††institutetext: aDepartment of Physics, University of Washington, Seattle,
WA, 98195-1560, USA.††institutetext: bHarvard University, 17 Oxford St.,
Cambridge, MA, 02138, USA.††institutetext: cDepartment of Physics and
Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver,
B.C. V6T 1Z1, Canada.

# Holographic BCFTs and Communicating Black Holes

Hao Genga, Severin Lüstb, Rashmish K. Mishrab and David Wakehamc<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the AdS/BCFT duality between two-dimensional conformal field theories
with two boundaries and three-dimensional anti-de Sitter space with two Karch-
Randall branes. We compute the entanglement entropy of a bipartition of the
BCFT, on both the gravity side and the field theory side. At finite
temperature this entanglement entropy characterizes the communication between
two braneworld black holes, coupled to each other through a common bath. We
find a Page curve consistent with unitarity. The gravitational result,
computed using double-holographically realized quantum extremal surfaces,
matches the conformal field theory calculation.
At zero temperature, we obtain an interesting extension of the AdS3/BCFT2
correspondence. For a central charge $c$, we find a gap
$(\frac{c}{16},\frac{c}{12})$ in the spectrum of the scaling dimension
$\Delta_{\text{bcc}}$ of the boundary condition changing operator (which
interpolates mismatched boundary conditions on the two boundaries of the
BCFT). Depending on the value of $\Delta_{\text{bcc}}$, the gravitational dual
is either a defect global AdS3 geometry or a single sided black hole, and in
both cases there are two Karch-Randall branes.

##  1 Introduction

Recent approaches to understanding the black hole information problem using
the AdS/CFT correspondence Penington:2019npb ; Almheiri:2019psf  emphasize the
role of entanglement islands in restoring the unitarity of the underlying
dynamics. Models of entanglement islands in higher dimensions are most cleanly
realized in the Karch-Randall (KR) braneworld Karch:2000ct ; Karch:2000gx ,
later refined to the so-called AdS/BCFT correspondence Takayanagi:2011zk ;
Fujita:2011fp . In these models Geng:2020qvw ; Almheiri:2019psy  (see also
Krishnan:2020fer ; Ling:2020laa ; Chen:2020hmv ; Chen:2020uac ; Chen:2020tes ;
Bousso:2020kmy ; Hernandez:2020nem ; Bhattacharya:2020uun ;
Chandrasekaran:2020qtn ; Chou:2020ffc ; Almheiri:2019hni ; Rozali:2019day ;
Bak:2020enw ; Cooper:2019rwk ; Laddha:2020kvp ; Chowdhury:2020hse ;
Basak:2020aaa ; Kawabata:2021hac ; Wang:2021woy ; Deng:2020ent ;
Caceres:2020jcn ; May:2021zyu ; Karlsson:2021vlh ; Karananas:2020fwx ;
Goto:2020wnk ; Matsuo:2020ypv ; Verheijden:2021yrb ; Anderson:2021vof ;
Bhattacharya:2021jrn ; Wang:2021mqq ; Aalsma:2021bit ; Kim:2021gzd ;
Hollowood:2021nlo ; Ghosh:2021axl ; Neuenfeld:2021bsb ; Kawabata:2021vyo ;
Reyes:2021npy ; Akal:2020twv ; Akal:2021foz  for related studies), the black
hole is living on the KR brane and is coupled to a finite temperature bath
modeled by a conformal field theory (CFT) on a manifold with a boundary. The
bath absorbs the radiation from the black hole and the entanglement island of
a subsystem of the bath is the component of its entanglement wedge
disconnected from it. The entanglement island lives on the KR brane and
contains the black hole interior.

In these early models, the bath was non-gravitating, but a recent
generalization of the KR construction involves gravitating baths Geng:2020fxl
. Several new features of entanglement islands emerge in this setup, and in
particular, it becomes possible to study the information transfer between two
black holes using the recently proposed wedge holography Bachas:2017rch ;
Akal:2020wfl ; Miao:2020oey . Here, “information transfer” is characterized by
the entanglement entropy between complementary subsystems of the wedge-
holographic field theory dual.

In Sully:2020pza ; Rozali:2019day , entanglement islands were studied for a
three-dimensional bulk geometry, with a non-gravitating bath. Interestingly,
in these cases the entanglement entropy can be computed using both the bulk
gravitational description and its dual boundary conformal field theory (BCFT)
description. On the gravity side, the entanglement entropy is computed using
the Ryu-Takayanagi (RT) surface Ryu:2006bv ; Ryu:2006ef , while on the CFT
side, using correlation functions of twist operators Calabrese:2009qy . These
two computations match each other under a natural extension of the conditions
for holographic CFTs (CFTs with a simple holographic dual- Einstein’s gravity)
to BCFTs Hartman:2013mia , namely large central charge and sparseness of the
spectrum of light operators.

In this paper, we merge these two lines of research, providing both a purely
field theoretic and a gravitational computation for two communicating black
holes. The black holes are coupled to a common non-gravitating bath, but from
the point of view of one of the black holes, the bath it is coupled to is
composed of a gravitating and a non-gravitating part. This is to be contrasted
with the original constructions Almheiri:2019psy ; Almheiri:2019yqk  when the
bath (from the point of view of the only black hole) was purely non-
gravitating, and the recent work Geng:2020fxl  where the bath (from the point
of view of one of the two black holes) was purely gravitating.

The field theory setup we consider is a two dimensional CFT living on a
strip111The strip is the time evolution of an interval. with generically
different conformal boundary conditions Cardy:2004hm  on the two edges. The
dual gravitational setup is a three-dimensional AdS bulk with two two-
dimensional KR branes, and with induced black holes living on the branes. The
asymptotic infinities of these braneworld black holes live on the asymptotic
conformal boundary of the three-dimensional bulk, but are separated by a
finite distance along the asymptotic boundary. This setup is to be contrasted
with the wedge holography proposal Akal:2020wfl  which takes the separation of
the two KR branes to zero, and for which the proposed field theory dual is
one-dimensional.

We will compute the entanglement entropy of a simple bipartition of the strip
in both its gravity description and the field theory description, and show
that they agree, under the usual conditions for holographic CFTs, namely
vacuum dominance for the correlator of twist operators. In particular, we are
not sensitive to microphysics beyond the proposed ansatz for the vacuum
sector. Interestingly, assigning mismatched boundary conditions on the two
edges of the strip will not only affect the two KR branes but also change the
geometry of the bulk dual itself. The reason for this is that mismatched
boundary conditions source a gradient for fields on the strip which results in
a non-vanishing kinetic energy, therefore changing the bulk geometry. This
insight allows us to generalize the AdS3/BCFT2 correspondence for a
(holographic) CFT2 living on a strip with generic conformal boundary
conditions.

To be more specific, we call the lowest energy eigenstate of the Hamiltonian
on the strip the ground state, and denote its energy by $E_{\text{gs}}$.
Denoting the central charge by $c$, $E_{\text{gs}}$ is generically larger than
$-\frac{c}{12}$, given different boundary conditions on the two ends. We will
distinguish this state from the state corresponding to the identity operator,
the vacuum state, which has energy $E=-\frac{c}{12}$. The difference between
the ground state energy $E_{\text{gs}}$ and the vacuum energy
$E=-\frac{c}{12}$ is given by $\Delta_{\text{bcc}}$, the scaling dimension of
the so-called boundary condition changing operator.

In the presence of an eternal black hole, the BCFT is in the thermal field
double (TFD) state Maldacena:2001kr  which has a finite temperature and the
effects of $\Delta_{\text{bcc}}$ are screened. At zero temperature, on the
other hand, we can determine the effect of a non-vanishing
$\Delta_{\text{bcc}}$ on the bulk geometry explicitly. Qualitatively different
behaviours arise depending on whether the ground state energy is positive or
negative, i.e. whether $\Delta_{bcc}$ is larger than or smaller than
$\frac{c}{12}$. We find that for a positive ground state energy the bulk
geometry is a single sided black hole with mass
$M=E_{\text{gs}}=\Delta_{\text{bcc}}-\frac{c}{12}>0$, while for a ground state
with negative energy $E_{\text{gs}}=\Delta_{\text{bcc}}-\frac{c}{12}<0$ it is
a global AdS3 geometry with a conical defect. Interestingly, in the second
case there appears to be a consistency condition which constraints
$\Delta_{bcc}$ to be smaller than $\frac{c}{16}$. Therefore, the holographic
duality suggests that there is a gap $(\frac{c}{16},\frac{c}{12})$ in the
spectrum of $\Delta_{bcc}$. Moreover, when $\Delta_{bcc}<\frac{c}{16}$, the
calculations of the entanglement entropy on the bulk and boundary sides can
only be consistently matched if the KR branes have non-positive tension,
predicting in turn also non-positive boundary entropies in the BCFT
description (note however that the overall entropy is still positive as it
must be).

We emphasize that in a given conformal field theory $\Delta_{bcc}$ can be in
principle computed if the boundary conditions are known. However,
$\Delta_{bcc}$ (as a function of the boundary conditions) is not universal but
depends on the microscopic details of the theory. In the gravitational
description, the boundary conditions are mapped to the tensions of the branes.
Still, we do not expect to be able to determine $\Delta_{bcc}$ in our bottom-
up dual gravitational model explicitly, as it does not contain any information
about the microscopic details of a specific boundary theory. Nonetheless, we
can choose $\Delta_{bcc}$ and the brane tensions independently (within the
previously mentioned constraints) to match our gravitational computations with
the field theory results.

This paper is organized as follows. In Sec. 2 we review relevant previous work
and set up the context for our study. In Sec. 3 we provide the calculations
for the entanglement entropy on the CFT side. In Sec. 4 we compute this
entanglement entropy on the gravity side and show the agreement with CFT side.
We conclude in Sec. 5 with discussions and possible future directions.

##  2 Review of previous work

In this section, we review the relevant background material Rozali:2019day ;
Sully:2020pza ; Akal:2020wfl ; Geng:2020fxl  and set up the context for our
calculations. We start with a lightning review of the recently proposed wedge
holography Akal:2020wfl ; Miao:2020oey  from the point of view of Karch-
Randall braneworlds Karch:2000gx ; Karch:2000ct  and the AdS/CFT
correspondence. Then we come to the system we want to study in this paper: two
communicating black holes. This system was first studied in higher dimensions
using holographic tools in Geng:2020fxl , an approach we review in detail. We
then discuss the BCFT calculation of entanglement entropy outlined in
Sully:2020pza .

###  2.1 Wedge holography

The AdS/CFT correspondence Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj
states that quantum gravity in an asymptotically AdSd+1 space is dual to a
$d$-dimensional conformal field theory living on its asymptotic boundary. This
asymptotic boundary is usually called the conformal boundary as the isometry
group of AdSd+1 is the conformal group of a $d$-dimensional conformal field
theory.

The observation that an AdSd+1 space can be foliated by an AdSd space (see
e.g. Karch:2000ct ) makes this correspondence even more powerful, and allows a
way to relate descriptions in spacetimes with a difference of two (spacelike)
dimensions. This realization has been fruitful in diverse setups, e.g. in the
context of asymptotic symmetries of AdS space Mishra:2017zan , and in the
context of black hole information paradox Almheiri:2019hni . A dictionary for
this codimension two holography duality was given in Akal:2020wfl ;
Miao:2020oey , using which observables of interest were calculated there.222It
is interesting to notice that Miao:2021ual  recently generalized these studies
to codimension-n holography.

Consider such a foliation of AdSd+1 space by AdSd space (represented by dashed
blue lines in Fig. 1a). These slices meet each other on the AdSd+1 conformal
boundary at the defect (the red dot in Fig. 1a), which is the conformal
boundary for each AdSd slice. To see this, it is simplest to start from the
standard Poincaré patch of the AdSd+1, with metric

$ds^{2}=\frac{-dt^{2}+dw^{2}+d\vec{x}^{2}+dz^{2}}{z^{2}}\,,\quad z>0\,,$ (1)

and perform the coordinate transformation

$w=u\cos\mu\,,\quad z=u\sin\mu\,.$ (2)

This yields the metric

$ds^{2}=\frac{1}{\sin^{2}\mu}\left(\frac{-dt^{2}+d\vec{x}^{2}+du^{2}}{u^{2}}+d\mu^{2}\right)\,,\qquad
0<\mu<\pi\,,\;u>0\>.$ (3)

Each constant $\mu$ slice (dashed blue lines in Fig. 1a) is a copy of AdSd
with conformal boundary at $u=0$. Then we can generate a wedge in the
$(d+1)$-dimensional bulk by putting two positive tension branes along the
$\mu=\theta_{1}$ and $\mu=\pi-\theta_{2}$ slices (the two black lines in Fig.
1a). These are called Karch-Randall branes Karch:2000ct ; Karch:2000gx , and
their tensions are determined by solving the Israel’s junction conditions
Kraus:1999it ; Takayanagi:2011zk ; Fujita:2011fp . In the bulk, the two KR
branes remove the portion of the spacetime behind them (gray region in Fig.
1a),333More precisely, in string theory constructions, the KR branes are the
locations where extra compact dimensions degenerate Chan:2000ms  and so are
also called the end-of-world (EOW) branes. See Uhlemann:2021nhu  for a recent
string theory realization of the wedge holography and communicating black
holes. giving us a wedge as the bulk.

$\theta_{1}$$\theta_{2}$$\bullet$$\circ$$w$$z$$\mu$$u$ (a) Wedge Holography

$\theta_{1}$$\theta_{2}$$\bullet$$\circ$$\bullet$$\circ$$L$ (b) UV Regularized
Wedge Holography

Figure 1: (a) An illustration of wedge holography: the bulk geometry is a part
of empty AdSd+1 between two end-of-world Karch-Randall branes (black lines),
meeting each other on the conformal boundary of AdSd+1 at their common
boundary (the red dot). The bulk shaded regions behind the branes are removed,
leaving only a wedge in the bulk. Wedge holography states that the
gravitational physics in the bulk wedge is dual to a $(d-1)$-dimensional
conformal field theory living on the red dot (the defect). (b) The wedge
holography proposal can be UV regularized by considering a BCFTd with two
boundaries (the two red dots). By sending the separation $L$ between the two
boundaries to zero, one goes back to the wedge.

The AdS/CFT correspondence for braneworlds ArkaniHamed:2000ds  tells us that
quantum gravity in the $(d+1)$-dimensional wedge is dual to UV cutoff CFTs
living on the two KR branes, coupled to dynamical gravity. Hence, this lower
dimensional dual description is also a theory of quantum gravity, with two
sectors each living on an AdSd space. Using the standard AdS/CFT
correspondence Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj , we can
further dualize this $d$-dimensional description to a $(d-1)$-dimensional CFT
living on the common conformal boundary of the branes, the defect itself. In
principle, this $(d-1)$-dimensional description has two sectors corresponding
to the two KR branes.

This co-dimension two wedge holography can be extrapolated from the standard
AdS/BCFT duality by first regularizing the UV, and then taking the regulator
to zero Akal:2020wfl . The most straightforward regularization separates the
two Karch-Randall branes by a distance $L$ (see Fig. 1b) on the conformal
boundary. In the Poincaré patch of the bulk AdSd+1, the two branes have the
embedding equations:

Left Brane: $\displaystyle w=u\cos\theta_{1}\,,\quad z=u\sin\theta_{1}\,,$
Right Brane: $\displaystyle w=-L-u\cos\theta_{2}\,,\quad z=u\sin\theta_{2}\,.$
(4)

The dual description is a $d$-dimensional CFT living on the conformal boundary
of the regulated wedge, with two parallel boundaries (the two red dots on Fig.
1b). We will study the entanglement entropies associated with a bipartition
which cuts this UV-regularizing interval into two, along a $(d-1)$-dimensional
sub-manifold parallel to the two defects.

###  2.2 Two communicating black holes

In Geng:2020fxl , a model of two communicating black holes was constructed by
placing two KR branes into an AdSd+1 black string with line element

$ds^{2}=\frac{1}{\sin^{2}\mu}\left(\frac{-h(u)\,dt^{2}+h(u)^{-1}\,du^{2}+d\vec{x}^{2}}{u^{2}}+d\mu^{2}\right),\quad
h(u)=1-\frac{u^{d-1}}{u_{h}^{d-1}}\,,$ (5)

where $u_{h}$ is the location of the horizon. Each of the branes at
$\mu=\theta_{1}$ and $\mu=\pi-\theta_{2}$ supports an AdSd planar black hole,
which in turn are coupled at their common conformal boundary and are in
thermal equilibrium with each other. Although in equilibrium, they exchange
information in the form of Hawking radiation, emitting quanta and swallowing
quanta from the other. From the viewpoint of each of the black holes, the bath
to which it is coupled is gravitating.

This communication between the black holes can be characterized by considering
the entanglement entropy of an internal bipartition. This internal bipartition
makes use of the fact that the defect Hilbert space can be factorized into two
sectors roughly corresponding to the two black holes. This entanglement
entropy was referred to as the L/R entropy in Ref. Geng:2020fxl  (see details
therein). To study the time evolution of the entanglement entropy, the time-
translation of the system has to be chosen such that the two disconnected
components of the system (corresponding to the two asymptotic boundaries in
the maximally extend bulk geometry) evolve in the same direction.444In the
bulk we considered an eternal AdSd+1 black string whose field theory dual is
known to be in a thermofield double state Maldacena:2001kr . We applied a
general trick used when studying time evolution of the thermofield double
state. By choosing the overall Hamiltonian $H$ to be $H_{1}+H_{2}$ rather than
$H_{1}-H_{2}$, there is a non-trivial time evolution. As noticed in
Geng:2020fxl , this entanglement entropy can be studied using the Ryu-
Takayanagi (RT) surface in the $(d+1)$-dimensional bulk, and the RT surface
has a phase transition between a connected phase and a disconnected phase.
This phase transition gives a Page curve for the L/R entropy, which is
consistent with the unitarity of the time evolution we chose.

However, it is not easy to give a calculable field theory model of this L/R
entropy, since the black string geometry does not have a natural UV
regularization preserving the conformal symmetry on the defect. In particular,
in Eq. (5) the translational symmetry of the AdS metric in Eq. (1) along the
$w$-coordinate is broken, and hence it is not possible anymore to embed two
branes with a separation as in Eq. (4) in a consistent way. Therefore, instead
of a black string, we will consider an eternal AdS3 BTZ black hole
Banados:1992wn  with line element

$ds^{2}=-\frac{h(z)}{z^{2}}dt^{2}+\frac{dz^{2}}{h(z)z^{2}}+\frac{dx^{2}}{z^{2}}\,,\quad
h(z)=1-\frac{z^{2}}{z_{h}^{2}}\,,$ (6)

where $z_{h}$ is the location of the horizon. Contrary to the higher-
dimensional case, in three dimensions KR branes can be consistently embedded
in such a black hole background, where the brane-localized action is just a
tension term. Moreover, now the two KR branes can be separated by a distance
on the conformal boundary. Each brane contains a two-dimensional black hole
induced from the bulk BTZ, a construction discussed in more detail later (Sec.
4). Thus, we have a conformal symmetry-preserving, UV-regularized description
of two communicating black holes. The field theory description consists of two
BCFT2s, each with two boundaries. These two BCFTs live on the conformal
boundary of the two copies of the system (since the maximal extension of the
Penrose diagram corresponding to the eternal black holes has two conformal
boundaries). On the time reflection-symmetric slice, these two BCFTs are in a
thermofield double (TFD) state.

Based on this observation, we will see that we can have a field theory
computation of the L/R entropy (see Sec. 3) which matches the gravity side
computation in Sec. 4. This provides an explicitly calculable field theory
model for two communicating black holes. Furthermore, it provides another
consistency check for the AdS/CFT correspondence and its possible
deformations.

###  2.3 BCFT calculations of entanglement entropy

In this section, we review the computation of entanglement entropy in two-
dimensional BCFTs, mostly following Sully:2020pza , and clarify some
subtleties. We start from the basic concepts of computing the entanglement
entropy using the replica trick. We list useful results for our computations
in the following sections and state our assumptions.

####  2.3.1 Boundary state at zero temperature

For the sake of convenience, we consider an Euclidean BCFT2 and firstly assume
that the bulk CFT is in the vacuum state. In this case, the CFT is living on
an upper half plane and the boundary condition on the real axis preserves half
of the conformal invariance in the bulk Cardy:2004hm . The Euclidean time $t$
is going along the horizontal axis (see Fig. 2). We want to compute the
entanglement entropy of the subsystem $\mathcal{A}$ in Fig. 2. Suppose that
the reduced density matrix of this subsystem is $\rho_{\mathcal{A}}$, then the
entanglement entropy can be computed by taking the limit $n\rightarrow 1$ of
its n-th Renyi entropy

$S_{\mathcal{A}}^{n}=-\frac{1}{n-1}\ln\Tr(\rho_{\mathcal{A}}^{n})\,.$ (7)

The trace $\Tr(\rho_{\mathcal{A}}^{n})$ in the above formula can be computed
by the standard replica trick Calabrese:2009qy . It is equivalent to the one-
point function of a twist operator $\Phi_{n}(z,\bar{z})$ inserted at the point
that separates $\mathcal{A}$ and its complement $\bar{\mathcal{A}}$ (the blue
cross in Fig. 2). This twist operator creates a branch cut on the upper half
plane (UHP) from the blue cross to infinity along the dashed black line in
Fig. 2.

$t$$\crossproduct$$\mathcal{A}$$\bar{\mathcal{A}}$$\ell_{\mathcal{A}}$ Figure
2: We consider a two dimensional boundary conformal field theory living on the
upper half plane (UHP). The bulk CFT is in the vacuum state, the boundary is
specified by the horizontal axis (in red) and the time direction is along the
horizontal direction. We consider a constant time slice (dashed black vertical
line) which defines the state we are looking at. For this state, we want to
compute the entanglement entropy of the region $\mathcal{A}$ (between the blue
cross and the boundary) with its complement $\bar{\mathcal{A}}$.

The twist operator is a primary operator Calabrese:2009qy  with conformal
dimensions

$h_{n}=\bar{h}_{n}=\frac{c}{24}\left(n-\frac{1}{n}\right).$ (8)

Therefore, we have the following formula for the entanglement entropy of the
subsystem $\mathcal{A}$:

$S_{\mathcal{A}}=\lim_{n\rightarrow
1}\frac{1}{1-n}\ln\langle\Phi_{n}(z,\bar{z})\rangle_{\text{UHP}}\,,$ (9)

where $(z,\bar{z})$ is the location of the boundary of $\mathcal{A}$ (the blue
cross in Fig. 2). This one-point function is fixed by conformal symmetry and
can be computed using the doubling trick Sully:2020pza  to map the BCFT on the
UHP to a chiral CFT on the whole complex plane $\mathbb{C}$. Doing this trick,
the one-point function for $\Phi_{n}(z,\bar{z})$ is equivalent to a two-point
function of a chiral primary field $\Phi_{n}(z)$ (with conformal dimensions
$h_{n}=\frac{c}{24}\left(n-\frac{1}{n}\right),\,\bar{h}_{n}=0$) where one
$\Phi_{n}$ is inserted at the blue cross and the other $\Phi_{n}$ at its
mirror point on the lower half plane,

$\langle\Phi_{n}(z,\bar{z})\rangle_{\text{UHP}}=\langle\Phi_{n}(z)\Phi_{n}(z^{*})\rangle_{\mathbb{C}}=\frac{\mathcal{A}_{\Phi_{n}}^{b}}{|z-z^{*}|^{2h_{n}}}\,,$
(10)

where $\mathcal{A}_{\Phi_{n}}^{b}$ is a normalization constant that will be
fixed soon Sully:2020pza .

This trick is equivalent to a boundary operator expansion (BOE) (see
Cardy:1996xt ; Wu:2020mxj  for explicit calculations of BOEs in simple models)
which states that on the UHP any operator $\mathcal{O}_{i}(z,\bar{z})$ can be
expanded as a series of boundary operators $\hat{O}_{I}(x)$,

$\mathcal{O}_{i}(z,\bar{z})=\sum_{J}\frac{\mathcal{B}_{i}^{bJ}}{(2y)^{\Delta_{i}-\Delta_{J}}}\tilde{\mathcal{C}}[y,\partial_{x}]\hat{\mathcal{O}}_{J}(x)\,.$
(11)

Here $\Delta_{i}$ is the conformal weight of $\mathcal{O}_{i}$
($h_{i}=\bar{h}_{i}=\frac{\Delta_{i}}{2}$), $\Delta_{J}$ is that of the
boundary primary operator $\hat{\mathcal{O}}_{J}(x)$ and we use the complex
coordinate $z=x+iy$ ($y>0$) on the UHP. The coefficients
$\mathcal{B}_{i}^{bJ}$ are called the BOE coefficients and they are determined
by the boundary condition and the structure of the parent CFT. The boundary
primary operator is normalized as

$\langle\hat{O}_{I}(x_{I})\hat{O}_{J}(x_{J})\rangle=\frac{G_{IJ}}{|x_{I}-x_{J}|^{2\Delta_{I}}}\,.$
(12)

By comparing Eq. (10) with the vacuum expectation value of Eq. (11), we see
that the only boundary operator in the BOE that appears in
$\left<\Phi_{n}\right>_{\text{UHP}}$ is the identity operator $\mathbf{1}$,
and

$\mathcal{A}_{\Phi_{n}}^{b}=\mathcal{B}_{\Phi_{n}\mathbf{1}}^{b}\,.$ (13)

Here the index $J$ in $B_{i}^{bJ}$ is lowered using $G_{IJ}$. Now we have an
exact result for the entanglement entropy of $\mathcal{A}$:

$S_{\mathcal{A}}=\frac{c}{6}\ln(\frac{2\ell_{\mathcal{A}}}{\epsilon})+\ln(g_{b})\,,$
(14)

where $\ell_{\mathcal{A}}=(z-\bar{z})/2i$ is the length of the subsystem
$\mathcal{A}$ and we have defined the boundary entropy term $\ln(g_{b})$
through the regularization

$\ln(g_{b})-\frac{c}{6}\ln(\epsilon)=\lim_{n\rightarrow
1}\frac{1}{1-n}\ln(B_{\Phi_{n}1}^{b})\,.$ (15)

Here $\epsilon$ is a UV cutoff. We note that this result matches the
holographic computation in Takayanagi:2011zk  and is of a universal form for
any BCFT with one boundary such as in Fig. 2.

####  2.3.2 Thermal field double state

We now consider the thermal field double (TFD) state of two BCFTs. We will
label the two BCFTs as L and R respectively (not to be confused with the L/R
entropy of the previous sections, the meaning should be clear by context). We
can prepare the TFD state using an Euclidean path integral. In the
gravitational picture, it corresponds to the Hartle-Hawking state in the
maximally extended two-sided geometry. In this Euclidean signature, the time
direction is periodic, and a time evolution for the L and R parts can be
chosen that makes the TFD state evolve non-trivially (see Fig. 3a). We would
like to calculate the entanglement entropy of the subsystem
$\mathcal{A}_{L}\cup\mathcal{A}_{R}$. As in the previous subsection, the
replica trick tells us that the question reduces to the computation of the
two-point function of a twist operator $\Phi_{n}$ and an anti-twist operator
$\bar{\Phi}_{n}$ inserted respectively at the points which separate
$\mathcal{A}_{L}(\mathcal{A}_{R})$ and its complement
$\bar{\mathcal{A}}_{L}(\bar{\mathcal{A}}_{R})$ (blue crosses in Fig. 3a).
Denoting the location of the crosses by $w_{L}$ and $w_{R}$, the relevant two-
point function is

$\langle\Phi_{n}(w_{R},\bar{w}_{R})\bar{\Phi}_{n}(w_{L},\bar{w}_{L})\rangle\,.$
(16)

The twist operator creates a branch cut from the right blue cross to infinity
along the right dashed green line and the anti-twist operator creates a branch
cut from the left blue cross to infinity along the left dashed green
line.555The twist operator and the anti-twist operator are related to each
other by a reflection (parity only in one spatial direction) transformation.
To compute the two-point function of the twist and anti-twist operators we can
map the configuration to the upper half plane in z coordinates using the
conformal transformation

$w=\frac{1}{z-\frac{i}{2}}-i\,.$ (17)

This transformation maps the boundary circle in $w$ coordinates (red circle in
Fig. 3a) to the real axis in $z$ and the infinity in $w$ to $z=i/2$. The blue
crosses are mapped to two point on the UHP. The branch cuts from the blue
crosses to infinity can be deformed to a horizontal branch cut connecting them
(the dashed green line in Fig. 3b) on the UHP.

BCFTLBCFTR$t_{R}$$t_{L}$$+$$+$$\mathcal{A}_{R}$$\mathcal{A}_{L}$$\bar{\mathcal{A}}_{R}$$\bar{\mathcal{A}}_{L}$
(a) The thermofield double state and its time evolution

$\crossproduct$$\crossproduct$_$\bar{\Phi}_{n}$__$\Phi_{n}$__$\bullet$__$\left(0,\frac{1}{2}\right)$_
(b) Conformal mapping of the region on left to a UHP

Figure 3: a) The two Euclidean BCFTs L and R in the TFD state: time evolution
is rotation with respect to the origin, hence the red circle is the time
evolution of the boundary. The two black dashed lines define the zero time
slice. Under the chosen time evolution, L and R evolve clockwise and counter-
clockwise respectively, as indicated. We are interested in the entanglement
entropy of the subsystem $\mathcal{A}=\mathcal{A}_{L}\cup\mathcal{A}_{R}$
corresponding to the solid green line segments. b) The UHP which results from
the conformal mapping of the region outside of the red circle in Fig. 3a. The
circular boundary is mapped to the real axis and infinity is mapped to
$\left(0,\frac{1}{2}\right)$. The location of twist operators is mapped to the
two blue crosses, separated in the horizontal direction. The branch cut is
mapped (and deformed) to the dashed green line connecting the two operators.

As a result, we have to compute the two-point function of a twist operator and
an anti-twist operator on the UHP:

$\langle\bar{\Phi}_{n}(z_{L},\bar{z}_{L})\Phi_{n}(z_{R},\bar{z}_{R})\rangle_{\text{UHP}}\,.$
(18)

When there is no boundary, this is just the two-point function of primary
operators which is totally fixed by kinematics:

$\langle\bar{\Phi}_{n}(z_{L},\bar{z}_{L})\Phi_{n}(z_{R},\bar{z}_{R})\rangle_{\mathbb{C}}=\frac{\epsilon^{2d_{n}}}{|z_{L}-z_{R}|^{2d_{n}}}\,,$
(19)

where the conformal weight $d_{n}$ equals to
$2h_{n}=2\bar{h}_{n}=\frac{c}{12}(n-\frac{1}{n})$ and we have set the
normalization to be $\epsilon^{2d_{n}}$, (where $\epsilon$ is a UV cutoff) for
reasons that will be clear when we compute the entanglement entropy.

However, when we have a boundary, there is an ambiguity. To see this, first
note that, away from the boundary, the BCFT has the same local structure as
its parent CFT. This means that the operator product expansion (OPE) for two
bulk operators is the same as in the parent CFT:

$\mathcal{O}_{i}(z_{1},\bar{z}_{1})\mathcal{O}_{j}(z_{2},\bar{z}_{2})=\sum_{k}\frac{\hat{\mathcal{C}}_{ij}^{k}}{|z_{1}-z_{2}|^{\Delta_{i}+\Delta_{j}-\Delta_{k}}}C_{\Delta_{i}\Delta_{j}\Delta_{k}}[z_{12},\bar{z}_{12},\partial_{2},\bar{\partial}_{2}]\mathcal{O}_{k}(z_{2},\bar{z}_{2})\,,$
(20)

for bulk primaries $\mathcal{O}_{k}$ and OPE coefficients
$\mathcal{\hat{C}}_{ij}^{k}$. To compute our two-point function, we can use
the OPE and compute the one-point functions of the primaries $\mathcal{O}_{k}$
appearing in the product, as per Eq. (10) and Eq. (11). We call this the bulk
channel. Alternatively, we can perform a BOE (Eq. (11)) first and then compute
the (normalized) boundary two-point functions. We call this the boundary
channel.

Consistency of the parent CFT and the boundary conditions tell us that these
two channels should give the same answer. This consistency requirement allows
us to simplify the computation of 2-point function in the case of a
holographic BCFT, which we now define.

First recall that for a holographic (two-dimensional) CFT we have
Hartman:2013mia :

  * •

The central charge $c$ is large.

  * •

The spectrum of light operators, of conformal dimension
$\mathcal{O}(c^{0})$,666In this paper, the $\mathcal{O}(\dots)$ symbol means
”of the order $\dots$”. is sparse.

The first condition tells us that in the OPE the operators of conformal
dimension $\mathcal{O}(c)$ or higher, which we refer to as heavy operators,
are suppressed (for technical details see Sully:2020pza ; Hartman:2013mia ;
Faulkner:2013ana ). The second condition tells us that the total contributions
to the OPE from the light operators is a multiplication of the contribution of
the identity operator. These conditions greatly simplify the OPE for
holographic CFTs because we only have to look at the fusion to the identity
operator.

For holographic BCFTs, the two assumptions are naturally generalized to
boundary operators Sully:2020pza , but some subtleties appear. First let us
consider the bulk channel. In the OPE, one-point functions of purely bulk
operators are now generically nonzero, and may compete with the suppressed OPE
contribution from the heavy operators when we compute a bulk two-point
function. However, this happens only when we are close enough to the boundary,
where the position-dependent part of the heavy operator one-point function
(calculated using the doubling trick in the previous subsection) is not small.

Now consider the boundary channel. The contribution of a boundary operator of
dimension $\Delta_{J}$ to the BOE of $\Phi$ scales as $y^{\Delta_{J}}$ (see
Eq. (11)). For large $y$, the contribution from heavy operators is not
suppressed anymore. Therefore, if we want to only look at the contribution of
the boundary identity operator, we must not be too far from the boundary.

In summary, the simplification on the calculation of the bulk two-point
function from using the conditions of holographic BCFTs is useful in the bulk
channel only when far away from the boundary, and in the boundary channel only
when close to the boundary. Consistency of the BCFT requires that the bulk
channel result must equal the boundary channel result.

Practically speaking, when we compute the bulk two-point function for a given
pair of insertions, we examine the identity block in both channels and simply
choose the larger one. This is justified because each contribution to the two-
point function from different blocks is positive and taking all corrections
into account should give the same result for both channels. Therefore, the
larger of the two identity blocks is a better approximation. In other words,
in the other channel with a smaller identity block contribution neglecting
everything except for the identity block gives rise to a larger error. This
takes into account the subtleties due to the appearance of the boundary.

As a result, for a holographic BCFT on an UHP we have the two-point function

$\langle\bar{\Phi}_{n}(z_{L},\bar{z}_{L})\Phi_{n}(z_{R},\bar{z}_{R})\rangle_{\text{UHP}}=\max\left(\frac{\epsilon^{2d_{n}}}{|z_{L}-z_{R}|^{2d_{n}}},\frac{\mathcal{B}_{\Phi_{n}1}^{b}\mathcal{B}_{\Phi_{n}1}^{b}}{|z_{L}-z_{L}^{*}|^{d_{n}}|z_{R}-z_{R}^{*}|^{d_{n}}}\right)\,,$
(21)

where the first (second) term is the contribution from bulk (boundary)
channel. We can immediately deduce the entanglement entropy, but we defer
details to the next section where we consider the more general case of two
boundaries. We briefly note, however, that the maximization prescription for
the two-point function gives a minimization prescription for the entanglement
entropy (due to the minus sign in Eq. (7)), consistent with the Ryu-Takayanagi
prescription in holographic computations Ryu:2006bv ; Ryu:2006ef .

##  3 The CFT computation

In this section, we discuss the two-dimensional CFT on a strip, and calculate
the entanglement entropy of a subsystem. Before moving to the finite
temperature case, we consider the zero temperature case first.

###  3.1 CFT with two boundaries, at zero temperature

We consider a two dimensional Euclidean CFT, living on a strip, with different
boundary conditions along the two boundaries. We denote these boundary
conditions as $a$ and $b$ respectively.

We can prepare a state of the system at a given time $t=0$ by a Euclidean path
integral from the past infinity to $t=0$. We are interested in the
entanglement entropy for a bipartition of the interval shown in Fig. 4a. As
reviewed in Sec. 2, this reduces to computing the one-point function of the
twist operator $\Phi_{n}$ inserted at the bipartition point.

$\crossproduct$$t$$a$$b$$\mathcal{A}$$\bar{\mathcal{A}}$$L$$\ell_{\mathcal{A}}$
(a) CFT on a strip with boundary conditions a, b.

$\crossproduct$$\circ$$b$$a$ (b) Conformal mapping of the strip on left to a
UHP.

$\crossproduct$$b$$a$ (c) Conformal mapping of the UHP on left to a UHP with a
wedge removed.

Figure 4: a) CFT on a strip with boundary conditions $a,b$. The time direction
is horizontal, and the black dashed line is a constant time slice ($t=0$). A
twist operator is inserted at the green cross, separating the subsystem
$\mathcal{A}$ and its complement $\bar{\mathcal{A}}$. b) The strip is now
conformally transformed to a UHP. The black circle at the origin is a boundary
condition changing operator $O_{b\rightarrow a}$. The twist operator is mapped
to the green cross. c) The configuration is finally mapped to a UHP with a
wedge removed: the twisted upper half plane (TUHP). The boundary consists of
two mismatched semi infinite lines, with boundary conditions $a$ and $b$. The
twist operator is again mapped to the green cross.

Let $(w,\bar{w})$ denote complex coordinates on the strip with width $L$ so
that $0<\imaginary(w)<L$. The subsystem $\mathcal{A}$ is of size
$\ell_{\mathcal{A}}<L$. Without loss of generality, we consider the zero
Euclidean time slice, so that our inserted twist operator
$\Phi_{n}(w,\bar{w})$ has $\real(w)=\real(\bar{w})=0$ and
$\imaginary(w)=-\imaginary(\bar{w})=\ell_{\mathcal{A}}$. Now we perform a
conformal transformation to map the strip with coordinates $(w,\bar{w})$ to
the UHP with coordinates $(z,\bar{z})$ via

$w=\frac{L}{\pi}\ln(z)\,.$ (22)

The boundary $a$ is mapped to the positive real axis, and $b$ to the negative
real axis, as depicted in Fig. 4b. The change in the boundary condition can be
achieved by inserting a boundary condition changing (bcc) operator
$O_{b\rightarrow a}$, which is a primary operator, at the origin, and another
bcc operator $O_{a\rightarrow b}$ at $\infty$ Cardy:2004hm .777Or
equivalently, it is achieved by inserting a boundary condition changing
operator $O_{b\leftarrow a}$ at the origin that changes the boundary condition
on the negative real axis from $a$ to $b$ and another $O_{a\leftarrow b}$ at
$\infty$.

Inserting these bcc operators is a quench which creates energy in the system.
They can be viewed as local scaling operators with respect to the (open
string) Hamiltonian $H=L_{0}$, performing radial evolution on the UHP
Cardy:2004hm . Transforming to the strip, the ground state energy of this
state, with respect to strip Hamiltonian, is
$E=(\pi/L)(L_{0}-\tfrac{c}{24})=(\pi/L)(h_{\text{bcc}}-c/24)$, where
$\Delta_{\text{bcc}}=2h_{\text{bcc}}$ is the scaling dimension of the bcc
operator. We can conformally map this UHP to another UHP with conical deficit
such that the energy excess $\Delta_{\text{bcc}}$ is eliminated. This is
achieved by yet another conformal transformation to coordinates
$(\tilde{z},\bar{\tilde{z}})$ given by

$\tilde{z}=z^{\alpha},\quad\alpha=\sqrt{1-\frac{12\Delta_{\text{bcc}}}{c}}\,,$
(23)

which transforms the holomorphic energy momentum tensor to

$\widetilde{T}(\tilde{z})=\left(\frac{dz}{d\tilde{z}}\right)^{2}T(z)+\frac{c}{12}\\{z,\tilde{z}\\}\,.$
(24)

The Schwarzian derivative $\\{z,\tilde{z}\\}$ can be computed as

$\\{z,\tilde{z}\\}=\frac{z^{\prime\prime\prime}(\tilde{z})}{z^{\prime}(\tilde{z})}-\frac{3}{2}\left(\frac{z^{\prime\prime}(\tilde{z})}{z^{\prime}(\tilde{z})}\right)^{2}=\frac{1}{2}\left(1-\frac{1}{\alpha^{2}}\right)\tilde{z}^{-2}\,,$
(25)

which gives us the holomorphic energy momentum tensor

$\widetilde{T}(\tilde{z})=\tilde{z}^{-2}\Bigg{[}\frac{1}{\alpha^{2}}z^{2}\,T(z)+\frac{c}{24}\left(1-\frac{1}{\alpha^{2}}\right)\Bigg{]}\,.$
(26)

On the $(\tilde{z},\bar{\tilde{z}})$ plane the configuration looks like Fig.
4c. We will call this geometry the twisted upper half plane (TUHP). We can
calculate the zeroth order generator of the Virasoro algebra $\tilde{L}_{0}$
as

$\begin{split}\widetilde{L}_{0}&=\int_{\mathcal{\widetilde{C}}}\frac{d\tilde{z}}{2\pi\alpha
i}\tilde{z}\,\widetilde{T}(\tilde{z})\\\ &=\int_{\mathcal{C}}\frac{dz}{2\pi
i}\Bigg{[}\frac{1}{\alpha^{2}}z\,T(z)+\frac{c}{24}\left(1-\frac{1}{\alpha^{2}}\right)\Bigg{]}\\\
&=\frac{1}{\alpha^{2}}\Big{[}L_{0}+\frac{c}{24}(\alpha^{2}-1)\Big{]}\>.\end{split}$
(27)

Here, in the first line, $\mathcal{\widetilde{C}}$ is the contour
$\tilde{z}=e^{i\tilde{\theta}}$, $\tilde{\theta}\in[0,2\pi\alpha]$ and
$\mathcal{C}$ is the unit circle $z=e^{i\theta},\theta\in[0,2\pi]$. As a
result, we have the transformed Hamiltonian $\widetilde{H}$ given as

$\begin{split}\alpha^{2}\widetilde{H}&=\alpha^{2}\frac{\pi}{L}\left(\widetilde{L}_{0}-\frac{c}{24}\right)\\\
&=\frac{\pi}{L}\left(L_{0}+\frac{c}{24}(\alpha^{2}-1)-\frac{c\alpha^{2}}{24}\right)\\\
&=\frac{\pi}{L}\left(L_{0}-\frac{c}{24}\right)\,,\end{split}$ (28)

which tells us that for $\widetilde{H}$ to be in its vacuum state, we need
$L_{0}=\frac{c}{24}(1-\alpha^{2})=\frac{\Delta_{\text{bcc}}}{2}$. This indeed
corresponds to the bcc operator insertion at the origin and the infinity of
the UHP.

Therefore, we see explicitly that the change of coordinates ensures that the
ground state energy from the bcc operator insertion and the contribution from
the conical deficit cancel. Thus, the one-point function for the twist
operator $\Phi_{n}(w,\bar{w})$ becomes

$\displaystyle\langle\Phi_{n}(w,\bar{w})\rangle_{\text{strip}}^{a,b}$
$\displaystyle=\absolutevalue{\frac{dz}{dw}}^{d_{n}}\langle O_{a\leftarrow
b}(\infty,\infty)\Phi_{n}(z,\bar{z})O_{b\leftarrow
a}(0,0)\rangle_{\text{UHP}}^{a}$ (29)
$\displaystyle=\absolutevalue{\frac{dz}{dw}}^{d_{n}}\absolutevalue{\frac{d\tilde{z}}{dz}}^{d_{n}}\langle\Phi_{n}(\tilde{z},\bar{\tilde{z}})\rangle_{\text{THUP}}^{a,b}$
$\displaystyle=\absolutevalue{\frac{\alpha\pi}{L}e^{\frac{\alpha\pi}{L}w}}^{d_{n}}\langle\Phi_{n}(\tilde{z},\bar{\tilde{z}})\rangle_{\text{THUP}}^{a,b}\,.$

Although the local scaling behavior of bcc operators have been trivialized
from a kinematic perspective, we still have two boundary conditions. This is
because the boundary condition changing operator is not a standard local
operator in CFT Cardy:2004hm  and the local physics near the boundary should
still be controlled by the corresponding boundary conditions. We know that on
the THUP we have two intersecting boundaries, so for a bulk CFT operator, we
have two boundary channels in which to perform BOE and theses two channles
should match.

For a generic CFT, we expect that the one-point function in the presence of
multiple boundaries is non-universal, since boundary operators associated with
$a$ will in turn be sourced by $b$. However, for a holographic CFT the
calculation simplifies due to the vacuum dominance. To be conservative, we
will treat this as a field theory ansatz for the holographic CFT on a strip,
and emphasize that a full understanding of these multi-boundary correlators
for a generic CFT remains an open question. This can be studied by
constructing a crossing equation for the matching of the two boundary channels
and looking for solutions of this crossing equation.

For the holographic BCFT we are interested in, we compute, as in Sec. 2.3, the
vacuum sector contributions from the two boundary channels and take the
maximum for the one-point function. As a result, we have

$\begin{split}\langle\Phi_{n}(w,\bar{w})\rangle_{\text{strip}}&=\absolutevalue{\frac{\alpha\pi}{L}e^{\frac{\alpha\pi}{L}w}}^{d_{n}}\max\\!\Bigg{(}\frac{\mathcal{B}^{a}_{\Phi_{n},1}}{\absolutevalue{2\imaginary\tilde{z}}^{d_{n}}},\frac{\mathcal{B}^{b}_{\Phi_{n},1}}{\absolutevalue{2\tilde{z}\sin(\alpha\pi-\arg\tilde{z})}^{d_{n}}}\Bigg{)}\\\
&=\absolutevalue{\frac{\alpha\pi}{L}e^{\frac{\alpha\pi}{L}w}}^{d_{n}}\max\\!\Bigg{(}\frac{\mathcal{B}^{a}_{\Phi_{n},1}}{\absolutevalue{2e^{\frac{\alpha\pi}{L}w}\sin\frac{\alpha\pi}{L}\imaginary
w}^{d_{n}}},\frac{\mathcal{B}^{b}_{\Phi_{n},1}}{\absolutevalue{2e^{\frac{\alpha\pi}{L}w}\sin\alpha(\pi-\frac{\pi}{L}\imaginary
w)}^{d_{n}}}\Bigg{)}\\\
&=\absolutevalue{\frac{\alpha\pi}{L}}^{d_{n}}\max\\!\Bigg{(}\frac{\mathcal{B}^{a}_{\Phi_{n},1}}{\absolutevalue{2\sin\frac{\alpha\pi}{L}\imaginary
w}^{d_{n}}},\frac{\mathcal{B}^{b}_{\Phi_{n},1}}{\absolutevalue{2\sin\alpha(\pi-\frac{\pi}{L}\imaginary
w)}^{d_{n}}}\Bigg{)}\,.\end{split}$ (30)

From Eq. (9), the entanglement entropy is

$\begin{split}S_{\mathcal{A}}=\min\Bigg{[}\frac{c}{6}\ln(\frac{2L}{\alpha\pi\epsilon}\sin\frac{\alpha\pi\ell_{\mathcal{A}}}{L})+\ln(g_{a}),\,\frac{c}{6}\ln(\frac{2L}{\alpha\pi\epsilon}\sin\frac{\alpha\pi(L-\ell_{\mathcal{A}})}{L})+\ln(g_{b})\Bigg{]}\,,\end{split}$
(31)

where we used the fact that $\imaginary w=\ell_{\mathcal{A}}$ and the
definition of boundary entropy in Eq. (15). So far, we have implicitly assumed
that $\alpha$ is real or $\Delta_{\text{bcc}}<\frac{c}{12}$. However,
generically $\alpha$ can be imaginary and this changes the scaling behavior of
the entanglement entropy with the subsystem size $\ell_{\mathcal{A}}$ from
$\sin\frac{\absolutevalue{\alpha}\pi\ell_{\mathcal{A}}}{L}$ to
$\sinh\frac{\absolutevalue{\alpha}\pi\ell_{\mathcal{A}}}{L}$. This tells us
that when $\Delta_{\text{bcc}}$ is large enough the field theory state is
locally thermalized.888This can be seen by observing that for a CFT2 in
thermal state, the entanglement entropy scales with the subsystem size $l$ as
$\ln(\frac{\beta}{\pi\epsilon}\sinh(\frac{\pi l}{\beta}))$ Ryu:2006ef ;
Calabrese:2009qy .  This provides an important hint about the dual
gravitational description.

Indeed, as we will see in the next section, on the gravity side the finite $L$
result will be reproduced by looking at a point particle in the global AdS3
(the defect AdS3 geometry) for real $\alpha$ and by considering an AdS3 black
hole for imaginary $\alpha$.

###  3.2 CFT with two boundaries, at non-zero temperature

We next consider the CFT living on the strip, now at a non-zero temperature.
As in subsection 2.3.2, we consider the thermal field double (TFD) state of
two BCFTs (L and R) each with two boundaries. Both BCFTs have the same
boundary conditions. This state can be easily prepared using the Euclidean
path integral, after making the time direction periodic (see Fig. 5). This
leads to considering the CFT on an annulus, with the inner and the outer
circles generated by the periodic time evolution of the two boundaries of the
CFT. We will denote the radii as $r_{I}$ and $r_{O}$ respectively (for inner
and outer). The length of the strip is given by $L=r_{O}-r_{I}$.

BCFTLBCFTR$t_{R}$$t_{L}$$+$$+$$\mathcal{A}_{R}$$\mathcal{A}_{L}$$\bar{\mathcal{A}}_{R}$$\bar{\mathcal{A}}_{L}$$a$$b$$L$$r_{O}$$r_{I}$
Figure 5: The two-dimensional CFT on a strip at a finite temperature: the two
BCFTs L and R are in the TFD state. Time evolution is rotation with respect to
the origin, and the red circles are the time evolution of the two boundaries.
The boundary conditions are specified as $a$ and $b$. The black dashed line is
the $t=0$ slice, and the choice of time evolution is indicated. We are
interested in the entanglement entropy of the subsystem
$\mathcal{A}=\mathcal{A}_{L}\cup\mathcal{A}_{R}$ corresponding to the solid
green line segments connecting the inner boundary and the blue crosses.

To proceed, we could naively attempt for a conformal mapping to a strip and
then use the technical results of the previous subsection. However, no
conformal transformation can accomplish this, because the annulus has a
modular parameter which is generally nonzero and should be preserved under any
conformal transformation Polchinski:1998rq . Consequently, there is no
conformal transformation from a generic annulus to a strip. It is still
possible to make progress if we focus on the high temperature limit.
Specifically, we can work in the limit of fixed $L=r_{O}-r_{I}$ and send the
radius of the smaller circle $r_{I}$ to be close to zero ($r_{I}/L\to 0$) in
Fig. 5. In this limit, we can conformally map the annulus (in $w$ coordinates)
to the UHP (in $z$ coordinates) by the transformation,

$w=r_{I}\left(\frac{1}{z-i/2}-i\right)\,.$ (32)

In the limit $r_{I}/L\rightarrow 0$, this transformation maps the inner
boundary of the annulus to the real axis and its outer boundary to $z=i/2$.
This is similar to Fig. 3b, but with an operator $\Phi_{a}$ inserted at
$z=i/2$, corresponding to the outer boundary of the annulus.

Now we focus on holographic CFTs which satisfy the assumptions reviewed in
Sec. 2. On the UHP there are various channels for the two-point function:

$\langle\Phi_{n}(z_{R},\bar{z}_{R})\bar{\Phi}_{n}(z_{L},\bar{z}_{L})\rangle_{\text{UHP},a}^{b}\,,$
(33)

where the upper index $b$ denotes the boundary condition $b$ along the real
axis and the lower index $a$ denotes the operator insertion $\Phi_{a}$ at
$z=i/2$. We refer to the one-point function with the shorthand
$\langle\Phi_{a}\rangle^{b}=\langle\Phi_{a}(z_{a},\bar{z}_{1})\rangle_{\text{UHP}}^{b}$.
Here we list the channels:

  * •

The boundary channel which, up to the one-point function
$\langle\Phi_{a}\rangle^{b}$, gives us

$\frac{\mathcal{B}_{\Phi_{n}1}^{b}\mathcal{B}_{\Phi_{n}1}^{b}}{|z_{L}-z_{L}^{*}|^{d_{n}}|z_{R}-z_{R}^{*}|^{d_{n}}}\,.$
(34)

  * •

The connected bulk channel which, up to the constant
$\langle\Phi_{a}\rangle^{b}$, yields

$\frac{\epsilon^{2d_{n}}}{|z_{L}-z_{R}|^{2d_{n}}}\,.$ (35)

  * •

The disconnected bulk channel which is given by the three-point function

$\langle\Phi_{n}(z_{L},\bar{z}_{L})\bar{\Phi}_{n}(z_{R},\bar{z}_{R})\Phi_{a}(z_{a},\bar{z}_{a})\rangle\,.$
(36)

The last channel is a correlator on the whole complex plane, but can be
reinterpreted as the boundary channel associated with $a$. This can be easily
computed using another conformal transformation to exchange the large and the
small circle.999More precisely, we map the larger circular boundary to the
real axis and the smaller one to the point $x=0,\,y=\frac{1}{2}$. This is
achieved by firstly transforming the annulus to another annulus such that the
two circular boundaries exchange ($w^{\prime}=\frac{r_{I}r_{O}}{w}$) and then
we do the same conformal transform as in Eq. (32). As reviewed in Sec. 2, for
holographic CFTs we just take the maximum of these vacuum contributions to
compute the entanglement entropy. Mapping back to the annulus, we get the
entanglement entropy:

$\begin{split}S_{\mathcal{A}}=\min\Bigg{[}\frac{c}{3}\ln(\frac{r^{2}-r_{I}^{2}}{r_{I}\epsilon})+2\ln(g_{b}),\,\frac{c}{3}\ln(\frac{\absolutevalue{w_{R}-w_{L}}}{\epsilon}),\,\frac{c}{3}\ln(\frac{r_{O}^{2}-r^{2}}{r_{O}\epsilon})+2\ln(g_{a})\Bigg{]}\,.\end{split}$
(37)

Taking $w_{L}=re^{i\theta},w_{R}=re^{i\pi-i\theta}$, and continuing back to
the Lorentzian time $t$ using $t=\frac{i\theta}{2\pi}\beta$, we get

$\begin{split}S_{\mathcal{A}}=\min\Bigg{[}\frac{c}{3}\ln(\frac{r^{2}-r_{I}^{2}}{r_{I}\epsilon})+2\ln(g_{b}),\,\frac{c}{3}\ln(\frac{2r\cosh\frac{2\pi
t}{\beta}}{\epsilon}),\,\frac{c}{3}\ln(\frac{r_{O}^{2}-r^{2}}{r_{O}\epsilon})+2\ln(g_{a})\Bigg{]}\,.\end{split}$
(38)

This gives us a Page curve for a typical $r_{I}<r<r_{O}$. The monotonically
increasing contribution from $\cosh(2\pi t/\beta)$ is the dominant
contribution at early times, but is switched off at late times due to the
$\min$ condition. We emphasize that this CFT calculation depends on taking the
high temperature limit, the holographic assumption of vacuum dominance, and
finally, the simple two-boundary ansatz for the BOE introduced in the previous
section.

##  4 The gravitational computation

In this section, we discuss the gravitational picture, and compute the
entanglement entropy, both at zero and non-zero temperature. For the zero
temperature case, the corresponding bulk geometry depends on whether the
parameter $\alpha$ is real or imaginary (or equivalently
$12\Delta_{\text{bcc}}$ is smaller or larger than the central charge $c$). For
the real case, the geometry is a point particle living on global AdS3. For the
imaginary case, the geometry is a one-sided AdS3 black hole. In both cases we
have two KR branes, separated by an arc. For the non-zero temperature case,
the bulk geometry is in general complicated. However, since we are restricting
to the high temperature limit, the geometry is simply a planar BTZ black hole
Maldacena:2001kr ; Banados:1992wn .

Before diving into the computation, we start with a brief review of the
realization of the quantum extremal surface prescription Engelhardt:2014gca
in the KR braneworld Geng:2020qvw ; Geng:2020fxl , which is relevant for a
holographic computation of the entanglement entropy.

###  4.1 Quantum extremal surface in Karch-Randall braneworld

In this subsection we give a short summary of double holography and quantum
extremal surfaces in the context of KR braneworlds. We begin with the
discussion for a single KR brane, which will then be extended to two branes
with a finite separation, the setup relevant in this paper.

In the simplest case with a single KR brane living on an asymptotically AdSd+1
space,101010Here we keep the dimension $d$ general. In the rest of the paper,
however, we exclusively consider the case $d=2$. there exist the following
three mutually dual descriptions which are depicted in Fig. 6a:

  1. (I)

Bulk Perspective: A $(d+1)$-dimensional gravitational theory on
$\mathcal{M}^{\prime}_{d+1}$, a part of an asymptotically AdSd+1 space,
containing a Karch-Randall brane $\mathcal{M}_{d}$ at which the bulk geometry
$\mathcal{M}^{\prime}_{d+1}$ ends.

  2. (II)

Brane Perspective: A CFTd living on the Karch-Randall brane $\mathcal{M}_{d}$
coupled to dynamical gravity together with a CFTd living on the asymptotic
boundary $\bar{\partial}{\mathcal{M}}^{\prime}_{d+1}$111111Here, to avoid
confusion, we use $\bar{\partial}$ to denote the asymptotic boundary and
$\partial$ to denote the actual topological boundary. of the cutoff bulk
AdSd+1. These two CFTs are connected with transparent boundary conditions on
their common boundary
$\mathcal{M}^{(0)}_{d-1}=\partial\bar{\partial}\mathcal{M}^{\prime}_{d+1}=\bar{\partial}\mathcal{M}_{d}$.

  3. (III)

Boundary Perspective: A BCFTd on $\bar{\partial}{\mathcal{M}}^{\prime}_{d+1}$
with a $(d-1)$-dimensional boundary $\mathcal{M}^{(0)}_{d-1}$ with conformal
boundary condition imposed.

Here, we emphasize that the geometry on the Karch-Randall brane is
asymptotically AdSd with asymptotic boundary
$\bar{\partial}{\mathcal{M}}_{d}$. These three descriptions are related by a
two-fold chain of the standard AdS/CFT correspondence Maldacena:1997re ;
Gubser:1998bc ; Witten:1998qj , justifying the term “double holography”.

Double holography is particularly helpful to give a holographic derivation and
interpretation of the island formula Almheiri:2019hni ; Geng:2020fxl .
Consider the entanglement entropy $S(\mathcal{R})$ of a subregion
$\mathcal{R}$ in the BCFTd in (III), disconnected from the boundary
$\bar{\partial}{\mathcal{M}}_{d}$. It can be computed using the quantum
corrected Ryu-Takayanagi formula or the quantum extremal surface
Lewkowycz:2013nqa ; Faulkner:2013ana ; Engelhardt:2014gca  in the
holographically dual description (II). This involves finding a subregion
$\mathcal{I}$ within the gravitating spacetime $\mathcal{M}_{d}$ which
extremizes the generalized entropy of the region $\mathcal{R}\cup\mathcal{I}$.
Moreover, if there are multiple such regions, we take the one that gives the
minimum value of the generalized entropy,

$S(\mathcal{R})=\operatorname*{min\,ext}\limits_{\mathcal{I}}\,S_{\mathrm{gen}}(\mathcal{R}\cup\mathcal{I})\,.$
(39)

Since $\mathcal{I}$ is disconnected from $\mathcal{R}$, it is called an
entanglement island. In this formula the generalized entropy functional
$S_{\mathrm{gen}}$ is given by

$S_{\mathrm{gen}}(\mathcal{R}\cup\mathcal{I})=\frac{A(\partial
I)}{4G_{d}}+S_{\mathrm{mat}}(\mathcal{R}\cup\mathcal{I})\,,$ (40)

where $A(\partial I)$ denotes the area of the boundary of $\mathcal{I}$ and
$G_{d}$ is Newton’s constant of the gravitational theory on $\mathcal{M}_{d}$.
Moreover, $S_{\mathrm{mat}}$ is the entanglement entropy of matter (usually
quantum field theories) in the region $\mathcal{R}\cup\mathcal{I}$. Employing
the holographic duality again, $S_{\mathrm{gen}}$ can be expressed in terms of
a classical RT surface $\gamma$ in the bulk geometrical description (I),

$S_{\mathrm{gen}}(\mathcal{R}\cup\mathcal{I})=\frac{A(\gamma)}{4G_{d+1}}\,.$
(41)

Here, $\gamma$ is a codimension-two minimal surface in
$\mathcal{M}^{\prime}_{d+1}$ connecting $\partial R$ and $\partial I$ and
$G_{d+1}$ is the $(d+1)$-dimensional bulk Newton’s constant. As a result, we
have the following formula to compute the entanglement entropy of the region
$\mathcal{R}$

$S(\mathcal{R})=\operatorname*{min\,ext}\limits_{\mathcal{I}}\,\frac{A(\gamma)}{4G_{d+1}}\,.$
(42)

This setup can easily be extended to the case with a second Karch-Randall
brane, a construction recently dubbed as “wedge holography” Akal:2020wfl ;
Miao:2020oey . The corresponding extension of the above prescription was
discussed in Geng:2020fxl  for the case that the two Karch-Randall branes
$\mathcal{M}^{a}_{d}$ and $\mathcal{M}^{b}_{d}$ share a common asymptotic
boundary
$\mathcal{M}^{(0)}_{d-1}=\bar{\partial}\mathcal{M}^{a}_{d}=\bar{\partial}\mathcal{M}^{b}_{d}$.
This implies that the $d$-dimensional BCFT in (III) degenerates to a
$(d-1)$-dimensional theory living on $\mathcal{M}^{(0)}_{d-1}$. This situation
is depicted in Fig. 6b.

In this paper, however, we consider a slightly different configuration and
allow for a finite spatial separation of the two branes at their boundaries
(as illustrated in Fig. 1). This implies that in the global coordinates the
branes can overlap. Therefore, the situation is slightly more complicated than
in the previous cases, see Fig. 6c for details. In particular, not only a part
of the asymptotic AdS spacetime gets removed, but also the parts of the branes
behind their intersection. Moreover, the BCFT in (III) stays $d$-dimensional,
but obtains another boundary: now it lives on a strip with boundaries
$\bar{\partial}\mathcal{M}^{a}_{d}$ and $\bar{\partial}\mathcal{M}^{b}_{d}$.

Altogether, we arrive at the following doubly holographic description:

  1. (I)

Bulk Perspective: A $(d+1)$-dimensional quantum gravitational theory on
$\mathcal{M}^{\prime}_{d+1}$, a part of an asymptotically AdSd+1 space,
containing two Karch-Randall branes. We denote their relevant parts in front
of the intersection by $\mathcal{M}^{a}_{d}$ and $\mathcal{M}^{b}_{d}$.

  2. (II)

Brane Perspective: Two UV-cutoff CFT${}_{d}\,$s coupled to gravity on
$\mathcal{M}^{a}_{d}$ and $\mathcal{M}^{b}_{d}$ together with a CFTd living on
the asymptotic boundary $\bar{\partial}\mathcal{M}^{\prime}_{d+1}$. These
CFT${}_{d}\,$s are connected with transparent boundary conditions on
$\mathcal{M}^{(0)}_{d-1}$=$\bar{\partial}\mathcal{M}^{a}_{d}\cup\bar{\partial}\mathcal{M}^{b}_{d}$
(see Fig. 6c).

  3. (III)

Boundary Perspective: A BCFTd on a $\bar{\partial}\mathcal{M}^{\prime}_{d+1}$
with two $(d-1)$-dimensional boundaries $\bar{\partial}\mathcal{M}^{a}_{d}$
and $\bar{\partial}\mathcal{M}^{b}_{d}$ where conformal boundary conditions
are imposed .

Now we want to compute the entanglement entropy associated with a bipartition
of the BCFTd in the description (III) such that the bipartition manifold
$\partial\mathcal{R}$ is parallel to $\bar{\partial}\mathcal{M}^{a}_{d}$ and
$\bar{\partial}\mathcal{M}^{b}_{d}$. This is exactly the entanglement entropy
we computed in Sec. 3 using field theory techniques. In this case, the quantum
corrected Ryu-Takayanagi formula or the quantum extremal surface is double-
holographically realized as

$S=\operatorname*{min\,ext}\limits_{\mathcal{I}}\,\frac{A(\gamma)}{4G_{d+1}}\,,$
(43)

where $\mathcal{I}$ is a disconnected region, from the asymptotic boundary
$\bar{\partial}\mathcal{M}^{\prime}_{d+1}$, that lives on either of the two
Karch-Randall branes and $\gamma$ is a minimal area surface in the bulk
$\mathcal{M}^{\prime}_{d+1}$ that connects the boundary of the island
$\partial\mathcal{I}$ and the bipartition manifold $\partial\mathcal{R}$.

---

(a)  (b)  (c)

Figure 6:  Double holography in global patch with (a) one Karch-Randall brane,
(b) two Karch-Randall branes with coinciding endpoints, and (c) two Karch-
Randall branes with spatially separated endpoints. The blue shaded regions
$\mathcal{M}^{\prime}_{d+1}$ contain a gravitational theory living on a part
of an asymptotically AdS space. The red lines represent CFTs on their
asymptotic boundaries $\bar{\partial}\mathcal{M}^{\prime}_{d+1}$. The green
lines $\mathcal{M}_{d}$, $\mathcal{M}^{a}_{d}$, and $\mathcal{M}^{b}_{d}$ are
(parts of) Karch-Randall branes hosting a UV-cutoff CFT coupled to gravity.
The black dots $\mathcal{M}^{(0)}_{d-1}$ are CFTs on the asymptotic boundaries
of the branes, implemented as boundary conditions of the higher-dimensional
CFTs. In each case all three descriptions are mutually holographically dual.
In this illustration we assume that the brane tensions are positive in (a) and
(b) but negative in (c). The reason for choosing negative brane tension for
(c) will be made clear in the next section.

###  4.2 Two boundaries at zero temperature

In this subsection, we calculate the holographic entanglement entropy of a
subinterval of the strip gravitationally. The system is at zero temperature.
As discussed in Sec. 3.1, we consider different boundary conditions on the two
edges of the strip. To determine the correct holographic dual, we note that
generically the ground state energy of the strip is bigger than the vacuum
value, due to the boundary conditions on the two ends being different. This
difference in the boundary conditions generically results in the fields
acquiring gradients between the edges, which contributes to the energy of the
system. In the CFT language, we have inserted a bcc operator so we are no
longer in the vacuum state, as elaborated in Sec. 3. From the state-operator
correspondence, the energy of this configuration is

$E_{\text{vac}}^{ab}=\frac{\pi}{2L}\left(\Delta_{\text{bcc}}-\frac{c}{12}\right)\,,$
(44)

where $\Delta_{\text{bcc}}$ is the conformal weight (or scaling dimension) of
the bcc operator $\mathcal{O}_{b\rightarrow a}$. We set $L=\pi/2$ for
convenience.

The AdS3/CFT2 correspondence Carlip:2005zn  tells us that the dual geometry is
of the form

$ds^{2}=-(r^{2}-8GM)dt^{2}+\frac{dr^{2}}{r^{2}-8GM}+r^{2}d\theta^{2}\,,$ (45)

where $\theta\in[0,2\pi)$ and

$M=E_{\text{vac}}^{ab}=\Delta_{\text{bcc}}-\frac{1}{8G}\,.$ (46)

Here we have set the curvature scale to unity and used the Brown-Henneaux
central charge $c=\frac{3}{2G}$ Brown:1986nw .

####  4.2.1 Bulk Geometry as Defect AdS3

Let us first consider the case when $\Delta_{\text{bcc}}<c/12$. In this case,
the mass parameter $M<0$, so this bulk dual contains a point particle, giving
rise to a conical defect. This geometry is conformally equivalent to a global
AdS3 geometry with a conical deficit:

$\begin{split}ds^{2}&=-\left(r^{2}+1-\frac{12\Delta_{\text{bcc}}}{c}\right)dt^{2}+\left(r^{2}+1-\frac{12\Delta_{\text{bcc}}}{c}\right)^{-1}dr^{2}+r^{2}\,d\theta^{2}\\\
&=-(\tilde{r}^{2}+1)d\tilde{t}^{2}+\frac{d\tilde{r}^{2}}{\tilde{r}^{2}+1}+\tilde{r}^{2}d\tilde{\theta}^{2}\,,\end{split}$
(47)

where we have defined the new “twisted” variables by $\tilde{t}=\mathcal{K}t$,
$\tilde{r}=r/\mathcal{K}$, and $\tilde{\theta}=\mathcal{K}\theta$, where

$\mathcal{K}=\sqrt{1-\frac{12\Delta_{\text{bcc}}}{c}}\,.$ (48)

Notice that the twisted angular coordinate $\tilde{\theta}$ is in the range
$[0,2\pi\mathcal{K}]$ with $\mathcal{K}<1$.

The bulk dual also contains two Karch-Randall branes corresponding to the
boundary conditions $a$ and $b$. To figure out the bulk embedding of these two
branes we go to the embedding space formulation of the AdS3 geometry,

$\begin{split}&ds^{2}=-dX_{0}^{2}-dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2}\,,\\\
&X_{0}^{2}+X_{1}^{2}-X_{2}^{2}-X_{3}^{2}=1\,.\end{split}$ (49)

For the twisted global patch Eq. (47), we have

$\begin{split}X_{0}&=\sqrt{1+\tilde{r}^{2}}\cos{\tilde{t}}\,,\\\
X_{1}&=\sqrt{1+\tilde{r}^{2}}\sin\tilde{t}\,,\\\
X_{2}&=\tilde{r}\cos\tilde{\theta}\,,\\\
X_{3}&=\tilde{r}\sin\tilde{\theta}\,,\end{split}$ (50)

where $\tilde{r}\in[0,\infty)$. Moreover, we notice that an AdS3 space can be
foliated by AdS2 slices (corresponding to constant $R$),

$\begin{split}X_{0}&=\cosh(R)\sqrt{1+\rho^{2}}\cos(t)\,,\\\
X_{1}&=\cosh(R)\sqrt{1+\rho^{2}}\sin(t)\,,\\\ X_{2}&=\cosh(R)\rho\,,\\\
X_{3}&=\sinh(R)\,,\end{split}$ (51)

for $\rho\in(-\infty,\infty)$. Comparing these two parametrizations we find

$\tan\tilde{\theta}=\frac{\tanh(R)}{\rho}\,,$ (52)

which tells us that each AdS2 slice is stretched between two boundary points
with an angular separation $\Delta\tilde{\theta}=\pi$. In terms of the
untwisted coordinates this implies $\Delta\theta=\pi/\mathcal{K}$, which is
larger than $\pi$ for $c/12>\Delta_{\text{bcc}}\neq 0$.

The two branes correspond to two such AdS2 slices, with the following
embeddings:

$\begin{split}\text{Brane
}a\colon&\tan\tilde{\theta}=\frac{\tanh(R_{a})}{\rho}\,,\\\ \text{Brane
}b\colon&\tan(\widetilde{\theta-\pi}+\pi)=\frac{\tanh(R_{b})}{\rho}\,,\end{split}$
(53)

where $\widetilde{\theta-\pi}=\tilde{\theta}-\pi\mathcal{K}$. The two AdS2
foliations in which brane $a$ and brane $b$ are constant $R$ slices are
oppositely oriented and not aligned. Therefore, we need to set $R_{a}>0$ and
$R_{b}<0$. For definiteness, we restrict the domain of the $\tan$ function to
$[0,\pi]\backslash\\{\frac{\pi}{2}\\}$. This implies that brane $a$ stretches
from $\theta=0$ to $\theta=\frac{\pi}{\mathcal{K}}$ and brane $b$ stretches
from $\theta=\pi$ to $\theta=\pi-\frac{\pi}{\mathcal{K}}$.121212 Notice, that
using this embedding, parts of brane $b$ are at negative values of
$\tilde{\theta}$. Exploiting the $2\pi\mathcal{K}$ period of $\tilde{\theta}$,
this segment can be mapped into the previously specified domain
$\tilde{\theta}\in[0,2\pi\mathcal{K}]$. Strictly speaking however, it is
better to describe the brane $b$ using a different fundamental domain for
$\tilde{\theta}$ and at the end exploiting the $2\pi\mathcal{K}$ period to
match with the $[0,2\pi\mathcal{K}]$ domain. This configuration is illustrated
in Fig. 7. To avoid self-intersection for each brane, we require that
$\mathcal{K}\geq\frac{1}{2}$ which translates to

$\Delta_{\text{bcc}}\leq\frac{c}{16}\,.$ (54)

We conclude that there is a gap $(\frac{c}{16},\frac{c}{12})$ in the spectrum
of $\Delta_{\text{bcc}}$.

Such branes satisfy Einstein’s equations if they have a specific tension $T$
which can be determined from the junction condition Kraus:1999it

$K_{\mu\nu}=Th_{\mu\nu}\,,$ (55)

where $h_{\mu\nu}$ is the induced metric on the brane and $K_{\mu\nu}$ is its
extrinsic curvature Wald:1984rg . The latter can be expressed as the
derivative of the induced metric with respect to the brane’s unit normal
vector $n^{\mu}$,

$K_{\mu\nu}=h_{\mu}^{\rho}h_{\nu}^{\sigma}\nabla_{\rho}n_{\sigma}\,.$ (56)

As explained above, we consider a situation where the bulk spacetime ends at
the branes and the branes are located such that the conformal boundary of the
bulk stretches from $\theta=0$ to $\theta=\pi$, i.e. from the first endpoint
of brane $a$ to the opposite-lying endpoint of brane $b$ (see Fig. 7). To
satisfy (55) the normal vector $n^{\mu}$ has to be chosen in such a way that
it points in the outward direction from the point of view of the bulk
spacetime ending on the branes. We can then compute $K_{\mu\nu}$ and
$h_{\mu\nu}$ explicitly and find

$T_{a}=-\tanh(R_{a})\qquad\text{and}\qquad T_{b}=\tanh(R_{b})\,.$ (57)

The relative minus sign comes from the fact that the two AdS2 foliations in
which brane $a$ and brane $b$ are described are oppositely oriented. For the
same reason, the brane embeddings Eq. (53) require $R_{a}>0$ and $R_{b}<0$, so
that both branes have negative tension.

To study the RT surface, we use again the AdS2 foliation Eq. (51). In these
coordinates, the bulk metric reads

$ds^{2}=dR^{2}+\cosh^{2}(R)\left(-(\rho^{2}+1)dt^{2}+\frac{d\rho^{2}}{\rho^{2}+1}\right)\,,$
(58)

and the two Karch-Randall branes are two constant $R$ slices. We let
$\rho=\sinh(\eta)$ and parametrize the RT surface by $\eta(R)$. The area
functional for the RT surface ending on $a$ then reads

$A_{a}=\int_{R_{a}}^{R_{\epsilon}}dR\sqrt{1+\cosh^{2}(R)\eta^{\prime}(R)^{2}}\,,$
(59)

where $R_{\epsilon}$ is a UV regulator. The resulting Euler-Lagrangian
equation can be integrated to

$C=\frac{\cosh^{2}(R)\eta^{\prime}(R)}{1+\cosh^{2}(R)\eta^{\prime}(R)^{2}}\,,$
(60)

for a constant $C$ determined by the boundary condition for the RT surface
near its end point on the brane $a$. This boundary condition, worked out in
Geng:2020fxl , is

$\left|\eta^{\prime}(R)\right|_{\text{brane }a}=0\,.$ (61)

Therefore, we have $C=0$ and the area

$A_{a}=R_{\epsilon}-R_{a}\,.$ (62)

This is an attempt to calculate the entanglement entropy of the boundary
subsystem $\mathcal{A}$ with size $\theta$. The UV regularization
$R_{\epsilon}$ depends on the size of this boundary subsystem, and can be
extracted by comparing Eqs. (50) and (51),

$1+\tilde{r}^{2}\sin^{2}(\tilde{\theta})=\cosh^{2}(R)\,.$ (63)

Hence, we have

$\frac{4}{\epsilon^{2}}\sin^{2}(\tilde{\theta})=e^{2R_{\epsilon}}\,,$ (64)

where we take $\tilde{r}=\epsilon^{-1}\rightarrow\infty$ and
$R_{\epsilon}\rightarrow\infty$. Using the RT formula Ryu:2006bv  and the
Brown-Henneaux central charge Brown:1986nw  we get a candidate expression for
the entanglement entropy of the boundary subsystem $\mathcal{A}$,

$S_{\mathcal{A}}^{a}=\frac{A_{a}}{4G}=\frac{c}{6}\ln\left(\frac{2}{\epsilon}\sin(\theta\mathcal{K})\right)+\ln(g_{a})\,,$
(65)

where we used the fact that $-R_{a}/4G=\ln(g_{a})$. This RT surface
corresponds to the solid gray line in Fig. 7.131313Some care is required here.
For negative-tension branes, we cut off the right part of the conformal
boundary in Fig. 7, and we have $R_{a}>0$ and $R_{b}<0$. Similarly, if we look
at the RT surface which ends on the brane $b$ we get

$A_{b}=R_{b}-R^{\prime}_{\epsilon}\,,$ (66)

and we have

$\frac{4}{\epsilon^{2}}\sin^{2}(\widetilde{\theta-\pi}+\pi)=e^{-2R^{\prime}_{\epsilon}}\,.$
(67)

Using $\frac{R_{b}}{4G}=\ln(g_{b})$ we get another candidate expression for
the entanglement entropy of the boundary subsystem $\mathcal{A}$,

$S_{\mathcal{A}}^{b}=\frac{c}{6}\ln\left(\frac{2}{\epsilon}\sin[(\pi-\theta)\mathcal{K}]\right)+\ln(g_{b})\,.$
(68)

This RT surface corresponds to the dashed gray line in Fig. 7.

In summary, for $\Delta_{\text{bcc}}<c/12$, we obtain the entanglement entropy

$\begin{split}S_{\mathcal{A}}=\min\Biggl{[}\frac{c}{6}\ln\left(\frac{2}{\epsilon}\sin(\theta\mathcal{K})\right)+\ln(g_{a}),\,\frac{c}{6}\ln\left(\frac{2}{\epsilon}\sin[(\pi-\theta)\mathcal{K}]\right)+\ln(g_{b})\Biggr{]}.\end{split}$
(69)

This exactly matches the result of our field theory calculation Eq. (31), with
the identifications $\pi\ell_{\mathcal{A}}/L=\theta$ and $\mathcal{K}=\alpha$.
We have previously established that both branes have negative tensions. Since
the brane tension determines the boundary entropy, this implies that both $\ln
g_{a}$ and $\ln g_{b}$ are also negative. Furthermore, it is important to
notice that the minimization prescription tells us that the actual RT surface
calculating the entanglement entropy never probes into the removed region.
This is provides a nontrivial consistency check of our setup (as depicted in
Fig. 7).

####  4.2.2 Bulk Geometry as a Single-sided Black Hole

For $\Delta_{\text{bcc}}>c/12$ one sees from the metric Eq. (45) that the bulk
dual is a single-sided black hole with mass $m=M+1/8G=\Delta_{\text{bcc}}$ and
zero spin. The entanglement entropy calculation is in parallel with the defect
case and the result can be simply obtained by analytic continuation of the
parameter $\mathcal{K}=\sqrt{1-12\Delta_{\text{bcc}}/c}$ from real to
imaginary. Consequently, one just has to change $\sin\to\sinh$, as on the
field theory side. Hence, for $\Delta_{\text{bcc}}>c/12$ we find the
entanglement entropy

$\begin{split}S_{\mathcal{A}}=\min\Biggl{[}\frac{c}{6}\ln\left(\frac{2}{\epsilon}\sinh(\theta\absolutevalue{\mathcal{K}})\right)+\ln(g_{a}),\,\frac{c}{6}\ln\left(\frac{2}{\epsilon}\sinh[(\pi-\theta)\absolutevalue{\mathcal{K}}]\right)+\ln(g_{b})\Biggr{]}\,,\end{split}$
(70)

which also exactly matches the field theory result, given the fact that
$\mathcal{K}=\alpha$.

$+$$\bullet$$\bullet$$a$$b$ Figure 7: A constant time slice of the point
particle (defect) bulk geometry Eq. (45) in the untwisted radial and angular
coordinates $(r,\theta)$ with two Karch-Randall branes. The blue circle is the
bulk conformal boundary. The two conformal boundaries $a$ and $b$ of the
branes (in red and green) are separated by $\pi$ in the coordinate $\theta$
($\theta_{a}=0$ and $\theta_{b}=\pi$). The black cross is an example
bipartition point on the boundary with two Ryu-Takayanagi surfaces ending on
the branes. The physical RT surface (solid) has always a smaller area than the
unphysical one (dashed) ending on the “hidden” part of the bulk.

###  4.3 Thermal field double state

In this subsection, we perform the holographic computation for the
entanglement entropy of the subsystem $\mathcal{A}$ at a finite temperature as
considered in Sec. 3.2. This is very similar to the left-right entanglement
entropy $S_{L/R}$ that was studied in Geng:2020fxl , and characterizes the
communication between two lower-dimensional black holes. The two black holes
are living on each of the branes, and are induced from the bulk BTZ black
hole:

$ds^{2}=-\frac{h(z)}{z^{2}}dt^{2}+\frac{dz^{2}}{h(z)z^{2}}+\frac{dx^{2}}{z^{2}}\,,\qquad
h(z)=1-\frac{z^{2}}{z_{h}^{2}}\,.$ (71)

To match the field theory computation, we focus on the high temperature phase
where the spatial direction $x$ is fully decompactified. We consider the
maximally extended black hole spacetime with two asymptotic (conformal)
boundaries, and the field theory duals are in the TFD state Maldacena:2001kr .
For simplicity, when we consider the branes and the RT surfaces ending on the
branes, we only consider a single side of the two-sided bulk geometry.

The two KR branes are both parametrized by $x(z)$ and are chosen to have a
separation $\Delta x=\ln(r_{O}/r_{I})$ on the conformal boundary $z=0$. For
such a brane $x(z)$, the outward normal vector is

$n_{\mu}=\frac{(0,-x^{\prime}(z),1)}{\sqrt{x^{\prime
2}(z)h(z)z^{2}+z^{2}}}\,.$ (72)

As before, the shape of the brane is determined by the boundary equations of
motion in the form of Israel’s junction condition Eq. (55). Using Eq. (56) we
find

$x^{\prime}(z)=\pm\frac{T}{\sqrt{1-T^{2}h(z)}}\,.$ (73)

This differential equation integrates to

$x(z)=x(0)\pm
z_{h}\operatorname{arcsinh}{\left(\frac{z\,T}{z_{h}\sqrt{1-T^{2}}}\right)}\,.$
(74)

Notice that this requires that $T^{2}<1$, consistent with the fact that KR
branes are subcritical Karch:2020iit .

For our purposes, we take both branes to have positive tension (not to be
confused with the defect case, where the tensions were negative), but one
takes the positive sign and the other one takes the negative sign in the above
solution,

$\displaystyle x_{L}(z)$
$\displaystyle=z_{h}\ln(\frac{r_{O}}{z_{h}})+z_{h}\operatorname{arcsinh}{\left(\frac{zT_{L}}{z_{h}\sqrt{1-T_{L}^{2}}}\right)}\,,$
(75) $\displaystyle x_{R}(z)$
$\displaystyle=z_{h}\ln(\frac{r_{I}}{z_{h}})-z_{h}\operatorname{arcsinh}{\left(\frac{zT_{R}}{z_{h}\sqrt{1-T_{R}^{2}}}\right)}\,.$

A typical brane configuration with such an embedding is illustrated in Fig. 8.

$\bullet$$a$$\bullet$$b$ Figure 8: A typical brane configuration on a constant
time $t=t_{0}$ slice in the presence of a bulk BTZ black hole. Both branes
have positive tension and are embedded according to Eq. (75). The shaded
regions are removed and the dashed black line is the planar black hole
horizon. We neglect the parts of the branes behind the black hole horizon
which are easily seen to be timelike.

We first study the RT surfaces that start from the conformal boundary and end
on one of the branes. As in Geng:2020fxl , this RT surface is time-
independent, therefore we can compute it on the zero time slice with the
metric

$ds^{2}|_{t=0}=\frac{dz^{2}}{h(z)z^{2}}+\frac{dx^{2}}{z^{2}}\,.$ (76)

To simplify the computation for these brane-ending RT surfaces, we perform the
coordinate transformation

$\tilde{x}=z_{h}e^{\frac{x}{z_{h}}}\sqrt{h(z)}\,,\qquad\tilde{z}=ze^{\frac{x}{z_{h}}}\,.$
(77)

This maps the zero time slice onto a constant time slice in the Poincare
patch,

$ds^{2}|_{t=0}=\frac{d\tilde{x}^{2}+d\tilde{z}^{2}}{\tilde{z}^{2}}\,.$ (78)

The branes become circles on this constant time slice, obeying

$\begin{split}\tilde{x}_{L}^{2}+\left(\tilde{z}-\frac{r_{O}T_{L}}{\sqrt{1-T_{L}^{2}}}\right)^{2}=\frac{r_{O}^{2}}{1-T_{L}^{2}}\,,\\\
\tilde{x}_{R}^{2}+\left(\tilde{z}+\frac{r_{I}T_{R}}{\sqrt{1-T_{R}^{2}}}\right)^{2}=\frac{r_{I}^{2}}{1-T_{R}^{2}}\,.\end{split}$
(79)

We notice from Eq. (77) that we require $\tilde{x}>0$, so strictly speaking
the branes are only arcs on these circles.

We take the bipartition point on the conformal boundary to be at
$x=z_{h}\ln(r/z_{h})$, or equivalently $\tilde{x}=r$. We parametrize the RT
surface ending on the left brane as $\tilde{x}(z)$ to obtain the area
functional

$A=\int_{0}^{\tilde{z}_{L}}\frac{d\tilde{z}}{\tilde{z}}\sqrt{\tilde{x}^{\prime}(\tilde{z})^{2}+1}\,,$
(80)

where $\tilde{z}_{L}$ is the $z$-coordinate of the end point on the brane.
This leads to an Euler-Lagrangian’s equation which can be integrated to

$\frac{1}{\tilde{z}_{*}}=\frac{1}{\tilde{z}}\frac{\tilde{x}^{\prime}}{\sqrt{\tilde{x}^{\prime
2}+1}}\,,$ (81)

where $\tilde{z}_{*}$ is the bulk point at which $\tilde{x}^{\prime}$ goes
from negative to positive. Integrating the equation of motion with the
boundary condition $\tilde{x}(0)=r$ we find

$(\tilde{x}-r-\tilde{z}_{*})^{2}+\tilde{z}^{2}=\tilde{z}_{*}^{2}\,,$ (82)

which is again a circle.

To pin down the values of $\tilde{z}_{*}$ and $\tilde{z}_{L}$, we need the
boundary condition of the RT surface on the left brane. This boundary
condition is determined by area minimization over all possible end points on
the brane. As suggested in Geng:2020fxl , this is most straightforwardly
determined by parametrizing the RT surface using its (normalized) proper
length $s$ ($0<s<1$). Now the area functional reads

$A=\int_{0}^{1}ds\frac{1}{\tilde{z}}\sqrt{\dot{\tilde{z}}^{2}+\dot{\tilde{x}}^{2}}\,.$
(83)

Assuming that the Euler-Lagrange equation is satisfied, the variation of the
action is

$\delta
A=\left.\frac{\dot{\tilde{z}}\delta{\tilde{z}}+\dot{\tilde{x}}\delta{\tilde{x}}}{\tilde{z}\sqrt{\dot{\tilde{z}}^{2}+\dot{\tilde{x}}^{2}}}\right|_{0}^{1}\,.$
(84)

Demanding that this vanishes, we find that the end-point of the RT surface for
the left brane obeys

$\tilde{x}^{\prime}(\tilde{z}_{L})=\frac{\dot{\tilde{x}}}{\dot{\tilde{z}}}=-\frac{\delta{\tilde{z}}}{\delta{\tilde{x}}}=-\left.\tilde{z}^{\prime}(\tilde{x})\right|_{\text{left
brane}}\,,$ (85)

where $\tilde{x}^{\prime}(\tilde{z}_{L})$ is the derivative of the RT surface
at its end-point. Thus, we can finally determine our parameters:

$\tilde{z}_{*}=\frac{r_{O}^{2}-r^{2}}{2r}\,,\qquad\tilde{z}_{L}=\frac{2r_{O}^{3}(1-T_{L})T_{L}}{\sqrt{1-T_{L}^{2}}(r_{O}^{2}(1-T_{L})+r^{2}(1+T_{L}))}\,.$
(86)

The regularized area is therefore

$\begin{split}A_{L}&=\int_{\epsilon}^{\tilde{z}_{*}}\frac{d\tilde{z}\,\tilde{z}_{*}}{\tilde{z}\sqrt{\tilde{z}_{*}^{2}-\tilde{z}^{2}}}+\int_{\tilde{z}_{L}}^{\tilde{z}_{*}}\frac{d\tilde{z}\,\tilde{z}_{*}}{\tilde{z}\sqrt{\tilde{z}_{*}^{2}-\tilde{z}^{2}}}\\\
&=\ln(\frac{r_{O}^{2}-r^{2}}{r_{O}\epsilon}\sqrt{\frac{1+T_{L}}{1-T_{L}}})\,.\end{split}$
(87)

The integrals are easiest to perform using the polar-parametrized equations
for circles. Similar calculations can be done for the right brane, and the
area of the corresponding brane-ending RT surface is

$A_{R}=\ln(\frac{r^{2}-r_{I}^{2}}{r_{I}\epsilon}\sqrt{\frac{1+T_{R}}{1-T_{R}}})\,.$
(88)

The brane tensions are related to the boundary entropies $\ln(g)$ as follows
Takayanagi:2011zk :

$\ln(g)=\frac{c}{6}\operatorname{arctanh}T=\frac{c}{6}\ln(\sqrt{\frac{1+T}{1-T}}).$
(89)

Thus, we can simplify the area results to

$A_{L}=\ln\left(\frac{r_{O}^{2}-r^{2}}{r_{O}\epsilon}\right)+\frac{6}{c}\ln(g_{L})\,,\qquad
A_{R}=\ln\left(\frac{r^{2}-r_{I}^{2}}{r_{I}\epsilon}\right)+\frac{6}{c}\ln(g_{R})\,.$
(90)

To get the entanglement entropy for our two-sided bipartition, we double the
areas of the brane ending RT surfaces.

Hence, using the RT formula, $c=3/2G$, and Eq. (90), the candidate
contributions to the entanglement entropy from these quantum extremal surfaces
read

$\begin{split}S_{1}&=\min\Biggl{[}\frac{2A_{L}}{4G},\,\frac{2A_{R}}{4G}\Biggr{]}=\min\Biggl{[}\frac{c}{3}\ln(\frac{r_{O}^{2}-r^{2}}{r_{O}\epsilon})+2\ln(g_{L}),\,\frac{c}{3}\ln(\frac{r^{2}-r_{I}^{2}}{r_{I}\epsilon})+2\ln(g_{R})\Biggr{]}.\end{split}$
(91)

However, there is another candidate entangling surface that stretches between
the bipartition points on the two asymptotic boundaries. This is called the
Hartman-Maldacena surface Hartman:2013qma  and it plays a central rule in the
recent computations of Page curves for black holes in the KR braneworld
Almheiri:2019psy ; Geng:2020qvw ; Geng:2020fxl .141414 This program has
recently been extended into de Sitter holography. In Geng:2021wcq  a page
curve for de Sitter space was obtained along similar lines.

In our case, the area of this surface is most easily obtained using embedding
space coordinates. Suppose that the bipartition point on one asymptotic
boundary has coordinates

$\tilde{z}=ze^{\frac{x}{z_{h}}}=\epsilon\,,\quad\tilde{x}=z_{h}e^{\frac{x}{z_{h}}}\sqrt{h(z)}=r\,.$
(92)

Then its embedding space coordinates read

$\begin{split}X_{0}&=\sqrt{\frac{z_{h}^{2}}{z^{2}}-1}\sinh(\frac{2\pi
t}{\beta})\,,\\\ X_{1}&=\frac{z_{h}}{z}\cosh(\frac{2\pi x}{\beta})\,,\\\
X_{2}&=\sqrt{\frac{z_{h}^{2}}{z^{2}}-1}\cosh(\frac{2\pi t}{\beta})\,,\\\
X_{3}&=\frac{z_{h}}{z}\sinh(\frac{2\pi x}{\beta})\,,\end{split}$ (93)

where the inverse temperature of the black hole is $\beta=2\pi z_{h}$. The
embedding space coordinate for the bipartition point on the other asymptotic
boundary is obtained by sending $t$ to $-t+\frac{i\beta}{2}$,151515This can be
seen by going to the Euclidean signature, rotating around the thermal circle,
and then returning to Lorentzian signature.

$\begin{split}X_{0}^{\prime}&=\sqrt{\frac{z_{h}^{2}}{z^{2}}-1}\sinh(\frac{2\pi
t}{\beta})\,,\\\ X_{1}^{\prime}&=\frac{z_{h}}{z}\cosh(\frac{2\pi
x}{\beta})\,,\\\
X_{2}^{\prime}&=-\sqrt{\frac{z_{h}^{2}}{z^{2}}-1}\cosh(\frac{2\pi
t}{\beta})\,,\\\ X_{3}^{\prime}&=\frac{z_{h}}{z}\sinh(\frac{2\pi
x}{\beta})\,.\end{split}$ (94)

As a result, we get the area of the Hartman-Maldacena (HM) surface as the
geodesic distance between these two points on the two asymptotic boundaries:

$\begin{split}A_{\text{HM}}&=\operatorname{arccosh}(X_{0}X_{0}^{\prime}+X_{1}X_{1}^{\prime}-X_{2}X_{2}^{\prime}-X_{3}X_{3}^{\prime})\\\
&=\operatorname{arccosh}\left(\frac{2z_{h}^{2}}{z^{2}}\cosh^{2}\left(\frac{2\pi
t}{\beta}\right)\right)\\\
&=\operatorname{arccosh}\left(\frac{2r^{2}}{\epsilon^{2}}\cosh^{2}\left(\frac{2\pi
t}{\beta}\right)\right)\\\
&=2\ln\left(\frac{2r}{\epsilon}\cosh\left(\frac{2\pi
t}{\beta}\right)\right),\end{split}$ (95)

where in the last step we used the fact that we are taking the limit
$\epsilon\rightarrow 0$. Therefore, using the RT formula and the Brown-
Henneaux central charge, its contribution to the entanglement entropy is

$S_{\text{HM}}=\frac{A_{\text{HM}}}{4G}=\frac{c}{3}\ln(\frac{2r}{\epsilon}\cosh\left(\frac{2\pi
t}{\beta}\right))\,.$ (96)

Putting all our candidate minimal surfaces together, we find an expression for
the holographic entanglement entropy capturing the communication between two
braneworld black holes:

$S=\min\Biggl{[}\frac{c}{3}\ln(\frac{r_{O}^{2}-r^{2}}{r_{O}\epsilon})+2\ln(g_{L}),\,\frac{c}{3}\ln(\frac{2r}{\epsilon}\cosh\left(\frac{2\pi
t}{\beta}\right)),\,\frac{c}{3}\ln(\frac{r^{2}-r_{I}^{2}}{r_{I}\epsilon})+2\ln(g_{R})\Biggr{]}.$
(97)

This precisely matches the result of our field theory calculation, Eq. (38).

It is straightforward to see that for a generic $r$ and boundary entropies
$g_{L}$ and $g_{R}$ the entanglement entropy, as a function of time, has a
rising part (coming from the $\cosh(2\pi t/\beta)$ and a constant part (coming
from the other two candidates, which ever is lower, for a given $r$). This is
consistent with the spirit of a Page curve (shown in Fig. 9a) and an
underlying unitarity of the dynamics, which prohibits entropy from
monotonically increasing with time. Another interesting feature is the
behavior of entropy as a function of the radius $r$. Since the system has a
finite length $L$, we expect the entropy to start and end at zero as we vary
the length of the subsystem (in one limit, the subsystem is the full system
and in other, the complement is the full system). However for given
$r_{I},r_{O}$, the time dependent $\cosh(2\pi t/\beta)$ part can contribute,
at least for small $t$. This makes the entropy as a function of $r$ also
depend on time, at least for initial times. The situation for three different
times $t_{1}<t_{2}<t_{3}$ is shown in Fig. 9b. However, it deserves to be
noticed that if $r$ is close enough to $r_{O}$ or $r_{I}$ or if the boundary
entropies $g_{L}$ and $g_{R}$ are small enough, the Hartman-Maldacena surface
is subdominant even at the very beginning. Hence in these cases we will see a
constant Page curve. This is consistent with the observation in Geng:2020fxl
that to see the time dependent part of the Page curve, the subsystem Hilbert
space has to be large enough.

(a)  (b)

Figure 9: (a) The entropy of a subsystem as a function of time (red solid
line). The green dashed line is the temperature dependent term, which scales
as $\cosh{2\pi t/\beta}$ while the horizontal lines correspond to the time
independent terms in the formula for the entropy, Eq. (97). (b) The entropy as
a function of the size of the subsystem, $r$. The gray lines correspond to the
time independent terms in Eq. (97), and are monotonically
increasing/decreasing with $r$. The green lines correspond to the time
dependent part, which increases as time increases ($t_{1}<t_{2}<t_{3}$). At
smaller times, this contributes non-trivially to the entropy (green solid and
green dashed lines), but is irrelevant at later times (green dotted line).

##  5 Conclusions and future directions

In this paper, we constructed a conformal field theory model for information
transfer between two braneworld black holes. The black holes are embedded in
two Karch-Randall branes, induced from a bulk AdS3 planar black hole geometry.
The field theory is a two-dimensional CFT living on a strip, with generically
different conformal boundary conditions on the two edges. Using the AdS/CFT
correspondence, these edges are dualized to two Karch-Randall branes.

We computed an entanglement entropy similar to the L/R entanglement entropy
introduced in Geng:2020fxl  to characterize the communication between the two
black holes. The calculations were performed on both the field theory side and
the gravitational side. We found perfect agreement between these descriptions,
and a Page curve in accord with the unitarity of the underlying dynamics was
obtained.

The setup introduced in Geng:2020fxl  can be obtained from the brane
configuration discussed here by sending the spatial displacement $L$ of the
brane endpoints on the conformal boundary to zero. It is hence a natural
question whether one can take the $L\rightarrow 0$ limit of our results in a
meaningful way. However, as far as our results for the entanglement entropy
are concerned, this limit is obstructed as it requires sending also the
regulator $\epsilon$ to zero. Moreover, we notice that in this limit the
boundary conformal field theory degenerates to a one-dimensional theory living
on the brane-endpoint. It is well known that such a theory, namely conformal
quantum mechanics, generically does not have normalizable energy eigenstates
deAlfaro:1976vlx ; BrittoPacumio:1999ax . We therefore do not expect that the
$L=0$ case can be straightforwardly recovered by naïvly taking the formal
$L\rightarrow 0$ limit of our expressions.

An interesting story appears in the study of the zero temperature case. To
perform this calculation on the field theory side, we used a conformal
transformation to eliminate the local scaling behaviour of the boundary
condition changing operator. This reduced the question to a one-point function
of a primary operator on the twisted upper half plane. Then, as it is standard
in holographic CFTs, we invoked vacuum dominance and simply maximized over BOE
channels for the two boundaries. On the gravity side, we followed the double-
holographically realized quantum extremal surface prescription to compute this
entanglement entropy in the AdS3 geometry with two Karch-Randall branes. We
found a precise match with the field theory result. This provides an extension
of the AdS3/BCFT2 correspondence.161616It deserves to be emphasized that for
the KR branes to model conformal boundary conditions it is essential that the
branes are non-dynamical (no DGP term Dvali:2000hr ) and that there are no
extra degrees of freedom localized on the branes. Otherwise the branes can
absorb and store energy from the bulk violating the conformal boundary
conditions. We thank the referee for pointing this out. However, we emphasize
that it would be interesting to repeat the analyses of Hartman:2013mia ;
Sully:2020pza  and determine the microscopic conditions which ensure the
vacuum dominance on the twisted upper half plane. Presumably, this would
involve constraints on the spectrum of excitations induced by the presence of
distinct boundaries.

We also discovered that in the conical defect geometry at zero temperature,
corresponding to $\Delta_{\text{bcc}}<c/12$, geometric constraints translate
into nontrivial spectral constraints. The requirement that the brane does not
self-intersect results in a gap in the allowed spectrum of bcc operators,
$(\tfrac{c}{16},\tfrac{c}{12})$.171717This is consistent with the bound
$h_{\text{bcc}}\geq c/24$ proposed in Miyaji:2021ktr . It would be interesting
to better understand the connection between the two approaches. Similarly, the
requirement that the branes subtend the correct angle forces their tension to
be negative to ensure that they do not wrap around the conical defect. These
implications from the simple holographic dual are somewhat surprising, and
deserve further exploration from the microscopic viewpoint. Along similar
lines, for a generic CFT living on the twisted-upper-half-plane the matching
between the two boundary channels provides an interesting bootstrap
constraint.181818During the final stages of our work, reference
Antunes:2021qpy  appeared, which studies some aspects of the bootstrap
constraint we propose here. We leave these problems to future work.

Moreover, as a consequence of the mismatching boundary condition or
equivalently of the different brane-tensions, the two branes intersect at an
angle in the bulk, as illustrated in Fig. 7. Typically, in theories which
allow for a quantum gravity UV-completion, there can be additional light
degrees of freedom localized at the intersection of branes. For example, in
string theory intersecting D-branes give rise to charged matter states
localized at the brane intersection, a fact which was often used in
phenomenological model building (see e.g. Ibanez:2012zz ). Such states are
evidently not visible in our low-energy effective action and therefore not
included in our computation, but one might speculate whether their inclusion
could have a qualitative effect on our results.

Finally, it should be emphasized once more that our extension of the
AdS3/BCFT2 correspondence has another interesting feature. The energy sourced
by a boundary condition changing operator warps the bulk in a scaling-
dimension dependent way, rather than simply cutting out portions of vacuum
AdS3 with Karch-Randall branes. This may therefore be a profitable setting in
which we can further study the connection between information transfer,
gravitational backreaction, and the black hole interior.

## Acknowledgements

We would like to thank Andreas Karch, Mark Van Raamsdonk and Lisa Randall for
comments on the draft, and Hong Liu, Suvrat Raju, Jamie Sully, Raman Sundrum,
Tadashi Takayanagi and Chirstoph Uhlemann for helpful discussions. SL is
supported by the US National Science Foundation grant NSF PHY-1915071. RKM is
supported by the National Science Foundation under Grant No. NSF PHY-1748958
and NSF PHY-1915071. DW is supported by an International Doctoral Fellowship
from the University of British Columbia. HG is very grateful to his parents
and recommenders.

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