††institutetext: Center for the Fundamental Laws of Nature, Harvard
University, 17 Oxford St., Cambridge, MA, 02138, USA.

# Entropy, Entanglement Islands and Swampland Bounds in the Karch-Randall
Braneworld

Hao Geng<EMAIL_ADDRESS>
###### Abstract

Motivated by the study of entanglement island in the Karch-Randall braneworld,
it has been conjectured and proven in general that entanglement island is not
consistent with long-range (massless) gravity. In this paper, we provide a
careful check of this conclusion in a model of massless gravity that is
constructed using the Karch-Randall braneworld. We show that there is indeed
no entanglement island and hence not a nontrivial Page curve due to the
diffeomorphism invariance if we are studying the correct question which is
though subtle. Moreover, we show that this conclusion is not affected by
deforming the setup with the Dvali–Gabadadze–Porrati (DGP) terms. Furthermore,
we show that the consistency of holography in this model will provide
nontrivial constraints to the DGP parameters. This study provides an example
that causality and holography in anti-de Sitter space can be used to constrain
low energy effective theories. At the end, we clarify several subtleties in
the braneworld gravity which are overlooked in the literature and we discuss
the robustness of the above results against possible coarse-graining protocols
to define a subregion in a gravitational theory.

##  1 Introduction

The Karch-Randall braneworld Karch:2000ct ; Karch:2000gx  considers a very
simple scenario— embedding a brane whose geometry is asymptotically AdSd in a
bulk spacetime with asymptotically AdSd+1 geometry. It however provides the
playground to study long-standing questions in quantum gravity such as the
black hole information paradox and it has taught us many important lessons
about quantum gravity. This is all due to its doubly holographic nature
Almheiri:2019hni ; Almheiri:2019psy ; Geng:2020qvw .

One example of the aforementioned progress is the calculation of the Page
curve of the black hole radiation. The original version of the Karch-Randall
braneworld provides the holographic dual of the setup that is used to define
and calculate the entanglement entropy of the black hole radiation (see
Fig.1). This setup considers coupling the asymptotically (large) AdSd black
hole spacetime to a thermal bath which is modeled by a d-dimensional conformal
field (CFTd) theory on a half Minkowski space. The coupling is achieved by
gluing the asymptotic boundary of the black hole spacetime along the boundary
of the half-space CFTd by imposing transparent boundary condition for the
energy flux. The black hole radiation can be modeled in this setup as a
subregion $\mathcal{R}$ of the bath CFTd. Then we can calculate the time-
dependence of this subregion entanglement entropy as the entanglement entropy
of the black hole radiation (see Fig.1). In the Karch-Randall braneworld, the
AdSd black hole can be thought of as the brane and the AdSd+1 bulk provides
the holographic dual of this d-dimensional setup. The aforementioned
entanglement entropy can be calculated in the AdSd+1 bulk using the Ryu-
Takayanagi surface Ryu:2006bv ; Takayanagi:2011zk . The calculation shows that
at late times the entanglement wedge of $\mathcal{R}$ contains a region
$\mathcal{I}$ on the brane that is disconnected from $\mathcal{R}$ and this
region is defined as the entanglement island of $\mathcal{R}$ and moreover the
entanglement entropy of $\mathcal{R}$ follows a unitary Page curve
Almheiri:2019psy ; Geng:2020qvw . In this calculation, the late time emergence
of the entanglement island is essential to have the unitary Page curve
Almheiri:2019psf ; Almheiri:2019hni ; Penington:2019npb . Nevertheless, the
original version of the Karch-Randall braneworld also captures another
important property of the d-dimensional setup we mentioned that is that the
graviton in the AdSd black hole spacetime becomes massive due to the bath
coupling Porrati:2001db . This can be seen in the Karch-Randall braneworld by
studying the Klauza-Klein spectrum of the AdSd+1 graviton Karch:2000ct .
Motivated by this observation it was conjectured and proved that entanglement
island cannot exist in massless gravity theories for a large class of
situations including asymptotically AdS spacetimes Geng:2020fxl ; Geng:2021mic
; Geng:2021hlu .

Interestingly, before the general proof was given in Geng:2021hlu , it was
shown in Geng:2021mic  that entanglement island doesn’t exist in a massless
gravity theory that is constructed as a modification of the original version
of the Karch-Randall braneworld. This was called wedge holography in
Akal:2020wfl . In this scenario we embed two Karch-Randall branes
$\mathcal{M}_{d}^{(L)}$ and $\mathcal{M}_{d}^{(R)}$ (i.e. with asymptotically
AdSd geometries) into an asymptotically AdSd+1 bulk such that the two branes
form the boundary of a wedge $\mathcal{W}_{d+1}$ (see Fig.2). This system is
described by the following action

$S_{1}=-\frac{1}{16\pi
G_{d+1}}\int_{\mathcal{W}_{d+1}}d^{d+1}x\sqrt{-g}(R-2\Lambda)-\frac{1}{8\pi
G_{d+1}}\int_{\mathcal{M}^{(L)}_{d}\cup M^{(R)}_{d}}d^{d}x\sqrt{-h}(K-T)\,,$
(1)

where $K$ is the trace of the extrinsic curvature on the brane (which in
general takes different values on the two branes), $h_{\mu\nu}$ is the induced
meric on the brane, $T$ is the tension of the brane (which in general takes
different values on the two branes) and $\Lambda=-\frac{d(d-1)}{2L^{2}}$
(later we will set the AdS length scale $L=1$ for convenience) is the bulk
cosmological constant. This system has three equivalent descriptions:

  1. 1.

Bulk description: Einstein gravity in an AdSd+1 space
$\mathcal{M}^{\prime}_{d+1}$ containing two AdSd branes
$\mathcal{M}_{d}^{(L)}$ and $\mathcal{M}_{d}^{(R)}$,which intersect each other
on the asymptotic boundary (we call the place they intersect as the defect and
it is (d-1)-dimensional) and therefore form the boundary of a wedge
$\mathcal{W}_{d+1}$ (see Fig.2);

  2. 2.

Intermediate description: Two $d$-dimensional CFTs coupled to gravity on
distinct asymptotically AdSd spaces $\mathcal{M}_{d}^{L}$ and
$\mathcal{M}_{d}^{R}$, with these systems being connected via a transparent
boundary condition at a defect
$\mathcal{M}_{d-1}^{(0)}=\partial\mathcal{M}_{d}^{L}=\partial\mathcal{M}_{d}^{R}$;

  3. 3.

Boundary description: A $(d-1)$-dimensional CFT on $\mathcal{M}_{d-1}^{(0)}$.

A consistent background of such system has to satisfies three sets of
differential equations– the bulk Einstein’s equation and the two brane
embedding equations with Neumann boundary condition for bulk metric
fluctuations near the brane. It was shown in Geng:2021mic  that there is no
entanglement island in one such consistent background that is relevant to the
study of black hole information paradox (as will be reviewed in Sec.2). The
essential reason was that now the subregion $\mathcal{R}$ is on the brane
$\mathcal{M}^{(R)}_{d}$ and due to diffeomorphism invariance of massless
gravity the subregion $\mathcal{R}$ itself should be determined dynamically.
It was shown in Geng:2021mic  that a natural dynamical principle gives a
vanishing $\mathcal{R}$ as well as its entanglement island $\mathcal{I}$.

Nevertheless, Miao:2022mdx ; Miao:2023unv  claimed that this conclusion that
there is no entanglement island in this wedge holography setup can change by
adding appropriately chosen DGP terms

$S_{DGP}=-\frac{1}{16\pi
G^{(L)}_{d}}\int_{\mathcal{M}^{(L)}_{d}}d^{d}x\sqrt{-h}R_{B}-\frac{1}{16\pi
G^{(R)}_{d}}\int_{\mathcal{M}^{(R)}_{d}}d^{d}x\sqrt{-h}R_{B}\,,$ (2)

where $R_{B}$ is the d-dimensional Ricci scalar on the brane. This
modification adds internal gravitational dynamics on the branes such that in
the intermediate picture the theory is no longer purely induced from the bulk.

The first aim of this paper is to carefully clarify the relevant question to
entanglement island we asked in Geng:2021mic  in each description of the wedge
holography setup. This can be used to show that there is indeed no
entanglement island with physically reasonable values of $G_{d}^{(L)}$ and
$G_{d}^{(R)}$. Moreover, this motivates another basic consideration that can
be used to further constrain the values of $G_{d}^{(L)}$ and $G_{d}^{(R)}$.
This consideration is based on the consistency of wedge holography in the
boundary description. As we will see that this result should be understood as
the statement that causality and holography in anti-de Sitter space can be
used to constrain low energy effective theories. This is the extension of the
earlier program of using holography to constrain low energy effective theories
in de Sitter space Geng:2019bnn  to anti-de Sitter space.111See
Montero:2018fns  for another example of using holography to constrain low
energy effective theories in anti-de Sitter space. The second aim of this
paper is to give a precise description of the intermediate picture and discuss
possible coarse-graining protocols under which we may be able to define a
subregion in a gravitational theory and hence may have entanglement island and
the problems of these protocols.

The paper is organized as follows. In Sec.2 we will review our study in
Geng:2021mic  to carefully clarify the questions we considered in each
description of the wedge holography. We will use this to show that there is in
fact no entanglement island in the setup considered in Miao:2022mdx ;
Miao:2023unv  due to diffeomorphism invariance. Then we show that the
consideration in the boundary description will give us two criteria to further
constrain the parameters in $G_{d}^{(L)}$ and $G_{d}^{(R)}$. In Sec.4 we will
derive the precise description of the intermediate description and show how we
could derive island formula in this description. Moreover, we discuss two
classes of coarse-graining protocols that is usually claimed to be able to
define a subregion in a gravitational universe and relax the diffeomorphism
invariance constraint on the existence of entanglement island in the
literature Ghosh:2021axl ; Dong:2020uxp . We show that it’s subtle for them to
be consistently defined and once they are consistently defined we either lose
entanglement island or have a nontrivial modification of the gravitational
theory. At the end, we will conclude in Sec.5.

$t$$t$$R_{I}$$R_{II}$ Figure 1: We show the Penrose diagram of an eternal
black in AdSd coupled to $d$-dimensional baths that is used in the calculation
of the Page curve of black hole radiation. The baths are the shaded green
region and the geometry of the baths are $d$-dimensional Minkowski space. We
specify the two red vertical lines as the conformal boundary of the AdSd black
hole. We choose time evolution as indicated in the diagram. We also specify
two Cauchy slices of this time evolution as the two blue curves and on each of
them we denote the subsystem $R=R_{I}\cup R_{II}$ in red. We emphasize that
the Cauchy slices of this time evolution all go through the bifurcation
horizon so they don’t touch the black interior.

##  2 No Entanglement Island in Massless Gravity

In this section, we review the result that there is no entanglement island in
the wedge holography model of massless gravity in Geng:2021mic . We will
clarify the question that we considered in the intermediate description and
show that introducing DGP terms doesn’t in fact change this conclusion. At the
end, we will consider the question in the boundary description and see that it
could be used to constrain DGP terms in wedge holography.

###  2.1 The Setup

As we discussed in the introduction, a consistent background where we could
study wedge holography should satisfy

$\delta S_{1}=0\,,$ (3)

where $S_{1}$ is given by Equ. (1) and the boundary condition for the metric
fluctuation is of the Neumann type:

$\delta g_{\mu\nu}\neq 0\,,\quad i=L,R\,$ (4)

where $n_{i}$ denotes the unit normal direction of the (two) brane(s). This
gives us three equations

$\begin{split}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=0\,,\quad
K_{\mu\nu}=(K-T)h_{\mu\nu}\,,\end{split}$ (5)

where the second equation should be satisfied for each brane with their own
values of tension and $h_{\mu\nu}$ is the induced metric on the brane.

A general solution of all these three equations is hard to find either due to
the complicated backreaction of the brane to the bulk geometry
Almheiri:2019psy  or the nonstationarity of the branes in a generic bulk
solution of the Einstein’s equation (the first equation) Kraus:1999it .
Nevertheless, it is known that a nice natural solution exists which can be
used to study black hole physics in wedge holography Geng:2021mic . It is
known as a black string. The bulk metric is given by

$ds^{2}=\frac{1}{u^{2}\sin^{2}\mu}\left[-h(u)dt^{2}+\frac{du^{2}}{h(u)}+d\vec{x}^{2}+u^{2}\,d\mu^{2}\right],\quad
h(u)=1-\frac{u^{d-1}}{u_{h}^{d-1}},$ (6)

where the polar coordinate $\mu$ takes its value in $(0,\pi)$,
$\\{\mu=0\\}\cup\\{\mu=\pi\\}$ corresponds to the asymptotic boundary and
$u_{h}$ is the parameter parametrizing the temperature of the black string.
The two branes are located at constant-$\mu$ hypersurfaces
$\mu=\mu_{L}<\frac{\pi}{2}$ and $\mu=\mu_{R}>\frac{\pi}{2}$ and their tensions
are given by

$T_{L}=(d-1)\cos\mu_{L}\,,\quad T_{R}=-(d-1)\cos\mu_{R}\,,$ (7)

which are both positive.

We can see that this (d+1)-dimensional geometry is a foliation of
d-dimensional AdS Schwartzschild black holes with a foliation parameter $\mu$.
The two branes will intersect each other at $u=0$ (a (d-1)-dimensional
manifold) on the asymptotic boundary. Since the two branes are of positive
tension, they will cutoff the bulk region from them to their closest part of
the asymptotic boundary ($\mu=0$ for left brane and $\mu=\pi$ for right
brane). Hence the left over region in the bulk is a (d+1)-dimensional wedge
bounded by the two branes and the geometries on the two branes are AdSd
Schwartzschild black holes. (Fig.2.) We notice that the intermediate picture
Penrose diagram of this case is different from the usual case with a flat bath
Fig.1.

$\mu_{L}$$\pi-\mu_{R}$$\bullet$$\circ$ Figure 2:  The black string in wedge
holography. We embed two KR branes in the setup as the two solid green lines.
The shaded region behind them is cutoff. The physical space in the bulk is the
wedge-shaped region bounded by the two branes. The two branes intersect at the
red defect on the asymptotic boundary of the bulk. The dashed black line is
the black string horizon separating the exterior and interior regions. The
dashed green lines are hypersurfaces of constant $\mu$. The geometry on each
constant-$\mu$ slice, including the two branes, is the AdSd Schwartzschild
black hole. We show one asymptotic exterior of the geometry and we always put
in mind that the geometry we consider is actually maximally extended which has
two exteriors.

###  2.2 The Question and the Result without DGP Terms

The question we asked in Geng:2021mic  is that in the intermediate picture
whether we could find entanglement island $\mathcal{I}$ on the left brane for
a well-defined subregion $\mathcal{R}$ on the right brane. To do so we
considered $\mathcal{R}$ to be of the type that is relevant to the computation
of Page curve in the Karch-Randall braneworld. That is that $\mathcal{R}$
should straddle the two exteriors of the black hole geometry on the right
brane (see Fig.3). Moreover, for $\mathcal{R}$ itself to be well-defined and
consistent with diffeomorphism invariance in massless gravity we defined it
dynamically in Geng:2021mic  by minimizing the entropy functional over both
$\mathcal{I}$ and $\mathcal{R}$:

$S=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I},\mathcal{R}\end{subarray}\,S_{\text{gen}}(\mathcal{R}\cup\mathcal{I})\,,\quad
S_{\text{gen}}(\mathcal{R}\cup\mathcal{I})=S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})\,.$
(8)

Here we notice that the generalized entropy doesn’t contain the usual area
term due to the fact that there is no internal gravitational dynamics on the
branes and, as we will discuss later (see Sec.4.1 for details), this would be
changed with DGP terms added. A more careful definition of $\mathcal{R}$ and
how Equ. (8) can be derived is discussed in Sec.4.2. The power of the Karch-
Randall braneworld models is that we can use the (d+1)-dimensional bulk to
compute the highly nontrivial quantity
$S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})$. It can be computed by the
Ryu-Takayanagi formula Ryu:2006bv  by looking for entangling surfaces that are
homologous to $\mathcal{R}\cup\mathcal{I}$ and using the one $\gamma$ that
minimizes the area functional to be the one that compute
$S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})$:

$S=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I},\mathcal{R}\end{subarray}\frac{A(\gamma)}{4G_{d+1}}\,,$ (9)

where $A(\gamma)$ is the area of the minimal surface $\gamma$. This formula
tells us that we have to further minimize over the possible ending points for
the area functional of minimal area surface $\gamma$ and the output of this
minimization is the entanglement island $\mathcal{I}$ and a well-defined
subregion $\mathcal{R}$.

As it is standard for disconnected boundary subregions in AdS/CFT
correspondence, there two possible topologies of the minimal area surface
$\gamma$. One gives a connected entanglement wedge of
$\mathcal{R}\cup\mathcal{I}$ in the bulk and the other gives a disconnected
one (see Fig.4). We call them $\gamma_{c}$ and $\gamma_{dc}$ respectively.
$\gamma_{c}$ has two components one in each exterior of the bulk wedge and
$\gamma_{dc}$ has two components one close to each brane and all go through
the interior of the black string (if $\mathcal{R}\cup\mathcal{I}$ is non-
empty). Moreover, $\gamma_{c}$ is confined to a constant time slice in $t$ of
Equ. (6) and $\gamma_{dc}$ is not confined to such a slice and its area
depends on $t$ where $t$ is the time-coordinate of the Cauchy slice
(constant-$t$ slice) we take to define the subregion $\mathcal{R}$ as shown in
the intermediate picture Penrose diagram Fig.3 (the corresponding diagram in
the bulk is in Fig.4). We proved in Geng:2021mic  that with the minimization
procedure in Equ. (9) done each component of $\gamma_{c}$ must be the black
string horizon and so the entanglement island $\mathcal{I}$ and $\mathcal{R}$
must both be empty. Hence there can be no entanglement island in this model.

Nevertheless, we should be careful about the value we get from Equ. (9) as it
computes the fine-grained entanglement entropy of the subregion $\mathcal{R}$.
If $\mathcal{R}$ is empty then $S$ should be zero but this is not consistent
with the result we would obtain if we use $\gamma_{c}$ as the union of two
horizons. To resolve this issue, we should also consider $\gamma_{dc}$. The
output of the minimization Equ. (9) for $\gamma_{dc}$ is that $\gamma_{dc}$
shrinks all the way to the union of two points and is hence empty which gives
$0$— the smallest possible value of $S$ according to Equ. (9). Therefore, we
still have the result that both $\mathcal{I}$ and $\mathcal{R}$ are empty and
we also have the physically consistent result that the resulting entanglement
entropy calculated by Equ. (9) is zero.

$t$$t$$\partial\mathcal{I}$$\bullet$$\partial\mathcal{I}$$\bullet$$t$$\partial\mathcal{R}$$\partial\mathcal{R}$$\bullet$$\bullet$
Figure 3: The Penrose diagram of the intermediate picture. We have two black
holes (i.e the geometry on the two branes) coupled to each other by gluing
them along their common asymptotic boundaries. The left asymptotic boundary of
the left black hole is glued with the right asymptotic boundary of the right
black hole. We take a constant time slice (the blue slices union the green and
orange slices) of the time evolution that we are considering in the
calculation of entanglement island and Page curve. The time is defined on the
asymptotic boundary (the defects). The orange region is a putative subregion
$\mathcal{R}$ and the green region is a putative entanglement island
$\mathcal{I}$ of the subregion $\mathcal{R}$. As opposed to the situation in
Fig.1 the bath (the green shaded region) is now an AdSd black hole.

$\mu_{L}$$\pi-\mu_{R}$$\bullet$$\circ$$\bullet$$\bullet$$\partial\mathcal{I}$$\partial\mathcal{R}$$\gamma_{c1}$$\mu_{L}$$\pi-\mu_{R}$$\bullet$$\circ$$\bullet$$\bullet$$\partial\mathcal{I}$$\partial\mathcal{R}$$\gamma_{c2}$
(a) The Connected Case

$\mu_{L}$$\pi-\mu_{R}$$\bullet$$\circ$$\bullet$$\bullet$$\partial\mathcal{I}$$\partial\mathcal{R}$$\gamma_{dcL}$$\mu_{L}$$\pi-\mu_{R}$$\bullet$$\circ$$\bullet$$\bullet$$\partial\mathcal{I}$$\partial\mathcal{R}$$\gamma_{dcR}$
(b) The Disconnected Case

Figure 4: The bulk diagram indicating the bulk configuration corresponding to
the constant time slice in Fig.3. The two horizons in each panel should be
identified as the horizon on each panel should be the bifurcation horizon. The
entangling surface $\gamma_{c}=\gamma_{c1}\cup\gamma_{c2}$ in the connected
case lives on this constant time slice (and we use solid red lines for them).
Nevertheless, the entangling surface
$\gamma_{dc}=\gamma_{dcL}\cup\gamma_{dcR}$ is not confined on this constant
time slice and is time-dependent (we draw them by dashed red lines to indicate
the fact that they are not confined on this diagram).

###  2.3 Adding DGP Terms Doesn’t Change the Result

If we have DGP terms Equ. (2) added to the system Equ. (1), the entropy
functional $S_{\text{gen}}(\mathcal{R}\cup\mathcal{I})$ would have additional
area terms than just
$S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})$.222Moreover, the brane
embedding equation will be different from the second equation in Equ. (5) with
a d-dimensional Einstein term added. However, it can be shown that the
constant$-\mu$ slices of Equ. (6) still satisfy the resulting brane embedding
equation with the tensions modified comparing to Equ. (7) Chen:2020hmv ;
Perez-Pardavila:2023rdz ; Geng:2023second . As a result, instead of Equ. (8)
we would have

$S=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I},\mathcal{R}\end{subarray}\,S_{\text{gen}}(\mathcal{R}\cup\mathcal{I})\,,\quad
S_{\text{gen}}(\mathcal{R}\cup\mathcal{I})=S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})+\frac{A(\partial\mathcal{I})}{4G_{d}^{(L)}}+\frac{A(\partial\mathcal{R})}{4G_{d}^{(R)}}\,,$
(10)

where $A(\partial\mathcal{I})$ and $A(\partial\mathcal{R})$ are the area of
the boundary of the entanglement island $\mathcal{I}$ and the subregion
$\mathcal{R}$. Similar to Equ. (9) this quantity can be computed purely
geometrically where we could use the Ryu-Takayanagi formula to replace
$S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})$ by
$\frac{A(\gamma)}{4G_{d+1}}$:

$S=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I},\mathcal{R}\end{subarray}\Bigg{(}\frac{A(\gamma)}{4G_{d+1}}+\frac{A(\partial\mathcal{I})}{4G_{d}^{(L)}}+\frac{A(\partial\mathcal{R})}{4G_{d}^{(R)}}\Bigg{)}\,,$
(11)

where $\gamma$ is a bulk minimal area surface that is homologous to
$\mathcal{R}\cup\mathcal{I}$ on the two branes. Again there are two possible
topologies of $\gamma$— $\gamma_{c}$ connects $\partial\mathcal{I}$ and the
left brane to $\partial\mathcal{R}$ on the right brane and $\gamma_{dc}$ has
two disconnected components homologous to $\mathcal{I}$ and $\mathcal{R}$
separately.

Nevertheless, for the interpretation of Equ. (11) as calculating entanglement
entropy we have to ensure that the functional to be minimized is positive for
any minimal area surface $\gamma$, otherwise the result of the minimization
would give us a negative value which cannot be interpreted as the entanglement
entropy of a quantum state. For such choices of the parameters $G_{d+1}$,
$G_{d}^{(L)}$ and $G_{d}^{(R)}$ the result of the minimization is again that
both $\mathcal{R}$ and $\mathcal{I}$ are empty (the orange and blue region in
Fig.3 all shrink to zero) and $S=0$. This tells us that there is no
entanglement island.

###  2.4 A Question in the Boundary Description

In Sec.2.2 and Sec.2.3, we studied a question that is relevant to entanglement
island in the intermediate picture of the wedge holography and we got
consistent results from holography with the general principle that
entanglement island is inconsistent with long-range gravity Geng:2021hlu .
However, as we reviewed in the introduction, the wedge holography has three
equivalent descriptions and for the internal consistency of wedge holography
we also have to make sure that the boundary description is self-consistent.

In the boundary description, the black string Equ. (6) with the the branes
Equ. (7) is described by the thermal-field double (TFD) state of two CFTd-1’s
each lives on one asymptotic defect Geng:2021mic . Moreover, the entanglement
entropy $S_{\text{defect}}$ between the two defects in this state is the
thermal entropy for each of the defects and it is captured by the black string
horizon $\mathcal{H}$ and is computed as

$S_{\text{defect}}=\frac{A_{\mathcal{H}}}{4G_{d+1}}+\frac{A_{\partial\mathcal{H}_{L}}}{4G_{d}^{(L)}}+\frac{A_{\partial\mathcal{H}_{R}}}{4G_{d}^{(R)}}\,,$
(12)

where $\partial\mathcal{H}_{L}$ and $\partial\mathcal{H}_{R}$ denotes the
induced horizons on the left and right branes respectively. In the absence of
DGP terms the expression only contains the first term. A basic consistency of
wedge holography is that the result Equ. (12) should be consistent with the
Ryu-Takayanagi formula. The Ryu-Takayanagi formula tells us that
$S_{\text{defect}}$ is computed by

$S=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\partial\gamma_{R},\partial\gamma_{L}\end{subarray}\Bigg{(}\frac{A(\gamma)}{4G_{d+1}}+\frac{A(\partial\gamma_{R})}{4G_{d}^{(L)}}+\frac{A(\partial\gamma_{L})}{4G_{d}^{(R)}}\Bigg{)}\,,$
(13)

where $\gamma$ is a bulk minimal area surface that connects the two branes and
homologous to the defect and $\partial\gamma_{R}$ and $\partial\gamma_{L}$ are
the cross-section of $\gamma$ with the right and left branes respectively (see
Fig.5). In other words, the consistency of wedge holography would require that
the output of the minimization in Equ. (13) is that the entangling surface
$\gamma$ should be the black string horizon $\mathcal{H}$.

For setups (combinations parameters $G_{d+1}$, $G_{d}^{(L)}$ and
$G_{d}^{(R)}$) where the the output of the minimization in Equ. (13) is not
the black string horizon $\mathcal{H}$, the wedge holography cannot be
consistently defined or these setups are not consistent with wedge holography.
Moreover, for these setups the entanglement wedge of the defect determined by
the RT surface $\gamma_{RT}$ would be smaller that the black string exterior
that include the defect we are considering. This is the same as saying that
the causal wedge of the defect is bigger than its entanglement wedge. Hence
the causality and holography are not consistent in these setups and these
setups should be in the swampland.

$\mu_{L}$$\pi-\mu_{R}$$\bullet$$\circ$$\gamma$$\bullet$$\bullet$$\partial\gamma_{L}$$\partial\gamma_{R}$$\bullet$$\bullet$$\partial\mathcal{H}_{L}$$\partial\mathcal{H}_{R}$$\mathcal{H}$
Figure 5: The calculation of the entanglement entropy of one of the defects.
The red surface $\gamma$ is a putative output of the minimization in Equ.
(13). For consistency of the wedge holography $\gamma$ should in fact be the
black string horizon i.e. the black dashed line.

##  3 Consistency of Wedge Holography and the Swampland Bounds

From our study in Sec.2.4 we see that the consistency of wedge holography
requires that in the background geometry Equ. (6) the black string horizon
$\mathcal{H}:u=u_{h}$ should compute the entanglement entropy between the two
defects in the TFD state. The formula of this entanglement entropy is given by
Equ. (12). Moreover, this result should be consistent with the Ryu-Takayanagi
formula which tells us that the black string horizon $\mathcal{H}$ should be
the solution of the minimization problem defined in Equ. (13).

These consistency conditions convey nontrivial messages as they input
important constrains on the DGP parameters $G_{d}^{(L)}$ and $G_{d}^{(R)}$.
Firstly, the entanglement entropy between the two defects has a clear
statistical meaning as the Hilbert spaces of the two defects are cleanly
factorized as a tensor product of two CFT${}_{\text{d-1}}$ Hilbert
spaces.333Here is a caveat that we are literally considering the Hilbert space
of a CFTd-1 which means that we shouldn’t take the infinite central charge
$c\rightarrow\infty$ limit. The reason is that in this limit the Hilbert space
is generated as a Fock space by single-trace operators and the resulting
theory doesn’t behave as a CFTd-1 due to the Hagedorn growth of the partition
function at high temperature Aharony:2003sx ; El-Showk:2011yvt . This tells us
that the entanglement entropy between them should be positive and finite (upto
an IR divergent factor due to the infinity volume). In other words the
algebras associated with the two defects are Type I Von-Neumann algebras
Haag:1992hx ; Witten:2018zxz ; Leutheusser:2021frk ; Leutheusser:2021qhd ;
Leutheusser:2022bgi ; Sorce:2023fdx ; Gomez:2023wrq  which again tells us that
the entanglement entropy between them is positive and finite. Moreover, the
requirement that the answer of the minimization problem Equ. (13) should be
the black string horizon $\mathcal{H}$ gives another constraint. This
requirement can be understood as a direct consequence of the entanglement
wedge reconstruction of holography which in our case says that the physics of
the defect is fully captured by the physics in the bulk region
$\Sigma_{\gamma_{\text{min}}}$ inclosed by the answer $\gamma_{\text{min}}$ of
the minimization problem Equ. (13) (the defect is on the boundary of
$\Sigma_{\gamma_{\text{min}}}$). This says that the bulk domain of dependence
of $\Sigma_{\gamma_{\text{min}}}$ should include or equal to the bulk domain
of dependence of the defect and meanwhile doesn’t intersect the bulk domain of
dependence of the other defect. In other words, this requirement comes from
the consistency between holography and causality. This condition exclusively
determines that $\gamma_{\text{min}}=\mathcal{H}$.

In summary, we have two swampland constraints on the DGP parameters
$G_{d}^{(L)}$ and $G_{d}^{(R)}$ from the consistency of wedge holography

  * •

Condition 1:
$S_{\text{defect}}=\frac{A_{\mathcal{H}}}{4G_{d+1}}+\frac{A_{\partial\mathcal{H}_{L}}}{4G_{d}^{(L)}}+\frac{A_{\partial\mathcal{H}_{R}}}{4G_{d}^{(R)}}\geq
0$ .

  * •

Condition 2: The answer of the minimization problem on the RHS of Equ. (13) is
the black string horizon i.e. $\gamma_{\text{min}}=\mathcal{H}$.

Here we quickly exploit the first constraint. In the background geometry Equ.
(6), we can compute

$\begin{split}&\frac{A_{\mathcal{H}}}{4G_{d+1}}+\frac{A_{\partial\mathcal{H}_{L}}}{4G_{d}^{(L)}}+\frac{A_{\partial\mathcal{H}_{R}}}{4G_{d}^{(R)}}\\\
=&V_{d-2}\Bigg{(}\frac{1}{4G_{d+1}}\int_{\mu_{L}}^{\mu_{R}}\frac{d\mu}{u_{h}^{d-2}\sin^{d}\mu}+\frac{1}{4G_{d}^{(L)}}\frac{1}{u_{h}^{d-2}\sin^{d-1}\mu_{L}}+\frac{1}{4G_{d}^{(R)}}\frac{1}{u_{h}^{d-2}\sin^{d-1}\mu_{R}}\Bigg{)}\,,\end{split}$
(14)

where $V_{d-2}$ is the transverse space volume and is always positive. Hence
we have the following constraint

$\frac{1}{4G_{d+1}}\int_{\mu_{L}}^{\mu_{R}}\frac{d\mu}{u_{h}^{d-2}\sin^{d}\mu}+\frac{1}{4G_{d}^{(L)}}\frac{1}{u_{h}^{d-2}\sin^{d-1}\mu_{L}}+\frac{1}{4G_{d}^{(R)}}\frac{1}{u_{h}^{d-2}\sin^{d-1}\mu_{R}}\geq
0\,.$ (15)

This constraint can be exploited numerically for a given dimension $d$. We
defer such an detailed exploration together with that of the Condition 2 to
Geng:2023second .

##  4 The Intermediate Picture and Coarse-Graining

In this section, we study more details of the intermediate picture. We firstly
show the precise description of the intermediate picture. Then we show how
Equ. (8) as well as Equ. (10) can be derived where the minimization over
$\mathcal{R}$ is a direct consequence of diffeomorphism invariance. At the
end, we discuss possible coarse-graining protocols that we can take
$\mathcal{R}$ in Equ. (8) and Equ. (10) as a fixed region (i.e. we don’t have
to minimize over $\mathcal{R}$) and the problem of these protocols.

###  4.1 The Precise Description of the Intermediate Picture

For the sake of convenience, let’s firstly consider the case without the DGP
terms which is the conventional description of the braneworld models
Randall:1999ee ; Randall:1999vf ; Karch:2000ct . Motivated by the early
studies of holography and braneworld Arkani-Hamed:2000ijo , it is usually
stated that the intermediate picture of the Karch-Randall braneworld is
described as a system of a conformal field theory with a UV-cutoff and coupled
to dynamical gravity in AdSd meanwhile this system in AdSd is coupled to a
d-dimensional bath glued along the conformal boundary of the AdSd. The bath
can be gravitational or nongravitational, for example in the wedge holography
model Equ. (1) the bath is gravitational and in the original version of Karch-
Randall braneworld Karch:2000ct  the bath is nongravitational.

This conclusion comes from the observations that first the Karch-Randall brane
cuts off part of the bulk geometry and second the boundary condition for bulk
metric fluctuation close to the brane is of the Neumann type Equ. (4).
Combined with the AdS/CFT correspondence Maldacena:1997re ; Gubser:1999vj ;
Witten:1998qj  the first observation tells us that the matter field on the
brane is dual to the bulk gravity and hence is a CFT with a UV cutoff and the
cutoff scale is set by the bulk brane position $\mu_{B}$. The second
obervation tells us that the metric on the brane is fluctuating i.e. the
intermediate picture is a gravitational theory.

Nonetheless, it is never made clear what this conclusion precisely means for
example given the CFTd that is dual to the bulk AdSd+1 gravity (without the
brane) then how one could get the theory on the brane (i.e. the intermediate
picture) when the brane is introduced.

The statement that the CFTd would then have a UV cutoff presumably means that
the matter theory in the intermediate picture is from an irrelevant
deformation of that CFTd and the cutoff scale denotes the strength of this
deformation. Moreover, such a deformation should be different from the usual
picture of the renormalization group (RG) flow where the UV cutoff scale comes
from momentum space Wilsonian procedure. There are two reasons that they are
different: 1) The Wilsonian procedure hides the UV details of the theory into
the effective coupling constants and the physics below the UV cutoff scale is
insensitive to this operation. This would predict that a light signal send
from the bulk to the brane will be reflected back to the bulk just like the
brane is not there.444More precisely, before the Wilsonian procedure there is
a deformation of the CFTd introduced. Such deformations in AdS/CFT are
generically multitrace deformations and their holographic duals are known
Witten:2001ua  as modifying the boundary conditions of bulk fields. This
clearly doesn’t make sense as this signal absorbing and emission procedure is
highly nonlinear in the intermediate picture and there is no reason that its
bulk dual would be as simple as the above expectation. 2) Fine-grained
quantities such as the replica entropy555The replica entropy is the entropy
calculated by the replica trick which however takes into account the conical
singularity of the replica manifold. For quantum field theory its value is
different from entanglement entropy if the theory is non-minimally coupled
with the metric. The replica entropy is called the black hole entropy in
Kabat:1995jq . The replica entropy is Wilsonian RG invariant but the
entanglement entropy is not Kabat:1995jq . is not sensitive to the Wilsonian
procedure Kabat:1995jq . This suggests that the subregion replica entropy is
invariant under the RG flow. This clearly contradicts the calculation based
the RT formula in the braneworld holography where the RT surface computing the
replica entropy666Analogous to the AdS/CFT results Lewkowycz:2013nqa , using
the replica trick and the holographic duality the replica entropy in the
intermediate picture is mapped to the area of the bulk RT surface with ending
points on the brane. never penetrates the brane but instead ending on it which
gives a cutoff scale dependent answer for the replica entropy. This issue is
partly resolved by the recent study of the holographic dual of the
$T\bar{T}$-deformation of CFT McGough:2016lol ; Gorbenko:2018oov ;
Caputa:2020lpa ; Ondo:2022zgf ; Kawamoto:2023wzj  where the CFT is deformed in
a controllable way and the entanglement entropy is shown to be non-invariant
under the deformation Donnelly:2018bef . Hence, the matter field in the
intermediate picture is the holographic dual of the cutoff AdSd+1 bulk and it
is described by a $T\bar{T}$-like deformation of the CFTd on the brane.

Furthermore, the statement that the brane part of the intermediate picture is
gravitational means that in the path integral description of the intermediate
picture the metric on the brane is fluctuating i.e. we are integrating over it
in the path integral. But it is not clear if this metric has its own kinetic
term (for example the Einstein-Hilbert term) or not. Interestingly, it was
proposed in Compere:2008us  that replacing Dirichlet boundary condition to
Neumann boundary condition in the AdS/CFT promotes the boundary metric to a
dynamical field in the path integral description but doesn’t add any kinetic
term of it. Hence, in our case there is no kinetic term for the metric in the
intermediate picture. This can be seen by the following consideration. We can
understand the perturbative Hilbert space (the Fock space) either from the
intermediate picture point of view or the bulk point of view. They are
equivalent since the intermediate picture and the bulk picture are dual to
each other. To construct this Hilbert space, we can do the perturbative
canonical quantization in the intermediate picture. When we consider the
situation where the metric is dynamical, if there is no kinetic term for the
metric then we don’t have creation and annihilation operators for the
graviton. Hence in this case we have equal number of creation and annihilation
operators to construct the Fock space representation of the perturbative
Hilbert space as in the case where the metric is not dynamical. Nevertheless,
if we have a kinetic term for the metric then we have creation and
annihilation operators for the graviton when the metric is dynamical.
Therefore, the (perturbative) Hilbert space would be larger than the case
where the metric is not dynamical. Meanwhile we can also construct the Hilbert
space by doing canonical quantization in the bulk picture. For the bulk
picture described by the action Equ. (1) with Neumann boundary condition Equ.
(4) for the bulk metric fluctuation, if we do canonical quantization then the
number of creation and annihilation operators associated with the bulk metric
is the same in the Dirichlet boundary case. The reason is that the Neumann
boundary condition in this case is equally as restrictive as the Dirichlet
boundary condition. This is because in the Neumann case the second equation in
Equ. (5) restricts the bulk metric fluctuation $\delta g_{\mu\nu}$ to satisfy
$\partial_{n}\delta g_{\mu\nu}-\frac{2T}{d-1}\delta g_{\mu\nu}=0$ near the
brane and in the Dirichlet case $\delta g_{\mu\nu}$ is restricted to be zero
near the brane so the two boundary conditions are just projecting into
different sets of bulk modes with equal dimensionality. We call these two sets
of modes $MD$ and $MN$. Furthermore, if we have a DGP term on the brane in the
bulk picture then the Hilbert space for the Dirichlet boundary condition case
is the same as the case without the DGP term as the bulk modes satisfying the
bulk equation of motion and the boundary condition are still $MD$.
Nevertheless, if we consider now the Neumann boundary condition case then the
boundary equation of motion for the bulk metric fluctuation $\delta
g_{\mu\nu}$ is naively given by the linearized equation

$\partial_{n}\delta g_{\mu\nu}-\frac{2T}{d-1}\delta
g_{\mu\nu}-\frac{G_{d+1}}{G_{d}}(\text{d-dimensional linearized Einstein's
equation for $\delta g_{\mu\nu}$})=0\,.$ (16)

So a bulk metric fluctuation $\delta g_{\mu\nu}$ can be expanded using the
modes from $MD\cup MN$. Let’s schematically denote such a fluctuation by

$\hat{\delta g_{\mu\nu}}=\hat{A}_{D}\delta g_{\mu\nu}^{D}+\hat{A}_{N}\delta
g_{\mu\nu}^{N}\,,$ (17)

where $\hat{A}_{D(N)}$ denotes the creation and annihilation operators in the
Dirichlet (Neumann) sector. This is the standard expression in the canonical
quantization. Nevertheless, we should be careful with the boundary equation of
motion Equ. (16) as obviously Equ. (17) doesn’t satisfy Equ. (16). More
precisely the left hand side of Equ. (16) reads (near the brane)

$\hat{A}_{D}\partial_{n}\delta
g_{\mu\nu}^{D}-\hat{A}_{N}\frac{G_{d+1}}{G_{d}}(\text{d-dimensional linearized
Einstein's equation for $\delta g_{\mu\nu}^{N}$})\,.$ (18)

There are two obstructions for this to be zero. The first is that
$\hat{A}_{D}$ and $\hat{A}_{N}$ are supposed to be independent operators. The
second is that in the one-brane case the bulk analysis implies that $\delta
g_{\mu\nu}^{N}$ are massive graviton modes Karch:2000ct  therefore

$(\text{d-dimensional linearized Einstein's equation for $\delta
g_{\mu\nu}^{N}$})=m^{2}\delta g_{\mu\nu}^{N}\,,$ (19)

where $m^{2}$ is the mass for the given mode $\delta g_{\mu\nu}^{N}$
(different modes $\delta g_{\mu\nu}^{N}$’s would have different mass). The
first obstruction is easy to understand as the equation of motion Equ. (16) is
actually an interacting equation (the interaction strength is controlled by
$\frac{G_{d}}{G_{d+1}}$) and for the (perturbative) canonical quantization (in
the weak interaction regime) we can forget about the interaction when we are
constructing the Fock space. Therefore, we can isolate the second order
derivative terms in Equ. (18) which is exactly the second term. Naively the
vanishing of this term is the interaction free equation of motion that is used
construct the Fock space. Nevertheless, the second obstruction tells us that
this expectation cannot be satisfied and the correct free equation of motion
we can use is a massive equation. Hence, in the one-brane case the DGP term
should be introduced together with a graviton mass term. This ensures Equ.
(19) which is a consistency condition. However, the choice of the mass
parameter is not arbitrary and it is controlled by the bulk analysis. For the
consistency with the picture that the graviton localized on the Karch-Randall
brane is the first massive mode $\delta g_{\mu\nu}^{N(1)}$ in $MN$
Karch:2000ct , we have to choose $m^{2}$ to be $m_{1}^{2}$ where $m_{n}^{2}$
($n=1,2,3\cdots$) is the mass square for the tower of modes in $MN$. This
tells us that the mode functions that we can use to construct the Fock space
are those from $MD\cup{\delta g_{\mu\nu}}^{N(1)}$. For each of them we have
the associated creation and annihilation operators. Hence now we see that the
Fock space (the perturbative Hilbert space) is larger than the Dirichlet
boundary condition case. This is also consistent with the result in the
intermediate picture (i.e. the modes are the original field theory (no metric
fluctuation) modes with the massive graviton modes (for a specific mass)).

From the above analysis, we see that the precise description of the
intermediate picture of the original version of the Karch-Randall braneworld
Equ. (1) is given by the following path integral for the brane part777For some
evidence of this result see Ondo:2022zgf ; Kawamoto:2023wzj .

$Z_{\text{intermediate}}^{\text{brane}}=\int[Dh_{\mu\nu}][D\phi_{\text{matter}}]_{h}e^{iS_{\text{CFT}+T\bar{T}\text{-like
deformation}}}[\phi;h]\,.$ (20)

Here we emphasize that the matter field measure also depends on the metric and
this is important for subtle quantum effect such as graviton mass
Geng:2023third . Again the full intermediate picture includes also a bath and
that can be easily included with Equ. (20) equipped. The background metric is
a solution of the induced gravitational equation

$\langle T_{\mu\nu}^{\text{matter}}\rangle=0\,,$ (21)

which is the saddle point approximation of the metric integration in Equ.
(20). Moreover, the DGP term on the brane can be now understood as adding a
kinetic term for the metric $h_{\mu\nu}$ in the intermediate picture (i.e. add
$\frac{1}{16\pi G_{d}}\int d^{d}x\sqrt{-g}R[h]$ to the exponent in Equ. (20))
meanwhile a proper graviton mass term should also be added for the consistency
with analysis in Karch:2000ct . Without the DGP term the replica entropy on
the brane is the matter field entanglement entropy (for a subregion which can
be defined diffeomorphism invariantly for example the exterior of the black
hole, see Sec.4.3).

###  4.2 The Derivation of the Modified Island Formula

In this section, we discuss the derivation of the formula Equ. (8) as well as
Equ. (10). For simplicity, we only consider the case without the DGP terms
i.e. Equ. (8). Then the result Equ. (10) in the case with the DGP terms is
easily obtained resorting to the calculation in Faulkner:2013ana .

Firstly, let’s consider the case of a fixed subregion $\mathcal{R}$. The
entanglement entropy of $\mathcal{R}$ can be calculated using the replica
trick

$S_{\mathcal{R}}=\lim_{n\rightarrow
1}\frac{1}{1-n}\ln\tr\big(\rho_{\mathcal{R}}^{n}\big{missing})=\lim_{n\rightarrow
1}\frac{1}{1-n}\ln Z_{n}\,,$ (22)

where $\rho_{\mathcal{R}}$ is the reduced density matrix of the subregion
$\mathcal{R}$. The appearance of the entanglement island is due to the the
replica wormhole contribution to the gravitational path integral
Almheiri:2019qdq  representation of the replicated partition function $Z_{n}$.
It is easy to see that we would get
$S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})$ with island included. The
question in our case is how to understand the
$\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I}\end{subarray}$. In the case that we have kinetic terms for the
metric it is the result of the saddle point approximation of the gravitational
path integral Lewkowycz:2013nqa ; Faulkner:2013ana ; Dong:2013qoa  where
$\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\ \mathcal{I}\end{subarray}$
comes from solving the resulting backreacted gravitational equation (and the
entropy functional will be $S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})$
together with additional geometric terms due to the nontrivial kinetic terms
for the metric in the total action). In our case, this can be understood by
the interesting property that the replica entropy is RG invariant Kabat:1995jq
. In other words, to compute $S_{\mathcal{R}}$ (with $\mathcal{R}$ fixed) we
can firstly integrating out the matter fields in the intermediate picture
which would give us an effective gravitational action $S_{eff}[h]$
Jacobson:1994iw .888For free fields $S_{eff}[h]$ can be computed by the heat
kernel method Solodukhin:2011gn . The resulting effective action is a higher
curvature gravity where the higher curvature expansion is controlled by the
cutoff scale of the matter fields. The leading order term is a cosmological
constant and the next the leading order term is the Einstein-Hilbert term
Solodukhin:2011gn . The RG invariance of the replica entropy tells us that
$S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})$ equals to the generalized Wald
entropy $S_{grav}(\mathcal{R}\cup\mathcal{I})$ of the region $R\cup I$
Dong:2013qoa  computed by $S_{eff}[h]$ as both of them are equal to the
replica entropy of the subregion $\mathcal{R}\cup\mathcal{I}$. Moreover, as a
gravitational theory the resulting $S_{\mathcal{R}}$ computed by $S_{eff}[h]$
is given by

$S_{\mathcal{R}}=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I}\end{subarray}S_{grav}(\mathcal{R}\cup\mathcal{I})\,,$ (23)

for the same reasons as in AdS/CFT Lewkowycz:2013nqa ; Faulkner:2013ana ;
Dong:2013qoa . So using the RG invariance of the replica entropy we have the
result

$S_{\mathcal{R}}=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I}\end{subarray}S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})\,.$
(24)

Here we emphasize that the right hand side is computed in the background
geometry as described in Fig.3 and it is a result of the saddle point
approximation of the path integral over metric as discussed at the end of
Sec.4.1.

Nevertheless, we’ve ignored the fact that $\mathcal{R}$ is a subregion in a
gravitational universe. The definition of $\mathcal{R}$ should be consistent
with diffeomorphism invariance and holography. A natural choice would be that
before we perform the path integral over the metric in the universe (i.e. the
gravitational bath) that contains $\mathcal{R}$ we define $\mathcal{R}$ for
each geometry by taking $\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{R}\end{subarray}$ of $S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})$
in that geometry. As a result, a diffeomorphism invariant quantity we get is

$S=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I},\mathcal{R}\end{subarray}S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})\,,$
(25)

and it is the entanglement entropy of the resulting radiation region
$\mathcal{R}_{\text{min}}$ from the extremization and minimization. The right
hand side in this formula is computed in the geometry of Fig.3 as a result of
the saddle approximation of the metric integration and
$\mathcal{R}_{\text{min}}$ lives on this geometry as well.

As we discussed in Sec.2.2 this formula tells us that both the entanglement
island $\mathcal{I}$ and the radiation region $\mathcal{R}$ are empty. This is
a direct consequence of diffeomorphism invariance.

###  4.3 Robustness of the Result Under Coarse Graining

There are possible coarse-graining protocols that we could have entanglement
island in the intermediate picture of the wedge holography model.

There are two types of such protocols Ghosh:2021axl ; Dong:2020uxp . The first
type of protocols is to turn off gravity on the universe that contains the
radiation region with the hope that we wouldn’t have to impose diffeomorphism
constraints and so able to specify arbitrary $\mathcal{R}$ Ghosh:2021axl .
Then the calculation reduces to the standard calculation of a nongravitational
bath Geng:2021mic  where the entanglement island and nontrivial Page curve
emerge. The second type of protocols is to specify $\mathcal{R}$ in a
diffeomorphism invariant way which however could give a nonvanishing result
for $\mathcal{R}$. An example is Dong:2020uxp  where the boundary
$\partial\mathcal{R}$ of $\mathcal{R}$ is defined by firstly choosing a time
band $\Gamma$ on the asymptotic boundary and then taking the domain of
dependence of the complement of $\mathcal{R}$ as the bulk (the bulk of the
gravitational bath) domain of dependence of this time band $\Gamma$. This is
done for each metric on the bath that we are integrating over (see Fig.6).
Then using the saddle point approximation for the metric integration we get
$\mathcal{R}$ as specified by the above protocol on the black hole geometry
(see Fig.7).

However, there are various problems with these coarse-graining protocols. The
first class requires a consistent decoupling of gravity in a gravitational
theory. In Einstein’s gravity this is usually claimed to be achieved by
sending the Newton’s constant $G_{N}$ to zero which will localize the path
integral over the metric to the metric that satisfies Einstein’s field
equations and with fluctuations suppressed.999Up to the subtlety of the
introduction of free graviton Donoghue:1994dn . In our case, the gravity is
induced and the induced Newton’s constant is controlled by the cutoff scale of
the matter field. Hence to decouple gravity in this way we have to tune up the
cutoff scale of the matter field. Hence, in the bulk description, the
consistent way to decouple gravity is to dial the bath brane all the way to
the conformal boundary. And we know that when the gravity is decoupled in the
bath in this way the graviton in the gravitational universe (the brane) that
is coupled to the bath becomes massive Porrati:2001db ; Karch:2000ct . The
second class relies on the specification of a boundary time band $\Gamma$
which however wouldn’t be consistent with holography. The reason is that due
to holography the boundary has a unitary dynamics with a time evolution
generator and so the time band essentially encode all the physics of the
boundary Marolf:2008mf ; Jacobson:2019gnm ; Raju:2020smc ; Chowdhury:2021nxw ;
Raju:2021lwh . Or in other words a consistent choice of the time band $\Gamma$
with holography would be to choose $\Gamma$ as the whole boundary. This will
again give $\mathcal{R}=\emptyset$.101010Including the Hamiltonian into the
algebra is important for holography otherwise many techniques in quantum
gravity which makes use of the existence of a time evolution, such as the
gravitational path integral, would breakdown.

$\Gamma$$\Gamma$$\partial\mathcal{R}$$\partial\mathcal{R}$$\bullet$$\bullet$
Figure 6: A demonstration of the protocol proposed in Dong:2020uxp  to specify
$\mathcal{R}$ in a gravitational universe diffeomorphism invariantly. The red
boundary lines are the fixed asymptotic boundaries where $\Gamma$ (the blue
intervals) is a time band and the bulk geometry is not fixed. The proposal is
that for each bulk geometry we specify $\mathcal{R}$ in the following way. We
first determine the bulk domain of dependence of $\Gamma$ (the bulk region
enclosed by $\Gamma$ and the green lines) whose boundary is
$\partial\mathcal{R}$ and the domain of dependence of $\mathcal{R}$ is
determined as shown in the figure as the interior of the orange diamond. Then
we have to integrate over the geometries.
$\Gamma$$\Gamma$$\partial\mathcal{R}$$\partial\mathcal{R}$$\bullet$$\bullet$
Figure 7: For simplicity, we only show the Penrose diagram of the
gravitational bath. The saddle point approximation of the metric integral Equ.
(21) determines the background geometry to be a black hole. The subregion
$\mathcal{R}$ in this diagram is determined by the protocol as discussed in
Fig.6.

##  5 Conclusion

In this paper, we provide a careful check of the conclusion that entanglement
island is not consistent with massless gravity in a model of massless gravity
constructed using the Karch-Randall braneworld- the wedge holography model.
This model is doubly holographic and it has three equivalent descriptions. We
discovered that the consistency of holography in this model provides
interesting bounds on the DGP parameters. Moreover, we provide a careful
analysis of the intermediate description of this doubly holographic model and
we see that it is described by an induced gravity theory with the matter field
as a $T\bar{T}$-like deformation of the CFTd that duals to the gravity in the
AdSd+1 bulk (with no brane). We also show that to match the bulk analysis a
graviton mass term should be introduced when we introduce a DGP term and the
mass square is given by that of the first normalizable KK mode for bulk
graviton with Neumann boundary near the brane. We used the intermediate
picture to show that entanglement island doesn’t exist due to diffeomorphism
invariance. At the end, we discussed two classes of coarse-graining protocols
that could relax the constraint from diffeomorphism invariance on the
existence of entanglement island. We found that defining them consistently is
subtle and once they are defined consistently we either lose entanglement
island or we modify the gravitaitional theory on the gravitational universe
(the brane)that is coupled to the bath to be massive .

## Acknowledgements

We thank Andreas Karch, Carlos Perez-Pardavila, Suvrat Raju, Lisa Randall,
Marcos Riojas, Sanjit Shashi and Merna Youssef for relevant collaborations. We
thank Amr Ahmadain, Juan Maldacena, Rongxin Miao, Rashimish K. Mishra and
Yasunori Nomura for relevant discussions. This work is supported by the grant
(272268) from the Moore Foundation “Fundamental Physics from Astronomy and
Cosmology”.

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