}_{\rm
UV}\,,\qquad\hat{F}=\frac{1+q}{1-q}\bigl{(}\hat{\chi}_{\rm
bulk}-\hat{\chi}_{\rm edge}\bigr{)}\,,$ (267)

Indeed since $\hat{F}\to F=\\{\hat{F}\\}_{+}$ in (366) affects just a finite
number of terms $c_{k}q^{k}\to c_{k}q^{-k}$, it does not alter the small-$t$
(UV) part of the integral. Moreover
$\hat{\chi}_{s}=\hat{\chi}_{s,\nu_{\phi}}-\hat{\chi}_{s,\nu_{\xi}}$, where
$\hat{\chi}_{s,\nu}$ is a massive spin-$s$ character. Thus the UV part may be
obtained simply by combining the results of (251) for general $\nu$ and $s$,
substituting the values $\nu_{\phi}$, $\nu_{\xi}$ set by (95).

$\bullet$ IR part: The IR part is the $\zeta^{\prime}$ part of (251), obtained
from the $q$-expansion of $F(q)$ in (266). This can be found in general by
using

$\displaystyle\frac{1+q}{1-q}\,\frac{q^{\Delta}}{(1-q)^{k}}=\sum_{n=0}^{\infty}P(n)\,q^{n+\Delta}\,,\qquad
P(n)=D_{n}^{k+2}\,,$ (268)

with $D_{n}^{k+2}$ the polynomial given in (209). For $k=0$, (263) is useful.
In particular, using the $\int^{\times}$ prescription (372), the IR
contribution from the last term in (266) is obtained by considering the $r\to
0$ limit of the bosonic formula in (263):

$\displaystyle\int^{\times}\frac{dt}{2t}\,\frac{1+q}{1-q}\bigl{(}-2\dim
G\bigr{)}\biggr{|}_{\rm IR}=\dim G\cdot\log(2\pi)\,.$ (269)

#### 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 (95) we read off $i\nu_{\phi}=\frac{3}{2}$,
$i\nu_{\xi}=\frac{5}{2}$, and from (102) we get

$\displaystyle\chi_{\rm bulk}=\bigl{[}\hat{\chi}_{\rm
bulk}\bigr{]}_{+}=\frac{10\,q^{3}-6\,q^{4}}{(1-q)^{3}}\,,\qquad\chi_{\rm
edge}=\bigl{[}\hat{\chi}_{\rm
edge}\bigr{]}_{+}=\frac{10\,q^{2}-2\,q^{3}}{1-q}\,.$ (270)

The small-$t$ expansion of the integrand in (266) is
$\frac{1}{2t}F=4\,t^{-5}-\frac{47}{3}\,t^{-3}-\frac{571}{45}\,t^{-1}+O(t^{0})$.
The coefficient of the log-divergent part of $\log Z_{\rm PI}$ is the
coefficient of $t^{-1}$:

$\displaystyle\log Z_{\rm PI}|_{\rm
log\,div}=-\frac{571}{45}\,\log\bigl{(}2e^{-\gamma}\epsilon^{-1}\bigr{)}\,,$
(271)

in agreement with Christensen:1979iy . The complete heat-kernel regularized UV
part of (251) can be read off directly from our earlier results for massive
spin-$s$ in $d=3$ as

$\displaystyle\log Z_{\rm PI}\bigr{|}_{\rm UV}$ $\displaystyle=\log Z_{\rm
PI}(s=2,\nu=\tfrac{3}{2}i)\bigr{|}_{\rm UV}\,-\,\log Z_{\rm
PI}(s=1,\nu=\tfrac{5}{2}i)\bigr{|}_{\rm UV}$
$\displaystyle=\frac{8}{3}\,\epsilon^{-4}-\frac{32}{3}\,\epsilon^{-2}-\frac{571}{45}\log\bigl{(}2e^{-\gamma}\epsilon^{-1}\bigr{)}+\frac{715}{48}\,.$
(272)

Here ${\cal M}=\frac{715}{48}$ is the “multiplicative anomaly” term. The
integrated heat kernel coefficients are similarly obtained from (260):
$\alpha_{0}=\frac{1}{3}$, $\alpha_{2}=-\frac{16}{3}$,
$\alpha_{4}=-\frac{571}{45}$.

The IR ($\zeta^{\prime}$) contributions from bulk and edge characters are
obtained from the expansions

$\displaystyle\frac{1+q}{1-q}\bigl{(}\chi_{\rm bulk}-\chi_{\rm
edge}\bigr{)}=\sum_{n}P_{\rm
b}(n)\,\bigl{(}10\,q^{3+n}-6\,q^{4+n}\bigr{)}-\sum_{n}P_{\rm
e}(n)\,\bigl{(}10\,q^{2+n}-2\,q^{3+n}\bigr{)}\,,$ (273)

where $P_{\rm b}(n)=D^{5}_{n}=\frac{1}{6}(n+1)(n+2)(2n+3)$, $P_{\rm
e}(n)=D^{3}_{n}=2n+1$. According to (251) this gives a contribution to $\log
Z_{\rm char}|_{\rm IR}$ equal to

$\displaystyle 5\,P_{\rm b}(\hat{\delta}-3)\,\zeta^{\prime}(0,3)-3\,P_{\rm
b}(\hat{\delta}-4)\,\zeta^{\prime}(0,4)-5\,P_{\rm
e}(\hat{\delta}-2)\,\zeta^{\prime}(0,2)+P_{\rm
e}(\hat{\delta}-3)\,\zeta^{\prime}(0,3)\,,$ (274)

where the polynomials are to be expanded in powers of $\hat{\delta}$, putting
$\hat{\delta}^{n}\zeta^{\prime}(0,\Delta)\equiv\zeta^{\prime}(-n,\Delta)$.
Working this out and adding the contribution (269), we find

$\displaystyle\log Z_{\rm char}\bigr{|}_{\rm IR}=-\log
2-\frac{47}{3}\,\zeta^{\prime}(-1)+\frac{2}{3}\,\zeta^{\prime}(-3)\,.$ (275)

Combining this with the UV part and reinstating $\ell$, we get272727This
splits as $\log Z_{\rm char}=10\,\log(2\pi)+\log Z_{\rm bulk}-\log Z_{\rm
edge}$ where $\log Z_{\rm
bulk}=\frac{8\ell^{4}}{3\epsilon^{4}}-\frac{8\ell^{2}}{3\epsilon^{2}}-\frac{331}{45}\log\frac{2e^{-\gamma}\ell}{\epsilon}+\frac{475}{48}\\\
-\frac{23}{3}\zeta^{\prime}(-1)+\frac{2}{3}\zeta^{\prime}(-3)-5\log(2\pi)$ and
$\log Z_{\rm
edge}=\frac{8\ell^{2}}{\epsilon^{2}}+\frac{16}{3}\log\frac{2e^{-\gamma}\ell}{\epsilon}-5+8\zeta^{\prime}(-1)+\log
2+5\log(2\pi)$.

$\displaystyle\log Z_{\rm char}=$
$\displaystyle\frac{8}{3}\,\frac{\ell^{4}}{\epsilon^{4}}-\frac{32}{3}\,\frac{\ell^{2}}{\epsilon^{2}}-\frac{571}{45}\log\frac{2e^{-\gamma}L}{\epsilon}$
$\displaystyle-\frac{571}{45}\log\frac{\ell}{L}+\frac{715}{48}-\log
2-\frac{47}{3}\zeta^{\prime}(-1)+\frac{2}{3}\zeta^{\prime}(-3)\,,$ (276)

where $L$ is an arbitrary length scale introduced to split off a finite part:

$\displaystyle\log Z_{\rm char}^{\rm
fin}=-\frac{571}{45}\log(\ell/L)+\frac{715}{48}-\log
2-\frac{47}{3}\zeta^{\prime}(-1)+\frac{2}{3}\zeta^{\prime}(-3)\,,$ (277)

To compute the group volume factor $Z_{G}$ in (266), we use (287) for
$G=SO(5)$ to get ${\rm vol}{(G)}_{{\rm c}}=\frac{2}{3}(2\pi)^{6}$, and
$\gamma=\sqrt{8\pi G_{\rm N}/4\pi\ell^{2}}$. Finally, $i^{-P}=i^{-(d+3)}=-1$.
Thus we conclude that the one-loop Euclidean path integral for Einstein
gravity on $S^{4}$ is

$\displaystyle Z_{\rm PI}=-\frac{(8\pi G_{\rm N}/4\pi\ell^{2})^{5}\,Z_{\rm
char}}{\frac{2}{3}(2\pi)^{6}}\,,$ (278)

where $Z_{\rm char}$ is given by (C.3).

#### Example: Einstein gravity on $S^{5}$

For $S^{5}$ an analogous (actually simpler) computation gives $Z_{\rm
PI}=i^{-7}Z_{G}Z_{\rm char}$ with

$\displaystyle\log Z_{\rm char}$
$\displaystyle=\frac{15\,\pi}{8}\,\frac{\ell^{5}}{\epsilon^{5}}-\frac{65\,\pi}{24}\,\frac{\ell^{3}}{\epsilon^{3}}-\frac{105\,\pi}{16}\,\frac{\ell}{\epsilon}+\frac{65\,\zeta(3)}{48\,\pi^{2}}+\frac{5\,\zeta(5)}{16\,\pi^{4}}+15\log(2\pi)$
(279) $\displaystyle\log Z_{G}$ $\displaystyle=\frac{15}{2}\log\frac{8\pi
G_{\rm N}}{2\pi^{2}\ell^{3}}-\log\frac{(2\pi)^{9}}{12}\,.$

###  C.4 Different regularization schemes

If we simply cut off the character integral at $t=\epsilon$, we get the
following instead of (251):

$\displaystyle\log Z_{\epsilon}=$
$\displaystyle\frac{1}{2}\sum_{\Delta}P_{\Delta}(\hat{\delta}-\Delta)\,\zeta^{\prime}(0,\Delta)+b_{d+1}(\nu)\log(e^{-\gamma}/\epsilon)+\sum_{k=0}^{d}\frac{b_{k}(\nu)}{d+1-k}\,\epsilon^{-(d+1-k)}\,,$
(280)

with $b_{k}(\nu)$ defined as before,
$\frac{1}{2t}F_{\nu}(t)=\sum_{k=0}^{d+1}b_{k}(\nu)\,t^{-(d+2-k)}\,+{\cal
O}(t^{0})$. Unsurprisingly, this differs from (251) only in its UV part, more
specifically in the terms polynomial in $\nu$, including the “multiplicative
anomaly” term discussed below (252). The transcendental ($\zeta^{\prime}$)
part and the $\log\epsilon$ coefficient remain unchanged. This remains true in
any other regularization.

If we stick with heat-kernel regularization but pick a different regulator
$f(\tau/\epsilon^{2})$ instead of $e^{-\epsilon^{2}/4\tau}$ (e.g. the
$f=(1-e^{-\tau\Lambda^{2}})^{k}$ 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 (261) with
$e^{-\epsilon^{2}/4\tau}\to f$. The relation $M(\epsilon)$ will depend on $f$,
but nothing else.

In dimensional regularization, some polynomial terms in $\nu$ 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 (280)) 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 (536) and (539), a portable covariant regularization scheme, like
heat-kernel regularization, must be applied consistently to each saddle. A
sphere-specific ad-hoc regularization as in (280) is not suitable for such
purposes.

##  Appendix 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
$S=(s_{1},\ldots,s_{r})$ 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

$\bullet$ $K=2r$:

$\displaystyle D^{K}_{S}={\cal N}_{K}^{-1}\prod_{1\leq i<j\leq
r}\bigl{(}\ell_{i}+\ell_{j}\bigr{)}\bigl{(}\ell_{i}-\ell_{j}\bigr{)}\,,\qquad\ell_{i}\equiv
s_{i}+\tfrac{K}{2}-i$ (281)

with ${\cal N}_{K}$ independent of $S$, hence fixed by $D^{K}_{0}=1$, i.e.
${\cal N}_{K}=\prod_{1\leq i<j\leq r}(K-i-j)(j-i)$.

$\bullet$ $K=2r+1$:

$\displaystyle D^{K}_{S}={\cal N}_{K}^{-1}\prod_{1\leq i\leq
r}(2\ell_{i})\prod_{1\leq i<j\leq
r}\bigl{(}\ell_{i}+\ell_{j}\bigr{)}\bigl{(}\ell_{i}-\ell_{j}\bigr{)}\,,\qquad\ell_{i}\equiv
s_{i}+\tfrac{K}{2}-i\,,$ (282)

where ${\cal N}_{K}$ is fixed as above: ${\cal N}_{K}=\prod_{1\leq i\leq
r}(K-2i)\prod_{1\leq i<j\leq r}(K-i-j)(j-i)$.

For convenience we list here some low-dimensional explicit expressions:

$\begin{array}[]{l|l|l|l}K&D_{s}^{K}&D_{n,s}^{K}&D^{K}_{k+\frac{1}{2},{\bf\frac{1}{2}}}\\\
\hline\cr 2&1&&1\\\ 3&2s+1&&2{k+1\choose 1}\\\
4&(s+1)^{2}&\left(n-s+1\right)\left(n+s+1\right)&2{k+2\choose 2}\\\
5&\frac{(s+1)(s+2)(2s+3)}{6}&\frac{\left(2n+3\right)\left(n-s+1\right)\left(n+s+2\right)\left(2s+1\right)}{6}&4{k+3\choose
3}\\\
6&\frac{(s+1)(s+2)^{2}(s+3)}{12}&\frac{\left(n+2\right){}^{2}\left(n-s+1\right)\left(n+s+3\right)\left(s+1\right){}^{2}}{12}&4{k+4\choose
4}\\\
7&\frac{(s+1)(s+2)(s+3)(s+4)(2s+5)}{120}&\frac{\left(n+2\right)\left(n+3\right)\left(2n+5\right)\left(n-s+1\right)\left(n+s+4\right)\left(s+1\right)\left(s+2\right)\left(2s+3\right)}{720}&8{k+5\choose
5}\\\
8&\frac{(s+1)(s+2)(s+3)^{2}(s+4)(s+5)}{360}&\frac{\left(n+2\right)\left(n+3\right){}^{2}\left(n+4\right)\left(n-s+1\right)\left(n+s+5\right)\left(s+1\right)\left(s+2\right){}^{2}\left(s+3\right)}{4320}&8{k+6\choose
6}\\\ \end{array}$ (283)

Here $(k+\frac{1}{2},{\bf\frac{1}{2}})$ means
$(s_{1},\ldots,s_{r})=(k+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2})$, i.e.
the spin $s=k+\frac{1}{2}$ representation.

For general $d\geq 3$, we can use (209) and (329) to compute

$\displaystyle
D^{K}_{s}=\mbox{\large$\binom{s+K-1}{K-1}-\binom{s+K-3}{K-1}$}\,,\qquad
D^{K}_{n,s}=D^{K}_{n}D^{K-2}_{s}-D^{K}_{s-1}D^{K-2}_{n+1}\,.$ (284)

Denoting 1 repeated $m$ times by $1^{m}$, e.g.
$(5,1^{2})=(5,1,1)={\tiny\Yvcentermath 1\yng(5,1,1)}$, we furthermore have

$\displaystyle
D_{1^{p}}^{d}=\mbox{\large$\binom{d}{p}$}\quad(p<\tfrac{d}{2}),\qquad
D^{2p}_{1^{p-1},\pm 1}=\mbox{\large$\tfrac{1}{2}\binom{2p}{p}$}\,,\qquad
D^{d+2}_{n,s,1^{m}}=D^{d+2}_{n}D^{d}_{s,1^{m}}-D^{d+2}_{s-1}D^{d}_{n+1,1^{m}}\,.$
(285)

###  D.2 Volumes

The volume of the unit sphere $S^{n}$ is

$\displaystyle\Omega_{n}\equiv{\rm
vol}(S^{n})=\frac{2\,\pi^{\frac{n+1}{2}}}{\Gamma\bigl{(}\frac{n+1}{2}\bigr{)}}\,=\,\frac{2\pi}{n-1}\cdot\Omega_{n-2}$
(286)

The volume of $SO(d+2)$ with respect to the invariant group metric normalized
such that minimal $SO(2)$ orbits have length $2\pi$ is

$\displaystyle{\rm vol}\bigl{(}SO(d+2)\bigr{)}_{{\rm c}}=\prod_{k=2}^{d+2}{\rm
vol}(S^{k-1})=\prod_{k=2}^{d+2}\frac{2\pi^{\frac{k}{2}}}{\Gamma(\frac{k}{2})}\,.$
(287)

This follows from the fact that the unit sphere $S^{n-1}=SO(n)/SO(n-1)$, which
implies ${\rm vol}(SO(n))_{{\rm c}}={\rm vol}(S^{n-1})\,{\rm
vol}(SO(n-1))_{{\rm c}}$ in the assumed normalization.

The volume of $SU(N)$ with respect to the invariant metric derived from the
matrix trace norm on the Lie algebra ${\rm su}(N)$ viewed as traceless
$N\times N$ matrices is (see e.g. Ooguri:2002gx )

$\displaystyle{\rm vol}\bigl{(}SU(N)\bigr{)}_{{\rm
Tr}_{N}}\,=\,\sqrt{N}\prod_{k=2}^{N}\frac{(2\pi)^{k}}{\Gamma(k)}\,=\,\sqrt{N}\,\frac{(2\pi)^{\frac{1}{2}(N-1)(N+2)}}{{\tt
G}(N+1)}\,.$ (288)

###  D.3 de Sitter and its Wick rotations to the sphere

Figure 14:  Penrose diagrams of dSd+1 and $S^{d+1}$ with coordinates 290, 292.
Each point corresponds to an $S^{d-1}$, contracted to zero size at thin-line
boundaries. a: Global dSd+1 in slices of constant $\bar{T}$. b: Wick rotation
of global dSd+1 to $S^{d+1}$. c: S/N = southern/northern static patch, F/P =
future/past wedge; slices of constant $T$ (gray) and $r$ (blue/red) = flows
generated by $H$. Yellow dot = horizon $r=1$. d: Wick-rotation of static patch
$S$ to $S^{d+1}$; slices of constant $\tau$ and constant $r$.

Global dSd+1 has a convenient description as a hyperboloid embedded in
${\mathbb{R}}^{1,d+1}$,

$\displaystyle X^{I}X_{I}\equiv\eta_{IJ}X^{I}X^{J}\equiv-
X_{0}^{2}+X_{1}^{2}+\cdots+X_{d+1}^{2}=\ell^{2}\,,\qquad
ds^{2}=\eta_{IJ}dX^{I}dX^{J}\,.$ (289)

Below we set $\boxed{\ell\equiv 1}$. The isometry group is $SO(1,d+1)$, with
generators $M_{IJ}=X_{I}\partial_{J}-X_{J}\partial_{I}$. Various coordinate
patches are shown in fig. 14a,c, with coordinates and metric given by

$\small\begin{array}[]{|l|l|l|l|}\hline\cr\text{co}&\text{embedding
}(X^{0},\ldots,X^{d+1})&\text{coordinate range}&\text{metric
}ds^{2}=\eta_{IJ}dX^{I}dX^{J}\\\ \hline\cr
G&(\sinh\bar{T},\cosh\bar{T}\,\bar{\Omega})&\bar{T}\in{\mathbb{R}},\,\bar{\Omega}\in
S^{d}&-d\bar{T}^{2}+\cosh^{2}\bar{T}\,d\bar{\Omega}^{2}\\\
S&(\sqrt{1-r^{2}}\sinh T,r\Omega,\sqrt{1-r^{2}}\cosh T)&T\in{\mathbb{R}},0\leq
r<1,\,\Omega\in
S^{d-1}&-(1-r^{2})dT^{2}+\frac{dr^{2}}{1-r^{2}}+r^{2}d\Omega^{2}\\\
F&(\sqrt{r^{2}-1}\cosh T,r\Omega,\sqrt{r^{2}-1}\sinh
T)&T\in{\mathbb{R}},\,r>1,\,\Omega\in
S^{d-1}&-\frac{dr^{2}}{r^{2}-1}+(r^{2}-1)dT^{2}+r^{2}d\Omega^{2}\\\
\hline\cr\end{array}$ (290)

illustrated in fig. 14a,c. $N$ is obtained from $S$ by $X^{d+1}\to-X^{d+1}$,
and $P$ from $F$ by $X^{0}\to-X^{0}$. The southern static patch $S$ is the
part of de Sitter causally accessible to an inertial observer at the south
pole of the global spatial $S^{d}$. The metric in this patch is static, with
the observer at $r=0$ and a horizon at $r=1$. The $SO(1,1)$ generator