aainstitutetext: Centre for Particle Theory, Department of Mathematical
Sciences, Durham University, South Road, Durham, DH1 3LE, UK.bbinstitutetext:
Departamento de Ciencias Físicas, Universidad Andres Bello, Sazié 2212,
Santiago de Chile.ccinstitutetext: Centro de Ciencias Exactas, Universidad del
Bío-Bío, Avda. Andrés Bello 720, 3800708, Chillán, Chile.

# Logarithmic Corrections, Entanglement Entropy,
and UV Cutoffs in de Sitter Spacetime

Gabriel Arenas–Henriquez b Felipe Diaz c Per Sundell  gabriel.arenas-
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

It has been argued that the entropy of de Sitter space corresponds to the
entanglement between disconnected regions computable by switching on a replica
parameter $q$ modeled by the quotient dS$/\mathbb{Z}_{q}$. Within this
framework, we show that the centrally-extended asymptotic symmetry algebra
near the cosmic horizon is a single copy of the Virasoro algebra. The
resulting density of states matches the semi-classical result of Gibbons and
Hawking up to an undetermined constant that is chosen to reproduce the
entanglement entropy previously found in the literature. It follows that the
logarithmic quantum corrections to the Cardy entropy reproduces the known one-
loop result computed in the bulk in the presence of a cutoff. The resulting
entanglement entropy follows the divergent area law, where the UV cutoff is
now a function of the replica parameter. Thus, as the near-horizon CFT fixes
the cutoff in units of the Planck scale, the model can be viewed as a probe
into whether the defect Hilbert space has a finite dimension; indeed, the
limit $q\to 0$, reproduces Banks’ formula. We also study the quantum
corrections of the effective description of the horizon entropy by means of
Liouville field theory, where the large $q$ limit corresponds to a realization
of dS3/CFT2 correspondence matching the logarithmic corrections to three-
dimensional de Sitter space obtained by computing the one-loop contribution to
the quantum gravity partition function in the round three-sphere.

##  1 Introduction and summary

Four-dimensional de Sitter spacetime, dS4, is a vacuum solution to Einstein
equations with a positive cosmological constant, with maximal isometry group,
$SO(1,4)$, which acts as a positive vacuum pressure. This solution can be
interpreted as a FLRW universe with positively curved spatial slices given by
round three-spheres and exponential scale factor. Current observations show
that our universe is described by a cosmological constant with positive small
value, and is expanding with accelerating pace
SupernovaCosmologyProject:1998vns ; SupernovaSearchTeam:1998fmf , such that at
late times the universe may end in an exponential growth which is
characterized by the dS4 geometry. Early periods of the universe may have been
dominated by an inflationary era after the big bang such that certain problems
in cosmology can be understood by means of this inflationary process
Linde:1981mu ; Linde:2007fr ; thus, the understanding of de Sitter spacetime
has a profound implication in the study of cosmology of our universe. One
consequence of this exponential growth is that a static observer inside a dS4
universe does not have access to the whole spacetime being limited by the
existence of a cosmological horizon. Interestingly, this horizon is a Killing
horizon which shares properties with the event horizon of a black hole. For
instance, it has been shown that there is an associated temperature
Figari:1975km  analogue to Hawking radiation Hawking:1975vcx , and therefore
an entropy that was found by Gibbons and Hawking Gibbons:1977mu  which follows
the area law PhysRevD.7.2333 . Nonetheless, in contrast with black hole
thermodynamics, these are observer dependent quantities, and due to the lack
of access to the asymptotic boundary, an static observer is limited in
constructing observables Witten:2001kn ; Banks:2002wr .

The understanding of de Sitter spacetime poses a challenge to quantum gravity,
and in particular to string and M theory, because of complications in
constructing stable de Sitter solutions in supergravity Maldacena:2000mw ;
Kachru:2003aw , which has stimulated the idea that de Sitter spacetime may
actually belong to an untractable swampland of a string theory landscape
Vafa:2005ui ; Garg:2018reu . As a matter of fact, the treatment of
asymptotically de Sitter backgrounds in the context of M-theory has lead to
the proposal that the dimension of the Hilbert space of a quantum theory of
gravity is finite Banks:2000fe  given by the exponential of the
Gibbons–Hawking entropy. This implies a problem for the quantization of
gravity with Einstein–Hilbert action as a starting point Witten:2001kn  .

In Carlip_1999  has been shown, using covariant phase-space methods, that the
constraint algebra of general relativity, upon taking a suitable set of
boundary conditions, gives rise to a centrally extended, affine algebra.
Particularly, if the boundary consists of a Killing horizon, it is possible to
relate this algebra with one copy of the Virasoro algebra. Therefore, one can
make use of conformal field theory (CFT) techniques in order to describe
underlying degrees of freedom of the horizon. In the case of the black hole
horizon, the Bekenstein–Hawking entropy matches the one of the horizon CFT
Cardy:1986ie . Remarkably, this result can be easily applied to the
cosmological horizon of de Sitter space, in which the Cardy entropy associated
to the horizon CFT renders the result obtained by Gibbons and Hawking. Thus,
by examining the asymptotic symmetry algebra of the horizon, one may obtain
non-trivial information about the density of states of a Killing horizon,
which thus extends the statistical mechanical approach to general relativity.
Moreover, computing Gaussian fluctuations, Carlip showed Carlip:2000nv  that
the density of states acquires an extra contribution resulting in a
logarithmic correction to the entropy of the horizon CFT. This reproduces the
lowest order quantum corrections to black hole entropy that has been computed
using other methods Kaul:2000kf ; Das:2001ic . In this approach, the
asymptotic algebra is constructed by considering diffeomorphisms that preserve
the asymptotic structure and have well defined asymptotic charges associated
to them. In this construction, the central charge appears with a free
parameter that can be fixed in order to recover the area law. This mechanism
has been used Mahapatra:2016iok  in the context of the AdS/CFT correspondence
such that the holographic Renyi entropy acquires an extra contribution that
imply a logarithmic divergence in the entanglement limit.

Recently, it has been argued Kabat:2002hj ; Narayan:2015vda ; Nomura:2017fyh ;
Narayan:2017xca ; Dong:2018cuv ; Arias:entanglement ; Arias:liouv ;
Narayan:2019pjl ; Fernandes:2019ige ; Narayan:2020nsc ; Geng:2020kxh ;
Manu:2020tty ; Goswami:2021ksw ; Susskind:2021esx ; Shaghoulian:2021cef ;
Shaghoulian:2022fop  by using different approaches, that de Sitter space
corresponds to a maximally entangled state incorporating pairings between
disconnected regions of spacetime, leading to a density matrix with an
associated Renyi entropy that is constant. Therefore, one can interpret the
Gibbons–Hawking entropy as the entanglement entropy between the entangled
regions. In Arias:entanglement ; Arias:liouv , the entanglement between
disconnected Rindler observers has been computed by using a single coordinate
patch that contains two causally disconnected Rindler observers, and
considering the reduced density matrix of one of them. This was achieved by
letting $\mathbb{Z}_{q}$ act on the doubled Rindler patch so as to produce an
orbifold geometry with two antipodal conical defects111In principle, one may
assign the two pole separate $q_{N}$ and $q_{S}$ parameters leading to a
Thurston’s spindle. In our work, we take $q_{N}=q_{S}=q$ such that the
resulting geometry can be described by means of a single quotient
dS$/\mathbb{Z}_{q}$ .. The parameter $q$ can be identified as the replica
parameter and the $q\to 1$ limit, referred to as the tensionless limit, leads
back to the smooth de Sitter geometry. This limit is nothing else than the
entanglement entropy limit of the Renyi entropy, which is bigger than the
Gibbons–Hawking entropy due to the presence of both patches considered
Arias:entanglement . Moreover, following Solodukhin:1998tc , a consistent
truncation of this construction onto Liouville field theory gives a
$q$-dependent central charge that vanishes in the tensionless limit, thus
having zero Cardy entropy. The associated Renyi entropy of the system equals
the Gibbons–Hawking entropy in agreement with previous results.

In this paper, we combine Carlip’s conformal description of the cosmological
horizon by imposing maximal entanglement requiring $q$-independent Renyi
entropy which leaves one free parameter that can be used to match the
entanglement entropy with the Gibbons–Hawking entropy at the semi-classical
level. To this end, we use of the Barnich–Brandt formalism Barnich:2001jy ;
Barnich:2006av  to compute the asymptotic charges using the Carlip–Silva
asymptotic generators Carlip:1999cy ; Silva:2002jq . This preserves the
asymptotic structure and produces finite charges whose algebra yields the same
non-trivial central charge as that found by Carlip Carlip:1999cy . Assuming
furthermore, the validity of Cardy entropy formula, the Gibbons–Hawking
entropy is recovered upon matching the free parameter. We then consider the
quantum corrections to the Cardy entropy found in Carlip:2000nv . It follows
that the Renyi entropy is no longer constant and diverges in the tensionless
limit as a direct consequence of the fact that the central charge vanishes in
this particular point. We also investigate other limits of the corrected Renyi
entropy: the min-entropy limit, which gives the larges eigenvalue of the
reduced density matrix; the max-entropy which gives the dimension of the
Hilbert space, and the von Neumann entropy, that is, the entanglement entropy.
We show that the latter now acquires a UV divergence in agreement with the
entanglement area law in quantum field theory (QFT) due to the short-range
interaction near the boundaries of the entanglement region Srednicki:1993im .
This divergence arises in the tensionless limit (that as aforementioned,
corresponds to the entanglement entropy limit), allowing us to introduce a UV
cutoff giving a physical interpretation for the orbifold parameter. The IR
finite term corresponds to the Gibbons–Hawking entropy, and we show that the
proposed connection between the dimension of the Hilbert space and the entropy
holds beyond the semiclassical order by noticing that the cutoff goes to
infinity in the max-entropy limit.

In Arias:liouv , it was shown that the large $q$ limit resembles the three-
dimensional global dS geometry with the defects corresponding to the conformal
future- and past-infinities ${\cal I}^{\pm}$. Therefore, the Liouville central
charge resembles the one found in Strominger:2001pn  of ${\cal I}^{\pm}$ by
considering an holographic CFT description of dS in three dimensions. The
corresponding Cardy entropy renders the Gibbons–Hawking entropy. We also
consider the logarithmic corrections to this scheme, showing that in the
$q\to\infty$ limit, the resulting min-entropy corresponds to the three-
dimensional de Sitter entropy with the quantum corrections recently found in
Anninos:2020hfj ; Coleman:2021nor .

##  2 Rindler horizons and defects

###  2.1 Undeformed de Sitter geometry

#### Global geometry.

Global de Sitter spacetime of signature $(1,d-1)$ and radius $\ell$, here
denoted by dSd, is the solution to Einstein’s field equations in $d$
dimensions with cosmological constant $\Lambda=(d-1)(d-2)/(2\ell^{2})$ given
by the induced metric $ds^{2}$ on the hyperboloid

$Y_{A}Y^{A}=\ell^{2}~{},\qquad A=0,1,\dots,d~{},$ (1)

in $(d+1)$-dimensional Minkowski spacetime, viz.

$\displaystyle ds^{2}=-dY_{0}^{2}+\sum_{i=1}^{d}dY_{i}^{2}~{}.$ (2)

The global geometry is maximally symmetric with isometry group $O(1,d-1)$, and
can be viewed as a foliation along a global time with the topology of
$\mathbb{R}$ with space-like leaves given by round $(d-1)$-spheres with two
conformal boundaries at $Y_{0}\to\pm\infty$, denoted by ${\cal{I}}^{\pm}_{d}$,
respectively. The intrinsic line-element can be coordinatized as

$\displaystyle ds^{2}=-dT^{2}+\ell^{2}\cosh^{2}(T/\ell)d\Omega_{d-1}^{2}\ ,$
(3)

using a global time coordinate $T\in]-\infty,+\infty[$, and the metric
$d\Omega_{d-1}^{2}$ of the round, unit $(d-1)$-sphere.

#### Causal patches.

The exponential inflation prevents a static observer, $O$, say, often referred
to as a Rindler observer, from communicating with the entire geometry, since
upon emitting light rays, these must return to $O$ prior to having reached
half-way across the spatial slice. The envelop of all such rays forms a
causally connected region, ${\mathfrak{R}}_{O}\subset{\rm dS}$, referred to as
a Rindler patch. Each such patch has $S^{2}$-topology and a metrically
bifurcated boundary

$\partial{\mathfrak{R}}_{O}={\cal H}_{-}\cup({\cal
H}_{\infty}\times\mathbb{R}_{-})\cup{\cal H}_{\infty}\cup({\cal
H}_{\infty}\times\mathbb{R}_{+})\cup{\cal H}_{+}\ ,\qquad{\cal
H}_{\pm}\cong{\cal H}_{\infty}\cong S^{d-2}\ ,$ (4)

where ${\cal H}_{\infty}$ are the cosmological horizons, with past and future
limits ${\cal H}_{-}$ and ${\cal H}_{+}$, respectively, residing as defects
inside ${\cal I}^{\pm}$. The interior of ${\mathfrak{R}}_{O}$ as well as the
branches of its bifurcated boundary are stabilized by

$SO(1,d)_{{\mathfrak{R}}}\cong SO(1,d-1)\ ,$ (5)

which together with the discrete time-reversal operation makes up the maximal
isometry group $O(1,d-1)$ of a Rindler patch.

#### Euclidean patch.

Treating de Sitter path integral in the Euclidean regime, the required Wick
rotation maps each Rindler patch into a round $S^{d}$, and the Euclidean field
theory has a Hawking temperature Figari:1975km

$\displaystyle T_{\rm H}=\frac{\hbar}{2\pi\ell}~{},$ (6)

with conjugated Gibbons–Hawking entropy Gibbons:1977mu

$\displaystyle S_{\rm GH}=\frac{A_{\cal H}}{4\hbar G_{d}}~{},$ (7)

where $A_{\cal H}$ is the area of the cosmological horizon, and $G_{d}$ is
Newton’s constant in $d$ dimensions.

#### Double Rindler patch.

From here on, we will work in four space-time dimensions, where $A_{\cal
H}=4\pi\ell^{2}$ . In order to exhibit entanglement, it is natural to
introduce coordinates adapted to unions
${\mathfrak{R}}_{S}\cup{\mathfrak{R}}_{N}$ of pair of causally disconnected
Rindler patches, which can be achieved by the embedding Arias:entanglement

$\displaystyle Y_{0}=$
$\displaystyle\mathchoice{{\hbox{$\displaystyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\textstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\scriptstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=6.14998pt,depth=-4.92001pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=4.7611pt,depth=-3.8089pt}}}\,\cos\theta\,\sinh(t/\ell)~{},\quad
Y_{1}=\mathchoice{{\hbox{$\displaystyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\textstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\scriptstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=6.14998pt,depth=-4.92001pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=4.7611pt,depth=-3.8089pt}}}\,\cos\theta\,\cosh(t/\ell)~{},$ (8)
$\displaystyle Y_{2}$ $\displaystyle=r\,\cos\theta~{},\quad
Y_{3}=\ell\sin\theta\cos\phi~{},\quad Y_{4}=\ell\sin\theta\sin\phi~{},$

where

$-\infty<t<\infty~{},\quad-\ell<r<\ell~{},\quad 0\leq\theta\leq\pi~{},\quad
0\leq\phi<2\pi~{},$ (9)

with $N$ and $S$ tracing out the worldlines $(r,\theta)=(0,0)$ and
$(r,\theta)=(0,\pi)$, respectively. The resulting intrinsic line element, viz.

$\displaystyle
ds^{2}\rvert_{{\mathfrak{R}}_{S}\cup{\mathfrak{R}}_{N}}=\ell^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})+\cos^{2}\theta\left(-f(r)dt^{2}+\frac{dr^{2}}{f(r)}\right)~{},\qquad
f(r)=1-\frac{r^{2}}{\ell^{2}}\ ,$ (10)

is a fibration over $S^{2}$ with fibers above the equator given by points, and
fibers above the strictly northern and southern hemispheres given by radially
extended, two-dimensional Rindler patches, denoted by ${\mathfrak{R}}_{2}$,
which are indeed contractible to points. This is illustrated in Figure 1 .

$\mathbf{\mathcal{I}^{+}}$$\mathbf{\mathcal{I}^{-}}$$\mathfrak{R}_{N}$$\mathfrak{R}_{S}$
Fig. 1:  Penrose diagram of de Sitter spacetime with two Rindler observers
${\mathfrak{R}}_{N}$ and ${\mathfrak{R}}_{S}$.
$\cal{I}^{\pm}$ corresponds to the past and future conformal infinities of the
global geometry.

We denote the northern and southern fibers by ${\mathfrak{R}}_{2}\pm$,
respectively, such that $N$’s worldline is located at $r=0$ in the
${\mathfrak{R}}^{+}_{2}$ above $\theta=0$, idem $S$ and $\theta=\pi$. The
stabilizers in $SO(1,4)$ of each embedded ${\mathfrak{R}}_{2}$-fiber is given
by

$SO(1,4)_{{\mathfrak{R}}_{2}^{\pm}}=SL(2,\mathbb{R})\times U(1)\ ,$ (11)

where the $SL(2,\mathbb{R})$-factor contains a $U(1)$ subgroup generated by
the time-translation vector field $\vec{\partial}_{t}$ , and the $U(1)$-factor
is generated by the rotational vector field $\vec{\partial}_{\phi}$222The
corresponding isometry group matches the one of the near horizon geometry of
an extremal Kerr black hole, such that the asymptotic symmetry group
generators of Guica:2008mu  can be used in order to reproduce the
Gibbons–Hawking entropy Diaz:2019khq ..

###  2.2 Orbifolding the horizon

We follow Arias:entanglement ; Arias:liouv , by considering the orbifold

$dS_{q}:=\widehat{({\mathfrak{R}}_{N}\cup{\mathfrak{R}}_{S})}_{q}/\mathbb{Z}_{q}\
,\qquad f=f\sim g\sim\mathbb{Z}_{q}\subset SO(1,3)\ ,$ (12)

with $q$-fold, branched cover
$\widehat{({\mathfrak{R}}_{N}\cup{\mathfrak{R}}_{S})}_{q}$ with a natural
$\mathbb{Z}_{q}$-action via a Killing vector field $\vec{J}$ of length
$g(\vec{J},\vec{J})=1$; as $\vec{J}$ generates a $U(1)$ subgroup of $SO(1,3)$,
the orbifold parameter can be extended from the positive integers to
$q\in\mathbb{R}_{+}$. The orbifold fixed points form a pair
$(\Sigma_{S},\Sigma_{N})$ of antipodal, sub-manifolds of co-dimension two,
referred to as defects, with induced ${\mathfrak{R}}_{2}$ metrics. The defects
generate $\delta^{2}$-function singularities in the Riemann curvature two-form
Fursaev:1995ef  proportional to the conical deficit $2\pi\left(1-1/q\right)$.
A well-posed variational principle requires to supplement the Einstein–Hilbert
action with extra terms localized at the defects that provide singular stress
tensors cancelling the singularities in the bulk Einstein tensor, such that
the field equations hold everywhere, even at the defects Gregory:1988xc . To
treat maximally symmetric defects, it suffices to dress the defects by
Nambu–Goto actions with a $q$-dependent tension ${\cal T}_{q_{\xi}}$,
resulting in a total action

$I_{q}=I_{\rm EH}+I_{\rm def}\ ,\qquad I_{\rm def}=\sum_{\xi={\rm
N,S}}I_{\xi}\ ,$ (13) $I_{\xi}=-{\cal
T}_{q_{\xi}}\int_{\Sigma_{\xi}}d^{2}\sigma\mathchoice{{\hbox{$\displaystyle\sqrt{-{\rm
det}(g|_{\Sigma_{\xi}})\,}$}\lower 0.9pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{-{\rm
det}(g|_{\Sigma_{\xi}})\,}$}\lower 0.9pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{-{\rm
det}(g|_{\Sigma_{\xi}})\,}$}\lower 0.9pt\hbox{\vrule
height=5.25pt,depth=-4.20003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{-{\rm
det}(g|_{\Sigma_{\xi}})\,}$}\lower 0.9pt\hbox{\vrule
height=3.75pt,depth=-3.00002pt}}}~{},\qquad{\cal
T}_{q_{\xi}}=\frac{1}{4G_{4}}\left(1-\frac{1}{q_{\xi}}\right)\ ,\qquad q_{\rm
N}=q_{\rm S}=q\ .$ (14)

Thus, the smooth de Sitter geometry is recovered in the $q\to 1$ limit,
referred as the tensionless limit. The dressed defects can thus be interpreted
Arias:liouv  as Rindler observers with mass proportional to $1-q$, stretched
out into strings (in the direction determined by $\vec{J}$) back-reacting to
the surrounding geometry. Choosing polar coordinates on $S^{2}$ such that
$\vec{J}=\vec{\partial}_{\phi}$, the orbifolded line element

$\displaystyle
ds^{2}_{q}=\cos^{2}\theta\left(-f(r)dt^{2}+\frac{dr^{2}}{f(r)}\right)+\ell^{2}\left(d\theta^{2}+\frac{\sin^{2}\theta}{q^{2}}d\phi^{2}\right)~{},$
(15)

where the ranges of all coordinates are as before, which thus cover the entire
$q$-deformed, double Rindler patch, with Rindler defects located at $\theta=0$
and $\theta=\pi$, respectively; it follows that

$\displaystyle
ds^{2}_{q}\big{\rvert}_{\Sigma_{N,S}}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}~{}.$
(16)

The $q$-deformed, double Rindler wedge has cosmological horizon given by a
bifurcated Killing horizon $S^{2}$-topology located at $r=\pm\ell$, where the
Killing vector field $\vec{\chi}:=\vec{\partial}_{t}$ passes from being time-
like to light-like, with $q$-independent surface gravity, viz.

$\chi^{2}:=ds^{2}_{q}(\vec{\chi},\vec{\chi})=\frac{\cos^{2}\theta}{\ell^{2}}\left(r^{2}-\ell^{2}\right)~{},\qquad\kappa^{2}:=-\lim_{\chi^{2}\to
0}\frac{\nabla_{\mu}\chi^{2}\nabla^{\mu}\chi^{2}}{4\chi^{2}}=\frac{1}{\ell^{2}}\
.$ (17)

The orbifolding procedure of the horizon is illustrated in Figure 2

••$\mathcal{H}\cong
S^{2}$$S^{2}/\mathbb{Z}_{q}$••$\Sigma_{N}$$\Sigma_{S}$$\ell_{q}=\ell/q$ Fig.
2:  The spherical geometry of the cosmological horizon ${\cal H}$ after the
orbifolding deforms to that of the spindle $S^{2}/\mathbb{Z}_{q}$ with radius
$\ell_{q}=\ell/q$. The set of fixed points $\Sigma_{N}$ and $\Sigma_{S}$
correspond to the defects located at $\theta=0,\pi$ respectively, both with
induced metric (16).

###  2.3 Semi-classical entropy computation

Using the formalism of Appendix A, we assign to each Rindler observer a
$q$-independent Hamiltonian $H_{\mathfrak{R}}$, and a reduced density matrix
$\rho_{\mathfrak{R}}:=e^{-H_{\mathfrak{R}}}$. Introducing a modular
temperature

$T_{q}:=q^{-1}\ ,$ (18)

treated as the conjugate variable of $H_{\mathfrak{R}}$, the resulting thermal
partition function

$\displaystyle{\cal Z}_{\mathfrak{R}}:={\rm
tr}_{H_{\mathfrak{R}}}\rho_{\mathfrak{R}}^{q}={\rm
tr}_{H_{\mathfrak{R}}}\,\exp\left\\{-qH_{\mathfrak{R}}\right\\}\ ,$ (19)

computed on the $q$-fold, branched cover $\widehat{S}^{4}_{q}$ of the
Euclideanized orbifold

$\displaystyle S^{4}_{q}:=\widehat{S}^{4}_{q}/\mathbb{Z}_{q}\ ,$ (20)

can be re-written using the Cardy–Calabrese formula Calabrese:2004eu

$\displaystyle{\cal Z}_{\mathfrak{R}}=\frac{{\cal
Z}[\widehat{S}_{q}^{4}]}{{\cal Z}[S^{4}]^{q}}=\left(\frac{{\cal
Z}[S_{q}^{4}]}{{\cal Z}[S^{4}]}\right)^{q}\ .$ (21)

In the semi-classical approximation,

$\displaystyle{\cal
Z}[S_{q}^{4}]\approx\exp\left\\{-\frac{1}{\hbar}I^{E}[S^{4}_{q}]\right\\}\
,\qquad{\cal
Z}[S^{4}]\approx\exp\left\\{-\frac{1}{\hbar}I^{E}[S^{4}]\right\\}~{},$ (22)

where the total Euclidean action

$\displaystyle I^{E}_{q}[S^{4}_{q}]=-\frac{q}{16\pi
G_{4}}\int_{S^{4}}d^{4}x\mathchoice{{\hbox{$\displaystyle\sqrt{{\rm
det}(g_{q})\,}$}\lower 0.9pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{{\rm
det}(g_{q})\,}$}\lower 0.9pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{{\rm
det}(g_{q})\,}$}\lower 0.9pt\hbox{\vrule
height=5.25pt,depth=-4.20003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{{\rm
det}(g_{q})\,}$}\lower 0.9pt\hbox{\vrule
height=3.75pt,depth=-3.00002pt}}}\left(R-2\Lambda\right)+I^{E}_{\rm
def}=\left(1-\frac{2}{q}\right)\frac{\pi\ell^{2}}{G_{4}}~{}.$ (23)

It follows that each $q$-deformed Rindler observer has the modular free energy

$\displaystyle F_{q}:=-\frac{1}{q}\log{\cal
Z}_{\mathfrak{R}}\approx\frac{1}{\hbar}(I^{E}_{\mathfrak{R}}[S^{4}/\mathbb{Z}_{q}]-I^{E}[S^{4}])=2\left(1-\frac{1}{q}\right)\frac{\pi\ell^{2}}{\hbar
G_{4}}~{},$

which is the quantity to be identified with the semi-classical entropy
obtained by computing the thermal partition function in reduced phase-spaces
attached to various regions of the Rindler patch. From the exact relation
between Renyi entropy and free energy,

$S_{q}=\left(1-\frac{1}{q}\right)^{-1}F_{q}~{},$ (24)

it follows that

$\displaystyle S_{q}\approx\frac{2\pi\ell^{2}}{\hbar G_{4}}=2S_{\rm
GH}~{},\qquad\forall q\geqslant 1\ ,$ (25)

whose $q$-independence can be interpreted as that the original pair of Rindler
observers formed a maximally entangled state in agreement with Kabat:2002hj ;
Nomura:2017fyh ; Dong:2018cuv ; Arias:entanglement ; Arias:liouv ;
Narayan:2019pjl ; Susskind:2021esx ; Shaghoulian:2021cef ; Shaghoulian:2022fop
, supporting the interpretation of that the Gibbons–Hawking entropy is due to
entanglement between causally disconnected Rindler patches333In Arias:liouv ,
the modular free energy was instead computed by means of a consistent
truncation of the total action to a Liouville theory on the defect, which
indeed yields a constant Renyi entropy in the semiclassical regime as well..

##  3 Near-Horizon algebra and Cardy entropy

###  3.1 Asymptotic charges

Carlip has shown Carlip:1999db ; Carlip:1999cy ; Carlip_1999  that general
relativity admits classes of asymptotic boundary conditions near Killing
horizons forming modules of centrally extended, asymptotic Virasoro charges.
Under the assumption of modular invariance, the asymptotic charges induce an
entropy obeying the area law.

In what follows, we shall assign the $q$-deformed Rindler patches classes of
boundary conditions at their cosmological horizons inducing Virasoro modules
with finite central charges for $q>1$ fixed up to one free parameter by
assumption of maximal entanglement. These considerations are facilitated by
the space-time covariant Barnich–Brandt–Compere (BBC) formalism Barnich:2001jy
; Barnich:2006av ; Barnich:2007bf ; Compere:2007az , which maps algebras of
asymptotic Killing vector fields (AKVs) $\vec{\zeta}$ stabilizing spaces of
metric fluctuations $h$ in the near-cosmic horizon region to charges

$\displaystyle{\cal Q}_{\vec{\zeta}}[h;g]=\frac{1}{8\pi
G_{4}}\oint\displaylimits_{{\cal H}}k_{\vec{\zeta}}[h,g]~{},$ (26)

where ${\cal H}$ denotes the cosmological horizon, and444Tensorial indices are
raised and lowered using the background metric $g$.

$\displaystyle k_{\vec{\zeta}}[h,g]=-\frac{1}{4}dx^{\alpha}\wedge
dx^{\beta}\epsilon_{\alpha\beta\mu\nu}\Big{[}$
$\displaystyle\zeta^{\nu}\nabla^{\mu}h^{\rho}{}_{\rho}-\zeta^{\nu}\nabla_{\sigma}h^{\sigma\mu}+\zeta_{\sigma}\nabla^{\nu}h^{\mu\sigma}$
(27)
$\displaystyle+\frac{1}{2}h^{\rho}{}_{\rho}\nabla^{\nu}\zeta^{\mu}-h^{\nu\sigma}\nabla_{\sigma}\zeta^{\mu}+\frac{1}{2}h^{\sigma\nu}\left(\nabla^{\mu}\zeta_{\sigma}+\nabla_{\sigma}\zeta^{\mu}\right)\Big{]}~{},$

is a background-covariant two-form, obeying the centrally extended, Dirac-
bracket algebra

$\displaystyle\left\\{{\cal Q}_{\vec{\zeta}},{\cal
Q}_{\vec{\zeta}^{\prime}}\right\\}={\cal
Q}_{[\vec{\zeta}^{\prime},\vec{\zeta}]}+\frac{1}{8\pi
G_{4}}\int\displaylimits_{\cal H}k_{\vec{\zeta}}[{\cal
L}_{\vec{\zeta}^{\prime}}g,g]~{},$ (28)

where ${\cal L}_{\vec{\zeta}}$ denotes the Lie derivative along $\vec{\zeta}$.
The two-form is a potential of the Noether three-form Julia:1998ys , which is
closed on the entire Rindler patch, including the defect, as a consequence of
the variational principle, which means that the central charge accounts for
those asymptotic degrees of freedom that extend smoothly into the entire
Rindler patch (including the defect).

A family of vectors satisfying the Diff$(S^{1})$ algebra (or de Witt algebra)

$\displaystyle i[\vec{\zeta}_{m},\vec{\zeta}_{n}]=(m-n)\vec{\zeta}_{m+n}~{},$
(29)

and that preserve the asymptotic symmetries of a Killing horizon, has been
found in Carlip:1999cy ; Silva:2002jq  and have the form of

$\displaystyle\zeta^{\mu}_{m}=T_{m}{\chi}^{\mu}+R_{m}\rho^{\mu}~{}.$ (30)

Here $\chi^{\mu}$ is the null vector defined in (17), while $R_{m}$ and
$T_{m}$ are coordinate-dependent functions obtained by requiring that the
structure of the horizon is preserved Carlip_1999 , i.e., $\chi^{\mu}$ remains
null at $r=\ell$, and

$\displaystyle\rho_{\mu}=-\frac{1}{2\kappa}\nabla_{\mu}\chi^{2}~{},$ (31)

is a vector normal to the horizon with vanishing norm at the horizon. To
obtain the Killing vectors $\vec{\zeta}$ , is enough to assume that, near the
horizon, they correspond to conformal Killing vectors Solodukhin:1998tc ;
Silva:2002jq , viz.

$\displaystyle\nabla_{\mu}\chi_{\nu}+\nabla_{\nu}\chi_{\mu}={\cal
O}(\chi^{2})g_{\mu\nu}+{\cal O}(\chi^{4})~{},$ (32)

such that a set of diffeomorphisms $\vec{\zeta}_{m}$ of (10) must satisfy

$\displaystyle\delta_{\zeta}\chi^{2}={}$ $\displaystyle{\cal O}(\chi^{4})~{},$
(33) $\displaystyle\delta_{\zeta}\rho_{\mu}={}$ $\displaystyle{\cal
O}(\chi^{2})\rho_{\mu}~{},$ (34)

that for (30) implies

$\displaystyle{\cal L}_{\chi}T_{n}+\kappa R_{n}={\cal
O}(\chi^{2})~{},\qquad\rho^{\mu}\nabla_{\mu}T_{n}={\cal O}(\chi^{2})T_{n}~{}.$
(35)

These asymptotic generators, indeed, satisfy a stronger condition imposed by
Carlip Carlip:1999cy

$\displaystyle\lim_{\chi^{2}\to
0}\frac{\chi^{\mu}\chi_{\nu}}{\chi^{2}}\nabla_{\mu}\zeta^{\nu}_{m}=\lim_{\chi^{2}\to
0}\chi^{\mu}\nabla_{\mu}\left(\frac{\chi_{\nu}\zeta^{\nu}_{m}}{\chi^{2}}\right)-\kappa\frac{\rho_{\mu}\zeta_{m}^{\mu}}{\chi^{2}}=0~{},$
(36)

guaranteeing that $\chi^{\mu}$ is a Killing vector in the horizon
neighborhood. Therefore, the condition (36) can be seen as the horizon analogs
to the fall-off conditions that one uses at infinity to obtain the asymptotic
symmetry algebra. Particularly, the previous condition is satisfied for (30)
with

$\displaystyle T_{m}=-\frac{\ell}{\alpha}\exp\left\\{im(\phi-\alpha
t/\ell)\right\\}~{},\qquad R_{m}=\frac{\alpha}{l\kappa}imT_{m}~{},$ (37)

where $\alpha$ is an undetermined real number, satisfying the de Witt algebra
(29) and preserving the asymptotic structure. The $\alpha$ parameter has been
chosen in order to reproduce the horizon entropy in Carlip:1998qw ;
Carlip:1999cy ; Carlip:1999db ; Carlip:2000nv ; Silva:2002jq . In the
following subsection, we will rescale $\alpha$ by requiring that the resulting
Renyi entropy describe a maximally mixed entangled state as obtained in
Dong:2018cuv ; Arias:entanglement , such that $\alpha$ depends on the orbifold
parameter $q$, giving a central charge that resembles the one found in
Arias:liouv .

###  3.2 Asymptotic Virasoro symmetries

The resulting symmetry group generated by the AKVs (30) is given by a single
copy of (28) with central extension

$\displaystyle\frac{1}{8\pi G_{4}}\int\displaylimits_{{\cal
H}}k_{\vec{\zeta}_{m}}[{\cal
L}_{\vec{\zeta}_{n}}g,g]=-i\left(\frac{\ell^{2}}{4q\alpha
G_{4}}\right)(\alpha^{2}m^{3}+2m)\delta_{m+n,0}~{}.$ (38)

Thus, promoting the Dirac bracket to a quantum commutator, viz.
$\\{\cdot,\cdot\\}\to{\textstyle{{\scriptstyle 1}\over{\scriptstyle
i\hbar}}}[\cdot,\cdot]$ , and defining quantum operators

$\displaystyle L_{m}:={\cal
Q}_{m}+\frac{3\ell^{2}}{8G_{4}}\frac{\alpha}{q}\delta_{m,0}~{},$ (39)

yields a chiral Virasoro algebra

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c_{q}}{12}m(m^{2}-1)\delta_{m+n,0}~{},$
(40)

with central charge

$\displaystyle c_{q}=12i\lim_{r\to\ell}{\cal Q}_{\vec{\zeta}_{m}}[{\cal
L}_{\vec{\zeta}_{-m}}g,g]\big{\rvert}_{m^{3}}=\frac{3\ell^{2}}{\hbar
G_{4}}\frac{\alpha}{q}~{}.$ (41)

###  3.3 Maximal entanglement condition

Assuming that the asymptotic charges give rise to a modular invariant thermal
quantum theory, the semi-classical approximation of the corresponding Virasoro
partition yields the Cardy entropy Cardy:1986ie

$\displaystyle S_{\rm C}\approx S_{\rm
C}^{(0)}:=2\pi\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{c_{q}\Delta}{6}\,}$}\lower
0.9pt\hbox{\vrule
height=12.34996pt,depth=-9.88002pt}}}{{\hbox{$\textstyle\sqrt{\frac{c_{q}\Delta}{6}\,}$}\lower
0.9pt\hbox{\vrule
height=8.67776pt,depth=-6.94225pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{c_{q}\Delta}{6}\,}$}\lower
0.9pt\hbox{\vrule
height=6.66664pt,depth=-5.33334pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{c_{q}\Delta}{6}\,}$}\lower
0.9pt\hbox{\vrule
height=6.66664pt,depth=-5.33334pt}}}=\frac{\pi^{2}}{3}c_{q}T\ ,$ (42)

where $T$, which is the temperature of the chiral modes near the horizon, has
been eliminated by means of

$\displaystyle\frac{\partial S_{\rm
C}}{\partial\Delta}=\frac{1}{T}~{}\quad\Rightarrow\quad\Delta={\textstyle{{\scriptstyle\pi^{2}}\over{\scriptstyle
6}}}c_{q}T^{2}\ .$ (43)

Viewing the orbifolding procedure as a computation in a quantum theory using
replicas, the Cardy entropy (42) is identified as the modular free energy
$F_{q}$ of the degrees of freedom in the $q$-deformed near-horizon region. Let
us redefine $\alpha=2\gamma(q-1)$ , where $\gamma>0$ is a finite undetermined
constant, by requiring that the original pair of Rindler observers formed a
maximally mixed entangled state, that is, that the corresponding Renyi entropy
$S_{q}$, which is given by Baez’ formula (24), is $q$-independent. Therefore,
the central charge, Cardy entropy, and Renyi entropy are given by

$\displaystyle c_{q}=\frac{6\ell^{2}}{\hbar
G_{4}}\gamma\left(1-\frac{1}{q}\right)\ ,\qquad S^{(0)}_{\rm
C}=\left(1-\frac{1}{q}\right)\gamma S_{\rm GH}~{},\qquad S^{(0)}_{q}=\gamma
S_{\rm GH}~{},\ $ (44)

respectively. When $\gamma=1$, the above result agrees with previous
computations Kabat:2002hj ; Nomura:2017fyh ; Dong:2018cuv ; Arias:entanglement
; Shaghoulian:2022fop . Therefore, the cosmological horizon of the deformed
geometry reduces to that of the smooth dS geometry, whose thermal properties
can thus be interpreted as being due to the entanglement between the causally
disconnected Rindler observers encoded into the two-dimensional thermal CFT
with modes having temperature (109) and central charge (41).

##  4 Logarithmic corrections and UV divergences

###  4.1 $q$-dependent bulk cut-offs

Working perturbatively inside the bulk, on a single Wick-rotated Rindler
patch, ${\mathfrak{R}}$, the free energy can be expanded as

${\cal Z}_{\mathfrak{R}}\approx\exp\left\\{-{\textstyle{{\scriptstyle
1}\over{\scriptstyle\hbar}}}I^{E}_{q}-\frac{1}{2}\log{\rm det}\nabla\right\\}\
,$ (45)

including the semi-classical Gibbons–Hawking entropy and its one-loop
correction. By our hypothesis, the thermal description of the near-horizon
entropy is captured by a chiral CFT with Bunch–Davies temperature and central
charge $c_{q}$ given in Eq. (44). We also recall that in a thermal replica
theory with normalized inverse temperature $q$, the Renyi entropy $S_{q}$ is
related via (24) to the moduluar, free energy $F_{q}$ in its turn given by

$F_{q}=-\frac{1}{q}\log{\cal Z}_{\mathfrak{R}}=S_{\rm C}~{};$ (46)

for further details, see Appendix A. Including Carlip’s correction to the
Cardy entropy Carlip:2000nv , the density of states for a two-dimensional,
thermal, chiral CFT can be written as

$\displaystyle\rho(\Delta)\approx\left(\frac{c}{96\Delta^{3}}\right)^{\frac{1}{4}}\exp\left\\{2\pi\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{c\Delta}{6}\,}$}\lower
0.9pt\hbox{\vrule
height=10.05553pt,depth=-8.04446pt}}}{{\hbox{$\textstyle\sqrt{\frac{c\Delta}{6}\,}$}\lower
0.9pt\hbox{\vrule
height=7.03888pt,depth=-5.63113pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{c\Delta}{6}\,}$}\lower
0.9pt\hbox{\vrule
height=5.02776pt,depth=-4.02223pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{c\Delta}{6}\,}$}\lower
0.9pt\hbox{\vrule
height=5.02776pt,depth=-4.02223pt}}}\right\\}\quad\leftrightarrow\quad\rho(T)\approx\frac{c}{144}(S_{\rm
C}^{(0)})^{-\frac{3}{2}}\exp\left\\{S_{\rm C}^{(0)}\right\\}~{}.$ (47)

Using the central charge $c_{q}$ for the asymptotic charges in the near-
horizon region, and the corresponding semi-classical Cardy entropy $S_{\rm
C}^{(0)}$, both given in (44), the logarithmically corrected Cardy entropy

$\displaystyle S_{{\rm C}}\approx S^{(0)}_{\rm C}-\frac{3}{2}\log S^{(0)}_{\rm
C}+\log\left(\frac{c}{144}\right)=\left(1-\frac{1}{q}\right)\gamma S_{\rm
GH}-\frac{3}{2}\log\left[\left(1-\frac{1}{q}\right)^{\frac{1}{3}}\gamma S_{\rm
GH}\right]~{},$ (48)

modulo constants, which thus contains a divergent quantum correction in the
tensionless limit. Following subsection 2.3, we can obtain the modular free
energy by inverting (24), which yields the Renyi entropy

$\displaystyle S_{q}={}$ $\displaystyle\gamma S_{\rm
GH}-\frac{3}{2}\left(\frac{q}{q-1}\right)\log\left[\left(1-\frac{1}{q}\right)^{\frac{1}{3}}\gamma
S_{\rm GH}\right]~{},$ (49)

with the following limits

$\displaystyle S_{\rm E}={}$ $\displaystyle\lim_{q\to 1}S_{q}=\gamma S_{\rm
GH}-\frac{3}{2}\lim_{q\to
1}\frac{q}{q-1}\log\left[\left(1-\frac{1}{q}\right)^{\frac{1}{3}}\gamma S_{\rm
GH}\right]~{},$ (50) $\displaystyle S_{\infty}={}$
$\displaystyle\lim_{q\to\infty}S_{q}=\gamma S_{\rm GH}-\frac{3}{2}\log\gamma
S_{\rm GH}~{},$ (51) $\displaystyle S_{0}={}$ $\displaystyle\lim_{q\to
0}S_{q}=\gamma S_{\rm GH}~{}.$ (52)

As we can see from (50), the entanglement entropy acquires an extra divergent
term coming from the 1-loop determinant and a finite term corresponding to the
Gibbons–Hawking entropy as previously computed in different contexts
Kabat:2002hj ; Nomura:2017fyh ; Dong:2018cuv ; Arias:entanglement ;
Arias:liouv ; Narayan:2019pjl ; Susskind:2021esx ; Shaghoulian:2021cef ;
Shaghoulian:2022fop . The extra logarithmic divergence can be rewritten as

$\displaystyle S_{\rm E}=\gamma\frac{A_{\cal
H}}{\delta_{q}^{2}}+\gamma\frac{A_{\cal H}}{4\hbar G_{4}}~{},$ (53)

where $\delta_{q}$ reads as

$\displaystyle\delta_{q}^{2}=\frac{2\gamma A_{\cal
H}}{3}\left(\frac{1-q}{q}\right)\log\left[\left(1-\frac{1}{q}\right)^{\frac{1}{3}}\frac{\gamma
A_{\cal H}}{4\hbar
G_{4}}\right]^{-1}\quad\leftrightarrow\quad\left(1-\frac{1}{q}\right)=\frac{\delta^{2}_{q}}{2\gamma
A_{\cal H}}W_{0}\left(\frac{128\hbar^{3}G_{4}^{3}}{\gamma^{2}A_{\cal
H}^{2}\delta^{2}_{q}}\right),$ (54)

where $W_{0}(x)$ is the Lambert $W$ function. We can see that in the
tensionless limit $q\to 1$,

$\displaystyle\lim_{q\to 1}\delta_{q}=0~{},$ (55)

giving the divergent area-law of entanglement entropy Bombelli:1986rw ;
Srednicki:1993im  where $\delta_{q}$ can be identified with the UV cutoff
appearing in the standard QFT description literature (see Appendix A).
Therefore, the cutoff is related to the geometric orbifold parameter which
acquires a physical interpretation in terms of energy scale. The finite
contribution corresponds to the Gibbons–Hawking entropy in agreement with
Nomura:2017fyh ; Dong:2018cuv ; Arias:conf . One can also compare the result
with the one-loop contributions to black-hole entropy obtained in
Solodukhin:1994yz  by considering conical singularities associated to time
coordinate. These conical singularities modifies the temperature of the black
hole giving quantum corrections to the entropy which contains two divergences;
one associated to interactions near the tip of the cone, and other related to
UV divergences. In order to obtain the correct result one needs to match the
regulators for both divergences. In particular, for the presented case, we
have only one regulator associated to both, the UV divergences and the conical
defects.

###  4.2 Dimensionality formulae for large and small $q$

The limits $q\to\infty$ and $q\to 0$ of the Renyi entropy, referred to,
respectively, as the min-entropy and the max-, or Hartley, entropy, yield
information of the dimensionality of the density matrix $\rho_{A}$
Headrick:2010zt ; Hung:2011nu , namely

$\displaystyle S_{\infty}={}$ $\displaystyle-\log\lambda_{1}~{},$ (56)
$\displaystyle S_{0}={}$ $\displaystyle\log{\cal D}~{},$ (57)

where $\lambda_{1}$ is the largest eigenvalue that is active in $\rho_{A}$,
and ${\cal D}$ corresponds to the number of non-vanishing elements of
$\rho_{A}$. In terms of (51) and (52) we get

$\displaystyle\lambda_{1}={}$ $\displaystyle(\gamma S_{\rm
GH})^{{\textstyle{{\scriptstyle 3}\over{\scriptstyle 2}}}}\exp\\{-\gamma
S_{\rm GH}\\}~{},$ (58) $\displaystyle{\cal D}={}$ $\displaystyle\exp\\{\gamma
S_{\rm GH}\\}~{},$ (59)

because of the large value of $S_{\rm GH}$ (for example, see Podolskiy:2018fwk
) we get $\lambda_{1}\ll 1$ which is consistent with the fact that
$S_{\infty}$ represents the minimum value of the Renyi entropy, and ${\cal
D}\gg 1$ gives the dimensionality of Hilbert space associated to a single
Rindler observer ${\cal H}_{A}$ such that it has dimension $\exp\\{\gamma
S\\}$, as is already proposed in Banks:2000fe ; Witten:2001kn  when
$\gamma=1$. The limit $q\to 0$ corresponds to the limit in which the cutoff
moves to the infrared $\delta\to\infty$ leading to the properties of dS
spacetime in the IR.

##  5 Codimension-2 holography in large $q$ limit

In Arias:liouv , it was observed that in the limit $q\rightarrow\infty$, the
orbifold dSq reduces to global, three-dimensional de Sitter spacetime (dS3)
and the defects are sent to ${\cal{I}}^{\pm}_{3}$. Moreover, it was shown that
Liouville theory on $\Sigma_{\xi}$ was a consistent truncation at the
classical level, and the associated central charge reproduces the one of (44)
for $\gamma=1/2$. Upon dimensional reduction of Newton’s constant, the sum of
the central charges of the two Rindler observers resembles the one appearing
in the context of dS/CFT correspondence Lin:1999gf ; Banados:1998tb ;
Strominger:2001pn . As a matter of fact, this can be viewed as a realization
of dS3/CFT2 holography starting from the higher-dimensional singular spacetime
dS${}_{4}/\mathbb{Z}_{q}$. Thus, we will extract information of the two-
dimensional CFTs living at the space-like boundaries of three-dimensional de
Sitter space ${\cal I}^{\pm}$. In particular, the quantum corrections to the
corresponding two-dimensional holographic CFT are thus given by the large-$q$
limits of those computed in the previous section, which remain finite, and
have been computed in Carlip:2000nv . The large-$q$ limit of the embedding
coordinates (8) is given by

$\displaystyle
Y_{0}=\mathchoice{{\hbox{$\displaystyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\textstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\scriptstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=6.14998pt,depth=-4.92001pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=4.7611pt,depth=-3.8089pt}}}\cos\theta\sinh(t/\ell)~{},\qquad
Y_{1}=\mathchoice{{\hbox{$\displaystyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\textstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=8.74889pt,depth=-6.99915pt}}}{{\hbox{$\scriptstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=6.14998pt,depth=-4.92001pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\ell^{2}-r^{2}\,}$}\lower
0.9pt\hbox{\vrule
height=4.7611pt,depth=-3.8089pt}}}\cos\theta\cosh(t/\ell)~{},$ $\displaystyle
Y_{2}=r\cos\theta~{},\qquad Y_{3}=\ell\sin\theta~{},\qquad Y_{4}=0~{},$ (60)

upon using the $\mathbb{Z}_{q}$ action, satisfying the hyperboloid constraint
$Y_{A}Y^{A}=\ell^{2}$ for $A=0,\dots,3$ , resulting in the induced line
element

$\displaystyle
ds^{2}=\cos^{2}\theta\left(-f(r)dt^{2}+\frac{dr^{2}}{f(r)}\right)+\ell^{2}d\theta^{2}~{}.$
(61)

As proposed in Arias:entanglement , the double Wick rotation $t\to
i\ell\hat{\varphi}~{},\theta\to i\hat{\tau}/\ell$ resembles dS3 in global
coordinates, which in our case the embbeding coordinates lead to the line
element

$\displaystyle
ds^{2}=-d\hat{\tau}^{2}+\cosh^{2}(\hat{\tau}/\ell)\left(\frac{dr^{2}}{f(r)}+\ell^{2}f(r)d\hat{\varphi}^{2}\right)~{},\qquad\hat{\tau}\in\mathbb{R}~{},\qquad\hat{\varphi}\in[0,2\pi]~{},$
(62)

where the line element at fixed $\hat{\tau}$ corresponds to a Euclidean
${\mathfrak{R}}_{2}$, that is, a round $S^{2}$. Thus, taking
$r=\ell\cos{\hat{\phi}}$, we get

$\displaystyle
ds^{2}=-d\hat{\tau}^{2}+\ell^{2}\cosh^{2}(\hat{\tau}/\ell)d\hat{\Omega}_{2}^{2}~{},\qquad\hat{\phi}\in[0,\pi]~{},$
(63)

where ${\rm
d}\hat{\Omega}_{2}^{2}=d\hat{\phi}^{2}+\sin^{2}\hat{\phi}~{}d\hat{\varphi}^{2}$
is the line element of the unit two-sphere. As expected, in the $q\to\infty$
limit, the minimal surfaces $\Sigma_{\xi}$ corresponds to the past and future
conformal infinities ${\cal I}^{\pm}_{3}$ of dS3. The large $q$ limit is
illustrated in Figure 3.

$S^{2}$$S^{2}/\mathbb{Z}_{q}$••$\ell_{q}=\frac{\ell}{q}$$\Sigma_{N}$$\Sigma_{S}$$q\to\infty$dS3$z$
Fig. 3:  The $q\to\infty$ limit of the geometry corresponds to the zero radius
$\ell_{q}$ limit where the spindle shrinks to a single traverse dimension
between the defects. The resulting geometry corresponds to global dS3
spacetime.

As in Arias:entanglement , the central charge (41) of the defects reproduces
the result of Strominger Strominger:2001pn  for dS3, which is

$\displaystyle
c({\Sigma_{N}\cup\Sigma_{S}})\stackrel{{\scriptstyle\begin{subarray}{c}q\to\infty\vspace{-1mm}\\\
{}\end{subarray}}}{{\longrightarrow}}c({\cal I}^{+}_{3}\cup{\cal
I}^{-}_{3})=\frac{3\ell}{2\hbar G_{3}}~{},$ (64)

by using the relation $G_{4}=4\ell G_{3}$ , and was shown to reproduce the
three-dimensional de Sitter entropy by taking the large $q$ limit of the
modular free energy of the total space given by the Cardy entropy of both
observers,

$\displaystyle S_{\rm
C}({{\mathfrak{R}}_{N}\cup{\mathfrak{R}}_{S}})\stackrel{{\scriptstyle\begin{subarray}{c}q\to\infty\vspace{-1mm}\\\
{}\end{subarray}}}{{\longrightarrow}}\frac{\pi\ell}{2\hbar G_{3}}~{}.$ (65)

Considering now the quantum corrections of the Cardy entropy on the defects,
obtained in section 4, in the large $q$ limit we get

$\displaystyle S_{\rm
C}({{\mathfrak{R}}_{N}\cup{\mathfrak{R}}_{S}})\stackrel{{\scriptstyle\begin{subarray}{c}q\to\infty\vspace{-1mm}\\\
{}\end{subarray}}}{{\longrightarrow}}\frac{\pi\ell}{2\hbar
G_{3}}-3\log\frac{\pi\ell}{2\hbar G_{3}}+{\rm const}~{},$ (66)

giving the recently obtained first quantum correction of three-dimensional de
Sitter Anninos:2020hfj ; Coleman:2021nor . In this limit the dual theory
becomes non-thermal, as $T=q^{-1}\to 0$ , in agreement with the interpretation
of Klemm:2001ea ; Klemm:2002ir ; Arias:entanglement  as dS3 entropy being the
Liouville momentum avoiding the problems pointed out in Dyson:2002nt . A
natural generalization of this construction would be the inclusion of higher
co-dimensional defects. This has been explored in Anti-de Sitter space by
considering a causal wedge of AdS with defects generalizing the idea of
AdS/CFT Akal:2020wfl ; Miao:2020oey ; Geng:2020fxl ; Miao:2021ual .

##  6 Conclusions and outlook

In this work, following the ideas of Arias:entanglement ; Arias:liouv , we
consider Einstein gravity on the quotient dS$/\mathbb{Z}_{q}$ with Nambu–Goto
improvement terms at defects so as to obey the variational principle
Fursaev:1995ef , such that the smooth de Sitter geometry re-appears in the
tensionless limit $q\to 1$. We find that the cosmological horizon is
stabilized by the asymptotic Killing vector fields of Carlip:1999cy ;
Silva:2002jq , inducing an asymptotic symmetry algebra à la Barnich–Brandt
Barnich:2001jy  given by a single copy of the Virasoro algebra with non-
trivial central charge. We fix the $q$-dependence of the central charge by
requiring maxima disorder, resulting in a $q$-dependent Cardy entropy. We also
choose the remaining free parameter, which was present already in the
construction Carlip:1999cy , by demanding constant Renyi entropy, which
reproduces the semi-classical result Arias:entanglement ; Arias:liouv ,
leading to the conclusion that the Gibbons–Hawking entropy can be interpreted
as the entanglement entropy between causally disconnected Rindler patches.
Following Carlip:2000nv , we consider Gaussian fluctuations in the density of
states in the thermal CFT describing the horizon, which results into finite
logarithmic corrections for $q>1$ . In the $q\to 1$ limit of the Renyi
entropy, namely, the tensionless limit, we obtain a divergent von Neumann
entropy satisfying the entanglement area law of QFTs Srednicki:1993im . We
regulate the UV divergence by introducing a cutoff given as a function of the
orbifold parameter $q$, allowing us to reinterpret this geometric construction
in terms of the energy scale of the theory. We also study the $q\to 0$ limit,
in which the cutoff moves into the IR, that gives the logarithm of the
dimension of the associated Hilbert space, and the $q\to\infty$ limit which
gives the larges eigenvalue of the reduced density matrix. The former indeed
reproduces Bank’s proposal Banks:2000fe  that a quantum theory of gravity in
an asymptotically de Sitter spacetime has a Hilbert space of finite dimension
that equals the logarithm of the Gibbons–Hawking entropy.

We have studied quantum logarithmic corrections of the Cardy entropy of the
Liouville field theory that was obtained in Arias:liouv  by considering a
reduction of the improved Einstein action to the defects. In the large $q$
limit, the central charges of this model reproduce the dS3/CFT2 results of
Strominger:2001pn , in which the central charges of the defects now correspond
to those of the conformal asymptopia, and the corresponding Cardy entropy
contains logarithmic corrections and reproduces those of three-dimensional de
Sitter quantum gravity found by computing the one-loop correction to its
partition function on the round three-sphere Anninos:2020hfj .

It would be interesting to match the results of the current work with those
found in Aros:2010jb , where it was found that the logarithmic correction to
entropy is a Noether charge given in terms of three ingredients: the horizon
Euler characteristic, the a-trace anomaly appearing the quantum corrections of
Einstein field equations, and the solution $\phi_{\cal H}$ to the
uniformization problem for $Q$-curvature malchiodi2007conformal .
Particularly, these corrections were computed for Schwarzschild and massless
topological AdS black holes. In the current work, we show that the latter
corresponds to the double Wick rotation of the static patch in dS spacetime.
Therefore, the solution for $\phi_{\cal H}$ is the same as in the case of
massless, topological black holes, while the Euler characteristic has opposite
sign. Since both are conformally flat backgrounds, it is possible to obtain
information about the trace anomaly on the four-dimensional dS background
using this formalism and comparing with our result.

Finally, similar spacetimes have been recently studied in supergravity with
negative cosmological constant. In particular, the resulting near-horizon
geometry of branes wrapping spindles corresponds to a direct product of the
type $AdS\times\Sigma$ where $\Sigma$ is a spindle Ferrero:2020laf ;
Ferrero:2021wvk ; Faedo:2021nub . For instance, it is argued that the near-
horizon limit of a M2-brane wrapped around a spindle is described by a well-
known classical solution, the Plebanski-Demianski metric Ferrero:2020twa ;
Cassani:2021dwa . Here, the tension of the cosmic string produces the
acceleration of the black hole Appels:2017xoe ; Gregory:2017ogk  which
interplays within the cosmological constant creating a Rindler horizon if the
acceleration is large enough. This second horizon resembles the
characteristics of the cosmological horizon as it disconnects the regions of
the spacetime preventing access to the AdS boundary. The thermodynamics of
these spacetimes seems to be well-understood in the regime where the
acceleration is slow and the Rindler horizon vanishes Appels:2017xoe ;
Anabalon:2018ydc ; Gregory:2019dtq ; Arenas-Henriquez:2022www . Nevertheless,
a consistent thermodynamic description of the accelerating horizon is still
unclear, and one could expect that using the asymptotic symmetry algebra of
this horizon as in section 3 could lead to a better understanding of
accelerating black holes and their holographic description. We leave this
question open for future work.

#### Acknowledgements.

We would like to thank C. Arias, R. Aros, A. Faraggi, C. Farías, C. Iazeolla,
K. Lara, N. Nguyen, M. Valenzuela, and B. Vallilo for enlightening discussions
on various topics related to this work. We also want to thank to K. Narayan
for his interesting comments regarding the notion of entanglement in de
Sitter. Ps acknowledges the kind support of the Centro de Ciencias Exactas at
Universidad del Bio-Bio. The work of Gah is funded by Becas Chile (ANID)
Scholarship N${}^{\rm o}~{}72200271$.

##  Appendix A Entropy functions

A key distinguishing feature of quantum mechanics vis-à-vis classical
mechanics is the notion of quantum entanglement, whereby a system built from
localizable degrees of freedom attached to two separate regions, $A$ and $B$,
say, of a manifold, or a set of points, is assigned a direct-product Hilbert
space $\mathfrak{H}_{A}\otimes\mathfrak{H}_{B}$, which contains _pure_ quantum
states 555In classical mechanics, the system is assigned a direct-product
symplectic manifold $M_{A}\times M_{B}$; probability distributions on this
space can describe mixed classical states in which the values observables in
$A$ and $B$ are correlated, but there is no classical counterpart of quantum
entanglement at the level of pure, classical states, which are given by delta
functions; indeed, pure, classical states remain pure as one of the symplectic
manifolds are integrated out. Moreover, defining a pure classical state as a
delta-sequence, its von Neumann entropy tends to minus infinity as the volume
of support goes to zero, manifesting a classical “catastrophe”. given by
superpositions that correlate observables confined to $A$ and $B$, that is,
the eigenvalues of operators of the form ${\cal O}_{A}\otimes{\rm Id}_{B}$ and
${\rm Id}_{A}\otimes{\cal O}_{B}$. Thus, an observer in one of the two
regions, $B$, say, measuring ${\cal O}_{B}$ by performing a von Neumann
measurement, which eventually produces a pure state in
$\mathfrak{H}_{A}\otimes\mathfrak{H}_{B}$ with fixed eigenvalue of ${\rm
Id}_{A}\otimes{\cal O}_{B}$, bears an impact on the spectrum of ${\cal
O}_{A}\otimes{\rm Id}_{B}$ in region $A$ (independently of the scale of the
spatial separation).

#### Renyi entropy and modular free energy.

Consider the reduced density matrix

$\rho_{A}:={\rm Tr}_{\mathfrak{H}_{B}}\,\rho_{AB}~{},$ (67)

whose spectral properties manifest the degree of entanglement of the original
pure state. These properties are encoded into the spectral function

$\displaystyle S_{q,A}:=\frac{1}{1-q}\log{\cal Z}_{A}\ ,\qquad{\cal
Z}_{A}:={\rm Tr}_{\mathfrak{H}_{A}}\,(\rho_{A})^{q}~{},\qquad{\rm Re}(q)>0\ ,$
(68)

known as the Renyi entropy; when restricted to the strictly positive integers,
the parameter $q$ is referred to as the replica number. The Renyi entropy is
$q$-independent iff the reduced density matrix describes an equiprobable
ensemble, in which case the original state is said to be maximally entangled
(with respect to the separation of the point set into $A\cup B$). The limits

$\displaystyle S_{{\rm E},A}:=\lim_{q\to 1}{S}_{q,A}\ ,\qquad
S_{0,A}:=\lim_{q\to 0}{S}_{q,A}\ ,\qquad
S_{\infty,A}:=\lim_{q\to\infty}{S}_{q,A}\ ,$ (69)

are referred to, respectively, as the entanglement, Hartley and minimum
entropies of $\rho_{A}$. From

$S_{{\rm E},A}=-{\rm Tr}_{\mathfrak{H}_{A}}\,\rho_{A}\,\log\rho_{A}\ ,$ (70)

it follows that the entanglement entropy is the von Neumann entropy of
$\rho_{A}$, which serves as a good measure of the degree of entanglement in
the original pure state666Viewing the reduction operations as processes taking
place in a modified, nonlinear version of quantum mechanics, one may think of
quantum dynamics acts as the fundamental source of (entanglement) entropy..
The Hartley entropy can be identified as

$S_{0,A}:=\log{\cal D}_{A}~{},$ (71)

where ${\cal D}_{A}$ is the number of non-zero eigenvalues of $\rho_{A}$, that
is, the dimension of the subspace of $\mathfrak{H}_{A}$ that participates in
the entanglement. Assuming that the reduced density matrix is in the image of
the exponential map, viz.

$\rho_{A}=:\exp\left\\{-H_{A}\right\\}~{},$ (72)

where $H_{A}$ is referred to as the modular Hamiltonian, one has

$S_{\infty,A}:=\lim_{q\to\infty}{S}_{q,A}\equiv-\log\lambda^{\rm max}_{A}~{},$
(73)

where $\lambda^{\rm max}_{A}$ is the largest eigenvalue of $\mathfrak{H}_{A}$.
The existence of such a Hamiltonian implies that the Renyi entropy can be
identified as

$S_{q,A}=\left(1-\frac{1}{q}\right)^{-1}F_{A}\ ,\qquad
F_{A}:=-\frac{1}{q}\log{\cal Z}_{A}\ ,$ (74)

where $F_{A}$ is thus the free energy of the corresponding thermal system with
temperature, entropy, and thermal energy given by

$\displaystyle T_{q}:=q^{-1}\ ,\qquad{S}_{A}:=(1-q\partial_{q})\log{\cal
Z}_{A}\ ,\qquad E_{A}=-\partial_{q}\log{\cal Z}_{A}~{}\ ,$ (75)

respectively (and one has $\lim_{q\to 1}{S}_{A}=S_{{\rm E},A}$); this system
is often referred to as the modular system, and $F_{A}$, $T_{q}$, $S_{A}$ and
$E_{A}$ as the modular free energy, temperature, modular entropy and modular
energy, respectively. The Renyi and modular entropies are related by

$\displaystyle{S}_{A}=q^{2}\partial_{q}\left(\frac{q-1}{q}S_{q,A}\right)~{}.$
(76)

Inverting this relation yields

$\displaystyle
S_{q,A}=\frac{q}{q-1}\int_{1}^{q}\frac{d\tilde{q}}{\tilde{q}^{2}}{S}_{A}~{}.$
(77)

#### Cardy–Calabrese formula and orbifolds.

In quantum field theories on metric backgrounds, the modular partition
function can be computed by letting $\rho_{A}\equiv\exp\\{-H_{A}\\}$ represent
the foliation of a smooth, Euclidean geometry $(M_{A},ds^{2})$, with
$M_{A}\supset A$, generated by $H_{A}$ in one unit of Euclidean time, with $A$
treated as a co-dimension one Cauchy surface; this yields the Cardy–Calabrese
formula

${\cal Z}_{A}=\frac{{{\cal
Z}}_{A}\left[\widehat{M}_{A,q},d\hat{s}^{2}\right]}{\left({\cal
Z}_{A}\left[{M}_{A},ds^{2}\right]\right)^{q}}~{},$ (78)

where ${{\cal Z}}_{A}\left[\widehat{M}_{A,q},ds^{2}\right]$ is computed on the
$q$-fold, branched, Euclidean cover geometry of $(M_{A},ds^{2})$, with
$(\widehat{M}_{A,1},d\hat{s}^{2})\equiv(M_{A},ds^{2})$, constructed by gluing
together $q$ copies of $(M_{A},ds^{2})$ along $A$, which yields a branched
geometry with a smooth metric $d\hat{s}^{2}$, and a natural
$\mathbb{Z}_{q}$-action with ramification surface given by the entangling
surface $\partial A$. Thus, the Renyi entropy

$S_{q,A}=\frac{1}{1-q}\left(\log{{\cal
Z}}_{A}\left[\widehat{M}_{A,q},d\hat{s}^{2}\right]-q\log{\cal
Z}_{A}\left[{M}_{A},ds^{2}\right]\right)~{}.$ (79)

In the semi-classical limit, one has

${{\cal
Z}}_{A}\left[\widehat{M}_{A,q},d\hat{s}^{2}\right]\approx\left.\exp\left\\{-\frac{1}{\hbar}I_{\rm
E}\left[\widehat{M}_{A,q},d\hat{s}^{2}\right]\right\\}\right|_{\rm Saddle}\ ,$
(80)

where $I_{\rm E}$ is the Euclidean action and the saddle-point requires a
suitable boundary condition; assuming locality, it follows that

$I_{\rm E}\left[\widehat{M}_{A,q},d\hat{s}^{2}\right]=qI_{\rm
E}\left[(\widehat{M}_{A,q},d\hat{s}^{2})/\mathbb{Z}_{q}\right]\ ,$ (81)

where the orbifold geometry

$(\widehat{M}_{A,q},d\hat{s}^{2})/\mathbb{Z}_{q}=(M_{A},ds^{2}_{q})\ ,$ (82)

that is, it is given by the manifold $M_{A}$ equipped with a non-smooth metric
$ds^{2}_{q}$ with a conical singularity of co-dimension two at $\partial A$
with a deficit angle $2\pi(1-q^{-1})$. The quantum corrections to Renyi
entropy contains short-distance divergences inside the bulk, that are
independent of entanglement data, and at the border of the traced-out region,
referred to as the entanglement surface; the latter exhibit universal scaling
behaviours leading to area laws for the entanglement entropy, viz.

$\displaystyle S_{{\rm E},A}=\gamma\frac{A(\partial
A)}{\delta^{d-2}}(1+o(\delta))\ ,\qquad\gamma\in\mathbb{R}^{+}~{},$ (83)

where $\delta$ is a UV cut-off, and $\gamma$ depends on the field content of
the theory.

At high temperature, the highest UV-finite contribution is dominated by the
thermal entropy of the subset $A$ Cardy:2014jwa , and consdering $B$ to be
empty, the entanglement entropy corresponds to the thermal entropy

$\displaystyle S_{{\rm E},A}=\beta\langle H_{A}\rangle+\log{\cal
Z}_{A}=\beta(E_{A}-F_{A})=S_{{\rm th},A}~{}.$ (84)

####  $q$-derivatives.

Defining

$\displaystyle\left(\frac{\partial F}{\partial
x}\right)_{q}\equiv\frac{F(qx)-F(x)}{qx-x}~{},$ (85)

referred to as $q$-derivative of $F(x)$, which recovers the standard
derivative wehen $q=1$ , the Renyi entropy can be seen as the
$q^{-1}$-derivative of the negative free energy baez2011renyi , viz.

$\displaystyle S_{q}=-\left(\frac{\partial F}{\partial T}\right)_{q^{-1}}~{}.$
(86)

This relation can be generalized by considering a re-scaled temperature

$\displaystyle T=\frac{T_{0}}{q}~{},$ (87)

where we normalize the partition function at $T_{0}$ as $\widehat{{\cal
Z}}(T_{0})=1$ ; then

$\displaystyle S_{q,A}=\frac{T\log\widehat{{\cal Z}}(T)-q\log\widehat{{\cal
Z}}(T_{0})}{T-T_{0}}~{},$ (88)

which by normalizing $T_{0}=1$ simplifies to Eq. (74) with $F(q)=F_{A}$ . In
other words, the Renyi entropy amounts to the maximum work that a system in
thermal equilibrium can perform by reducing its temperature by a factor of
$q^{-1}$, divided by the change in temperature.

#### Disorder and von Neumnann entropy.

The von Neumann entropy has a natural thermodynamic interpretation as the
equilibrium value of a disorder functional. Let us refer to a set
$\\{\sigma_{\xi}\\}_{\xi=1}^{N}\equiv\Sigma$ of $N$ labelled elements, or
micro states, as an $N$-state system. Let $\mathbb{R}^{N}_{+}$ be the
$N$-dimensional real cone. A map $\rho:\Sigma\to\mathbb{R}^{N}_{+}$ obeying

$\sum_{\xi=1}^{N}\rho(\sigma_{\xi})=1\ ,$ (89)

is referred to as a macro-state. Each disjoint union
$\Sigma=\Sigma^{\prime}_{1}\cup\Sigma^{\prime}_{2}$ induces a 2-state system
$\check{\Sigma}:=\\{\Sigma^{\prime}_{1},\Sigma^{\prime}_{2}\\}$, whose micro
states are thus given by the two disjoint subsystems, with a macro state
$\check{\rho}$ defined by

$\check{\rho}(\Sigma^{\prime}_{i})=\sum_{\sigma\in\Sigma^{\prime}_{i}}\rho(\sigma)\
,\qquad i=1,2\ .$ (90)

The disorder functional $D:\rho\to{\mathbb{R}}_{+}$ is defined by requiring
that

$D\left[\rho\right]=\check{\rho}(\Sigma^{\prime}_{1})D\left[\frac{1}{\check{\rho}(\Sigma^{\prime}_{1})}\rho|_{\Sigma^{\prime}_{1}}\right]+\check{\rho}(\Sigma^{\prime}_{2})D\left[\frac{1}{\check{\rho}(\Sigma^{\prime}_{2})}\rho|_{\Sigma^{\prime}_{2}}\right]+D\left[\check{\rho}\right]\
.$ (91)

for any disjoint union, that is, the disorder of $\rho$ is given by the
weighted contributions from the two subsystems plus an extra contribution
arising due to forming the disjoint union, referred to as the mixing entropy.
The latter is clearly negligible in the large $N$ limit, but significant after
repeated partitions down to a large number of subsystems of size of order one,
and the sole source of entropy in a process setting out from separate micro
states. It follows that

$D[\rho]=k\sum_{\xi}\rho(\sigma_{\xi})\log\frac{1}{\rho(\sigma_{\xi})}\ ,$
(92)

where $k$ is an undetermined constant.

Classical thermodynamics amounts to maximizing the disorder in the space of
macro-states while keeping expectation values of observables fixed; in the
absence of any constraints, the extremal macro-state is the equiprobable
distribution of micro states, while the presence of constraints imply that the
extremal macro-states are given by Boltzmann distributions with conjugate
variables arising as Lagrange multipliers. An interesting question is thus how
to modify the above notions such that the resulting extremization procedure
leads to Renyi entropy.

##  Appendix B Asymptotic charges

Let $M$ be a manifold with interior $M^{\prime}$ and boundary $\partial M$
admitting a one-parameter foliation with leafs $\Sigma$ for a canonical
formulation, and denote by ${\rm Vect}(M;\partial M)$ the algebra of vector
fields on $M$ that stabilizes $\partial M$. Let i) $g$ be an on-shell metric
that equips $\partial M$ with an asymptotic structure $[g]|_{[\partial M]}$;
ii) ${\cal C}$ be a space of classical solutions $\hat{g}:=g+h$ that are
smooth in the interior of $M$, containing $\hat{g}=g$; and iii) $\mathfrak{K}$
be a Lie algebra represented by subalgebra of ${\rm Vect}(M;\partial M)$,
referred to as the AKVs, that stabilizes ${\cal C}$. The canonical formalism
Barnich:2001jy ; Silva:2002jq  assigns the system asymptotic charges

$\widehat{{\cal
Q}}^{(\epsilon)}_{\vec{\zeta}}[\hat{g}]=\oint_{\Sigma_{\epsilon}}\hat{k}_{\vec{\zeta}}[\hat{g}]\
,$ (93)

where $\vec{\zeta}$ is the vector field representing $\zeta\in\mathfrak{K}$,
and $\Sigma_{\epsilon}\subset M\stackrel{{\scriptstyle\epsilon\to
0}}{{\longrightarrow}}\Sigma$, that are diffeomorphism invariant, that is

$\int_{M}\left(({\cal
L}_{\vec{v}}\hat{g})_{\mu\nu}\frac{\delta}{\delta\hat{g}_{\mu\nu}}+({\cal
L}_{\vec{v}}\vec{\zeta})^{\mu}\frac{\delta}{\delta(\vec{\zeta})^{\mu}}\right)\widehat{{\cal
Q}}^{(\epsilon)}_{\vec{\zeta}}=0\ ,$ (94)

for smooth $\vec{v}$ on $M$, i.e.

$\int_{M}({\cal
L}_{\vec{v}}\hat{g})_{\mu\nu}\frac{\delta}{\delta\hat{g}_{\mu\nu}}\widehat{{\cal
Q}}^{(\epsilon)}_{\vec{\zeta}}=-\widehat{{\cal
Q}}^{(\epsilon)}_{[\vec{v},\vec{\zeta}]}\ ,$ (95)

such that if

$\widehat{\cal Q}_{\vec{\zeta}}:=\lim_{\epsilon\to 0}\widehat{{\cal
Q}}^{(\epsilon)}_{\vec{\zeta}}\ ,$ (96)

are finite, then they are moment functionals for a projective representation
of $\mathfrak{K}$ in ${\cal C}$ by means of a Dirac bracket
$\\{\cdot,\cdot\\}$, in which $g$ is treated as a constant background, viz.

$\\{\widehat{\cal Q}_{\vec{\zeta}},h\\}:={\cal L}_{\vec{\zeta}}h\ .$ (97)

It follows that $\\{\widehat{{\cal Q}}_{\vec{\zeta}},\hat{g}\\}={\cal
L}_{\vec{\zeta}}\hat{g}-{\cal L}_{\vec{\zeta}}g$, which yields

$\\{\widehat{{\cal Q}}_{\vec{\zeta}},\widehat{{\cal
Q}}_{\vec{\zeta}^{\prime}}\\}\equiv\int_{M}(\widehat{{\cal
Q}}_{\vec{\zeta}})_{\mu\nu}\frac{\delta}{\delta\hat{g}_{\mu\nu}}\widehat{{\cal
Q}}_{\vec{\zeta}}=\widehat{{\cal
Q}}_{[\vec{\zeta}^{\prime},\vec{\zeta}]}-\int_{M}({\cal
L}_{\zeta}g])_{\mu\nu}\frac{\delta}{\delta\hat{g}_{\mu\nu}}\widehat{{\cal
Q}}_{\vec{\zeta}^{\prime}}\ ,$ (98)

upon using (95) and when combined with (96). Assuming fall-off conditions on
$h$ such that

$\widehat{{\cal Q}}_{\vec{\zeta}}[\hat{g}]=\widehat{{\cal
Q}}_{\vec{\zeta}}[g]+{{\cal Q}}_{\vec{\zeta}}[h;g]\ ,\qquad{{\cal
Q}}_{\vec{\zeta}}[h;g]:=\oint_{\Sigma}k_{\vec{\zeta}}[h;g]\ ,$ (99)

where $k_{\vec{\zeta}}[h;g]$ is linear in $h$, one has

$\\{{{\cal Q}}_{\vec{\zeta}},{{\cal Q}}_{\vec{\zeta}^{\prime}}\\}={{\cal
Q}}_{[\vec{\zeta}^{\prime},\vec{\zeta}]}+C(\vec{\zeta},\vec{\zeta}^{\prime};g)\
,$ (100)

where the central term

$C(\vec{\zeta},\vec{\zeta}^{\prime};g)=\widehat{{\cal
Q}}_{[\vec{\zeta}^{\prime},\vec{\zeta}]}[g]+\oint_{\Sigma}k_{\vec{\zeta}}[{\cal
L}_{\vec{\zeta}^{\prime}}g;g]\ $ (101)

which is an anti-symmetric bi-linear form on $\mathfrak{K}$ built from
background charges and the charges of ${\cal L}_{\vec{\zeta}}g$ treated as
fluctuations; as a simple sign check, one has

$\\{{\cal Q}_{\vec{\zeta}},\underbrace{\\{{\cal
Q}_{\vec{\zeta}^{\prime}},h\\}}_{{\cal L}_{\vec{\zeta}^{\prime}}h}\\}+\\{{\cal
Q}_{\vec{\zeta}^{\prime}},\underbrace{\\{h,{\cal Q}_{\vec{\zeta}}\\}}_{-{\cal
L}_{\vec{\zeta}}h}\\}+\\{h,\underbrace{\\{{\cal Q}_{\vec{\zeta}},{\cal
Q}_{\vec{\zeta}^{\prime}}\\}}_{{\cal
Q}_{[\vec{\zeta}^{\prime},\vec{\zeta}]}+C}\\}={\cal
L}_{\vec{\zeta}^{\prime}}\\{{\cal Q}_{\vec{\zeta}},h\\}-{\cal
L}_{\vec{\zeta}}\\{{\cal Q}_{\vec{\zeta}^{\prime}},h\\}-{\cal
L}_{[\vec{\zeta}^{\prime},\vec{\zeta}]}h\ ,$ (102)

i.e. the algebra is indeed consistent with the Jacobi identity. While the
computation of the central charge is based on self-consistency, the existence
of the projective representation requires a duality relation Barnich:2001jy
between the scaling behaviours of elements in ${\cal C}$ and $\mathfrak{K}$ in
the asymptotic region, referred to as the asymptotic boundary conditions
(ABC). Relaxing the ABC on ${\cal C}$, enlarges ${\rm Vect}(M;\partial
M)_{\cal C}$ and restricts ${\cal C}^{\ast}_{\widehat{\cal Q}}$, where ${\rm
Vect}(M;\partial M)_{\cal C}$ and ${\cal C}^{\ast}_{\widehat{\cal Q}}$,
respectively, denote the stabilizer and ${\cal Q}$-dual of ${\cal C}$ in ${\rm
Vect}(M;\partial M)$; thus, imposing ABC such that

${\rm Vect}(M;\partial M)_{\cal C}={\cal C}^{\ast}_{\widehat{\cal Q}}\ ,$
(103)

yields symplectic spaces of boundary states with finite central charges
containing the orbit of $g$ under the $\mathfrak{K}$-action. In a near-horizon
geometry, the null Killing vector field $\vec{\chi}$ (NKV) induces an algebra
$\mathfrak{K}_{\vec{\chi}}$ of AKVs that preserve the NKV, in its turn
inducing ${\cal
C}_{\vec{\chi}}=(\mathfrak{K}_{\vec{\chi}})^{\ast}_{\widehat{\cal Q}}$.

##  Appendix C Bunch–Davies vacuum

Considering the deformed background (15), we can characterize macroscopic
thermal properties of a entangled state that is encoded in the Boltzmann
factor $\exp\\{-\omega/T_{dS}\\}$ where $\omega$ corresponds to the energy
eigenstate with temperature $T_{dS}$. Near to the defects, the line element
(15)

$\displaystyle ds^{2}\approx-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+\dots~{},$ (104)

and taking the near horizon limit $r\to\ell$, by considering the dimensionless
Rindler coordinates $\tau=\ell t$ and $\Upsilon^{2}=2(1-r/\ell)$ ,

$\displaystyle
ds^{2}\approx-\Upsilon^{2}d\tau^{2}+d\Upsilon^{2}~{},\qquad\Upsilon\ll 1~{}.$
(105)

Considering a scalar field Bunch:1978yq  on the near-to-defect background
(104) expanded in eigenstates with energy $\omega$

$\displaystyle\Phi(t,r)=\sum_{n}\phi_{n}(r)\exp\\{-i\omega
t\\}~{},\qquad\omega>0~{},$ (106)

and taking the near horizon limit, the mode expansion in terms of the Rindler
coordinates (105) renders

$\displaystyle\Phi(\tau,\Upsilon)=\sum_{n}\phi_{n}(\Upsilon)\exp\\{-iN\tau\\}~{},\qquad
N\equiv\ell\omega~{},$ (107)

which turns the Boltzmann factor near the horizon to

$\displaystyle\exp\left\\{-\frac{\omega}{T_{dS}}\right\\}=\exp\left\\{-\frac{N}{T}\right\\}~{},$
(108)

where thus $N$ corresponds to the number of states with temperature

$\displaystyle T=\frac{1}{2\pi}~{},$ (109)

referred as the Bunch–Davies vacuum or Euclidean vacuum Bunch:1978yq . We
follow to use the temperature of modes near the horizon (109) in the Cardy
formula.

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