aainstitutetext: Stanford Institute for Theoretical Physics, 382 Via Pueblo,
Stanford CA 94305bbinstitutetext: Enrico Fermi Institute & Kadanoff Center for
Theoretical Physics, University of Chicago, Chicago, IL 60637,
USAccinstitutetext: Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA,
UKddinstitutetext: Centro Atómico Bariloche and CONICET, S.C. de Bariloche,
Río Negro, R8402AGP, Argentina

#  de Sitter Microstates from $T\bar{T}+\Lambda_{2}$ and the Hawking-Page
Transition

Evan Coleman b Edward A. Mazenc a Vasudev Shyam a Eva Silverstein a,c Ronak M
Soni d Gonzalo Torroba a and Sungyeon Yang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We obtain microstates accounting for the Gibbons-Hawking entropy in $dS_{3}$,
along with a subleading logarithmic correction, from the solvable
$T\bar{T}+\Lambda_{2}$ deformation of a seed CFT with sparse light spectrum.
The microstates arise as the dressed CFT states near dimension $\Delta=c/6$,
associated with the Hawking-Page transition; they dominate the real spectrum
of the deformed theory. We exhibit an analogue of the Hawking-Page transition
in de Sitter. Appropriate generalizations of the $T\bar{T}+\Lambda_{2}$
deformation are required to treat model-dependent local bulk physics
(subleading at large central charge) and higher dimensions. These results add
considerably to the already strong motivation for the continued pursuit of
such generalizations along with a more complete characterization of $T\bar{T}$
type theories, building from existing results in these directions.

##  1 Introduction: from Bekenstein-Hawking to Gibbons-Hawking via Hawking-
Page

It is of considerable interest to formulate quantum gravity consistently with
the observed accelerated expansion of the universe. Recent years have seen
renewed progress toward this goal stemming in part from a simple
generalization GST ; LLST  of the $T\bar{T}$ deformation Zamolodchikov:2004ce
; Smirnov:2016lqw ; Dubovsky:2012wk ; Cavaglia:2016oda ; Dubovsky:2017cnj ;
Gorbenko:2018oov  applied to AdS/CFT duality, as pioneered in McGough:2016lol
; Kraus:2018xrn ; Hartman:2018tkw . In this correspondence, patches of AdS or
dS spacetime are reconstructed from a deformed holographic seed CFT formulated
on the boundary of a patch; energy levels and various entanglement entropies
accessible in the gravitational sector agree precisely on the two sides
Donnelly:2018 ; Donnelly:2018bef ; GST ; LLST . The global cosmological
spacetime may be obtained by sewing together such building blocks. This led to
a recent calculation of the Gibbons-Hawking entropy $S_{GH}$ in Shyam:2021ciy
, reproducing Dong:2018cuv  and a logarithmic correction computed recently in
Anninos:2020hfj .

$S=S_{GH}-3\log(S_{GH})+\dots$ (1)

In the present work, we define a version of this deformation which provides a
microstate count for the de Sitter observer patch, reproducing the corrected
Gibbons-Hawking entropy Gibbons:1977mu  (1) in a more direct way.
Specifically, we construct microstates of the de Sitter observer patch from
the microstates of a particular black hole in AdS. The generalization GST ;
LLST  of the $T\bar{T}$ deformation accommodating de Sitter spacetime (known
as the $T\bar{T}+\Lambda_{2}$ deformation) is defined by

$\frac{\partial}{\partial\lambda}\,\log Z=-2\pi\,\int
d^{2}x\,\sqrt{g}\,\langle T\bar{T}\rangle+\frac{1-\eta}{2\pi\lambda^{2}}\int
d^{2}x\,\sqrt{g}$ (2)

with $Z$ the partition function of the theory and where $T\bar{T}$ is defined
as the quadratic combination

$T\bar{T}\equiv\frac{1}{8}(T_{ab}T^{ab}-(T^{a}_{a})^{2})$ (3)

of components of the energy-momentum tensor $T_{ab}$.

The prescription (2) is applied piecewise, beginning with a pure $T\bar{T}$
deformation ($\eta=1$) from the seed CFT theory. Once $\lambda$ is nonzero, it
is possible to start a new segment of the trajectory with nonzero $\eta-1$
beginning at some finite value $\lambda=\lambda_{0}$ (say taking $\eta=-1$),
as sketched in figure 1. This is universal and solvable in the same sense as
the original $T\bar{T}$ theory, enabling a calculation of the deformed
(‘dressed’) energy spectrum for all energy levels.

In our new example of such a combined $T\bar{T}\to T\bar{T}+\Lambda_{2}$
deformation, the microstates associated with the de Sitter observer patch are
dressed microstates of BTZ black holes at the $\Delta\simeq c/6$ energy level
of the seed CFT of central charge $c\gg 1$ ($\Delta$ denoting the operator
dimension).111As we will describe in the bulk of the paper, the precise
statement of the relevant energy levels is as a band of order 1 around
$\Delta=c/6$ Hartman_2014 ; Mukhametzhanov:2019pzy .  In the dual
gravitational description, the trajectory corresponds to bringing in the AdS
boundary until it skirts the horizon of the $\Delta\simeq c/6$ BTZ black hole,
where the horizon becomes indistinguishable from the horizon of the dS static
patch. At this value of $\lambda=\lambda_{0}$, we adjoin a
$T\bar{T}+\Lambda_{2}$ segment of the trajectory, building up the static patch
of dS, bounded by a worldline at its observer position; see Fig. 1.222For
earlier work on holography and the observer patch see e.g. Anninos:2011af .
Throughout this process, the entire spectrum of microstates accounting for the
Gibbons-Hawking entropy are retained as real energy levels. This
microcanonical state count has a counterpart in the canonical ensemble, where
the $\Delta=c/6$ energy level corresponds to the Hawking-Page transition in
the seed AdS/CFT system. In the course of our analysis below, we find an
analogue of the Hawking-Page transition in de Sitter.

Figure 1:  A schematic of our prescription for holographic reconstruction of
the static patch, including a microstate count, for a given deformation
parameter $y=\lambda/L^{2}$. This may be summarized as follows, with details
in the bulk of the paper. On the left, we obtain the indicated patch
containing the cosmic horizon by dressing the $\Delta\simeq c/6$ microstates
comprising a BTZ black hole at the Hawking-Page transition and switching
between the $T\bar{T}$ and $T\bar{T}+\Lambda_{2}$ trajectories at a value
$y_{0}$ of the deformation parameter such that the BTZ and Cosmic Horizons
degenerate to indistinguishable near-horizon regions. On the right we depict
the trajectory described in GST LLST  which formulates the complementary patch
for the same boundary geometry. The extrinsic curvature of the boundary, and
hence the square root term in the dressed energy formula, are equal and
opposite for the two patches. As contributions to the thermal partition
function at fixed boundary geometry, these dominate for complementary ratios
of $(\beta/L)^{2}=y/y_{0}-1$, providing a de Sitter analogue of the Hawking-
Page transition. In both cases, we can capture the bulk of the static patch by
continuing the trajectory in the manner indicated by the arrow. The Pole patch
with a Dirichlet boundary condition always excludes the cosmic horizon and
does not directly account for the microstates, in keeping with the sparse set
of real dressed energies in that case. However, for the cosmic horizon patch –
the main subject of the current paper – proceeding with the trajectory to
$y_{final}\gg 1$ formulates the static patch with boundary at the observer
position at the North Pole.

This derivation of de Sitter microstates is our main result, recovering the
connection between the Gibbons-Hawking entropy and the $\Delta=c/6$ states
associated with the Hawking-Page transition that was anticipated in
Dong:2018cuv  and recovered with the first $1/c$ correction in Shyam:2021ciy .
As we will see, there are different versions of such a trajectory with
distinct properties, having to do with choices of integration constants
arising in the differential equation for the dressed energy levels derived
from (2), related to the role (or not) of modular transformations in the
theory. In holography, the prescription (2) captures the pure gravitational
sector of the theory – the black hole spectrum in AdS and the observer patch
in dS. The physics of other energy levels is subdominant at large $c$ and
model-dependent, as we will describe in our analysis below. A complete
description of subleading excitations (such as local bulk matter) requires
additions to the prescription (2) including those analyzed for bulk scalar
fields in Hartman:2018tkw . We will also explain the relation of the present
results with those of GST ; LLST ; Shyam:2021ciy .

##  2 $T\bar{T}+\Lambda_{2}$ trajectories, patchwise holography, and
microstates

In the pure gravity sector, finite-cutoff holography in $AdS_{3}$ and $dS_{3}$
has recently been shown, starting with the work of McGough:2016lol ;
Kraus:2018xrn ; GST  to be dual to certain deformations of a holographic CFT.
Although cosmological spacetimes do not contain time-like boundaries,333With
the exception of specific UV-complete solutions with end-of-the-world
codimension-one orientifold planes such as Polchinski:1998rq  and relatives
such as Horava:1996ma . cut-off patches of spacetime are useful as building
blocks for a formulation of the global spacetime (the full spacetime being
obtained by sewing such patches together with an integral over their bounding
geometry).444For a different approach involving finite patches of spacetime
see e.g. Banks:2018ypk . Moreover, in the special case that the finite patch
is the domain accessible to an observer, bounded by their trajectory and
conditioned on their existence and experience,555We thank Zhenbin Yang for
suggesting this perspective in related discussions. The fixed metric condition
at the Dirichlet wall below corresponds to conditioning on the metric
experienced by this observer. the dual formulation may provide more than just
a building block. The prescriptions McGough:2016lol ; Kraus:2018xrn ; GST ;
LLST  along with the new versions we introduce in the present work provide
concrete holographic duals for finite patches, applicable to both situations.
In this section, we give a quick review of the connection, generalizing it
appropriately for our present purposes.

At the end of this section, we will get to the advertised microstate
construction obtained by judicious choice of $T\bar{T}+\Lambda_{2}$
trajectory. We will focus on the dressing of the $\Delta\simeq c/6$ band of
energy levels to the de Sitter ground state. The role of other levels depends
on additional specifications in the trajectory, which we will elucidate for
two versions in the subsequent sections §3-5. In fact, as we will see those
prescriptions essentially reduce to the micro-states we identify in the
present section.

###  2.1 General formulation

A $T\bar{T}$-deformed CFT is defined by the differential equation, and
boundary condition,

$\frac{\partial}{\partial\lambda}\,\log Z=-2\pi\,\int
d^{2}x\,\sqrt{g}\,\langle T\bar{T}\rangle+\frac{1-\eta}{2\pi\lambda^{2}}\int
d^{2}x\,\sqrt{g}\;,\qquad Z_{\lambda=0,\eta=1}=Z_{CFT},$ (4)

where $Z(\lambda,g)$ is the partition function of the theory on a manifold
with metric $g$, the $T\bar{T}$ operator is the composite of the stress tensor
defined above in (3), and (4) is expressed in Euclidean signature.666Appendix
A collects our notation and conventions. We will be particularly interested in
the solvable case where the theory is formulated on a flat space, with $Z$
depending on the circumference $L$ of the cylinder; in the case of the
canonical ensemble at finite temperature it depends on both $L$ and the
Euclidean time period $\beta$ (in addition to its dependence on the
deformation parameter $\lambda$). We sometimes denote the second term here as
$\Lambda_{2}$ (distinct from the bulk $3d$ cosmological constant $\Lambda_{3}$
whose sign will correspond to $-$sign($\eta$)). $Z_{CFT}$ is the partition
function of the ‘seed’ CFT that we deform by this procedure. As we will
explain shortly, one can define a deformed theory by joining together
solutions to this equation with different values of $\eta$, given appropriate
boundary conditions on each segment of the trajectory.

This deformation is most rigorously defined for all QFTs on flat two-
dimensional manifolds Zamolodchikov:2004ce ; Cardy:2018sdv , though progress
has been made on curved spaces GST ; Mazenc:2019cfg  and higher dimensions
Hartman:2018tkw  in perturbation theory. Here, we work only with holographic
seed CFTs, where there is a large-$c$ and large ’t Hooft coupling semi-
classical limit, for which the theory has additional bulk locality properties
and enhanced theoretical control. This entails a sparse light spectrum in the
sense of Hartman_2014 ; Mukhametzhanov:2019pzy , which guarantees a Cardy
entropy count beginning with the energy level corresponding to operator
dimension $\Delta=c/6$. We will be most interested in universal aspects of the
physics, captured by the Einstein gravitational sector which well describes
black hole solutions in AdS and the de Sitter cosmic horizon. At the same
time, we will keep track of where this description becomes inadequate, as it
does for energy levels arising from model-dependent particle states in the
sparse low-energy part of the seed CFT spectrum.

As indicated in (4), a differential equation is not sufficient to define a
theory; we also need a boundary condition for each segment of the trajectory.
For $\eta=1$, when the second term vanishes, there is an obvious boundary
condition at $\lambda=0$,

$Z_{\eta=1,\lambda}\xrightarrow{\lambda\to 0}Z_{CFT}.$ (5)

Once $\lambda$ is nonzero, we may split off a trajectory including the
$\Lambda_{2}$ deformation with $\eta=-1$ starting from any nonzero value
$\lambda_{0}$.

Before continuing, it is useful to incorporate the fact that the physics
depends only on the appropriate dimensionless quantities. The deformed path
integral has the scale-invariance

$Z[\lambda,\lambda_{0},g_{2}]=Z\left[e^{2\Omega}\lambda,e^{2\Omega}\lambda_{0},e^{2\Omega}g_{2}\right],$
(6)

where $\Omega$ is a constant. We will mainly be interested in the case when
the manifold is a cylinder with circumference $L$ (or a torus with cycle
lengths $\beta,L$ in some Euclidean calculations). Let us introduce the
shorthand notation

$y=\frac{\lambda}{L^{2}},\quad y_{0}=\frac{\lambda_{0}}{L^{2}}\,.$ (7)

It is useful to think of the trajectory as obtained by varying the
dimensionless variable $y$. As we will review shortly, the gravity side
holographically starts by bringing in the boundary, most directly changing $L$
McGough:2016lol ; Kraus:2018xrn ; GST ; LLST , while the description natural
in the 2d language involves changing $\lambda$ as in the defining equation (4)
while working on a fixed cylinder; in both descriptions $y$ changes along the
trajectory.

Given a nonzero value of $y_{0}$, we have a boundary condition

$Z_{\eta\neq
1,y=y_{0}}=Z_{\eta=1,y=y_{0}}~{}~{}~{}~{}{\text{(general~{}case)}}\,.$ (8)

In saying this, we must keep in mind that the simple formulation as written in
(4) pertains holographically only to universal (‘pure gravity’) quantities in
the theory. Below we will describe in detail the implementation of the
boundary conditions at the level of the appropriate energy levels in the
system.

Previous work GST ; LLST  focused on the case where we begin this
$T\bar{T}+\Lambda_{2}$ trajectory at $y_{0}\gg 1$. The $\Lambda_{2}$ term
disappears at $\lambda\to\infty$, and we may set a boundary condition in this
limit

$\lim_{\lambda\to\infty}Z_{\eta\neq
1,\lambda}=\lim_{\lambda\to\infty}Z_{\eta=1,\lambda}~{}~{}~{}~{}{\text{(previous~{}work)}}\,.$
(9)

In the present work, we will introduce a different deformed theory in which
the $T\bar{T}+\Lambda_{2}$ trajectory splits off from the pure $T\bar{T}$
trajectory well before $y=\lambda/L^{2}$ reaches infinity. As motivated above
in the introduction, this will yield a microstate count for the de Sitter
static patch by dressing CFT states at dimension $\Delta=h+\bar{h}=c/6$ to the
$dS_{3}$ static patch, obtained via a splitting off point $y_{0}=3/\pi c^{2}$.

As pioneered in Zamolodchikov:2004ce ; Smirnov:2016lqw ; Cavaglia:2016oda ,
the dressed energy levels comprise a key computable quantity in the deformed
theory.777See e.g. Cardy:2019qao  and Kruthoff:2020hsi  for interesting
studies of the behavior of additional quantities (correlation functions and
wavefunctions) under the deformation. The original derivation proceeds by
expressing the deformation in terms of the interaction Hamiltonian, deriving a
differential equation for the energy levels (applied level by level).
Equivalent results arise from an inverse Laplace transform of the partition
function as in Shyam:2021ciy  and from the $2d$ gravity reformulation of the
theory Dubovsky:2017cnj ; Dubovsky:2018bmo ; Mazenc:2019cfg . With
$\Lambda_{2}\propto 1-\eta$ included, the differential equation for the energy
levels becomes (see Appendix A of GST )

$\pi
y\mathcal{E}(y)\mathcal{E}^{\prime}(y)-\mathcal{E}^{\prime}(y)+\frac{\pi}{2}\mathcal{E}(y)^{2}=\frac{1-\eta}{2\pi
y^{2}}+2\pi^{3}J^{2}$ (10)

where we defined the dimensionless dressed energy

${\cal E}=EL\,.$ (11)

For a seed CFT,

${\cal E}\big{|}_{y=0,\eta=1}={\cal
E}_{CFT}=2\pi\left(\Delta-\frac{c}{12}\right)\,.$ (12)

Again here, a given $T\bar{T}+\Lambda_{2}$ theory will entail particular
boundary conditions on this equation.

The general solution to the differential equation takes the form

$\mathcal{E}(y)=\frac{1}{\pi
y}\left(1\pm\sqrt{\eta-4C_{1}y+4\pi^{4}J^{2}y^{2}}\right)\,,$ (13)

with the integration constant $C_{1}$ and the branch of the square root
determined appropriately for each segment of the trajectory (meaning the
initial segment consisting of the $T\bar{T}$ deformation alone, and the final
segment deforming by $T\bar{T}+\Lambda_{2}$ starting from $y_{0}$). Here $J$
is the spin; in terms of left and right scaling dimension $(h,\tilde{h})$, we
have $\Delta=h+\tilde{h}$, $J=h-\tilde{h}$.

For quantities admitting a smooth ‘pure gravity’ description – which are the
only ones captured by the pure $T\bar{T}$ and $T\bar{T}+\Lambda_{2}$
deformations – we will require continuity in the energy at $y=y_{0}$ (in
addition to the boundary condition matching the seed at $y=0$ (10)). We do not
generally require continuity of this formula for all energy levels, for the
following reason. In general, in order to capture local bulk matter in detail,
the prescription for the deformation (2) will require additional model-
dependent specifications such as those carried out in the scalar sector in
Hartman:2018tkw . For such model-dependent contributions to the spectrum,
subdominant for $c\gg 1$, the differential equation for the energies (10) and
its solutions (13) require additional contributions; it is only those
corrected formulas that must be continuous. Happily, the entropically dominant
states – those of our main interest – are captured by the solvable and
universal pure $T\bar{T}+\Lambda_{2}$ deformation.

In the remainder of this section, we will focus on this universal piece of the
physics, associated with the observer patch of the $dS_{3}$ vacuum as dressed
$\Delta=c/6$ BTZ black holes. In subsequent sections, we will lay out separate
versions of the complete theory, accounting for the fate of all energy levels
(with due regard to the appropriate level of model-dependence in $dS_{3}$
holography).

###  2.2 Gravity side and dictionary

At this point, it is useful to review the basic idea of the map to the gravity
side in the holographic correspondence developed in McGough:2016lol ;
Kraus:2018xrn ; GST ; LLST . The differential equation for the energy levels
is equivalent to the Einstein equations in a bounded patch of (A)dS spacetime.
This connection proceeds starting from the bulk gravity action (in Lorentzian
signature)

$\displaystyle I$ $\displaystyle=\frac{1}{16\pi
G_{N}}\int_{\mathcal{M}_{3}}d^{3}x\sqrt{-g}\left(R+\frac{2}{\ell^{2}}\eta\right)+\frac{1}{8\pi
G_{N}}\int_{\partial\mathcal{M}_{3}}d^{2}x\sqrt{-g}\left(K-\frac{1}{\ell}\right)$
(14)

with boundary condition

$ds^{2}_{\mathcal{M}_{3}}\Big{|}_{\partial\mathcal{M}_{3}}=ds^{2}_{\mathcal{M}_{2}}\,.$
(15)

Specializing to the case where the boundary metric is a flat cylinder of
proper circumference $L$, the differential equation for the energy given above
in (10) follows from the Einstein equations McGough:2016lol ; Kraus:2018xrn ,
once we identify the energy with the Brown-York energy Brown:1977sj .888with
similar statements holding for the curved case perturbatively around large $c$
The Einstein equations can be processed as reviewed in the appendix of GST  to
yield the equation (10) for

${\cal E}=L\int_{0}^{L}dx\,T^{t}_{t}$ (16)

in terms of cylinder coordinates $-dt^{2}+dx^{2}$, with the Brown-York stress-
energy tensor given by

$T_{\mu\nu}={\frac{2}{\sqrt{-g}}\frac{\delta I_{\text{on-shell}}}{\delta
g^{\mu\nu}}}=\frac{1}{8\pi
G_{N}}\left(K_{\mu\nu}-g_{\mu\nu}K+\frac{1}{\ell}g_{\mu\nu}\right)\,$ (17)

in terms of the extrinsic curvature $K_{\mu\nu}$ of the Dirichlet wall. In
this correspondence, the map of parameters is

$c=\frac{3\ell}{2G_{N}},~{}~{}~{}\lambda=8G_{N}\ell\,.$ (18)

In this correspondence, different choices of branch and integration constant
in the solutions (13) correspond to different choices of the patch and its
excitations (insofar as they are contained in the ‘pure gravity’ sector of the
theory). This will be a key technical point of our work, which we will spell
out in detail in what follows for the cases of interest. In particular, the
choice of branch of the square root in (13) corresponds to the sign of the
extrinsic curvature $K$. This is not immediately obvious from inspection, but
can be seen from the derivations in McGough:2016lol ; Kraus:2018xrn ; GST  (in
which the non-square-root term is tied to a relatively inconsequential
counterterm).999This counterterm is chosen first of all to match the one
required for AdS/CFT Balasubramanian:1999re  in the limit $y\to 0$, and for
simplicity we maintain the same value for it throughout the trajectories we
consider here. As such, opposite signs of the square root correspond to
complementary ways to fill in the bulk patch starting from the same Dirichlet
wall. To anticipate a little, this is the difference between the patches
depicted in figure 1.

In order to capture a particular bulk patch as a gravitational representation
of a dressed energy level in this way, we must specify the full trajectory
including the choices of branch and integration constant. At this point,
rather than continuing with the generalities, we will shortly spell this out
for several cases of interest.

To proceed to that, let us record metrics for AdS and dS vacua and for BTZ
black holes (with these and particle states related by various
identifications). First, the locally-$AdS_{3}$ metric

$ds_{3}^{2}=-\left(\frac{r^{2}-r_{h}^{2}}{l^{2}}\right)d\tau^{2}+\left(\frac{l^{2}}{r^{2}-r_{h}^{2}}\right)dr^{2}+r^{2}d\phi^{2},\qquad
r_{h}^{2}\geq-\ell^{2},\ \phi\sim\phi+2\pi$ (19)

captures the $AdS_{3}$ global vacuum and spinless BTZ black holes. The case
$r_{h}^{2}=-\ell^{2}$ is global $AdS$. In terms of the black hole mass,
$8G_{N}M=r_{h}^{2}/\ell^{2}$, we have $\ell M=\Delta-c/12$, leading to the
dictionary (adding to (18) above)

$\Delta-\frac{c}{12}=\frac{r_{h}^{2}}{8G_{N}\ell}=\frac{1}{2\pi}{\cal
E}_{y=0}$ (20)

where the last subscript indicates the dimensionless energy ${\cal E}=EL$ in
the seed CFT. As mentioned above, the spectrum of the holographic seed theory
breaks up into the following windows (see for instance Hartman_2014 ;
Mukhametzhanov:2019pzy ):

$\displaystyle\Delta$ $\displaystyle<$
$\displaystyle\frac{c}{12}:~{}{\text{sparse~{}spectrum
~{}(particle~{}states)}}$ $\displaystyle\Delta$ $\displaystyle\geq$
$\displaystyle\frac{c}{6}:S\simeq
S_{Cardy}=2\pi\sqrt{\frac{c}{3}\left(\Delta-\frac{c}{12}\right)}$ (21)

(in between which is a window of light BTZ black hole states). More precisely,
the detailed CFT states are generically non-degenerate, split by energy
differences of order $\exp(-S)$. This enables a continuum approximation to the
density of states; in recent work, the count of states given by $S_{Cardy}$
Mukhametzhanov:2019pzy  was shown to apply to an order one band (i.e. order
$c^{0}$) around a fixed $\Delta\sim{\cal O}(c)$ level. We note from (18), (20)
a relation which will play a role in our analysis (following Dong:2018cuv ):

$\Delta=\frac{c}{6}\,\,\implies\,\,r_{h}=\ell\,.$ (22)

We will see below in §4 that this energy level has a special place in the
spectrum associated with modular transformations and the Hawking-Page
transition. These results are general; one may consider as an example of a
seed theory the D1-D5 CFT, whose black hole entropy count is known
microphysically Strominger:1996sh  and for which an uplift to de Sitter was
constructed in Dong:2010pm .

Secondly, the static patch metric for $dS_{3}$ is

$ds_{3}^{2}=-\left(\frac{\ell^{2}-r^{2}}{{\ell^{2}}}\right)d\tau^{2}+\left(\frac{{\ell^{2}}}{\ell^{2}-r^{2}}\right)dr^{2}+r^{2}d\phi^{2}\,.$
(23)

In both cases, various further identifications yield additional states; e.g.
restricting the period of $\phi$ in (23) describes the metric sourced by a
particle at the pole at $r=0$ GST , and more elaborate identifications
introduce spinning states (see e.g. Bousso:2001mw ).

###  2.3  Trajectories and horizon entropy

Now that we have laid out the formalism on both sides of the duality, let us
explain the key examples of interest here. In brief: we will review how the
original version of the $T\bar{T}+\Lambda_{2}$ trajectory captures a patch of
spacetime not containing the cosmic horizon but containing the observer
trajectory (say the North Pole) within the static patch. The new version
captures the complementary part of the static patch, including the cosmic
horizon. The latter exhibits the microstates (with the correct count) as
appropriate dressed energy levels of the seed CFT.

####  2.3.1 Previous Version: The Pole Patch

First we will quickly review the original prescription carried out in GST ;
LLST . There, we work with the boundary condition (9) for the
$T\bar{T}+\Lambda_{2}$ part of the trajectory, meaning we continue with the
pure $T\bar{T}$ trajectory until $y=\lambda/L^{2}$ reaches the matching point
$y_{0}\gg 1$. This first segment of the trajectory, applied to the ground
state, entails bringing in the boundary of global $AdS_{3}$ – prescribing a
Dirichlet wall at $r=r_{c}=L/2\pi$ and filling in the region $r<r_{c}$ until
the bounding cylinder is very small, with the patch inside of it being
indistinguishable from a similar patch centered on an observer worldline
(e.g.the North Pole $r=0$) in de Sitter spacetime. Then beginning the
$\eta=-1$ part of the trajectory, with the appropriate choice of branch and
integration constants, formulates a patch of $dS_{3}$ with $r\leq r_{c}$. The
energy formula for this case is given by

${\cal
E}=\frac{L^{2}}{\pi\lambda}\left(1-\sqrt{\eta-\frac{4\pi^{2}\lambda}{L^{2}}\left(\Delta-\frac{c}{12}\right)+4\pi^{4}J^{2}\frac{\lambda^{2}}{L^{4}}}\right)\,.$
(24)

This produces the correct CFT energy at $y=\lambda/L^{2}\to 0$, $\eta=1$, and
at the matching between $T\bar{T}$ and $T\bar{T}+\Lambda_{2}$ the energies are
continuous since $y=\lambda/L^{2}\to\infty$.

Note that the gravity-side patch that is thus formulated does not contain the
cosmic horizon at $r=\ell$. For $r_{c}\to\ell$, $\Delta=0$ this (almost)
captures the static patch of de Sitter, but without the cosmic horizon given
this order of limits, a distinction that will greatly concern us in this work.
What this corresponds to in the deformed-CFT dual is the following: in the
family of (A)dS patches we just reviewed that were formulated in the original
trajectory GST LLST , only the particle states (2.2) survive in the real
spectrum. This is not enough to account directly for the Gibbons-Hawking
microstates. That fact agrees with the absence of the cosmic horizon (CH) in
the patch.

We are now ready to present the essence of the new trajectory, one that
formulates the complementary patch containing the cosmic horizon, with a
concommitant count of microstates.

####  2.3.2 New Version: The Cosmic Horizon (CH) Patch

The branching-off point $y_{0}$ for the $T\bar{T}+\Lambda_{2}$ part of the
trajectory may be chosen differently from that in the previous section, so
long as the energy levels that are accessible in the model-independent pure-
gravity sector of the theory are continuous in the transition from $\eta=1$
($\Lambda_{3}<0$) to $\eta=-1$ ($\Lambda_{3}>0$). Moreover, as we stressed
above in (13), the solution space of the differential equation for the
energies has choices of integration constant $C_{1}$ and a choice of the
branch of the square root (in principle, these could be chosen separately
level by level).

In this section, we will see that this freedom enables the construction of a
consistent class of trajectories which formulate the cosmic horizon (CH)
region of the static patch as a set of dressed states in the band of energies
around $\Delta=c/6$. At the end of the trajectory, this captures the entire
static patch, bounded by the observer worldline at (say) the North Pole of the
bulk de Sitter spacetime.

Let us begin with a standard observation — the similarity between the near-
horizon regions in a black hole spacetime (19) with $r_{h}=l$ and the $dS_{3}$
static patch (23) with the same curvature radius $\ell$. In the black hole
case, we make the coordinate transformation

$\displaystyle r\equiv r_{h}+\delta r=\ell+\delta r,\qquad\delta
r=\frac{w^{2}}{2\ell}\ll r_{h}=\ell,$ (25)

to find that the black hole metric (19) gets transformed into a nearly flat
near-horizon region

$ds_{3}^{2}=-\frac{w^{2}}{\ell^{2}}d\tau^{2}+dw^{2}+\ell^{2}d\phi^{2}+\dots$
(26)

Similarly, in the $dS_{3}$ vacuum we take

$r\equiv\ell-\delta r,\qquad\delta r=\frac{w^{2}}{2\ell}\ll\ell\,.$ (27)

Keeping one more contribution, for both cases the near-horizon metric takes
the form

$ds_{3}^{2}=-\frac{w^{2}}{\ell^{2}}d\tau^{2}+dw^{2}+(\ell^{2}+\eta
w^{2})d\phi^{2}.$ (28)

where as usual $\eta=1$ for the $r_{h}=\ell$ BTZ case and $\eta=-1$ for the
$dS_{3}$ vacuum. Thus we find find exactly the same near-horizon metric (26)
at leading order in an expansion in $\delta r=w^{2}/2\ell$. Note that the
value $r_{h}=\ell$ at which these match corresponds to the energy level
$\Delta=c/6$ (22) Dong:2018cuv .

Recall that the bulk intuition for the boundary condition (9) was precisely
that the spacetime near $r=0$ is nearly flat in either global $AdS_{3}$ (19)
or the static patch of $dS_{3}$ (23), i.e. it becomes independent of the bulk
cosmological constant in the relevant limit; in that case the system reverted
to a nearly flat bulk spacetime contained within a small bounding cylinder. It
is quite analogous that the near-horizon region boils down similarly to a
nearly flat patch of spacetime. The extrinsic curvature $K_{\mu\nu}$ and hence
the Brown-York stress energy (17) agree at leading order in this approximation
(when we drop the $\eta$-dependent term in (28)).

In particular, the extrinsic curvature of the metric (28) approaches zero in
this region:

$K_{\mu\nu}=\frac{1}{2}\partial_{w}g_{\mu\nu}\sim{\cal O}(w)\to 0$ (29)

This non-trivial agreement at leading order in $w$ suggests that there is a
new class of trajectories that may be defined, with a consistent boundary
condition at a finite value of $y_{0}$, in which the transition from $\eta=1$
to $\eta=-1$ occurs when the system formulates equivalently the near-horizon
region of either the $\Delta=c/6$ black hole or the $dS_{3}$ static patch.

As explained above, the square root term in the energy formula (given in
general form in (13)) is proportional to the extrinsic curvature of the
boundary on the gravity side. The matching of the two segments of the
trajectory, at the indistinguishable horizon on the gravity side, should occur
for the value of $y_{0}$ where this square root term vanishes. At that point,
we are free to continue with the $T\bar{T}+\Lambda_{2}$ trajectory with either
sign of the square root, still maintaining continuity of the energy.

Indeed, if we choose

$y_{0}=\frac{3}{c\pi^{2}}$ (30)

then we see that the square root term in the $\eta=1$ (pure $T\bar{T}$) energy
formula (24) vanishes at $y\simeq y_{0}$, for a band of energy levels within
order one of $\Delta=c/6$.101010 Below in §3 will spell out the role of this
band of split levels around $\Delta=c/6$.

At that point, we may start our $\eta=-1$ segment of the trajectory, with
energy levels of the form (13), matching the $T\bar{T}$-dressed $\Delta\simeq
c/6$ level at $y=y_{0}$ with an energy level that similarly has vanishing
square root.

Consider again the energy formula (24) for $\eta=-1$ and $\Delta=0$. This is
the Brown-York energy of a cutoff region of the static patch not containing
the cosmic horizon (the pole patch in the right hand picture in figure 1). We
can immediately write down the energy formula for a $T\bar{T}+\Lambda_{2}$
trajectory which builds up the patch complementary to this region, which we
will call the cosmic horizon (CH) patch. It is given simply by flipping the
sign of the square root:

${\cal
E}=\frac{L^{2}}{\pi\lambda}\left(1{\textbf{+}}\sqrt{-1-\frac{4\pi^{2}\lambda}{L^{2}}\left(-\frac{c}{12}\right)}\right)=\frac{1}{\pi
y}\left(1{\textbf{+}}\sqrt{-1+\frac{y}{y_{0}}}\right)\,$ (31)

since this corresponds to flipping the sign of the extrinsic curvature, thus
capturing the complementary fill-in of the bulk region starting from the
Dirichlet wall cutoff surface. We see that also for the cosmic horizon patch
when $y=\lambda/L^{2}$ reaches $y_{0}$ (30), the square root vanishes.

Putting all this together we may prescribe a combined trajectory as follows.
First, we do a $T\bar{T}$ trajectory as in (24), increasing $y$ from $y=0$
until $y$ reaches $y_{0}$ (30). Then we evolve with $T\bar{T}+\Lambda_{2}$
increasing $y$ from its initial value of $y_{0}$ and choosing the branch of
the square root indicated in (31).

As advertised, this smoothly dresses the $\Delta\simeq c/6$ BTZ black hole to
the dS vacuum. We can identify the microstates comprising the $dS_{3}$
observer patch as given explicitly by the dressed $\Delta\simeq c/6$ states.
The Cardy entropy of these states, matching the horizon entropy
$2\pi\ell/4G_{N}$, carries through the entire deformation since this pertains
energy level by energy level.111111This is sometimes described as an
integrable deformation, though we stress that the seed theory need not be an
integrable QFT (and our seed holographic CFTs of interest are not integrable).

Moreover, the agreement extends at least to the first logarithmic correction
(1) in the entropy Anninos:2020hfj  at the level of pure gravity, as already
derived in Shyam:2021ciy . This follows immediately because the same result
$S=A/4G_{N}-3\log(A/4G_{N})$ arises in the AdS case as computed by Sen and
Carlip Sen:2012dw , Carlip_2000 . This density of states of the $\Delta\simeq
c/6$ band of energy levels goes along for the ride in the $T\bar{T}$ and
$T\bar{T}+\Lambda_{2}$ trajectories, for the pure gravity subsector of the
theory. It would be very interesting to extend this to all orders as in
Anninos:2020hfj  (for a pure gravitational subsector of the theory) and
explore the role of edge mode degrees of freedom as in Anninos:2021ihe .

Here we focused on the dressing of a particular band of energy levels near
$\Delta=c/6$. At the level of the $T\bar{T}+\dots$ deformed CFT, there are
multiple ways to extend this readily to account for other energy levels. For
the purpose of $dS_{3}$ holography, we will describe the relevant prescription
next in §3. In this context, it is the energy level we already described which
contains the information about the theory that is accessible at the level of
the pure gravity prescription we have formulated. In our prescription in §3,
the higher energy levels will not remain in the real spectrum of the theory;
the total dimension of the real Hilbert space captures the $dS_{3}$ Gibbons
Hawking entropy (in addition to a log correction Anninos:2020hfj ;
Shyam:2021ciy ). In order to capture the physics of the model-dependent energy
levels with $\Delta<c/6$, which are generically subleading at large $c$, we
generally require additions to the specifications of the trajectory (as laid
out in some cases in Hartman:2018tkw ). As such, the energy levels in this
window $\Delta<c/6$ need not be (and are not) continuous in the transition
from between $\eta=1$ and $\eta=-1$.

As it turns out, there is another way to generalise the energy level dressing
(31) of $\Delta\simeq c/6$ to other levels in the seed, based on matching the
bulk geometries for black holes with $r_{h}\neq\ell$ following the same steps
as (25)–(28). We present this in §5. Interestingly, levels with $\Delta>c/6$
dress to locally $dS_{3}$ geometries with undesirable conical excesses; but
these levels are screened out with a Hagedorn transition.

A standard result from the original $T\bar{T}$ studies on a two-dimensional
cylinder is the super-luminal propagation of perturbations. On the gravity
side of the holographic version laid out in McGough:2016lol , this is clear
from the statement that perturbations travel more quickly in the bulk bounded
by a finite cylinder (rather than along the boundary). It is not a pathology,
being consistent with causality in the gravity theory, but does complicate the
analysis of causal and entanglement wedges in the theory LLST
.121212Interestingly, this issue does not arise for the case where the
boundary theory lives on $dS_{2}$ GST , which can be similarly understood on
both sides of the duality. We may ask the same question for
$T\bar{T}+\Lambda_{2}$ on the cylinder. This retains the faster-in-the-bulk
propagation for pole patch formulated by the original trajectory GST ; LLST .
By the same token, it is sub-luminal, faster to propagate on the boundary, for
the cosmic horizon patch. One can show that this lines up with the behavior of
the boundary theory including for entanglement-based measures of propagation
speed Shaghoulian:2021cef ; CSY .

Finally, without getting into details, we can describe the Hawking-Page
generalization rather simply with the aid of figure 1: below the transition,
the spatial circle size $L$ can shrink to zero in the bulk, while above the
transition it is $\beta$ that can shrink to zero, producing a horizon with a
finite area. These features are evident in the distinction between the non-CH
patch and the CH patch in de Sitter: in the non-CH patch the circle shrinks to
zero at the north pole, while in the CH patch we get a horizon (where $\beta$
shrinks to zero in the interior but $L$ stays finite, taking the value
$2\pi\ell$). Finally, we note that one may formulate a thermofield-double
system as in figure 3 and join the non-CH and CH patches together, integrating
over their identified boundaries, to obtain a formulation of global $dS_{3}$
(consistent with its full symmetry group). We will comment on this further
below, see figure 3.

##  3 Further details of $dS_{3}$ holography and microstates from
$T\bar{T}+\Lambda_{2}$

Recall first that ref. Dong:2018cuv  showed that the Cardy entropy of the
$\Delta=c/6$ energy level matches the dS Gibbons-Hawking entropy (understood
as an entanglement entropy between the two sectors of the dS/dS
correspondence) if we identify the neck of this BTZ black hole with the dS
neck. This extends universally to higher dimensions, where the agreement works
for AdS black holes at the Hawking-Page transition. The way that the
$\sinh^{2}(w/\ell)$ warp factor of AdS/dS morphs into the $\sin^{2}(w/\ell)$
warp factor of dS/dS Alishahiha:2004md  was clarified in GST ; LLST , by
showing that the $\Lambda_{2}$ addition to $T\bar{T}$ gives a holographic
description of the bulk dS rather than bulk AdS geometry in various patches.
This used the prescription reviewed above in Sec. 2.3.1, joining the
trajectories at $\lambda/L^{2}\gg 1$. But the count of microstates here has
remained obscure given the fact that the vacuum is dressed to the vacuum, and
the $\Delta=c/6$ energy level (which contains the GH number of states) becomes
complex in the dressed energy formula. Furthermore, the remaining step of
joining the two dS/dS sectors by path-integrating over the metric at the
central slice was not taken in GST ; LLST .

In this section we present more details of a prescription which provides a
more direct count of microstates for the dS entropy as described above.
Instead of joining the $T\bar{T}$ and $T\bar{T}+\Lambda_{2}$ trajectories at
$\lambda/L^{2}\to\infty$, we join them at a horizon — the $\Delta=c/6$ (HP)
BTZ horizon to the dS static patch horizon. That is, we bring in the boundary
on the AdS side, stopping at the value of $\lambda/L^{2}$ for which the
Dirichlet wall skirts the horizon of the HP black hole. The patch captured by
the near horizon region of the BTZ HP black hole is indistinguishable from the
near-horizon region of the de Sitter static patch. We start the
$T\bar{T}+\Lambda_{2}$ deformation from this value of $\lambda/L^{2}$, and
move the boundary back out, obtaining instead the dS static patch. This
deformation uses the opposite sign of the square root compared to GST ; LLST ,
since the bulk patch we formulate is on the other side of the boundary. The
full static patch is obtained as a region bounded by a small cylinder at the
pole (the region outside the cylinder).

###  3.1 Energy Levels Beyond $\Delta\simeq c/6$

If we move off along the $T\bar{T}+\Lambda_{2}$ trajectory starting at some
nonzero value $y_{0}$, the dressed energy levels now take the form

${\cal E}=\frac{1}{\pi
y}\left(1\mp\sqrt{\eta+\frac{y}{y_{0}}(1-\eta)-4\pi^{2}y\left(\Delta-\frac{c}{12}\right)+4\pi^{4}y^{2}J^{2}}\right)\,,$
(32)

for states accessible at the level of pure gravity, where we have fixed the
integration constant $C_{1}$ of (13) in terms of the joining point $y_{0}$. If
we kept the same branch of the square root, this formula would be manifestly
continuous for all energy levels and all angular momenta, since the $\eta$
dependence drops out at $y=y_{0}$. However, the pressure component is not
continuous in this case.

We want to require continuity of the dressed energy for the entropically
dominating HP level $\Delta=c/6$, $J=0$. Also, since the Cardy count comes
from an $O(1)$ window Mukhametzhanov:2019pzy

$\frac{c}{6}<\Delta\leq\frac{c}{6}+\delta$ (33)

with $\delta\sim 1$, we need to make sure that the corresponding dressed
states stay real. These requirements are satisfied if the matching point
corresponds to the horizon of the BTZ black hole with

$\Delta_{*}=\frac{c}{6}+\delta\,,$ (34)

namely

$y_{0}=\frac{1}{4\pi^{2}(\Delta_{*}-c/12)}\,.$ (35)

The minus branch from $\eta=+1$ is matched to the plus branch for $\eta=-1$
and this results in a continuous stress-tensor across $y=y_{0}$ for this
level.

The resulting dressed energy for $y>y_{0}$ and $J=0$ becomes

$\mathcal{E}_{\eta=-1}=\frac{1}{\pi
y}\left(1+\sqrt{-1+4\pi^{2}y\left(\Delta_{*}-\frac{c}{12}+(\Delta_{*}-\Delta)\right)}\,\right)\,.$
(36)

The upper limit $\Delta=\Delta_{*}$ gives a stress tensor that matches
continuously by construction at $y=y_{0}$. For the remaining states in the
window (33) there are $1/c$-suppressed discontinuities in the stress-
energy.131313These states translate into particle states with $O(1)$ negative
masses, corresponding to small conical excesses which are degenerate with
$1/c$ corrections including quantum effects. Therefore the Cardy entropy is
not modified at leading order in $c$. States with $\Delta>\Delta_{*}$ have
complex dressed energies at $y=y_{0}$; they are truncated from the spectrum.
We focused here on $J=0$, but it was shown in Hartman_2014  that nontrivial
spins do not change the basic structure of the spectrum and entropy count.

This achieves our goal, providing an explicit microstate count for the de
Sitter entropy in terms of dressed BTZ black hole states with real dressed
energies.

###  3.2 The sparse light spectrum (subleading at large $c$)

Consider for completeness now the dressed $\Delta<c/6$ states. These are
sparse in the holographic seed CFT and correspond to model-dependent particle
states in the bulk theory. In the $T\bar{T}$ part of the trajectory, when we
deform until $y=y_{0}$ (30) we reach the horizon of the $\Delta=c/6$ black
hole states. As such, in the case of the $\Delta<c/6$ states we do not reach
the particle position before moving out with $T\bar{T}+\Lambda_{2}$. Thus on
the gravity side, these states entail a domain wall between bulk regions with
$\Lambda_{3}<0$ and $\Lambda_{3}>0$. We very schematically depict a time-slice
of this model-dependent state in Figure 2. However, in order to capture that
accurately, one would require additions to the trajectory going beyond pure
gravity (building from the results in Hartman:2018tkw ).

Figure 2:  A schematic of a time slice of the bulk dual for dressed
$\Delta<c/6$ states. The spatial geometry piecewise forms a cone. A domain
wall sits between bulk regions with different signs of $\Lambda_{3}$.
Deforming with $T\bar{T}+\Lambda_{2}$ to $y>y_{0}$ has the effect of
increasing the volume of the positive-$\Lambda_{3}$ domain. The radius of the
internal circle decreases away from the wall at $y=y_{0}$ on both sides,
unlike in the diagram.

###  3.3 Lazarus States: $\Delta>c/6$ at large $y$

An interesting note regarding the formula (32) is that the energies are real
for

$\Delta<\Delta_{max}\approx\frac{c}{4}-\frac{1}{4\pi^{2}y}.$ (37)

For $y=y_{0}$, the RHS is $c/6$, as required by consistency with the previous
discussion. But the value of $\Delta_{max}$ _increases_ with $y$.

This means that as we increase $y$ to $y_{0}$, flip $\eta$ and continue along
the trajectory, many states with $\Delta>c/6$ go complex and then become real
again — they are resurrected, as in the Biblical story of Lazarus. We do not
currently have a proposed bulk interpretation for these states. Of course, if
we maintain unitarity throughout this process by restricting the theory to its
real spectrum at every step, there is no reason to include them again.

##  4 Modular Behavior And A Role for the Hawking-Page Transition

The dressed $\Delta\simeq c/6$ band of energies accounting for the microstates
has a special status in the seed CFT: this energy level corresponds to the
Hawking-Page transition Hartman:2013qma ; Dong:2018cuv . In this section, we
note that the role of these microstates could be motivated starting from the
original trajectory GST ; LLST  (reviewed in §2.3.1) and applying a remnant of
modular invariance in our system.

We may work in a canonical or microcanonical ensemble, with fluctuations
suppressed at large $c$. In the gravity variables, these are spelled out in
the Brown-York papers Brown:1992br  and Brown:1992bq . Consider a partition
function for a two-dimensional theory on a torus of size $L\times\beta$. For
simplicity, let us set the spin to $0$ and take the sides of the torus to be
orthogonal. The modular S-transformation takes this torus to one of dimensions
$\beta\times L$. This is a symmetry, due to the fact that there is no special
choice of time on this Euclidean torus,

$Z_{\lambda}(L,\beta)=Z_{\lambda}(\beta,L).$ (38)

This trivial property, which is just a special case of the diffeomorphism-
invariance of the theory, is called S-invariance.141414In some of the
literature, the S-transformation is defined not just as a diffeomorphism but
as a diffeomorphism followed by a rescaling $L\times\beta\to L\times L/\beta$,
following the CFT literature. That is a symmetry if we also take
$\lambda\to\lambda/(\beta/L)^{2}$. The final answers don’t depend on which
symmetry we work with. See e.g. Datta:2018thy ; Aharony:2018bad ;
Aharony:2018ics  for previous studies of modular properties of $T\bar{T}$
deformed theories. As the deformation (4) is diffeomorphism-invariant, we
expect it to also be S-invariant, at least before discarding the complex
energy levels (at which point the theory reduces to ordinary $0+1$-dimensional
quantum mechanics). In fact, even restricting attention to the real energy
levels leaves us with a residual S-invariance as follows.

In large $c$ field theories with a sparse spectrum, for $\beta$ sufficiently
close to $L$, the partition function takes the form

$\log Z\simeq\max\left\\{-\beta E_{vac}(L)\ ,\ -LE_{vac}(\beta)\right\\}$ (39)

to leading order at large $c$ Hartman_2014 . Starting from this seed CFT, the
$T\bar{T}$ and $T\bar{T}+\Lambda_{2}$ deformations proceed energy level by
energy level Smirnov:2016lqw ; GST ; LLST . Clearly, the first term dominates
if $L<\beta$ and the second if $L>\beta$.

It is interesting to consider the bulk dual corresponding to the two terms. To
do so, we have to embed the torus into the Euclidean version of the static
patch metric (23) (where the Euclidean time $\tau_{E}$ is constrained to have
a periodicity of $2\pi\ell$ by smoothness). From the bulk metric, an $r=r_{c}$
surface is a torus of dimensions $2\pi r_{c}\times
2\pi\sqrt{\ell^{2}-r_{c}^{2}}$. By the matching conditions (18), this
constrains the boundary modular parameter to be

$\left(\beta/L\right)^{2}=\frac{y}{y_{0}}-1.$ (40)

So we see that, unlike a standard two-dimensional theory, the modular
parameter and the coupling cannot be varied independently.151515This is the
case if we restrict to global $dS_{3}$. More generally, we may allow conical
deficits, in which case (40) becomes an inequality — owing to the fact that
conical excesses are not allowed. This is still surprising from the two-
dimensional point of view.

In the recent work Shyam:2021ciy , the second term of (39) in the
$T\bar{T}+\Lambda_{2}$ deformation, in the limit $\beta/L\to 0$, accounts for
the leading (and first subleading) contribution to the Gibbons-Hawking
entropy. So, let us restrict attention to the second term, where the
contractible cycle in the bulk is identified with the time cycle in the
boundary. Assuming then that $\beta<L$, we can write (39) as

$\log Z|_{\beta<L}\simeq-LE_{vac}(\beta)=S_{Cardy}(\Delta=c/6)-\beta
E_{\Delta=c/6}(L)$ (41)

with $E_{vac}$ given by the dressed energy (24) with $\Delta=0$, and
$E_{\Delta=c/6}$ the $T\bar{T}+\Lambda_{2}$ dressed energy (32) corresponding
to the cosmic horizon patch. The crucial second equality follows from (40). A
proof of (41) is explained in more detail in Appendix C.

Note that $E_{vac}$ is the vacuum energy in the trajectory of LLST  (discussed
in §2.3.1), whereas $E_{\Delta=c/6}$ is an energy in the new trajectory
defined in §2.3.2. In §2, these two trajectories were introduced merely as two
versions of the $T\bar{T}+\Lambda_{2}$ theory, but now we see that the
partition function somehow puts them in the same theory, as two different
sectors. The vacuum state in the old §2.3.1 trajectory seems to be mapped to
the $\Delta=c/6$ state in the new §2.3.2 trajectory. To be clear, in this
section we have not allowed arbitrary energies to appear — thanks to the
constraint (40). In that sense, we have discussed a one-dimensional theory
which has only certain states and cannot have full modular S-invariance.
However, because of the large $c$ thermodynamic limit, the partition function
in the CH phase is dominated by states with $\Delta\simeq c/6$; thus it is
reasonable for the vacuum to map to only this window of states in some remnant
of S-invariance that shows up in a partition function constrained by (40).
Thus, demanding that the theory defined by the energy levels in LLST  satisfy
this constrained version of S-invariance automatically implies that the theory
also contains energy levels corresponding to the Gibbons-Hawking entropy.

This is analogous to the Hawking-Page transition in AdS spacetime. For the
Cosmic Horizon patch, considered in Euclidean signature, the thermal circle
shrinks smoothly in the interior (at the horizon). The complementary Pole
Patch has the spatial circle shrinking smoothly at the pole. On the gravity
side, we have defined a Dirichlet boundary condition that the boundary be a
torus of given dimensions and the path integral is a sum over all 3d manifolds
that satisfy this boundary condition. So, we find

$\displaystyle\log Z=-\min(I_{P}\ ,\ I_{CH})=-\frac{\beta
L}{\pi\lambda}\min\left\\{1-\frac{\beta}{L}\ ,\ 1-\frac{L}{\beta}\right\\}$
(42)

where $I_{P}$ and $I_{CH}$ are the classical Euclidean actions for the Pole
and Cosmic Horizon patches. We supply the details of this calculation in
Appendix B. We see explictly that the pole (CH) saddle dominates when
$\beta>L$ ($\beta<L$).

In the way that the two saddles correspond to a choice of contractible
boundary cycle, the role of the two terms here is the same as the standard
Hawking-Page transition in AdS/CFT. However, there is an important difference:
in the present case we don’t have a tower of energy levels, as shown by our
energy formula. To see that this is consistent with the formulas above, we
merely have to take the $\beta\to 0$ limit of (42) to find the dimension of
the Hilbert space Caputa:2020fbc . We find that the action approaches
$1/(\pi\sqrt{y_{0}y})=\sqrt{c/(3y)}$.

Finally, let us note that we may formulate a thermofield double state of two
copies of our system,

$|TFD\rangle=\sum_{n}e^{-\beta E_{n}/2}|E_{n,L}\rangle|E_{n,R}\rangle$ (43)

whose norm is $Z(\beta)$. The corresponding bulk dual is as in figure 3. Note
however that, because of (40), we cannot describe the CH patch with a boundary
arbitrarily close to the poles in the canonical ensemble. This patch was,
however, formulated above in §2-3 in the microcanonical ensemble, leading to
our microstate count for arbitrary $y\in(y_{0},\infty)$. In the canonical
ensemble, on the other hand, since this saddle only dominates when $\beta<L$,
we find that the boundary of the CH patch must satisfy

$y<y_{max,CH}=2y_{0}=\frac{6}{c\pi^{2}}\qquad\Rightarrow\qquad
r_{c}^{2}>\frac{\ell^{2}}{2}.$ (44)

Interestingly, since this is the limit $\beta=L$, the boundary of the pole
patch must satisfy $r_{c}^{2}<\ell^{2}/2$ for the dominant contribution in the
canonical ensemble.

Figure 3: The two phases of the doubled system, including the entangled cosmic
horizon patches in the right panel. In that case, the upper and lower
triangles should be reconstructable by bulk evolution. One may treat these
patches as conditioned on an observer at the Dirichlet wall boundary, or
alternatively use them as building blocks, joining these patches together at
their common boundary, integrating over its geometry, to obtain global
$dS_{3}$ with its symmetries. It would be interesting to connect the full
nonlinearity of the deformed theory as a function of the fundamental seed CFT
variables to the notions of hyper-scrambling and complexity suggested recently
in Susskind:2021esx ; Chapman:2021eyy .

##  5 An Alternative Deformation with a Hagedorn Transition

In this section, we will explore another version of the $T\bar{T}+\Lambda_{2}$
deformed CFT which again makes use of the freedom to choose the integration
constant and branch of the square root in the energy formula (13). This
version will exhibit a Hagedorn transition at the would-be Hawking-Page
transition point $\beta=L$, leading to different physics above that scale.
This version agrees with the above prescription in §3 for the microstates of
interest — at least in the near-horizon regime —, but rather than turning the
higher levels complex (so dropped in a unitary theory), it prescribes a high-
energy spectrum that leads to a Hagedorn divergence in the naive partition
function above the dressed $\Delta=c/6$ energy level. It may be that this
Hagedorn behavior similarly screens the system from unphysical higher energy
levels — indeed in the fundamental string version of a Hagedorn transition,
there is a winding string condensate that has been argued to reduce the number
of degrees of freedom in the system Atick:1988si . In this work we will not
pursue this transition further, returning our attention to the microstates
captured in the prescription of §2-3. The simplest way to think about this new
theory is to imagine following the steps of the matching in (25)–(28) that
related the $dS_{3}$ vacuum to the $r_{h}=\ell$ BTZ black hole, but for a
black hole with $r_{h}>\ell$. The only role that the precise value of $r_{h}$
played in the matching above was in ensuring that on the $dS$ side the horizon
had the right area — which is fixed by the global smoothness of the solution,
So, the cost we pay for increasing $r_{h}$ on the BTZ side is to introduce a
pair of conical _excesses_ at the poles of $dS$; the Penrose diagram remains
identical to that in Fig. 1.

We find for the energies

$\mathcal{E}=\frac{1}{\pi
y}\left\\{1+\sqrt{-1+4\pi^{2}y\left(\Delta-\frac{c}{12}\right)+4\pi^{4}J^{2}y^{2}}\right\\}\,.$
(45)

In terms of a trajectory, this corresponds to flipping the value of $\eta$ at
a state-dependent value

$y_{0}=\frac{1}{4\pi^{2}(\Delta-c/12)}\,.$ (46)

The bulk description of these states in the case $J=0$ is the patch

$\displaystyle
ds^{2}=-\left(1-\frac{\rho^{2}}{\ell^{2}}\right)d\tau^{2}+\frac{d\rho^{2}}{1-\frac{\rho^{2}}{\ell^{2}}}+\rho^{2}d\phi^{2},\qquad\qquad\rho\in\left[\rho_{c},\ell\right],\,\phi\sim\phi+2\pi\mu,$
(47)

where

$\displaystyle\rho_{c}=\frac{\ell}{\mu}\sqrt{\frac{3}{c\pi^{2}y}}=\frac{L}{2\pi\mu},\qquad\qquad\mu=\frac{1}{\sqrt{(4\pi^{2}y)(c/12)-(\beta/L)^{2}}}.$
(48)

The parameter $\mu$, the weight of the conical defect, can be fixed by
calculating the Brown-York stress tensor, giving

$\Delta-\frac{c}{12}=\frac{c}{12}\mu^{2}\qquad\qquad\left(\Delta>\frac{c}{12}\right).$
(49)

For $\Delta>c/6$, this would correspond to a conical excess sourced at
$\rho=0$. By increasing $\Delta$, this would formally give an arbitrarily
negative mass source at this point in the bulk. The reason this is not an out-
and-out inconsistency is that these negative-mass sources are outside the
patch described by our putative boundary theory. Regardless, since our
interest is a theory of $dS_{3}$ we would hope that they are excluded in some
fashion.

As it happens, this may very well be the case. To see this, let us ask whether
there is any reason to consider this alternative theory at all, given the
double strangeness of a state-dependent $y_{0}$161616We should clarify that
this is an aesthetic problem; a theory with the above energy levels is a
perfectly valid solution to the flow equation. It is also connected to the
seed CFT in a novel and interesting way. and conical excesses. The answer is
that these conical excesses naturally arise when trying to extend the
considerations of §4 beyond boundary modular parameters satisfying the global
$dS_{3}$ constraint (40).171717The weight of the conical singularity in the
canonical ensemble is  $\mu^{2}=\frac{1}{(y/y_{0})-(\beta/L)^{2}}.$ Happily,
studying this partition function further reveals the existence of a Hagedorn
transition that intervenes before these naive (and problematic) states enter.

Indeed, let us check the Hagedorn temperature for the naive spectrum (45). The
partition function takes the form181818The density of states is the Cardy
density of states in terms of the seed dimension $\Delta$; this can be checked
by an inverse Laplace transform of $I_{CH}$ on a background that includes
these excesses.

$Z(\beta,L)=\sum_{\Delta}\rho(\Delta)\exp(-\beta{\cal E}/L)$ (50)

where in this expression $\rho(\Delta)$ is the number of states at the energy
level $\frac{2\pi}{L}(\Delta-\frac{c}{12})$ in the seed theory (this number
stays constant throughout the deformation at the level of pure gravity). In
order to locate the Hagedorn transition temperature, we must check convergence
of this expression. For fixed deformation parameter $y$ and a given $\beta$,
the question of convergence is determined by the behavior of the summand (or
integrand in a continuum treatment) at large $\Delta$, where the density of
states take the Cardy form (as it does for all levels starting at
$\Delta=c/6)$, and where the $\Delta$ term inside the square root dominates in
the energies (45). So the summand/integrand boils down to

$\exp\left(2\pi\sqrt{\frac{c}{3}\left(\Delta-\frac{c}{12}\right)}-2\frac{\beta}{L}\sqrt{\frac{\Delta-\frac{c}{12}}{y}}\right)$
(51)

meaning that there is a Hagedorn transition,

$Z<\infty\qquad\text{when}\qquad\left.\frac{\beta}{L}>\frac{\beta}{L}\right|_{Hagedorn}=\pi\sqrt{\frac{cy}{3}}=\sqrt{\frac{y}{y_{0}}}.$
(52)

Since we don’t have the constraint (40) in this case, this system exhibits a
Hawking-Page transition at

$I_{CH}<I_{P}\qquad\text{when}\qquad\left.\frac{\beta}{L}\right|_{Hawking-
Page}<1.$ (53)

Since we want to describe the CH patch, we need to check that the inequality
(52) is consistent with this one. Clearly, the answer to this question depends
on $y$. In the trajectory defined here, $y$ ranges from $y_{0}$ where we
switch to the $T\bar{T}+\Lambda_{2}$ part of the trajectory, to some
$y_{final}$ which we will take to be state-independent.

First, note that if we took $y_{final}\to\infty$ (to capture the full static
patch with identifications) it would lead to a vanishing Hagedorn temperature.
To avoid this, let us consider a $y_{final}$ which is common to all levels,
and as small as possible given that. This is a $y_{f}$ which is equal to the
$y_{0}$ value for the $\Delta=c/6$ level,

$y_{final}\equiv
y_{0}=\frac{1}{4\pi^{2}(\frac{c}{6}-\frac{c}{12})}=\frac{3}{c\pi^{2}}$ (54)

and we find

$\left.\frac{\beta}{L}\right|_{Hagedorn}=\pi\sqrt{\frac{c~{}y_{final}}{3}}=\pi\sqrt{\frac{c~{}y_{0}}{3}}=1=\left.\frac{\beta}{L}\right|_{HP}\,.$
(55)

As a result, the partition function of this version of a $2d$
$T\bar{T}+\Lambda_{2}$ trajectory simply does not converge above the
HP=Hagedorn transition, indicating a phase transition.191919Indeed, this value
of $y_{final}$ is minimal: from this analysis, the partition function
converges for $\frac{\beta}{L}\geq\pi\sqrt{\frac{c\lambda}{3L^{2}}}$. Since we
used S-invariance in the approach of this section, we also need to demand this
inequality under the exchange $\beta\leftrightarrow L$,
$\frac{L}{\beta}\geq\pi\sqrt{\frac{c\lambda}{3\beta^{2}}}$. This last
inequality requires $\frac{c\,\pi^{2}}{3}y\leq 1\;\Rightarrow\;y\leq y_{0}$
This phase transition is complicated in the analogous case of fundamental
string theory Atick:1988si , but it may provide a self-censorship of the
problematic high energy spectrum (45).

Further, since we are interested in describing the CH patch with no conical
excesses, we must compare with the global $dS_{3}$ condition (40). Clearly,

$\left.\frac{\beta}{L}\right|_{Hagedorn}=\sqrt{\frac{y}{y_{0}}}>\left.\frac{\beta}{L}\right|_{global}=\sqrt{\frac{y}{y_{0}}-1},$
(56)

and so the partition function doesn’t converge in the requisite regime. It is
of course possible that the scale of new physics is at high enough temperature
that the global $dS_{3}$ regime isn’t screened, but we cannot check this with
the tools at our disposal.

Thus, we are led to focus on the prescription in §2-3 accounting for the
microstates as dressed $\Delta=c/6$ levels (related to modular transformations
and the Hawking-Page transition as summarized in §4).

##  6 Discussion

In this work, we exhibited a concrete set of microstates accounting for the
Gibbons-Hawking entropy $S_{GH}$. The microstates, a band of energy levels
obtained as dressed $\Delta\simeq c/6$ states in the seed CFT, are accessible
at the level of pure gravity, which is formulated by the
$T\bar{T}+\Lambda_{2}$ deformation. This theory also reconstructs the de
Sitter geometry as in GST LLST . The finiteness of $S_{GH}$ fits with the
finiteness of the real dressed spectrum of the theory. The results extend to
corrections such as those treated in Anninos:2020hfj ; Shyam:2021ciy  as a
dressing of the corresponding AdS/CFT results in Sen:2012dw ; it will be
interesting to connect also to recent studies such as Anninos:2021ihe
,hikida2021holography  identifying microphysical degrees of freedom from
another approach, as well as top-down methods such as Dong:2010pm .

The theoretical control we have in this work is based on the universality and
solvability of the $T\bar{T}+\Lambda_{2}$ theory, in the same sense as in the
original $T\bar{T}$ deformation Zamolodchikov:2004ce ; Smirnov:2016lqw ;
Dubovsky:2018bmo . This universal and solvable theory captures the physics
that is available in the pure gravity sector. A full accounting of finer
structure requires additional generalizations, as introduced e.g. for bulk
scalar fields in Hartman:2018tkw . Even at the level of pure $T\bar{T}$ and
$T\bar{T}+\Lambda_{2}$, although calculable in terms of the dressed energy
spectrum, these theories remain somewhat mysterious. We may regard the
holographic matching of energies and entropies – even in the case of bulk de
Sitter spacetime – as ample motivation to improve our understanding of this
space of deformed theories. It explicitly captures essential features of the
physics, but it may have limitations in its regime of validity; we comment on
this further below in the context of late-time physics in cosmological quantum
gravity.

###  6.1 Remarks and future directions

Finally, in the remainder of this section we will comment on additional
aspects of our theory and future directions.

####  6.1.1 Other dimensionalities

We have exhibited the Gibbons-Hawking microstates specifically for a $dS_{3}$
bulk theory in the bulk of this paper. At least at large $N$, similar methods
apply in other dimensionalities: the $\Lambda_{2}$ deformation generalizes to
a similar $\Lambda_{d}$ contribution to the appropriate analogue of the
$T\bar{T}$ deformation, providing a building block for formulating $dS_{d+1}$.
Analogues of the $T\bar{T}$ deformation in other dimensionalities have been
treated in a number of works including Hartman:2018tkw ; Gross:2019ach ;
Iliesiu:2020zld ; Stanford:2020qhm ; Griguolo:2021wgy .

In all dimensionalities it was noted in Dong:2018cuv  that black holes at the
Hawking-Page transition at large $c$ account for the Gibbons-Hawking entropy
for de Sitter with the same horizon size; moreover at the HP level the warp
factor in the $AdS_{d+1}$ geometry matches that of $dS_{d+1}$, with both
foliated by $dS_{d}$ slices. The static patch Hamiltonian is the modular
Hamiltonian for the dS/dS correspondence Alishahiha:2004md ; GST ; LLST , so
the calculations are closely related. It would be interesting to analyze this
further.202020including the question of the fate of modular invariance
properties – which played a role at intermediate steps in our analysis – in
higher dimensions Shaghoulian:2016gol  It is particularly important to study
the case of $dS_{4}$ and incorporate the physics of black holes in de Sitter,
a topic studied recently in e.g. Maldacena:2019cbz ; Susskind:2021dfc  with an
eye toward holography.

####  6.1.2 Chaos and complexity

It was recently suggested in Susskind:2021esx  that a dual to de Sitter
requires a hyper-scrambling behavior, and must generate rapidly growing
complexity.212121The infinite complexity of de Sitter surprising nobody who
has worked on the subject. To the extent that this follows from the gravity-
side calculation given there and in Chapman:2021eyy , this was connected to
the Hamiltonian needing to be non-k-local in terms of the fundamental degrees
of freedom. This means that at fixed number of degrees of freedom, the
Hamiltonian must contain more than a certain monomial power of the fundamental
degrees of freedom Susskind:2021esx .

In this regard, let us note that the $T\bar{T}+\Lambda_{2}$ deformation is
irreducibly nonlinear (and hence non-k-local) in terms of the seed CFT degrees
of freedom. With $\eta=-1$ (corresponding to nonzero $\Lambda_{2}$), the
square root formula for the dressed energies – as well as the dressed
classical action and Hamiltonian – does not admit an expansion in the CFT
variables. (During the earlier $T\bar{T}$ segment of the trajectory, the
Taylor expansion of the square root is available, and has radius of
convergence 1, becoming highly nonlinear as we approach the matching point.)

Since the $T\bar{T}+\Lambda_{2}$ theory embodies the geometry of de Sitter
space, having passed significant and detailed tests in matching the Brown-York
energies and entanglement entropies GST ; LLST , it would be interesting to
probe it further using these ideas about scrambling and chaos. This approach
is versatile and applies to truly large-radius gravity, e.g. for the seed CFT
being the D1-D5 system, although with additional elements beyond
$T\bar{T}+\Lambda_{2}$ Hartman:2018tkw  required for approximate bulk
locality.

However, in this analysis we should take into account the subtleties with
arbitrarily late time physics in cosmological quantum gravity. We will comment
on that briefly in §6.1.4.

####  6.1.3 Joining the cosmic horizon and pole patches: global bulk $dS$ and
its symmetries

As described above, the culmination of the new trajectory we defined here is
to move the boundary of the cosmic horizon patch to the pole. This yields in
the bulk the full static patch, with boundary at the observer position. This
may be viewed as a formulation conditioned on such an observer. But it is also
possible to join together the cosmic horizon and pole patches, identifying
their shared boundary and path integrating over its geometry. In the doubled
system depicted in figure 3, this joined system covers global $dS_{3}$. As
such, it should contain the full symmetries of global de Sitter at the level
of the combined $T\bar{T}+\Lambda_{2}$ systems. Understanding the re-emergence
of the full symmetry group is an important direction, at the appropriate level
of approximation (given the possibility of non-perturbative decays of de
Sitter spacetime).

####  6.1.4 Late times

There are strong indications that de Sitter eventually decays at late times, a
manifest feature of string compactifications (including those built by
uplifting AdS/CFT models Dong:2010pm ; DeLuca:2021pej ) since the effective
potential decays to zero at weak curvature. This is not immediately evident in
the $T\bar{T}+\Lambda_{2}$ theory, which is solvable at finite $c$. But this
is not a contradiction: the details of the decay arise in the bulk scalar
sector of the theory, which is model-dependent and involves the further
generalization of $T\bar{T}$ formulated e.g. in Hartman:2018tkw . Both the
covariant entropy bound Bousso:1999xy  and string-theoretic counts of
microphysical degrees of freedom Dong:2011uf  indicate that the dimension of
the Hilbert space relevant to the full decaying system increases,
asymptotically approaching infinity. It would be interesting to extend the
$T\bar{T}+\Lambda_{2}$ approach to include this late-time physics.

Even before considering this more general late-time behavior, it is
interesting to consider situations where approximately de Sitter expansion
does not extend arbitrarily far in the past. For example, depending on the
quantum gravitational resolution of spacelike singularities, it may happen
that consistent histories exist which do not include the de Sitter neck (where
its spatial sphere reaches its minimal value at the moment of time symmetry),
and the system instead starts from a spacelike classically singular initial
condition. Such cases, if they are consistent in quantum gravity, may require
additional states with large entropy.222222Recent works exploring interesting
aspects of the particular case of no boundary wavefunctions, and their
implications for de Sitter entropy include Penington:2019kki  and Chen:2020tes
.

In any case, for the present work we focused on the observer patch in
$dS_{3}$, whose finite entropy (1) precisely matches the real spectrum of our
simple solvable $T\bar{T}+\Lambda$ deformation. This motivates further work in
many directions.

## Acknowledgements

We would like to thank Victor Gorbenko for many discussions, including
stressing early on the solvability and universality of $T\bar{T}+\Lambda_{2}$
on the cylinder, and for comments on the manuscript. We would also like to
thank Edgar Shaghoulian and Zhenbin Yang for useful discussions.

The work of E.S. is supported in part by a Simons Investigator award and
National Science Foundation grant PHY-1720397. V.S. is supported by The Branco
Weiss Fellowship - Society in Science, administered by the ETH Zürich. G.T. is
supported by CONICET, CNEA, UNCuyo and by ANPCYT PICT grant 2018-2517. E.A.C.
is supported by the US NSF Graduate Research Fellowship under Grant
DGE-1656518. E.A.M is supported by a Kadanoff fellowship at the Kadanoff
Center of the University of Chicago, and would also like to acknowledge the
support of the Jones Endowment for Physics Research. RMS is supported by STFC
consolidated grant ST/T000694/1, the Isaac Newton Trust Grant “Quantum
Cosmology and Emergent Time” and the AFOSR grant “Tensor Networks and
Holographic Spacetime.”

##  Appendix A Conventions

We consider the following conventions, in Euclidean signature (euclidean signs
as in Kraus:2018xrn ):

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{g}}\frac{\delta}{\delta g^{\mu\nu}}\,\log Z$
$\displaystyle T\bar{T}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{8}(T_{ab}T^{ab}-(T^{a}_{a})^{2})$
$\displaystyle\frac{\partial}{\partial\lambda}\,\log Z$ $\displaystyle=$
$\displaystyle-2\pi\,\int d^{2}x\,\sqrt{g}\,\langle
T\bar{T}\rangle+\frac{1-\eta}{2\pi\lambda^{2}}\int d^{2}x\,\sqrt{g}$
$\displaystyle T^{\mu}_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{c}{24\pi}R^{(2)}-4\pi\lambda
T\bar{T}+\frac{1}{\pi\lambda}(1-\eta)$ $\displaystyle\mathcal{E}$
$\displaystyle=$ $\displaystyle
EL=(-T^{\tau}_{\tau})L^{2}=\frac{L^{2}}{\pi\lambda}\left(1\mp\sqrt{\eta-4\pi^{2}\frac{\lambda}{L^{2}}\left(\Delta-\frac{c}{12}\right)+4\pi^{4}J^{2}\frac{\lambda^{2}}{L^{4}}}\,\right)$
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
h+\tilde{h}\;,\;J=h-\tilde{h}\,.$ (57)

Here $L=2\pi r$ is the length of the circle, the energy density
$\rho=-T^{\tau}_{\tau}$, and $\lambda$ has energy dimension $-2$. We will
often employ the dimensionless combination

$y=\frac{\lambda}{L^{2}}\,,$ (58)

which transforms under modular transformations. In particular, the
S-transformation having $L$ explicit is just $\beta\leftrightarrow L$, where
$\beta$ is the length of the thermal circle; so
$y\to\frac{L^{2}}{\beta^{2}}y$.

In Lorentzian signature $(-,+)$ this is:

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{2}{\sqrt{-g}}\frac{\delta}{\delta g^{\mu\nu}}\,\log
S_{eff}$ $\displaystyle T\bar{T}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{8}(T_{ab}T^{ab}-(T^{a}_{a})^{2})$
$\displaystyle-\frac{\partial}{\partial\lambda}\,S_{eff}$ $\displaystyle=$
$\displaystyle-2\pi\,\int d^{2}x\,\sqrt{-g}\,\langle
T\bar{T}\rangle+\frac{1-\eta}{2\pi\lambda^{2}}\int d^{2}x\,\sqrt{-g}$
$\displaystyle T^{\mu}_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{c}{24\pi}R^{(2)}-4\pi\lambda
T\bar{T}+\frac{1}{\pi\lambda}(1-\eta)$ $\displaystyle\mathcal{E}$
$\displaystyle=$ $\displaystyle
EL=(+T^{\tau}_{\tau})L^{2}=\frac{L^{2}}{\pi\lambda}\left(1\mp\sqrt{\eta-4\pi^{2}\frac{\lambda}{L^{2}}\left(\Delta-\frac{c}{12}\right)+4\pi^{4}J^{2}\frac{\lambda^{2}}{L^{4}}}\,\right)$
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
h+\tilde{h}\;,\;J=h-\tilde{h}\,.$ (59)

where again $L=2\pi r$ is the length of the circle, $\lambda$ has energy
dimension $-2$, and we will define $y=\lambda/L^{2}$. The energy density in
Lorentzian signature is now $\rho=+T^{\tau}_{\tau}$.

The energies in the seed CFT are

${\cal E}_{0}=E_{0}L={2\pi}(\Delta-\frac{c}{12})\,.$ (60)

##  Appendix B Euclidean actions for the Pole patch and the Cosmic Horizon
patch: exchange of dominance at $\beta=L$

In this appendix, we compute the Euclidean gravitational actions for the pole
patch and the Cosmic Horizon patch, and show that the two saddles switch
dominance at $\beta=L$.

The Euclidean version of the action (14) is

$I_{E}=-\frac{1}{16\pi
G_{N}}\int_{\mathcal{M}_{3}}d^{3}x\sqrt{g}\left(R+\frac{2}{\ell^{2}}\eta\right)-\frac{1}{8\pi
G_{N}}\int_{\partial\mathcal{M}_{3}}d^{2}x\sqrt{g}\left(K-\frac{1}{\ell}\right).$
(61)

The Pole patch metric with $\tau_{E}=i\tau$ is

$ds_{3}^{2}=\left(\frac{\ell^{2}-r^{2}}{{\ell^{2}}}\right)d\tau_{E}^{2}+\left(\frac{{\ell^{2}}}{\ell^{2}-r^{2}}\right)dr^{2}+r^{2}d\phi^{2},$

with

$r\in\left[0,r_{c}\right],\quad\tau_{E}\sim\tau_{E}+\beta_{c},\quad\phi\sim\phi+2\pi$
(62)

where

$r_{c}=\ell\sqrt{\frac{3}{c\pi^{2}y}}=\frac{L}{2\pi},\quad\beta_{c}=\frac{\beta}{\sqrt{1-\frac{3}{c\pi^{2}y}}}=\frac{\beta}{\sqrt{1-\frac{r_{c}^{2}}{\ell^{2}}}}.$
(63)

From the near-horizon behavior derived above in (28) we see that
$\beta_{c}=2\pi\ell$.

The extrinsic curvature at the boundary $r=r_{c}$ is

$\displaystyle K_{{\tau_{E}}{\tau_{E}}}=-\frac{r_{c}}{\ell^{2}}$
$\displaystyle\sqrt{1-\frac{r_{c}^{2}}{\ell^{2}}},\quad
K_{\phi\phi}=r_{c}\sqrt{1-\frac{r_{c}^{2}}{\ell^{2}}},\quad
K=\frac{1-\frac{2r_{c}^{2}}{\ell^{2}}}{r_{c}\sqrt{1-\frac{r_{c}^{2}}{\ell^{2}}}}$
(64)
$\displaystyle\sqrt{g}\left(K-\frac{1}{\ell}\right)=1-\frac{2r_{c}^{2}}{\ell^{2}}-\frac{r_{c}}{\ell}\sqrt{1-\frac{r_{c}^{2}}{\ell^{2}}}\,.$
(65)

The first term in (61) is

$-\frac{1}{16\pi
G_{N}}\int_{0}^{\beta_{c}}d\tau_{E}\int_{0}^{2\pi}d\phi\int_{0}^{r_{c}}dr\,\frac{4r}{\ell^{2}}=-\frac{\beta_{c}}{4G_{N}}\frac{r_{c}^{2}}{\ell^{2}}$
(66)

and the second term in (61) is

$-\frac{1}{8\pi
G_{N}}\int_{0}^{\beta_{c}}d\tau_{E}\int_{0}^{2\pi}d\phi\sqrt{g}\left(K-\frac{1}{\ell}\right)=-\frac{\beta_{c}}{4G_{N}}\left(1-\frac{2r_{c}^{2}}{\ell^{2}}-\frac{r_{c}}{\ell}\sqrt{1-\frac{r_{c}^{2}}{\ell^{2}}}\right).$
(67)

Adding the two terms, we get

$I_{Pole}=\frac{1}{4G_{N}}\frac{\beta}{\sqrt{1-\frac{r_{c}^{2}}{\ell^{2}}}}\left(\frac{r_{c}}{\ell}\sqrt{1-\frac{r_{c}^{2}}{\ell^{2}}}-\left(1-\frac{r_{c}^{2}}{\ell^{2}}\right)\right)=\frac{\beta
r_{c}}{4G_{N}\ell}\left(1-\sqrt{\frac{\ell^{2}}{r_{c}^{2}}-1}\right).$ (68)

Using the dictionary (18) and (63), we get

$I_{Pole}=\frac{\beta
L}{\pi\lambda}\left(1-\sqrt{\frac{c\pi^{2}\lambda}{3L^{2}}-1}\right)=\frac{\beta
L}{\pi\lambda}\left(1-\sqrt{\frac{c\pi^{2}y}{3}-1}\right)=\beta E_{vac}(L)$
(69)

for the pole patch. Note that, when the pole patch can be embedded in global
$dS_{3}$, (40) gives us $I_{P}=\frac{\beta L(1-\beta/L)}{\pi\lambda}$.

The Cosmic Horizon patch captures the other side of the Dirichlet wall. Here
we use a different coordinate notation ($r\to\rho$) to distinguish our two
cases.

$ds^{2}=\left(\frac{\ell^{2}-\rho^{2}}{\ell^{2}}\right)d\tau_{E}^{2}+\left(\frac{\ell^{2}}{\ell^{2}-\rho^{2}}\right)d\rho^{2}+\rho^{2}d\phi^{2}$

with

$\rho\in\left[\rho_{c},\ell\right],\quad\tau_{E}\sim\tau_{E}+\beta_{c},\quad\phi\sim\phi+2\pi$
(70)

where

$\displaystyle\rho_{c}=\ell\sqrt{\frac{3}{c\pi^{2}y}}=\frac{L}{2\pi},\qquad\beta_{c}=\frac{\beta}{\sqrt{1-\frac{\rho_{c}^{2}}{\ell^{2}}}}\,.$
(71)

For the CH patch, the extrinsic curvature at $\rho=\rho_{c}$ has different
signs from (64):

$\displaystyle K_{{\tau_{E}}{\tau_{E}}}=\frac{\rho_{c}}{\ell^{2}}$
$\displaystyle\sqrt{1-\frac{\rho_{c}^{2}}{\ell^{2}}},\quad
K_{\phi\phi}=-\rho_{c}\sqrt{1-\frac{\rho_{c}^{2}}{\ell^{2}}},\quad
K=\frac{-1+\frac{2\rho_{c}^{2}}{\ell^{2}}}{\rho_{c}\sqrt{1-\frac{\rho_{c}^{2}}{\ell^{2}}}}$
(72)
$\displaystyle\sqrt{g}\left(K-\frac{1}{\ell}\right)=-1+\frac{2\rho_{c}^{2}}{\ell^{2}}-\frac{\rho_{c}}{\ell}\sqrt{1-\frac{\rho_{c}^{2}}{\ell^{2}}}\,.$
(73)

The first term in (61) for the CH patch is

$-\frac{1}{16\pi
G_{N}}\int_{0}^{\beta_{c}}d\tau_{E}\int_{0}^{2\pi}d\phi\int_{\rho_{c}}^{\ell}d\rho\,\frac{4\rho}{\ell^{2}}=-\frac{\beta_{c}}{4G_{N}}\frac{\ell^{2}-\rho_{c}^{2}}{\ell^{2}}$
(74)

and the second term is

$-\frac{1}{8\pi
G_{N}}\int_{0}^{\beta_{c}}d\tau_{E}\int_{0}^{2\pi}d\phi\sqrt{g}\left(K-\frac{1}{\ell}\right)=-\frac{\beta_{c}}{4G_{N}}\left(-1+\frac{2\rho_{c}^{2}}{\ell^{2}}-\frac{\rho_{c}}{\ell}\sqrt{1-\frac{\rho_{c}^{2}}{\ell^{2}}}\right)\,.$
(75)

Adding the two terms,

$I_{CH}=\frac{\beta_{c}}{4G_{N}}\left(-\frac{\rho_{c}^{2}}{\ell^{2}}+\frac{\rho_{c}}{\ell}\sqrt{1-\frac{\rho_{c}^{2}}{\ell^{2}}}\right)=\frac{\beta\rho_{c}}{4G_{N}\ell}\left(1-\sqrt{\left(1-\frac{\rho_{c}^{2}}{\ell^{2}}\right)^{-1}-1}\right)\,.$
(76)

Again, using the dictionary (18) and (71) with the relation
$(\beta/L)^{2}=y/y_{0}-1$, we get:

$I_{CH}=\frac{\beta
L}{\pi\lambda}\left(1-\sqrt{\frac{c\pi^{2}y}{3(\beta/L)^{2}}-1}\right)=\frac{\beta
L}{\pi\lambda}\left(1-\sqrt{\frac{c\pi^{2}\lambda}{3\beta^{2}}-1}\right)=LE_{vac}(\beta).$
(77)

where we used the relation:

$\left(1-\frac{\rho_{c}^{2}}{\ell^{2}}\right)^{-1}=\frac{y}{y-y_{0}}=\frac{y}{y_{0}(\beta/L)^{2}}.$

Again, in the case in which the bulk can be embedded in global $dS_{3}$, we
find by (40) that $I_{CH}=\frac{\beta L(1-L/\beta)}{\pi\lambda}$.

From (69) and (77), we can see that the Pole patch dominates for $\beta>L$ for
which $(I_{Pole}<I_{CH})$, and the CH patch dominates otherwise.

##  Appendix C Proof of relationship (41)

To prove (41), we first recall that

$\displaystyle S_{Cardy}(\Delta=c/6)$
$\displaystyle=2\pi\sqrt{\frac{c}{3}\left(\frac{c}{6}-\frac{c}{12}\right)}=\frac{\pi
c}{3}=\frac{1}{\pi y_{0}}$ $\displaystyle-\beta E_{\Delta=c/6}(L)$
$\displaystyle=-\frac{\beta
L}{\pi\lambda}\left(1+\sqrt{-1+\frac{\pi^{2}c}{3}y}\right)=-\frac{\beta/L}{\pi
y}\left(1+\sqrt{-1+\frac{y}{y_{0}}}\right)$ $\displaystyle-LE_{vac}(\beta)$
$\displaystyle=-\frac{\beta
L}{\pi\lambda}\left(1-\sqrt{-1+\frac{\pi^{2}cy}{3}\frac{L^{2}}{\beta^{2}}}\right)=-\frac{\beta/L}{\pi
y}\left(1-\sqrt{-1+\frac{y}{y_{0}}\frac{L^{2}}{\beta^{2}}}\right)\,.$

Using (40), we can rewrite these as:

$\displaystyle S_{Cardy}(\Delta=c/6)$ $\displaystyle=\frac{1}{\pi
y}\,\frac{y}{y_{0}}=\frac{1}{\pi y}\left(1+\frac{\beta^{2}}{L^{2}}\right)$
(78) $\displaystyle-\beta E_{\Delta=c/6}(L)$
$\displaystyle=-\frac{\beta/L}{\pi y}\left(1+\frac{\beta}{L}\right)$
$\displaystyle-LE_{vac}(\beta)$ $\displaystyle=-\frac{\beta/L}{\pi
y}\left(1-\frac{L}{\beta}\right)=\frac{1}{\pi
y}\left(1-\frac{\beta}{L}\right)$

Now, from (78), we have that

$S_{Cardy}(\Delta=c/6)-\beta E_{\Delta=c/6}(L)=\frac{1}{\pi
y}\left(1+\frac{\beta^{2}}{L^{2}}-\frac{\beta^{2}}{L^{2}}-\frac{\beta}{L}\right)=\frac{1}{\pi
y}\left(1-\frac{\beta}{L}\right)=-LE_{vac}(\beta),$ (79)

which is (40).

## References

  * (1) V. Gorbenko, E. Silverstein and G. Torroba, _dS/dS and $\textit{T}\overline{T}$_, _JHEP_ 03 (2019) 085 [1811.07965].
  * (2) A. Lewkowycz, J. Liu, E. Silverstein and G. Torroba, _$\textit{T}\overline{T}$ and ee, with implications for (A)dS subregion encodings_, _JHEP_ 04 (2020) 152 [1909.13808].
  * (3) A.B. Zamolodchikov, _Expectation value of composite field T anti-T in two-dimensional quantum field theory_ , hep-th/0401146.
  * (4) F.A. Smirnov and A.B. Zamolodchikov, _On space of integrable quantum field theories_ , _Nucl. Phys._ B915 (2017) 363 [1608.05499].
  * (5) S. Dubovsky, R. Flauger and V. Gorbenko, _Solving the Simplest Theory of Quantum Gravity_ , _JHEP_ 09 (2012) 133 [1205.6805].
  * (6) A. Cavaglia, S. Negro, I.M. Szecsenyi and R. Tateo, _$T\bar{T}$ -deformed 2D Quantum Field Theories_, _JHEP_ 10 (2016) 112 [1608.05534].
  * (7) S. Dubovsky, V. Gorbenko and M. Mirbabayi, _Asymptotic fragility, near AdS 2 holography and $T\overline{T}$_, _JHEP_ 09 (2017) 136 [1706.06604].
  * (8) V. Gorbenko, E. Silverstein and G. Torroba, _dS/dS and $T\overline{T}$_, _JHEP_ 03 (2019) 085 [1811.07965].
  * (9) L. McGough, M. Mezei and H. Verlinde, _Moving the CFT into the bulk with $T\overline{T}$_, _JHEP_ 04 (2018) 010 [1611.03470].
  * (10) P. Kraus, J. Liu and D. Marolf, _Cutoff AdS 3 versus the $T\overline{T}$ deformation_, _JHEP_ 07 (2018) 027 [1801.02714].
  * (11) T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, _Holography at finite cutoff with a $T^{2}$ deformation_, 1807.11401.
  * (12) W. Donnelly and V. Shyam, _Entanglement entropy and $\textit{T}\overline{T}$ deformation_, _Phys. Rev. Lett._ 121 (2018) 131602 [1806.07444].
  * (13) W. Donnelly and V. Shyam, _Entanglement entropy and $T\overline{T}$ deformation_, 1806.07444.
  * (14) V. Shyam, _$\textit{T}\overline{T}+\mathrm{\Lambda}_{2}$ deformed cft on the stretched $\textit{dS}_{3}$ horizon_, 2106.10227.
  * (15) X. Dong, E. Silverstein and G. Torroba, _De sitter holography and entanglement entropy_ , _JHEP_ 07 (2018) 050 [1804.08623].
  * (16) D. Anninos, F. Denef, Y.T.A. Law and Z. Sun, _Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions_ , 2009.12464.
  * (17) G.W. Gibbons and S.W. Hawking, _Cosmological Event Horizons, Thermodynamics, and Particle Creation_ , _Phys. Rev. D_ 15 (1977) 2738.
  * (18) T. Hartman, C.A. Keller and B. Stoica, _Universal spectrum of 2d conformal field theory in the large c limit_ , _Journal of High Energy Physics_ 2014 (2014) .
  * (19) B. Mukhametzhanov and A. Zhiboedov, _Modular invariance, tauberian theorems and microcanonical entropy_ , _JHEP_ 10 (2019) 261 [1904.06359].
  * (20) D. Anninos, S.A. Hartnoll and D.M. Hofman, _Static Patch Solipsism: Conformal Symmetry of the de Sitter Worldline_ , _Class. Quant. Grav._ 29 (2012) 075002 [1109.4942].
  * (21) J. Polchinski, _String theory. Vol. 1: An introduction to the bosonic string_ , Cambridge University Press (2007).
  * (22) P. Horava and E. Witten, _Eleven-dimensional supergravity on a manifold with boundary_ , _Nucl. Phys. B_ 475 (1996) 94 [hep-th/9603142].
  * (23) T. Banks and W. Fischler, _The holographic spacetime model of cosmology_ , _Int. J. Mod. Phys. D_ 27 (2018) 1846005 [1806.01749].
  * (24) J. Cardy, _The $\textit{T}\overline{T}$ deformation of quantum field theory as random geometry_, _JHEP_ 10 (2018) 186 [1801.06895].
  * (25) E.A. Mazenc, V. Shyam and R.M. Soni, _A $\textit{T}\overline{T}$ Deformation for Curved Spacetimes from 3d Gravity_, 1912.09179.
  * (26) J. Cardy, _$T\bar{T}$ deformation of correlation functions_, _JHEP_ 12 (2019) 160 [1907.03394].
  * (27) J. Kruthoff and O. Parrikar, _On the flow of states under $T\overline{T}$_, 2006.03054.
  * (28) S. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, _$T\overline{T}$ partition function from topological gravity_, _JHEP_ 09 (2018) 158 [1805.07386].
  * (29) L.S. Brown and J.P. Cassidy, _Stress Tensors and their Trace Anomalies in Conformally Flat Space-Times_ , _Phys. Rev._ D16 (1977) 1712.
  * (30) V. Balasubramanian and P. Kraus, _A Stress tensor for Anti-de Sitter gravity_ , _Commun. Math. Phys._ 208 (1999) 413 [hep-th/9902121].
  * (31) A. Strominger and C. Vafa, _Microscopic origin of the Bekenstein-Hawking entropy_ , _Phys. Lett. B_ 379 (1996) 99 [hep-th/9601029].
  * (32) X. Dong, B. Horn, E. Silverstein and G. Torroba, _Micromanaging de Sitter holography_ , _Class. Quant. Grav._ 27 (2010) 245020 [1005.5403].
  * (33) R. Bousso, A. Maloney and A. Strominger, _Conformal vacua and entropy in de Sitter space_ , _Phys. Rev. D_ 65 (2002) 104039 [hep-th/0112218].
  * (34) A. Sen, _Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions_ , _JHEP_ 04 (2013) 156 [1205.0971].
  * (35) S. Carlip, _Logarithmic corrections to black hole entropy, from the cardy formula_ , _Classical and Quantum Gravity_ 17 (2000) 4175–4186.
  * (36) D. Anninos and E. Harris, _Three-dimensional de Sitter horizon thermodynamics_ , 2106.13832.
  * (37) E. Shaghoulian, _The central dogma and cosmological horizons_ , 2110.13210.
  * (38) E.A. Coleman, R.M. Soni and S. Yang, _In Preparation_ , .
  * (39) M. Alishahiha, A. Karch, E. Silverstein and D. Tong, _The dS/dS correspondence_ , _AIP Conf. Proc._ 743 (2005) 393 [hep-th/0407125].
  * (40) T. Hartman and J. Maldacena, _Time evolution of entanglement entropy from black hole interiors_ , _JHEP_ 05 (2013) 014 [1303.1080].
  * (41) J.D. Brown and J.W. York, Jr., _Quasilocal energy and conserved charges derived from the gravitational action_ , _Phys. Rev. D_ 47 (1993) 1407 [gr-qc/9209012].
  * (42) J.D. Brown and J.W. York, Jr., _The Microcanonical functional integral. 1\. The Gravitational field_ , _Phys. Rev. D_ 47 (1993) 1420 [gr-qc/9209014].
  * (43) S. Datta and Y. Jiang, _$\textit{T}\overline{T}$ deformed partition functions_, _JHEP_ 08 (2018) 106 [1806.07426].
  * (44) O. Aharony, S. Datta, A. Giveon, Y. Jiang and D. Kutasov, _Modular invariance and uniqueness of $\textit{T}\overline{T}$ deformed cft_, _JHEP_ 01 (2019) 086 [1808.02492].
  * (45) O. Aharony, S. Datta, A. Giveon, Y. Jiang and D. Kutasov, _Modular covariance and uniqueness of $\textit{J}\overline{T}$ deformed cfts_, _JHEP_ 01 (2019) 085 [1808.08978].
  * (46) P. Caputa, J. Kruthoff and O. Parrikar, _Building Tensor Networks for Holographic States_ , _JHEP_ 05 (2021) 009 [2012.05247].
  * (47) L. Susskind, _Entanglement and Chaos in De Sitter Holography: An SYK Example_ , 2109.14104.
  * (48) S. Chapman, D.A. Galante and E.D. Kramer, _Holographic Complexity and de Sitter Space_ , 2110.05522.
  * (49) J.J. Atick and E. Witten, _The Hagedorn Transition and the Number of Degrees of Freedom of String Theory_ , _Nucl. Phys. B_ 310 (1988) 291.
  * (50) Y. Hikida, T. Nishioka, T. Takayanagi and Y. Taki, _Holography in de sitter space via chern-simons gauge theory_ , 2021.
  * (51) D.J. Gross, J. Kruthoff, A. Rolph and E. Shaghoulian, _$T\overline{T}$ in AdS2 and Quantum Mechanics_, 1907.04873.
  * (52) L.V. Iliesiu, J. Kruthoff, G.J. Turiaci and H. Verlinde, _JT gravity at finite cutoff_ , _SciPost Phys._ 9 (2020) 023 [2004.07242].
  * (53) D. Stanford and Z. Yang, _Finite-cutoff JT gravity and self-avoiding loops_ , 2004.08005.
  * (54) L. Griguolo, R. Panerai, J. Papalini and D. Seminara, _Nonperturbative effects and resurgence in jackiw-teitelboim gravity at finite cutoff_ , _Phys. Rev. D_ 105 (2022) 046015 [2106.01375].
  * (55) E. Shaghoulian, _Modular Invariance of Conformal Field Theory on $S^{1}\texttimes S^{3}$ and Circle Fibrations_, _Phys. Rev. Lett._ 119 (2017) 131601 [1612.05257].
  * (56) J. Maldacena, G.J. Turiaci and Z. Yang, _Two dimensional Nearly de Sitter gravity_ , _JHEP_ 01 (2021) 139 [1904.01911].
  * (57) L. Susskind, _Black Holes Hint Towards De Sitter-Matrix Theory_ , 2109.01322.
  * (58) G.B. De Luca, E. Silverstein and G. Torroba, _Hyperbolic compactification of M-theory and de Sitter quantum gravity_ , 2104.13380.
  * (59) R. Bousso, _A Covariant entropy conjecture_ , _JHEP_ 07 (1999) 004 [hep-th/9905177].
  * (60) X. Dong, B. Horn, S. Matsuura, E. Silverstein and G. Torroba, _FRW solutions and holography from uplifted AdS/CFT_ , _Phys. Rev._ D85 (2012) 104035 [1108.5732].
  * (61) G. Penington, S.H. Shenker, D. Stanford and Z. Yang, _Replica wormholes and the black hole interior_ , 1911.11977.
  * (62) Y. Chen, V. Gorbenko and J. Maldacena, _Bra-ket wormholes in gravitationally prepared states_ , _JHEP_ 02 (2021) 009 [2007.16091].