Holographic BCFT with a Defect on the

End-of-the-World Brane

Masamichi Miyaji1,2 and Chitraang Murdia3,4

1 Institute for Advanced Research, Nagoya University,
Nagoya, Aichi 464-8601, Japan
2Department of Physics, Nagoya University,
Nagoya, Aichi 464-8602, Japan

3 Berkeley Center for Theoretical Physics, Department of Physics,
University of California, Berkeley, CA 94720, USA
4 Theoretical Physics Group, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA

###### Abstract

In this paper, we propose a new gravity dual for a $2$d BCFT with two
conformal boundaries by introducing a defect that connects the two End-of-the-
World branes. We demonstrate that the BCFT dual to this bulk model exhibits a
richer lowest spectrum. The corresponding lowest operator dimension can
continuously interpolate between $0$ and $\frac{c}{24}$. This range was
inaccessible to the conventional AdS/BCFT model with distinct boundary
conditions. We compute the holographic entanglement entropy and find that it
exhibits three different phases, one of which breaks the time reflection
symmetry. We also construct a wormhole saddle, analogous to a $3$d replica
wormhole, which connects different boundaries through the AdS bulk. This
saddle is present only if the BCFT is non-unitary and is always subdominant
compared to the disconnected saddle.

###### Contents

  1. 1 Introduction
  2. 2 Gravity Dual
    1. 2.1 Gravity Dual of BCFT with a Corner
  3. 3 BCFT on a Strip
    1. 3.1 Bulk Geometry
    2. 3.2 Euclidean Action
    3. 3.3 BCFT Spectrum
    4. 3.4 Spectral Gap in Liouville Theory
  4. 4 Entanglement Entropy and Entanglement Island
    1. 4.1 Three Phases of the RT Surface
    2. 4.2 Interpretation as Entanglement Island
  5. 5 Three-dimensional Wormhole Saddles
  6. 6 Discussion
  7. A Extention of Boundary Conformal Transformation to Bulk

##  1 Introduction

AdS/BCFT [1, 2, 3] studies the gravity dual of boundary conformal field theory
[4, 5]. The simplest, bottom-up model proposed for AdS/BCFT is a spacetime
terminating at the End-of-the-World (EOW) brane [2, 3, 6]. The EOW brane is
anchored to the BCFT boundary. This bottom-up model captures some qualitative
aspects of stringy models for AdS/BCFT [7, 8, 9], and has served a useful toy
model for black hole evaporation [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20],
interpreted as doubly-holographic brane-world models [21, 22, 23, 24]. On the
other hand, this simplest model of AdS/BCFT is known to have several atypical
features in the boundary operator spectrum amongst all holographic BCFTs, such
as fine-tuned boundary operator spectrum [25] and the absence of interactions
between distinct EOW branes. The lattter, which is the main focus of this
paper, results in a fixed gap between the lowest eigenvalues of the BCFT with
two distinct boundaries and the BCFT with identical boundaries [26].

In this paper, we modify the conventional AdS/BCFT model by allowing the EOW
branes with different tensions to be connected at defects. Clearly, this
defect introduces an interaction between these EOW branes. We find that this
interaction leads to several novel results:

  * •

The defect modifies the lowest eigenvalue of the conventional AdS/BCFT model.
In fact, the lowest eigenvalue in our refined model continuously interpolates
between the identical boundary BCFT and the aforementioned spectral gap. We
will also add a bulk conical defect in the gravity side, corresponding to a
boundary operator [27, 28, 29, 30].

  * •

The bulk theory we propose can also be considered as a gravity dual of a BCFT
with corners [31, 32]. In this duality, the corner on the boundary is in
direct correspondence with the defect on the EOW branes.

  * •

The holographic entanglement entropy in our model exhibits three different
phases. This entanglement entropy can be seen as a toy model of a matter state
prepared by cosmological spacetimes [33, 34]. Interestingly, we have a phase
which breaks the time reflection symmetry, so the corresponding entanglement
entropy is closely related to the pseudo entropy [35].

  * •

The defect enables us to construct a wormhole saddle which connects multiple
AdS boundaries, analogous to the replica wormhole with EOW branes considered
in [13]. In our model, the wormhole saddle is allowed only when the BCFT is
non-unitary, and this saddle is always subdominant compared to the factorized
saddle without any wormholes.

The organization of this paper is as follows. In section 2, we present our
refined AdS/BCFT model and its connection to BCFT with a corner. In section 3,
we study the spectrum of BCFT on a strip, confirming that it has a richer
spectrum in comparison with the conventional AdS/BCFT model. In section 4, we
compute the holographic entanglement entropy and describe its three phases. In
section 5, we construct wormhole geometries using our model. In section 6, we
conclude with discussions and future directions.

##  2 Gravity Dual

In this section, we describe our proposed AdS/BCFT model in which the EOW
branes are connected at a defect. We use our proposal to find the bulk dual of
a BCFT with a corner.

In the original AdS/BCFT proposal[2, 3], the holographic BCFT was dual to an
asymptotically AdS spacetime $M$ with an EOW brane $\Sigma$ anchored to the
BCFT boundary. This EOW brane might contain matter fields and have a non-
trivial gravitational action. We take the bulk theory to be Einstein gravity

$I_{\text{EH}}=-\frac{1}{16\pi
G_{N}}\int_{M}\sqrt{g}\left(R-2\Lambda\right)-\frac{1}{8\pi
G_{N}}\int_{N}\sqrt{h}K,$ (2.1)

where $N$ is the AdS boundary.

The simplest EOW brane action is given by the Gibbons-Hawking boundary term
plus the tension term,

$I_{\text{ETW}}=-\frac{1}{8\pi G_{N}}\int_{\Sigma}\sqrt{h}K+\frac{1}{8\pi
G_{N}}\int_{\Sigma}\sqrt{h}T-\frac{1}{8\pi
G_{N}}\int_{\Gamma_{\Sigma,N}}\sqrt{g_{\Gamma_{\Sigma,N}}}\left(\pi-\theta_{(\Sigma,N)}\right).$
(2.2)

Here $h_{ab}$ is the induced metric and $K_{ab}$ is the extrinsic curvature
defined using the outgoing normal vector. We also need to include the Hayward
term for $\Gamma_{\Sigma,N}$, the corner between AdS boundary $N$ and the EOW
brane $\Sigma$, with internal angle $\theta_{(\Sigma,N)}$. This action leads
to the equation of motion for the EOW brane

$\displaystyle K_{ab}=(K-T)h_{ab}.$ (2.3)

The boundary entropy $S_{B}$ in $2$d BCFT is defined in terms of the disk
partition function via $Z_{\text{Disk}}=e^{S_{B}}$ [36], which counts the
number of degrees of freedom on the boundary. The tension $T$ of the EOW brane
is related to the boundary entropy as

$S_{B}=\frac{c}{6}\operatorname{arctanh}(\ell T),$ (2.4)

where $\ell$ is the AdS radius [37]. In this simple model, there are no
interactions and no intersections between EOW branes dual to distinct BCFT
boundaries where as for identical boundaries, the EOW branes can be smoothly
connected. This simple interaction leads to a simple value for the lowest
eigenvalue [26] that is highly non-generic, which we will discuss in section
3.

The new ingredient introduced in this paper is such an interaction in the form
of a defect which glues two distinct EOW branes. Since the dual spacetime
terminates at some finite depth, the spectrum is no longer that simple, and
indeed we have a new lowest eigenvalue as we show later. The simplest action
for this defect is given by

$I_{\text{defect}}=-\frac{1}{8\pi
G_{N}}\int_{\Gamma_{(a,b)}}\sqrt{g_{\Gamma_{(a,b)}}}\left(\theta_{0:(a,b)}-\theta_{(a,b)}\right).$
(2.5)

Here $\Gamma_{(a,b)}$ is the defect which glues two EOW branes $\Sigma_{a}$
and $\Sigma_{b}$, and $\theta_{(a,b)}$ is the internal angle of $M$ at
$\Gamma_{(a,b)}$, see Fig. 1. Also, $\theta_{0:(a,b)}-\pi$ can be regarded as
the tension of the defect, The action in (2.5) is called Hayward term [38]
when $\theta_{0:(a,b)}=\pi$, and has been considered in different AdS/BCFT
contexts in [6, 39, 40]. This Hayward term can be obtained by taking a
singular limit of the Gibbons-Hawking-York term where the boundary has a sharp
corner.

Figure 1: Sketch of our proposed gravity dual of BCFT. $N$ is AdS boundary
where the BCFT lives. $\Sigma_{a}$ and $\Sigma_{b}$ are the EOW branes, and
$\Gamma_{(a,b)}$ is the defect connecting them.

The total action is the sum of these three contributions,

$I_{\text{total}}=I_{\text{EH}}+I_{\text{ETW}}+I_{\text{defect}}.$ (2.6)

It is worth mentioning that this $I_{\text{total}}$ has UV divergences coming
from the region near the AdS boundary. To obtain a finite answer, these
divergences need to be cured by introducing a short distance cutoff at the
boundary and including appropriate counterterms to cancel the divergence.

The variation of $I_{\text{total}}$ is

$\delta I_{\text{total}}=-\frac{1}{16\pi
G_{N}}\int_{M}\sqrt{g}\left(R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda
g_{\mu\nu}\right)\delta g^{\mu\nu}\\\ -\sum_{i=a,b}\frac{1}{8\pi
G_{N}}\int_{\Sigma_{i}}\sqrt{h}\left(K_{\alpha\beta}-Kh_{\alpha\beta}+T_{i}h_{\alpha\beta}\right)\delta
h^{\alpha\beta}-\frac{1}{8\pi
G_{N}}\int_{N}\sqrt{h}\left(K_{\alpha\beta}-Kh_{\alpha\beta}\right)\delta
h^{\alpha\beta}\\\ -\frac{1}{8\pi
G_{N}}\int_{\Gamma_{(a,b)}}\left(\theta_{0:(a,b)}-\theta_{(a,b)}\right)\delta\sqrt{g_{\Gamma_{(a,b)}}}-\sum_{i=a,b}\frac{1}{8\pi
G_{N}}\int_{\Gamma_{(i,N)}}\left(\pi-\theta_{(i,N)}\right)\delta\sqrt{g_{\Gamma_{(i,N)}}}.$
(2.7)

In AdS/BCFT, the metric can fluctuate freely on $\Sigma_{i}$, but not at AdS
boundary $N$, therefore

$\begin{cases}\delta h^{\alpha\beta}_{N}=0,&\delta
h^{\alpha\beta}_{\Sigma_{i}}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{free},\\\
\delta\sqrt{g_{\Gamma_{(i,N)}}}=0,&\delta\sqrt{g_{\Gamma_{(a,b)}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{free}.\end{cases}$ (2.8)

These boundary conditions at $\Sigma_{i}$ and $\Gamma_{(a,b)}$ result in the
following equations of motion

$\displaystyle K_{\alpha\beta}=(K-T_{i})h_{\alpha\beta},$ (2.9)
$\displaystyle\theta_{0:(a,b)}-\theta_{(a,b)}=0.$ (2.10)

These two equations determine the shapes of the EOW branes $\Sigma_{i}$ and of
the corner $\Gamma_{(a,b)}$. On the other hand, the above boundary conditions
do not induce equations of motion at the AdS boundary and $\Gamma_{(a,N)}$
because of the Dirichlet boundary condition.

###  2.1 Gravity Dual of BCFT with a Corner

In this subsection, we examine the gravity dual of BCFT with a corner as a
simple example. This dual is given by a vacuum AdS spacetime with two EOW
branes glued at a defect.

Consider the Poincare background for vacuum AdS3

$\mathrm{d}s^{2}=\frac{\ell^{2}}{z^{2}}(\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2})$
(2.11)

with $\ell$ being the AdS length. We restrict to the region

$D:=\\{u=re^{\mathrm{i}\theta}\in\mathbb{C}|r\geq 0,\leavevmode\nobreak\
0\leq\theta\leq\gamma_{0}\\}$ (2.12)

on the AdS boundary with $u:=x+\mathrm{i}y$. Since this region has a corner at
the origin, the bulk geometry is dual to BCFT with a corner. The bulk
spacetime we consider is a spacetime bounded by two EOW branes anchored to
$\partial D$. The equation of motion for the EOW brane fixes its shape to be a
plane in $(x,y,z)$ coordinates. Fig. 2 shows the bulk geometry with these two
intersecting EOW branes. Since the bulk metric is the Poincare metric, the
Fefferman-Graham expansion tells us that the stress tensor $\langle
T_{uu}\rangle$ vanishes away from the boundaries.

Figure 2: Two EOW branes embedded in Poincare AdS. They intersect in the bulk
at an internal angle $\theta_{(1,2)}$ and on the boundary at an internal angle
$\gamma_{0}$. The bulk region is dual to BCFT on the cornered region $N=D$.

Let us first assume that the two EOW branes have equal tensions, so the
boundary entropy for the two boundaries are identical. The dimensionless
tension $\mathcal{T}_{a}=\ell T_{a}$ of the EOW brane is related to the
internal angle $\theta_{(a,N)}$ between the brane and the AdS boundary as

$\mathcal{T}_{a}=-\cos\theta_{(a,N)}.$ (2.13)

Note that the allowed values of the tension are given by $|\mathcal{T}|\leq
1$.

The internal angle $\gamma_{0}$ at AdS boundary and the internal angle
$\theta_{(1,2)}$ between EOW branes are related as

$\gamma_{0}=\begin{cases}\operatorname{arccos}\left(\frac{\cos\theta_{(1,2)}+\mathcal{T}^{2}}{1-\mathcal{T}^{2}}\right)&\quad
2\operatorname{arcsin}|\mathcal{T}|<\theta_{(1,2)}<\pi\\\
2\pi-\operatorname{arccos}\left(\frac{\cos\theta_{(1,2)}+\mathcal{T}^{2}}{1-\mathcal{T}^{2}}\right)&\quad\pi<\theta_{(1,2)}<2\pi-2\operatorname{arcsin}|\mathcal{T}|\end{cases}.$
(2.14)

Since the configuration of EOW brane is a plane in the $(x,y,z)$ coordinates,
$\theta_{(1,2)}$ is simply the angle of intersection of two planes.

For arbitrary tensions $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, we have

$\gamma_{0}=\begin{cases}\operatorname{arccos}\left(\frac{\cos\theta_{(1,2)}+\mathcal{T}_{1}\mathcal{T}_{2}}{\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}}\right),\\\
\quad\quad\quad\quad\operatorname{arccos}\left(\text{\footnotesize${\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}-\mathcal{T}_{1}\mathcal{T}_{2}}$
}\right)<\theta_{(1,2)}<\pi-\operatorname{arccos}\left(\text{\footnotesize${\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}+\mathcal{T}_{1}\mathcal{T}_{2}}$}\right),\\\
2\pi-\operatorname{arccos}\left(\frac{\cos\theta_{(1,2)}+\mathcal{T}_{1}\mathcal{T}_{2}}{\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}}\right),\\\
\quad\quad\quad\quad\pi+\operatorname{arccos}\left(\text{\footnotesize${\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}+\mathcal{T}_{1}\mathcal{T}_{2}}$
}\right)<\theta_{(1,2)}<2\pi-\operatorname{arccos}\left(\text{\footnotesize${\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}-\mathcal{T}_{1}\mathcal{T}_{2}}$}\right).\end{cases}$
(2.15)

In the following sections, we will use this geometry to construct the gravity
dual of BCFT on a strip.

##  3 BCFT on a Strip

In this section, we construct the gravity dual of BCFT on an infinite
Euclidean strip by using our refined model for AdS/BCFT. We suppose that
conformal boundary conditions $a$ and $b$ have been imposed on the two
boundaries of this strip.

Let us first review the conventional AdS/BCFT model which does not have any
defects connecting the EOW branes. When the two boundary conditions are
identical, the dual is given by thermal AdS with the EOW branes being
connected without any defect (if the two boundaries are sufficiently close to
each other) [2, 3, 26]. In this case, the lowest eigenvalue was found to be
$E_{(a,b)}^{\text{BCFT}}=-\frac{\pi c}{24\Delta x}$ where $\Delta x$ is the
width of the strip. This spectrum corresponds to the conformal dimension of
the boundary condition changing operator being $h^{\text{bcc}}=0$.

When the boundary entropy of two boundary conditions are different, the dual
geometry is given by two disconnected EOW branes in the Poincare background.
As the result, the lowest eigenvalue for the BCFT was found to be
$E_{(a,b)}^{\text{BCFT}}=0$, which corresponds to
$h^{\text{bcc}}=\frac{c}{24}$. These two results imply that there is no
operator with conformal dimension between $h^{\text{bcc}}=0$ and
$h^{\text{bcc}}=\frac{c}{24}$ in the conventional AdS/BCFT model.

In the following subsections, we generalize this simple spectrum by
introducing a defect which can connect the EOW branes, even if they have
different tensions, or equivalently, different boundary entropies. This will
allow us to obtain any value for the lowest operator dimension between
$h^{\text{bcc}}=0$ and $h^{\text{bcc}}=\frac{c}{24}$.

###  3.1 Bulk Geometry

We are interested in constructing the gravity dual of an infinite Euclidean
strip of width $\Delta x$. Let us start by considering the thermal AdS3
geometry with two EOW branes connected through a defect. The metric of the
thermal AdS3 without any branes is

$\mathrm{d}s^{2}=\ell^{2}\left(\frac{\mathrm{d}\tau^{2}}{z_{0}^{2}\chi^{2}}+\frac{\mathrm{d}\chi^{2}}{h(\chi)\chi^{2}}+\frac{h(\chi)}{\chi^{2}}\mathrm{d}\phi^{2}\right).$
(3.1)

Here $h(\chi)=(1-\chi^{2})$, with $0<\chi\leq 1$. We denote the periodicity of
the Euclidean time $\tau$ as $T_{\mathrm{BCFT}}^{-1}=2\pi z_{0}\chi_{H}$ which
is assumed to be large compared to $z_{0}$ so that the thermal AdS saddle is
dominant compared to Euclidean black hole. The $\phi$ coordinate has period
$2\pi$ so that there is no conical defect at $\chi=1$. Note that $z_{0}$ sets
the length scale for the BCFT.

Let us first embed a single EOW brane without any defects in this geometry.
Since the brane satisfies the equation of motion in (2.9), its profile is

$\phi(\chi)=\phi(0)\pm\operatorname{arctan}\text{\small$\frac{\mathcal{T}\chi}{\sqrt{h(\chi)-\mathcal{T}^{2}}}$},$
(3.2)

in terms of the dimensionless tension $\mathcal{T}$. As explained in appendix
A, the bulk coordinate transformation in (A.2) can be used to show that this
configuration is equivalent to EOW brane with tension $\mathcal{T}$ anchored
to a line in Poincare AdS. The deepest point that the EOW brane reaches before
turning back towards the AdS boundary is

$\chi=\chi_{0}(\mathcal{T}):=\sqrt{1-\mathcal{T}^{2}}.$ (3.3)

Next we consider two EOW branes $\Sigma_{1}$ and $\Sigma_{2}$, with
dimensionless tensions $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$, respectively.
We assume that these tensions satisfy
$|\mathcal{T}_{2}|\leq|\mathcal{T}_{1}|$, without loss of generality. The
branes are anchored to the AdS boundary at
$\Gamma_{1,N}=\\{\chi=0,\,\phi=\pi-\alpha\\}$ and
$\Gamma_{2,N}=\\{\chi=0,\,\phi=\pi+\alpha\\}$ respectively, with some
$\alpha\in(0,\pi)$. As mentioned earlier, $M$ denotes the bulk region between
these branes. Its boundary $\partial M$ consists of the two EOW branes
$\Sigma_{1}$ and $\Sigma_{2}$, along with a region on the AdS boundary,
$N=\\{\chi=0,\pi-\alpha\leq\phi\leq\pi+\alpha\\}$. From the perspective of the
AdS boundary, the angular size of $N$ is $\Delta\phi=2\alpha$. The width of
the strip is $\Delta x=2\alpha z_{0}$.

We want to calculate the angle $\theta_{(1,2)}$ at which the two EOW branes
intersect in the bulk. Depending on the value of $\alpha$, the bulk geometries
are qualitatively different, so we need to treat the various cases separately.

As an illustration, let us consider the case where
$\mathcal{T}_{1},\mathcal{T}_{2}>0$ and $\alpha$ is small, in particular,
$0\leq\alpha\leq\frac{\pi}{4}+\frac{1}{2}\operatorname{arctan}\frac{\mathcal{T}_{2}\sqrt{1-\mathcal{T}_{1}^{2}}}{\sqrt{\mathcal{T}_{1}^{2}-\mathcal{T}_{2}^{2}}}$.
We will state the results for the other cases momentarily. A constant $\tau$
slice of the bulk geometry in this case is shown in Fig. 3. The profiles of
the EOW branes in the intersection region are

$\phi_{1}(\chi)=-\alpha+\operatorname{arctan}\text{\small$\frac{\mathcal{T}_{1}\chi}{\sqrt{h(\chi)-\mathcal{T}_{1}^{2}}}$},\quad\quad\phi_{2}(\chi)=\alpha-\operatorname{arctan}\text{\small$\frac{\mathcal{T}_{2}\chi}{\sqrt{h(\chi)-\mathcal{T}_{2}^{2}}}$}.$
(3.4)

Figure 3: A constant $\tau$ slice of thermal AdS with two EOW branes,
$\Sigma_{1}$ and $\Sigma_{2}$, meeting at the corner $\Gamma_{(1,2)}$.

To obtain the intersection point $\chi_{*}$, we set
$\phi_{1}(\chi_{*})=\phi_{2}(\chi_{*})$ to get an equation for $\chi_{*}$

$\displaystyle\operatorname{arctan}\text{\small$\frac{\mathcal{T}_{1}\chi_{*}}{\sqrt{h(\chi_{*})-\mathcal{T}_{1}^{2}}}$}+\operatorname{arctan}\text{\small$\frac{\mathcal{T}_{2}\chi_{*}}{\sqrt{h(\chi_{*})-\mathcal{T}_{2}^{2}}}$}=2\alpha.$
(3.5)

This $\chi_{*}$ increases monotonically with $\alpha$. The upper bound on
$\alpha$ for this case is attained when $\chi_{*}$ equals the maximum value
$\chi_{0}(\mathcal{T}_{1})=\sqrt{1-\mathcal{T}_{1}^{2}}$ 111This maximal value
corresponds to the deepest point on $\Sigma_{1}$ because we have assumed that
$|\mathcal{T}_{1}|>|\mathcal{T}_{2}|$, which implies that
$\chi_{0}(\mathcal{T}_{1})<\chi_{0}(\mathcal{T}_{2})$., and the explicit value
is
$\alpha=\frac{\pi}{4}+\frac{1}{2}\operatorname{arctan}\frac{\mathcal{T}_{2}\sqrt{1-\mathcal{T}_{1}^{2}}}{\sqrt{\mathcal{T}_{1}^{2}-\mathcal{T}_{2}^{2}}}$
as stated above. The angle of intersection between the two EOW branes is

$\theta_{(1,2)}=\operatorname{arccos}\text{\small$\frac{\sqrt{(h(\chi_{*})-\mathcal{T}_{1}^{2})(h(\chi_{*})-\mathcal{T}_{1}^{2})}-\mathcal{T}_{1}\mathcal{T}_{2}}{h(\chi_{*})}$}.$
(3.6)

Now we state the result for general $\alpha$. Assuming that the tensions
satisfy the condition $|\mathcal{T}_{2}|\leq|\mathcal{T}_{1}|$, the equation
to obtain the intersection point is

$\begin{cases}\Big{|}\operatorname{arctan}\frac{\mathcal{T}_{1}\chi_{*}}{\sqrt{h(\chi_{*})-\mathcal{T}_{1}^{2}}}+\operatorname{arctan}\frac{\mathcal{T}_{2}\chi_{*}}{\sqrt{h(\chi_{*})-\mathcal{T}_{2}^{2}}}\Big{|}=2\alpha,&0<\alpha<\frac{\pi}{4}+\frac{\operatorname{sgn}(\mathcal{T}_{1})}{2}\operatorname{arctan}\frac{\mathcal{T}_{2}\sqrt{1-\mathcal{T}_{1}^{2}}}{\sqrt{\mathcal{T}_{1}^{2}-\mathcal{T}_{2}^{2}}},\\\
\Big{|}\operatorname{arctan}\frac{\mathcal{T}_{1}\chi_{*}}{\sqrt{h(\chi_{*})-\mathcal{T}_{1}^{2}}}-\operatorname{arctan}\frac{\mathcal{T}_{2}\chi_{*}}{\sqrt{h(\chi_{*})-\mathcal{T}_{2}^{2}}}\Big{|}=\pi-2\alpha,&\frac{\pi}{4}+\frac{\operatorname{sgn}(\mathcal{T}_{1})}{2}\operatorname{arctan}\frac{\mathcal{T}_{2}\sqrt{1-\mathcal{T}_{1}^{2}}}{\sqrt{\mathcal{T}_{1}^{2}-\mathcal{T}_{2}^{2}}}<\alpha<\frac{\pi}{2},\\\
\Big{|}\operatorname{arctan}\frac{\mathcal{T}_{1}\chi_{*}}{\sqrt{h(\chi_{*})-\mathcal{T}_{1}^{2}}}-\operatorname{arctan}\frac{\mathcal{T}_{2}\chi_{*}}{\sqrt{h(\chi_{*})-\mathcal{T}_{2}^{2}}}\Big{|}=2\alpha-\pi,&\frac{\pi}{2}<\alpha<\frac{3\pi}{4}-\frac{\operatorname{sgn}(\mathcal{T}_{1})}{2}\operatorname{arctan}\frac{\mathcal{T}_{2}\sqrt{1-\mathcal{T}_{1}^{2}}}{\sqrt{\mathcal{T}_{1}^{2}-\mathcal{T}_{2}^{2}}},\\\
\Big{|}\operatorname{arctan}\frac{\mathcal{T}_{1}\chi_{*}}{\sqrt{h(\chi_{*})-\mathcal{T}_{1}^{2}}}+\operatorname{arctan}\frac{\mathcal{T}_{2}\chi_{*}}{\sqrt{h(\chi_{*})-\mathcal{T}_{2}^{2}}}\Big{|}=2\pi-2\alpha,&\frac{3\pi}{4}-\frac{\operatorname{sgn}(\mathcal{T}_{1})}{2}\operatorname{arctan}\frac{\mathcal{T}_{2}\sqrt{1-\mathcal{T}_{1}^{2}}}{\sqrt{\mathcal{T}_{1}^{2}-\mathcal{T}_{2}^{2}}}<\alpha<\pi,\end{cases}$
(3.7)

and the angle of intersection is

$\theta_{(1,2)}=\begin{cases}\operatorname{arccos}\left(\frac{\sqrt{(h(\chi_{*})-\mathcal{T}_{1}^{2})(h(\chi_{*})-\mathcal{T}_{2}^{2})}-\mathcal{T}_{1}\mathcal{T}_{2}}{h(\chi_{*})}\right),&0<\alpha<\frac{\pi}{4}+\frac{\operatorname{sgn}(\mathcal{T}_{1})}{2}\operatorname{arctan}\frac{\mathcal{T}_{2}\sqrt{1-\mathcal{T}_{1}^{2}}}{\sqrt{\mathcal{T}_{1}^{2}-\mathcal{T}_{2}^{2}}},\\\
\operatorname{arccos}\left(-\frac{\sqrt{(h(\chi_{*})-\mathcal{T}_{1}^{2})(h(\chi_{*})-\mathcal{T}_{2}^{2})}+\mathcal{T}_{1}\mathcal{T}_{2}}{h(\chi_{*})}\right),&\frac{\pi}{4}+\frac{\operatorname{sgn}(\mathcal{T}_{1})}{2}\operatorname{arctan}\frac{\mathcal{T}_{2}\sqrt{1-\mathcal{T}_{1}^{2}}}{\sqrt{\mathcal{T}_{1}^{2}-\mathcal{T}_{2}^{2}}}<\alpha<\frac{\pi}{2},\\\
2\pi-\operatorname{arccos}\left(-\frac{\sqrt{(h(\chi_{*})-\mathcal{T}_{1}^{2})(h(\chi_{*})-\mathcal{T}_{2}^{2})}+\mathcal{T}_{1}\mathcal{T}_{2}}{h(\chi_{*})}\right),&\frac{\pi}{2}<\alpha<\frac{3\pi}{4}-\frac{\operatorname{sgn}(\mathcal{T}_{1})}{2}\operatorname{arctan}\frac{\mathcal{T}_{2}\sqrt{1-\mathcal{T}_{1}^{2}}}{\sqrt{\mathcal{T}_{1}^{2}-\mathcal{T}_{2}^{2}}},\\\
2\pi-\operatorname{arccos}\left(\frac{\sqrt{(h(\chi_{*})-\mathcal{T}_{1}^{2})(h(\chi_{*})-\mathcal{T}_{2}^{2})}-\mathcal{T}_{1}\mathcal{T}_{2}}{h(\chi_{*})}\right),&\frac{3\pi}{4}-\frac{\operatorname{sgn}(\mathcal{T}_{1})}{2}\operatorname{arctan}\frac{\mathcal{T}_{2}\sqrt{1-\mathcal{T}_{1}^{2}}}{\sqrt{\mathcal{T}_{1}^{2}-\mathcal{T}_{2}^{2}}}<\alpha<\pi.\end{cases}$
(3.8)

Here, $\operatorname{sgn}(x):=\frac{x}{|x|}$ is the sign function. Note that
these results are valid even when one or both tensions are negative.

The equation of motion (2.10) fixes the angle of intersection to be
$\theta_{(1,2)}=\theta_{0}$. This determines the value of $\alpha$, and hence,
the geometry in terms of $\theta_{0}$ and the tensions of EOW branes. Indeed,
substituting $\theta_{(1,2)}=\theta_{0}$ in (3.8), we get

$\chi_{*}=\sqrt{1-\text{\small$\frac{\mathcal{T}_{1}^{2}+\mathcal{T}_{2}^{2}+2\mathcal{T}_{1}\mathcal{T}_{2}\cos\theta_{0}}{\sin^{2}\theta_{0}}$}}.$
(3.9)

Also, the value of $\alpha$ that corresponds to this $\theta_{0}$ is

$\alpha_{0}=\begin{cases}\frac{1}{2}\operatorname{arccos}\left(\frac{\cos\theta_{0}+\mathcal{T}_{1}\mathcal{T}_{2}}{\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}}\right),\\\
\quad\quad\quad\quad\operatorname{arccos}\left(\text{\footnotesize${\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}-\mathcal{T}_{1}\mathcal{T}_{2}}$
}\right)<\theta_{0}<\pi-\operatorname{arccos}\left(\text{\footnotesize${\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}+\mathcal{T}_{1}\mathcal{T}_{2}}$}\right),\\\
\pi-\frac{1}{2}\operatorname{arccos}\left(\frac{\cos\theta_{0}+\mathcal{T}_{1}\mathcal{T}_{2}}{\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}}\right),\\\
\quad\quad\quad\quad\pi+\operatorname{arccos}\left(\text{\footnotesize${\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}+\mathcal{T}_{1}\mathcal{T}_{2}}$
}\right)<\theta_{0}<2\pi-\operatorname{arccos}\left(\text{\footnotesize${\sqrt{(1-\mathcal{T}_{1}^{2})(1-\mathcal{T}_{2}^{2})}-\mathcal{T}_{1}\mathcal{T}_{2}}$}\right).\end{cases}$
(3.10)

Note that (3.9) and (3.10) are symmetric in $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$, therefore the formula applies even when
$|\mathcal{T}_{1}|<|\mathcal{T}_{2}|$. Moreover, this result for $\alpha_{0}$
is identical to (2.15) if we identify $\gamma_{0}$ with $2\alpha_{0}$. In
fact, the two bulk geometries can be identified by a bulk coordinate
transformation which maps the infinite strip $N$ to the cornered region $D$,
as explained in appendix A.

(a) Equal tension $\mathcal{T}_{1}=\mathcal{T}_{2}$

(b) Fixed $\mathcal{T}_{2}$

(c) Fixed $\mathcal{T}_{1}$

Figure 4: Plot of $\alpha_{0}$ as a function of $\theta_{0}$ for various
values of tensions.

Fig. 4 shows $\alpha_{0}$ as we vary $\theta_{0}$ for various values of the
tensions based on (3.10). When the two tensions are equal
$\mathcal{T}_{1}=\mathcal{T}_{2}$, then $\theta_{0}$ is a continuous function
of $\alpha_{0}$. In this case, (3.9) and (3.10) simplify to

$\chi_{*}=\sqrt{1-\text{\small$\frac{\mathcal{T}^{2}}{\sin^{2}(\theta_{0}/2)}$}},$
(3.11)

and

$\alpha_{0}=\begin{cases}\frac{1}{2}\operatorname{arccos}\left(\frac{\cos\theta_{0}+\mathcal{T}^{2}}{1-\mathcal{T}^{2}}\right),&\quad
2\operatorname{arcsin}|\mathcal{T}|<\theta_{0}\leq\pi,\\\
\pi-\frac{1}{2}\operatorname{arccos}\left(\frac{\cos\theta_{0}+\mathcal{T}^{2}}{1-\mathcal{T}^{2}}\right),&\quad\pi\leq\theta_{0}<2\pi-2\operatorname{arcsin}|\mathcal{T}|.\end{cases}$
(3.12)

If we also set $\theta_{0}=\pi$, then we get $\alpha_{0}=\frac{\pi}{2}$. The
dual bulk geometry has a smooth EOW brane geometry without any defect, which
is the same as the conventional AdS/BCFT model. Conversely, we find that
$\theta_{0}=\pi$ is possible only if $\mathcal{T}_{1}=\mathcal{T}_{2}$.

When $\mathcal{T}_{1}\neq\mathcal{T}_{2}$, the curve for $\theta_{0}$ has a
discontinuity at $\alpha_{0}=\pi/2$. While all other values for $\alpha_{0}$
are valid, $\alpha_{0}=\pi/2$ is disallowed. This is because as we send
$\theta_{0}$ to its corresponding value, the defect approaches the boundary
and one of the EOW branes disappears. For $\alpha_{0}=\pi/2$, we only have one
EOW brane and no defect, so this value needs to be excluded.

#### Conical Defect in the Bulk

So far, the geometries we have considered have a defect only on the EOW brane.
We can construct more general geometries by including a bulk conical defect as
well. First, we consider the metric (3.1), and embed the EOW branes whose
profiles are given by (3.2) with internal angle $\theta_{(1,2)}$. A bulk
conical defect located at $\chi=1$ with deficit angle of $\Delta\phi$ can be
introduced by identifying $\phi=\phi_{0}$ and $\phi=\phi_{0}+\Delta\phi$ in
the geometry (3.1). Here, $\phi_{0}$ is chosen such that the region given by
$\phi\in[\phi_{0},\phi_{0}+\Delta\phi_{0}]$ is contained within the spacetime
$M$ which is bounded by EOW branes. This is possible if and only if

$\mathcal{T}_{1},\leavevmode\nobreak\ \mathcal{T}_{2}>0,\quad\quad\quad\quad
2\alpha_{0}\geq\Delta\phi.$ (3.13)

The first condition $\mathcal{T}_{1},\leavevmode\nobreak\ \mathcal{T}_{2}>0$
is required for $\chi=1$ to be contained in the geometry, and the second
condition prevents self-intersection of the EOW branes [41]. For more recent
analysis, see [27, 28, 29, 30].

###  3.2 Euclidean Action

In this subsection, we compute the Euclidean action for the bulk geometry dual
to the strip. This action is directly related to the lowest eigenvalue of the
BCFT as we shall see in the next subsection.

First, consider the case with no conical defect in the bulk. After an explicit
computation, we find that the total action given by (2.6) is

$I_{\text{total}}=\frac{\ell\chi_{H}}{2G_{N}}\left[-\frac{\alpha_{0}}{\epsilon^{2}}+\frac{1}{2\epsilon}\sum_{a=1}^{2}\left(\frac{\mathcal{T}_{a}}{\sqrt{1-\mathcal{T}_{a}^{2}}}-\operatorname{arccos}\mathcal{T}_{a}\right)\right],$
(3.14)

where we have introduced the short-distance cutoff $\epsilon$ at the AdS
boundary to regulate the UV divergences.

To cancel these divergences, we need to include boundary counter terms. These
counter terms must be covariant and local, and are chosen to cancel the
divergences. In our case,

$\begin{split}I_{\text{ct}}&=\frac{1}{8\pi G_{N}\ell}\int_{N}\sqrt{h_{\partial
M}}+\sum_{a}\frac{\operatorname{arccos}\mathcal{T}_{a}}{8\pi
G_{N}}\int_{\Gamma_{(a,N)}}\sqrt{g_{\Gamma_{(a,N)}}}\\\
&=\frac{\ell\chi_{H}}{2G_{N}}\left[\alpha_{0}\left(\frac{1}{\epsilon^{2}}-\frac{1}{2}\right)-\frac{1}{2\epsilon}\sum_{a=1}^{2}\frac{\mathcal{T}_{a}}{\sqrt{1-\mathcal{T}_{a}^{2}}}\right]+\frac{\ell\chi_{H}}{4G_{N}\epsilon}\sum_{a=1}^{2}\operatorname{arccos}\mathcal{T}_{a}.\end{split}$
(3.15)

On including these counter terms, we obtain the Euclidean action

$I_{E}=-\frac{\ell\chi_{H}}{4G_{N}}\alpha_{0}=-\frac{c}{6\pi\xi}\alpha_{0}^{2}.$
(3.16)

Here we have defined the aspect ratio $\xi:=\Delta x\cdot T_{\mathrm{BCFT}}$
in order to have scale invariant expressions. We have also used that the width
of the strip is $\Delta x=2\alpha_{0}z_{0}$ and the central charge is
$c=\frac{3\ell}{2G_{N}}$.

As an example, let us consider the case $\mathcal{T}_{1}=\mathcal{T}_{2}$ and
$\alpha_{0}=\frac{\pi}{2}$, for which the EOW branes do not have a defect.
Then the action is given by

$I_{E}^{\mathrm{con}}=-\frac{\pi c}{24\xi},$

where the superscript denotes that this is the answer in the connected phase
of the conventional AdS/BCFT model.

Let us generalize the above result to geometries with bulk conical defects,
assuming $\mathcal{T}_{1},\leavevmode\nobreak\ \mathcal{T}_{2}>0$. The action
can be obtained by subtracting off the contribution corresponding to the bulk
portion that get removed due to the conical defect. The result is

$I_{E}=-\frac{\ell\chi_{H}}{4G_{N}}\left(\alpha_{0}-\frac{\Delta\phi}{2}\right)=-\frac{c}{6\pi\xi}\left(\alpha_{0}-\frac{\Delta\phi}{2}\right)^{2}.$
(3.17)

Here we have used that the width of the strip is $\Delta
x=(2\alpha_{0}-\Delta\phi)z_{0}$.

###  3.3 BCFT Spectrum

In this subsection, we examine the spectrum of the BCFT on a strip of width
$\Delta x$. This BCFT is dual to the bulk geometry that is bounded by two EOW
branes that are connected at a defect and it also has a conical defect in the
bulk. For this purpose, we consider the partition function
$Z_{a,b}^{\text{BCFT}}(T_{\text{BCFT}}^{-1},\Delta x)$, where
$T_{\text{BCFT}}^{-1}$ is the periodicity in $\tau$ direction which we will
take to be infinitely large. In this limit, we have

$Z_{a,b}^{\text{BCFT}}(T_{\text{BCFT}}^{-1},\Delta x)\underset{\Delta x\cdot
T_{\text{BCFT}}\to 0}{\xrightarrow{\hskip
56.9055pt}}e^{-E_{a,b}^{\text{BCFT}}\cdot T_{\text{BCFT}}^{-1}},$ (3.18)

where $E_{a,b}^{\text{BCFT}}$ is the lowest eigenvalue of the BCFT Hamiltonian
$H_{a,b}^{\text{BCFT}}$. In general, the corrections to the above equation are
exponentially suppressed. Using this lowest eigenvalue, we can define the
spectral gap as

$\Delta
E_{a,b}^{\text{BCFT}}:=E_{a,b}^{\text{BCFT}}-\frac{1}{2}E_{a,a}^{\text{BCFT}}-\frac{1}{2}E_{b,b}^{\text{BCFT}}.$
(3.19)

Suppose the EOW branes have dimensionless tensions $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$. Since $Z\approx e^{-I_{E}}$ and using (3.17), the lowest
eigenvalue is

$E_{a,b}^{\text{BCFT}}=-\frac{c}{6\pi\Delta
x}\left(\alpha_{0}-\frac{\Delta\phi}{2}\right)^{2}.$ (3.20)

The condition for the boundary changing operator to be the identity operator
is

$\alpha_{0}-\frac{\Delta\phi}{2}=\frac{\pi}{2},$

so the corresponding lowest eigenvalue is

$E_{a,b}^{\text{BCFT}}=-\frac{\pi c}{24\Delta x}.$ (3.21)

This situation includes the conventional model where the EOW branes are
connected and have identical tensions, and there is no bulk conical defect. In
particular, when the two boundary conditions are identical, we have
$E_{a,a}^{\text{BCFT}}=-\frac{\pi c}{24\Delta x}$.

The lowest eigenvalue of any unitary BCFT must be higher than this value.
Therefore, if we demand that the BCFT is unitary, the following condition
needs to be satisfied

$0\leq\alpha_{0}-\frac{\Delta\phi}{2}\leq\frac{\pi}{2}.$ (3.22)

Therefore, for a fixed $\alpha_{0}$, the spectrum obtained for an arbitrary
$\Delta\phi$ subject to (3.22) satisfies

$-\frac{\alpha_{0}^{2}c}{6\pi\Delta x}\leq E_{a,b}^{\text{BCFT}}\leq 0.$
(3.23)

We note that (3.23) generalizes the lowest eigenvalue dictated by the
conventional AdS/BCFT model. In the conventional AdS/BCFT, neither
intersections nor interactions between distinct EOW branes are allowed, so
such EOW branes are disconnected. As the consequence, the lowest eigenvalue is
given by

$E_{a,b}^{\text{BCFT}}=0.$ (3.24)

The gravity dual that we have constructed using the defect on EOW branes
generalizes this lowest eigenvalue to go below (3.24), for any choice of the
two boundary entropies. Note that when two boundary conditions are identical,
the lowest eigenvalue is given by the identity operator, so the dual geometry
corresponds to $\theta_{0}=\pi$ with $\Delta\phi=0$ and $\alpha_{0}=\pi/2$.

Finally, we note that $\alpha_{0}=\pi/2$ is allowed only if
$\mathcal{T}_{1}=\mathcal{T}_{2}$, so there is no longer a defect on the EOW
branes. This implies that we can attain the lowest eigenvalue (3.21) only when
the boundary entropy of the two EOW branes are identical. In this case, the
bulk conical defect can reproduce the spectrum

$-\frac{\pi c}{24\Delta x}\leq E_{a,b}^{\text{BCFT}}\leq 0.$ (3.25)

For $\mathcal{T}_{1}\neq\mathcal{T}_{2}$, the corresponding $\alpha_{0}$ can
be made arbitrarily close to $\pi/2$ by tuning $\theta_{0}$, but equality
cannot be achieved.

To summarize, a key advantage of our model is that it have rich spectrum even
if two boundary entropies are distinct, as opposed to the convectional
AdS/BCFT model which has fixed lowest eigenvalue (3.24).

#### CFT Stress Tensor for Thermal AdS

The above results for the lowest eigenvalue can also be derived by analyzing
the metric near the AdS boundary, without knowing the details of the bulk
geometry. Rewriting the thermal AdS3 metric in Fefferman-Graham coordinate, we
have

$\mathrm{d}s_{\text{FG}}^{2}=\frac{\ell^{2}}{r^{2}}\left(dr^{2}+\frac{1+2r^{2}+r^{4}}{2z_{0}^{2}}\mathrm{d}\tau^{2}+\frac{1-2r^{2}+r^{4}}{2}\mathrm{d}\phi^{2}\right),$
(3.26)

with $\chi=\frac{2r}{1+r^{2}}$. Then the dual CFT stress tensor is given by
[42]

$\langle T_{\tau\tau}\rangle=-\langle T_{xx}\rangle=\frac{\ell}{16\pi
G_{N}}\frac{1}{z_{0}^{2}}=\frac{c}{6\pi}\frac{\left(\alpha_{0}-\frac{\Delta\phi}{2}\right)^{2}}{(\Delta
x)^{2}}.$ (3.27)

The lowest eigenvalue can be obtained by integrating the CFT stress tensor on
the width of the strip,

$E_{a,b}^{\text{BCFT}}=-\int_{-\Delta x/2}^{\Delta x/2}\mathrm{d}x\,\langle
T_{\tau\tau}\rangle=-\frac{c}{6\pi\Delta
x}\left(\alpha_{0}-\frac{\Delta\phi}{2}\right)^{2},$ (3.28)

which matches with our previous result.

###  3.4 Spectral Gap in Liouville Theory

In this section, we show that Liouville theory with ZZ boundaries [43] gives
us a concrete example for the BCFT spectral gap $\Delta E_{a,b}^{\text{BCFT}}$
that is consistent with our gravity dual. For comparison, the spectral gap for
our gravity model without bulk conical defects is

$\Delta E_{a,b}^{\text{BCFT}}=\frac{\pi c}{24\Delta
x}\left(1-\frac{\alpha_{0}^{2}}{(\pi/2)^{2}}\right)\in\left[0,\frac{\pi
c}{24\Delta x}\right].$ (3.29)

The central charge for the Liouville theory is $c=1+6Q^{2}$ where
$Q=b+b^{-1}$. The semiclassical limit $c\to\infty$ can be obtained by taking
$b\to 0$. The degenerate representations appear at conformal dimensions

$\Delta_{(m,n)}:=\frac{Q^{2}}{4}-\frac{(m/b+nb)^{2}}{4}$ (3.30)

and the corresponding degenerate characters are

$\chi_{m,n}(\tau)=\frac{q^{-(m/b+nb)^{2}/4}-q^{-(m/b-nb)^{2}/4}}{\eta(\tau)},\quad\quad\quad\quad\quad\quad
q:=e^{2\pi i\tau}.$ (3.31)

For non-negative integers $m$ and $n$, the conformal dimension
$\Delta_{(m,n)}$ is non-negative only if

$(m,n)\in\\{(1,1),(1,0)\\}\cup\left\\{(0,n)\Big{|}n<\frac{1}{b^{2}}\right\\}$
(3.32)

This conformal dimension ranges between
$0\leq\Delta_{(m,n)}\leq\frac{Q^{2}}{4}=\frac{c-1}{24}$.

The inner product between two ZZ boundary states $|B_{(m,n)}\rangle$ is given
by [43]

$\langle B_{(m,n)}|e^{-\beta
H/2}|B_{(m^{\prime},n^{\prime})}\rangle=\sum_{k=0}^{\min(m,m^{\prime})-1}\sum_{l=0}^{\min(n,n^{\prime})-1}\chi_{m+m^{\prime}-2k-1,n+n^{\prime}-2l-1}\left(\frac{\mathrm{i}}{\beta
T_{\text{BCFT}}}\right).$ (3.33)

When we take the limit $\beta T_{\text{BCFT}}\to 0$, the term with $k=l=0$
dominates, so

$\frac{\langle B_{(m,n)}|e^{-\Delta
xH}|B_{(m^{\prime},n^{\prime})}\rangle}{\sqrt{\langle B_{(m,n)}|e^{-\Delta
xH}|B_{(m,n)}\rangle\langle B_{(m^{\prime},n^{\prime})}|e^{-\Delta
xH}|B_{(m^{\prime},n^{\prime})}\rangle}}\underset{\Delta x\cdot
T_{\text{BCFT}}\to 0}{\xrightarrow{\hskip 56.9055pt}}e^{-\Delta
E_{(m,n),(m^{\prime},n^{\prime})}^{\text{BCFT}}\cdot T_{BCFT}^{-1}},$ (3.34)

where

$\Delta E_{(m,n),(m^{\prime},n^{\prime})}^{\text{BCFT}}:=\frac{\pi}{4\Delta
x}\left((n-n^{\prime})b+\frac{m-m^{\prime}}{b}\right)^{2}.$ (3.35)

This spectral gap ranges between

$0\leq\Delta
E_{(m,n),(m^{\prime},n^{\prime})}^{\text{BCFT}}\leq\frac{\pi(c-1)}{24\Delta
x},$ (3.36)

which is indeed the spectrum gap found in our AdS/BCFT model with defects on
EOW branes in the large $c$ limit. However, note that

$E_{(m,n),(m,n)}^{\text{BCFT}}=\frac{\pi}{4\Delta
x}\left((2n-1)b+\frac{2m-1}{b}\right)^{2},$ (3.37)

is distinct from (3.21) although this amplitude is between identical states.
This mismatch disallows us from interpreting the Liouville theory with ZZ
boundaries as the usual BCFT.

##  4 Entanglement Entropy and Entanglement Island

In this section, we study the holographic entanglement entropy in our model.
We use the Ryu–Takayanagi (RT) prescription to compute this entanglement
entropy. We restrict to the case with no bulk conical defect.

We start by considering a normalized boundary state for a conformal boundary
condition $B_{a}$ prepared on a circle with circumference
$T_{\text{BCFT}}^{-1}$,

$|B_{a}(\Delta x/2)\rangle=\frac{e^{-\frac{1}{2}H_{\text{CFT}}\Delta
x}|B_{a}\rangle}{\sqrt{\langle B_{a}|e^{-H_{\text{CFT}}\Delta
x}|B_{a}\rangle}}.$ (4.1)

This state is often used to model an initial state for a global quantum
quench[44, 45].

In the following, we consider the entanglement entropy of a superposition
state

$|\Psi\rangle:=\sum_{a}c_{a}|B_{a}(\Delta x/2)\rangle,$ (4.2)

where the $c_{a}$’s are complex coefficients. We restrict to the case where
the boundary entropy $S_{B_{a}}$ is equal to a fixed value $S_{B}$ (or it is
in a narrow window around $S_{B}$) for the boundary conditions $B_{a}$
appearing in $|\Psi\rangle$. Let $N_{S_{B}}$ be the number of such boundary
states.

We assume that the inner products $\langle B_{a}(\Delta x/2)|B_{b}(\Delta
x/2)\rangle$ have holographic duals with a defect connecting the EOW branes.
For simplicity, we assume that $\theta_{(a,a)}=\pi$ and
$\theta_{(a,b)}=\theta_{0}$ when $B_{a}\neq B_{b}$. Under these assumptions,
the geometries with $\theta_{(a,b)}=\theta_{0}$ dominate the gravitational
computation for $\langle B_{a}(\Delta x/2)|B_{b}(\Delta x/2)\rangle$ if

$\sum_{a=1}^{N_{S_{B}}}|c_{a}|^{2}\ll\left(\sum_{\begin{subarray}{c}a,b=1\\\
a\neq
b\end{subarray}}^{N_{S_{B}}}c_{a}^{*}c_{b}\right)\exp\left[-\frac{c}{6\pi\Delta
x}\left(\frac{\pi^{2}}{4}-\alpha_{0}^{2}\right)\right].$ (4.3)

If all the $c_{a}$’s are approximately equal, then this condition can be
satisfied only if $N_{S_{B}}\gg\exp\left[\frac{c}{6\pi\Delta
x}\left(\frac{\pi^{2}}{4}-\alpha_{0}^{2}\right)\right]$. Since the gravity
dual for $\langle\Psi|\Psi\rangle$ is given by the geometry with
$\theta_{(a,b)}=\theta_{0}$ in this case, the RT surfaces in this geometry
computes the entanglement entropy of the state $|\Psi\rangle$.

One can also understand this RT surface in terms of the _pseudo entropy_ [35].
The pseudo entropy is given by the von Neumann entropy
$S_{P}(A)=-\operatorname{Tr}[X_{A}\log X_{A}]$ of a normalized transition
matrix $X_{A}$. In our case, this transition matrix is

$X_{A}:=\frac{\operatorname{Tr}_{A}\big{[}|B_{a}(\Delta x/2)\rangle\langle
B_{b}(\Delta x/2)|\big{]}}{\operatorname{Tr}_{AA^{c}}\big{[}|B_{a}(\Delta
x/2)\rangle\langle B_{b}(\Delta x/2)|\big{]}}.$ (4.4)

###  4.1 Three Phases of the RT Surface

In this subsection, we will use the RT surface prescription to compute the
entanglement entropy for a subregion [44, 45] of the state $|\Psi\rangle$ or
the pseudo entropy for $X_{A}$. We assume the tensions of the EOW branes are
equal $T_{1}=T_{2}=T$, and $\mathcal{T}$ is the corresponding dimensionless
tension. Also, the internal angle at the defect is $\theta_{0}$. We assume the
bulk conical defect is absent for simplicity.

Recall that our BCFT is defined on the strip

$N=\\{(\tau,\,x=z_{0}\phi)\leavevmode\nobreak\ |\leavevmode\nobreak\
0\leq\tau\leq T^{-1}_{\mathrm{BCFT}},\leavevmode\nobreak\
\pi-\alpha_{0}\leq\phi\leq\pi+\alpha_{0}\\},$ (4.5)

so the width of the strip is $\Delta x=2z_{0}\alpha_{0}$. On this strip, we
consider the subregion

$A=\\{(\tau,\,x=z_{0}(\pi-\sigma))\leavevmode\nobreak\ |\leavevmode\nobreak\
0\leq\tau\leq|A|\\},$ (4.6)

with a particular value of $\sigma\in[0,\alpha_{0})$. Since $\tau$ is periodic
with periodicity $T^{-1}_{\mathrm{BCFT}}$ and $|\Psi\rangle$ is a pure state,
it is sufficient to consider the case $|A|<T^{-1}_{\mathrm{BCFT}}/2$. There
are three different phases of the RT surface as shown in Fig. 5.

(a) Case 1: Thermal Phase

(b) Case 2: Boundary Phase

(c) Case 3: Defect Phase

Figure 5: Three phases of the RT surfaces for the boundary subregion $A$.

#### Case 1: Thermal Phase

In this case, the RT surface is a single connected surface. This surface is
shown in Fig. 5(a) and we label it as $\gamma_{1}$. This phase is realized
when the subregion $A$ is sufficiently small. We will see that the
entanglement entropy is extensive in this case.

The bulk geometry has reflection symmetry about the $\phi=\pi-\sigma$ plane,
so the $\gamma_{1}$ lies on this plane. This surface is given by the geodesic

$\sqrt{1-\chi^{2}}=\sqrt{1-\chi_{0}^{2}}\cosh\text{\small$\frac{\tau-\tau_{0}}{z_{0}}$}$
(4.7)

with the parameters being

$\tau_{0}=\frac{|A|}{2},\quad\quad\chi_{0}=\tanh\frac{|A|}{2z_{0}}.$ (4.8)

Note that $(\tau,\chi,\phi)=(\tau_{0},\chi_{0},\pi-\sigma)$ is the deepest
point on this geodesic. If the maximum depth $\chi_{0}$ is sufficiently large,
$\gamma_{1}$ will be cut by the EOW branes in the negative tension case.
Therefore, the thermal phase with $\mathcal{T}<0$ exists only if
$\mathcal{T}\geq-\frac{\tan(\alpha_{0}-\sigma)\operatorname{sech}(|A|/2z_{0})}{\sqrt{\tan(\alpha_{0}-\sigma)^{2}+\tanh^{2}(|A|/2z_{0})}}$.

The holographic entanglement entropy is

$S(A)=\frac{\mathcal{A}(\gamma_{1})}{4G_{N}}=\frac{c}{3}\log\frac{2}{\epsilon}+\frac{c}{3}\log\sinh(|A|/2z_{0}).$
(4.9)

Since this result is independent of $\sigma$, analytically continuing it to
real time does not give rise to any time dependence. For large $|A|$, this
entropy demonstrates the volume law

$S(A)\approx\frac{c}{3}\log\frac{2}{\epsilon}+\frac{c|A|}{6z_{0}}-\frac{c}{3}\log
2,$ (4.10)

at an inverse effective temperature

$\beta_{|\Psi\rangle}=2\pi z_{0}=\frac{\pi}{2\alpha_{0}}(2\Delta x).$ (4.11)

There is a prefactor $\frac{\pi}{2\alpha_{0}}$ here, which _lowers_ the
effective temperature for the same $\Delta x$ for a given
$\alpha_{0}\leq\pi/2$ as compared to the conventional case for which
$\alpha_{0}=\pi/2$.

#### Case 2: Boundary Phase

In this phase, the RT surface has two disconnected pieces that end on the EOW
branes. This surface is shown in Fig. 5(b) and we label it as $\gamma_{2}$.
This phase is preferred over the thermal phase for sufficiently large $|A|$.
We will see that the entanglement entropy is intensive in this case.

Each piece of $\gamma_{2}$ lies on a constant $\tau$ surface because of the
$\tau$ reflection symmetry. These pieces is given by the geodesic

$\sqrt{1-\chi^{2}}\cos(\phi-\pi+\sigma+\phi_{0})=\cos\phi_{0}.$ (4.12)

Here we have determined the parameter $\phi_{0}=\alpha_{0}-\sigma$ by
requiring that the geodesic intersects the EOW brane orthogonally. The
endpoints of $\gamma_{2}$, lying on the EOW brane $\Sigma_{1}$, are given by
$\chi_{\text{ep}}=\sqrt{1-\mathcal{T}^{2}}\sin(\alpha_{0}-\sigma)$.

If $\alpha_{0}\leq\pi/2$, or equivalently, $\theta_{0}\leq\pi$, this geodesic
exists for all values of $\sigma$. If $\alpha_{0}\geq\pi/2$, the critical
value corresponds to the endpoint lying on the defect. This is given by
$\chi_{\text{ep}}=\chi_{*}=\sqrt{1-\mathcal{T}^{2}\csc^{2}(\frac{\theta_{0}}{2})}$,
so the condition for the existence of this geodesic is

$\alpha_{0}-\sigma\leq\operatorname{arccos}\text{\small$\frac{\mathcal{T}\left|\cot(\frac{\theta_{0}}{2})\right|}{\sqrt{1-\mathcal{T}^{2}}}$}.$
(4.13)

In particular, this surface exists for $\sigma=0$ if and only if
$\alpha_{0}\leq\pi/2$.

The holographic entanglement entropy is

$S_{A}=\frac{\mathcal{A}(\gamma_{2})}{4G_{N}}=\frac{c}{3}\log\frac{2}{\epsilon}+\frac{c}{3}\log\text{\small$\sin(\alpha_{0}-\sigma)$}+2S_{B}.$
(4.14)

Here $S_{B}$ is the boundary entropy

$S_{B}=\frac{c}{6}\operatorname{arctanh}\mathcal{T},$ (4.15)

which can also be obtained using the disk partition function [3]. Note that
there is an extremal but not minimal surface $\gamma_{2}^{\prime}$ ending on
the other EOW brane $\Sigma_{2}$, i.e., the EOW brane anchored at
$\phi=\pi+\alpha_{0}$. For this surface, we have

$\frac{\mathcal{A}(\gamma_{2}^{\prime})}{4G_{N}}=\frac{c}{3}\log\frac{2}{\epsilon}+\frac{c}{3}\log\text{\small$\sin(\alpha_{0}+\sigma)$}+2S_{B}.$
(4.16)

The surface $\gamma_{2}^{\prime}$ exists when
$\alpha_{0}+\sigma\leq\operatorname{arccos}\text{\small$\frac{\mathcal{T}\left|\cot(\frac{\theta_{0}}{2})\right|}{\sqrt{1-\mathcal{T}^{2}}}$}$.

When we analytically continue this Euclidean entanglement entropy to real
time, it is important to consider both $\mathcal{A}(\gamma_{2})$ and
$\mathcal{A}(\gamma_{2}^{\prime})$, as well as cases where the two pieces of
the RT surface ends on distinct EOW branes. To evaluate the entanglement
entropy of the state $e^{-\mathrm{i}H^{\text{CFT}}t}|\Psi\rangle$, we
analytically continue $\frac{\mathcal{A}(\gamma_{2})}{4G_{N}}$ and
$\frac{\mathcal{A}(\gamma_{2}^{\prime})}{4G_{N}}$ by taking
$\sigma\to-\mathrm{i}\frac{t}{z_{0}}$. After this substitution, these two
extremal areas are complex conjugates of the other,
$\mathcal{A}(\gamma_{2})=\overline{\mathcal{A}(\gamma_{2}^{\prime})}$. The
total areas in the other two cases are purely real and are equal to
${\mathrm{Re}}\left[\mathcal{A}(\gamma_{2})\right]={\mathrm{Re}}\left[\mathcal{A}(\gamma_{2}^{\prime})\right]$.
Therefore, the real time entanglement entropy is given by this real piece,

$S_{A}={\mathrm{Re}}\left[\frac{\mathcal{A}(\gamma_{2})}{4G_{N}}\right]=\frac{c}{3}\log\frac{2}{\epsilon}+\frac{c}{6}\log\text{\small$\frac{\cosh(2t/z_{0})-\cos(2\alpha_{0})}{2}$}+2S_{B}-\frac{c}{3}\log
2.$ (4.17)

At late times $t\gg z_{0}$, this entanglement entropy is

$S_{A}\approx\frac{c}{3}\log\frac{2}{\epsilon}+\frac{ct}{3z_{0}}+2S_{B}.$
(4.18)

Hence, at late time satisfying

$t>\frac{|A|}{2}-\frac{6z_{0}S_{B}}{c},$ (4.19)

the thermal phase will be favored over the boundary phase.

#### Case 3: Defect Phase

In this phase, the RT surface has two disconnected pieces that end on the
defect. This surface is shown in Fig. 5(c) and we label it as $\gamma_{3}$. As
in the boundary phase, the entropy is intensive. This phase is always
subdominant compared to the boundary entropy phase if the latter exists.
Therefore, this phase is realized only when the subregion is sufficiently
large and the boundary entropy phase is absent. For this reason, when
$\alpha_{0}\leq\pi/2$, this phase is irrelevant.

As earlier, each piece of $\gamma_{3}$ lies on a constant $\tau$ surface.
These pieces are given by the geodesic

$\sqrt{1-\chi^{2}}\cos(\phi-\pi+\sigma+\phi_{0})=\cos\phi_{0},$ (4.20)

Here we have determined the parameter
$\phi_{0}=\operatorname{arctan}\frac{\sin(\theta_{0}/2)+\mathcal{T}\cos\sigma}{\mathcal{T}\sin\sigma}$
by requiring that the geodesic passes through the defect.

If $\mathcal{T}>0$ this geodesic is guaranteed to exist. However, when
$\mathcal{T}<0$ this geodesic exists only if $\sigma\leq
2\operatorname{arctan}\left(\text{\small$\tan\left(\frac{\theta_{0}}{4}\right)\sqrt{\frac{\sin(\theta_{0}/2)+\mathcal{T}}{\sin(\theta_{0}/2)-\mathcal{T}}}$}\right)$.
This implies that we this phase always exists around $\sigma=0$.

The holographic entanglement entropy is

$S_{A}=\frac{\mathcal{A}(\gamma_{3})}{4G_{N}}=\frac{c}{3}\log\frac{2}{\epsilon}+\frac{c}{3}\log\text{\small$\frac{\sin(\theta_{0}/2)+\mathcal{T}\cos\sigma}{\sin(\theta_{0}/2)+\mathcal{T}}$}+2S_{D}.$
(4.21)

Here we have defined

$S_{D}:=\frac{c}{12}\log\text{\small$\frac{\sin(\theta_{0}/2)+\mathcal{T}}{\sin(\theta_{0}/2)-\mathcal{T}}$},$
(4.22)

and we call it the defect entropy. This defect entropy can be considered as a
generalization of the boundary entropy $S_{B}$ and it reduces to $S_{B}$ if we
make the defect disappear by setting $\theta_{0}=\pi$.

As mentioned earlier, the boundary phase is favored over this phase,
$\mathcal{A}(\gamma_{2})<\mathcal{A}(\gamma_{3})$, if both phases exist. This
is because $\gamma_{2}$ is minimal amongst all surface that end on the EOW
brane.

The real time entanglement entropy is given by analytically continuing
$\sigma\to-\mathrm{i}\frac{t}{z_{0}}$, so

$S_{A}=\frac{c}{3}\log\frac{2}{\epsilon}+\frac{c}{3}\log\left(\text{\small$\frac{\mathcal{T}\cosh(t/z_{0})+\sin(\theta_{0}/2)}{\mathcal{T}+\sin(\theta_{0}/2)}$}\right)+2S_{D}.$
(4.23)

Note that this result is manifestly real unlike case 2. At late times $t\gg
z_{0}$, this entanglement entropy is

$S_{A}\approx\frac{c}{3}\log\frac{2}{\epsilon}+\frac{ct}{3z_{0}}+2S_{D}-\frac{c}{3}\log\text{\small$\frac{2\left(\mathcal{T}+\sin(\theta_{0}/2)\right)}{\mathcal{T}}$}.$
(4.24)

It is also worth mentioning that the $\gamma_{3}$ is minimal but not extremal
because of the singular nature of the defect. One can imagine a scenario when
there is a matter theory on the EOW brane which smoothly interpolates between
the two different tensions of the EOW branes, and the defect is realized as
the sharp limit. In this case the RT surface is extremal, and the distinction
between cases $2$ and $3$ disappears.

###  4.2 Interpretation as Entanglement Island

In this subsection we explain and interpret the results in the previous
section in terms of the island formula. We interpret the state $|\Psi\rangle$
as a field theoretic wavefunction being prepared by a matter field that lives
on the EOW brane. In other words, the EOW brane is now the spacetime on which
a quantum state $|\Psi\rangle$ is prepared [33, 34]. We are interested in the
entanglement entropy of a subregion $A$ of the state $|\Psi\rangle$. We assume
that the matter theory is a holographic BCFT to simplify the analysis [46].
When the effective temperature of $|\Psi\rangle$ is sufficiently low, the EOW
branes in the bulk are disconnected. When the temperature is sufficiently
high, the EOW branes are connected via a defect, creating a closed universe
that terminates at this defect.

The three phases of the RT surface in the previous section can be interpreted
as the phases of entanglement entropy of $A$ [34]. When the subregion is
sufficiently small, the entanglement entropy is given by the thermal answer
from matter theory. This corresponds to case $1$. When the subregion is large
enough so that the naive thermal entropy is much larger than the boundary
entropy, then an entanglement island is formed on the EOW brane (case $2$) or
on the defect (case $3$).

In case $2$, the two pieces of the RT surface can end on the same EOW brane or
on different EOW branes. However, the “island” in the latter case cuts through
the defect on the EOW brane, so the corresponding entropy increases by the
boundary entropy of the defect coming from the matter field on the EOW branes.
Consequently, it can be argued that this configuration is subdominant when we
are looking for the island. As the result, the island lies only on one of the
two EOW branes. The bulk matter wavefunction on such an island is a transition
matrix, so the matter part of the generalized entropy is given by the pseudo
entropy. It is expected that there is a generalization of entanglement wedge
reconstruction to such cases, although there is no quantum information
theoretic foundations for this claim yet.

In case $3$, the RT surface ends on the defect. This can be interpreted as an
island lying exclusively on the defect. Therefore, a portion of the defect is
included in the entanglement wedge of the subregion $A$.

##  5 Three-dimensional Wormhole Saddles

In this section, we construct a connected bulk geometry with multiple AdS
boundaries using our model of EOW branes and defects. This geometry is similar
to the replica wormhole in $2$d gravity with EOW branes [13]. However, we will
be interested in defects with $\theta_{0}>\pi$, so these defects have negative
energy.

One way to add such negative energy in the bulk is by adding non-local
interaction between different boundaries [47], which can originate from
integrating out “fast” degrees of freedom. Such non-local interaction between
boundaries can make a non-traversable Einstein-Rosen bridge traversable, so
that different boundaries can communicate with each other [47, 48]. We
interpret our model in this section as a realization of such non-local
interaction between BCFTs, and the non-unitarity of the BCFTs with
$\theta_{0}>\pi$ comes from such non-local interactions obtained by
integrating out these fast degrees of freedom.

Let us consider two BCFTs that live on two different strips with widths
$\Delta x_{1}$ and $\Delta x_{2}$ respectively. They have the same periodicity
$T_{\mathrm{BCFT}}^{-1}$ in $\tau$ direction, so the aspect ratios are
$\xi_{1}=\Delta x_{1}\cdot T_{\mathrm{BCFT}}$ and $\xi_{2}=\Delta x_{2}\cdot
T_{\mathrm{BCFT}}$. Fig. 6(a) shows a constant $\tau$ slice for this setup.

(a) Boundary Setup

(b) Factorized Geometry

(c) Wormhole Geometry

Figure 6: Constant $\tau$ slices showing two BCFTs along with the disconnected
and connected dual geometries. The factorized geometry is the leading
contribution to the product of partition functions where as the wormhole
geometry is subleading. The latter arises from non-unitarity of the BCFTs with
$\theta_{0}>\pi$.

We assume that the tensions satisfy $T_{1}=T_{3}$ and $T_{2}=T_{4}$ and that
the angle of intersection between the $\Sigma_{1},\,\Sigma_{3}$ EOW branes and
the $\Sigma_{2},\,\Sigma_{4}$ EOW branes is $\theta_{0}>\pi$. This assumption
is required to get connected geometries. it implies that
$\alpha_{0}>\frac{\pi}{2}$ and that the BCFTs are non-unitary.

The factorized geometry is shown in Fig. 6(b). The Euclidean action for this
geometry is the sum of the two corresponding actions

$I_{E}^{\mathrm{f}}=-\frac{c}{6\pi}\frac{\alpha_{0}^{2}}{\xi_{1}}-\frac{c}{6\pi}\frac{\alpha_{0}^{2}}{\xi_{2}}.$
(5.1)

where $\alpha_{0}$ is given by (3.10). The connected wormhole geometry is
shown in Fig. 6(c). It is obtained by stitching together two geometries, each
with a boundary angle $2\alpha_{0}$. The total boundary angle for this
wormhole geometry is $2\alpha_{\mathrm{w}}=(4\alpha_{0}-2\pi)$, where we have
subtracted a $2\pi$ to account for the connection. The Euclidean action for
the wormhole geometry is

$I_{E}^{\mathrm{w}}=-\frac{c}{6\pi\xi_{\mathrm{w}}}\alpha_{\mathrm{w}}^{2}=-\frac{c}{6\pi}\frac{(2\alpha_{0}-\pi)^{2}}{\xi_{1}+\xi_{2}},$
(5.2)

where $\xi_{\mathrm{w}}=\xi_{1}+\xi_{2}$ is the total aspect ratio for the
wormhole.

The factorized geometry strictly dominates over the wormhole geometry, since

$I_{E}^{\mathrm{f}}=-\frac{c\alpha_{0}^{2}}{6\pi}\left(\frac{1}{\xi_{1}}+\frac{1}{\xi_{2}}\right)\leq-\frac{c\alpha_{0}^{2}}{6\pi}\frac{4}{\xi_{1}+\xi_{2}}<-\frac{c(2\alpha_{0}^{2}-\pi)^{2}}{6\pi}\frac{1}{\xi_{1}+\xi_{2}}=I_{E}^{\mathrm{w}}.$
(5.3)

Here we have used the arithmetic mean-harmonic mean inequality to establish
the first inequality.

More generally, if we have $n$ disconnected boundary regions with aspect
ratios $\xi_{i}=\Delta x_{i}\cdot T_{\mathrm{BCFT}}$, then the Euclidean
actions for the factorized geometry is

$I_{E}^{\mathrm{f}}=-\frac{c}{6\pi}\sum_{i=1}^{n}\frac{(\alpha|_{\theta_{0}})^{2}}{\xi_{i}},$
(5.4)

and for the fully connected wormhole geometry is

$I_{E}^{\mathrm{w}}=-\frac{c}{6\pi}\frac{(n\alpha|_{\theta_{0}}-(n-1)\pi)^{2}}{\sum_{i=1}^{n}\xi_{i}},\quad\quad.$
(5.5)

This fully connected geometry exists only for
$\alpha_{0}\geq\frac{(n-1)\pi}{n}$. From these results, we can conclude that
any wormhole geometry is subdominant compared to the corresponding factorized
geometry, even though we have considered a non-unitary BCFT. Recall that
without non-unitarity, it is not even possible to construct a connected
saddle. Interestingly, in order to increase the number of boundary components,
we need larger a $\alpha_{0}$, which means that we need to make the BCFT
“more” non-unitary.

##  6 Discussion

In this paper, we have developed a generalization of the conventional AdS/BCFT
model by including defects that connect EOW branes with different tensions.
This construction enables us to study a BCFT whose lowest eigenvalue can be
tuned arbitrarily close to that of the identity operator. This construction is
particularly useful when the boundary entropies of the two boundaries are
distinct, i.e., when the conventional model has a restricted lowest
eigenvalue.

We conclude with some remarks and possible future directions. The construction
in our model is based on $3$d gravity, and we expect that the generalization
to higher dimensions should be straightforward. Although we have calculated
the real time entropy in section 4, we did not study the corresponding real
time geometry. It would be interesting to develop this analytic continuation.
It would also be interesting to generalize our defect on the EOW branes to a
smooth configuration. We can imagine a model in which the tension of the EOW
brane is position dependent, and in particular it interpolates smoothly
between the two BCFT boundaries. we would like to understand the
interpretation of the boundary entropy in this configuration, for which a
specific example is given by the defect entropy in (4.21).

The connection between unitarity and connected geometries in section 5 remains
mysterious and interesting. We should emphasize that although the connected
saddle exists in the non-unitary case, this saddle is always subleading
compared to the disconnected saddle.

### Acknowledgements

We thank Yuya Kusuki and Tadashi Takayanagi for useful discussions and
comments. C.M. is supported in part by the U.S. Department of Energy, Office
of Science, Office of High Energy Physics under QuantISED Award DE-SC0019380
and contract DE-AC02-05CH11231.

##  Appendix A Extention of Boundary Conformal Transformation to Bulk

In this appendix, we examine the bulk coordinate transformation from the
Poincare patch to another coordinate patch. This transformation is dual to the
boundary conformal transformation from a plane to a $2$d patch. In particular,
we find the map between vacuum AdS and thermal AdS.

The Poincare AdS3 metric is

$\mathrm{d}s^{2}=\frac{\ell^{2}}{z^{2}}\left(-2\mathrm{d}v_{+}\mathrm{d}v_{-}+\mathrm{d}z^{2}\right).$
(A.1)

Here we have used the light cone coordinate $v_{\pm}:=\frac{1}{\sqrt{2}}(t\pm
r)$ for the Minkowski space
$\mathrm{d}s^{2}=-2\mathrm{d}v_{+}\mathrm{d}v_{-}=-\mathrm{d}t^{2}+\mathrm{d}r^{2}$
at the boundary of AdS.

Suppose we have a conformal transformation $v_{\pm}=f_{\pm}(u_{\pm})$ that
acts on the boundary. We can extend this to a bulk coordinate transformation
via [49]

$\begin{cases}v_{\pm}=f_{\pm}(u_{\pm})+\frac{2\zeta^{2}f^{\prime}_{\pm}(u_{\pm})^{2}f^{\prime\prime}_{\mp}(u_{\mp})}{8f_{+}^{\prime}(u_{+})f_{-}^{\prime}(u_{-})-\zeta^{2}f_{+}^{\prime\prime}(u_{+})f_{-}^{\prime\prime}(u_{-})},\\\
z=\frac{8\zeta(f^{\prime}_{+}(u_{+})f^{\prime}_{-}(u_{-}))^{3/2}}{8f_{+}^{\prime}(u_{+})f_{-}^{\prime}(u_{-})-\zeta^{2}f_{+}^{\prime\prime}(u_{+})f_{-}^{\prime\prime}(u_{-})}.\end{cases}$
(A.2)

where $\zeta$ is the new radial coordinate. This bulk transformation maps the
Poincare AdS3 metric to

$\mathrm{d}s^{2}=\ell^{2}\left(\frac{\mathrm{d}\zeta^{2}}{\zeta^{2}}+L_{+}\mathrm{d}u_{+}^{2}+L_{-}\mathrm{d}u_{-}^{2}-\left(\frac{2}{\zeta^{2}}+\frac{\zeta^{2}}{2}L_{+}L_{-}\right)\mathrm{d}u_{+}\mathrm{d}u_{-}\right).$
(A.3)

Here $L_{\pm}$ are related to the Schwarzian of the conformal transformation
and are given by

$L_{\pm}:=-\frac{1}{2}\\{f_{\pm}(u_{\pm}),u_{\pm}\\}=\frac{3f_{\pm}^{\prime\prime
2}-2f_{\pm}^{\prime}f_{\pm}^{\prime\prime\prime}}{4f_{\pm}^{\prime 2}}.$ (A.4)

Hence, the CFT stress tensor in the transformed coordinates is

$T_{u_{\pm}u_{\pm}}=\frac{c}{12\pi}L_{\pm},\quad\quad\quad\quad
T_{u_{+}u_{-}}=0.$ (A.5)

The analytical continuation of $v_{\pm}$ to imaginary time is

$v_{\pm}=\frac{t\pm r}{\sqrt{2}}\to\frac{-\mathrm{i}\tau\pm r}{\sqrt{2}}.$
(A.6)

Similarly, for
$\mathrm{d}s^{2}=-2\mathrm{d}x_{+}\mathrm{d}x_{-}=-\mathrm{d}t_{x}^{2}+z_{0}^{2}\mathrm{d}\phi^{2}$,
we have

$u_{\pm}=\frac{t_{u}\pm z_{0}\phi}{\sqrt{2}}\to\frac{-\mathrm{i}\tau_{u}\pm
z_{0}\phi}{\sqrt{2}}.$ (A.7)

In these complexified coordinates, the conformal transformation can be
rewritten as $v=r+\mathrm{i}\tau=g(u),\leavevmode\nobreak\
\bar{v}=r-\mathrm{i}\tau=\bar{g}(\bar{u})$ with
$u:=z_{0}\phi+\mathrm{i}\tau_{u},\leavevmode\nobreak\
\bar{u}:=z_{0}\phi-\mathrm{i}\tau_{u}$. The metric is

$\mathrm{d}s^{2}=\ell^{2}\left(\frac{\mathrm{d}\zeta^{2}}{\zeta^{2}}+L\mathrm{d}u^{2}+\bar{L}\mathrm{d}\bar{u}^{2}+\left(\frac{1}{\zeta^{2}}+\zeta^{2}L\bar{L}\right)\mathrm{d}u\mathrm{d}\bar{u}\right).$
(A.8)

where we have defined
$L:=\frac{1}{2}\\{g(u),u\\}=-\frac{1}{4}\\{f_{-}(u_{-}),u_{-}\\}$ and
$\bar{L}:=-\frac{1}{2}\\{\bar{g}(\bar{u}),\bar{u}\\}=-\frac{1}{4}\\{f_{+}(u_{+}),u_{+}\\}$.

### Mapping a Plane to a Cylinder

We consider a conformal transformation from a cylinder to a plane,

$v=z_{0}e^{-\frac{\mathrm{i}}{z_{0}}u},\quad\quad\bar{v}=z_{0}e^{\frac{\mathrm{i}}{z_{0}}\bar{u}}.$
(A.9)

Here we have defined the coordinates on the cylinder as

$u:=z_{0}\phi+\mathrm{i}\tau=-\sqrt{2}u_{-},\quad\bar{u}:=z_{0}\phi-\mathrm{i}\tau=\sqrt{2}u_{+},$
(A.10)

and the coordinates on the plane as

$v:=x-\mathrm{i}y=-\sqrt{2}v_{-},\quad\bar{v}:=x+\mathrm{i}y=\sqrt{2}v_{+}.$
(A.11)

This conformal transformation can be also be expressed as
$v_{\pm}=\pm\frac{z_{0}}{\sqrt{2}}e^{\frac{\mathrm{i}\sqrt{2}}{z_{0}}u_{\pm}}$.

Applying (A.2) to this conformal transformation, we obtain the bulk
transformation between the Poincare metric,
$\mathrm{d}s^{2}=\frac{\ell^{2}}{s^{2}}(\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2})$,
and thermal AdS,
$\mathrm{d}s^{2}=\ell^{2}\left(\frac{\mathrm{d}\tau^{2}}{z_{0}^{2}\chi^{2}}+\frac{\mathrm{d}\chi^{2}}{h(\chi)\chi^{2}}+\frac{h(\chi)\mathrm{d}\phi^{2}}{\chi^{2}}\right)$,
to be

$\begin{cases}v_{\pm}=\pm\frac{z_{0}}{\sqrt{2}}\sqrt{h(\chi)}e^{\mathrm{i}\sqrt{2}u_{\pm}/z_{0}},\\\
z=z_{0}e^{i(u_{+}+u_{-})/(\sqrt{2}z_{0})}\chi.\end{cases}$ (A.12)

where we have related $\chi=\frac{4z_{0}\zeta}{\zeta^{2}+4z_{0}^{2}}$. This
can also be expressed as

$\begin{cases}x=z_{0}e^{\tau/z_{0}}\sqrt{h(\chi)}\cos\phi,\\\
y=z_{0}e^{\tau/z_{0}}\sqrt{h(\chi)}\sin\phi,\\\
z=z_{0}e^{\tau/z_{0}}\chi,\end{cases}$ (A.13)

For this transformation,

$L_{\pm}=-\frac{1}{2z_{0}^{2}},$ (A.14)

so the stress tensor is given by

$T_{\tau\tau}=-\frac{T_{\phi\phi}}{z_{0}^{2}}=\frac{c}{24\pi z_{0}^{2}}.$
(A.15)

##

## References

  * [1] Andreas Karch and Lisa Randall, “Open and closed string interpretation of SUSY CFT’s on branes with boundaries,” JHEP 06, 063 (2001), arXiv:hep-th/0105132
  * [2] Tadashi Takayanagi, “Holographic Dual of BCFT,” Phys. Rev. Lett. 107, 101602 (2011), arXiv:1105.5165 [hep-th]
  * [3] Mitsutoshi Fujita, Tadashi Takayanagi, and Erik Tonni, “Aspects of AdS/BCFT,” JHEP 11, 043 (2011), arXiv:1108.5152 [hep-th]
  * [4] John L. Cardy, “Conformal Invariance and Surface Critical Behavior,” Nucl. Phys. B 240, 514–532 (1984)
  * [5] John L. Cardy, “Boundary conformal field theory,” (11 2004), arXiv:hep-th/0411189
  * [6] Masahiro Nozaki, Tadashi Takayanagi, and Tomonori Ugajin, “Central Charges for BCFTs and Holography,” JHEP 06, 066 (2012), arXiv:1205.1573 [hep-th]
  * [7] Marco Chiodaroli, Eric D’Hoker, and Michael Gutperle, “Simple Holographic Duals to Boundary CFTs,” JHEP 02, 005 (2012), arXiv:1111.6912 [hep-th]
  * [8] Ofer Aharony, Leon Berdichevsky, Micha Berkooz, and Itamar Shamir, “Near-horizon solutions for D3-branes ending on 5-branes,” Phys. Rev. D 84, 126003 (2011), arXiv:1106.1870 [hep-th]
  * [9] Marco Chiodaroli, Eric D’Hoker, and Michael Gutperle, “Holographic duals of Boundary CFTs,” JHEP 07, 177 (2012), arXiv:1205.5303 [hep-th]
  * [10] Geoffrey Penington, “Entanglement Wedge Reconstruction and the Information Paradox,” JHEP 09, 002 (2020), arXiv:1905.08255 [hep-th]
  * [11] Ahmed Almheiri, Netta Engelhardt, Donald Marolf, and Henry Maxfield, “The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,” JHEP 12, 063 (2019), arXiv:1905.08762 [hep-th]
  * [12] Ahmed Almheiri, Raghu Mahajan, Juan Maldacena, and Ying Zhao, “The Page curve of Hawking radiation from semiclassical geometry,” JHEP 03, 149 (2020), arXiv:1908.10996 [hep-th]
  * [13] Geoff Penington, Stephen H. Shenker, Douglas Stanford, and Zhenbin Yang, “Replica wormholes and the black hole interior,” (11 2019), arXiv:1911.11977 [hep-th]
  * [14] Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini, “Replica Wormholes and the Entropy of Hawking Radiation,” JHEP 05, 013 (2020), arXiv:1911.12333 [hep-th]
  * [15] Moshe Rozali, James Sully, Mark Van Raamsdonk, Christopher Waddell, and David Wakeham, “Information radiation in BCFT models of black holes,” JHEP 05, 004 (2020), arXiv:1910.12836 [hep-th]
  * [16] Hong Zhe Chen, Zachary Fisher, Juan Hernandez, Robert C. Myers, and Shan-Ming Ruan, “Information Flow in Black Hole Evaporation,” JHEP 03, 152 (2020), arXiv:1911.03402 [hep-th]
  * [17] Hao Geng and Andreas Karch, “Massive islands,” JHEP 09, 121 (2020), arXiv:2006.02438 [hep-th]
  * [18] Hong Zhe Chen, Robert C. Myers, Dominik Neuenfeld, Ignacio A. Reyes, and Joshua Sandor, “Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane,” JHEP 10, 166 (2020), arXiv:2006.04851 [hep-th]
  * [19] Hong Zhe Chen, Robert C. Myers, Dominik Neuenfeld, Ignacio A. Reyes, and Joshua Sandor, “Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane,” JHEP 12, 025 (2020), arXiv:2010.00018 [hep-th]
  * [20] Jorrit Kruthoff, Raghu Mahajan, and Chitraang Murdia, “Free fermion entanglement with a semitransparent interface: the effect of graybody factors on entanglement islands,” SciPost Phys. 11, 063 (2021), arXiv:2106.10287 [hep-th]
  * [21] Lisa Randall and Raman Sundrum, “A Large mass hierarchy from a small extra dimension,” Phys. Rev. Lett. 83, 3370–3373 (1999), arXiv:hep-ph/9905221
  * [22] Lisa Randall and Raman Sundrum, “An Alternative to compactification,” Phys. Rev. Lett. 83, 4690–4693 (1999), arXiv:hep-th/9906064
  * [23] Andreas Karch and Lisa Randall, “Locally localized gravity,” JHEP 05, 008 (2001), arXiv:hep-th/0011156
  * [24] Steven B. Giddings, Emanuel Katz, and Lisa Randall, “Linearized gravity in brane backgrounds,” JHEP 03, 023 (2000), arXiv:hep-th/0002091
  * [25] Wyatt Reeves, Moshe Rozali, Petar Simidzija, James Sully, Christopher Waddell, and David Wakeham, “Looking for (and not finding) a bulk brane,” JHEP 12, 002 (2021), arXiv:2108.10345 [hep-th]
  * [26] Masamichi Miyaji, Tadashi Takayanagi, and Tomonori Ugajin, “Spectrum of End of the World Branes in Holographic BCFTs,” JHEP 06, 023 (2021), arXiv:2103.06893 [hep-th]
  * [27] Taishi Kawamoto, Takato Mori, Yu-ki Suzuki, Tadashi Takayanagi, and Tomonori Ugajin, “Holographic local operator quenches in BCFTs,” JHEP 05, 060 (2022), arXiv:2203.03851 [hep-th]
  * [28] Yuya Kusuki, “Analytic bootstrap in 2D boundary conformal field theory: towards braneworld holography,” JHEP 03, 161 (2022), arXiv:2112.10984 [hep-th]
  * [29] Yuya Kusuki, “Semiclassical Gravity from Averaged Boundaries in two-dimensional BCFTs,” (6 2022), arXiv:2206.03035 [hep-th]
  * [30] Yuya Kusuki and Zixia Wei, “In Preparation,”
  * [31] John L Cardy, “Conformal invariance and statistical mechanics,” Les Houches 40 (1988)
  * [32] Yosuke Imamura, Hiroshi Isono, and Yutaka Matsuo, “Boundary states in open string channel and CFT near corner,” Prog. Theor. Phys. 115, 979–1002 (2006), arXiv:hep-th/0512098
  * [33] Yiming Chen, Victor Gorbenko, and Juan Maldacena, “Bra-ket wormholes in gravitationally prepared states,” JHEP 02, 009 (2021), arXiv:2007.16091 [hep-th]
  * [34] Masamichi Miyaji, “Island for gravitationally prepared state and pseudo entanglement wedge,” JHEP 12, 013 (2021), arXiv:2109.03830 [hep-th]
  * [35] Yoshifumi Nakata, Tadashi Takayanagi, Yusuke Taki, Kotaro Tamaoka, and Zixia Wei, “New holographic generalization of entanglement entropy,” Phys. Rev. D 103, 026005 (2021), arXiv:2005.13801 [hep-th]
  * [36] Ian Affleck and Andreas W. W. Ludwig, “Universal noninteger ’ground state degeneracy’ in critical quantum systems,” Phys. Rev. Lett. 67, 161–164 (1991)
  * [37] Tatsuo Azeyanagi, Andreas Karch, Tadashi Takayanagi, and Ethan G. Thompson, “Holographic calculation of boundary entropy,” JHEP 03, 054 (2008), arXiv:0712.1850 [hep-th]
  * [38] G. Hayward, “Gravitational action for space-times with nonsmooth boundaries,” Phys. Rev. D 47, 3275–3280 (1993)
  * [39] Tadashi Takayanagi and Kotaro Tamaoka, “Gravity Edges Modes and Hayward Term,” JHEP 02, 167 (2020), arXiv:1912.01636 [hep-th]
  * [40] Ibrahim Akal, Yuya Kusuki, Tadashi Takayanagi, and Zixia Wei, “Codimension two holography for wedges,” Phys. Rev. D 102, 126007 (2020), arXiv:2007.06800 [hep-th]
  * [41] Hao Geng, Severin Lüst, Rashmish K. Mishra, and David Wakeham, “Holographic BCFTs and Communicating Black Holes,” jhep 08, 003 (2021), arXiv:2104.07039 [hep-th]
  * [42] Sebastian de Haro, Sergey N. Solodukhin, and Kostas Skenderis, “Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence,” Commun. Math. Phys. 217, 595–622 (2001), arXiv:hep-th/0002230
  * [43] Alexander B. Zamolodchikov and Alexei B. Zamolodchikov, “Liouville field theory on a pseudosphere,” , 280–299(1 2001), arXiv:hep-th/0101152
  * [44] Pasquale Calabrese and John L. Cardy, “Evolution of entanglement entropy in one-dimensional systems,” J. Stat. Mech. 0504, P04010 (2005), arXiv:cond-mat/0503393
  * [45] Pasquale Calabrese and John Cardy, “Entanglement entropy and conformal field theory,” J. Phys. A 42, 504005 (2009), arXiv:0905.4013 [cond-mat.stat-mech]
  * [46] James Sully, Mark Van Raamsdonk, and David Wakeham, “BCFT entanglement entropy at large central charge and the black hole interior,” JHEP 03, 167 (2021), arXiv:2004.13088 [hep-th]
  * [47] Ping Gao, Daniel Louis Jafferis, and Aron C. Wall, “Traversable Wormholes via a Double Trace Deformation,” JHEP 12, 151 (2017), arXiv:1608.05687 [hep-th]
  * [48] Juan Maldacena and Xiao-Liang Qi, “Eternal traversable wormhole,” (4 2018), arXiv:1804.00491 [hep-th]
  * [49] Matthew M. Roberts, “Time evolution of entanglement entropy from a pulse,” JHEP 12, 027 (2012), arXiv:1204.1982 [hep-th]