N}}\,[\bar{\xi}^{\prime},\bar{\xi}]_{\rm L}\,,$ (409)

where $G_{\rm N}$ is the Newton constant. This can be checked by starting from
the Einstein-Hilbert action, expanding to quadratic order (see e.g.
Goon:2018fyu  for convenient and reliable explicit expressions in dSd+1), and
making the appropriate convention rescalings. This is the Einstein gravity
analog of (402). To be more concrete, let us consider the ambient space
description of the sphere $S^{d+1}$, i.e. $X^{I}X_{I}=1$ with
$X\in{\mathbb{R}}^{d+2}$. Then the basis of Killing vectors
$M_{IJ}\equiv-(X_{I}\partial_{J}-X_{J}\partial_{I})/\sqrt{16\pi G_{\rm N}}$
satisfy our standard ${\rm so}(d+2)$ commutation relations
$[\\![M_{12},M_{23}]\\!]=M_{13}$ etc, hence by (408), $\langle
M_{12}|M_{12}\rangle_{c}=1$. The path integral metric (397) on the other hand
corresponds to the invariant bilinear
$\langle\bar{\xi}|\bar{\xi}^{\prime}\rangle_{\rm
PI}=\frac{M^{4}}{2\pi}\,\int\bar{\xi}\cdot\bar{\xi}^{\prime}$, so $\langle
M_{12}|M_{12}\rangle_{\rm PI}=\frac{M^{4}}{2\pi}\frac{1}{16\pi G_{\rm
N}}\int_{S^{d+1}}(X_{1}^{2}+X_{2}^{2})=\frac{M^{4}}{2\pi}\frac{1}{16\pi G_{\rm
N}}\frac{2}{d+2}{\rm vol}(S^{d+1})$. Thus we obtain the following relation
between PI and canonical metrics and volumes for $G=SO(d+2)$:

$\displaystyle ds^{2}_{\rm PI}=\frac{A_{d-1}}{4G_{\rm
N}}\frac{1}{d(d+2)}\frac{M^{4}}{2\pi}\,ds^{2}_{\rm
c}\qquad\Rightarrow\qquad\frac{{\rm vol}{(G)}_{\rm PI}}{{\rm vol}{(G)}_{{\rm
c}}}=\left(\frac{A_{d-1}}{4G_{\rm
N}}\frac{1}{d(d+2)}\frac{M^{4}}{2\pi}\right)^{\frac{1}{2}\dim G}$ (410)

where $\dim G=\frac{1}{2}(d+2)(d+1)$, $A_{d-1}={\rm vol}(S^{d-1})$ as in
(407), and we again used ${\rm vol}(S^{d+1})=\frac{2\pi}{d}{\rm
vol}(S^{d-1})$. Combining this with (395), we get our desired result:

$\displaystyle Z_{G}=\frac{\gamma^{\dim G}}{{\rm vol}{(G)}_{{\rm
c}}}\,,\qquad\qquad\gamma\equiv\sqrt{\frac{8\pi G_{\rm N}}{A_{d-1}}}\,.$ (411)

#### Higher-spin gravity

We follow the same template for the higher-spin case. In the interacting
higher-spin theory, the Killing tensors generate a subalgebra of the nonlinear
gauge algebra, with bracket $[\\![\cdot,\cdot]\\!]$ inherited from the gauge
algebra, just like in the Yang-Mills and Einstein examples, except the gauge
algebra is much more complicated in the higher-spin case. Fortunately it is
not necessary to construct the exact gauge algebra to determine the Killing
tensor algebra: it suffices to determine the lowest order deformation of the
linearized gauge transformation (94) fixed by the transverse-traceless cubic
couplings of the theory Joung:2013nma . The Killing tensor algebra includes in
particular an ${\rm so}(d+2)$ subalgebra, that is to say an algebra of the
same general form (409) as in Einstein gravity, with some constant appearing
on the right-hand side determined by the spin-2 cubic coupling in the TT
action. We define the “Newton constant” $G_{\rm N}$ of the higher-spin theory
to be this constant, that is to say we read off $G_{\rm N}$ from the ${\rm
so}(d+2)$ Killing vector subalgebra by writing it as

$\displaystyle[\\![\bar{\xi},\bar{\xi}^{\prime}]\\!]=\sqrt{16\pi G_{\rm
N}}\,[\bar{\xi}^{\prime},\bar{\xi}]_{\rm L}\,.$ (412)

The standard Killing vector basis is then again given by
$M_{IJ}\equiv-(X_{I}\partial_{J}-X_{J}\partial_{I})/\sqrt{16\pi G_{\rm N}}$,
satisfying $[\\![M_{12},M_{23}]\\!]=M_{13}$ etc.

It was argued in Joung:2013nma  that for the most general set of consistent
parity-preserving cubic interactions, assuming the algebra does not split as a
direct sum of subalgebras, i.e. assuming the algebra is simple, there exists
an up to normalization unique invariant bilinear form
$\langle\cdot|\cdot\rangle_{c}$ on the Killing tensor algebra. We fix its
normalization again by requiring the standard ${\rm so}(d+2)$ Killing vectors
$M_{IJ}$ have unit norm,

$\displaystyle\langle M_{12}|M_{12}\rangle_{c}\equiv 1\,.$ (413)

Expressed in terms of the bilinears
$\langle\bar{\xi}_{s-1}|\bar{\xi}_{s-1}\rangle_{\rm
PI}=\frac{M^{4}}{2\pi}\,\int\bar{\xi}_{s-1}\cdot\bar{\xi}_{(s-1)}$
corresponding to (397), the invariant bilinear on the Killing tensor algebra
takes the general form

$\displaystyle\langle\bar{\xi}|\bar{\xi}^{\prime}\rangle_{c}=\sum_{s}B_{s}\,\langle\bar{\xi}_{s-1}|\bar{\xi}_{s-1}^{\prime}\rangle_{\rm
PI}\,,$ (414)

where $B_{s}$ are certain constants fixed in principle by the algebra. The
arguments given in Joung:2013nma  moreover imply that up to overall
normalization, the coefficients $B_{s}$ are independent of the coupling
constants in the theory. More specifically, adapted (with some work, as
described below) to our setting and conventions, and correcting for what we
believe is a typo in Joung:2013nma , the coefficients are
$B_{s}\propto(2s+d-4)(2s+d-2)$. We confirmed this by comparison to
Sleight:2016xqq , where the invariant bilinear form for minimal Vasiliev
gravity in AdSd+1, dual to the free $O(N)$ model, was spelled out in detail,
building on Joung:2013nma ; Joung:2014qya ; Sleight:2016dba . Analytically
continuing to positive cosmological constant, implementing their ambient space
$X$-contractions by a Gaussian integral, and reducing this integral to the
sphere by switching to spherical coordinates, the expression in
Sleight:2016xqq  can be brought to the form (414). This transformation almost
completely cancels the factorials in the analogous coefficients $b_{s}$ in
Sleight:2016xqq , reducing to the simple $B_{s}\propto(2s+d-4)(2s+d-2)$. (The
alternating signs of Sleight:2016xqq  are absent here due to the analytic
continuation to positive cc.) Taking into account our normalization
prescription (413) (which is different from the normalization chosen in
Sleight:2016xqq ), we thus get

$\displaystyle\bigl{\langle}\bar{\xi}|\bar{\xi}\bigr{\rangle}_{{\rm
c}}=\frac{2\pi}{M^{4}}\cdot\frac{4G_{\rm
N}}{A_{d-1}}\sum_{s}(2s+d-4)(2s+d-2)\,\bigl{\langle}\bar{\xi}^{(s-1)}\bigl{|}\bar{\xi}^{(s-1)}\bigr{\rangle}_{\rm
PI}\,,$ (415)

with $A_{d-1}={\rm vol}(S^{d-1})$ as before. In view of the independence of
the coefficients $B_{s}$ of the couplings within the class of theories
considered in Joung:2013nma , i.e. all parity-invariant massless higher-spin
gravity theories consistent to cubic order, this result is universal, valid
for this entire class.

As before for Einstein gravity and Yang Mills, from (415) we get the ratio

$\displaystyle\frac{{\rm vol}{(G)}_{\rm PI}}{{\rm vol}{(G)}_{{\rm
c}}}=\prod_{s}\left(\frac{M^{4}}{2\pi}\cdot\frac{A_{d-1}}{4G_{\rm
N}}\cdot\frac{1}{\bigl{(}2s+d-4\bigr{)}\bigl{(}2s+d-2\bigr{)}}\right)^{\frac{1}{2}N^{\rm
KT}_{s-1}}\,.$ (416)

Combining this with (395) we see that, rather delightfully, all the
unpleasant-looking factors cancel, leaving us with

$\displaystyle\boxed{\quad Z_{G}=\frac{\gamma^{\dim G}}{{\rm vol}{(G)}_{{\rm
c}}}\,\,,\qquad\gamma\equiv\sqrt{\frac{8\pi G_{\rm N}}{A_{d-1}}}\quad}$ (417)

This takes exactly the same form as the Einstein gravity result (411) except
$G$ is now the higher-spin symmetry group rather than the $SO(d+2)$ spin-$2$
symmetry group.

The cancelation of the UV divergent factors $M$ is as expected from
consistency with locality. The cancelation of the $s$-dependent factors on the
other hand seems surprising, in view of the different origin of the numerator
(spectrum of quadratic action) and the denominator (invariant bilinear form on
higher spin algebra of interactions). Apparently the former somehow knows
about the latter. We do not see an obvious reason why this is the case,
although the simplicity and universality of the result suggests we should, and
that this entire section should be replaceable by a one-line argument. Perhaps
it is obvious in a frame-like formalism.

#### Newton constant from central charge

Recall that the Newton constant $G_{\rm N}$ appearing in (417) was defined by
the ${\rm so}(d+2)$ algebra (412) in our normalization conventions. An
analogous definition can be given in dSd+1 or AdSd+1 where the algebra becomes
${\rm so}(1,d+1)$ resp. ${\rm so}(2,d)$. Starting from this definition,
$G_{\rm N}$ can also be formally related to the Cardy central charge $C$ of a
putative434343There is no assumption whatsoever this CFT actually exists. One
just imagines it exists and uses the formal holographic dictionary to infer
the two-point function of this imaginary CFT’s stress tensor. In dS, this
“dual CFT” can be thought of as computing the Hartle-Hawking wave function of
the universe Maldacena:2002vr . boundary CFT for AdS or dS, defined as the
coefficient of the CFT 2-point function of the putative energy-momentum
tensor. With our definition of $G_{\rm N}$, the computation of Kovtun:2008kw
remains unchanged, so we can just copy the result obtained there:

$\displaystyle\boxed{\quad C=\frac{(\pm
1)^{\frac{d-1}{2}}\,\Gamma(d+2)}{(d-1)\,\Gamma\bigl{(}\frac{d}{2}\bigr{)}^{2}}\cdot\frac{A_{d-1}}{8\pi
G_{\rm N}}\quad}$ (418)

where as before $A_{d-1}=2\pi^{d/2}\ell^{d-1}/\Gamma(\frac{d}{2})$, and $\pm
1$ = $+1$ for AdS and $-1$ for dS. The central charge of $N$ free real scalars
equals $C=\frac{d}{2(d-1)}\,N$ in the conventions used here. Note that (418)
reduces to the Brown-Henneaux formula $C=3\ell/2G_{\rm N}$ for $d=2$. In
Anninos:2011ui  it was argued that the Hartle-Hawking wave function of minimal
Vasiliev gravity in dS4 is perturbatively computed by a $d=3$ CFT of $N$ free
Grassmann scalars. This CFT has central charge $C=-\frac{3}{4}N$, hence
according to (418), $G_{\rm N}=2^{5}/\pi N$ and $\gamma=\sqrt{2G_{\rm
N}}=8/\sqrt{\pi N}$.

The final result of this appendix, putting everything together, is stated in
(112).

##  Appendix H One-loop and exact results for 3D theories

###  H.1 Character formula for $Z_{\rm PI}^{(1)}$

For $d=2$, i.e. dS3 / $S^{3}$, some of the generic-$d$ formulae in sections 4
and 5 become a bit degenerate, requiring separate discussion. One reason $d=2$
is a bit more subtle is that the spin-$s$ irreducible representation of
$SO(2)$ actually comes in two distinct chiral versions $\pm s$, as do the
corresponding $SO(1,3)$ irreducible representations $(\Delta,\pm s)$. Likewise
the field modes of a spin $s$ field in the path integral on $S^{3}$ split into
chiral irreps $(n,\pm s)$ of $SO(4)$. The dimensions $D^{2}_{s}=D^{2}_{-s}=1$
and $D^{4}_{n,s}=D^{4}_{n,-s}=(1+n-s)(1+n+s)$ of the $SO(2)$ and $SO(4)$
irreps are correctly reproduced by the Weyl dimension formula (281), rather
than (209). It should however be kept in mind that the single-particle Hilbert
space of for instance a massive spin-$s\geq 1$ Pauli-Fierz field on dS3
carries both helicity versions $(\Delta,\pm s)$ of the massive spin-$s$
SO(1,3) irrep, hence the character $\chi$ to be used in expressions for
$Z_{\rm PI}$ in this case is
$\chi=\chi_{+s}+\chi_{-s}=2\chi_{+s}=2(q^{\Delta}+q^{2-\Delta})/(1-q)^{2}$. On
the other hand for a real scalar field, we just have
$\chi=\chi_{0}=(q^{\Delta}+q^{2-\Delta})/(1-q)^{2}$.

For massless higher-spin gauge fields of spin $s\geq 2$, a similar reasoning
implies we should include an overall factor of 2 in (374)-(375). For an $s=1$
Maxwell field on the other hand, we get a factor of 2 in the first term but
not in the second term (since the gauge parameter/ghost field is a scalar).
The proper massless spin-$s$ bulk and edge characters are then obtained from
these by the polar term flip (377) as usual. This results in

$\displaystyle\chi_{\rm bulk,s}=0\quad(s\geq 2)\,,\qquad\chi^{(s=1)}_{\rm
bulk}=\frac{2q}{(1-q)^{2}}\,,\qquad\chi_{\rm edge,s}=0\quad(\mbox{all }s)\,,$
(419)

expressing the absence of propagating degrees of freedom (i.e. particles) for
massless spin-$s\geq 2$ fields on dS3.

This can also be derived more directly from the general path integral formula
(355), taking into account the $\pm s$ doubling. In particular for massless
$s\geq 2$, (365) gets replaced by

$\displaystyle\hat{F}_{s}$
$\displaystyle=\sum_{n\geq-1}\Theta(1+n)\,2D^{4}_{n,s}\bigl{(}q^{s+n}+q^{2-s+n}\bigr{)}-\sum_{n\geq-1}\Theta(1+n)\,2D^{4}_{n,s-1}\bigl{(}q^{s+1+n}+q^{1-s+n}\bigr{)},$
(420)

which matters for the $n=-1$ term because $\Theta(0)\equiv\frac{1}{2}$. For
$s=1$, we get instead

$\displaystyle\hat{F}_{1}$
$\displaystyle=\sum_{n\geq-1}\Theta(1+n)\,2D^{4}_{n,1}\bigl{(}q^{1+n}+q^{1+n}\bigr{)}-\sum_{n\geq
0}D^{4}_{n,0}\bigl{(}q^{2+n}+q^{n}\bigr{)}\,.$ (421)

For $s\geq 2$, the computation of $Z_{\rm char}$ and $Z_{G}$ remains
essentially unchanged. For $s=1$ there are some minor changes. The edge
character in (375) acquires an extra $q^{0}$ term in $d=2$ because
$q^{s+d-3}=q^{0}$, so the map $\hat{\chi}_{\rm edge}\to[\hat{\chi}_{\rm
edge}]_{+}$ gets an extra $-1$ subtraction, as a result of which the factor
$-2$ in (376) becomes a $-3$. Relatedly we get an extra $q^{0}$ term in
$q^{2s+d-4}+q^{2s+d-2}+2=q^{2}+3$ in (394), and we end up with
$Z_{G}=\tilde{\gamma}^{\dim G}/{\rm vol}{(G)}_{{\rm c}}$ with
$\tilde{\gamma}=g\ell/\sqrt{A_{1}}=g\sqrt{\ell}/\sqrt{2\pi}$ instead of (407).
Everything else remains the same.

Finally, the phase $i^{-P_{s}}$ (367) is somewhat modified. For $s\geq 2$,
from (420),

$\displaystyle
P_{s}=-\sum_{n=-1}^{s-3}\Theta(1+n)\,2D_{n,s}^{4}-\sum_{n=-1}^{s-2}\Theta(1+n)\,2D_{n,s-1}^{4}=\frac{1}{3}(2s-3)(2s-1)(2s+1)$
(422)

Note that $P_{2}=5$, in agreement with Polchinski:1988ua . $P_{1}=0$ as
before, since there are no negative modes.

#### Conclusion

The final result for $Z^{(1)}_{\rm PI}=Z_{G}Z_{\rm char}$ in dS3 replacing
(112)-(113) is:

  * •

For Einstein and HS gravity theories with $s\geq 2$,

$\displaystyle Z_{\rm PI}^{(1)}=i^{-P}\,\frac{\gamma^{\dim G}}{{\rm
vol}{(G)}_{{\rm c}}}\cdot Z_{\rm char}\,,\qquad Z_{\rm char}=e^{-2\dim
G\int^{\times}\frac{dt}{2t}\,\frac{1+q}{1-q}}=(2\pi)^{\dim G}e^{-\dim G\cdot
c\,\ell\,\epsilon^{-1}}\,,$ (423)

where as before $\gamma=\sqrt{8\pi G_{\rm N}/A_{1}}=\sqrt{4G_{\rm N}/\ell}$,
$P=\sum_{s}P_{s}$, and ${\rm vol}{(G)}_{{\rm c}}$ is the volume with respect
to the metric for which the standard $\rm so(4)$ generators $M_{IJ}$ have norm
1. We used (269) to evaluate $Z_{\rm char}$. The coefficient $c$ of the
linearly divergent term is an order 1 constant depending on the regularization
scheme. (For the heat kernel regularization of appendix C, following section
C.3, $c=\frac{3\pi}{4}$. For a simple cutoff at $t=\epsilon$ as in section
C.4, $c=2$.) The finite part is

$\displaystyle\boxed{\quad Z^{(1)}_{\rm
PI,fin}=i^{-P}\,\frac{(2\pi\gamma)^{\dim G}}{{\rm vol}{(G)}_{{\rm
c}}}\,,\qquad\gamma=\sqrt{\frac{8\pi G_{\rm N}}{2\pi\ell}}\quad}$ (424)

For example for Einstein gravity with $G=SO(4)$, we get

$\displaystyle Z^{(1)}_{\rm
PI,fin}=i^{-5}\,\frac{(2\pi\gamma)^{6}}{(2\pi)^{4}}=-i\,4\pi^{2}\gamma^{6}\,.$
(425)

  * •

For Yang-Mills theories with gauge group $G$ and coupling constant $g$, we get

$\displaystyle Z_{\rm PI}^{(1)}=\frac{\tilde{\gamma}^{\dim G}}{{\rm
vol}{(G)}_{{\rm c}}}\cdot e^{\dim
G\int^{\times}\frac{dt}{2t}\,\frac{1+q}{1-q}\left(\frac{2q}{(1-q)^{2}}-3\right)}\,,\qquad\tilde{\gamma}=\frac{g\sqrt{\ell}}{\sqrt{2\pi}}\,.$
(426)

Using (251), (269), the finite part evaluates to

$\displaystyle Z^{(1)}_{\rm PI,fin}=\frac{(2\pi g\sqrt{\ell}\,Z_{1})^{\dim
G}}{{\rm vol}{(G)}_{{\rm c}}}\,,\qquad
Z_{1}=e^{-\frac{\zeta(3)}{4\pi^{2}}}\,.$ (427)

As in (112), ${\rm vol}{(G)}_{{\rm c}}$ is the volume of $G$ with respect to
the metric defined by the trace appearing in the Yang-Mills action. As a
check, for $G=U(1)$ we have ${\rm vol}{(G)}_{{\rm c}}=2\pi$, so
$Z=g\,e^{-\zeta(3)/4\pi^{2}}\sqrt{\ell}$ in agreement with Klebanov:2011td
eq. (3.25).

  * •

We could also consider the Chern-Simons partition function on $S^{3}$,

$\displaystyle Z_{k}=\int{\cal D}A\,e^{i\,k\,S_{\rm CS}[A]}\,,\qquad S_{\rm
CS}[A]\equiv\frac{1}{4\pi}\int{\rm Tr}(A\wedge dA+\frac{2}{3}A\wedge A\wedge
A)\,,$ (428)

with $k>0$ suitably quantized ($k\in{\mathbb{Z}}$ for $G=SU(N)$ with ${\rm
Tr}$ the trace in the $N$-dimensional representation). Because in this case
the action is first order in the derivatives and not parity-invariant, it
falls outside the class of theories we have focused on in this paper. It is
not too hard though to generalize the analysis to this case. The main
difference with Yang-Mills is that $\chi_{\rm bulk}=0=\chi_{\rm edge}$: like
in the $s\geq 2$ case, the $s=1$ Chern-Simons theory has no particles. The
function $\hat{F}_{1}$ is no longer given by the Maxwell version (421), but
rather by (420), except without the factors of $2$, related to the fact that
the CS action is first order in the derivatives. This immediately gives
$F_{1}=\hat{F}_{1}=-2\,\frac{1+q}{1-q}$. The computation of the volume factor
is analogous to our earlier discussions. The result (in canonical framing
Witten:1989ti ) is

$\displaystyle Z^{(1)}_{k}=\frac{\tilde{\gamma}^{\dim G}}{{\rm vol}(G)_{\rm
Tr}}\,e^{-2\dim G\int^{\times}\frac{dt}{2t}\frac{1+q}{1-q}}\,,\qquad
Z^{(1)}_{k,\rm fin}=\frac{(2\pi\tilde{\gamma})^{\dim G}}{{\rm vol}(G)_{\rm
Tr}}\,,\qquad\tilde{\gamma}=\frac{1}{\sqrt{k}}\,,$ (429)

where ${\rm vol}(G)_{\rm Tr}$ is the volume with respect to the metric defined
by the trace appearing in the Chern-Simons action (428). This agrees with the
standard results in the literature, nicely reviewed in section 4 of
Marino:2011nm .

###  H.2 Chern-Simons formulation of Einstein gravity

3D Einstein gravity can be reformulated as a Chern-Simons theory Witten:1988hc
; Achucarro:1986uwr . Although well-known, we briefly review some of the basic
ingredients and conceptual points here to facilitate the discussion of the
higher-spin generalization in section H.3. A more detailed review of certain
aspects, including more explicit solutions, can be found in section 4 of
Cotler:2019nbi . Explicit computations using the Chern-Simons formulation of
$\Lambda>0$ Euclidean quantum gravity with emphasis on topologies more
sophisticated than the sphere can be found in Carlip:1992wg ;
Guadagnini:1994ahx ; Castro:2011xb .

#### Lorentzian gravity

For the Lorentzian theory with positive cosmological constant, amplitudes are
computed by path integrals $\int{\cal D}A\,e^{iS_{L}}$ with real Lorentzian
$SL(2,{\mathbb{C}})$ Chern-Simons action Witten:1989ip

$\displaystyle S_{L}=(l+i\kappa)S_{\rm CS}[A_{+}]+(l-i\kappa)S_{\rm
CS}[A_{-}]\,,\quad\qquad A_{+}^{*}=A_{-},$ (430)

where $S_{\rm CS}$ is as in (428) with $A_{\pm}$ an ${\rm
sl}(2,{\mathbb{C}})$-valued connection and ${\rm Tr}={\rm Tr}_{2}$. The
vielbein $e$ and spin connection $\omega$ are the real and imaginary parts of
the connection:

$\displaystyle A_{\pm}=\omega\pm ie/\ell\,,\qquad ds^{2}=2\,{\rm
Tr}_{2}\,e^{2}=\eta_{ij}e^{i}e^{j}\,.$ (431)

For the last equality we decomposed $e=e^{i}L_{i}$ in a basis $L_{i}$ of ${\rm
sl}(2,{\mathbb{R}})$, say

$\displaystyle(L_{1},L_{2},L_{3})\equiv(\tfrac{1}{2}\sigma_{1},\tfrac{1}{2}i\sigma_{2},\tfrac{1}{2}\sigma_{3})\,\qquad\Rightarrow\qquad\eta_{ij}\equiv
2\,{\rm Tr}_{2}(L_{i}L_{j})={\rm diag}(1,-1,1)\,.$ (432)

Note that $[L_{i},L_{j}]=-\epsilon_{ijk}L^{k}$ with
$L^{k}\equiv\eta^{kk^{\prime}}L_{k^{\prime}}$. When $l=0$, the action reduces
to the firs-order form of the Einstein action with Newton constant $G_{\rm
N}=\ell/4\kappa$ and cosmological constant $\Lambda=1/\ell^{2}$. The equations
of motion stipulate $A_{\pm}$ must be flat connections:

$\displaystyle dA_{\pm}+A_{\pm}\wedge A_{\pm}=0\,,$ (433)

equivalent with the Einstein gravity torsion constraint (with
${\omega^{i}}_{j}\equiv\eta^{il}\epsilon_{ljk}\omega^{k}$) and the Einstein
equations of motion Witten:1988hc . Turning on $l$ deforms the action by
parity-odd terms of gravitational Chern-Simons type. This does not affect the
equations of motion (433). We can take $l\geq 0$ without loss of generality.
The part of the action multiplied by $l$ has a discrete ambiguity forcing $l$
to be integrally quantized, like $k$ in (428). Summarizing,

$\displaystyle 0\leq l\in{\mathbb{Z}}\,\,,\qquad 0<\kappa=\frac{2\pi\ell}{8\pi
G_{\rm N}}\in{\mathbb{R}}\,,$ (434)

#### dS3 vacuum solution

A flat connection corresponding to the de Sitter metric can be obtained as
follows. (We will be brief because the analog for the sphere below will be
simpler and make this more clear.) Define $Q(X)\equiv
2\,(X^{4}L_{4}+iX^{i}L_{i})$ with $L_{4}\equiv\frac{1}{2}{\bf 1}$ and note
that $\det Q=X_{4}^{2}+\eta_{ij}X^{i}X^{j}=:\eta_{IJ}X^{I}X^{J}$, so ${\cal
M}\equiv\\{X|\det Q(X)=1\\}$ is the dS3 hyperboloid, and $Q$ is a map from
${\cal M}$ into $SL(2,{\mathbb{C}})$. Its square root $h\equiv Q^{1/2}$ is
then a map from ${\cal M}$ into $SL(2,{\mathbb{C}})/{\mathbb{Z}}_{2}\simeq
SO(1,3)$, so $A_{+}\equiv h^{-1}dh$ is a flat ${\rm
sl}(2,{\mathbb{C}})$-valued connection on ${\cal M}$. Moreover on ${\cal M}$
we have $Q^{*}=Q^{-1}$, so $h^{*}=h^{-1}$, $A_{-}=A_{+}^{*}=-(dh)h^{-1}$, and
$ds^{2}=-\frac{1}{2}\ell^{2}\,{\rm
Tr}(A_{+}-A_{-})^{2}=-\frac{1}{2}\ell^{2}\,{\rm
Tr}\,(Q^{-1}dQ)^{2}=\ell^{2}\eta_{IJ}dX^{I}dX^{J}$, which is the de Sitter
metric of radius $\ell$ on ${\cal M}$.

#### Euclidean gravity

Like the Einstein-Hilbert action — or any other action for that matter — (430)
may have complex saddle points, that is to say flat connections $A_{\pm}$
which do not satisfy the reality constraint (430), or equivalently solutions
for which some components of the vielbein and spin connection are not real. Of
particular interest for our purposes is the solution corresponding to the
round metric on $S^{3}$. This can be obtained from the dS3 solution as usual
by a Wick rotation of the time coordinate. Given our choice of ${\rm
sl}(2,{\mathbb{R}})$ basis (432), this means $X^{2}\to-iX^{2}$. At the level
of the vielbein $e=e^{i}L_{i}$ such a Wick rotation is implemented as
$e^{2}\to-ie^{2}$. Similarly, recalling
$\omega_{ij}=\epsilon_{ijk}\omega^{k}$, the spin connection
$\omega=\omega^{i}L_{i}$ rotates as $\omega^{1}\to i\omega^{1}$,
$\omega^{3}\to i\omega^{3}$. Equivalently, $A_{\pm}\to(\omega^{i}\pm
e^{i}/\ell)S_{i}$ where $S_{i}\equiv\frac{1}{2}i\sigma_{i}$. Notice the
$S_{i}$ are the generators of ${\rm su}(2)$, satisfying
$[S_{i},S_{j}]=-\epsilon_{ijk}S_{k}$, $-2\,{\rm
Tr}_{2}(S_{i}S_{j})=\delta_{ij}$ and $S_{i}^{\dagger}=-S_{i}$. Thus the
Lorentzian metric $\eta_{ij}$ gets replaced by the Euclidean metric
$\delta_{ij}$, the Lorentzian ${\rm sl}(2,{\mathbb{C}})={\rm so}(1,3)$ reality
condition gets replaced by the Euclidean ${\rm su}(2)\oplus{\rm su}(2)={\rm
so}(4)$ reality condition, and the Lorentzian path integral $\int{\cal
D}A\,e^{iS_{L}}$ becomes a Euclidean path integral $\int{\cal
D}A\,e^{-S_{E}}$, where $S_{E}\equiv-iS_{L}$ is the Euclidean action:

$\displaystyle S_{E}=(\kappa-il)\,S_{\rm CS}[A_{+}]-(\kappa+il)\,S_{\rm
CS}[A_{-}]\,,\qquad A_{\pm}^{\dagger}=-A_{\pm}\,.$ (435)

This can be interpreted as the Chern-Simons formulation of Euclidean Einstein
gravity with positive cosmological constant. The ${\rm su}(2)\oplus{\rm
su}(2)$-valued connection $(A_{+},A_{-})$ encodes the Euclidean vielbein, spin
connection and metric as

$\displaystyle A_{\pm}=\omega\pm e/\ell=(\omega^{i}\pm
e^{i}/\ell)S_{i}\,,\qquad S_{i}\equiv\tfrac{1}{2}i\sigma_{i}\,,\qquad
ds^{2}=-2\,{\rm Tr}_{2}\,e^{2}=\delta_{ij}e^{i}e^{j}\,.$ (436)

The Euclidean counterpart of the reality condition of the Lorentzian action is
that $S_{E}$ gets mapped to $S_{E}^{*}$ under reversal of orientation.
Reversal of orientation maps $S_{\rm CS}[A]\to-S_{\rm CS}[A]$, and in addition
here it also exchanges the $\pm$ parts of the decomposition ${\rm so}(4)={\rm
su}(2)_{+}\oplus{\rm su}(2)_{-}$ into self-dual and anti-self-dual parts, that
is to say it exchanges $A_{+}\leftrightarrow A_{-}$. Thus orientation reversal
maps $S_{E}\to-(\kappa-il)\,S_{\rm CS}[A_{-}]+(\kappa+il)\,S_{\rm
CS}[A_{+}]=S_{E}^{*}$, as required.

#### Round sphere solutions

Parametrizing $S^{3}$ by $g\in SU(2)\simeq S^{3}$, it is easy to write down a
flat ${\rm su}(2)\oplus{\rm su}(2)$ connection yielding the round metric of
radius $\ell$:

$\displaystyle(A_{+},A_{-})=(g^{-1}dg,0)\qquad\Rightarrow\qquad
e/\ell=\tfrac{1}{2}g^{-1}dg=\omega\,,\quad ds^{2}=-\tfrac{1}{2}\ell^{2}\,{\rm
Tr}(g^{-1}dg)^{2}\,.$ (437)

The radius can be checked by observing that along an orbit
$g(\varphi)=e^{\varphi S_{3}}$, we get $g^{-1}dg=d\varphi\,S_{3}$ so
$ds=\frac{1}{2}\ell d\varphi$ and the orbit length is
$\int_{0}^{4\pi}ds=2\pi\ell$. The on-shell action is $S_{E}=-\frac{\kappa-
il}{12\pi}\int_{S^{3}}{\rm Tr}_{2}(g^{-1}dg)^{3}=-\frac{2(\kappa-
il)}{3\pi\ell^{3}}\int e^{i}e^{j}e^{k}\,{\rm
Tr}_{2}(S_{i}S_{j}S_{k})=-\frac{\kappa-il}{6\pi\ell^{3}}\int
e^{i}e^{j}e^{k}\,\epsilon_{ijk}=-2\pi(\kappa-il)$, so

$\displaystyle\exp(-S_{E})=\exp\bigl{(}2\pi\kappa+2\pi
il\bigr{)}=\exp\Bigl{(}\frac{2\pi\ell}{4G_{\rm N}}\Bigr{)}\,,$ (438)

where we used (434). This reproduces the standard Gibbons-Hawking result
PhysRevD.15.2752  for dS3.

More generally we can consider flat connections of the form
$(A_{+},A_{-})=(h_{+}^{-1}dh_{+},h_{-}^{-1}dh_{-})$ with
$h_{\pm}=g^{n_{\pm}}$, where $n_{\pm}\in{\mathbb{Z}}$ if we take the gauge
group to be $G=SU(2)\times SU(2)$. These are all related to the trivial
connection $(0,0)$ by a large gauge transformation $g\in
S^{3}\to(h_{+},h_{-})\in G$. All other flat connections on $S^{3}$ are
obtained from these by gauge transformations continuously connected to the
identity, which are equivalent to diffeomorphisms and vielbein rotations
continuously connected to the identity in the metric description Witten:1988hc
. Large gauge transformations on the other hand are in general not equivalent
to large diffeomorphisms. Indeed,

$\displaystyle e^{-S_{E}}=e^{2\pi n\kappa+2\pi i\tilde{n}l}=e^{2\pi
n\kappa}\,,\qquad n\equiv n_{+}-n_{-},\quad\tilde{n}\equiv n_{+}+n_{-}\,,$
(439)

so evidently different values of $n$ are physically inequivalent. Conversely,
for a fixed value of $n$ but different values of $\tilde{n}$, we get the same
metric, so these solutions are geometrically equivalent. In particular the
$n=1$ solutions all produce the same round metric (437). For $n=0$, the metric
vanishes. For $n<0$, we get a vielbein with negative determinant. Only
vielbeins with positive determinant reproduce the Einstein-Hilbert action with
the correct sign, so from the point of view of gravity we should discard the
$n<0$ solutions. Finally the cases $n>1$ correspond to a metric describing a
chain of $n$ spheres connected by throats of zero size, presumably more
appropriately thought of as $n$ disconnected spheres.

#### Euclidean path integral

The object of interest to us is the Euclidean path integral $Z=\int{\cal
D}A\,e^{-S_{E}[A]}$, defined perturbatively around an $n=n_{+}-n_{-}=1$ round
sphere solution
$(\bar{A}_{+},\bar{A}_{-})=(g^{-n_{+}}dg^{n_{+}},g^{-n_{-}}dg^{n_{-}})$, such
as the $(1,0)$ solution (437). Physically, this can be interpreted as the all-
loop quantum-corrected Euclidean partition function of the dS3 static patch.
For simplicity we take $G=SU(2)\times SU(2)$, so $n_{\pm}\in{\mathbb{Z}}$ and
we can formally factorize $Z$ as an $SU(2)_{k_{+}}\times SU(2)_{k_{-}}$ CS
partition function where $k_{\pm}=l\pm i\kappa$, with $l\in{\mathbb{Z}}^{+}$
and $\kappa\in{\mathbb{R}}^{+}$:

$\displaystyle Z=\int_{(n_{+},n_{-})}\\!\\!\\!{\cal D}A\,e^{ik_{+}S_{\rm
CS}[A_{+}]+ik_{-}S_{\rm CS}[A_{-}]}=Z_{\rm
CS}\bigl{(}SU(2)_{k_{+}}|\bar{A}_{n_{+}}\bigr{)}\,Z_{\rm
CS}\bigl{(}SU(2)_{k_{-}}|\bar{A}_{n_{-}}\bigr{)}\,,$ (440)

Here the complex-$k$ CS partition function $Z_{\rm
CS}(SU(2)_{k}|\bar{A}_{m})\equiv\int_{m}{\cal D}A\,e^{ikS_{\rm CS}[A]}$ is
defined perturbatively around the critical point $\bar{A}=g^{-m}dg^{m}$. It is
possible, though quite nontrivial in general, to define Chern-Simons theories
at complex level $k$ on general 3-manifolds $M_{3}$ Witten:2010cx ;
Gukov:2016njj . Our goal is less ambitious, since we only require a
perturbative expansion of $Z$ around a given saddle, and moreover we restrict
to $M_{3}=S^{3}$. In contrast to generic $M_{3}$, at least for integer $k$,
the CS action on $S^{3}$ has a unique critical point modulo gauge
transformations, and its associated perturbative large-$k$ expansion is not
just asymptotic, but actually converges to a simple, explicitly known
function: in canonical framing Witten:1989ti ,

$\displaystyle Z_{\rm
CS}\bigl{(}SU(2)_{k}|\bar{A}\bigr{)}_{0}=\sqrt{\tfrac{2}{2+k}}\,\sin\bigl{(}\tfrac{\pi}{2+k}\bigr{)}\,e^{i(2+k)S_{\rm
CS}[\bar{A}]}\qquad(k\in{\mathbb{Z}}^{+})\,.$ (441)

The dependence on the choice of critical point $\bar{A}=g^{-m}dg^{m}$ actually
drops out for integer $k$, as $S_{\rm CS}[\bar{A}]=-2\pi m\in
2\pi{\mathbb{Z}}$. We have kept it in the above expression to because this is
no longer the case for complex $k$. Analytic continuation to $k_{\pm}=l\pm
i\kappa$ with $l\in{\mathbb{Z}}^{+}$ and $\kappa\in{\mathbb{R}}^{+}$ in (440)
then gives:

$\displaystyle Z_{0}$
$\displaystyle=\Bigl{|}\sqrt{\tfrac{2}{2+l+i\kappa}}\,\sin\bigl{(}\tfrac{\pi}{2+l+i\kappa}\bigr{)}\Bigr{|}^{2}\,e^{2\pi
n\kappa-2\pi
i\,\tilde{n}(2+l)}=\Bigl{|}\sqrt{\tfrac{2}{2+l+i\kappa}}\sin\bigl{(}\tfrac{\pi}{2+l+i\kappa}\bigr{)}\cdot
e^{\pi\kappa}\Bigr{|}^{2}\,.$ (442)

#### Framing dependence of phase and one-loop check

For a general choice of $S^{3}$ framing with $SO(3)$ spin connection
$\hat{\omega}$, (441) gets replaced by Witten:1989ti

$\displaystyle Z_{\rm
CS}\bigl{(}SU(2)_{k}|\bar{A}\bigr{)}=\exp\bigl{(}\tfrac{i}{24}c(k)I(\hat{\omega})\bigr{)}\,Z_{\rm
CS}\bigl{(}SU(2)_{k}|\bar{A}\bigr{)}_{0}\,,\qquad
c(k)=3\bigl{(}1-\tfrac{2}{2+k}\bigr{)}\,,$ (443)

where $I(\hat{\omega})=\frac{1}{4\pi}\int{\rm Tr}_{3}(\hat{\omega}\wedge
d\hat{\omega}+\frac{2}{3}\hat{\omega}\wedge\hat{\omega}\wedge\hat{\omega})$ is
the gravitational Chern-Simons action. The action $I(\hat{\omega})$ can be
defined more precisely as explained under (2.22) of Witten:2007kt , by picking
a 4-manifold $M$ with boundary $\partial M=S^{3}$ and putting

$\displaystyle I(\hat{\omega})\equiv I_{M}\equiv\frac{1}{4\pi}\int_{M}{\rm
Tr}(R\wedge R)\,,$ (444)

where $R$ is the curvature form of $M$,
${R^{\mu}}_{\nu}=\frac{1}{2}{R^{\mu}}_{\nu\rho\sigma}dx^{\rho}\wedge
dx^{\sigma}$. Taking $M$ to be a flat 4-ball $B$, the curvature vanishes so
$I_{B}=0$, corresponding to canonical framing. Viewing $B$ as a 4-hemisphere
with round metric has ${\rm Tr}(R\wedge R)=0$ pointwise so again $I_{B}=0$.
Gluing any other 4-manifold $M$ with boundary $S^{3}$ to $B$, we get a closed
4-manifold $X=M-B$, with $I_{M}-I_{B}=\frac{1}{4\pi}\int_{X}{\rm Tr}(R\wedge
R)=2\pi p_{1}(X)$, where $p_{1}(X)$ is the Pontryagin number of $X$. According
to the Hirzebruch signature theorem, the signature
$\sigma(X)=b_{2}^{+}-b_{2}^{-}$ of the intersection form of the middle
cohomology of $X$ equals $\frac{1}{3}p_{1}(X)$. Therefore, for any choice of
$M$,

$\displaystyle I_{M}=6\pi r\,,\qquad r=\sigma(X)\in{\mathbb{Z}}\,.$ (445)

For example $r=1$ for $X={\mathbb{CP}}^{2}$ and $r=p-q$ for
$X=p{\mathbb{CP}}^{2}\\#q\overline{{\mathbb{CP}}}^{2}$. Thus for general
framing, (443) becomes $Z_{\rm CS}(k|m)=Z_{\rm
CS}(k|m)_{0}\,\exp\bigl{(}r\,c(k)\,\tfrac{i\pi}{4}\bigr{)}$ and (442) becomes

$\displaystyle
Z_{r}=e^{ir\phi}\,\Bigl{|}\sqrt{\tfrac{2}{2+l+i\kappa}}\sin\bigl{(}\tfrac{\pi}{2+l+i\kappa}\bigr{)}e^{\pi\kappa}\Bigr{|}^{2}\,,\qquad
r\in{\mathbb{Z}}\,,$ (446)

where, using $c(k)=3\bigl{(}1-\tfrac{2}{2+k}\bigr{)}$, the phase is given by

$\displaystyle
r\phi=r\bigl{(}c(l+i\kappa)+c(l-i\kappa)\bigr{)}\frac{\pi}{4}=r\bigl{(}1-\tfrac{2(2+l)}{(2+l)^{2}+\kappa^{2}}\bigr{)}\frac{3\pi}{2}\,.$
(447)

In the weak-coupling limit $\kappa\to\infty$,

$\displaystyle Z_{r}\to(-i)^{r}\,\frac{2\pi^{2}}{\kappa^{3}}\cdot
e^{2\pi\kappa}\,.$ (448)

Using (434) and taking into account that we took $G=SU(2)\times SU(2)$ here,
the absolute value agrees with our general one-loop result (424) in the metric
formulation, with the phase $(-i)^{r}$ matching Polchinski’s phase
$i^{-P}=i^{-5}=-i$ in (112) for odd framing $r$.444444Strictly speaking for
$r=1$ mod 4, but $i^{P}$ vs $i^{-P}$ in (112) is a matter of conventions, so
there is no meaningful distinction we can make here. We do not have any useful
insights into why (or whether) CS framing and the phase $i^{-P}$ might have
anything to do with each other, let alone why odd but not even framing should
reproduce the phase of Polchinski:1988ua . Perhaps different contour rotation
prescriptions as those assumed in Polchinski:1988ua  might reproduce the
canonically framed ($r=0$) result in the metric formulation of Euclidean
gravity. We leave these questions open.

Comparison to previous results: The Chern-Simons formulation of gravity was
applied to calculate Euclidean $\Lambda>0$ partition functions in
Carlip:1992wg ; Guadagnini:1994ahx ; Castro:2011xb . The focus of these works
was on summing different topologies. Our one-loop (448) in canonical framing
agrees with Carlip:1992wg  up to an unspecified overall normalization constant
in the latter, agrees with $Z(S^{3})/Z(S^{1}\times S^{2})$ in
Guadagnini:1994ahx  combining their eqs. (13),(32), and disagrees with eq.
(4.39) in Castro:2011xb , $Z^{(1)}(S^{3})=\pi^{3}/(2^{5}\kappa)$.

###  H.3 Chern-Simons formulation of higher-spin gravity

The $SL(2,{\mathbb{C}})$ Chern-Simons formulation of Einstein gravity (430)
has a natural extension to an $SL(n,{\mathbb{C}})$ Chern-Simons formulation of
higher-spin gravity — the positive cosmological constant analog of the
theories studied e.g. in Blencowe:1988gj ; Bergshoeff:1989ns ; Castro:2011fm ;
Perlmutter:2012ds ; Campoleoni:2010zq ; Ammon:2011nk . The Lorentzian action
is

$\displaystyle S_{L}=(l+i\kappa)\,S_{\rm CS}[{\cal A}_{+}]+(l-i\kappa)\,S_{\rm
CS}[{\cal A}_{-}]\,,\qquad{\cal A}_{+}^{*}={\cal A}_{-}\,,$ (449)

where $S_{\rm CS}[{\cal A}]=\frac{1}{4\pi}\int{\rm Tr}_{n}\bigl{(}{\cal
A}\wedge d{\cal A}+\tfrac{2}{3}{\cal A}\wedge{\cal A}\wedge{\cal A}\bigr{)}$,
an ${\cal A}$ is an ${\rm sl}(n,{\mathbb{C}})$-valued connection,
$\kappa\in{\mathbb{R}}^{+}$ and $l\in{\mathbb{Z}}^{+}$. The corresponding
Euclidean action $S_{E}=-iS_{L}$ extending (435) is given by

$\displaystyle S_{E}=(\kappa-il)\,S_{\rm CS}[{\cal A}_{+}]-(\kappa+il)\,S_{\rm
CS}[{\cal A}_{-}]\,,\qquad{\cal A}_{\pm}^{\dagger}=-{\cal A}_{\pm}\,,$ (450)

where ${\cal A}_{\pm}$ are now independent ${\rm su}(n)$-valued connections.

####  H.3.1 Landscape of dS3 vacua

The solutions $A$ of the original ($n=2$) Einstein gravity theory can be
lifted to solutions ${\cal A}=R(A)$ of the extended ($n>2$) theory by choosing
an embedding $R$ of ${\rm sl}(2)$ into ${\rm sl}(n)$. More concretely, such
lifts are specified by picking an $n$-dimensional representation $R$ of ${\rm
su}(2)$,

$\displaystyle R=\oplus_{a}{\bf m}_{a}\,,\qquad\sum_{a}m_{a}=n\,,\qquad{\cal
S}_{i}=R(S_{i})=\oplus_{a}J^{(m_{a})}_{i}\,.$ (451)

Here $J^{(m)}_{i}$ are the standard anti-hermitian spin $j=\frac{m-1}{2}$
representation matrices of ${\rm su}(2)$, satisfying the same commutation
relations and reality properties as the spin-$\frac{1}{2}$ generators $S_{i}$
in (436). Then the matrices ${\cal L}_{i}\equiv R(L_{i})$ with the $L_{i}$ as
in (432) are real, generating the corresponding $n$-dimensional representation
of ${\rm sl}(2,{\mathbb{R}})$. The Casimir eigenvalue of the spin
$j=\frac{m-1}{2}$ irrep is $j(j+1)=\frac{1}{4}(m^{2}-1)$, so

$\displaystyle{\rm Tr}_{n}({\cal S}_{i}{\cal
S}_{j})=-\tfrac{1}{2}T_{R}\,\delta_{ij}\,,\qquad{\rm Tr}_{n}({\cal L}_{i}{\cal
L}_{j})=\tfrac{1}{2}T_{R}\,\eta_{ij}\,,\qquad
T_{R}\equiv\frac{1}{6}\sum_{a}m_{a}(m_{a}^{2}-1)\,;$ (452)

A general $SL(2,{\mathbb{C}})$ connection $A=A^{i}L_{i}$ has curvature
$dA+A\wedge
A=\bigl{(}dA^{i}-\frac{1}{2}{\epsilon^{i}}_{jk}A^{j}A^{k}\bigr{)}L_{i}$, and
an $SL(n,{\mathbb{C}})$ connection of the form ${\cal A}=R(A)=A^{i}{\cal
L}_{i}$ has curvature $d{\cal A}+{\cal A}\wedge{\cal
A}=\bigl{(}dA^{i}-\frac{1}{2}{\epsilon^{i}}_{jk}A^{j}A^{k}\bigr{)}{\cal
L}_{i}$, hence ${\cal A}=R(A)$ solves the equations of motion of the extended
$SL(n,{\mathbb{C}})$ theory iff $A$ solves the equations of motion of the
original Einstein $SL(2,{\mathbb{C}})$ theory. In other words, restricting to
connections ${\cal A}=A^{i}{\cal L}_{i}$ amounts to a consistent truncation,
which may be interpreted as the gravitational subsector of the $n>2$ theory.
Substituting ${\cal A}=R(A)$ into the action (449) gives the consistently
truncated action

$\displaystyle S_{L}=(l+i\kappa)T_{R}\,S_{\rm
CS}[A_{+}]+(l-i\kappa)T_{R}\,S_{\rm CS}[A_{-}]\,,\qquad A_{+}^{*}=A_{-}\,,$
(453)

which is of the exact same form as the original Einstein CS gravity theory
(430), except $l+i\kappa$ is replaced by $(l+i\kappa)T_{R}$. Thus we can
naturally interpret the components $A^{i}_{\pm}$ again as metric/vielbein/spin
connection degrees of freedom, just like in (431), i.e.
$A^{i}_{\pm}=\omega^{i}\pm ie^{i}/\ell$, $ds^{2}=\eta_{ij}e^{i}e^{j}$, and the
lift ${\cal A}=R(A)$ of the original solution $A$ corresponding to the dS3
metric again as a solution corresponding to the dS3 metric. The difference is
that the original relation (434) between $\kappa$ and $\ell/G_{\rm N}$ gets
modified to

$\displaystyle\kappa\,T_{R}=\frac{2\pi\ell}{8\pi G_{\rm N}}\,.$ (454)

Since $\kappa$ is fixed, this means the dimensionless ratio $\ell/G_{\rm N}$
depends on the choice of $R$. Thus the different solutions ${\cal A}=R(A)$ of
the $SL(n,{\mathbb{C}})$ theory can be thought of as different de Sitter vacua
of the theory, labeled by $R$, with different values of the curvature radius
in Planck units $\ell/G_{\rm N}$. These are the dS analog of the AdS vacua
discussed in Ammon:2011nk . The total number of vacua labeled by
$R=\oplus_{a}{\bf m}_{a}$ equals the number of partitions of
$n=\sum_{a}m_{a}$,

$\displaystyle{\cal N}_{\rm vac}\sim e^{2\pi\sqrt{n/6}}\,\qquad(n\gg 1)\,.$
(455)

For, say, $n\sim 2\times 10^{5}$, this gives ${\cal N}_{\rm vac}\sim
10^{500}$.

Analogous considerations hold for the Euclidean version of the theory. For
example the round sphere solution (437) is lifted to

$\displaystyle({\cal A}_{+},{\cal
A}_{-})=\bigl{(}R(A_{+}),0\bigr{)}=\bigl{(}R(g)^{-1}dR(g),0\bigr{)},\qquad
R(e^{\alpha^{i}S_{i}})\equiv e^{\alpha^{i}{\cal S}_{i}}\,,$ (456)

with the sphere radius $\ell$ in Planck units given again by (454). The tree-
level contribution of the solution (456) to the Euclidean path integral is

$\displaystyle\exp(-S_{E})=\exp\bigl{(}2\pi\kappa\,T_{R}\bigr{)}=\exp\Bigl{(}\frac{2\pi\ell}{4\,G_{\rm
N}}\Bigr{)}\,.$ (457)

Note that ${\cal S}^{(0)}\equiv-S_{E}=\frac{2\pi\ell}{4G_{\rm N}}$ is the
usual dS3 Gibbons-Hawking horizon entropy PhysRevD.15.2752 . Its value ${\cal
S}^{(0)}=2\pi\kappa\,T_{R}$ depends on the vacuum $R=\oplus_{a}{\bf m}_{a}$
through $T_{R}$ as given by (452). The vacuum $R$ maximizing $e^{-S_{E}}$
corresponds to the partition of $n=\sum_{a}m_{a}$ maximizing $T_{R}$. Clearly
the maximum is achieved for $R={\bf n}$:

$\displaystyle\max_{R}\,T_{R}=T_{\bf n}=\frac{n(n^{2}-1)}{6}\,.$ (458)

The corresponding embedding of ${\rm su}(2)$ into ${\rm su}(n)$ is called the
“principal embedding”. Thus the “principal vacuum” maximizes the entropy at
$S_{\rm GH,{\bf n}}=\frac{1}{6}n(n^{2}-1)\,2\pi\kappa$, exponentially
dominating the Euclidean path integral in the semiclassical (large-$\kappa$)
regime. In the remainder we focus on the Euclidean version of the theory.

####  H.3.2 Higher-spin field spectrum and algebra

Of course for $n>2$, there are more degrees of freedom in the $2(n^{2}-1)$
independent components of ${\cal A}_{\pm}$ than just the 3+3 vielbein and spin
connection degrees of freedom ${\cal A}^{i}_{\pm}{\cal S}_{i}$. The full set
of fluctuations around the vacuum solution can be interpreted in a metric-like
formalism as higher-spin field degrees of freedom. The precise spectrum
depends on the vacuum $R$. For the principal vacuum $R={\bf n}$, we get the
higher-spin vielbein and spin connections of a set of massless spin-$s$ fields
of $s=2,3,\ldots,n$, as was worked out in detail for the AdS analog in
Campoleoni:2010zq . Indeed ${\rm su}(n)$ decomposes under the principally
embedded ${\rm su}(2)$ subalgebra into spin-$r$ irreps, $r=1,2,\ldots,n-1$,
generated by the traceless symmetric products ${\cal S}_{i_{1}\cdots i_{r}}$
of the generators ${\cal S}_{i}$. As reviewed in Joung:2014qya , this means we
can identify the ${\rm su}(n)_{+}\oplus{\rm su}(n)_{-}$ Lie algebra of the
theory (450) with the higher-spin algebra ${\rm hs}_{n}({\rm
su}(2))_{+}\oplus{\rm hs}_{n}({\rm su}(2))_{-}$, where ${\rm
su}(2)_{+}\oplus{\rm su}(2)_{-}={\rm so}(4)$ is the principally embedded
gravitational subalgebra. In the metric-like formalism the spin-$r$ generators
correspond to (anti-)self-dual Killing tensors of rank $r$. These are the
Killing tensors of massless symmetric spin-$s$ fields with $s=r+1$. As a
check, recall the number of (anti-)self-dual rank $r$ Killing tensors on
$S^{3}$ equals $D^{4}_{r,\pm r}=2r+1$, correctly adding up to

$\displaystyle\sum_{\pm}\sum_{r=1}^{n-1}D^{4}_{r,\pm
r}=2\sum_{r=1}^{n-1}(2r+1)=2(n^{2}-1)\,.$ (459)

For different choices of embedding $R$, we get different ${\rm su}(2)$
decompositions of ${\rm su}(n)$. For example for $n=12$, while the principal
embedding $R={\bf 12}$ considered above gives the ${\rm su}(2)$ decomposition
${\bf 143}_{{\rm su}(12)}={\bf 3}+{\bf 5}+{\bf 7}+{\bf 9}+{\bf 11}+{\bf
13}+{\bf 15}+{\bf 17}+{\bf 19}+{\bf 21}+{\bf 23}$, taking $R={\bf 6}\oplus{\bf
4}\oplus{\bf 2}$ gives ${\bf 143}_{{\rm su}(12)}=2\cdot{\bf 1}+7\cdot{\bf
3}+8\cdot{\bf 5}+6\cdot{\bf 7}+3\cdot{\bf 9}+{\bf 11}$. Interpreting these as
Killing tensors for $n_{s}$ massless spin-$s$ fields, we get for the former
$n_{2}=1,n_{3}=1,\ldots,n_{12}=1$, and for the latter
$n_{1}=2,n_{2}=7,n_{3}=8,n_{4}=6,n_{5}=3,n_{6}=1$. The tree-level entropy
${\cal S}^{(0)}=2\pi\ell/4G_{\rm N}$ for $R={\bf 12}$ is ${\cal
S}^{(0)}=286\cdot 2\pi\kappa$, and for $R={\bf 6}\oplus{\bf 4}\oplus{\bf 2}$
it is ${\cal S}^{(0)}=46\cdot 2\pi\kappa$.

####  H.3.3 One-loop Euclidean path integral from metric-like formulation

In view of the above higher-spin interpretation of the theory, we can apply
our general massless HS formula (424) with $G=SU(n)\times SU(n)$ to obtain the
one-loop contribution to the Euclidean path integral (for $l=0$). In
combination with (454) this takes the form

$\displaystyle Z^{(1)}_{\rm PI}=i^{-P}\,\frac{(2\pi\gamma)^{\dim G}}{{\rm
vol}{(G)}_{{\rm c}}}\,,\qquad\gamma=\sqrt{\frac{8\pi G_{\rm
N}}{2\pi\ell}}=\frac{1}{\sqrt{\kappa\,T_{R}}}\,.$ (460)

Recall that ${\rm vol}{(G)}_{{\rm c}}$ is the volume of $G$ with respect to
the metric normalized such that $\langle M|M\rangle_{\rm c}=1$, where $M$ is
one of the standard ${\rm so}(4)={\rm su}(2)\oplus{\rm su}(2)$ generators,
which we can for instance take to be the rotation generator $M={\cal
S}_{3}\oplus{\cal S}_{3}$. In the context of Chern-Simons theory, it is more
natural to consider the volume ${\rm vol}(G)_{{\rm Tr}_{n}}$ with respect to
the metric defined by the trace appearing in the Chern-Simons action (450).
Using the definition of $T_{R}$ in (453), we see the trace norm of $M$ is
$\langle M|M\rangle_{{\rm Tr}_{n}}=-2\,{\rm Tr}_{n}({\cal S}_{3}{\cal
S}_{3})=T_{R}=T_{R}\langle M|M\rangle_{\rm c}$, hence ${\rm vol}(G)_{{\rm
Tr}_{n}}=(\sqrt{T_{R}})^{\dim G}\,{\rm vol}{(G)}_{{\rm c}}$. Note that upon
substituting this in (460), the $T_{R}$-dependent factors cancel out. Finally,
using (422), we get
$P=\sum_{s=2}^{n}P_{s}=\frac{1}{3}(2s-3)(2s-1)(2s+1)=\frac{2}{3}n^{2}(n-1)(n+1)-(n^{2}-1)$.
Because $(n-1)\cdot n\cdot(n+1)$ is divisible by 3, the first term is an
integer, and moreover a multiple of $8$ because either $n^{2}$ or $(n+1)(n-1)$
is a multiple of $4$. Hence $i^{-P}=i^{(n^{2}-1)}$, which equals $-i$ for even
$n$ and $+1$ for odd $n$. Thus we get

$\displaystyle Z^{(1)}_{\rm PI}=i^{n^{2}-1}\,\frac{(2\pi\tilde{\gamma})^{\dim
G}}{{\rm vol}(G)_{{\rm
Tr}_{n}}}\,,\qquad\quad\tilde{\gamma}\equiv\frac{1}{\sqrt{\kappa}}\,.$ (461)

####  H.3.4 Euclidean path integral from CS formulation

As in the $SU(2)\times SU(2)$ Einstein gravity case, we can derive an all-loop
expression for the Euclidean partition function $Z(R)$ of the $SU(n)\times
SU(n)$ higher-spin gravity theory (450) expanded around a lifted round sphere
solution $\bar{\cal A}=R(\bar{A})$ such as (456), by naive analytic
continuation of the exact $SU(n)_{k_{+}}\times SU(n)_{k_{-}}$ partition
function on $S^{3}$ to $k_{\pm}=l\pm i\kappa$, paralleling (440) and the
subsequent discussion there. The $SU(n)_{k}$ generalization of the canonically
framed $SU(2)_{k}$ result (441) as spelled out e.g. in Marino:2004uf ;
Periwal:1993yu  is

$\displaystyle Z_{\rm CS}(SU(n)_{k}|\bar{\cal
A})_{0}=\frac{1}{\sqrt{n}}\frac{1}{(n+k)^{\frac{n-1}{2}}}\prod_{p=1}^{n-1}\Bigl{(}2\sin\frac{\pi
p}{n+k}\Bigr{)}^{(n-p)}\cdot e^{i(n+k)S_{\rm CS}[\bar{\cal A}]}\,.$ (462)

The corresponding higher-spin generalization of (442) is therefore

$\displaystyle
Z(R)_{0}=\biggl{|}\frac{1}{\sqrt{n}}\frac{1}{(n+l+i\kappa)^{\frac{n-1}{2}}}\prod_{p=1}^{n-1}\Bigl{(}2\sin\frac{\pi
p}{n+l+i\kappa}\Bigr{)}^{(n-p)}\,\biggr{|}^{2}\,\cdot e^{2\pi\kappa T_{R}}\,.$
(463)

Physically this can be interpreted as the all-loop quantum-corrected Euclidean
partition function of the dS3 static patch in the vacuum labeled by $R$. The
analog of the result (443) for more general framing $I_{M}$ is

$\displaystyle Z_{\rm CS}(SU(n)_{k}|\bar{\cal
A})=\exp\bigl{(}\tfrac{i}{24}c(k)I_{M}\bigr{)}\,Z_{\rm CS}(SU(n)_{k}|\bar{\cal
A})_{0}\,,\qquad c(k)=(n^{2}-1)\bigl{(}1-\tfrac{n}{n+k}\bigr{)}\,,$ (464)

hence the generalization of (446) for arbitrary framing $I_{M}=6\pi r$,
$r\in{\mathbb{Z}}$, is

$\displaystyle Z(R)_{r}=e^{ir\phi}\,Z(R)_{0}\,,$ (465)

where
$\phi=\bigl{(}c(l+i\kappa)+c(l-i\kappa)\bigr{)}\frac{\pi}{4}=\bigl{(}1-\frac{2(n+l)}{(n+l)^{2}+\kappa^{2}}\bigr{)}(n^{2}-1)\frac{\pi}{2}$.
In the limit $\kappa\to\infty$,

$\displaystyle Z(R)_{r}\to
i^{r(n^{2}-1)}\,\frac{1}{n}\frac{1}{\kappa^{n-1}}\prod_{r=1}^{n-1}\Bigl{(}\frac{2\pi
r}{\kappa}\Bigr{)}^{2(n-r)}=i^{r(n^{2}-1)}\left(\frac{2\pi}{\sqrt{\kappa}}\right)^{2(n^{2}-1)}\biggl{(}\frac{1}{\sqrt{n}}\prod_{s=2}^{n}\frac{\Gamma(s)}{(2\pi)^{s}}\biggr{)}^{2}\,.$
(466)

Recognizing $n^{2}-1=\dim SU(n)$ and
$\sqrt{n}\prod_{s=2}^{n}(2\pi)^{s}/\Gamma(s)={\rm vol}(SU(n))_{{\rm Tr}_{n}}$
(288), we see this precisely reproduces the one-loop result (461). Like in the
original $n=2$ case, the phase again matches for odd framing $r$. (The
agreement at one loop can also be seen more directly by a slight variation of
the computation leading to (429).) This provides a nontrivial check of our
higher-spin gravity formula (424) and more generally (112).

####  H.3.5 Large-$n$ limit and topological string description

In generic dSd+1 higher-spin theories, $\dim G=\infty$. To mimic this case,
consider the $n\to\infty$ limit of $SU(n)\times SU(n)$ dS3 higher-spin theory
with $l=0$. A basic observation is that the loop expansion is only reliable
then if $n/\kappa\ll 1$. Using (454), this translates to
$T_{R}\,n\ll\frac{\ell}{G_{\rm N}}$ For the exponentially dominant principal
vacuum $R={\bf n}$, this becomes $n^{4}\ll\ell/G_{\rm N}$ while at the other
extreme, for the nearly-trivial $R={\bf 2}\oplus{\bf 1}\oplus\cdots\oplus{\bf
1}$, this becomes $n\ll\ell/G_{\rm N}$. Either way, for fixed $\ell/G_{N}$,
the large-$n$ limit is necessarily strongly coupled, and the one-loop formula
(460), or equivalently (461) or (466), becomes unreliable. Indeed, according
to this formula, $\log Z^{(1)}\sim\log\left(\frac{n}{\kappa}\right)\cdot
n^{2}$ in this limit, whereas the exact expression (463) actually implies
$\log Z^{\rm{(loops)}}\to 0$.

In fact, the partition function does have a natural weak coupling expansion in
the $n\to\infty$ limit — not as a 3D higher-spin gravity theory, but rather as
a topological string theory. $U(n)_{k}$ Chern-Simons theory on $S^{3}$ has a
description Witten:1992fb  as an open topological string theory on the
deformed conifold $T^{*}S^{3}$ with $n$ topological D-branes wrapped on the
$S^{3}$, and a large-$n$ ’t Hooft dual description Gopakumar:1998ki  as a
closed string theory on the resolved conifold. Both descriptions are reviewed
in Marino:2004uf , whose notation we follow here. The string coupling constant
is $g_{s}=2\pi/(n+k)$ and the Kähler modulus of the resolved conifold is
$t=\int_{S^{2}}J+iB=ig_{s}n=2\pi in/(n+k)$. Under this identification,

$\displaystyle Z_{\rm CS}(SU(n)_{k})_{0}=\sqrt{\tfrac{n+k}{n}}\,Z_{\rm
CS}(U(n)_{k})_{0}=\sqrt{\tfrac{2\pi i}{t}}\,Z_{\rm
top}(g_{s},t)\equiv\tilde{Z}_{\rm top}(g_{s},t)\,.$ (467)

Thus we can write the $SU(n)_{l+i\kappa}\times SU(n)_{l-i\kappa}$ higher-spin
Euclidean gravity partition function (463) expanded around the round $S^{3}$
solution $\bar{\cal A}=R(\bar{A})$ as

$\displaystyle\boxed{Z(R)_{0}=\left|\tilde{Z}_{\rm top}(g_{s},t)\,e^{-\pi
T_{R}\cdot 2\pi i/g_{s}}\right|^{2}}$ (468)

where $T_{R}$ was defined in (452), maximized for $R={\bf n}$ at $T_{\bf
n}=\frac{1}{6}n(n^{2}-1)$, and

$\displaystyle g_{s}=\frac{2\pi}{n+l+i\kappa}\,,\qquad t=ig_{s}n=\frac{2\pi
in}{n+l+i\kappa}\,.$ (469)

Note that $t$ takes values inside a half-disk of radius $\frac{1}{2}$ centered
at $t=i\pi$, with ${\rm Re}\,t>0$. The higher-spin gravity theory (or the open
string theory description on the deformed conifold) is weakly coupled when
$\kappa\gg n$, which implies $|t|\ll 1$. In the free field theory limit
$\kappa\to\infty$, we get $g_{s}\sim-2\pi i/\kappa\to 0$ and $t\sim 2\pi
n/\kappa\to 0$, which is singular from the closed string point of view. In the
’t Hooft limit $n\to\infty$ with $t$ kept finite, the closed string is weakly
coupled and sees a smooth geometry. The earlier discussed Vasiliev-like limit
$n\to\infty$ with $l=0$ and $\ell/G_{\rm N}\sim T_{\bf n}\kappa\sim
n^{3}\kappa$ fixed, infinitely strongly coupled from the 3D field theory point
of view, maps to $g_{s}\sim 2\pi/n\to 0$ and $t\sim 2\pi i+2\pi\kappa/n\to
2\pi i$, which is again singular from the closed string point of view,
differing from the 3D free field theory singularity by a mere $B$-field
monodromy, reflecting the more general $n\leftrightarrow l+i\kappa$,
$t\leftrightarrow 2\pi i-t$ level-rank symmetry.

##  Appendix I Quantum dS entropy: computations and examples

Here we provide the details for section 8.

###  I.1 Classical gravitational dS thermodynamics

####  I.1.1 3D Einstein gravity example

For concreteness we start with pure 3D Einstein gravity as a guiding example,
but we will phrase the discussion so generalization will be clear. The
Euclidean action in this case is

$S_{E}[g]=\frac{1}{8\pi G}\int
d^{3}x\sqrt{g}\Bigl{(}\Lambda-\frac{1}{2}R\Bigr{)}\,,$ (470)

with $\Lambda>0$. The tree-level contribution to the entropy (153) is

$\displaystyle{\cal S}^{(0)}=\log{\cal Z}^{(0)}\,,\qquad{\cal
Z}^{(0)}=\mbox{\Large$\int$}_{\\!\\!\\!\text{tree}}\,{\cal
D}g\,e^{-S_{E}[g]}\,.$ (471)

The dominant saddle of (471) is a round $S^{3}$ metric $g_{\ell}$ of radius
$\ell\\!=\\!\ell_{0}$ minimizing $S_{E}(\ell)\equiv S_{E}[g_{\ell}]$:

$\displaystyle{\cal
Z}^{(0)}=\mbox{\Large$\int$}_{\\!\\!\\!\text{tree}}\,d\ell\,e^{-S_{E}(\ell)}\,,\qquad
S_{E}(\ell)=\frac{2\pi^{2}}{8\pi G}\bigl{(}\Lambda\ell^{3}-3\ell\bigr{)}\,,$
(472)

where $\int_{\rm tree}$ means evaluation at the saddle point, here at the on-
shell radius $\ell=\ell_{0}$:

$\displaystyle\partial_{\ell}S_{E}(\ell_{0})=0\qquad\Rightarrow\qquad\Lambda=\frac{1}{\ell_{0}^{2}}\,,\qquad{\cal
S}^{(0)}=-S_{E}(\ell_{0})=\frac{2\pi\ell_{0}}{4G}\,,$ (473)

reproducing the familiar area law ${\cal S}^{(0)}=A/4G$ for the horizon
entropy.

We now recast the above in a way that will allow us to make contact with the
formulae of section 7.1 and will naturally generalize beyond tree level in a
diffeomorphism-invariant way. To this end we define an “off-shell” tree-level
partition function at fixed (off-shell) volume $V$:

$\displaystyle
Z^{(0)}(V)\equiv\mbox{\Large$\int$}_{\\!\\!\\!\text{tree}}\,d\sigma\,\mbox{\Large$\int$}_{\\!\\!\\!\text{tree}}\,{\cal
D}g\,e^{-S_{E}[g]+\sigma(\int\\!\sqrt{g}-V)}\,.$ (474)

Evaluating the integral is equivalent to a constrained extremization problem
with Lagrange multiplier $\sigma$ enforcing the constraint
$\int\\!\sqrt{g}=V$. The dominant saddle is the round sphere $g=g_{\ell}$ of
radius $\ell(V)$ fixed by the volume constraint:

$\displaystyle Z^{(0)}(V)=e^{-S_{E}(\ell)}\,,\qquad 2\pi^{2}\ell^{3}=V\,.$
(475)

Paralleling (135) and (136), we define from this an off-shell energy density
and entropy,

$\displaystyle\rho^{(0)}$ $\displaystyle\equiv-\partial_{V}\log
Z^{(0)}=-{\tfrac{1}{3}\ell\partial_{\ell}\log
Z^{(0)}}\,/\,{V}=\bigl{(}\Lambda-\ell^{-2}\bigr{)}/8\pi G$ (476)
$\displaystyle S^{(0)}$
$\displaystyle\equiv\bigl{(}1-V\partial_{V}\bigr{)}\log
Z^{(0)}=\bigl{(}1-\tfrac{1}{3}\ell\partial_{\ell}\bigr{)}\log
Z^{(0)}=\frac{2\pi\ell}{4G}\,.$

$\rho^{(0)}$ is the sum of the positive cosmological constant and negative
curvature energy densities. $S^{(0)}$ is independent of $\Lambda$. It is the
Legendre transform of $\log Z^{(0)}$:

$\displaystyle S^{(0)}=\log Z^{(0)}+V\rho^{(0)}\,,\qquad d\log
Z^{(0)}=-\rho^{(0)}dV\,,\qquad dS^{(0)}=Vd\rho^{(0)}\,,$ (477)

Note that evaluating $\int_{\rm tree}d\sigma$ in (474) sets
$\sigma=-\partial_{V}\log Z^{(0)}=\rho^{(0)}(V)$. On shell,

$\displaystyle\rho^{(0)}(\ell_{0})=0\,,\qquad{\cal S}^{(0)}=\log
Z^{(0)}(\ell_{0})=S^{(0)}(\ell_{0})=\frac{2\pi\ell_{0}}{4G}\,.$ (478)

Paralleling (137), the differential relations in (477) can be viewed as the
first law of tree-level de Sitter thermodynamics. We can also consider
variations of coupling constants such as $\Lambda$. Then $d\log
Z^{(0)}=-\rho^{(0)}dV-\frac{1}{8\pi G}Vd\Lambda$,
$dS^{(0)}=Vd\rho^{(0)}-\frac{1}{8\pi G}Vd\Lambda$. On shell, $d{\cal
S}^{(0)}=-\frac{V_{0}}{8\pi G}\,d\Lambda$.

####  I.1.2 General $d$ and higher-order curvature corrections

The above formulae readily extend to general dimensions and to gravitational
actions $S_{E}[g]$ with general higher-order curvature corrections. Using that
$R_{\mu\nu\rho\sigma}=(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho})/\ell^{2}$
for the round454545By virtue of its $SO(d+2)$ symmetry, the round sphere
metric $g_{\ell}$ with $\Omega_{d+1}\ell^{d+1}=V$ is a saddle of (474).
Spheres of dimension $\geq 5$ admit a plentitude of Einstein metrics that are
not round 10.4310/jdg/1214431962 ; Bohm:1998vc ; Gibbons:2002th ; Boyer:2003pe
, but as explained e.g. in Gibbons:2011dh , by Bishop’s theorem bishop , these
saddles are subdominant in Einstein gravity. In the large-size limit, higher-
order curvature corrections are small, hence the round sphere dominates in
this regime.  $S^{d+1}$, $Z^{(0)}(V)$ (474) can be evaluated explicitly for
any action. It takes the form

$\displaystyle\log Z^{(0)}(V)=-S_{E}[g_{\ell}]=\frac{\Omega_{d+1}}{8\pi
G}\bigl{(}-\Lambda\ell^{d+1}+\tfrac{d(d+1)}{2}\,\ell^{d-1}+\cdots\bigr{)}\,,\qquad\Omega_{d+1}\ell^{d+1}=V\,,$
(479)

where $+\cdots$ is a sum of $R^{n}$ higher-order curvature corrections
$\propto\ell^{-2n}$ and $\Omega_{d+1}=(\ref{volSn})$. The off-shell energy
density and entropy are defined as in (476)

$\displaystyle\rho^{(0)}$
$\displaystyle={-\tfrac{1}{d+1}\ell\partial_{\ell}\log
Z^{(0)}}\,/\,{V}=\bigl{(}\Lambda-\tfrac{d(d-1)}{2}\,\ell^{-2}+\cdots\bigr{)}/8\pi
G$ (480) $\displaystyle S^{(0)}$
$\displaystyle=\bigl{(}1-\tfrac{1}{d+1}\ell\partial_{\ell}\bigr{)}\log
Z^{(0)}=\frac{A}{4G}\bigl{(}1+\cdots\bigr{)}\,.$

where $A=\Omega_{d-1}\ell^{d-1}$ and $+\cdots$ are $1/\ell^{2n}$ curvature
corrections. The on-shell radius $\ell_{0}$ solves $\rho^{(0)}(\ell_{0})=0$,
most conveniently viewed as giving a parametrization $\Lambda(\ell_{0})$.

As an example, consider the general action up to order $R^{2}$ written as

$\displaystyle S_{E}=\frac{1}{8\pi
G}\int\\!\\!\sqrt{g}\Bigl{(}\Lambda-\tfrac{1}{2}R-l_{s}^{2}\bigl{(}\lambda_{C^{2}}C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}+\lambda_{R^{2}}R^{2}+\lambda_{E^{2}}E^{\mu\nu}E_{\mu\nu}\bigr{)}\Bigr{)}\,,$
(481)

where $E_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{d+1}R\,g_{\mu\nu}$,
$C_{\mu\nu\rho\sigma}$ is the Weyl tensor, $l_{s}$ is a length scale and the
$\lambda_{i}$ are dimensionless. The Weyl tensor vanishes on the round sphere
and $R_{\mu\nu}=d\,g_{\mu\nu}/\ell^{2}$, hence

$\displaystyle\log Z^{(0)}=\frac{\Omega_{d+1}}{8\pi
G}\bigl{(}-\Lambda\,\ell^{d+1}+\tfrac{1}{2}d(d+1)\,\ell^{d-1}+\lambda_{R^{2}}\,d^{2}(d+1)^{2}\,l_{s}^{2}\,\ell^{d-3}\,\bigr{)}\,,$
(482)

For example for $d=2$,

$\displaystyle\log
Z^{(0)}=\frac{\pi}{4G}\Bigl{(}-\Lambda\,\ell^{3}+3\,\ell+\frac{36\,l_{s}^{2}\lambda_{R^{2}}}{\ell}\Bigr{)}\,,$
(483)

hence, using (480) and $\rho^{(0)}(\ell_{0})=0$,

$\displaystyle{\cal
S}^{(0)}=S^{(0)}(\ell_{0})=\frac{2\pi\ell_{0}}{4G}\Bigl{(}1+\frac{24\,l_{s}^{2}\lambda_{R^{2}}}{\ell_{0}^{2}}\Bigr{)}\,,\qquad\Lambda=\frac{1}{\ell_{0}^{2}}\Bigl{(}1-\frac{12\,l_{s}^{2}\lambda_{R^{2}}}{\ell_{0}^{2}}\Bigr{)}\,.$
(484)

####  I.1.3 Effective field theory expansion and field redefinitions

Curvature corrections such as those considered above naturally appear as terms
in the derivative expansion of low-energy effective field theories of quantum
gravity, with $l_{s}$ the characteristic length scale of UV-completing physics
and higher-order curvature corrections terms suppressed by higher powers of
$l_{s}^{2}/\ell^{2}\ll 1$. The action (481) is then viewed as a truncation at
order $l_{s}^{2}$, and (484) can be solved perturbatively to obtain $\ell_{0}$
and ${\cal S}^{(0)}$ as a function of $\Lambda$.

Suppose someone came up with some fundamental theory of de Sitter quantum
gravity, producing both a precise microscopic computation of the entropy and a
precise low-energy effective action, with the large-$\ell_{0}/l_{s}$ expansion
reproduced as some large-$N$ expansion. At least superficially, the higher-
order curvature-corrected entropy obtained above looks like a Wald entropy
Wald:1993nt ; Iyer:1994ys . In the spirit of for instance the nontrivial
matching of $R^{2}$ corrections to the macroscopic BPS black hole entropy
computed in LopesCardoso:1998tkj  and the microscopic entropy computed from
M-theory in Maldacena:1997de , it might seem then that matching microscopic
$1/N$-corrections and macroscopic $l_{s}^{2}/\ell_{0}^{2}$-corrections to the
entropy such as those in (484) could offer a nontrivial way of testing such a
hypothetical theory.

However, this is not the case. Unlike the Wald entropy, there are no charges
$Q$ (such as energy, angular momentum or gauge charges) available here to give
these corrections physical meaning as corrections in the large-$Q$ expansion
of a function $S(Q)$. Indeed, the detailed structure of the $l_{s}/\ell_{0}$
expansion of ${\cal S}^{(0)}=S^{(0)}(\ell_{0})$ has no intrinsic physical
meaning at all, because all of it can be wiped out by a local metric field
redefinition, order by order in $l_{s}/\ell_{0}$, bringing the entropy to pure
Einstein area law form, and leaving only the value of ${\cal S}^{(0)}$ itself
as a physically meaningful, field-redefinition invariant, dimensionless
quantity.

This is essentially a trivial consequence of the fact that in perturbation
theory about the round sphere, the round sphere itself is the unique solution
to the equations of motion. Let us however recall in more detail how this
works at the level of local field redefinitions, and show how this is
expressed at the level of $\log Z^{(0)}(\ell)$, as this will be useful later
in interpreting quantum corrections. For concreteness, consider again (481)
viewed as a gravitational effective field theory action expanded to order
$l_{s}^{2}R^{2}$. Under a local metric field redefinition

$\displaystyle g_{\mu\nu}\to g_{\mu\nu}+\delta
g_{\mu\nu}+O(l_{s}^{4})\,,\qquad\delta g_{\mu\nu}\equiv
l_{s}^{2}\bigl{(}u_{0}\,\Lambda\,g_{\mu\nu}+u_{1}\,R\,g_{\mu\nu}+u_{2}\,R_{\mu\nu}\bigr{)}\,,$
(485)

where the $u_{i}$ are dimensionless constants, the action transforms as

$\displaystyle S_{E}\to S_{E}+\frac{1}{16\pi
G}\int\\!\\!\sqrt{g}\,\bigl{(}R^{\mu\nu}-\tfrac{1}{2}Rg^{\mu\nu}+\Lambda
g^{\mu\nu}\bigr{)}\,\delta g_{\mu\nu}\,+\,O(l_{s}^{4})\,,$ (486)

shifting $\lambda_{R^{2}},\lambda_{E^{2}}$ and rescaling $G,\Lambda$ in (481).
A suitable choice of $u_{i}$ brings $S_{E}$ to the form

$\displaystyle S_{E}=\frac{1}{8\pi
G}\int\\!\\!\sqrt{g}\Bigl{(}\\!\Lambda^{\prime}-\tfrac{1}{2}R-l_{s}^{2}\lambda_{C^{2}}C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}+O(l_{s}^{4})\\!\Bigr{)},\qquad\Lambda^{\prime}=\Lambda\bigl{(}1-\tfrac{4(d+1)^{2}}{(d-1)^{2}}\,\lambda_{R^{2}}l_{s}^{2}\Lambda\bigr{)}.$
(487)

Equivalently, this is obtained by using the $O(l_{s}^{0})$ equations of motion
$R_{\mu\nu}=\frac{2}{d-1}\Lambda g_{\mu\nu}$ in the $O(l_{s}^{2})$ part of the
action. Since $\lambda^{\prime}_{R^{2}}=0$, the entropy computed from this
equivalent action takes a pure Einstein area law form ${\cal
S}^{(0)}=\Omega_{d+1}\ell_{0}^{\prime\,d-1}/4G$, with
$\ell_{0}^{\prime}=\sqrt{d(d-1)/2\Lambda^{\prime}}$. The on-shell value ${\cal
S}^{(0)}$ itself remains unchanged of course under this change of variables.

In the above we picked a field redefinition keeping $G^{\prime}=G$. Further
redefining $g_{\mu\nu}\to\alpha\,g_{\mu\nu}$ leads to another equivalent set
of couplings $G^{\prime\prime},\Lambda^{\prime\prime},\ldots$ rescaled with
powers of $\alpha$ according to their mass dimension. We could then pick
$\alpha$ such that instead $\Lambda^{\prime\prime}=\Lambda$, or such that
$\ell_{0}^{\prime\prime}=\ell_{0}$, now with $G^{\prime\prime}\neq G$. If we
keep $\ell_{0}^{\prime\prime}=\ell_{0}$, we get

$\displaystyle{\cal
S}^{(0)}=\frac{\Omega_{d+1}\ell_{0}^{d-1}}{4G^{\prime\prime}}\,,\qquad\Lambda^{\prime\prime}=\frac{d(d-1)}{2\ell_{0}^{2}}\,,$
(488)

where for example in $d=2$ starting from (484),
$G^{\prime\prime}=G\bigl{(}1-24\,\lambda_{R^{2}}l_{s}^{2}/\ell_{0}^{2}+O(l_{s}^{4})\bigr{)}$.

At the level of $\log Z^{(0)}(\ell)$ in (482) the metric redefinition (485)
amounts to a radius redefininition $\ell\to\ell\,f(\ell)$ with
$f(\ell)=1+l_{s}^{2}\bigl{(}v_{10}\Lambda+v_{11}\ell^{-2}\bigr{)}+O(l_{s}^{4})$.
For suitable $v_{i}$ this brings $\log Z^{(0)}$ and therefore ${\cal S}^{(0)}$
to pure Einstein form. E.g. for the $d=2$ example (483),

$\displaystyle\ell=\bigl{(}1-12\,\lambda_{R^{2}}l_{s}^{2}\bigl{(}\Lambda+\ell^{\prime\,-2}\bigr{)}\bigr{)}\ell^{\prime}\qquad\Rightarrow\qquad\log
Z^{(0)}=\frac{\pi}{4G}\bigl{(}-\Lambda^{\prime}\,\ell^{\prime\,3}+3\,\ell^{\prime}+O(l_{s}^{4})\bigr{)}\,.$
(489)

The above considerations generalize to all orders in the $l_{s}$ expansion.
$R^{n}$ corrections to $\log Z^{(0)}$ are $\propto(l_{s}/\ell)^{2n}$ and can
be removed order by order by a local metric/radius redefinition

$\ell\to\alpha\,f(\ell)\,\ell\,,\qquad
f(\ell)=1+l_{s}^{2}\bigl{(}v_{10}\Lambda+v_{11}\ell^{-2}\bigr{)}+l_{s}^{4}\bigl{(}v_{20}\Lambda^{2}+v_{21}\Lambda\ell^{-2}+v_{22}\ell^{-4}\bigl{)}\,+\cdots$
(490)

bringing $\log Z^{(0)}$ and thus ${\cal S}^{(0)}$ to Einstein form to any
order in the $l_{s}$ expansion.

In $d=2$, the Weyl tensor vanishes identically. The remaining higher-order
curvature invariants involve the Ricci tensor only, so can be removed by field
redefinitions, reducing the action to Einstein form in general. Thus in $d=2$,
${\cal S}^{(0)}$ is the only tree-level invariant in the theory, i.e. the only
physical coupling constant. In the Chern-Simons formulation of H.2, ${\cal
S}^{(0)}=2\pi\kappa$. In $d\geq 3$, there are infinitely many independent
coupling constants, such as the Weyl-squared $\lambda_{C^{2}}$ in (481), which
are not picked up by ${\cal S}^{(0)}$, but are analogously probed by
invariants ${\cal S}^{(0)}_{M}=\log{\cal Z}^{(0)}[g_{M}]=-S_{E}[g_{M}]$ for
saddle geometries $g_{M}$ different from the round sphere. We comment on those
and their role in the bigger picture in section I.5.

The point of considering quantum corrections to the entropy ${\cal S}$ is that
these include nonlocal contributions, not removable by local redefinitions,
and thus, unlike the tree-level entropy ${\cal S}^{(0)}$, offering actual data
quantitatively constraining candidate microscopic models.

###  I.2 Quantum gravitational thermodynamics

The quantum off-shell partition function $Z(V)$ generalizing the tree-level
$Z^{(0)}(V)$ (474) is defined by replacing $\int_{\text{tree}}{\cal
D}g\to\int{\cal D}g$ in that expression:464646$Z(V)$ is reminiscent of but
different from the fixed-volume partition function considered in the 2D
quantum gravity literature, e.g. (2.20) in Ginsparg:1993is . The latter would
be defined as above but with $\frac{1}{2\pi i}\int_{i{\mathbb{R}}}d\sigma$
instead of $\int_{\text{tree}}d\sigma$, constraining the volume to $V$,
whereas $Z(V)$ constrains the expectation value of the volume to $V$.

$\displaystyle
Z(V)\equiv\mbox{\Large$\int$}_{\\!\\!\\!\text{tree}}\,d\sigma\,\mbox{\Large$\int$}{\cal
D}g\,e^{-S_{E}[g]+\sigma(\int\\!\sqrt{g}-V)}\,.$ (491)

The quantum off-shell energy density and entropy generalizing (476) are

$\displaystyle\rho(V)\equiv-\partial_{V}\log Z\,,\qquad
S(V)\equiv\bigl{(}1-V\partial_{V})\log Z\,.$ (492)

$S$ is the Legendre transform of $\log Z$:

$\displaystyle S=\log Z+V\rho\,,\qquad d\log Z=-\rho\,dV\,,\qquad
dS=Vd\rho\,.$ (493)

Writing $e^{-\Gamma(V)}\equiv Z(V)$, the above definitions imply that as a
function of $\rho$,

$\displaystyle
S(\rho)=\log\mbox{\Large$\int$}_{\\!\\!\\!\text{tree}}\,dV\,e^{-\Gamma(V)+\rho
V}=\log\mbox{\Large$\int$}{\cal D}g\,e^{-S_{E}[g]+\rho\int\\!\sqrt{g}}$ (494)

hence $S(\rho)$ is the generating function for moments of the volume. In
particular

$\displaystyle
V=\bigl{\langle}\mbox{\large$\int$}\\!\sqrt{g}\bigr{\rangle}_{\rho}=\partial_{\rho}S(\rho)$
(495)

is the expectation value of the volume in the presence of a source $\rho$
shifting the cosmological constant $\frac{\Lambda}{8\pi
G}\to\frac{\Lambda}{8\pi G}-\rho$. $\Gamma(V)$ can be viewed as a quantum
effective action for the volume, in the spirit of the QFT 1PI effective action
weinberg_1996 ; Vilkovisky:1984st ; Barvinsky:1985an  but taking only the
volume off-shell. At tree level it reduces to $S_{E}[g_{\ell}]$ appearing in
(479). At the quantum on-shell value
$V=\bar{V}=\bigl{\langle}\mbox{\large$\int$}\\!\sqrt{g}\bigr{\rangle}_{0}$,

$\displaystyle\rho(\bar{V})=0\,,\qquad{\cal S}=\log Z(\bar{V})=S(\bar{V})\,.$
(496)

It will again be convenient to work with a linear scale variable $\ell$
instead of $V$, defined by

$\displaystyle\Omega_{d+1}\ell^{d+1}\equiv V\,,$ (497)

Since the mean volume
$V=\bigl{\langle}\int\\!\\!\sqrt{g}\bigr{\rangle}_{\rho}$ is diffeomorphism
invariant, (497) gives a manifestly diffeomorphism-invariant definition of the
“mean radius” $\ell$ of the fluctuating geometry. Given $Z(\ell)\equiv
Z(V(\ell))$, the off-shell energy density and entropy are then computed as

$\displaystyle\rho(\ell)=-\frac{\frac{1}{d+1}\ell\partial_{\ell}\log
Z}{V}\,,\qquad S(\ell)=\bigl{(}1-\tfrac{1}{d+1}\ell\partial_{\ell}\bigr{)}\log
Z\,.$ (498)

The quantum on-shell value of $\ell$ is denoted by $\bar{\ell}$ and satisfies
$\rho(\bar{\ell})=0$.

The magnitude of quantum fluctuations of the volume about its mean value is
given by $\delta
V^{2}\equiv\bigl{\langle}\bigl{(}\int\\!\\!\sqrt{g}-V\bigr{)}^{2}\bigr{\rangle}_{\rho}=S^{\prime\prime}(\rho)=1/\Gamma^{\prime\prime}(V)=1/\rho^{\prime}(V)=V/S^{\prime}(V)$.
At large $V$, $\delta V/V\propto 1/\sqrt{S}$.

###  I.3 One-loop corrected de Sitter entropy

The path integral (491) for $\log Z$ can be computed perturbatively about its
round sphere saddle in a semiclassical expansion in powers of $G$. To leading
order it reduces to $\log Z^{(0)}$ defined in (474). For 3D Einstein gravity,

$\displaystyle\log Z(V)=\log Z^{(0)}(V)+O(G^{0})=\frac{\Omega_{3}}{8\pi
G}\bigl{(}-\Lambda\,\ell^{3}+3\,\ell\bigr{)}\,+\,O(G^{0})\,,$ (499)

To compute the one-loop $O(G^{0})$ correction, recall that evaluation of
$\int_{\rm tree}d\sigma$ in (491) is equivalent to extremization with respect
to $\sigma$, which sets $\sigma=-\partial_{V}\log Z(V)=\rho(V)$ and

$\displaystyle\log Z(V)=\log\mbox{\Large$\int$}{\cal
D}g\,e^{-S_{E}[g]+\rho(V)(\int\\!\sqrt{g}-V)}\,.$ (500)

To one-loop order, we may replace $\rho$ by its tree-level approximation
$\rho^{(0)}=-\partial_{V}\log Z^{(0)}$. By construction this ensures the round
sphere metric $g=g_{\ell}$ of radius $\ell(V)$ given by (497) is a saddle.
Expanding the action to quadratic order in fluctuations about this saddle then
gives a massless spin-2 Gaussian path integral of the type solved in general
by (112), or more explicitly in (148)-(149). For 3D Einstein gravity, using
(149),

$\displaystyle\log Z=-\Bigl{(}\frac{\Lambda}{8\pi
G}+c_{0}^{\prime}\Bigr{)}\Omega_{3}\ell^{3}+\Bigl{(}\frac{1}{8\pi
G}+c_{2}^{\prime}-\frac{3}{4\pi\epsilon}\Bigr{)}3\Omega_{3}\ell-3\log\frac{2\pi\ell}{4G}+5\log(2\pi)+O(G)$
(501)

Here $c_{0}^{\prime}$ and $c_{2}^{\prime}$ arise from $O(G^{0})$ local
counterterms

$\displaystyle S_{E,\rm
ct}=\int\\!\\!\sqrt{g}\Bigl{(}c_{0}^{\prime}-\frac{c_{2}^{\prime}}{2}R\Bigr{)}\,,$
(502)

split off from the bare action (470) to keep the tree-level couplngs $\Lambda$
and $G$ equal to their “physical” (renormalized) values to this order. We
define these physical values as the coefficients of the local terms
$\propto\ell^{3},\ell$ in the $V\to\infty$ asymptotic expansion of the quantum
$\log Z(V)$. That is to say, we fix $c_{0}^{\prime}$ and $c_{2}^{\prime}$ by
imposing the renormalization condition

$\displaystyle\log Z(V)=\frac{\Omega_{3}}{8\pi
G}\bigl{(}-\Lambda\,\ell^{3}+3\,\ell\,\bigr{)}+\cdots\qquad(V\to\infty)\,.$
(503)

This renormalization prescription is diffeomorphism invariant, since $Z(V)$,
$V$ and $\ell$ were all defined in a manifestly diffeomorphism-invariant way.
In (501) it fixes $c_{0}^{\prime}=0$, $c_{2}^{\prime}=\frac{3}{4\pi\epsilon}$,
hence $\log Z(\ell)=\log Z^{(0)}+\log Z^{(1)}+O(G)$, where

$\displaystyle\log Z^{(1)}=-3\log\frac{2\pi\ell}{4G}+5\log(2\pi)\,.$ (504)

We can express the renormalization condition (503) equivalently as

$\displaystyle\boxed{\log Z^{(1)}=\log Z_{\rm PI}^{(1)}+\log Z_{\rm
ct}\,,\qquad\lim_{\ell\to\infty}\partial_{\ell}\log Z^{(1)}=0}$ (505)

where $\log Z_{\rm ct}=-S_{E,\rm ct}[g_{\ell}]$ with $g_{\ell}$ the round
sphere metric of volume $V$. On $S^{3}$ we have $\log Z_{\rm
ct}=c_{0}\ell^{3}+c_{2}\ell$, and the $\ell\to\infty$ condition fixes $c_{0}$
and $c_{2}$. Recalling (135), we can physically interpret this as requiring
the renormalized one-loop Euclidean energy $U^{(1)}$ of the static patch
vanishes in the $\ell\to\infty$ limit.

For general $d$, the UV-divergent terms in $\log Z_{\rm PI}^{(1)}$ come with
non-negative powers $\propto\ell^{d+1-2n}$, canceled by counterterms
consisting of $n$-th order curvature invariants. For example on $S^{5}$, $\log
Z_{\rm ct}=c_{0}\ell^{5}+c_{2}\ell^{3}+c_{4}\ell$. In odd $d+1$, the
renormalization prescription (505) then fixes the $c_{2n}$. In even $d+1$,
$\log Z_{\rm ct}$ has a constant term $c_{d+1}$, which is not fixed by (505).
As we will make explicit in examples later, it can be fixed by
$\lim_{\ell\to\infty}Z^{(1)}=0$ for massive field contributions, and for
massless field contributions by minimal subtraction at scale $L$,
$c_{d+1}=-\alpha_{d+1}\log(M_{\epsilon}L)$,
$M_{\epsilon}=2e^{-\gamma}/\epsilon$ (261), with $L\partial_{L}\log Z=0$, i.e.
$L\partial_{L}\log Z^{(0)}=\alpha_{d+1}$.

The renormalized off-shell $\rho$ and $S$ are obtained from $\log Z$ as in
(498). For 3D Einstein,

$\displaystyle\rho^{(1)}=\frac{1}{2\pi^{2}\ell^{3}}\,,\qquad
S^{(1)}=-3\log\frac{2\pi\ell}{4G}+5\log(2\pi)+1\,.$ (506)

The on-shell quantum dS entropy ${\cal S}=\log Z(\bar{\ell})=S(\bar{\ell})$
(496) is

$\displaystyle\boxed{{\cal
S}=S(\bar{\ell})=S^{(0)}(\bar{\ell})+S^{(1)}(\bar{\ell})\,+\,O(G)}$ (507)

where $\bar{\ell}$ is the quantum mean radius satisfying
$\rho(\bar{\ell})\propto\partial_{\ell}\log Z(\bar{\ell})=0$. For 3D Einstein,

$\displaystyle{\cal
S}=\frac{2\pi\bar{\ell}}{4G}-3\log\frac{2\pi\bar{\ell}}{4G}+5\log(2\pi)+1\,+\,O(G)\,,\qquad\Lambda=\frac{1}{\bar{\ell}^{2}}-\frac{4G}{\pi\bar{\ell}^{3}}+O(G^{2})\,.$
(508)

Alternatively, ${\cal S}$ can be expressed in terms of the tree-level
$\ell_{0}$, $\rho^{(0)}(\ell_{0})\propto\partial_{\ell}\log
Z^{(0)}(\ell_{0})=0$, using ${\cal S}=\log Z^{(0)}(\bar{\ell})+\log
Z^{(1)}(\bar{\ell})+O(G)$, $\bar{\ell}=\ell_{0}+O(G)$ and Taylor expanding in
$G$:

$\displaystyle\boxed{{\cal S}=\log Z(\bar{\ell})=S^{(0)}(\ell_{0})+\log
Z^{(1)}(\ell_{0})\,+\,O(G)}$ (509)

This form would be obtained from (153) by a more standard computation. For 3D
Einstein,

$\displaystyle{\cal
S}=\frac{2\pi\ell_{0}}{4G}-3\log\frac{2\pi\ell_{0}}{4G}+5\log(2\pi)\,+\,O(G)\,,\qquad\Lambda=\frac{1}{\ell_{0}^{2}}\,.$
(510)

The equivalence of (507) and (509) can be checked directly here noting
$\bar{\ell}=\ell_{0}-\frac{2}{\pi}G\,+\,O(G^{2})$, so
$\frac{2\pi\bar{\ell}}{4G}=\frac{2\pi\ell_{0}}{4G}-1+O(G)$. The $-1$ cancels
the $+1$ in (508), reproducing (510).

More generally and more physically, the relation between these two expressions
can be understood as follows. At tree level, the entropy equals the geometric
horizon entropy $S^{(0)}(\ell_{0})$, with radius $\ell_{0}$ such that the
geometric energy density $\rho^{(0)}$ vanishes. At one loop, we get additional
contributions from quantum field fluctuations. The UV contributions are
absorbed into the gravitational coupling constants. The remaining IR
contributions shift the entropy by $S^{(1)}$ and the energy density by
$\rho^{(1)}$. The added energy backreacts on the fluctuating geometry: its
mean radius changes from $\ell_{0}$ to $\bar{\ell}$ such that the geometric
energy density changes by $\delta\rho^{(0)}=-\rho^{(1)}$, ensuring the total
energy density vanishes. This in turn changes the geometric horizon entropy by
an amount dictated by the first law (477),

$\displaystyle\delta S^{(0)}=V_{0}\,\delta\rho^{(0)}=-V_{0}\,\rho^{(1)}\,.$
(511)

We end up with a total entropy ${\cal
S}=S^{(0)}(\bar{\ell})+S^{(1)}=S^{(0)}(\ell_{0})-V_{0}\,\rho^{(1)}+S^{(1)}=S^{(0)}(\ell_{0})+\log
Z^{(1)}$, up to $O(G)$ corrections, relating (507) to (509). (See also fig.
21.)

More succinctly, obtaining (509) from (507) is akin to obtaining the canonical
description of a thermodynamic system from the microcanonical description of
system + reservoir. The analog of the canonical partition function is
$Z^{(1)}=e^{S^{(1)}-V_{0}\,\rho^{(1)}}$, with $-V_{0}\,\rho^{(1)}$ capturing
the reservoir (horizon) entropy change due to energy transfer to the system.

###  I.4 Examples

####  I.4.1 3D scalar

Figure 21:  One-loop contributions to the dS3 entropy from metric and scalars
with $\eta=1,\frac{1}{4},\frac{5}{4}$, i.e. $\xi=0,\frac{1}{8},-\frac{1}{24}$.
Blue dotted line = renormalized entropy $S^{(1)}$. Green dotted line = horizon
entropy change $\delta S^{(0)}=2\pi\delta\ell/4G=-V\rho^{(1)}$ due to quantum
backreaction $\ell_{0}\to\bar{\ell}=\ell_{0}+\delta\ell$, as dictated by first
law. Solid red line = total $\delta{\cal S}=S^{(1)}-V\rho^{(1)}=\log Z^{(1)}$.
The metric contribution is negative within the semiclassical regime of
validity $\ell\gg G$. The renormalized scalar entropy and energy density are
positive for $m\ell\gg 1$, and for all $m\ell$ if $\eta=1$. If $\eta\\!>\\!1$
and $\ell_{0}\to\ell_{*}\equiv\frac{\sqrt{\eta-1}}{m}$, the correction
$\delta\ell\sim-\frac{G}{3\pi}\frac{\ell_{*}}{\ell_{0}-\ell_{*}}\to-\infty$,
meaning the one-loop approximation breaks down. The scalar becomes tachyonic
beyond this point. If a $\phi^{4}$ term is included in the action, two new
dominant saddles emerge with $\phi\neq 0$.

An example with matter is 3D Einstein gravity + scalar $\phi$ as in (140).
Putting $\xi\equiv\frac{1-\eta}{6}$,

$\displaystyle S_{E}[g,\phi]=\frac{1}{8\pi
G}\int\\!\\!\sqrt{g}\,\bigl{(}\Lambda-\tfrac{1}{2}R\bigr{)}\,+\,\frac{1}{2}\int\\!\\!\sqrt{g}\,\phi\bigl{(}-\nabla^{2}+m^{2}+\tfrac{1-\eta}{6}R\,\bigr{)}\phi\,,$
(512)

The metric contribution to $\log Z^{(1)}$ remains $\log Z^{(1)}_{\rm
metric}=-3\log\frac{2\pi\ell}{4G}+5\log(2\pi)$ as in (504). The scalar $Z_{\rm
PI}^{(1)}$ was given in (143). Its finite part is

$\displaystyle\log Z^{(1)}_{\rm
PI,fin,scalar}=\frac{\pi\nu^{3}}{6}-\sum_{k=0}^{2}\frac{\nu^{k}}{k!}\,\frac{{\rm
Li}_{3-k}(e^{-2\pi\nu})}{(2\pi)^{2-k}}\,,\qquad\nu\equiv\sqrt{m^{2}\ell^{2}-\eta}\,.$
(513)

The polynomial $\log Z_{\rm ct}(\ell)=c_{0}\ell^{3}+c_{2}\ell$ corresponding
to the counterterm action (502) is fixed by the renormalization condition
(505), resulting in

$\displaystyle\log Z^{(1)}_{\rm scalar}=\log Z^{(1)}_{\rm
PI,fin,scalar}-\frac{\pi}{6}\,m^{3}\ell^{3}+\frac{\pi\eta}{4}\,m\ell\,.$ (514)

The finite polynomial cancels the local terms $\propto\ell^{3},\ell$ in the
large-$\ell$ asymptotic expansion of the finite part: $\log Z_{\rm
scalar}^{(1)}=\frac{\pi\eta^{2}}{16}(m\ell)^{-1}+\frac{\pi\eta^{3}}{96}(m\ell)^{-3}+\cdots$
when $m\ell\to\infty$. The $(m\ell)^{-{2n-1}}$ terms have the
$\ell$-dependence of $R^{n}$ terms in the action and can effectively be
thought of as finite shifts of higher-order curvature couplings in the
$m\ell\gg 1$ regime. In the opposite regime $m\ell\ll 1$, IR bulk modes of the
scalar becomes thermally activated and $\log Z^{(1)}_{\rm scalar}$ ceases to
have a local expansion. In particular in the minimally-coupled case $\eta=1$,

$\displaystyle\log Z^{(1)}_{\rm scalar}\simeq-\log(m\ell)\qquad(m\ell\to
0)\,.$ (515)

The total energy density is $\rho=-\frac{1}{3}\ell\partial_{\ell}\log
Z/V=\frac{1}{8\pi G}\bigl{(}\Lambda-\ell^{-2}\bigr{)}+1+\rho^{(1)}_{\rm
scalar}$ where

$\displaystyle V\rho^{(1)}_{\rm
scalar}=-\frac{\pi}{6}(m\ell)^{2}\nu\coth(\pi\nu)+\frac{\pi}{6}(m\ell)^{3}-\frac{\pi\eta}{12}\,m\ell\,.$
(516)

The on-shell quantum dS entropy is given to this order by (509) or by (507) as

$\displaystyle{\cal S}={\cal S}^{(0)}+{\cal S}^{(1)}=S^{(0)}(\ell_{0})+\log
Z^{(1)}=S^{(0)}(\ell_{0})-V\rho^{(1)}+S^{(1)}=S^{(0)}(\bar{\ell})+S^{(1)}\,,$
(517)

where $\ell_{0}^{-2}=\Lambda=\bar{\ell}^{-2}-8\pi G\,\rho^{(1)}(\bar{\ell})$
and $S^{(1)}=S_{\rm PI,fin}^{(1)}+\frac{1}{6}\pi\eta\,m\ell$, with the scalar
contribution to $S_{\rm PI,fin}^{(1)}$ given by the finite part of (145). Some
examples are shown in fig. 21.

For a massless scalar, $m=0$, the renormalized scalar one-loop correction to
${\cal S}$ is a constant independent of $\ell_{0}$ given by (513) evaluated at
$\nu=\sqrt{-\eta}$, and $\rho^{(1)}_{\rm scalar}=0$. For example for a
massless conformally coupled scalar, $\eta=\frac{1}{4}$, $Z^{(1)}_{\rm
scalar}=\frac{3\,\zeta(3)}{16\pi^{2}}-\frac{\log(2)}{8}$.

####  I.4.2 3D massive spin $s$

The renormalized one-loop correction ${\cal S}^{(1)}_{s}=\log Z^{(1)}_{s}$ to
the dS3 entropy from a massive spin-$s$ field is obtained similarly from (90):

$\displaystyle{\cal S}^{(1)}_{s}=\log Z^{(1)}_{s,\rm bulk}-\log Z^{(1)}_{s,\rm
edge}\,,$ (518)

where $\log Z^{(1)}_{s,\rm bulk}$ equals twice the contribution of an
$\eta=(s-1)^{2}$ scalar as given in (514), while the edge contribution is,
putting $\nu\equiv\sqrt{m^{2}\ell^{2}-(s-1)^{2}}$,

$\displaystyle\log Z^{(1)}_{s,\rm
edge}=s^{2}\bigl{(}\pi(m\ell-\nu)-\log(1-e^{-2\pi\nu})\bigr{)}\,.$ (519)

The edge contribution to (518) is manifestly negative. It dominates the bulk
part, and increasingly so as $s$ grows. Examples are shown in fig. 22.

Figure 22:  Contributions to the dS3 entropy from massive spin $s=1,2,3$
fields, as a function of $m\ell_{0}$, with coloring as in fig. 21.
Singularities = Higuchi bound, as discussed under (145).

####  I.4.3 2D scalar

As mentioned below (505), the counterterm polynomial $\log Z_{\rm ct}^{(0)}$
has a constant term in even spacetime dimensions $d+1$, which is not fixed yet
by the renormalization prescription given there. Let us consider the simplest
example: a $d=1$ scalar with action (140). Denoting

$\displaystyle{\tt f}(\nu)\equiv\sum_{\pm}\zeta^{\prime}(-1,\tfrac{1}{2}\pm
i\nu)\mp i\nu\zeta^{\prime}(0,\tfrac{1}{2}\pm i\nu)\,,\,,$ (520)

and $M_{\epsilon}={2e^{-\gamma}}/{\epsilon}$ as in (261), we get from (253)
with $\nu\equiv\sqrt{m^{2}\ell^{2}-\eta}$, $\eta\equiv\tfrac{1}{4}-2\xi$,

$\displaystyle\log Z_{\rm
PI}^{(1)}=\bigl{(}2\epsilon^{-2}-m^{2}\log(M_{\epsilon}\ell)+m^{2}\bigr{)}\,\ell^{2}+\bigl{(}\eta+\tfrac{1}{12}\bigr{)}\log\bigl{(}M_{\epsilon}\ell\bigr{)}-\eta+{\tt
f}(\nu)\,,$ (521)

In the limit $m\ell\to\infty$, using the asymptotic expansion of the Hurwitz
zeta function474747See e.g. appendix A of Dib:2000hd ,

$\displaystyle\log Z_{\rm
PI}^{(1)}=\bigl{(}2\epsilon^{-2}-m^{2}\log(M_{\epsilon}/m)-\tfrac{1}{2}m^{2}\bigr{)}\,\ell^{2}+(\eta+\tfrac{1}{12})\log(M_{\epsilon}/m)\,+\,O((m\ell)^{-2})\,.$
(522)

Notice the $\log\ell$ dependence apparent in (521) has canceled out. The
counterterm action to this order is again of the form (502), corresponding to
$\log Z_{\rm ct}=4\pi(-c_{0}^{\prime}\ell^{2}+c_{2}^{\prime})$. The
renormalization condition (505) fixes $c_{0}^{\prime}$ but leaves
$c_{2}^{\prime}$ undetermined. Its natural extension here is to pick
$c_{2}=4\pi c_{2}^{\prime}$ to cancel off the constant term as well, that is

$\displaystyle
c_{2}=-(\eta+\tfrac{1}{12})\log(M_{\epsilon}/m)\qquad\Rightarrow\qquad\lim_{\ell\to\infty}\log
Z^{(1)}=0\,,$ (523)

ensuring the tree-level $G$ equals the renormalized Newton constant to this
order, as in (503). The renormalized scalar one-loop contribution to the off-
shell partition function is then

$\log
Z^{(1)}=\bigl{(}\tfrac{3}{2}-\log(m\ell)\bigr{)}(m\ell)^{2}+(\eta+\tfrac{1}{12})\log(m\ell)-\eta+{\tt
f}(\nu)\,.$ (524)

In the large-$m\ell$ limit, $\log
Z^{(1)}=\frac{240\,\eta^{2}+40\,\eta+7}{960}(m\ell)^{-2}+\cdots>0$, while in
the small-$m\ell$ limit

$\displaystyle\log
Z^{(1)}\simeq\bigl{(}\eta+\tfrac{1}{12}\bigr{)}\log(m\ell)\quad(\eta<\tfrac{1}{4})\,,\qquad\log
Z^{(1)}\simeq\bigl{(}\tfrac{1}{4}+\tfrac{1}{12}-1\bigr{)}\log(m\ell)\quad(\eta=\tfrac{1}{4})\,.$
(525)

The extra $-\log(m\ell)$ in the minimally-coupled case $\eta=\frac{1}{4}$ is
the same as in (515) and has the same thermal interpretation. The energy
density is $\rho^{(1)}=-\frac{1}{2}\ell\partial_{\ell}\log Z^{(1)}/V$ with
$V=4\pi\ell^{2}$:

$\displaystyle
V\rho^{(1)}=-\tfrac{1}{2}(\eta+\tfrac{1}{12})+\tfrac{1}{2}(m\ell)^{2}\bigl{(}2\log(m\ell)-\mbox{$\sum_{\pm}$}\psi^{(0)}\bigl{(}\tfrac{1}{2}\pm
i\nu\bigr{)}\bigr{)}$ (526)

In the massless case $m=0$, $\nu=\sqrt{-\eta}$ is $\ell$-independent, and we
cannot use the asymptotic expansion (522), nor the renormalization
prescription (523). Instead we fix $c_{2}$ by minimal subtraction, picking a
reference length scale $L$ and putting (with
$M_{\epsilon}=\frac{2e^{-\gamma}}{\epsilon}$ (261) as before)

$\displaystyle c_{2}(L)\equiv-(\eta+\tfrac{1}{12})\log(M_{\epsilon}L)\,,$
(527)

The renormalized $G$ then satisfies $\partial_{L}(\frac{4\pi}{8\pi
G}+c_{2})=0$, i.e. $L\partial_{L}\frac{1}{2G}=\eta+\frac{1}{12}$, and

$\displaystyle\log Z^{(1)}=(\eta+\tfrac{1}{12})\log(\ell/L)-\eta+{\tt
f}\bigl{(}\sqrt{-\eta}\bigr{)}\,,\qquad
V\rho^{(1)}=-\tfrac{1}{2}(\eta+\tfrac{1}{12})\,.$ (528)

The total $\log Z=\frac{1}{2G}(-\Lambda\ell^{2}+1)+\log Z^{(1)}$ is of course
independent of the choice of $L$.

####  I.4.4 4D massive spin $s$

Figure 23:  Edge contributions to the dS4 entropy from massive spin $s=1,2,3$
fields, as a function of $m\ell_{0}$, with coloring as in fig. 21. The
Higuchi/unitarity bound in this case is
$(m\ell_{0})^{2}-(s-\frac{1}{2})^{2}>-\frac{1}{4}$.

4D massive spin-$s$ fields can be treated similarly, starting from (C.2). In
particular the edge contribution $\log Z^{(1)}_{\rm edge}$ equals minus the
$\log Z^{(1)}$ of $D_{s-1}^{5}=\frac{1}{6}s(s+1)(2s+1)$ scalars on $S^{2}$,
computed earlier in (524), with the same $\nu=\sqrt{m^{2}\ell^{2}-\eta_{s}}$
as the bulk spin-$s$ field, which according to (79) means
$\eta_{s}=(s-\frac{1}{2})^{2}$. The corresponding contribution to the
renormalized energy density is $\rho^{(1)}_{\rm
edge}=-\frac{1}{4}\ell\partial_{\ell}\log Z^{(1)}_{\rm edge}/V$ with
$V=\Omega_{4}\ell^{4}$, so $V\rho^{(1)}_{\rm edge}$ equals
$-\frac{1}{2}D_{s-1}^{5}$ times the scalar result (526).

As in the $d=2$ case, the renormalized one-loop edge contribution ${\cal
S}^{(1)}_{\rm edge}$ to the entropy is negative and dominant. Some examples
are shown in fig. 23.

####  I.4.5 Graviton contribution for general $d$

For $d\geq 3$, UV-sensitive terms in the loop expansion renormalize higher-
order curvature couplings in the gravitational action, prompting the inclusion
of such terms in $S_{E}[g]$. Some caution is in order then if we wish to apply
(112) or (148)-(149) to compute $\log Z^{(1)}$. The formula (112) for $Z_{\rm
PI}^{(1)}$ depends on $\gamma=\sqrt{8\pi G_{\rm N}/A_{d-1}}$, gauge-
algebraically defined by (109) and various normalization conventions. We
picked these such that in pure Einstein gravity, $\gamma=\sqrt{8\pi
G/A(\ell_{0})}$, $\ell_{0}=\sqrt{d(d-1)/2\Lambda}$, with $G$ and $\Lambda$
read off from the gravitational Lagrangian. However this expression of
$\gamma$ in terms of Lagrangian parameters will in general be modified in the
presence of higher-order curvature terms. This is clear from the discussion in
I.1.3, and (488) in particular. Since $\gamma_{0}$ is field-redefinition
invariant, and since after transforming to a pure Einstein frame we have
$\gamma_{0}=\sqrt{2\pi/{\cal S}^{(0)}}$, with the right hand side also
invariant, we have in general (for Einstein + perturbative higher-order
curvature corrections)

$\displaystyle\gamma=\sqrt{{2\pi}/{{\cal S}^{(0)}}}\,.$ (529)

From (148) we thus get (ignoring the phase)

$\displaystyle{\cal S}={\cal S}^{(0)}-\frac{D_{d}}{2}\log{\cal
S}^{(0)}+\alpha^{(2)}_{d+1}\log\frac{\ell_{0}}{L}+K_{d+1}\,+\,O\bigl{(}1/{\cal
S}^{(0)}\bigr{)}$ (530)

where $D_{d}=\frac{(d+2)(d+1)}{2}$, $\alpha_{d+1}^{(2)}=0$ for even $d$ and
given by (116) for odd $d$, and $K_{d+1}$ a numerical constant obtained by
evaluating (115). For odd $d$ the constant in the counterterm $\log Z_{\rm
ct}(\ell)$ is fixed by minimal subtraction at a scale $L$,
$c_{d+1}(L)\equiv-\alpha_{d+1}\log(M_{\epsilon}L)$, with
$M_{\epsilon}=2e^{-\gamma}/\epsilon$ determined by the heat kernel regulator
as in (261), and $L\partial_{L}{\cal S}=0$, i.e. $L\partial_{L}{\cal
S}^{(0)}=\alpha_{d+1}^{(2)}$. Explicitly for $d=2,3,4$, using (149), (12)

$\begin{array}[]{l|l}d&{\cal S}\\\ \hline\cr 2&{\cal S}^{(0)}-3\log{\cal
S}^{(0)}+5\log(2\pi)\\\ 3&{\cal S}^{(0)}-5\log{\cal
S}^{(0)}-\frac{571}{90}\log(\frac{\ell_{0}}{L})-\log(\frac{8\pi}{3})+\frac{715}{48}-\frac{47}{3}\zeta^{\prime}(-1)+\frac{2}{3}\zeta^{\prime}(-3)\\\
4&{\cal S}^{(0)}-\frac{15}{2}\log{\cal
S}^{(0)}+\log(12)+\frac{27}{2}\log(2\pi)+\frac{65\,\zeta(3)}{48\,\pi^{2}}+\frac{5\,\zeta(5)}{16\,\pi^{4}}\end{array}$
(531)

For a $d=3$ action (481) up to $O(l_{s}^{2}R^{2})$, with dots denoting
$O(l_{s}^{4})$ terms,

$\displaystyle{\cal
S}^{(0)}=\frac{\pi}{G}\left(\ell_{0}^{2}+{48\,\lambda_{R^{2}}\,l_{s}^{2}}+\cdots\right)\,,\qquad\Lambda=\frac{3}{\ell_{0}^{2}}+\cdots\,.$
(532)

where
$L\partial_{L}\lambda_{R^{2}}=-\frac{G}{48\,\pi\,l_{s}^{2}}\cdot\frac{571}{45}$.
Putting $L=\ell_{0}$, and defining the scale $\ell_{R^{2}}$ by
$\lambda_{R^{2}}(\ell_{R^{2}})=0$,

$\displaystyle{\cal S}={\cal S}^{(0)}-5\log{\cal
S}^{(0)}+K_{4}\,+\,O\bigl{(}1/{\cal S}^{(0)}\bigr{)}\,,\qquad{\cal
S}^{(0)}=\frac{\pi\ell_{0}^{2}}{G}-\frac{571}{45}\log\frac{\ell_{0}}{\ell_{R^{2}}}+\cdots\,.$
(533)

The constant $K_{4}$ could be absorbed into $\lambda_{R^{2}}$ at this level.
Below, in (539), we will give it relative meaning however, by considering
saddles different from the round $S^{4}$.

###  I.5 Classical and quantum observables

Here we address question 3 in our list below (153). To answer this, we need
“observables” of the $\Lambda>0$ Euclidean low-energy effective field theory
probing independent gravitational couplings (for simplicity we restrict
ourselves to purely gravitational theories here), i.e. diffeomorphism and
field-redefinition invariant quantities, analogous to scattering amplitudes in
asymptotically flat space. For this to be similarly useful, an infinite amount
of unambiguous data should be extractable, at least in principle, from these
observables.

As discussed above, ${\cal S}^{(0)}=\log{\cal Z}^{(0)}=-S_{E}[g_{\ell_{0}}]$
invariantly probes the dimensionless coupling given by
$\ell_{0}^{d-1}/G\propto 1/G\Lambda^{{(d-1)}/2}$ in Einstein frame. The
obvious tree-level invariants probing different couplings in the gravitational
low-energy effective field theory are then the analogous ${\cal
S}^{(0)}_{M}\equiv\log{\cal Z}^{(0)}_{M}=-S_{E}[g_{M}]$ evaluated on saddles
$g_{M}$ different from the round sphere, in the parametric $\ell_{0}\gg l_{s}$
regime of validity of the effective field theory, with $g_{M}$ asymptotically
Einstein in the $\ell_{0}\to\infty$ limit. These are the analogs of tree-level
scattering amplitudes. The obvious quantum counterparts are the corresponding
generalizations of ${\cal S}$, i.e. ${\cal S}_{M}\equiv\log{\cal Z}_{M}$
evaluated in large-$\ell_{0}$ perturbation theory about the saddle $g_{M}$.
These are the analogs of quantum scattering amplitudes. Below we make this a
bit more concrete in examples.

#### 3D

In $d=2$, the Weyl tensor vanishes identically, so higher-order curvature
invariants involve $R_{\mu\nu}$ only and can be removed from the action by a
field redefinition in large-$\ell_{0}$ perturbation theory, reducing it to
pure Einstein form in general. As a result, ${\cal S}^{(0)}$ is the only
independent invariant in pure 3D gravity, all $g_{M}$ are Einstein, and the
${\cal S}^{(0)}_{M}$ are all proportional to ${\cal S}_{0}\equiv{\cal
S}^{(0)}_{S^{3}}$.

As discussed under (168), the quantum ${\cal S}={\cal S}_{S^{3}}$ takes the
form

${\cal S}={\cal S}_{0}={\cal S}_{0}-3\log{\cal
S}_{0}+5\log(2\pi)+\mbox{$\sum_{n}$}\,c_{n}\,{\cal S}_{0}^{-2n}$ (534)

The corrections terms in the expansion are all nonlocal (no odd powers of
$\ell_{0}$), and the coefficients provide an unambiguous, infinite data set.

#### odd D $\geq$ 5

In 5D gravity, there are infinitely many independent coupling constants. There
are also infinitely many different $\Lambda>0$ Einstein metrics on $S^{5}$,
including a discrete infinity of Böhm metrics with $SO(3)\times SO(3)$
symmetry Bohm:1998vc  amenable to detailed numerical analysis Gibbons:2002th ,
and 68 Sasaki-Einstein families with moduli spaces up to real dimension 10
Boyer:2003pe . Unlike the round $S^{5}$, these are not conformally flat, and
thus, unlike ${\cal S}^{(0)}$, the corresponding ${\cal S}^{(0)}_{M}$ will
pick up couplings such as the Weyl-squared coupling $\lambda_{C^{2}}$ in
(481). It is plausible that this set of known Einstein metrics (perturbed by
small higher-order corrections to the Einstein equations of motion at finite
$\ell_{0}$) more than suffices to invariantly probe all independent couplings
of the gravitational action, delivering moreover infinitely many quantum
observables ${\cal S}_{M}$, providing an infinity of unambiguous low-energy
effective field theory data to any order in perturbation theory, without ever
leaving the sphere — at least in principle.

The landscape of known $\Lambda>0$ Einstein metrics on odd-dimensional spheres
becomes increasingly vast as the dimension grows, with double-exponentially
growing numbers Boyer:2003pe . For example there are at least 8610 families of
Sasaki-Einstein manifolds on $S^{7}$, spanning all 28 diffeomorphism classes,
with the standard class admitting a 82-dimensional family, and there are at
least $10^{828}$ distinct families of Einstein metrics on $S^{25}$, featuring
moduli spaces of dimension greater than $10^{833}$.

#### 4D

4D gravity likewise has infinitely many independent coupling constants. It is
not known if $S^{4}$ has another Einstein metric besides the round sphere. In
fact the list of 4D topologies known to admit $\Lambda>0$ Einstein metrics is
rather limited 2008arXiv0810.4830A : $S^{4}$, $S^{2}\times S^{2}$,
${\mathbb{CP}}^{2}$, and the connected sums
${\mathbb{CP}}^{2}\\#k\overline{{\mathbb{CP}}^{2}}$, $1\leq k\leq 8$. However
for $k\geq 5$ these have a moduli space of nonzero dimension Tian:1987tw ;
Tian:1990wd , which might suffice to probe all couplings. (The moduli space
would presumably be lifted at sufficiently high order in the $l_{s}$ expansion
upon turning on higher-order curvature perturbations.)

Below we illustrate in explicit detail how the Weyl-squared coupling can be
extracted from suitable linear combinations of pairs of ${\cal S}_{M}^{(0)}$
with $M\in\\{S^{4},S^{2}\times S^{2},{\mathbb{CP}}^{2}\\}$, and how a suitable
linear combination of all three can be used to extract an unambiguous linear
combination of the constant terms arising at one loop.

The Weyl-squared coupling $\lambda_{C^{2}}$ in
$S_{E}[g]=\mbox{(\ref{SERiem2})}+\cdots$ is invisible to ${\cal S}^{(0)}$
(532) but it is picked up by ${\cal S}_{M}^{(0)}$ by $M=S^{2}\times S^{2}$:

$\displaystyle{\cal S}^{(0)}_{S^{2}\times
S^{2}}=\frac{2}{3}\cdot\frac{\pi}{G}\bigl{(}\ell_{0}^{2}+48\,\lambda_{R^{2}}\,l_{s}^{2}+16\,\lambda_{C^{2}}\,l_{s}^{2}\,+\,\cdots\bigr{)}\,,$
(535)

with the dots denoting $O(l_{s}^{4})$ terms and
$\ell_{0}=\sqrt{3/\Lambda}+\cdots$ as in (532). Physically, ${\cal
S}^{(0)}_{S^{2}\times S^{2}}$ is the horizon entropy of the ${\rm
dS}_{2}\times S^{2}$ static patch, i.e. the Nariai spacetime between the
cosmological and maximal Schwarzschild-de Sitter black hole horizons, both of
area $A=\frac{1}{3}\cdot 4\pi\ell_{0}^{2}$. Comparing to (532), the linear
combination

$\displaystyle{\cal S}^{(0)}_{C^{2}}\equiv 3\,{\cal S}^{(0)}_{S^{2}\times
S^{2}}-2\,{\cal S}^{(0)}=\frac{32\pi
l_{s}^{2}}{G}\bigl{(}\,\lambda_{C^{2}}\,+\,\cdots\bigr{)}$ (536)

extracts the Weyl-squared coupling of $S_{E}[g]$. Analogously, for the
Einstein metric on ${\mathbb{CP}}^{2}$, we get $\tilde{\cal
S}^{(0)}_{C^{2}}\equiv 8\,{\cal S}^{(0)}_{{\mathbb{CP}}^{2}}-6\,{\cal
S}^{(0)}=\frac{48\pi l_{s}^{2}}{G}(\lambda_{C^{2}}+\cdots)$. Then

$\displaystyle{\cal S}^{(0)}_{\rm cub}\equiv 2\,\tilde{\cal
S}^{(0)}_{C^{2}}-3\,{\cal S}^{(0)}_{C^{2}}=16\,{\cal
S}^{(0)}_{{\mathbb{CP}}^{2}}-9\,{\cal S}^{(0)}_{S^{2}\times S^{2}}-6\,{\cal
S}^{(0)}\,=0\,+\,\cdots\,,$ (537)

which extracts some curvature-cubed coupling in the effective action.

To one loop, the quantum ${\cal S}_{M}=\log{\cal Z}_{M}$ can be expressed in a
form paralleling (530):

$\displaystyle{\cal S}_{M}={\cal S}^{(0)}_{M}-\frac{D_{M}}{2}\log{\cal
S}_{M}^{(0)}+\alpha_{M}\log\frac{\ell_{0}}{L}+K_{M}\,+\,\cdots\,,$ (538)

where $D_{M}$ is the number of Killing vectors of $M$: $D_{S^{4}}=10$,
$D_{S^{2}\times S^{2}}=6$, $D_{{\mathbb{CP}}^{2}}=8$, and $\alpha_{M}$ can be
obtained from the local expressions in Christensen:1979iy :
$\alpha_{S^{4}}=-\frac{571}{45}$, $\alpha_{S^{2}\times S^{2}}=-\frac{98}{45}$,
$\alpha_{{\mathbb{CP}}^{2}}=-\frac{359}{60}$. Computing the constants $K_{M}$
generalizing $K_{S^{4}}$ given in (531) would require more work. Moreover,
computing them for one or two saddles would provide no unambiguous information
because they may be absorbed into $\lambda_{C^{2}}$ and $\lambda_{R^{2}}$.
However, since there only two undetermined coupling constants at this order,
computing them for all three does provide unambiguous information, extracted
by the quantum counterpart of (537):

$\displaystyle{\cal S}_{\rm cub}\,\equiv\,16\,{\cal
S}_{{\mathbb{CP}}^{2}}-9\,{\cal S}_{S^{2}\times S^{2}}-6\,{\cal
S}_{S^{4}}=-7\,\log{\cal S}^{(0)}+16\,K_{{\mathbb{CP}}^{2}}-9\,K_{S^{2}\times
S^{2}}-6\,K_{S^{4}}+\cdots$ (539)

The $\log(\ell_{0}/L)$ terms had to cancel in this linear combination because
the tree-level parts at this order cancel by design and $L\partial_{L}{\cal
S}_{\rm cub}=0$.

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