aainstitutetext:  The Institute of Mathematical Sciences, IV Cross Road,
C.I.T. Campus, Taramani, Chennai, India 600113bbinstitutetext:  Homi Bhabha
National Institute, Training School Complex, Anushakti Nagar, Mumbai, India
400094

# Half-sided Translations and Islands

Krishna Jalan  Roji Pius<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recently Liu and Leutheusser argued that the operators in the interior of a
black hole can be reconstructed by the half-sided evolution of the non-local
operators in the boundary CFT which are dual to the bulk operators outside the
black hole horizon. However, for an AdS2 black hole in equilibrium with a
finite temperature bath, the continuous exchange of Hawking quanta, emitted by
the black hole, and the bath quanta give rise to an information paradox. The
resolution of this paradox requires transferring the ownership of the black
hole interior from the boundary CFT to the bath at late time. As a result, the
reconstruction method proposed by Liu and Leutheusser must cease to
reconstruct the black hole interior after the Page time using the boundary
CFT. In this paper, we demonstrate this by showing that the relevant half-
sided translations are trivial at late time.

##  1 Introduction

One of the major goal of quantum gravity is the description of the physics
near the singularity of a blackhole. The AdS/CFT duality provides a conformal
field theory (CFT) description of quantum gravity in asymptotically Anti-de
Sitter (AdS) spacetimes Maldacena:1997re . In principle, we can use the dual
boundary CFT to describe the physics near the singularity of an AdS black
hole. However, we must reconstruct the bulk degrees of freedom inside the
horizon of a black hole in terms of the boundary degrees of freedom
Banks:1998dd ; Hamilton:2006az ; Hamilton:2006fh ; Hamilton:2005ju ;
Hamilton:2007wj ; Balasubramanian:1999ri ; Bena:1999jv  in order to study the
physics inside a black hole using AdS/CFT. There are two interesting proposals
for the reconstruction of bulk fields inside a black hole horizon. The
proposal of Papadodimas and Raju requires introducing a “mirror operator ” for
each of the operators in the dual boundary CFT. The field inside the black
hole horizon is then expressed as a sum of linearly independent operators and
its mirror operators with appropriate weight factors Papadodimas:2012aq ;
Papadodimas:2013jku . On the other hand, the method proposed by Liu and
Leutheusser reconstructs bulk operators inside the horizon by the half-sided
evolution of the operators in the boundary CFT dual to bulk operators that are
outside the horizon Leutheusser:2021qhd ; Leutheusser:2021frk .

Both proposals are claimed to be capable of reconstructing the bulk fields
inside the horizon of one-sided black holes and eternal black holes. For an
eternal AdS2 black hole, the black hole interior belongs to the entanglement
wedge of the left and right boundaries. Therefore, the reconstruction of
fields behind the horizon of an eternal AdS2 black hole requires both the
right and the left dual CFTs denoted as CFTR, CFTL respectively
Maldacena:2001kr . The interior reconstruction using the mirror operators is
consistent with this demand, because the construction of the mirror operators
require the information of the entangled state of the dual left and right CFTs
that corresponds to the eternal black hole. Since the modular operators acts
both on the left and right CFTs, the reconstruction method using the half-
sided translation also depends on the left and right CFTs.

The interior of an eternal black hole does not always belong to the
entanglement wedge of the left and right dual CFTs. Such a situation is
present in an eternal AdS2 black hole in equilibrium with a finite temperature
bath. The continuous exchange of Hawking quanta emitted by black hole and the
bath quanta builds an ever-growing entanglement between the black hole and the
bath and give rise to a linear growth of von Neumann entropy for the black
hole with respect to the boundary time 111However, the classical geometry of
the eternal black hole is unaffected by this exchange due to the vanishing net
flow of stress tensor across the cut out boundary of the AdS2 region..
Eventually, the entropy of the black hole will grow beyond the thermodynamic
entropy of the black hole. The resulting information paradox can be resolved
by transferring the ownership of the interior of the black hole from the
combined system of CFTL and CFTR after the black hole Page time to the bath
Almheiri:2019yqk . Hence, the black hole interior can not be reconstructed
using the left and the right CFTs after the Page time. The goal of this paper
is to derive this fact using the bulk reconstruction proposal of Liu and
Leutheusser.

In this paper, we show that at late time, the half-sided translation defined
using the algebra of observables restricted to the gravitational region
outside the right horizon of the black hole is trivial. The bulk operators in
the right/left side of the black hole can be expressed as non-local operators
in CFTR/CFTL. Moreover, a bulk half-sided modular translation is dual to an
appropriate boundary half-sided modular translation. Therefore, our result can
be interpreted as the inability of the Liu-Leutheusser proposal to reconstruct
the interior bulk operators using boundary CFTs, the CFTR and the CFTL after
the Page time. This is in perfect agreement with the general principles of the
entanglement wedge reconstruction Dong:2016eik  and the discussion in
Engelhardt:2023xer .

##  2 The half-sided translation and the in-falling

The main ingredient of the Liu-Leutheusser proposal for reconstructing the
bulk operators in the black hole interior is the half-sided translation. In
this section, we shall briefly discuss the definition of a half-sided
translation, and describe how a primary field in an arbitrary CFT defined on
(1+1)-Minkowski spacetime can be translated along a null direction using the
half-sided translation.

###  2.1 The half-sided translation

Half-sided translation is an element of a continuous one-parametric unitary
group defined with respect to a Type III1 von Neumann algebra $\mathcal{M}$,
having a cyclic and separating vector $|\Omega\rangle$, and a subalgebra
$\mathcal{N}$ having the same cyclic and separating vector. Since
$|\Omega\rangle$ is a seprating vector of $\mathcal{M}$, it is not annihilated
by any of the operators in $\mathcal{M}$. The cyclic property of
$|\Omega\rangle$ enables us to construct a dense subspace of the Hilbert space
associated with $\mathcal{M}$ by acting the elements of $\mathcal{M}$ on
$|\Omega\rangle$. An element $U(s)$ of this unitary group is called a half-
sided translation. It has the following properties:

$\displaystyle i.$ $\displaystyle\quad
U(s)|\Omega\rangle=|\Omega\rangle\qquad\forall s\in\mathbb{R},$ $\displaystyle
ii.$ $\displaystyle\quad
U(s)=e^{-\text{i}Gs},~{}where~{}G~{}is~{}a~{}Hermitian~{}positive~{}operator,$
$\displaystyle iii.$ $\displaystyle\quad
U^{\dagger}(s)\mathcal{M}U(s)\subset\mathcal{M},\quad\forall s\leq 0,$
$\displaystyle iv.$
$\displaystyle\quad\mathcal{N}=U(-1)^{\dagger}\mathcal{M}U(-1).$ (1)

This structure exists only if $\mathcal{M}$ is a type III1 von Neumann algebra
borchers1998half . The half-sided translation $U(s)$ has the following
expression Borchers:2000pv  in terms of the modular operators
$\Delta_{\mathcal{M}}$ and $\Delta_{\mathcal{N}}$ associated with the algebras
$\mathcal{M}$ and $\mathcal{N}$ respectively:

$U(s)=\Delta_{\mathcal{M}}^{-\text{i}t}\Delta_{\mathcal{N}}^{\text{i}t},\quad\quad
s=e^{-2\pi t}-1\quad\forall~{}t\leq 0.$ (2)

Modular operator $\Delta_{\mathcal{M}}$ has the property that it preserves the
operator algebra $\mathcal{M}$, i.e. its action on any operator in
$\mathcal{M}$ produces another operator in $\mathcal{M}$. Moreover, it keeps
the cyclic and the separating vector invariant. Its acts both on $\mathcal{M}$
and it’s commutant $\mathcal{M}^{\prime}$, the set of all observables that
commute with all the elements in $\mathcal{M}$. In general,
$\Delta_{\mathcal{M}}$ can not be factorised into two operators
$\bar{\Delta}_{\mathcal{M}}$ and $\hat{\Delta}_{\mathcal{M}}$ which acts only
on $\mathcal{M}$ and $\mathcal{M}^{\prime}$ respectively.

Modular operator $\Delta_{\mathcal{M}}$ can be expressed as the exponential of
the modular Hamiltonian $\mathcal{K}_{\mathcal{M}}$ of the algebra
$\mathcal{M}$, $\Delta_{\mathcal{M}}=e^{-\mathcal{K}_{\mathcal{M}}}$. Since
$\mathcal{N}$ is a subalgebra of $\mathcal{M}$, their modular Hamiltonians
satisfy the following algebra Borchers:2000pv

$\left[\mathcal{K}_{\mathcal{M}},\mathcal{K}_{\mathcal{N}}\right]=-2\pi\text{i}\left(\mathcal{K}_{\mathcal{M}}-\mathcal{K}_{\mathcal{N}}\right).$
(3)

Using (2), (3) and the Baker-Campbell-Hausdorff formula we can then express
$U(s)$ as follows

$U(s)=e^{-\text{i}sG},\quad\quad s=e^{-2\pi t}-1,$ (4)

where the generator of the half-sided translation is given by

$G=\frac{1}{2\pi}(\mathcal{K}_{\mathcal{M}}-\mathcal{K}_{\mathcal{N}}).$

###  2.2 The null-translations in (1+1)-Minkowski spacetime

In this subsection, we shall demonstrate the the half-sided translations can
be used to translate a field in (1+1)-dimensional Minkowski spacetime along
null directions. Consider an arbitrary CFT having central charge $c$ in in
(1+1)-dimensional Minkowski spacetime. Assume that $\mathcal{M}$ is the set of
all CFT observables restricted to the right Rindler wedge of the Minkowski
spacetime, and $\mathcal{N}$ is the set of all observables restricted to a
subregion of the right Rindler wedge as shown in the figure 1. The cyclic and
separating state is given by the Minkowski vacuum. Unlike in the general case,
for an algebra $\mathcal{W}$ associated with a wedge region, it is possible to
factorise the modular operator $\Delta_{\mathcal{W}}$ into a piece that acts
on $\mathcal{W}$ and another piece that acts only on its commutant
$\mathcal{W}^{\prime}$. Since $\mathcal{M}$ is a wedge algebra, the modular
operator takes the form

$\Delta_{\mathcal{M}}=\hat{\Delta}_{\mathcal{M}}\otimes\bar{\Delta}_{\mathcal{M}},$
(5)

where $\bar{\Delta}_{\mathcal{M}}$ acts only on the operators in $\mathcal{M}$
and $\hat{\Delta}_{\mathcal{M}}$ acts only on the operators in
$\mathcal{M}^{\prime}$. The operator $\bar{\Delta}_{\mathcal{M}}$ and
$\hat{\Delta}^{-1}_{\mathcal{M}}$ can be understood as the reduced density
matrices $\rho_{\mathcal{M}}$ of the region $\mathcal{M}$ and
$\rho_{\mathcal{M}^{\prime}}$ of the region $\mathcal{M}^{\prime}$
respectively. Therefore, the modular operator $\Delta_{\mathcal{M}}$ is given
by

$\Delta_{\mathcal{M}}=\rho^{-1}_{\mathcal{M}^{\prime}}\otimes\rho_{\mathcal{M}}.$
(6)

Likewise, the modular operator $\Delta_{\mathcal{N}}$ becomes

$\Delta_{\mathcal{N}}=\rho^{-1}_{\mathcal{N}^{\prime}}\otimes\rho_{\mathcal{N}}.$
(7)

Hence, using (2) the half-sided translation operator can be expressed as

$U(s)=\rho_{\mathcal{M}^{\prime}}^{\text{i}t}\rho^{-\text{i}t}_{\mathcal{N}^{\prime}}\otimes\rho^{-\text{i}t}_{\mathcal{M}}\rho_{\mathcal{N}}^{\text{i}t},\quad\quad
s=e^{-2\pi t}-1\quad\forall~{}t\leq 0.$ (8)

$y^{-}=-\infty$$y^{+}=\infty$$y^{+}=-\infty$$y^{-}=\infty$$y^{-}=\infty$$\psi(P)$$\psi(P^{\prime})$$\mathcal{N}$$\mathcal{N}^{\prime}$$P_{\mathcal{M}}$$P_{\mathcal{N}}$$y^{+}=0$$y^{-}=0$
Figure 1: The half-sided translation is defined using the von Neumann algebra
$\mathcal{M}$ restricted to the right Rindler wedge and the subalgebra
$\mathcal{N}$ restricted to the blue shaded region of the (1+1)-dimensional
Minkowski spacetime. It translates a field $\psi$ from point $P$ to
$P^{\prime}$ along a null ray away from the Rindler horizon for $s<0$. This
half-sided translation is generated by the difference between two boost
generators $\mathcal{K_{M}}$, the modular Hamiltonian associated with the
algebra $\mathcal{M}$ and $\mathcal{K_{N}}$, the modular Hamiltonian
associated with the subalgebra $\mathcal{N}$. The blue dotted arrow shows the
trajectory of $\mathcal{K_{M}}$, the boost that fixes the point
$P_{\mathcal{M}}$, and the gray dotted arrow shows the trajectory of
$\mathcal{K_{N}}$, the boost that fixes the point $P_{\mathcal{N}}$. The
difference $\mathcal{K_{M}}-\mathcal{K_{N}}$ generates translation along null
direction represented by the red line.

The density matrix $\rho_{\mathcal{M}}$ can be prepared by performing the CFT
path integral over a complex plane, obtained by the Euclidean continuation of
the (1+1)-dimensional Minkowski spacetime, with a cut from $P_{\mathcal{M}}$
to $\infty$. The point $P_{\mathcal{M}}$ is the intersection point of the
regions where the operator algebras $\mathcal{M}$ and $\mathcal{M}^{\prime}$
are defined, figure 1. Similarly, the density matrix
$\rho_{\mathcal{M}^{\prime}}$ is obtained by performing the CFT path integral
over the plane with a cut from point $P_{\mathcal{M}}$ to $-\infty$. Hence,
the path integral representation of $\rho_{\mathcal{M}}^{\text{i}t}$ is given
by the Euclidean CFT path integral in a wedge having clockwise angle
$\theta=\text{i}t$. One of the edge of this wedge matches with the part of the
Cauchy slice between the points $P_{\mathcal{M}}$ and $\infty$. Therefore, the
operator $\rho_{\mathcal{M}}^{\text{i}t}$ replaced by the path integral over a
wedge Witten:2018zxz . Interestingly, the path integral over a wedge can be
replaced by the following conical twist field operator

$\mathcal{T}_{\theta}(z_{\mathcal{M}},\bar{z}_{\mathcal{M}})=J\left(\theta,z_{\mathcal{M}},\bar{z}_{\mathcal{M}}\right)~{}e^{\theta\int_{\mathcal{C}}(z-z_{{\mathcal{M}}})T(z)dz+c.c.},$
(9)

where $T(z)$ is the holomorphic stress-tensor of the CFT, $\mathcal{C}$ is a
curve that connects the points $P_{\mathcal{M}}$ and ${\infty}$, and
$(z_{\mathcal{M}},\bar{z}_{\mathcal{M}})$ are the complexified coordinates of
the point $P_{\mathcal{M}}$ Castro-Alvaredo:2017wzf . The factor
$J\left(\theta,z_{\mathcal{M}},\bar{z}_{\mathcal{M}}\right)$ guarantees that
the conical twist field is a spinless primary field with scaling dimension
$2h_{\theta}$, where $h_{\theta}$ is given by

$h_{\theta}=\frac{c}{24}\left(\frac{2\pi+\theta}{2\pi}-\frac{2\pi}{2\pi+\theta}\right).$
(10)

$\psi(P)$$\mathcal{C}_{1}$$\mathcal{C}_{2}$$\mathcal{C}_{3}$$\mathcal{C}_{4}$$P_{\mathcal{M}}$$P_{\mathcal{N}}$
Figure 2: The Euclidean half-sided translation is generated by enclosing the
operator $\psi(z_{P}\bar{z}_{P})$ inside the contours
$\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ and $\mathcal{C}_{4}$ used
for defining the conical twist fields.

The factor $J\left(\theta,z_{\mathcal{M}},\bar{z}_{\mathcal{M}}\right)$ does
not play a significant role in our discussion. Hence, we will not describe
more about it. More about
$J\left(\theta,z_{\mathcal{M}},\bar{z}_{\mathcal{M}}\right)$ can be found in
Castro-Alvaredo:2017wzf . For $\theta=2\pi m,m\in\mathbb{Z}$, the conical
twist field reduces to the usual twist field. Therefore, the Euclidean
continuation of the half-sided operator is given by

$U(s)=\mathcal{T}_{-\theta}(z_{\mathcal{M}^{\prime}},\bar{z}_{\mathcal{M}^{\prime}})\mathcal{T}_{\theta}(z_{\mathcal{N}^{\prime}},\bar{z}_{\mathcal{N}^{\prime}})\mathcal{T}_{\theta}(z_{\mathcal{M}},\bar{z}_{\mathcal{M}})\mathcal{T}_{-\theta}(z_{\mathcal{N}},\bar{z}_{\mathcal{N}}),\quad\quad
s=e^{2\pi\text{i}\theta}-1.$ (11)

Let us consider the half-sided translation of the field
$\psi(z_{P},\bar{z}_{P})$ in the complex plane.

$\displaystyle\psi_{s}(z_{P},\bar{z}_{P})=U^{\dagger}(s)\psi(z_{P},\bar{z}_{P})U(s).$
(12)

The coordinates $(z_{P},\bar{z}_{P})$ is the Euclidean continuation of the
coordinate $(y_{P}^{+},y_{P}^{-})$ of the point $P$ in the right Rindler wedge
where the field $\psi$ is placed. Using (10) and (11) we can express the half-
sided translated operator $\psi_{s}(z_{P},\bar{z}_{P})$ as

$\displaystyle\psi_{s}(z_{P},\bar{z}_{P})$
$\displaystyle=e^{-\theta\int_{\mathcal{C}_{1}+\mathcal{C}_{4}}(z-z_{{\mathcal{N}}})T(z)dz+c.c.}e^{-\theta\int_{\mathcal{C}_{2}+\mathcal{C}_{3}}(z-z_{{\mathcal{M}}})T(z)dz+c.c.}\psi(z_{P},\bar{z}_{P}),$
$\displaystyle=e^{-\theta\oint_{\mathcal{C}_{P}}(z-z_{{\mathcal{N}}})T(z)dz+c.c.}e^{\theta\oint_{\mathcal{C}_{P}}(z-z_{{\mathcal{M}}})T(z)dz+c.c.}\psi(z_{P},\bar{z}_{P}),$
(13)

where $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ and $\mathcal{C}_{4}$
are the contours shown in figure 2, and $\mathcal{C}_{P}$ is a counter-
clockwise contour enclosing the point $z=z_{P}$. We can simplify the product
of the operators $e^{A+c.c}$ and $e^{B+c.c}$, where

$A=-\theta\oint_{\mathcal{C}_{P}}dz(z-z_{\mathcal{N}})T(z),\qquad\qquad
B=\theta\oint_{\mathcal{C}_{P}}dz(z-z_{\mathcal{M}})T(z),$

using the Baker-Campbell-Hausdorff formula. Since the commutator of $A$ and
$B$ is given by

$[A,B]=2\pi\text{i}\theta(A+B),$ (14)

we can use the following special case of Baker-Campbell-Hausdorff (BCH)
formula Van-Brunt:2015ala

$e^{A}e^{B}=e^{\frac{(e^{2\pi\text{i}\theta}-1)}{\theta}(A+B)},\quad
if\quad[A,B]=2\pi\text{i}\theta(A+B),$ (15)

to obtain that

$\displaystyle\psi_{s}(z_{P},\bar{z}_{P})$
$\displaystyle=e^{s\oint_{\mathcal{C}_{P}}(z_{{\mathcal{M}}}-z_{{\mathcal{N}}})T(z)dz+c.c.}\psi(z_{P},\bar{z}_{P}),$
$\displaystyle=\psi(z_{P}+s(z_{{\mathcal{M}}}-z_{{\mathcal{N}}}),\bar{z}_{P}+s(\bar{z}_{{\mathcal{M}}}-\bar{z}_{{\mathcal{N}}})).$
(16)

Therefore, the Lorentzian half-sided translation operator is given by

$U(s)=e^{\text{i}s\left\\{(y^{+}_{\mathcal{M}}-y^{+}_{\mathcal{N}})P^{-}+(y^{-}_{\mathcal{M}}-y^{-}_{\mathcal{N}})P^{+}\right\\}},$
(17)

where $(P^{-},P^{+})$ are the generators of null translations. Since
$y^{+}_{\mathcal{M}}=y^{+}_{\mathcal{N}}=0$, the resulting half-sided
translation for $s>0$ is an in-falling across the Rindler horizon.

##  3 The in-falling across the black hole horizon

$\psi(P)$$\psi(P^{\prime})$$\mathcal{N}$$P_{\mathcal{M}}$$P_{\mathcal{N}}$$w^{+}=0$$w^{-}=0$
Figure 3: The half-sided translation defined using the von Neumann algebra
$\mathcal{M}$ restricted to the right exterior of the eternal black hole and
the subalgebra $\mathcal{N}$ restricted to the blue shaded region translates a
field $\psi$ from point $P$ to $P^{\prime}$ along the outward direction for
$s<0$. The green lines indicate the horizon and the red dashed lines indicates
the singularity of the black hole. $(w^{+},w^{-})$ are the Kruskal
coordinates. The blue dotted arrow shows the trajectory generated by the
modular Hamiltonian $\mathcal{K_{M}}$, the boost that fixes the point
$P_{\mathcal{M}}$, and the gray dotted arrow shows the trajectory generated by
the modular Hamiltonian $\mathcal{K_{N}}$, the boost that fixes the point
$P_{\mathcal{N}}$.

In this section, we shall discuss the in-falling of a field across the event
horizon of an AdS2 eternal black hole solution of JT gravity. We choose the
von Neumann algebra of observables $\mathcal{M}$ to be the algebra of
observables restricted to the right side of the black hole exterior, and
$\mathcal{N}$ to be the algebra of observables restricted to a subregion in
the right exterior, as shown in figure 3. The cyclic and the separating state
is the Hartle-Hawking state prepared using the bulk Euclidean path integral.
The associated half-sided translation $U(s)$ can be determined, if we can
compute the modular operators $\Delta_{\mathcal{M}}$ and
$\Delta_{\mathcal{N}}$. By assuming the factorisation of the Hilbert space, we
can define a modular operator in terms of the density matrices associated with
the operator algebra and its commutant using (5).

The density matrix associated with an operator algebra can be defined using
the path integral over the Euclidean geometry of the black hole with a cut
that corresponds to the part of the Cauchy slice which belongs to the region
in which the elements of the operator algebra are defined. The Euclidean
geometry of AdS2 eternal black hole is a hyperbolic disc. Therefore, the path
integral representation of the modular operators $\Delta_{\mathcal{M}}$ can be
obtained by performing the path integral of the JT gravity coupled the matter
CFT over two hyperbolic discs each with a cut, one from $P_{\mathcal{M}}$ to
$B_{R}$, a point in the boundary of the disc and the another from
$P_{\mathcal{M}}$ to $B_{L}$, another point in the boundary which is on the
other side of the boundary of the disc compared to the point $B_{R}$.
Similarly, the path integral representation of the modular operator
$\Delta_{\mathcal{N}}$ can be obtained by performing the path integral over a
pair of hyperbolic discs each with a cut, one from $P_{\mathcal{N}}$ to
$B_{R}$, and the another from $P_{\mathcal{N}}$ to $B_{L}$. Therefore, the
Euclidean path integral representation of $\Delta^{\text{i}t}_{\mathcal{M}}$
is given by the path integral over a pair of hyperbolic wedges having angle
$\theta$, one with an edge that connects points $P_{\mathcal{M}}$ and $B_{R}$,
and another with an edge that connects points $P_{\mathcal{M}}$ and $B_{L}$,
where $\theta=\text{i}t$. The dominant contribution to this gravity path
integral comes from the Hayward term in JT gravity Arias:2021ilh  associated
with the corners of the wedges. It is given by

$\Delta^{\theta}_{\mathcal{M}}=B_{\theta}e^{\frac{\theta}{8\pi
G_{N}}\Phi_{R}(w_{\mathcal{M}},\bar{w}_{\mathcal{M}})}e^{-\frac{\theta}{8\pi
G_{N}}\Phi_{L}(w_{\mathcal{M}},\bar{w}_{\mathcal{M}})},$ (18)

where $B_{\theta}$ contains the boundary terms. Here $(w,\bar{w})$ is the
Euclidean Kruskal coordinates. The Euclidean metric takes the following form
in this coordinates

$ds^{2}_{eBH}=-\frac{4dwd\bar{w}}{(1-w\bar{w})^{2}}.$ (19)

The dilaton operator $\Phi_{R}(w_{\mathcal{M}},\bar{w}_{\mathcal{M}})$ has the
following expression Callebaut:2018xfu

$\displaystyle\frac{\Phi_{R}(w_{\mathcal{M}},\bar{w}_{\mathcal{M}})-\Phi_{0}(w_{\mathcal{M}},\bar{z}_{\mathcal{M}})}{8\pi
G_{N}}=$
$\displaystyle\int_{\mathcal{C}_{\mathcal{M}}}\frac{\left(w-w_{{\mathcal{M}}}\right)\left(1-w\bar{w}_{\mathcal{M}}\right)}{\left(1-w_{\mathcal{M}}\bar{w}_{\mathcal{M}}\right)}T(w)dw+c.c.,$
(20)

where $\Phi_{0}(w_{\mathcal{M}},\bar{w}_{\mathcal{M}})$ is the dilaton profile
for vanishing matter CFT stress-tensor and $\mathcal{C}_{\mathcal{M}}$ is
curve that connects $P_{\mathcal{M}}$ and $B_{R}$. Similarly, the dilaton
$\Phi_{L}(w_{\mathcal{M}},\bar{w}_{\mathcal{M}})$ has the following expression

$\displaystyle\frac{\Phi_{L}(w_{\mathcal{M}},\bar{w}_{\mathcal{M}})-\Phi_{0}(w_{\mathcal{M}},\bar{w}_{\mathcal{M}})}{8\pi
G_{N}}=$
$\displaystyle\int_{\mathcal{C}^{\prime}_{\mathcal{M}}}\frac{\left(w-w_{{\mathcal{M}}}\right)\left(1-w\bar{w}_{\mathcal{M}}\right)}{\left(1-w_{\mathcal{M}}\bar{w}_{\mathcal{M}}\right)}T(w)dw+c.c.,$
(21)

where $\mathcal{C}^{\prime}_{\mathcal{M}}$ is curve that connects
$P_{\mathcal{M}}$ and $B_{L}$. Finally, the Euclidean continuation of the
half-sided translation operator is given by

$U(s)=J_{\theta}e^{\theta\int_{\mathcal{C}_{\mathcal{M}}}\frac{\left(w-w_{{\mathcal{M}}}\right)\left(1-w\bar{w}_{\mathcal{M}}\right)}{\left(1-w_{\mathcal{M}}\bar{w}_{\mathcal{M}}\right)}T(w)dw+c.c.}e^{-\theta\int_{\mathcal{C}_{\mathcal{N}}}\frac{\left(w-w_{{\mathcal{N}}}\right)\left(1-w\bar{w}_{\mathcal{N}}\right)}{\left(1-w_{\mathcal{N}}\bar{w}_{\mathcal{N}}\right)}T(w)dw+c.c.}$
(22)

where $J_{\theta}$ is a CFT stress-tensor independent factor, and
$s=e^{2\pi\text{i}\theta}-1$.

$\psi(P)$$\mathcal{C}_{1}$$\mathcal{C}_{2}$$\mathcal{C}_{3}$$\mathcal{C}_{4}$$P_{\mathcal{M}}$$P_{\mathcal{N}}$
Figure 4: The Euclidean half-sided translation in the euclidean continuation
of the eternal black hole geometry is generated by enclosing the operator
$\psi(w_{P},\bar{w}_{P})$ inside the contours
$\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ and $\mathcal{C}_{4}$.

Let us study the action of $U(s)$ on a field $\psi(w_{P},\bar{w}_{P})$ placed
at a point $P$ in the Euclidean continuation of the eternal black hole
geometry, see figure 4. The half-sided translated field
$\psi_{s}(z_{P},\bar{z}_{P})$ is as follows

$\displaystyle\psi_{s}(w_{P},\bar{w}_{P})$
$\displaystyle=U^{\dagger}(s)\psi(w_{P},\bar{w}_{P})U(s),$
$\displaystyle=e^{-\theta\oint_{\mathcal{C}_{P}}\frac{\left(w-w_{{\mathcal{N}}}\right)\left(1-w\bar{w}_{\mathcal{N}}\right)}{\left(1-w_{\mathcal{N}}\bar{w}_{\mathcal{N}}\right)}T(w)dw+c.c.}e^{\theta\oint_{\mathcal{C}_{P}}\frac{\left(w-w_{{\mathcal{M}}}\right)\left(1-w\bar{w}_{\mathcal{M}}\right)}{\left(1-w_{\mathcal{M}}\bar{w}_{\mathcal{M}}\right)}T(w)dw+c.c.}\psi(w_{P},\bar{w}_{P}),$
(23)

where $\mathcal{C}_{P}$ is a counter-clockwise contour enclosing the point
$w=w_{P}$. Using the following commutator Das:2020goe

$\displaystyle\left[\oint_{\mathcal{C}_{P}}dw\frac{\left(w-w_{{\mathcal{N}}}\right)\left(1-w\bar{w}_{\mathcal{N}}\right)}{\left(1-w_{\mathcal{N}}\bar{w}_{\mathcal{N}}\right)}T(w),\oint_{\mathcal{C}_{P}}dv\frac{\left(v-w_{{\mathcal{M}}}\right)\left(1-v\bar{w}_{\mathcal{M}}\right)}{\left(1-w_{\mathcal{M}}\bar{w}_{\mathcal{M}}\right)}T(v)\right]$
$\displaystyle=2\pi\text{i}\oint_{\mathcal{C}_{P}}dw\left\\{\frac{\left(w-w_{{\mathcal{N}}}\right)\left(1-w\bar{w}_{\mathcal{N}}\right)}{{\left(1-w_{\mathcal{N}}\bar{w}_{\mathcal{N}}\right)}}-\frac{\left(w-w_{{\mathcal{M}}}\right)\left(1-w\bar{w}_{\mathcal{M}}\right)}{{\left(1-w_{\mathcal{M}}\bar{w}_{\mathcal{M}}\right)}}\right\\}T(w)$

and the BCH formula (15) we can obtain the following simplified expression

$\displaystyle\psi_{s}(w_{P},\bar{w}_{P})=e^{s\oint_{\mathcal{C}_{P}}dw\left\\{\frac{\left(w-w_{{\mathcal{N}}}\right)\left(1-w\bar{w}_{\mathcal{N}}\right)}{{\left(1-w_{\mathcal{N}}\bar{w}_{\mathcal{N}}\right)}}-\frac{\left(w-w_{{\mathcal{M}}}\right)\left(1-w\bar{w}_{\mathcal{M}}\right)}{{\left(1-w_{\mathcal{M}}\bar{w}_{\mathcal{M}}\right)}}\right\\}T(w)+c.c.}\psi(w_{P},\bar{w}_{P}).$
(25)

$\psi(P)$$\psi(P^{\prime})$$\psi(\tilde{P})$$\psi(P^{\prime})$$\psi(P)$$\psi(\tilde{P}^{\prime})$$\mathcal{N}$$P_{\mathcal{M}}$$P_{\mathcal{N}}$$w^{+}=0$$w^{-}=0$
Figure 5: The half-sided translation defined using the von Neumann algebra
$\mathcal{M}$ restricted to the right exterior of the eternal black hole and
the subalgebra $\mathcal{N}$ restricted to the blue shaded region translates a
field $\psi$ from a point $P$ outside the horizon to a point $P^{\prime}$ in
the black hole interior. The blue dotted arrow shows the trajectory generated
by the half-sided translation. The black dotted arrow shows the trajectory of
a field $\psi(\tilde{P})$ placed on a late time Cauchy slice.

This allows us to identify the generator of the half-sided translation defined
using the von Neumann algebra $\mathcal{M}$ restricted to the right exterior
of the eternal black hole and the subalgebra $\mathcal{N}$, with
$w^{-}_{\mathcal{M}}=w^{+}_{\mathcal{M}}=w^{+}_{\mathcal{N}}=0$. It is given
by

$\displaystyle G$ $\displaystyle=w^{-}_{{\mathcal{N}}}\int
dw^{-}~{}T_{--}(w^{-})+w^{-}_{\mathcal{N}}\int
dw^{+}~{}(w^{+})^{2}T_{++}(w^{+})$ (26)

The half-sided translation generated using this generator changes the position
of an operator placed at $(w_{P}^{+},w_{P}^{-})$ as follows

$(w_{P}^{+},w_{P}^{-})\rightarrow\left(\frac{w_{P}^{+}}{1-sw^{-}_{\mathcal{N}}w_{P}^{+}},w_{P}^{-}-sw^{-}_{\mathcal{N}}\right).$

Since $w^{-}_{\mathcal{N}}$ is negative, this change represents an in-falling
across the horizon for $s>0$. The trajectory of this in-falling is shown in
figure 5. It is clear from the figure that the operators near the singularity
of the black hole can be reconstructed by the half-sided evolution of fields
placed on late time Cauchy slices.

##  4 The half-sided translations and the islands

The half-sided translation translates a field outside the horizon to the
inside. As a result, the degrees of freedom inside the horizon can be
reconstructed using the degrees of freedom in the gravitational region outside
the horizon with the help of half-sided translation. However, according to the
entanglement wedge reconstruction principle it is not possible to reconstruct
the degrees of freedom inside an AdS2 eternal black hole in equilibrium with a
finite temperature non-gravitating bath using the degrees of freedom in the
left and right gravitational regions outside the horizon, after the Page time.
This is a consequence of the appearance of a nontrivial island that includes
the black hole interior for the bath after the Page time Almheiri:2019yqk .
The island rule Almheiri:2019hni  then demands the failure of the half-sided
translation to take a field in the gravitational regions outside the horizon
to the interior, after the Page time. In this section, we show that after the
Page time, the action of the half-sided translation on a field in the
gravitational regions outside the black hole horizon is trivial, and thus
cease to translate a field in the gravitational region outside the event
horizon to the black hole interior.

###  4.1 An eternal black hole in equilibrium with a finite temperature bath

$w^{+}=\infty,~{}y_{R}^{+}=\infty$$w^{-}=-\infty,~{}y_{R}^{-}=-\infty$$y_{L}^{-}=\infty,~{}w^{-}=\infty$$y_{L}^{+}=-\infty,~{}w^{+}=-\infty$$w^{+}=0,~{}y_{L}^{+}=\infty\qquad\quad~{}y_{R}^{+}=-\infty$$y_{L}^{-}=-\infty\qquad
w^{-}=0,~{}y_{R}^{-}=\infty$$w^{+}w^{-}=-e^{\frac{4\pi\epsilon}{\beta}}$$w^{+}w^{-}=-e^{\frac{4\pi\epsilon}{\beta}}$
Figure 6: An eternal black hole in equilibrium with a finite temperature bath
can be described using a plane with light cone coordinates $(w^{+},w^{-})$.
The right/left Rindler wedge describes the right/left side of the black
exterior coupled to the right/left bath having flat metric. The right/left
Rindler wedge can be covered using the light cone coordinates
$(y_{R/L}^{+},y_{R/L}^{-})$. The lines $w^{+}w^{-}=0$ are the future and the
past horizons of the eternal black hole. The interface between the black hole
and the bath satisfy the equation
$w^{+}w^{-}=-e^{\frac{4\pi\epsilon}{\beta}}$. The red dashed hyperbola
represents the singularity on which the dilaton profile vanishes.

Consider an eternal black hole solution in JT gravity coupled to a non-
gravitating bath. Assume that both the bath and the black hole have the same
inverse temperature $\beta$. Since the net flow of stress-energy tensor across
the interface between the black hole and the bath is zero, the classical
geometry of the black hole spacetime is the same as that of an isolated
eternal black hole. We can describe the geometry of this glued spacetime as a
region in a plane with light cone coordinates $(w^{+},w^{-})$ Almheiri:2019yqk
, see figure 6. The right Rindler wedge in the $w$-plane is the union of the
right exterior of the black hole and the bath. Similarly, the union of the
left side of the black hole and the bath forms the left Rindler wedge. The
$(y_{R}^{+},y_{R}^{-})$ coordinates that cover the right Rindler wedge and
$(y_{L}^{+},y_{L}^{-})$ coordinates that cover the left Rindler wedge are
related to the $(w^{+},w^{-})$ coordinates as follows

$w^{\pm}=\pm e^{\pm\frac{2\pi y_{R}^{\pm}}{\beta}},\qquad w^{\pm}=\mp
e^{\mp\frac{2\pi y_{L}^{\pm}}{\beta}}.$ (27)

In the $w$-plane the black hole horizon is given by the equation
$w^{+}w^{-}=0$, and the singularity lies on the hyperbola on which the dilaton
profile vanishes. The dilaton profile in this coordintes takes the following
form

$\phi(w^{+},w^{-})=\phi_{0}+\frac{2\pi\phi_{r}}{\beta}\frac{1-w^{+}w^{-}}{1+w^{+}w^{-}},$
(28)

where $\phi_{0}$ give rise to the extremal entropy and
$\frac{\phi_{r}}{\epsilon}$ is the boundary value of the difference
$\phi(w^{+},w^{-})-\phi_{0}$ at the boundary of the black hole spacetime given
by the equation

$w^{+}w^{-}=-e^{\frac{4\pi\epsilon}{\beta}}.$ (29)

Here $\epsilon$ is a real parameter that specifies the location of the cut-out
boundary. The metric in the gravitational region is given by

$ds^{2}_{BH}=\frac{4dw^{+}dw^{-}}{(1+w^{+}w^{-})^{2}},$ (30)

and the metric in the bath region is given by

$ds^{2}_{B}=\frac{\beta^{2}}{4\pi^{2}\epsilon^{2}}dw^{+}dw^{-}.$ (31)

###  4.2 The appearance of an island

The classical geometry of an eternal black hole in equilibrium with a thermal
bath is the same as that of an isolated eternal black hole. However, quantum
mechanically the two systems are significantly different. This is due to the
continuous exchange of the Hawking radiation and the radiation from the bath.
The exchange leads to an ever-growing entanglement between the black hole and
the bath. This can be seen by computing the entanglement entropy of the bath.
Assume that the CFT living in the glued spacetime has central charge $c$. Then
the entropy of the bath can be calculated using the replica method
Calabrese:2004eu , and is given by

$\displaystyle S_{\text{bath}}(u)=S_{\text{BH}}(u)$
$\displaystyle=\frac{c}{3}~{}\text{ln}\left(\frac{\beta}{\pi\epsilon}\text{cosh}\left(\frac{2\pi
u}{\beta}\right)\right)\xrightarrow{u>>\beta}\frac{2\pi c}{3\beta}u.$ (32)

The entanglement entropy of the bath/black hole increases linearly at late
time. Such an unbounded growth of entanglement entropy makes it bigger than
the thermal entropy of the black hole at sufficiently large times. The
paradoxical growth of the von Neumann entropy can be resolved by including the
non-perturbative gravitational correction (replica wormhole) to the
entanglement entropy after the Page time, the time at which the von Neumann
entropy of the black hole is equal to its Bekenstein-Hawking entropy
Almheiri:2019yqk . This additional contribution to the replica path integral
introduces a non-trivial entanglement wedge island, containing the black hole
interior, for the bath. Consequently, after the Page time the degrees of
freedom inside the black hole can not be reconstructed using the degrees of
freedom outside the horizon restricted to the gravitational region, i.e.
inside the left and the right cut-out AdS2 boundaries.

###  4.3 The nature of half-sided translation and the JT gravity saddle

$w^{+}=\infty$$w^{-}=-\infty$$w^{-}=\infty$$w^{+}=-\infty$$\psi(P)$$\mathcal{N}$$\mathcal{M}$$P_{\mathcal{M}}$$P_{\mathcal{N}}$$w^{+}=0$$w^{-}=0$$w^{+}w^{-}=-e^{\frac{4\pi\epsilon}{\beta}}$$w^{+}w^{-}=-e^{\frac{4\pi\epsilon}{\beta}}$
Figure 7:  The half-sided translation is defined using the von Neumann algebra
$\mathcal{M}$ restricted to the gravitational region in the right exterior of
the eternal black hole and the subalgebra $\mathcal{N}$ restricted to the blue
shaded region.

The appearance of an island for the bath after the Page time demands that the
half-sided translation must cease to translate a field in the gravitational
region outside the horizon to the interior. Otherwise, the degrees of freedom
inside the black hole can be reconstructed using the degrees of freedom in the
dual boundary CFT by performing half-sided translations, which is a violation
of the entanglement wedge bulk reconstruction theorem Dong:2016eik . The
nature of the half-sided translation after the Page time can be investigated
using the path integral description of the action of half-sided translations
on a local operator.

Let $\mathcal{M}$ be the von Neumann algebra of observables defined in the
right exterior of the black hole within the gravitational region and
$\mathcal{N}$ be the subalgebra restricted to the blue shaded region shown in
figure 7. The cyclic and the separating state $|\Omega\rangle$ can be prepared
by performing the Euclidean path integral. Since the path integral
representations for reduced density matrices are readily available, the path
integral representation of the associated half-sided translation $U(s)$ can be
obtained by using the expression (2) in terms of the modular operators

$\Delta_{\mathcal{M}}=\rho_{\mathcal{M}^{\prime}}^{-1}\otimes\rho_{\mathcal{M}}$

and

$\Delta_{\mathcal{N}}=\rho_{\mathcal{N}^{\prime}}^{-1}\otimes\rho_{\mathcal{N}}.$

Here, we have assumed the factorisation of the Hilbert space
$\mathcal{H}=\mathcal{H}_{\mathcal{M}^{\prime}}\otimes\mathcal{H}_{\mathcal{M}}$.

Consider an arbitrary correlation function

$\langle\Omega|\cdots
U^{\dagger}(s)\psi(w_{P}^{+},w^{-}_{P})U(s)\cdots|\Omega\rangle$

that involves a half-sided translated operator
$\psi_{s}(P)=U^{\dagger}(s)\psi(P)U(s)$. The dots represent all other
operators away from the point $P$ where the operator $\psi$ is inserted. We
assume that the point $P$ is in the region associated with the operator
algebra $\mathcal{M}$, see figure 7. Using the expression (8), we can write
down the Euclidean continuation of this correlation function as follows

$\displaystyle\langle\Omega|\cdots
U(-s)\psi(w_{P},\bar{w}_{P})U(s)\cdots|\Omega\rangle$
$\displaystyle=\langle\Omega|\cdots\rho^{\theta}_{\mathcal{N}^{\prime}}\otimes\rho^{-\theta}_{\mathcal{N}}\rho^{-\theta}_{\mathcal{M}^{\prime}}\otimes\rho^{\theta}_{\mathcal{M}}\psi(w_{P},\bar{w}_{P})\rho^{-\theta}_{\mathcal{M}}\otimes\rho^{\theta}_{\mathcal{M}^{\prime}}\rho^{\theta}_{\mathcal{N}}\otimes\rho^{-\theta}_{\mathcal{N}^{\prime}}\cdots|\Omega\rangle$
(33)

where $\theta=-\frac{\text{i}}{2\pi}\text{ln}(1+s)$. Initially we assume
$\theta$ to be an integer $n$, and at the end of the manipulation we
analytically continue $n$ back to an arbitrary real number.

The path integral representation of the modular operator
$\Delta_{\mathcal{M}}=\rho_{\mathcal{M}^{\prime}}^{-1}\otimes\rho_{\mathcal{M}}$
is obtained by considering a pair of $w$-planes identified at the origin with
a cut that connects the points $B_{u}^{R}$ and $P_{\mathcal{M}}$ of the first
$w$-plane, the origin of the $w$-planes, and the point $B_{u}^{L}$ of the
second $w$-plane. The points $B_{u}^{R}$ and $B_{u}^{L}$ in the Euclidean
$w$-plane after the analytic continuation of the Euclidean time
$\tau\to\tau+\text{i}u$ correspond to the points in the Lorentzian $w$-plane
where the equal time slice $\Sigma_{u}$ intersects the right and the left cut-
out AdS2 boundaries respectively. The equal time slice $\Sigma_{u}$,
parameterised by the boundary time $u$, contains the point $P$ at which the
field $\psi$ is placed. The Loretzian coordinates of the point $B_{u}^{L}$ and
$B_{u}^{R}$ are given by

$\displaystyle(w^{+}_{B_{u}^{L}},w^{-}_{B_{u}^{L}})$
$\displaystyle=\left(-e^{-\frac{2\pi(u-\epsilon)}{\beta}},e^{\frac{2\pi(u+\epsilon)}{\beta}}\right),$
$\displaystyle(w^{+}_{B_{u}^{R}},w^{-}_{B_{u}^{R}})$
$\displaystyle=\left(e^{\frac{2\pi(u-\epsilon)}{\beta}},-e^{\frac{-2\pi(u+\epsilon)}{\beta}}\right)$
(34)

Similarly, the path integral representation of the modular operator
$\Delta_{\mathcal{N}}=\rho_{\mathcal{N}^{\prime}}^{-1}\otimes\rho_{\mathcal{N}}$
can also be constructed. Then, by following the ideas described in
Lewkowycz:2013nqa , within a correlation function the insertion of the half-
sided translated operator

$\rho^{n}_{\mathcal{N}^{\prime}}\otimes\rho^{-n}_{\mathcal{N}}\rho^{-n}_{\mathcal{M}^{\prime}}\otimes\rho^{n}_{\mathcal{M}}\psi(w_{P},\bar{w}_{P})\rho^{-n}_{\mathcal{M}}\otimes\rho^{n}_{\mathcal{M}^{\prime}}\rho^{n}_{\mathcal{N}}\otimes\rho^{-n}_{\mathcal{N}^{\prime}}$

can be replaced by a path integral of JT gravity coupled to matter. Then the
nature of the half-sided translation can be identified by determining the JT
gravity path integral saddle with the least free energy.

###  4.4 The free energy of the trivial and the wormhole saddle

$P_{\mathcal{M}}$$P_{\mathcal{N}}$$B^{R}_{u}$$B^{L}_{u}$$P$$P_{\mathcal{M}}$$\rho_{\mathcal{M}}$$P_{\mathcal{M}}$$\rho^{-1}_{\mathcal{M}^{\prime}}$$P_{\mathcal{M}}$$\rho_{\mathcal{M}}$$P_{\mathcal{M}}$$\rho^{-1}_{\mathcal{M}^{\prime}}$$P_{\mathcal{M}}$$\rho^{-1}_{\mathcal{M}}$$P_{\mathcal{M}}$$\rho_{\mathcal{M}^{\prime}}$$P_{\mathcal{M}}$$\rho^{-1}_{\mathcal{M}}$$P_{\mathcal{M}}$$\rho_{\mathcal{M}^{\prime}}$$P_{\mathcal{N}}$$\rho^{-1}_{\mathcal{N}}$$P_{\mathcal{N}}$$\rho_{\mathcal{N}^{\prime}}$$P_{\mathcal{N}}$$\rho^{-1}_{\mathcal{N}}$$P_{\mathcal{N}}$$\rho_{\mathcal{N}^{\prime}}$$P_{\mathcal{N}}$$\rho^{-1}_{\mathcal{N}}$$P_{\mathcal{N}}$$\rho_{\mathcal{N}^{\prime}}$$P_{\mathcal{N}}$$\rho^{-1}_{\mathcal{N}}$$P_{\mathcal{N}}$$\rho_{\mathcal{N}^{\prime}}$
Figure 8: The trivial saddle is obtained by cyclically gluing the $w$-planes
along the cuts having same colours. The resulting Riemann surface viewed as an
(8n+1)-sheeted geometry over the $w$-plane has branch points at locations
$B_{u}^{R}$ and $B_{u}^{L}$ on the $w$-plane. The figure describes the gluing
for $n=2$

The dominant saddle in the gravitational path integral can be determined by
computing the free energy of various saddles.

The trivial saddle: It is obtained by sewing the $8n+1$ number of $w$-planes
as shown in figure 8. The resulting Riemann surface has eight conical
singularities at points $B_{u}^{R}$ and $B_{u}^{L}$ on the $w$-plane. The
contribution to the free energy of a CFT with central charge $c$ from a
conical singularity having excess angle $\gamma$ is given by

$\Delta
F\approx\frac{c(\gamma+2\pi)}{24\pi}\left(1-\left(\frac{2\pi}{(\gamma+2\pi)}\right)^{2}\right)\text{ln}~{}L,$
(35)

where $L$ is the characteristic length of the geometry. This result can be
found studying the infinitesimal Weyl transformation Cardy:1988tk . Assume
that the characteristic length $L$ of the Riemann surface changes to
$L+\lambda L$ under an infinitesimal Weyl transformation of the metric
$g_{\mu\nu}$ on the surface to $g_{\mu\nu}+\lambda g_{\mu\nu}$, where
$\lambda$ is an infinitesimal real number. The contribution $\Delta F$ to the
free energy $F$ of the CFT having central charge $c$ defined on the Riemann
surface, due to the conical singularity at $w=w_{c}$ having excess angle
$\gamma$ satisfies the following relation

$L\frac{\partial\Delta F}{\partial L}=\frac{1}{4\pi}\int_{D_{w_{c}}}\langle
T_{\mu\nu}\rangle g^{\mu\nu}d^{2}w,$ (36)

where $D_{w_{c}}$ is the close neighbourhood of the point $w=w_{c}$. Then the
result (35) can be obtained using the fact that expectation value of the
stress tensor near $w=w_{c}$ is given by

$\langle
T_{\mu\nu}\rangle=\frac{c}{24(w-w_{c})^{2}}\left(1-\left(\frac{2\pi}{(\gamma+2\pi)}\right)^{2}\right).$
(37)

The contribution to the free energy of the correlation function (4.3) from the
trivial saddle is given by the sum of the contributions to the free energy
from all eight conical singularities having excess angle $2\pi n$. Therefore,
the free energy of the trivial saddle is given by

$\Delta
F_{T}\approx\frac{2(n+1)c}{3}\left(1-\frac{1}{(n+1)^{2}}\right)\text{ln}\left(\frac{\beta}{\pi\epsilon}\text{cosh}\left(\frac{2\pi
u}{\beta}\right)\right).$ (38)

Here we declared the characteristic length for each of the disconnected
Riemann surface as the coordinate distance between the conical singularities (
the distance between the points $B_{u}^{L}$ and $B_{u}^{R}$) multiplied by the
product of the square root of the Weyl factors at $B_{u}^{L}$ and $B_{u}^{R}$,
i.e.

$L=\sqrt{\frac{(w^{+}_{B_{u}^{R}}-w^{+}_{B_{u}^{L}})(w^{-}_{B_{u}^{R}}-w^{-}_{B_{u}^{L}})}{(1+w^{+}_{B_{u}^{R}}w^{-}_{B_{u}^{R}})(1+w^{+}_{B_{u}^{L}}w^{-}_{B_{u}^{L}})}}$
(39)

Note that $L$ changes under a Weyl transformation and is symmetric with
respect to the exchange of the conical singularities. Therefore, the
approximate free energy of the trivial saddle that describe the action of the
half-sided translation $U(s)$ for large $u$ is given by

$\mathcal{F}_{T}(u)\approx\frac{4\pi(n+1)c}{3\beta}\left(1-\frac{1}{(n+1)^{2}}\right)u.$
(40)

$\tilde{P}^{W}_{\mathcal{M}}$$P^{W}_{\mathcal{M}}$$\tilde{P}^{W}_{\mathcal{N}}$$P^{W}_{\mathcal{N}}$$B^{R}_{u}$$B^{L}_{u}$$P^{W}$$P^{W,L}_{\mathcal{M}}$$\tilde{P}^{W,L}_{\mathcal{M}}$$P^{W,L}_{\mathcal{M}}$$\tilde{P}^{W,L}_{\mathcal{M}}$$\tilde{P}^{W,R}_{\mathcal{M}}$$P^{W,R}_{\mathcal{M}}$$\tilde{P}^{W,R}_{\mathcal{M}}$$P^{W,R}_{\mathcal{M}}$$\tilde{P}^{W,L}_{\mathcal{N}}$$P^{W,L}_{\mathcal{N}}$$\tilde{P}^{W,L}_{\mathcal{N}}$$P^{W,L}_{\mathcal{N}}$$\tilde{P}^{W,R}_{\mathcal{N}}$$P^{W,R}_{\mathcal{N}}$$P^{W,R}_{\mathcal{N}}$$\tilde{P}^{W,R}_{\mathcal{N}}$
Figure 9:  A wormhole geometry that has the topology of a sphere with $8n+1$
number of boundaries. Here $n$ is chosen to be 2. All the cuts with the same
colour are being cyclicly glued. The dotted circles are the closed geodesics
on the hyperbolic cylinders that extend to the boundaries of the surface.

The wormhole saddle: A non-trivial saddle can have bridges connecting varies
sheets that are being glued along the cuts. Our aim is to compute the free
energy of a wormhole geometry that has the topology of a sphere with $8n+1$
number of boundaries. The cuts present in the boundary regions of the wormhole
geometry are glued as shown in figure 9. Since the matter CFT is not coupled
to the JT gravity dilaton, the wormhole must have hyperbolic metric on it. It
is a well know fact in hyperbolic geometry that gluing of two hyperbolic
surfaces produces another hyperbolic surface if and only if they are glued
along a closed geodesic or along a part of the open geodesic that connects two
geodesics on the hyperbolic surface imayoshi2012introduction . This implies
that a cut on the wormhole geometry must be an open geodesic that connects a
boundary of the surface and the nearest closed geodesic. As a result, the
gluing of these cuts as shown in figure 9 produces a hyperbolic Riemann
surface.

Interestingly, the computation of JT gravity coupled to matter path integral
can be simplified by considering another equivalent representation of the
wormhole geometry. In the equivalent representation, the entire geometry does
not have hyperbolic metric, instead different parts of the geometry has
hyperbolic metrics. However, the hyperbolic metric on two different parts
having common interface does not agree each other. Consequently, the metric on
the equivalent representation is not hyperbolic. Nevertheless, this
representation has the following nice feature. The wormhole geometry with this
metric can be obtained by gluing $8n$ number of one punctured hyperbolic discs
having one cut on it, a one punctured sphere with 8 cuts on it and a
hyperbolic sphere with $8n+1$ punctures by identifying the local coordinates
around the punctures via the plumbing fixture construction. The cuts are
appropriately identified as shown in figure 9. Below we shall briefly explain
the plumbing fixture construction. More detailed description of this
construction can be found in obitsu2008grafting ; Moosavian:2017qsp ;
Moosavian:2017uso .

$P_{\mathcal{M}}^{\prime\prime}$$P_{\mathcal{M}}^{\prime}$$P_{\mathcal{N}}^{\prime\prime}$$P_{\mathcal{N}}^{\prime}$$B^{R}_{u}$$B^{L}_{u}$$P$$P^{\prime}_{\mathcal{M}}$$P^{\prime\prime}_{\mathcal{M}}$$P^{\prime}_{\mathcal{M}}$$P^{\prime\prime}_{\mathcal{M}}$$P^{\prime}_{\mathcal{M}}$$P^{\prime\prime}_{\mathcal{M}}$$P^{\prime}_{\mathcal{M}}$$P^{\prime\prime}_{\mathcal{M}}$$P^{\prime}_{\mathcal{N}}$$P^{\prime\prime}_{\mathcal{N}}$$P^{\prime}_{\mathcal{N}}$$P^{\prime\prime}_{\mathcal{N}}$$P^{\prime\prime}_{\mathcal{N}}$$P^{\prime}_{\mathcal{N}}$$P^{\prime}_{\mathcal{N}}$$P^{\prime\prime}_{\mathcal{N}}$
Figure 10: A wormhole geometry can be constructed by gluing the $w$-planes and
the hyperbolic $8n+1$ punctured sphere by identifying the coordinates around
the punctures on the sphere and the origin of the $w$-planes via the the
plumbing construction. The plumbing fixture sewing removes region inside the
dotted circles and identify the dotted circles on the disc and the dotted
circles on the $8n+1$ punctured sphere. After an appropriate Weyl rescaling of
the metric on the glued surface matches with metric on the hyperbolic surface
in figure 9. Moreover, the dotted circles in this figure become the geodesics
on the hyperbolic surface, the dotted circles in figure 9. All the cuts with
the same colour are being cyclicly glued.

The plumbing fixture construction sews two Riemann surfaces $\mathcal{S}_{1}$
and $\mathcal{S}_{2}$ by identifying the neighbourhood around the punctures
$p_{1}$ on $\mathcal{S}_{1}$ and $p_{2}$ on $\mathcal{S}_{2}$, see figure 11.
Assume that the local coordinate around the puncture on $p_{1}$ on
$\mathcal{S}_{1}$ is $w_{p_{1}}$ and the local coordinate around the puncture
on $p_{2}$ on $\mathcal{S}_{2}$. The neighbourhood of a puncture on a
hyperbolic Riemann surface takes the form of a cusp. Identifying the local
coordinate $w_{p_{1}}$ to $w_{p_{2}}$ using the following relation

$w_{p_{1}}w_{p_{2}}=t,\qquad 0\leq|t|<1,$ (41)

sews two such cusps, one from $\mathcal{S}_{1}$ and the other from
$\mathcal{S}_{2}$. The plumbing-fixture identification (41) removes the
neighbourhood of the punctures within the cusps bounded by the curves
$w_{p_{1}}=\sqrt{t}$ and $w_{p_{2}}=\sqrt{t}$ and sews them along the curves
$w_{p_{1}}=\sqrt{t}$ and $w_{p_{2}}=\sqrt{t}$. We shall denote the new Riemann
surface produced by the sewing as $\mathcal{S}_{t}$. The local expression for
the hyperbolic metric around the puncture $p_{i}$ in terms of the local
coordinate $w_{p_{i}}$ is given by

$ds^{2}=\left(\frac{|d\zeta|}{|\zeta|\ln|\zeta|}\right)^{2},\qquad\zeta=w_{p_{i}}\qquad
i=1,2.$ (42)

Using this local expression for hyperbolic metric it is straightforward to
verify the absence of any geodesic in the neighbourhood of an arbitrary
puncture on a hyperbolic Riemann surface. It is a well known fact in
hyperbolic geometry that we can obtain a hyperbolic surface by gluing two
hyperbolic surfaces provided the component surfaces are glued along a geodesic
imayoshi2012introduction . This implies that the curves $w_{p_{1}}=\sqrt{t}$
and $w_{p_{2}}=\sqrt{t}$ are not geodesics. The metric on the glued Riemann
surface has constant curvature $-1$ everywhere except along the curve
$w_{p_{1}}=w_{p_{2}}=\sqrt{t}$. Therefore, the resulting surface
$\mathcal{S}_{t}$ cannot be a hyperbolic Riemann surface. However, this metric
can be made smooth and can be related to the hyperbolic metric on the surface
by an appropriate Weyl rescaling obitsu2008grafting . The length $l$ of the
curve $w_{p_{1}}=w_{p_{2}}=\sqrt{t}$ computed using the resulting hyperbolic
metric for infinitesimal values of $|t|$ can be approximated as

$l=-\frac{2\pi^{2}}{\text{ln}|t|}.$ (43)

$p_{1}$$p_{2}$ Figure 11: The plumbing fixture gluing of two Riemann surfaces
$\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ by identifying the neighbourhood
around the punctures $p_{1}$ on $\mathcal{S}_{1}$ and $p_{2}$ on
$\mathcal{S}_{2}$.

The usefulness of the plumbing fixture description of the wormhole geometry is
the following. It allows us to approximate the CFT path integral over the
wormhole geometry as the product of the CFT path integral over the $w$-planes
with modified cut interval and the CFT path integral over the punctured
hyperbolic Riemann surface for infinitesimal values of the plumbing fixture
parameter $|t|$. As explained above, one of the end point of the cuts in the
wormhole geometry with hyperbolic metric are on the geodesics which are mapped
to the curves

$w\bar{w}=t_{i}=e^{-\frac{2\pi^{2}}{l_{i}}}$

on the $i^{th}$ copy of the $w$-plane. This implies that the end points of the
cuts on the $w$-planes act as the parameters of the moduli space of the
wormhole geometry, just like the geodesic lengths $l_{i}$, the Fenchel-Nielsen
parameters.

Suppose that the cuts on the trivial saddle $B^{L}_{u}P_{\mathcal{M}}\cup
P_{\mathcal{M}}B^{R}_{u}$, $B^{L}_{u}P_{\mathcal{N}}\cup
P_{\mathcal{N}}B^{R}_{u}$ are mapped to the cuts
$B^{L}_{u}\tilde{P}^{W,L}_{\mathcal{M}}\cup P^{W,R}_{\mathcal{M}}B^{R}_{u}$,
$B^{L}_{u}\tilde{P}^{W,L}_{\mathcal{N}}\cup P^{W,R}_{\mathcal{N}}B^{R}_{u}$ on
the wormhole geometry. See figure 9. In the plumbing fixture description of
the wormhole geometry, assume that the cuts
$B^{L}_{u}\tilde{P}^{W,L}_{\mathcal{M}}\cup P^{W,R}_{\mathcal{M}}B^{R}_{u}$,
$B^{L}_{u}\tilde{P}^{W,L}_{\mathcal{N}}\cup P^{W,R}_{\mathcal{N}}B^{R}_{u}$
are being mapped to the cuts $B^{L}_{u}P^{\prime\prime}_{\mathcal{M}}\cup
P^{\prime}_{\mathcal{M}}B^{R}_{u}$,
$B^{L}_{u}P^{\prime\prime}_{\mathcal{N}}\cup
P^{\prime}_{\mathcal{N}}B^{R}_{u}$. See figure 10. We shall denote the
coordinates of the point
$P_{\mathcal{M}}^{\prime\prime},P_{\mathcal{M}}^{\prime},P_{\mathcal{N}}^{\prime\prime},P_{\mathcal{N}}^{\prime}$
by
$\left(w_{\mathcal{M}}^{\prime\prime},\bar{w}_{\mathcal{M}}^{\prime\prime}\right),\left(w_{\mathcal{M}}^{\prime},\bar{w}_{\mathcal{M}}^{\prime}\right),\left(w_{\mathcal{N}}^{\prime\prime},\bar{w}_{\mathcal{N}}^{\prime\prime}\right),\left(w_{\mathcal{N}}^{\prime},\bar{w}_{\mathcal{N}}^{\prime}\right)$
respectively. Then the discussion so far concludes that finding the locations
of the points
$P_{\mathcal{M}}^{\prime\prime},P_{\mathcal{M}}^{\prime},P_{\mathcal{N}}^{\prime\prime},P_{\mathcal{N}}^{\prime}$
is equivalent to identifying the moduli parameters of the wormhole.
Interestingly, the points or equivalently the moduli parameters of the
wormhole saddle can be determined by minimising the free energy of the
wormhole.

The conical excess angle at
$P_{\mathcal{M}}^{D,L_{i}},P_{\mathcal{M}}^{D,R_{i}},P_{\mathcal{N}}^{D,L_{i}},P_{\mathcal{N}}^{D,R_{i}}$
leads to two kinds of contributions to the free energy of the wormhole. They
are the CFT contribution to the free energy from the conical excess angles and
the area term, the gravity contribution due to the Hayward term in the JT
gravity action from the conical excess angles Arias:2021ilh . Adding all these
contributions gives the following leading term in the free energy of the
wormhole saddle at late time

$\displaystyle\mathcal{F}_{W}(u)$
$\displaystyle\approx\frac{n}{2G_{N}}\left\\{\phi\left({w_{\mathcal{M}}^{\prime\prime}}^{+},{w_{\mathcal{M}}^{\prime\prime}}^{-}\right)+\phi\left({w_{\mathcal{M}}^{\prime}}^{+},{w_{\mathcal{M}}^{\prime}}^{-}\right)+\phi\left({w_{\mathcal{N}}^{\prime\prime}}^{+},{w_{\mathcal{N}}^{\prime\prime}}^{-}\right)+\phi\left({w_{\mathcal{N}}^{\prime}}^{+},{w_{\mathcal{N}}^{\prime}}^{-}\right)\right\\}$
$\displaystyle+\frac{c(n+1)}{12}\left(1-\frac{1}{(n+1)^{2}}\right)\text{ln}\left(\frac{(e^{\frac{2\pi
u}{\beta}}-w^{\prime+}_{\mathcal{M}})(e^{-\frac{2\pi
u}{\beta}}+w^{\prime-}_{\mathcal{M}})(e^{\frac{2\pi
u}{\beta}}-w^{\prime+}_{\mathcal{N}})(e^{-\frac{2\pi
u}{\beta}}+w^{\prime-}_{\mathcal{N}})}{(1+w^{\prime+}_{\mathcal{M}}w^{\prime-}_{\mathcal{M}})(1+w^{\prime+}_{\mathcal{N}}w^{\prime-}_{\mathcal{N}})}\right)$
$\displaystyle+\frac{c(n+1)}{12}\left(1-\frac{1}{(n+1)^{2}}\right)\text{ln}\left(\frac{(e^{-\frac{2\pi
u}{\beta}}+w^{\prime\prime+}_{\mathcal{M}})(e^{\frac{2\pi
u}{\beta}}-w^{\prime\prime-}_{\mathcal{M}})(e^{-\frac{2\pi
u}{\beta}}+w^{\prime\prime+}_{\mathcal{N}})(e^{\frac{2\pi
u}{\beta}}-w^{\prime\prime-}_{\mathcal{N}})}{(1+w^{\prime\prime+}_{\mathcal{M}}w^{\prime\prime-}_{\mathcal{M}})(1+w^{\prime\prime+}_{\mathcal{N}}w^{\prime\prime-}_{\mathcal{N}})}\right).$
(44)

Minimising the free energy with respect to the points
$P^{\prime}_{\mathcal{M}},P^{\prime\prime}_{\mathcal{M}},P^{\prime}_{\mathcal{N}}$
and $P^{\prime\prime}_{\mathcal{N}}$ gives that

$\displaystyle
w^{\prime+}_{\mathcal{M}}=w^{\prime+}_{\mathcal{N}}\approx\frac{G_{N}\beta}{6\pi\phi_{r}}e^{\frac{2\pi
u}{\beta}}\qquad
w^{\prime-}_{\mathcal{M}}=w^{\prime-}_{\mathcal{N}}\approx-\frac{G_{N}\beta}{6\pi\phi_{r}}e^{-\frac{2\pi
u}{\beta}},$ $\displaystyle
w^{\prime\prime+}_{\mathcal{M}}=w^{\prime\prime+}_{\mathcal{N}}\approx-\frac{G_{N}\beta}{6\pi\phi_{r}}e^{\frac{-2\pi
u}{\beta}}\qquad
w^{\prime\prime-}_{\mathcal{M}}=w^{\prime\prime-}_{\mathcal{N}}\approx\frac{G_{N}\beta}{6\pi\phi_{r}}e^{\frac{2\pi
u}{\beta}}.$ (45)

Thus we find that the point $P^{\prime}_{\mathcal{M}}$ is same as
$P^{\prime}_{\mathcal{N}}$, and the point $P^{\prime\prime}_{\mathcal{M}}$ is
same as $P^{\prime\prime}_{\mathcal{N}}$. Moreover, both the points are close
to the black hole horizon. This implies that the plumbing fixture parameter
$|t|$ that describe the wormhole geometry is of the order of $G_{N}^{2}$.
Hence, the free energy of the wormhole saddle is given by

$\mathcal{F}_{W}\approx\frac{2n}{G_{N}}\left(\phi_{0}+\frac{2\pi\phi_{r}}{\beta}\right).$
(46)

The linear rise of free energy of the trivial saddle given by (40) will become
more than the constant free energy (46) of the wormhole saddle at late time.
Therefore, the dominant saddle in the path integral description of the action
of $U(s)$ at late time is the wormhole saddle.

###  4.5 The island and the trivial half-sided translation

$P^{\prime}_{\mathcal{M}}$$P^{\prime}_{\mathcal{N}}$$B^{R}_{u}$$B^{L}_{u}$$P$$\mathcal{C}_{2}$$\mathcal{C}_{3}$$\mathcal{C}_{1}$$\mathcal{C}_{4}$
Figure 12: The path integral over $w$-planes can be replaced by the product of
exponential of dilaton operators defined with respect to the points
$P_{\mathcal{M}}^{\prime}$ and $P_{\mathcal{N}}^{\prime}$ and curves
$\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ and $\mathcal{C}_{4}$.

At late time, the euclidean wormhole is the dominant saddle of the path
integral that describe the action of the half-sided translation. Hence, we can
determine the nature of the half-sided translation at late time by
investigating the path integral of CFT defined on the wormhole geometry. As
discussed in the previous subsection, the plumbing fixture parameters that
describe the wormhole geometry are all infinitesimal. Therefore, the CFT path
integral over the wormhole geometry can be factorised into the path integral
over $8n+1$ number of $w$-planes having cuts that are appropriately sewed as
shown in figure 10, and the path integral over the hyperbolic sphere with
$8n+1$ number of punctures. The path integral over $8n+1$ number of $w$-planes
can be replaced by the product of exponential of dilaton operators defined
with respect to the points
$P_{\mathcal{M}}^{\prime\prime},P_{\mathcal{M}}^{\prime},P_{\mathcal{N}}^{\prime\prime}$
and $P_{\mathcal{N}}^{\prime}$. Therefore, the expression for half-sided
translation $U(s)$ on a local operator $\psi(P)$ placed at the point $P$ at
late time is given by

$\displaystyle U(s)=$ $\displaystyle
e^{\frac{n}{4G_{N}}\left(\frac{\oint_{\mathcal{C}_{2}+\mathcal{C}_{3}}dz(w-w^{\prime}_{\mathcal{N}})(w+\bar{w}^{\prime}_{\mathcal{N}})T(w)+c.c}{w^{\prime}_{\mathcal{N}}+\bar{w}^{\prime}_{\mathcal{N}}}\right)}e^{-\frac{n}{4G_{N}}\left(\frac{\oint_{\mathcal{C}_{1}+\mathcal{C}_{4}}dw(w-w^{\prime}_{\mathcal{M}})(w+\bar{w}^{\prime}_{\mathcal{M}})T(w)+c.c}{w^{\prime}_{\mathcal{M}}+\bar{w}^{\prime}_{\mathcal{M}}}\right)},$
(47)

where $n=-\frac{\text{i}}{2\pi}\text{ln}(1+s)$. The curves
$\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ and $\mathcal{C}_{4}$ are as
shown in figure 12. However, $P^{\prime}_{\mathcal{N}}$ is equal to
$P^{\prime}_{\mathcal{M}}$ for the wormhole saddle. Setting
$\left(w^{\prime}_{\mathcal{N}},\bar{w}^{\prime}_{\mathcal{N}}\right)=\left(w^{\prime}_{\mathcal{M}},\bar{w}^{\prime}_{\mathcal{M}}\right)$
makes the right hand side of the above equation identity. This proves that the
half-sided translation $U(s)$ at late time is trivial. Consequently, a field
inside the black hole horizon can not be reconstructed in terms of the degrees
of freedom in the gravitational region outside the horizon using half-sided
translations. This is consistent with the fact that after the Page time, the
black hole interior is contained in the island region of the bath entanglement
wedge Almheiri:2019yqk .

##  5 Discussion

In this paper, we showed that after Page time, the half-sided modular
translation operator defined using the algebra of observables restricted to
the gravitational region of black hole becomes trivial. The bulk half-sided
modular translations can also be interpreted as appropriate boundary half-
sided modular translations. Therefore, our result can be interpreted as the
inability of the Liu-Leutheusser proposal to reconstruct the interior bulk
operators using the boundary CFTs, the CFTR and the CFTL after Page time. This
implies, after Page time, the black hole interior does not belong to the
entanglement wedge of the dual boundary CFTs Almheiri:2019yqk . Thus the
result upholds the general principles of the entanglement wedge reconstruction
Dong:2016eik  and agrees with the discussion in Engelhardt:2023xer .

Let us briefly mention some of the directions that deserve immediate
attention. It is known that non-trivial half-sided translation exists only if
the von Neumann operator algebra is type III1 Wiesbrock:1992mg . Hence, we
expect that the algebra of observables restricted to the gravitational region
outside the horizon of the black hole in equilibrium with a bath have type
III1 algebra before Page time. However, our result suggests that after Page
time the algebra of observables must cease to be a type III1 algebra. This can
be checked by finding the spectrum of the logarithm of the modular operator of
the operator algebra. If the spectrum does not cover the full real axis, then
the operator algebra of our interest is not of type III1
connes1973classification ; connes1974caracterisation . Another interesting
direction is to check that after Page time whether half-sided translation
allows us to express a field inside the black hole interior in terms of
degrees of freedom of bath. We hope to report on this soon.

## Acknowledgements

We would like to thank Debodirna Ghosh for the initial collaboration and the
discussions. We are indebted to Sujay Ashok and Alok Laddha for the numerous
illuminating discussions and the comments. We also thank Abhijit Gadde, Shiraz
Minwalla, Sitender Kashyap, Venkata Suryanarayana Nemani, Suvrat Raju, Pratik
Rath, Ashoke Sen, Sandip Trivedi and Amitabh Virmani for the discussions.

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