YITP-22-56

IPMU22-0032

Brane Dynamics of Holographic BCFTs

Keisuke Izumia,b, Tetsuya Shiromizub,a, Kenta Suzukic,

Tadashi Takayanagic,d,e and Norihiro Tanahashif

a Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan
b Department of Mathematics, Nagoya University, Nagoya 464-8602, Japan
cCenter for Gravitational Physics and Quantum Information (CGPQI),
Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
dInamori Research Institute for Science,
620 Suiginya-cho, Shimogyo-ku, Kyoto 600-8411 Japan
eKavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa, Chiba 277-8582, Japan
fDepartment of Physics, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551,
Japan

In this paper we study various dynamical aspects of the AdS/BCFT
correspondence in higher dimensions. We study properties of holographic stress
energy tensor by analyzing the metric perturbation in the gravity dual. We
also calculate the stress energy tensor for a locally excited state on a half
plane in a free scalar CFT. Both of them satisfy a reflective boundary
condition that is expected for any BCFTs. We also study the behavior of the
scalar field perturbation in the AdS/BCFT setup and show that they also show
complete reflections. Moreover, we find that the entanglement entropy of a
BCFT computed from the AdS/BCFT matched with that calculated from the Island
formula, which supports the Island/BCFT correspondence in higher dimensions.
Finally we show how we can calculate one point functions in a BCFT in our
gravity dual.

###### Contents

  1. 1 Introduction
  2. 2 Stress energy tensors in BCFTs
    1. 2.1 Free scalar BCFTd
    2. 2.2 Holographic stress energy tensor
    3. 2.3 Holographic stress energy tensor in AdS${}_{3}/$BCFT2
  3. 3 Gravitational dynamics in higher dimensional AdS/BCFT
    1. 3.1 Setup
    2. 3.2 Perturbed equation in the bulk
    3. 3.3 Junction condition
    4. 3.4 Massive tensor on AdS boundary
    5. 3.5 Holographic stress energy tensor in AdS/BCFT
    6. 3.6 Case with $d=3$
  4. 4 Scalar field excitations in higher dimensional AdS/BCFT
    1. 4.1 Massive scalar field
    2. 4.2 Scalar field reflection at the boundary
  5. 5 Island/BCFT correspondence in higher dimensions
    1. 5.1 Entanglement entropy from island formula
    2. 5.2 Entanglement entropy in BCFT from AdS/BCFT
  6. 6 One point functions in higher dimensional AdS/BCFT
    1. 6.1 The model
    2. 6.2 Free massless case
    3. 6.3 Free massive case
    4. 6.4 Boundary one-point function
  7. 7 Conclusions and Discussions
  8. A A class of explicit metric perturbations at $d=3$
    1. A.1 Perturbative solutions to Einstein equation
    2. A.2 Comparison with AdS4 black brane solution
    3. A.3 Full solutions with boundary condition imposed and relation to the TT gauge result
  9. B Minimal surface in AdSd+1

##  1 Introduction

A boundary conformal field theory (BCFT) is a conformal field theory (CFT)
defined on a manifold with boundaries such that a part of conformal symmetry
is preserved by the boundaries [1, 2, 3, 4]. Recently, a class of gravity
duals of BCFTs, called the AdS/BCFT [5, 6, 7] (for an earlier model refer to
[8]), have been actively studied. One reason for this is that the AdS/BCFT
makes calculations of the entanglement entropy much more tractable. Moreover
it provides useful setups which model black hole evaporation processes from
which we can derive the Page curve [9], where the Island prescription [10, 11]
is expected to be realized via the brane-world holography [12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23]. Refer to e.g.[24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] for applications of
AdS/BCFT to black hole information problem.

The AdS/BCFT has also been successfully applied to many other problems. For
example, this includes condensed matter and field theoretic aspects of chaotic
BCFTs [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57], studies of boundary
renormalization group flow and boundary entropy, [58, 59, 60, 61], holographic
calculations of the computational complexity [62, 63, 64, 65, 66, 67, 68, 69]
and cosmological models [70, 71, 72, 73, 74]. String theory embeddings have
been studied in [75, 76, 77, 78, 79, 80, 81, 82, 83]. Moreover, the AdS/BCFT
correspondence was generalized in [84, 85] to construct gravity duals of
higher codimension holography. Refer also to [86] for an analysis of AdS/BCFT
with one loop quantum corrections and to [87] for an appearance of non-
locality in the brane-world description of AdS/BCFT.

The basic idea of AdS/BCFT is to simply extend the BCFT towards the bulk AdS
by introducing the end of the world-brane (EOW brane) such that the EOW brane
at the AdS boundary coincides with the boundary of the BCFT. The
AdS${}_{d+1}/$BCFTd argues that a $d$ dimensional BCFT (BCFTd) is dual to the
$d+1$ dimensional gravity with a negative cosmological constant on the region
surrounded by the boundary where the BCFT is situated and the the EOW brane,
as sketched in Fig. 1.

Figure 1: A sketch of AdS/BCFT construction. The gravity dual of a BCFT is
given by the red colored region which is surrounded by the boundary where the
BCFT is situated (blown colored) and the the end of the world-brane (EOW
brane, purple colored).

The EOW brane is defined by a Neumann boundary condition and has a tension
$\sigma$. Therefore in general, its gravitational backreaction deforms the
geometry. This problem of backreaction can be solved analytically in the
lowest dimensional AdS/BCFT for $d=2$, i.e. AdS${}_{3}/$BCFT2. This is because
any solutions in the three dimensional pure gravity locally coincide with the
AdS3, reflecting the infinite dimensional Virasoro symmetry in the dual BCFT.
Accordingly, various dynamical aspects of AdS${}_{3}/$BCFT2 have been studied
quite well. On the other hand, in higher dimensions $d\geq 3$, there have been
little results known for the dynamics of higher dimensional AdS/BCFT. The
purpose of this paper is to extensively study dynamical aspects of the higher
dimensional AdS/BCFT from several perspectives. We will study dynamical
properties of stress energy tensor in BCFTs by analyzing metric perturbations
in the AdS/BCFT and also by directly calculating a simple free scalar BCFT.
This is also interesting from the brane-world interpretation. In the AdS/BCFT,
gravity is localized on the EOW brane and this mode is dual to the boundary
degrees of freedom of the BCFT, while the bulk gravity is dual to the bulk
degrees of freedom of the BCFT. One basic question is whether the AdS/BCFT
correctly explains the complete reflection of the energy and momentum flux in
BCFTs, which is not so obvious from the profile of the gravity dual geometry.
A similar question will also be considered in a scalar field excitation in the
bulk. We will also study how one point functions of scalar operators in BCFTs
can be reproduced from the gravity dual calculations, generalizing the
previous result in [40] for $d=2$.

The triality between (i) a $d$ dimensional BCFT, (ii) its gravity dual in
$d+1$ dimension via the AdS${}_{d+1}/$BCFTd and (iii) its brane-world
interpretation, leads to a purely $d$ dimensional duality relation that a $d$
dimensional BCFT is equivalent to a $d$ dimensional CFT on a half plane,
coupled to $d$ dimensional gravity on AdSd, which is called Island/BCFT
correspondence [40]. As we mentioned, this duality can be derived from known
conjectures. However, we are still far from a complete justification of the
Island/BCFT correspondence. The recent paper [40] examined this problem and
presented quantitative evidences in the lowest dimension $d=2$. In this paper,
we would like to examine the Island/BCFT correspondence by calculating
entanglement entropy in higher dimensions $d\geq 3$.

The rest of the paper is organized as follows. In section 2, we present field
theory results of stress energy tensor in BCFTs. We also give gravity dual
results for two dimensional BCFTs via AdS${}_{3}/$BCFT2. In section 3, we
analyze metric perturbations in AdS${}_{d+1}/$BCFTd for higher dimensions
$d\geq 3$. We work out the behavior of holographic stress energy tensor. In
section 4, we analyze scalar field excitations in AdS/BCFT setup and show that
the scalar wave is completely reflected. In section 5, we give evidences of
the Island/BCFT correspondence by comparing the calculation of entanglement
entropy in a holographic BCFT and in a CFT coupled to gravity. In section 6,
we give a prescription of computing one point functions in higher dimensional
AdS/BCFT. In section 7, we summarize our conclusions. In appendix A, we
present a class of explicit solutions of metric perturbations for $d=3$. In
appendix B, we show our detailed analysis of minimal surfaces in AdSd+1.

Note added: When we were completing this draft, we became aware of the
independent work [88], which has a partial overlap with ours in that the
metric perturbations in a higher dimensional AdS/BCFT were analyzed.

##  2 Stress energy tensors in BCFTs

We consider a $d$ dimensional conformal field theory (CFTd) on a manifold with
$d-1$ dimensional boundaries. In particular, we impose those classes of
boundary conditions which preserve a part of conformal invariance. This is
called a boundary conformal field theory (BCFTd). The CFTd originally has the
conformal symmetry $SO(d,2)$ in Lorentzian signature (or $SO(d+1,1)$ in
Euclidean signature) and the symmetry preserved by BCFTd is its subgroup
$SO(d-1,2)$ (or $SO(d,1)$). The main purpose of this paper is to explore
various dynamical aspects of gravity duals of BCFTs. In this section, as a
preparation of our later arguments, we would like to explain what we expect as
the behavior of the stress energy tenor in BCFTs. We will also describe the
gravity results for BCFT2, leaving the gravity analysis in higher dimensions
left as the main problems discussed in later section.

###  2.1 Free scalar BCFTd

In this section, we consider the simplest example of BCFTd given by the
conformally coupled real free scalar theory

$\displaystyle I\,=\,\frac{1}{2}\int
d^{d}x\left[\partial_{a}\phi\partial^{a}\phi\,+\,\frac{d-2}{4(d-1)}\,R_{0}\phi^{2}\right]\,,$
(2.1)

where $R_{0}$ is the background scalar curvature and we assume an Euclidean
signature. We consider this theory on a flat $d$ dimensional Euclidean
space111 Since we consider a flat space, $R_{0}=0$. However, this terms is
needed to derive the conformal stress energy tensor.  $\xi^{a}=(x^{i},w)$
where $i=0,1,\cdots,d-2$, and place a boundary at $w=0$, so that the BCFT
lives on the half plane $w>0$. The stress energy tensor of this theory is
given by

$\displaystyle{\cal
T}_{ab}\,=\,\frac{d}{2(d-1)}\partial_{a}\phi\partial_{b}\phi-\frac{\delta_{ab}}{2(d-1)}(\partial\phi)^{2}-\frac{d-2}{2(d-1)}(\partial_{a}\partial_{b}\phi)\phi+\frac{(d-2)\delta_{ab}}{2d(d-1)}(\partial^{2}\phi)\phi.$
(2.2)

With this expression of the stress energy tensor, it is traceless ${\cal
T}^{a}_{a}=0$ and, provided the equation of motion $\partial^{2}\phi=0$, it is
conserved $\partial^{a}{\cal T}_{ab}=0$.

Now, we would like to compute the one-point function of this stress energy
tensor. This can be done by using the two-point function of the scalar fields,
which is given by [3]

$\displaystyle\qquad\big{\langle}\phi(w_{1},\vec{x}_{1})\phi(w_{2},\vec{x}_{2})\big{\rangle}$
(2.3)
$\displaystyle=\,\frac{1}{(d-2)S_{d}}\Bigg{(}\frac{1}{\big{(}(w_{1}-w_{2})^{2}+|\vec{x}_{1}-\vec{x}_{2}|^{2}\big{)}^{\frac{d-2}{2}}}+\frac{(-1)^{\eta}}{\big{(}(w_{1}+w_{2})^{2}+|\vec{x}_{1}-\vec{x}_{2}|^{2}\big{)}^{\frac{d-2}{2}}}\Bigg{)}\,,$

where $S_{d}=2\pi^{d/2}/\Gamma(d/2)$. The boundary coalition parameter $\eta$
is $\eta=0$ for Neumann boundary condition at $w=0$:
$\partial_{w}\phi(0,\vec{x})=0$, or $\eta=1$ for Dirichlet boundary condition
at $w=0$: $\phi(0,\vec{x})=0$. First using the expression (2.2) and this two-
point function (2.1), one can explicitly check that the one-point function of
the stress energy tensor in the vacuum state indeed vanishes everywhere

$\displaystyle\big{\langle}{\cal T}_{ab}(w,\vec{x})\big{\rangle}\,=\,0\,.$
(2.4)

Next, we consider an excited state $|\Psi_{\phi}\rangle$ which is created from
the vacuum by inserting a single operator located at $w=l$ and $\vec{x}=0$.
However, since the local operator excitation is singular, we make a
regularization by introducing an infinitesimally small imaginary time
evolution by $\epsilon$ [89] (refer to [90, 91] for gravity duals) as follows

$\displaystyle|\Psi_{\phi}\rangle=e^{-\epsilon
H}\phi(w=l,\vec{x}=0)|0\rangle,$ (2.5)

which is equivalent to shifting the value of $x_{0}$ into $x_{0}=-\epsilon$.
This regularization leads to a finite inner product

$\displaystyle\langle\Psi_{\phi}|\Psi_{\phi}\rangle=\frac{1}{(d-2)S_{d}}(2\epsilon)^{-(d-2)}.$
(2.6)

However, note that it is also meaningful to consider a fine value of
$\epsilon$ which defines a class of excitation states. Thus below we allow
$\epsilon$ to take any positive values.

The one-point function of the stress energy tensor in this state is expressed
as

$\displaystyle\big{\langle}{\cal T}_{ab}(w,\vec{x})\big{\rangle}_{\phi}\,$
$\displaystyle\equiv\,\frac{\big{\langle}\Psi_{\phi}|{\cal
T}_{ab}(w,\vec{x})|\Psi_{\phi}\big{\rangle}}{\langle\Psi_{\phi}|\Psi_{\phi}\rangle}$
(2.7)
$\displaystyle=\,(d-2)S_{d}(2\epsilon)^{d-2}\,\big{\langle}\phi(l,\epsilon,0)|{\cal
T}_{ab}(w,\vec{x})|\phi(l,-\epsilon,0)\big{\rangle}\,,$

and this one-point function can be computed from the four-point function of
the scalar fields (we ignore the Wick contraction of $\phi(w_{1},\vec{x}_{1})$
and $\phi(w_{2},\vec{x}_{2})$ as this leads to the vacuum expectation value of
stress energy tensor)

$\displaystyle\quad\
\big{\langle}\phi(l,\epsilon,0)\phi(w_{1},\tau_{1},x_{1})\phi(w_{2},\tau_{2},x_{2})\phi(l,-\epsilon,0)\big{\rangle}$
(2.8)
$\displaystyle=\,\frac{1}{(d-2)^{2}S_{d}^{2}}\,\sum_{\pm,\pm,\pm}\frac{(\pm
1)^{\eta}}{\big{(}(w_{1}\pm
l)^{2}+(\tau_{1}\pm\epsilon)^{2}+(x_{1})^{2}\big{)}^{\frac{d-2}{2}}\big{(}(w_{2}\pm
l)^{2}+(\tau_{2}\mp\epsilon)^{2}+(x_{2})^{2}\big{)}^{\frac{d-2}{2}}}\,,$

where we introduced the expression $\vec{x}_{1,2}=(\tau_{1,2},x_{1,2})$ and
the summation contains eight terms for all possible combinations of the
$\pm$’s in the denominator. However, the $\mp$ sign in
$(\tau_{2}\mp\epsilon)^{2}$ is not independent but the sign must be opposite
from the sign in $(\tau_{1}\pm\epsilon)^{2}$. The numerator is $(+1)^{\eta}$
when the two signs in $(w_{1}\pm l)^{2}$ and $(w_{2}\pm l)^{2}$ are same, and
it is $(-1)^{\eta}$ when the two signs in $(w_{1}\pm l)^{2}$ and $(w_{2}\pm
l)^{2}$ are opposite. Then by taking the derivative with respect to
$(w_{1,2},\tau_{1,2},x_{1,2})$ following (2.2), we can find the expectation
value of stress energy tensors. Since the full expression of this one-point
function of the stress energy tensor is quite lengthy, we do not write down
all components explicitly. Instead we will present the expression of ${\cal
T}_{ab}$ at the boundary $w=0$ by choosing the dimension $d=3$ and Neumann
boundary condition $\eta=0$. They are given as follows:

$\displaystyle{\cal T}_{\tau\tau}|^{d=3}_{w=0}$
$\displaystyle=\frac{4L^{2}\\!(\\!L^{2}\\!+\\!\tau^{2}\\!+\\!x^{2})^{2}\epsilon\\!-\\!8\left(\\!L^{4}+L^{2}(\\!-7\tau^{2}\\!+\\!3x^{2}\\!)\\!+\\!2\\!(\\!\tau^{4}\\!-\\!4\tau^{2}x^{2}\\!+\\!x^{4}\\!)\right)\epsilon^{3}\\!-4(\\!7L^{2}\\!-\\!8\tau^{2}\\!+\\!8x^{2}\\!)\epsilon^{5}\\!-\\!16\epsilon^{7}\\!}{16\pi^{2}(L^{2}+x^{2}+(\tau-\epsilon)^{2})^{\frac{5}{2}}(L^{2}+x^{2}+(\tau+\epsilon)^{2})^{\frac{5}{2}}},$
$\displaystyle{\cal T}_{xx}|^{d=3}_{w=0}$
$\displaystyle=\frac{4L^{2}\\!(\\!L^{2}\\!+\\!\tau^{2}\\!+\\!x^{2})^{2}\epsilon\\!+\\!8\left(2L^{4}\\!+\\!\tau^{4}\\!-\\!10\tau^{2}x^{2}+x^{4}+L^{2}(\tau^{2}\\!+\\!3x^{2}\\!)\right)\epsilon^{3}+4(5L^{2}\\!-\\!4\tau^{2}\\!+\\!4x^{2}\\!)\epsilon^{5}\\!+\\!8\epsilon^{7}}{16\pi^{2}(L^{2}+x^{2}+(\tau-\epsilon)^{2})^{\frac{5}{2}}(L^{2}+x^{2}+(\tau+\epsilon)^{2})^{\frac{5}{2}}},$
$\displaystyle{\cal T}_{ww}|^{d=3}_{w=0}$
$\displaystyle=\frac{-8L^{2}\\!(\\!L^{2}\\!+\\!\tau^{2}\\!+\\!x^{2})^{2}\epsilon\\!+\\!8\left(-L^{4}\\!-8L^{2}\tau^{2}+(\tau^{2}+x^{2})^{2}\right)\epsilon^{3}+8(L^{2}\\!-\\!2\tau^{2}\\!+\\!2x^{2}\\!)\epsilon^{5}\\!+\\!8\epsilon^{7}}{16\pi^{2}(L^{2}+x^{2}+(\tau-\epsilon)^{2})^{\frac{5}{2}}(L^{2}+x^{2}+(\tau+\epsilon)^{2})^{\frac{5}{2}}},$
(2.9)

and

$\displaystyle{\cal T}_{\tau x}|^{d=3}_{w=0}\,=\,\frac{48\tau
x(L^{2}-\tau^{2}+x^{2}+\epsilon^{2})\epsilon^{3}}{16\pi^{2}(L^{2}+x^{2}+(\tau-\epsilon)^{2})^{\frac{5}{2}}(L^{2}+x^{2}+(\tau+\epsilon)^{2})^{\frac{5}{2}}}\,,$
$\displaystyle{\cal T}_{\tau w}|^{d=3}_{w=0}\,=\,{\cal
T}_{xw}|^{d=3}_{w=0}\,=\,0\,.$ (2.10)

In general, the stress energy tensor in BCFT satisfies the boundary condition
[92]:

$\displaystyle{\cal T}_{wi}|_{w=0}=0,\ \ (i=0,1,2,\cdots,d-2),$ (2.11)

while other components of stress tensor are non-vanishing at the boundary in
general. In the bulk, it satisfies as usual the traceless and conservation
condition

$\displaystyle{\cal T}^{a}_{a}=0,$ $\displaystyle\partial^{a}{\cal T}_{ab}=0.$
(2.12)

Notice that this boundary condition (2.11) means that the energy flux and
momentum flux are completely reflected at the boundary. Indeed the total
energy and momentum are conserved

$\displaystyle\frac{d}{dt}\int dx^{d-2}dw{\cal T}_{0i}=0,\ \ \
(i=0,1,\cdots,d-2)$ (2.13)

owing to the conservation law (2.12) and the boundary condition (2.11).

###  2.2 Holographic stress energy tensor

In the AdS/CFT, the stress energy tensor can be computed from the behavior of
the metric near the AdS boundary, so called holographic stress energy tensor
[93, 94]. When the CFT is defined on a flat space, which we always assume in
this paper, its takes a simple form as we will explain below. We can write the
metric of $d+1$ dimensional asymptotically Poincare AdS background in the
Fefferman-Graham expansion form:

$\displaystyle ds^{2}=\frac{dz^{2}+g_{ab}(\xi,z)d\xi^{a}d\xi^{b}}{z^{2}},$
$\displaystyle g_{ab}\simeq\eta_{ab}+\frac{16\pi G_{N}}{d}z^{d}~{}{\cal
T}_{ab}(\xi)+O(z^{d+1})\ \ (z\to 0),$ (2.14)

where $\xi^{a}\ \ (a=0,1,2,\cdots,d-1)$ is the coordinate of the flat space on
which the CFT is defined. In the above expansion, ${\cal T}_{ab}(\xi)$ is the
holographic stress energy tensor, normalized such that it is computed from the
variation of the CFT action $S_{CFT}$ with respect to the background metric
$g_{ab}$

$\displaystyle{\cal T}_{ab}(\xi)=-\frac{2}{\sqrt{-g}}\frac{\delta
S_{CFT}}{\delta g^{ab}}.$ (2.15)

The bulk Einstein equation leads to the traceless condition and the
conservation law (2.12).

In this paper we will consider the BCFT defined on the $d$ dimensional half
plane $\xi_{d-1}(\equiv w)>0$. Then from the standard BCFT result[3, 4, 92],
we expect the boundary condition for the stress tensor at $w=0$ (2.11). Note
that we can still apply the calculation of holographic stress energy tensor
(2.14) even for BCFTs. Refer to [7, 95, 96, 97] for earlier calculations.

###  2.3 Holographic stress energy tensor in AdS${}_{3}/$BCFT2

The gravity dual for BCFT2 based on AdS/BCFT [5, 6] is exceptionally simple
and the computations of stress energy tensor is straightforward as opposed to
those in the higher dimensional BCFTs. Therefore we would like to briefly
present this lower dimensional analysis below.

Consider the pure Einstein gravity with negative cosmological constant.
Solutions to the Einstein equation are all locally equivalent to the pure
AdS3. Therefore we start with the Poincaré metric in AdS3 (we set $U=T-W$ and
$V=T+W$)

$\displaystyle ds^{2}=\frac{d\eta^{2}-dUdV}{\eta^{2}},$ (2.16)

which is dual to the vacuum state in a CFT2. In this paper, we set the AdS
radius to unity for simplicity. In AdS/BCFT, the gravity dual of BCFT is given
by considering the end of the world-brane (EOW brane) $Q$ such that it gives
an extension of the boundary of BCFT toward the bulk AdS and such that it
satisfied the Neumann boundary condition

$\displaystyle K_{\mu\nu}-Kh_{\mu\nu}+\frac{\sigma}{2}h_{\mu\nu}=0,$ (2.17)

where $h_{\mu\nu}$ and $K_{\mu\nu}$ are the induced metric and the extrinsic
curvature (with the out-going normal vector) of the EOW brane $Q$. The
parameter $\sigma$ is the tension of the EOW brane. A canonical solution is
the hyperplane given by

$\displaystyle W+\lambda\eta=0,$ (2.18)

where the parameter $\lambda$ is related to the tension of the EOW brane via

$\displaystyle\lambda=\frac{\sigma}{\sqrt{4-\sigma^{2}}}.$ (2.19)

The AdS/BCFT argues that the gravity dual of BCFT vacuum on the right half
plane $W\geq 0$ is given by the region $W+\lambda\eta\geq 0$ in the AdS3
(2.16).

To construct the gravity dual for generic excited states, we consider a
conformal transformation of the form (we set $u=t-w$ and $v=t+w$):

$\displaystyle\tilde{u}=p(u),\ \ \ \tilde{v}=q(v).$ (2.20)

Via the AdS/CFT, this is dual to the following coordinate transformation in
AdS3

$\displaystyle\begin{split}U&=p(u)+\frac{2z^{2}(p^{\prime})^{2}q^{\prime\prime}}{4p^{\prime}q^{\prime}-z^{2}p^{\prime\prime}q^{\prime\prime}},\\\
V&=q(v)+\frac{2z^{2}(q^{\prime})^{2}p^{\prime\prime}}{4p^{\prime}q^{\prime}-z^{2}p^{\prime\prime}q^{\prime\prime}},\\\
\eta&=\frac{4z(p^{\prime}q^{\prime})^{3/2}}{4p^{\prime}q^{\prime}-z^{2}p^{\prime\prime}q^{\prime\prime}},\end{split}$
(2.21)

which is known as the so-called Ba$\tilde{\text{n}}$ados map [98, 99, 52]. The
metric expressed in terms of the new coordinates $(u,v,z)$ looks like

$\displaystyle ds^{2}=\frac{dz^{2}}{z^{2}}+{\cal T}_{++}(u)(du)^{2}+{\cal
T}_{--}(v)(dv)^{2}-\left(\frac{1}{z^{2}}+z^{2}{\cal T}_{++}(u){\cal
T}_{--}(v)\right)dudv,$ (2.22)

where

$\displaystyle{\cal
T}_{++}(u)=\frac{3(p^{\prime\prime})^{2}-2p^{\prime}p^{\prime\prime\prime}}{4p^{\prime
2}},\qquad{\cal
T}_{--}(v)=\frac{3(q^{\prime\prime})^{2}-2q^{\prime}q^{\prime\prime\prime}}{4q^{\prime
2}}$ (2.23)

are proportional to the chiral and anti-chiral part of the holographic stress
energy tensor, respectively.

We consider a BCFT defined on the right half $w\geq 0$ of a two dimensional
plane. If we require that after the conformal map (2.20) of this right half
plane, we have the same geometry (i.e. right half plane), we need to require
that $p$ and $q$ are the same function: $p(t)=q(t)$. In this case it is easy
to find222This relation can be violated if $p\neq q$ and this is realized when
the boundary has a non-trivial time dependent profile. Such a BCFT is known as
a moving mirror (see e.g.[34, 36]). In this paper, we always assume that the
boundary of the background spacetime is time-independent and is just given by
a straight line or plane.

$\displaystyle{\cal T}_{++}(u)={\cal T}_{--}(v),$ (2.24)

on the boundary $w=0$. Therefore the energy flux is conserved. In this way,
the perfect reflection of energy flux is clear in the gravity dual also.

##  3 Gravitational dynamics in higher dimensional AdS/BCFT

Now we would like to move on to our main problem, higher dimensional AdS/BCFT.
Here we investigate the metric perturbations for the standard AdS/BCFT setup
in $d+1$ dimensions, depicted in Fig. 2. This allows up to calculate the
holographic stress energy tensor in the dual BCFTd.

Figure 2: A sketch of AdS/BCFT setup.

First, let us present the original background for which we perform the
perturbation. Consider the ($d+1$) dimensional Poincare AdS

$\displaystyle ds^{2}=\frac{dz^{2}+dw^{2}+\sum_{i=0}^{d-2}dx_{i}^{2}}{z^{2}}.$
(3.1)

Via the coordinate transformation

$\displaystyle z=\frac{y}{\cosh\rho},\qquad w=y\tanh\rho,$ (3.2)

we obtain the following metric in terms of the hyperbolic slice of AdSd:

$\displaystyle
ds^{2}=d\rho^{2}+\cosh^{2}\rho\left(\frac{dy^{2}+\sum_{i=0}^{d-2}dx_{i}^{2}}{y^{2}}\right).$
(3.3)

The ground state of a BCFT defined on the half plane $w>0$ in the coordinate
$(\tau,x,w)$ [5, 6] is dual to the part of AdSd+1 defined by (refer to Fig. 2)

$\displaystyle w+\lambda z>0,$ (3.4)

where $\lambda$ is related to the brane tension $\sigma$ of the end of the
world-brane via

$\displaystyle\sigma=\frac{2(d-1)\lambda}{\sqrt{1+\lambda^{2}}}.$ (3.5)

It is straightforward to confirm that this satisfies the required boundary
condition (2.17). In the hyperbolic slicing coordinates, the brane is located
at $\rho=\rho_{*}$ (with $\rho_{*}<0$), where the brane location $\rho_{*}$ is
related to $\lambda$ by

$\displaystyle\lambda\,=\,-\sinh\rho_{*}\,.$ (3.6)

Below we study the metric perturbations around this background solution,
assuming a pure gravity theory with a negative cosmological constant. Our
convention of coordinate labeling is as follows. $(M,N.\cdots)$ and
$(\mu,\nu,\cdots)$ are indices for $(d+1)$-dimensional and $d$-dimensional
directions without $\rho$ coordinate. Variables and operators with tilde are
those with respect to the $d$-dimensional background metric $\gamma_{\mu\nu}$
(see Eq. (3.7)). Prime ′ denotes a derivative w.r.t. $\rho$. Equality with hat
($\hat{=}$) means that the equation holds only on the brane.

###  3.1 Setup

The metric of $(d+1)$-dimensional bulk is

$\displaystyle
ds^{2}_{AdS_{d+1}}=d\rho^{2}+\left(\mathfrak{a}^{2}(\rho)\gamma_{\mu\nu}+h_{\mu\nu}\right)dx^{\mu}dx^{\nu},$
(3.7)

where

$\displaystyle\mathfrak{a}(\rho):=\ell\cosh\left(\frac{\rho}{\ell}\right)$
(3.8)

and $\gamma_{\mu\nu}$ is the $d$-dimensional AdS metric with the unit
curvature. The Riemann curvature of the background metric (i.e. for the case
with $h_{\mu\nu}=0$) becomes

$\displaystyle
R^{(0)}_{MNAB}=-\frac{1}{\ell^{2}}\left(g_{MA}g_{NB}-g_{MB}g_{NA}\right).$
(3.9)

When the Einstein equation with the cosmological constant $\Lambda$ gives this
solution, $\Lambda$ should be

$\displaystyle\Lambda=-\frac{d(d-1)}{2\ell^{2}}.$ (3.10)

For the perturbation $h_{\mu\nu}$, we use the following gauge condition:

$\displaystyle\tilde{\nabla}^{\mu}h_{\mu\nu}=0,\qquad h^{\mu}{}_{\mu}=0.$
(3.11)

Note that it gives

$\displaystyle\nabla_{M}h^{M}{}_{N}=0.$ (3.12)

This gauge condition is called transverse-traceless (TT) gauge, which is often
used for the analysis of gravitational waves in the vacuum and of
gravitational perturbations in braneworld model [15, 100, 101].

###  3.2 Perturbed equation in the bulk

The perturbation of the bulk Ricci tensor is calculated as

$\displaystyle R_{MN}$ $\displaystyle=$ $\displaystyle
R^{(0)}_{MN}+\frac{1}{2}\left(\nabla_{A}\nabla_{M}h^{A}{}_{N}+\nabla_{A}\nabla_{N}h^{A}{}_{M}-\Box
h_{MN}-\nabla_{M}\nabla_{N}h\right)+{\cal O}\left(h_{MN}^{2}\right)$ (3.13)
$\displaystyle=$
$\displaystyle-\frac{d}{\ell^{2}}\left(g^{(0)}_{MN}+h_{MN}\right)-\frac{1}{2}\Box
h_{MN}-\frac{1}{\ell^{2}}h_{MN}+{\cal O}\left(h_{MN}^{2}\right).$

Then vacuum equation gives the perturbation equation in the bulk:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
R_{MN}-\frac{1}{2}Rg_{MN}+\Lambda g_{MN}\qquad\left(=:G_{MN}+\Lambda
g_{MN}\right)$ (3.14) $\displaystyle=$
$\displaystyle\left(-\frac{d}{\ell^{2}}+\frac{1}{2}\frac{d(d+1)}{\ell^{2}}-\frac{1}{2}\frac{d(d-1)}{\ell^{2}}\right)\left(g^{(0)}_{MN}+h_{MN}\right)-\frac{1}{2}\left(\Box+\frac{2}{\ell^{2}}\right)h_{MN}+{\cal
O}\left(h_{MN}^{2}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\Box+\frac{2}{\ell^{2}}\right)h_{MN}+{\cal
O}\left(h_{MN}^{2}\right),$

where $\delta\left(\cdots\right)$ means the first order terms of
$\left(\cdots\right)$ with respect to $h_{MN}$. Therefore, the bulk equation
of the metric perturbation becomes

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(\Box+\frac{2}{\ell^{2}}\right)h_{\mu\nu}$ (3.15)
$\displaystyle=$ $\displaystyle
h_{\mu\nu}^{\prime\prime}+(d-4)\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}h_{\mu\nu}^{\prime}-2(d-2)\left(\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}\right)^{2}h_{\mu\nu}-2\left(\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}\right)^{\prime}h_{\mu\nu}+\mathfrak{a}^{-2}\left(\tilde{\Box}+2\right)h_{\mu\nu}$
$\displaystyle=$
$\displaystyle\mathfrak{a}^{2}\left(H_{\mu\nu}^{\prime\prime}+d\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}H_{\mu\nu}^{\prime}\right)+\left(\tilde{\Box}+2\right)H_{\mu\nu}$
$\displaystyle=$
$\displaystyle\mathfrak{a}^{2}\left(H_{\mu\nu}^{\prime\prime}+dl\tanh^{2}\left(\frac{\rho}{\ell}\right)H_{\mu\nu}^{\prime}+\cosh^{-2}\left(\frac{\rho}{\ell}\right)\left(\tilde{\Box}+2\right)H_{\mu\nu}\right),$

where

$\displaystyle h_{MN}=:\mathfrak{a}^{2}H_{MN}$ (3.16)

and

$\displaystyle\Box:=g^{(0)MN}\nabla_{M}\nabla_{N},\qquad\tilde{\Box}:=\gamma^{\mu\nu}\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}.$
(3.17)

Note that $\Box$ includes covariant derivatives, and thus, the operation to
$h_{\mu\nu}$ is different from that to scalar field. Note that “$2$” in
$\left(\tilde{\Box}+2\right)$ is required for spin-2 fields, which is not
mass. Due to the curvature of AdS, this “$2$” is generated from the kinetic
term of spin-2 action. Equation (3.15) can be solved with the separation of
the variables.

Hereinafter, we set $\ell=1$. Solutions could be written in separable form
$H_{\mu\nu}=\sum_{i}{\mathcal{R}}^{(i)}(\rho)Y^{(i)}_{\mu\nu}(x^{\mu})$. With
this separable form of $H_{\mu\nu}$, Eq. (3.15) becomes the following two
different equations with eigenvalues $\lambda_{\rho}^{(i)}$

$\displaystyle{\mathcal{R}}^{(i)}_{,\rho\rho}+d\,\tanh\rho{\mathcal{R}}^{(i)}_{,\rho}=-\frac{\lambda^{(i)}_{\rho}}{\cosh^{2}\rho}{\mathcal{R}}^{(i)},$
(3.18)
$\displaystyle\left(\tilde{\Box}+2-\lambda^{(i)}_{\rho}\right)Y^{(i)}_{\mu\nu}=0.$
(3.19)

This decomposition is interpreted as a Kaluza-Klein reduction; the equation
for extra-dimension Eq. (3.18) fixes the Kaluza-Klein (KK) mass as
$(m^{(i)}_{KK})^{2}=\lambda^{(i)}_{\rho}$, while Eq. (3.19) is $d$-dimensional
equation for spin-2 field with (KK) mass $(m^{(i)}_{KK})^{2}$ on the
$d$-dimensional AdS spacetime. Hereinafter, we omit the superscript $(i)$ for
the simple notation, when the difference of $(i)$ is not important.

Let us solve Eq. (3.18) first. Equation (3.18) can be written as

$\displaystyle{\cal
L}_{\mathcal{R}}{\mathcal{R}}=\left(1-\zeta^{2}\right)^{-\frac{d}{2}}\lambda_{\rho}{\mathcal{R}},$
(3.20) $\displaystyle{\cal
L}_{\mathcal{R}}:=-\frac{d}{d\zeta}\left(1-\zeta^{2}\right)^{-\frac{d}{2}+1}\frac{d}{d\zeta},$
(3.21) $\displaystyle\zeta:=\tanh\rho.$ (3.22)

Solutions of Eq. (3.18) are expressed by Ferrers’ associated Legendre
functions333Note that it is different from Hobson’s associated Legendre
functions. $P^{\mu}_{l}(\zeta)$ and $Q^{\mu}_{l}(\zeta)$;

$\displaystyle{\mathcal{R}}=\left[c_{1}P^{\mu}_{l}(\zeta)+c_{2}Q^{\mu}_{l}(\zeta)\right](1-\zeta^{2})^{\frac{d}{4}},$
(3.23)

where

$\displaystyle
l=\nu-\frac{1}{2},\qquad\mu=\frac{d}{2},\qquad\nu=\frac{1}{2}\sqrt{(d-1)^{2}+4\lambda_{\rho}}.$
(3.24)

Since, near the AdS boundary $\zeta\to 1$ $(\rho\to\infty)$,
$P^{\mu}_{l}(\zeta)$ and $Q^{\mu}_{l}(\zeta)$ behave as

$\displaystyle
P^{\mu}_{l}(\zeta)\,\simeq\,(1-\zeta)^{-\frac{\mu}{2}}\frac{2^{\frac{\mu}{2}}e^{i\pi\mu}}{\Gamma(1-\mu)},$
(3.25) $\displaystyle
2\sin(\pi\mu)Q^{\mu}_{l}(\zeta)\,\simeq\,\pi\cos(\pi\mu)P^{\mu}_{l}(\zeta),$
(3.26)

the Dirichlet boundary condition on the AdS boundary gives the relation
between $c_{1}$ and $c_{2}$;

$\displaystyle 2\sin(\pi\mu)c_{1}=-\pi\cos\left(\pi\mu\right)c_{2}.$ (3.27)

Note that, because of $\mu=d/2$ (See Eq. (3.24)), Eq. (3.27) gives

$\displaystyle{\mathcal{R}}=\begin{cases}P^{\mu}_{l}(\zeta)(1-\zeta^{2})^{\frac{d}{4}}&\mbox{(if
$d$ is even)}\\\ Q^{\mu}_{l}(\zeta)(1-\zeta^{2})^{\frac{d}{4}}&\mbox{(if $d$
is odd)}\end{cases}.$ (3.28)

However, with the form of Eq. (3.23), both cases can be analysed
simultaneously and if the bulk field has mass, which we will face on in Sec.
4, the mode function ${\mathcal{R}}$ has a generic form of Eq. (3.23).
Therefore, we use Eq. (3.23), although it can be expressed with the simple
form (3.28). For $l=\mu-n$ where $n$ is any positive integer, Eq. (3.28)
becomes zero. This is because, in the definition of associated Legendre
functions, there exists a constant factor which goes to zero for $l\to\mu-n$.
To avoid this situation, we use $P^{-\mu}_{l}$. Mode function ${\mathcal{R}}$
expressed in Eq. (3.23) with the boundary condition (3.27) is proportional to
$P^{-\mu}_{l}$ for general $\mu$, not only for $\mu=d/2$. For $l=\mu-n$, its
proportional constant becomes zero, but $P^{-\mu}_{l}$ is non-zero and a bulk
solution satisfying the Dirichlet boundary condition on AdS boundary.
Therefore, $P^{-\mu}_{l}$ is a unified expression which can be used for any
$d$ and any $l$. Hence, we write the bulk mode function as

$\displaystyle\qquad\qquad{\mathcal{R}}\,=\,B_{0}P^{-\mu}_{l}(\zeta)(1-\zeta^{2})^{\frac{d}{4}}\,,\qquad\mbox{(for\,
any\, $d$)}$ (3.29)

where $B_{0}$ is a normalization constant.

Another boundary, namely junction condition on the brane, will be discussed in
the next section, which specifies the value of $\lambda_{\rho}$.

###  3.3 Junction condition

The boundary condition on the brane, that is, the junction condition (2.17) or
equally

$\displaystyle
K^{\mu}{}_{\nu}-K\delta^{\mu}{}_{\nu}\,\hat{=}\,-\frac{1}{2}\sigma\delta^{\mu}{}_{\nu}$
(3.30)

is required locally. In the derivation of the junction condition for the
metric perturbations, it is better to use the local coordinate based on the
brane called Gaussian normal coordinate. Note that the brane position can be
perturbed, that is, it does not generically exist on $\rho=\rho_{*}$ surface
in the TT gauge, where $\rho_{*}$ is a constant. By the coordinate
transformation $\bar{\rho}=\rho+\varphi(x^{\mu})$ from the TT gauge, we can
set the brane to be on $\bar{\rho}=$ const. surface. This coordinate
transformation generates the non-zero $(\bar{\rho},x^{\mu})$-components of
metric perturbations. Considering the coordinate transformation

$\displaystyle\bar{x}^{\nu}=x^{\nu}+\xi^{\nu},$ (3.31)

we can remove the cross components. Then, the Gaussian normal coordinate
$(\bar{\rho},\bar{x}^{\mu})$ is achieved [101]. The relation between the
metric perturbations $\bar{h}_{MN}$ in the Gaussian normal coordinate and
$h_{MN}$ in TT gauge is

$\displaystyle\bar{h}_{MN}=h_{MN}+\nabla_{M}\xi_{N}+\nabla_{M}\xi_{N},$ (3.32)

where $\xi^{\rho}=\varphi$. The Gaussian normal coordinate is chosen such that
the $(\rho\rho)$ and $(\rho\mu)$ components of the metric perturbation are
zero, as with the TT gauge condition. The former

$\displaystyle\bar{h}_{\rho\rho}=h_{\rho\rho}(=0)$ (3.33)

gives

$\displaystyle\partial_{\rho}\xi_{\rho}=0.$ (3.34)

Therefore, $\xi_{\rho}$ depends only on $x^{\mu}$, which is consistent with

$\displaystyle\xi^{\rho}=\xi_{\rho}=\varphi(x^{\mu}).$ (3.35)

The latter one, namely vanishing of $(\rho\mu)$ component, means

$\displaystyle\bar{h}_{\rho\mu}=h_{\rho\mu}(=0).$ (3.36)

This gives

$\displaystyle
0=\nabla_{\rho}\xi_{\mu}+\nabla_{\mu}\xi_{\rho}=\partial_{\rho}\xi_{\mu}+\partial_{\mu}\xi_{\rho}-2\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}\xi_{\mu}.$
(3.37)

The solution for $\xi^{\mu}(=g^{(0)\mu M}\xi_{M})$ is

$\displaystyle\xi^{\mu}=-\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}\gamma^{\mu\nu}\partial_{\nu}\varphi+\chi^{\mu}(x^{\alpha})$
(3.38)

where $\chi^{\mu}$ is an integration constant.Then, the relation in the
$(\mu\nu)$ component is

$\displaystyle\bar{h}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
h_{\mu\nu}+\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}$ (3.39)
$\displaystyle=$ $\displaystyle
h_{\mu\nu}+2\mathfrak{a}\mathfrak{a}^{\prime}\left(\gamma_{\mu\nu}\varphi-\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}\varphi\right)+2\mathfrak{a}^{2}\tilde{\nabla}_{(\mu}\left(\gamma_{\nu)\alpha}\chi^{\alpha}\right).$

The junction condition (3.30) for the leading order gives

$\displaystyle\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}\,\hat{=}\,-\frac{\sigma}{2(d-1)},$
(3.40)

where $\sigma$ is the brane tension. This fixes the background position of
brane, which is consistent with Eq. (3.5).

The perturbed junction condition in the Gaussian normal coordinate is written
as

$\displaystyle\frac{1}{2}\bar{h}_{\mu\nu}^{\prime}\,\hat{=}\,-\frac{\sigma}{2(d-1)}\bar{h}_{\mu\nu}.$
(3.41)

Substituting the background equation (3.40) into the above, we have

$\displaystyle\bar{h}_{\mu\nu}^{\prime}-2\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}\bar{h}_{\mu\nu}\,\hat{=}\,0.$
(3.42)

From the relation between the metric perturbations in the Gaussian normal
coordinate and TT gauge (3.39), the perturbed junction condition for the TT
gauge is obtained as

$\displaystyle
h_{\mu\nu}^{\prime}-2\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}h_{\mu\nu}\,\hat{=}\,\frac{2}{l^{2}}\left(\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}\varphi-\gamma_{\mu\nu}\varphi\right).$
(3.43)

The trace of this equation gives

$\displaystyle\left(\tilde{\Box}-d\right)\varphi\,\hat{=}\,0$ (3.44)

and the traceless part becomes

$\displaystyle
h_{\mu\nu}^{\prime}-2\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}h_{\mu\nu}\,\hat{=}\,2\left(\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}-\frac{1}{d}\tilde{\Box}\gamma_{\mu\nu}\right)\varphi.$
(3.45)

This boundary condition says that the brane bending $\varphi$ excites a metric
perturbation in the bulk. This mode is called a brane bending mode.

Let us analyze the brane bending mode first. Note that divergence of the right
hand side in Eq. (3.45) becomes

$\displaystyle\tilde{\nabla}^{\mu}\left(\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}-\frac{1}{d}\tilde{\Box}\gamma_{\mu\nu}\right)\varphi$
$\displaystyle=$
$\displaystyle\left(\frac{d-1}{d}\tilde{\nabla}_{\nu}\tilde{\Box}+\tilde{R}_{\nu}{}^{\alpha}\tilde{\nabla}_{\alpha}\right)\varphi$
(3.46) $\displaystyle=$
$\displaystyle\frac{d-1}{d}\tilde{\nabla}_{\nu}\left(\tilde{\Box}-d\right)\varphi.$

With Eq. (3.44), Eq. (3.46) shows that the divergence of the right hand side
of Eq. (3.45) becomes zero. Moreover, the right hand side of Eq. (3.45) is
traceless. Hence, it is a transverse-traceless tensor:

$\displaystyle
Y^{(\varphi)}_{\mu\nu}=\left(\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}-\frac{1}{d}\tilde{\Box}\gamma_{\mu\nu}\right)\varphi$
(3.47)

satisfies

$\displaystyle\tilde{\nabla}^{\mu}Y^{(\varphi)}_{\mu\nu}=0,\qquad
Y^{(\varphi)}_{\mu}{}^{\mu}=0.$ (3.48)

This means that a scalar mode with $m^{2}=d$ can generate spin-2 (transverse-
traceless 2-tensor)444 Then, helicity-0 mode of spin-2 can be expressed by
scalar as Eq. (3.48). This property is used in partially massless gravity
discussed in cosmology [102]. The corresponding mass of spin-2 in
$4$-dimensional de Sitter is “2”, which coincides with Higuchi mass.  [103].
The corresponding mass in spin-2 field can be understood from the following
calculation;

$\displaystyle\left(\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}-\frac{1}{d}\tilde{\Box}\gamma_{\mu\nu}\right)\left(\tilde{\Box}-d\right)\varphi$
$\displaystyle=$
$\displaystyle\left(\tilde{\Box}+2+(d-2)\right)\left(\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}-\frac{1}{d}\tilde{\Box}\gamma_{\mu\nu}\right)\varphi$
(3.49) $\displaystyle=$
$\displaystyle\left(\tilde{\Box}+2+(d-2)\right)Y^{(\varphi)}_{\mu\nu},$

that is, the corresponding mass for spin-2 is given by

$\displaystyle m_{2}^{2}=\lambda_{\rho}=-(d-2).$ (3.50)

Therefore, for the brane bending mode, we have $l=\mu-2$. Since the
$d$-dimensional eigenfunction $Y_{\mu\nu}$ of the brane bending mode is fixed
by $Y_{\mu\nu}^{(\varphi)}$, the mode decomposition corresponding to the brane
bending is written as

$\displaystyle
h^{\varphi}_{\mu\nu}=\mathfrak{a}^{2}{\mathcal{R}}^{(\varphi)}Y_{\mu\nu}^{(\varphi)}=\mathfrak{a}^{2}{\mathcal{R}}^{(\varphi)}\left(\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}-\frac{1}{d}\tilde{\Box}\gamma_{\mu\nu}\right)\varphi.$
(3.51)

Then, the junction condition (3.45) gives

$\displaystyle{{\mathcal{R}}^{(\varphi)}}^{\prime}\,\hat{=}\,2.$ (3.52)

We will see that there always exists a mode satisfying this junction
condition.

Since the perturbative analysis here is linear, any solution is written with
superposition of homogeneous solutions and non-homogeneous solutions. The
source generating non-homogeneous solutions is only the brane bending. Let us
next derive the homogeneous solutions. With the brane position fixed
($\varphi=0$), Eq. (3.43) becomes

$\displaystyle
h_{\mu\nu}^{\prime}-2\frac{\mathfrak{a}^{\prime}}{\mathfrak{a}}h_{\mu\nu}\,\hat{=}\,0.$
(3.53)

The above equation (3.53) means

$\displaystyle{\mathcal{R}}^{\prime}\,\hat{=}\,0,$ (3.54)

that is, the Neumann boundary condition is imposed for ${\mathcal{R}}$ on the
brane. With the form of Eq. (3.29) and an identity for the associated Legendre
function, Eq. (3.54) is written as

$\displaystyle 0$ $\displaystyle\hat{=}$
$\displaystyle\frac{d}{d\zeta}\left(1-\zeta^{2}\right)^{\frac{d}{4}}P^{-\mu}_{l}(\zeta)$
(3.55) $\displaystyle=$
$\displaystyle-\frac{d}{2}\left(1-\zeta^{2}\right)^{\frac{d}{4}-1}\zeta
P^{-\mu}_{l}(\zeta)+\left(1-\zeta^{2}\right)^{\frac{d}{4}}\frac{d}{d\zeta}P^{-\mu}_{l}(\zeta)$
$\displaystyle=$
$\displaystyle-\left(1-\zeta^{2}\right)^{\frac{d-2}{4}}P^{-\mu+1}_{l}(\zeta).$

We suppose that the brane exists at $\zeta=\zeta_{*}$ and then, the Neumann
boundary condition (3.55) gives vanishing of $P^{-\mu+1}_{l}(\zeta)$ there,
which is written as

$\displaystyle
0=P^{-\mu+1}_{l}(\zeta_{*})=\frac{\sqrt{2}\left(\sin\theta\right)^{-(\mu-1)}}{\sqrt{\pi}{\Gamma\left({\mu-\frac{1}{2}}\right)}}\int_{0}^{\theta}\frac{\cos\left(\left(l+\frac{1}{2}\right)\psi\right)}{\left(\cos\psi-\cos\theta\right)^{\frac{3}{2}-\mu}}d\psi,$
(3.56)

where $\cos\theta:=\zeta_{*}=\tanh\rho_{*}$. This equation is generically
difficult to solve for $l$, but in the case with $d=3$ it is solvable. We will
analyse the case with $d=3$ in Sec. 3.6.

Let us go back to the junction condition for the brane bending mode (3.52),
which is written as

$\displaystyle
2\,\hat{=}\,{{\mathcal{R}}^{(\varphi)}}^{\prime}=-B_{1}\left(1-\zeta^{2}\right)^{\frac{d-2}{4}}P^{-\mu+1}_{\mu-2}(\zeta).$
(3.57)

Therefore, if $P^{-\mu+1}_{\mu-2}(\zeta)$ is nonzero for any $\zeta$, we have
a constant $B_{1}$ satisfying the above equation wherever the brane is.
Moreover, other modes with the same $\lambda_{\rho}$ as brane bending mode
never exist, because then junction condition (3.56) is not satisfied.
$P^{-\mu+1}_{l}(\zeta)$ is

$\displaystyle
P^{-\mu+1}_{l}(\zeta)=\frac{1}{\Gamma(\mu)}\left(\frac{1+\zeta}{1-\zeta}\right)^{\frac{1-\mu}{2}}F\left(-l,l+1,\mu;\frac{1-\zeta}{2}\right),$
(3.58)

where $F$ is the Gauss hypergeometric function. Euler’s transformation gives

$\displaystyle
F\left(-l,l+1,\mu;\frac{1-\zeta}{2}\right)=\left(\frac{1+\zeta}{2}\right)^{\mu-1}F\left(\mu+l,\mu-l-1,\mu;\frac{1-\zeta}{2}\right)$
$\displaystyle\hskip
14.22636pt=\left(\frac{1+\zeta}{2}\right)^{\mu-1}\frac{\Gamma(\mu)}{\Gamma(\mu+l)\Gamma(\mu-l-1)}\sum_{n=0}^{\infty}\frac{\Gamma(\mu+l+n)\Gamma(\mu-l-1+n)}{\Gamma(\mu+n)}\frac{(1-\zeta)^{n}}{2^{n}n!}.$
(3.59)

For $-1<\zeta<1$, if $l$ satisfies $\mu-l-1>0$, each term in the sum is
positive. Therefore, Eq. (3.59) does not have zero. This leads to the
positivity (that is, nonzero) of $P^{-\mu+1}_{l}(\zeta)$. For $\mu-l-1=0$,
$n=0$ term in the sum diverges but the divergence is canceled with the factor
$1/\Gamma(\mu-l-1)$. Then, $n=0$ term becomes positive and other terms in the
sum are also. Therefore, for $\mu-l-1=0$, $P^{-\mu+1}_{l}(\zeta)$ is positive.
This means for modes satisfying the junction condition (3.54), $l$ is smaller
larger than $\mu-1$, which gives

$\displaystyle\lambda_{\rho}>0,\ \ \ \ \nu>\frac{d-1}{2}.$ (3.60)

For the brane bending mode, $l$ is $\mu-2$, which satisfies $\mu-l-1\geq 0$.
Thus, the brane bending mode exists wherever the brane is, and no other modes
with the same $\lambda_{\rho}$ of the brane bending mode exist.

###  3.4 Massive tensor on AdS boundary

Here, we discuss the tensor $Y^{(i)}_{\mu\nu}$, which is a symmetric tensor on
the $d$-dimensional AdS spacetime

$\displaystyle d\tilde{s}^{2}$ $\displaystyle=$
$\displaystyle\gamma_{\mu\nu}dx^{\mu}dx^{\nu}$ (3.61) $\displaystyle=$
$\displaystyle\frac{1}{y^{2}}\left(dy^{2}+\eta_{ij}dq^{i}dq^{j}\right),$

and satisfies Eq. (3.19) and the transverse-traceless conditions

$\displaystyle\tilde{\nabla}^{\mu}Y^{(i)}_{\mu\nu}=0,\qquad
Y^{(i)\mu}{}_{\mu}=0,$ (3.62)

which can be written as

$\displaystyle\partial^{i}Y_{iy}=-\partial_{y}Y_{yy}+\frac{d-2}{y}Y_{yy},$
(3.63)
$\displaystyle\partial^{j}Y_{ji}=-\partial_{y}Y_{yi}+\frac{d-2}{y}Y_{yi},$
(3.64) $\displaystyle Y^{i}{}_{i}=-Y_{yy}.$ (3.65)

With these conditions, Eq. (3.19) is written as

$\displaystyle\Box_{0}Y_{yy}-\frac{d-2}{y}\partial_{y}Y_{yy}-\frac{\lambda_{\rho}}{y^{2}}Y_{yy}=0,$
(3.66)
$\displaystyle\Box_{0}Y_{yi}-\frac{d-4}{y}\partial_{y}Y_{yi}-\frac{d-2+\lambda_{\rho}}{y^{2}}Y_{yi}-\frac{2}{y}\partial_{i}Y_{yy}=0,$
(3.67)
$\displaystyle\Box_{0}Y_{ij}-\frac{d-6}{y}\partial_{y}Y_{ij}-\frac{2d-6+\lambda_{\rho}}{y^{2}}Y_{ij}-\frac{2}{y}\left(\partial_{i}Y_{yj}+\partial_{j}Y_{yi}\right)+\frac{2}{y^{2}}Y_{yy}\eta_{ij}=0,$
(3.68)

where $\Box_{0}$ is the D’Alembertian of the flat metric, that is,

$\displaystyle\Box_{0}:=\partial_{y}^{2}+\partial_{i}\partial^{i}.$ (3.69)

By virtue of the symmetry in the $(d-1)$-dimensional spacetime labeled by
$(i,j,\cdots)$, these dynamical degrees of freedom are decomposed by helicity.
$Y_{yy}$ is expressed only by helicity-0. $Y_{yi}$ and $Y_{ij}$ are decomposed
as

$\displaystyle Y_{yi}=\partial_{i}\phi+V_{i},$ (3.70) $\displaystyle
Y_{ij}=\partial_{i}\partial_{j}\psi+\eta_{ij}\chi+\partial_{i}W_{j}+\partial_{j}W_{i}+M_{ij}$
(3.71)

with

$\displaystyle\partial^{i}V_{i}=\partial^{i}W_{i}=0,\qquad\partial^{i}M_{ij}=0,\qquad
M^{i}{}_{i}=0.$ (3.72)

Here, $\phi$, $\psi$ and $\chi$ are helicity-0 modes, $V_{i}$ and $W_{i}$ are
helicity-1 modes, and $M_{ij}$ is a helicity-2 mode. Equations (3.66)-(3.68)
can be solved separately for each helicity. The transverse-traceless
conditions (3.63)-(3.65) give the relations among $Y_{yy}$, $\phi$, $\psi$ and
$\chi$ and between $V_{i}$ and $W_{i}$, and we will see that each number of
dynamical helicity-0, 1 1nd 2 modes is one. Let us derive the solution for
each helicity.

#### Helicity-0

Since a helicity-0 mode includes $Y_{yy}$ component, we first solve Eq.
(3.66). The solution is obtained as

$\displaystyle
Y_{yy}^{(0)}=y^{\frac{d-1}{2}}\left[b_{0}J_{\nu}(\sqrt{-\boldsymbol{k}^{2}}y)+c_{0}Y_{\nu}(\sqrt{-\boldsymbol{k}^{2}}y)\right]\left[e^{i\boldsymbol{k\cdot
x}}+\mbox{c.c.}\right],$ (3.73)

where $b_{0}$ and $c_{0}$ are constants and $\nu$ is defined in Eq. (3.24).
The decaying solution would be chosen, that is, we set $c_{0}$ to be zero. The
helicity-0 component of $Y_{yi}$ is expressed only by $\phi$ and it is
obtained by Eq. (3.63) as

$\displaystyle
Y_{yi}^{(0)}=\frac{\partial_{i}}{\boldsymbol{k}^{2}}\left(\partial_{y}Y_{yy}^{(0)}-\frac{d-2}{y}Y_{yy}^{(0)}\right).$
(3.74)

Similarly, $Y_{ij}$ components are obtained from Eqs. (3.64) and (3.65) as

$\displaystyle
Y_{ij}^{(0)}=\frac{1}{\boldsymbol{k}^{2}}\partial_{i}\partial_{j}Y_{yy}^{(0)}+\frac{1}{(\boldsymbol{k}^{2})^{2}}\left((d-1)\partial_{i}\partial_{j}+\boldsymbol{k}^{2}\eta_{ij}\right)$
$\displaystyle\hskip
85.35826pt\times\left[-\frac{1}{y}\partial_{y}Y_{yy}^{(0)}+\left((d-1)+\frac{\lambda_{\rho}}{d-2}\right)\frac{1}{y^{2}}Y_{yy}^{(0)}\right].$
(3.75)

These solution (3.74) and (3.75) are consistent with Eqs. (3.67) and (3.68).

The asymptotic behavior of $Y_{yy}$ in the boundary limit $y\to 0$ becomes

$\displaystyle
Y_{yy}^{(0)}=b_{0}y^{\frac{d-1}{2}}J_{\nu}(\sqrt{-\boldsymbol{k}^{2}}y)\left[e^{i\boldsymbol{k\cdot
x}}+\mbox{c.c.}\right]\sim y^{\frac{d-1}{2}+\nu}.$ (3.76)

Then, the asymptotic behavior of other components are estimated as555 In this
estimate, nontrivial cancellation is not taken into account. We will see soon
later that, with a special value of $\lambda_{\rho}$, the cancellation occurs.

$\displaystyle Y_{yi}^{(0)}\sim y^{\frac{d-3}{2}+\nu},\qquad
Y_{ij(T)}^{(0)}\sim y^{\frac{d-1}{2}+\nu},\qquad Y_{(ij)}^{(0)}\sim
y^{\frac{d-5}{2}+\nu},$ (3.77)

where $Y_{ij(T)}^{(0)}$ and $Y_{(ij)}^{(0)}$ are the trace and traceless parts
of $Y_{ij}^{(0)}$, respectively.

#### Helicity-1

The helicity-1 mode has no $Y_{yy}$ component and thus Eq. (3.66) is trivially
satisfied. Equation (3.67) gives a solution

$\displaystyle
Y_{yi}^{(1)}=y^{\frac{d-3}{2}}\left[b_{1}J_{\nu}(\sqrt{-\boldsymbol{k}^{2}}y)+c_{1}Y_{\nu}(\sqrt{-\boldsymbol{k}^{2}}y)\right]\left[e^{i\boldsymbol{k\cdot
x}}+\mbox{c.c.}\right]E_{i}(\boldsymbol{k},s),$ (3.78)

where $b_{1}$ and $c_{1}$ are constants and $E_{i}(\boldsymbol{k},s)$ is basis
vectors satisfying

$\displaystyle\boldsymbol{k}^{i}E_{i}(\boldsymbol{k},s)=0,\qquad
E^{i}(\boldsymbol{k},s)E_{i}(\boldsymbol{k},s^{\prime})=\delta^{ss^{\prime}}.$
(3.79)

Here, $s$ is the label representing the different basis vectors. The number of
its degrees of freedom is $(d-2)$ because the index $i$ is for
$(d-1)$-dimensional space and we have the transverse constraint, the first
equation of Eq. (3.79). It may be required to choose the decaying mode, that
is, $c_{1}=0$. The helicity-1 mode of $Y_{ij}$ components is obtained from Eq.
(3.64) as

$\displaystyle
Y_{ij}^{(1)}=\frac{1}{\boldsymbol{k}^{2}}\left(\partial_{y}-\frac{d-2}{y}\right)\left(\partial_{i}Y_{yj}^{(1)}+\partial_{j}Y_{yi}^{(1)}\right).$
(3.80)

This solution satisfies Eq. (3.68).

The asymptotic behavior of each component in the limit $y\to 0$ is estimated
as

$\displaystyle Y_{yi}^{(1)}\sim y^{\frac{d-3}{2}+\nu},\qquad Y_{ij}^{(1)}\sim
y^{\frac{d-5}{2}+\nu}.$ (3.81)

#### Helicity-2

For helicity-2 modes,666 Note that, for $d=3$, helicity-2 modes does not
exist.  $Y_{yy}$ and $Y_{yi}$ components vanish. Only Eq. (3.68) is nontrivial
and it gives

$\displaystyle
Y_{ij}^{(2)}=y^{\frac{d-5}{2}}\left[b_{2}J_{\nu}(\sqrt{-\boldsymbol{k}^{2}}y)+c_{2}Y_{\nu}(\sqrt{-\boldsymbol{k}^{2}}y)\right]\left[e^{i\boldsymbol{k\cdot
x}}+\mbox{c.c.}\right]E_{ij}(\boldsymbol{k},s),$ (3.82)

where $b_{1}$ and $c_{1}$ are constants, and $E_{i}(\boldsymbol{k},s)$ is
basis of symmetric tensor satisfying

$\displaystyle\boldsymbol{k}^{i}E_{ij}(\boldsymbol{k},s)=0,\qquad
E^{i}{}_{i}(\boldsymbol{k},s)=0,\qquad
E^{ij}(\boldsymbol{k},s)E_{ij}(\boldsymbol{k},s^{\prime})=\delta^{ss^{\prime}}.$
(3.83)

Here, $s$ is the label for different basis of helicity-2 modes. The number of
its degrees of freedom can be counted as follows; $E_{ij}$ is a symmetric
tensor in the $(d-1)$ dimensional spacetime. We have the transverse and
traceless conditions, which are the first and the second equations of Eq.
(3.83), respectively. Then the dynamical number of degrees of freedom for a
helicity-2 mode is $d(d-1)/2-(d-1)-1=d(d-3)/2$. The decaying modes would be
chosen, that is, $c_{2}=0$, and its asymptotic behavior is

$\displaystyle Y_{ij}^{(2)}\sim y^{\frac{d-5}{2}+\nu}.$ (3.84)

The degrees of freedom that we obtained above is 1 for helicity-0, $(d-2)$ for
helicity-1 and $d(d-3)/2$ for helicity-2. The total is $(d+1)(d-2)/2$, which
is consistent with the number of degrees of freedom for $d$-dimensional
transverse-traceless tensor, that is, $(d+1)d/2-d-1$ where $(d+1)d/2$ is the
number of components of symmetric 2-tensor and $d$ and 1 correspond to the
transverse and traceless conditions.

From the estimates (3.76) and (3.77), if $Y_{yy}$ component exists, the
asymptotic behavior of $Y_{yi}$ seems to be one order lower than that of
$Y_{yy}$. However, there is an exception; we can see from Eq. (3.63) that, if
$Y_{yy}$ asymptotically behave as $Y_{yy}\sim y^{d-2}$, the corresponding
contribution in the right hand side of Eq. (3.63) is cancelled and the
asymptotic behavior of $Y_{yi}$ becomes lower than the naive estimate through
Eq. (3.77). Indeed, it occurs in the case of

$\displaystyle\frac{d-1}{2}+\nu=d-2\quad\Leftrightarrow\quad\lambda_{\rho}=-(d-2).$
(3.85)

This value of $\lambda_{\rho}$ corresponds to that of the brane bending mode
(3.50). As explained in the previous section, the brane bending mode produces
a transeverse-traceless mode even though it is a scalar mode. It appears as a
helicity-0 mode of spin-2 and it does not excite $Y_{yi}$ and $Y_{yy}$ much,
because it is originally scalar.

Let us see that the brane bending mode corresponds to the helicity-0 mode of
spin-2 with $\lambda_{\rho}=-(d-2)$. The brane bending mode $\varphi$
satisfies Eq. (3.44), a solution of which is written as

$\displaystyle\varphi=y^{\frac{d-1}{2}}\left[b_{\varphi}J_{\frac{d+1}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)+c_{\varphi}Y_{\frac{d+1}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)\right]\left[e^{i\boldsymbol{k\cdot
x}}+\mbox{c.c.}\right],$ (3.86)

where $b_{\varphi}$ and $c_{\varphi}$ are constants. The transverse-traceless
tensor generating with $\varphi$ is written as

$\displaystyle Y^{(\varphi)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}-\frac{1}{d}\tilde{\Box}\gamma_{\mu\nu}\right)\varphi$
(3.87) $\displaystyle=$
$\displaystyle\left(\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}-\gamma_{\mu\nu}\right)\varphi,$

where we used Eq. (3.44). Its $(yy)$-component becomes

$\displaystyle Y^{(\varphi)}_{yy}$ $\displaystyle=$
$\displaystyle\left(\tilde{\nabla}_{y}\tilde{\nabla}_{y}-\gamma_{yy}\right)\varphi$
(3.88) $\displaystyle=$
$\displaystyle\left(\partial_{y}^{2}+\frac{1}{y}\partial_{y}-\frac{1}{y^{2}}\right)y^{\frac{d-1}{2}}\left[b_{\varphi}J_{\frac{d+1}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)+c_{\varphi}Y_{\frac{d+1}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)\right]\left[e^{i\boldsymbol{k\cdot
x}}+\mbox{c.c.}\right]$ $\displaystyle=$
$\displaystyle-\boldsymbol{k}^{2}y^{\frac{d-1}{2}}\left[b_{\varphi}J_{\frac{d-3}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)+c_{\varphi}Y_{\frac{d-3}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)\right]\left[e^{i\boldsymbol{k\cdot
x}}+\mbox{c.c.}\right].$

One can see that this corresponds to Eq. (3.73) with $\lambda_{\rho}=-(d-2)$.
Due to the transverse-traceless conditions, other components are uniquely
fixed, and thus, the brane bending mode is confirmed to be the helicity-0 mode
with $\lambda_{\rho}=-(d-2)$. This behaves $y^{d-2}$ in the limit $y\to 0$.

To see the asymptotic behavior of other components, we derive their exact
forms. $(yi)$-component is obtained as

$\displaystyle Y^{(\varphi)}_{yi}$ $\displaystyle=$
$\displaystyle\tilde{\nabla}_{y}\tilde{\nabla}_{i}\varphi$ (3.89)
$\displaystyle=$
$\displaystyle\partial_{i}\left(\partial_{y}+\frac{1}{y}\right)y^{\frac{d-1}{2}}\left[b_{\varphi}J_{\frac{d+1}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)+c_{\varphi}Y_{\frac{d+1}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)\right]\left[e^{i\boldsymbol{k\cdot
x}}+\mbox{c.c.}\right]$ $\displaystyle=$
$\displaystyle\sqrt{-\boldsymbol{k}^{2}}y^{\frac{d-1}{2}}\left[b_{\varphi}J_{\frac{d-1}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)+c_{\varphi}Y_{\frac{d-1}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)\right]\partial_{i}\left[e^{i\boldsymbol{k\cdot
x}}+\mbox{c.c.}\right].$

This behaves $y^{d-1}$ in the limit $y\to 0$, which is one order higher than
that of $Y^{(\varphi)}_{yy}$, that is, $Y^{(\varphi)}_{yi}$ decays faster than
$Y^{(\varphi)}_{yy}$. The trace part of $(ij)$-component can be easily
obtained from the traceless condition (3.65), which asymptotic order is the
same as that of $Y^{(\varphi)}_{yy}$. The traceless part of $(ij)$-component
is calculated as

$\displaystyle Y^{(\varphi)}_{(ij)}$ $\displaystyle=$
$\displaystyle\left(\tilde{\nabla}_{i}\tilde{\nabla}_{j}-\frac{1}{d-1}\eta_{ij}\eta^{kl}\tilde{\nabla}_{k}\tilde{\nabla}_{l}\right)\varphi$
(3.90) $\displaystyle=$ $\displaystyle
y^{\frac{d-1}{2}}\left[b_{\varphi}J_{\frac{d+1}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)+c_{\varphi}Y_{\frac{d+1}{2}}(\sqrt{-\boldsymbol{k}^{2}}y)\right]$
$\displaystyle\hskip
56.9055pt\times\left(\partial_{i}\partial_{j}-\frac{1}{d-1}\eta_{ij}\eta^{kl}\partial_{k}\partial_{l}\right)\left[e^{i\boldsymbol{k\cdot
x}}+\mbox{c.c.}\right].$

This behaves $y^{d}$ in the limit $y\to 0$.

###  3.5 Holographic stress energy tensor in AdS/BCFT

Now we would like to study the behavior of holographic stress tensor in the
light of the analysis of metric perturbations in the previous subsections. The
holographic stress energy tensor in AdS${}_{d+1}/$BCFTd can be computed as

$\displaystyle{\cal T}_{ab}=\frac{d}{16\pi G_{N}}\lim_{z\to
0}g_{ab}(x,w,z)z^{2-d},$ (3.91)

where $g_{ab}$ is the metric perturbation in the asymptotically Poincare AdS
space (2.14) and $x$ denotes $d-1$ dimensional coordinate
$(x_{0},x_{1},\cdots,x_{d-2})$, such that
$(\xi_{0},\cdots,\xi_{d-2},\xi_{d-1})=(x_{0},\cdots,x_{d-2},w)$. Note that
this holographic stress tensor satisfies the traceless condition and momentum
conservation (2.12) owing to the Einstein equation [93, 94].

Then the holographic stress tensor can also be evaluated from the hyperbolic
slice metric by using the coordinate transformation (3.2) as follows

$\displaystyle{\cal T}_{ab}=\frac{d}{16\pi
G_{N}}\lim_{\rho\to\infty}\left[\left(\frac{e^{\rho}}{2y}\right)^{d-2}h_{ab}(x,y)\right]\Biggr{|}_{y=w},$
(3.92)

where we identify $\xi_{d-1}(=w)$ with $y$ because we take $\rho\to\infty$
limit. As we performed in the previous subsections, the metric perturbation
$h_{\mu\nu}$ can be computed by imposing the Dirichlet boundary condition
$h_{\mu\nu}=O(e^{-\rho})$ in the boundary limit $\rho\to\infty$ and the
Neumann boundary condition (2.17) on the EOW brane.

For $d=2$, i.e. AdS${}_{3}/$BCFT2, the only possible metric perturbation is
the helicity $0$ mode with $\lambda_{\rho}=0$, which is the brane bending
mode. This leads to the $O(1)$ metric perturbation in the AdS boundary limit
$z\to 0$. This corresponds to the perturbations proportional to ${\cal
T}_{++}$ and ${\cal T}_{--}$ in the asymptotically AdS3 metric (2.22). Thus,
the present metric perturbation analysis simply reproduces the result in
section 2.3.

In higher dimensions $d\geq 3$, there are helicity 0, 1 and 2 modes as we have
seen. (Helicity 2 mode does not appear for $d=3$.) In the BCFT boundary limit
$y\to 0$, these modes in general behave as (note $i,j=0,1,\cdots,d-2$)

$\displaystyle\mbox{helicity 0}:\ \ Y_{yy}=O(y^{\frac{d-1}{2}+\nu}),\ \ \
Y_{yi}=O(y^{\frac{d-3}{2}+\nu}),\ \ \ Y_{(ij)}=O(y^{\frac{d-5}{2}+\nu}),$
$\displaystyle\mbox{helicity 1}:\ \ Y_{yy}=0,\ \ \
Y_{yi}=O(y^{\frac{d-3}{2}+\nu}),\ \ \ Y_{(ij)}=O(y^{\frac{d-5}{2}+\nu}),$
$\displaystyle\mbox{helicity 2}:\ \ Y_{yy}=0,\ \ \ Y_{yi}=0,\ \ \
Y_{(ij)}=O(y^{\frac{d-5}{2}+\nu}),$ (3.93)

where $\nu=\sqrt{\frac{(d-1)^{2}}{4}+\lambda_{\rho}}$. The indices $(ij)$
denotes the traceless symmetric part. By imposing the boundary conditions, the
value of $\lambda_{\rho}$ gets quantized.

As we have noted before, we only allow the values of $\nu$ which satisfy the
previous constraint (3.60) with one exception, namely the brane bending mode.
In the helicity 0 perturbation, the lowest mode $\lambda_{\rho}=-(d-2)$, which
is the brane bending mode, anomalously appears and this shows the following
peculiar scaling different from (3.5):

$\displaystyle Y_{yy}=O(y^{d-2}),\ \ \ Y_{yi}=O(y^{d-1}),\ \ \
Y_{(ij)}=O(y^{d}).$ (3.94)

Thus in this lowest mode, both ${\cal T}_{yy}$ and the trace part of ${\cal
T}_{ij}$ take finite values, while we have ${\cal T}_{wi}={\cal T}_{(ij)}=0$
at the BCFT boundary $w=0$. Notice that this is the only mode which gives non-
vanishing ${\cal T}_{yy}$ and the trace part of ${\cal T}_{ij}$.

For higher modes of the helicity 0 perturbation follows the standard scaling
(3.5). This leads to the following behavior of stress tensor near the boundary
$w\to 0$:

$\displaystyle{\cal T}_{ww}=O(w^{-\frac{d-3}{2}+\nu}),\ \ \ {\cal
T}_{wi}=O(w^{-\frac{d-1}{2}+\nu}),\ \ \ {\cal
T}_{ij}=O(w^{-\frac{d+1}{2}+\nu}).$ (3.95)

Owing to the constraint (3.60), the boundary condition $T_{wi}=0$ at $w=0$ is
always satisfied as expected for BCFTs.

Similarly, we can read off the stress energy tensor for the helicity 1 and 2
perturbations. Again we find the behavior $T_{wi}=0$ at $w=0$ due to the
constraint (3.60). In this way, we can confirm the expected boundary condition
in BCFTs (2.12), which basically argues the complete reflection at the
boundary of a given BCFT on a half plane $w>0$. Notice also that in the bulk
of BCFT, there are $\frac{(d-2)(d+1)}{2}$ independent components of stress
tensor after we imposed the conservation and traceless condition. This agrees
with the number of independent metric perturbations.

###  3.6 Case with $d=3$

As an explicit example, below we focus on $d=3$ and study the perturbation
modes in detail. The helicity 2 mode is absent at $d=3$.

For $d=3$, $\mu$ is $3/2$ and thus the denominator of the integrand of Eq.
(3.56) becomes unity and it becomes

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\left(\sin\theta\right)^{-\frac{1}{2}}}{\sqrt{\pi}{\Gamma\left({1}\right)}}\int_{0}^{\theta}\cos\left(\left(l+\frac{1}{2}\right)\psi\right)d\psi$
(3.96) $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\left(\sin\theta\right)^{-\frac{1}{2}}\sin\left(\left(l+\frac{1}{2}\right)\theta\right)}{\sqrt{\pi}\left(l+\frac{1}{2}\right)}$

and the solution for $l$ is

$\displaystyle l=\frac{\pi}{\theta}a-\frac{1}{2}\qquad(a\in\mathbb{N}).$
(3.97)

(Here we use the definition that $\mathbb{N}$ does not include zero.) From Eq.
(3.24), $\lambda_{\rho}$ is written as

$\displaystyle\left(m_{KK}^{2}=\right)\lambda_{\rho}=\left(\frac{2l+1}{2}\right)^{2}-1=\left(\frac{\pi}{\theta}a\right)^{2}-1.$
(3.98)

Since $\theta$ takes a value in the range $0<\theta<\pi$, $\lambda_{\rho}$ is
positive, which is consistent with Eq. (3.60).

The previous analysis leads to the expression of perturbative solutions:

$\displaystyle h_{ab}=\sqrt{\cosh\rho}\cdot
P^{-\frac{3}{2}}_{\sqrt{1+\lambda_{\rho}}-\frac{1}{2}}(\tanh\rho)\cdot
Y_{ab}(\tau,x,y),$ (3.99)

where we set $(x^{0},x^{1},x^{2})=(\tau,x,y)$. Note that the Legendre function
behaves like

$\displaystyle
P^{-\frac{3}{2}}_{\sqrt{1+\lambda_{\rho}}-\frac{1}{2}}(\tanh\rho)=O(e^{-\frac{3\rho}{2}}),\
\ \ (\rho\to\infty),$ (3.100)

so that it fits nicely with the Dirichlet boundary condition. The Neumann
boundary condition leads to the quantization (3.98) in the $\rho$ direction,
which leads to

$\displaystyle\nu=\sqrt{1+\lambda_{\rho}}=\frac{\pi a}{\theta},\ \ \
(a=1,2,\cdots),$ (3.101)

where we also impose the condition (3.60) i.e. $\nu>1$. Here $\theta$, which
takes values in the range $0<\theta<\pi$, is the angle of the EOW brane.
$\sigma>0\ (\rho_{*}<0)$, $\sigma=0\ (\rho_{*}=0)$ and $\sigma<0\
(\rho_{*}>0)$ correspond to $\theta>\frac{\pi}{2}$, $\theta=\frac{\pi}{2}$,
and $\theta<\frac{\pi}{2}$, respectively.

We have the other extra mode, the brane bending mode. For the brane bending
mode, we have $\sqrt{1+\lambda_{\rho}}=0$, which corresponds to $a=0$ case of
Eq. (3.101). Note again that the brane bending mode generates only helicity 0
mode, and helicity 1 modes are absent for $a=0$. Below we study the behavior
of the corresponding stress tensor for each value of $a$.

Let us start with the anomalous mode of the helicity 0 perturbation at $a=0$
or equally $\lambda_{\rho}=-1$, which is the brane bending mode. For any
values of tension or angle $\theta$, we find

$\displaystyle\sqrt{\cosh\rho}\cdot
P^{-\frac{3}{2}}_{-\frac{1}{2}}(\tanh\rho)$
$\displaystyle\propto\left[\cosh\rho\left(1+2\sinh\rho\arctan\left[\tanh\frac{\rho}{2}\right]\right)-\frac{\pi}{2}\sinh\rho\cosh\rho\right]\sim_{\rho\to\infty}e^{-\rho},$
$\displaystyle Y_{yy}\propto y,\ \ \ \ Y_{yi}\propto y^{2},\ \ \ \
Y_{ij(T)}\propto y,\ \ \ \ Y_{(ij)}\propto y^{3},\ \ \ (y\to 0),$ (3.102)

where $Y_{ij(T)}$ shows the trace part of $Y_{ij}$. For this mode, which
corresponds to the brane bending mode, the stress tensor behaves in the
boundary limit $w\to 0$ as follows

$\displaystyle\mbox{Brane bending mode\ ($a=0)$}:{\cal T}_{yy}=\mbox{finite},\
\ \ {\cal T}_{wi}=0,\ \ \ {\cal T}_{ij}=\mbox{finite}.$ (3.103)

In particular, when $Y_{yy}$ does not depend on $x^{i}$, we can set
$Y_{yi}=Y_{(ij)}=0$. In this case the solution describes the constant stress
energy tensor which satisfies $T_{ww}=-2T_{\tau\tau}=-2T_{xx}$ with all other
components vanishing. This corresponds to the $U_{1}$ and $U_{2}$ mode (A.3)
in the appendix, by turning on the $\varphi(x)$ mode in (3.35). As an
illustration, we show the time evolution of a wave packet of the brane bending
mode and corresponding ${\cal T}_{yy}$ in Fig. 5 (see also section 4.2).

For $a\geq 1$, the helicity 0 mode looks like

$\displaystyle\sqrt{\cosh\rho}\cdot P^{-\frac{3}{2}}_{\frac{\pi
a}{\theta}-\frac{1}{2}}(\tanh\rho)\sim_{\rho\to\infty}e^{-\rho},$
$\displaystyle Y_{yy}\propto y^{\frac{\pi a}{\theta}+1},\ \ \ \ Y_{yi}\propto
y^{\frac{\pi a}{\theta}},\ \ \ \ Y_{(ij)}\propto y^{\frac{\pi a}{\theta}-1}.\
\ \ (y\to 0).$ (3.104)

This leads to the following behavior of stress tensor in the boundary limit
$w\to 0$:

$\displaystyle\mbox{helicity 0}:{\cal T}_{ww}\propto w^{\frac{\pi
a}{\theta}},\ \ \ {\cal T}_{wi}\propto w^{\frac{\pi a}{\theta}-1},\ \ \ {\cal
T}_{ij}\propto w^{\frac{\pi a}{\theta}-2}.\ \ \ (a=1,2,3,\cdots).$ (3.105)

We can analyze the helicity 1 similarly. Their stress tensor behaves like

$\displaystyle\mbox{helicity 1}:{\cal T}_{ww}=0,\ \ \ {\cal T}_{wi}\propto
w^{\frac{\pi a}{\theta}-1},\ \ \ {\cal T}_{ij}\propto w^{\frac{\pi
a}{\theta}-2}.\ \ \ $ (3.106)

Note that owing to the condition (3.60) i.e. $\frac{\pi a}{\theta}>1$, we find
$T_{wi}=0$ at the boundary $w=0$ for both the helicity 0 result (3.105) and
the helicity 1 one (3.106), as expected.

It is also intriguing to look into the tensionless case $\sigma=0$ i.e.
$\theta=\frac{\pi}{2}$ specially. In this case, both the helicity 0 mode
(3.105) and helicity 1 one (3.106) at $a=1$ leads to finite values of stress
tensor ${\cal T}_{ij}$ in addition to the brane bending mode (3.103).
Moreover, for any $a$, the power of $w$ of the stress tensor in the limit
$w\to 0$ is always integer at $\sigma=0$. All of the above analysis at $d=3$
perfectly matches with the direct perturbation analysis in appendix A as in
(A.17).

##  4 Scalar field excitations in higher dimensional AdS/BCFT

In the previous section, we have studied metric perturbations in the AdS/BCFT
setup, where the EOW brane plays a very important role. As a next step, we
would like to analyze the dynamical properties of matter fields in the
presence of EOW brane. In this section, as such an example, we study the free
massive scalar fields in AdSd+1 with the EOW brane and their reflection at the
brane.

###  4.1 Massive scalar field

A massive scalar field in higher dimensional AdS/BCFT setting can be treated
in a parallel manner to the gravitational wave discussed in the previous
section. The equation of motion of a massive scalar field on the spacetime
(3.3) is given by

$0=\left(\square-m^{2}\right)\phi(\rho,x^{\mu})=\frac{1}{\cosh^{2}\rho}\left[y^{2}\Box_{0}-(d-2)y\phi_{,y}\right]+\phi_{,\rho\rho}+d\,\tanh\rho\phi_{,\rho}-m^{2}\phi~{},$
(4.1)

where $\Box_{0}$ is the flat D’Alembertian (3.69). This equation may be solved
using separation of variable by writing the general solution as
$\phi={\mathcal{R}}(\rho)Y(x^{\mu})$, where ${\mathcal{R}}(\rho)$ and
$Y(x^{\mu})$ obey

$\displaystyle{\mathcal{R}}_{,\rho\rho}+d\,\tanh\rho{\mathcal{R}}_{,\rho}-m^{2}{\mathcal{R}}=-\frac{\lambda_{\rho}}{\cosh^{2}(\rho)}{\mathcal{R}}~{},$
(4.2)
$\displaystyle\Box_{0}Y-\frac{d-2}{y}\partial_{y}Y-\frac{\lambda_{\rho}}{y^{2}}Y=0~{}.$
(4.3)

The ${\mathcal{R}}$ equation is modified from Eq. (3.18) just by the mass
term, and the $Y$ equation is the same as Eq. (3.66) for $Y_{yy}$.

Because the separated equations (4.2), (4.3) are similar to those for the
gravitational perturbations, their general solutions can be immediately found
from the results in the previous section. Introducing $\zeta=\tanh\rho$ as we
did in Eq. (3.22), we find that the solution to Eq. (4.2) for the bulk
direction $\rho$ satisfying the Dirichlet boundary condition at the AdSd+1
boundary $\zeta\to 1$ is given by Eq. (3.29), where the index $\mu$ of the
associated Legendre functions is promoted to

$\mu=\frac{1}{2}\sqrt{d^{2}+4m^{2}}~{},$ (4.4)

while the other indices $l,\nu$ are the same as those in Eq. (3.24). $\mu$ is
guaranteed to be a real number if the Breitenlohner-Freedman bound
$m^{2}\geq-\frac{1}{4}d^{2}$ is satisfied. Equation (4.3) for the AdSd
direction is the same as Eq. (3.66), then its solution is given by Eq. (3.73).
If we assume the Dirichlet condition at the AdS boundary $y=0$ on each AdSd
slice $\rho=\text{const.}$, the decaying (normalizable) solution with
$c_{0}=0$ will be taken as the solution.

We assume that the scalar field minimally couples with the EOW brane,777Later
in section 6, we will introduce a non-trivial coupling between the scalar
field and the EOW brane so that the dual scalar operator has a non-zero
expectation value. Here we assume that a scalar field whose dual operator has
a vanishing one point function i.e. $a_{0}=0$. and then the scalar field
satisfies the Neumann boundary condition $\phi_{,\zeta}\hat{=}0$ at the brane
located at $\zeta=\zeta_{*}$. The separation constant $\lambda_{\rho}$ is
fixed by solving Eq. (4.2) as an eigenvalue equation imposing the Dirichlet
condition at the AdSd+1 boundary $\zeta=1$ and the Neumann condition at the
brane $\zeta=\zeta_{*}$, as argued in Sec. 3.3. Using the solution (3.29), the
condition to fix $\lambda_{\rho}$ in terms of $\zeta_{*}$ is given by

$0\hat{=}\frac{d}{d\zeta}\left(1-\zeta^{2}\right)^{\frac{d}{4}}P^{-\mu}_{l}(\zeta)=-\frac{d}{2}\left(1-\zeta^{2}\right)^{\frac{d}{4}-1}\zeta
P^{-\mu}_{l}(\zeta)+\left(1-\zeta^{2}\right)^{\frac{d}{4}}\frac{d}{d\zeta}P^{-\mu}_{l}(\zeta)~{}.$
(4.5)

This expression can be further simplified and reduces to Eq. (3.55) for a
massless scalar field $m^{2}=0$, while for general $m^{2}\neq 0$ we need to
use Eq. (4.5). Once $\lambda_{\rho}$ is fixed, then $Y(x^{\mu})$ is given by
Eq. (3.73) with $c_{0}=0$. As an example, in Fig. 3 we show the first three
eigenfunctions for $d=3,m^{2}=0$ and the brane located at
$\zeta=\zeta_{*}=-1/2$, corresponding to the cases $a=1,2,3$ of Eq. (3.97).

Figure 3: The first three eigenfunctions ${\mathcal{R}}(\zeta)$ for
$d=3,m^{2}=0$ and the brane located at $\zeta=\zeta_{*}=-1/2$, corresponding
to $\theta=2\pi/3$. ${\mathcal{R}}(\zeta)$ is normalized so that
${\mathcal{R}}(\zeta_{*})=1$.

Any solution $\phi(\rho,x^{\mu})$ can be expressed in terms of the general
solutions ${\mathcal{R}}(\rho),Y(x^{\mu})$ constructed above. In principle, a
given initial data of the scalar field distribution may be decomposed into
mode solutions given above, and then the time dependence scalar field will be
described analytically.

###  4.2 Scalar field reflection at the boundary

As an example of nontrivial dynamics realized in our setup, in this section
let us construct bulk dual of waves that propagate toward and reflect at the
boundary in a BCFT. We model such waves by normalizable modes of a bulk scalar
field, which may be identified as the expectation value of a scalar operator
in the BCFT. A reflecting solution can be constructed by taking an appropriate
superposition of the mode solutions constructed in the previous section, but
instead of that we directly solve Eq. (4.1) as a time-evolution equation.

Expressing a solution as
$\phi={\mathcal{R}}(\rho)Y(t,y)e^{i\mathbf{k}\cdot\mathbf{x}}$, the scalar
field equation of motion (4.1) is separated into the $\rho$ equation (4.2) and
an equation for $Y(t,y)$ given by

$y^{2}\left(-Y_{,tt}+Y_{,yy}\right)-(d-2)yY_{,y}-\left(\lambda_{\rho}+\mathbf{k}^{2}y^{2}\right)Y=0~{}.$
(4.6)

Once an initial data for $Y(t,r)$ on a time slice is given, Eq. (4.6) fixes
the time evolution of $\phi$ for given parameters $d,\mathbf{k}$ and
$\lambda_{\rho}$. We consider solutions satisfying the Dirichlet condition at
the boundary of AdSd+1, ${\mathcal{R}}(\rho=\infty)=0$, which corresponds to
the normalizable mode. Then the eigenvalue $\lambda_{\rho}$ is fixed by
specifying the brane position $\rho_{*}$ and the mode number for
${\mathcal{R}}(\rho)$. Because (the real part of)
$Y(t,r)e^{i\mathbf{k}\cdot\mathbf{x}}$ is proportional to the normalizable
mode in the $\rho$ direction, its value may be interpreted as the expectation
value $\langle\phi\rangle$ for a scalar operator in the BCFT realized on AdSd.

We numerically solve Eq. (4.6) to construct time-dependent solutions $Y(t,y)$.
In order to conduct a stable numerical calculation, it is useful to introduce
the ingoing Eddington-Finkelstein coordinate $v=t-y$, with which the metric
(3.3) is expressed as

$ds^{2}=d\rho^{2}+\cosh^{2}\rho\left(\frac{-dv^{2}-2dvdy+d\mathbf{x}^{2}}{y^{2}}\right)$
(4.7)

and Eq. (4.6) becomes an equation for $Y(v,y)$ given by

$y^{2}\left(-2Y_{,vy}+Y_{,yy}\right)-(d-2)y\left(-Y_{,v}+Y_{,y}\right)-\left(\lambda_{\rho}+\mathbf{k}^{2}y^{2}\right)Y=0~{}.$
(4.8)

In this coordinate, ingoing and outgoing null rays are expressed by
$v=\text{const}$ and $v+2y=\text{const}$, respectively.

The setting for the numerical solution construction is as follows. We take an
initial time slice given by $0\leq y\leq y_{\text{fin}}$ at $v=0$, and solve
Eq. (4.8) in the domain of dependence of the initial time slice, which is
given by $v\in[0,v_{\text{fin}}]$ and $y\in[0,y_{\text{fin}}-\frac{1}{2}v]$
with $v_{\text{fin}}=2y_{\text{fin}}$. On the initial time slice, we specify
the profile of $Y(v,y)$ as

$Y(v=0,y)=\exp\left[-\frac{1}{2}\left(\frac{y-y_{0}}{s}\right)^{2}\right]~{},$
(4.9)

which describes a gaussian wave packet in the bulk that propagates toward the
boundary of AdSd at $y=0$. We also need to specify the boundary condition at
the AdSd boundary at $y=0$. We assume that $Y(y)$ behaves as the normalized
mode in AdSd, hence we impose the Dirichlet condition $Y(y=0)=0$. In the
numerical calculation, for simplicity we instead impose this condition at
$y=\epsilon>0$, where the cutoff $\epsilon$ is taken to be sufficiently small.

(a) Time evolution of $Y$

(b) Near reflection point

Figure 4: Numerical solution for $d=3,m^{2}=0,\mathbf{k}=0,\lambda_{\rho}=5/4$
and initial wave packet with $y_{0}=4,s=1/8$. In panel (a), only a part of the
numerical domain for $0<t<12$ is shown. Panel (b) shows the solution near the
reflection point $(t,y)\simeq(8,0)$.

In Fig. 4, we show an example of numerical solution in the case
$d=3,m^{2}=0,\mathbf{k}=0$ for the initial wave packet with $y_{0}=4,s=1/8$.
We have chosen ${\mathcal{R}}(\rho)$ to be the lowest mode ($a=1$) for the
brane located at $\zeta_{*}=-1/2$, for which $\lambda_{\rho}=5/4$ and
$\theta=2\pi/3$. For numerical calculation we used $y_{\text{fin}}=12$ and
$\epsilon=10^{-4}$. We can observe that the scalar field wave is almost
completely reflected by the AdSd boundary at $y=0$, and it approximately
follows null lines given by $t\pm y=\text{const}$.

(a) ${\mathcal{R}}^{(\varphi)}(\zeta)$ (b) $Y_{yy}^{(\varphi)}$ near
reflection point

Figure 5: The dynamics of the brane bending mode in $d=3$ for the brane
located at $\zeta=\zeta_{*}=-1/2$. Panel (a) shows
${\mathcal{R}}^{(\varphi)}(\zeta)$ analyzed in Sec. 3.3. Panel (b) shows the
time evolution of $Y_{yy}^{(\varphi)}$ with $\mathbf{k}=0$, for which
$m^{2}=0,\lambda_{\rho}=-1$. The initial field configuration is the same as
that for Fig. 4.

The equation governing the $Y_{yy}$ component of the gravitational
perturbation, Eq. (3.66), is the same as that for the scalar field (4.3).
Hence the numerical method introduced above can be straightforwardly applied
to the dynamics of $Y_{yy}$. As an example, in Fig. 5 we show the time
evolution of the
$h_{yy}=\mathfrak{a}^{2}{\mathcal{R}}^{(\varphi)}(\rho)Y^{(\varphi)}_{yy}$ of
the brane bending mode analysed in Sec. 3 in the case
$d=3,m^{2}=0,\mathbf{k}=0$ and $\zeta_{*}=-1/2$. Panel (a) shows the profile
of ${\mathcal{R}}^{(\varphi)}(\rho)$, which is given by Eq. (3.29) with the
junction condition (3.57) imposed at $\zeta=\zeta_{*}$. Panel (b) shows the
time evolution of $Y_{yy}^{(\varphi)}$, which obeys Eq. (4.3) with
$\lambda_{\rho}=-1$, for the initial wave packet with $y_{0}=4,s=1/8$. As
argued in Sec. 3.6, $Y_{yy}^{(\varphi)}$ is proportional to the holographic
stress energy tensor component ${\cal T}_{yy}$. The qualitative feature of its
time evolution is similar to those for the massless scalar field shown in Fig.
4(b). The wave approximately follows a null line reflecting at the AdSd
boundary $y=0$, while the shape of the wave packet after the reflection is
slightly distorted compared to that of the scalar field due to the difference
in the value of $\lambda_{\rho}$.

In this section, we showed time evolution of the scalar field and the scalar
part of the gravitational perturbations assuming that the mode function in the
bulk direction $\rho$ has only one mode given by Eq. (3.29) for given $l$ and
$\mu$. This field profile corresponds to a plane wave that has a
$y=\text{const.}$ surface as the wavefront and propagates toward the $y$
direction keeping its profile in the $\rho$ direction invariant. In the bulk
gravity viewpoint, this structure is the origin of the almost complete
reflection at the AdSd boundary observed in Figs. 4 and 5. The time evolution
around and after the reflection may be qualitatively different for waves with
more general shapes, and it would be interesting to study the dynamics of such
waves with general profiles and their interpretations in the BCFT viewpoint.
We will address such issues in the future work.

##  5 Island/BCFT correspondence in higher dimensions

Another important aspect of the AdS/BCFT correspondence is that it has the
third description where the $d$ dimensional CFT on a half plane $w>0$ is
coupled to $d$ dimensional gravity on AdSd which extends in $w<0$. This
description is qualitatively expected from the brane-world holography [12, 13,
14, 8, 16, 17, 18, 19, 20, 21, 22, 23] and has been beautifully applied to the
black hole information problem in the light of the Island formula [10, 11, 9].
This triality includes a purely $d$ dimensional statement that a $d$
dimensional BCFT is equivalent to a $d$ dimensional CFT on a half plane,
coupled to $d$ dimensional gravity on AdSd, which is called Island/BCFT
correspondence [40]. Even though this duality can be formally obtained by
combining the AdS/BCFT and brane-world holography, it is fair to say that the
precise and quantitative justification of the Island/BCFT correspondence is
still poorly understood. In the recent paper [40], we examined this problem
and presented quantitative evidences in the lowest dimension $d=2$. Below we
would like to test the Island/BCFT correspondence via computations of
entanglement entropy in higher dimensions $d>2$. Since we do not have powerful
controls of entanglement entropy, as opposed to the $d=2$ case studied in
[40], we will focus on the area law term contributions.

###  5.1 Entanglement entropy from island formula

Consider a $d$ dimensional BCFT on a half plane $w>0$ in the flat Lorentzian
spacetime

$\displaystyle ds_{d}^{2}\,=\,-dt^{2}+dw^{2}+\sum_{i=1}^{d-2}dx_{i}^{2}\,,$
(5.1)

and take the subregion $B$ as a half ball

$\displaystyle w^{2}\,+\,\sum_{i=1}^{d-2}x_{i}^{2}\,\leq\,l^{2}\,,\qquad
w>0\,.$ (5.2)

We call its complement the subsystem $A$ for which we consider the
entanglement entropy, denoted by $S_{A}$. We couple this BCFTd to the AdSd
gravity at $w=0$, so that we have AdSd in $w<0$ and BCFTd in $w>0$. Then the
corresponding metric of AdSd is given by

$\displaystyle
ds_{d}^{2}\,=\,\frac{\delta^{2}}{(\delta-w)^{2}}\left(-dt^{2}+dw^{2}+\sum_{i=1}^{d-2}dx_{i}^{2}\right)\,,\qquad(w<0),$
(5.3)

where in this AdSd spacetime, $\delta$ can be understood as an UV cutoff. By
the following coordinate transformation

$\displaystyle\delta-w\,$ $\displaystyle=\,r\cos\theta\,,$ (5.4)
$\displaystyle x_{i}\,$
$\displaystyle=\,r\sin\theta\cos\phi_{i}\prod_{n=1}^{i-1}\sin\phi_{n}\,,\qquad(i=1,\cdots,d-3)$
(5.5) $\displaystyle x_{d-2}\,$
$\displaystyle=\,r\sin\theta\prod_{n=1}^{d-3}\sin\phi_{n}\,,$ (5.6)

we can also bring the metric into the spherical coordinates

$\displaystyle
ds_{d}^{2}\,=\,\frac{\delta^{2}}{r^{2}\cos\theta^{2}}\left(-dt^{2}+dr^{2}+r^{2}d\Omega_{d-2}^{2}\right)\,.$
(5.7)

In such a system of a CFT coupled to gravity, the entanglement entropy $S_{A}$
can be computed by applying the Island prescription [10, 11, 9]. Namely, the
entanglement entropy is obtained by the formula (refer to Fig. 6)

$\displaystyle S_{A}=\mbox{Ext}_{Is}\left[S^{(G)}_{\partial Is}+S^{(Q)}_{A\cup
Is}\right].$ (5.8)

Here $Is$ denotes the Island which is a region in the AdSd which is originally
taken to be arbitrary under the condition that it connects with the region $A$
at the boundary $w=0$ as in Fig. 6. The entropy $S^{(G)}$ denotes the
classical gravity contribution, which is the area of the boundary of Island
$\partial Is$ divided by $4G_{N}$ in Einstein gravity. In our case, we expect
that the brane-world gravity is purely induced from quantum corrections of
matter fields and thus we simply set $S^{(G)}=0$. In other words, we can set
$\frac{1}{G^{(d)}_{N}}=0$ before we integrate over the matter fields, where
$G^{(d)}_{N}$ is the $d$ dimensional Newton constant. On the other hand,
$S^{(Q)}$ describes all contributions from the quantum fields to entanglement
entropy, where the subsystem is taken to be an union of $A$ and the Island.
Finally we extremize the total entropy by allowing to change the shape of the
Island. This is the Island prescription in our setup. In our higher
dimensional case $d>2$, we expect that $S^{(Q)}$ follows the area law [104,
105] and thus the leading divergent contribution looks like

$\displaystyle S^{(Q)}_{A\cup
Is}\simeq\mbox{Ext}_{Is}\left[\gamma\cdot\frac{\mbox{Area}(\partial
Is)}{\epsilon^{d-2}}\right]+S_{0}+O(\epsilon^{-(d-4)}),$ (5.9)

where $S_{0}$ represents the area law contribution from the subsystem $A$,
being proportional to $\frac{l^{d-2}}{\epsilon^{d-2}}$. The extremization
procedure is equivalent to finding an extremal surface as $\partial Is$. For
static backgrounds, which we focus in this section, this becomes a minimal
surface on a constant time slice.

Figure 6: Island setup in BCFT3

We parameterize a co-dimension two extremal surface on a constant time slice
in the AdSd, we parameterize the surface by

$\displaystyle r\,=\,r(\theta,\phi_{1},\cdots,\phi_{d-3})\,,$ (5.10)

with a boundary condition $r=l$ at $w=0$. Then, the induced metric on the
surface is given by

$\displaystyle
ds_{d-2}^{2}=\frac{\delta^{2}}{\cos^{2}\theta}\left[\left(\frac{\partial\log
r}{\partial\theta}d\theta+\sum_{i=1}^{d-3}\,\frac{\partial\log
r}{\partial\phi_{i}}d\phi_{i}\right)^{2}\,+\,d\Omega_{d-2}^{2}\right]\,.$
(5.11)

For the above induced metric, we denote the contribution from
$d\Omega_{d-2}^{2}$ by $(h_{d-2}^{(0)})_{\mu\nu}$ and the contribution from
$dr^{2}$ by $(\tilde{h}_{d-2})_{\mu\nu}$ as

$\displaystyle\big{(}h_{d-2}\big{)}_{\mu\nu}\,=\,\frac{\delta^{2}}{\cos^{2}\theta}\left[\big{(}h_{d-2}^{(0)}\big{)}_{\mu\nu}\,+\,\big{(}\tilde{h}_{d-2}\big{)}_{\mu\nu}\right]\,,$
(5.12)

where

$\displaystyle\big{(}h_{d-2}^{(0)}\big{)}_{\mu\nu}\,$
$\displaystyle=\,\begin{pmatrix}1&0&\cdots&0\\\ 0&\sin^{2}\theta&\cdots&0\\\
\vdots&\vdots&&\vdots\\\
0&0&\cdots&\sin^{2}\theta\prod_{i=1}^{d-4}(\sin\phi_{i})^{2}\end{pmatrix}\,,$
(5.13) $\displaystyle\big{(}\tilde{h}_{d-2}\big{)}_{\mu\nu}\,$
$\displaystyle=\,v_{\mu}\,(v^{T})_{\nu}\,,$ (5.14) $\displaystyle
v^{T}_{\mu}\,$ $\displaystyle=\,\left(\frac{\partial\log
r}{\partial\theta}\,,\,\frac{\partial\log
r}{\partial\phi_{1}}\,,\,\cdots\,,\,\frac{\partial\log
r}{\partial\phi_{d-3}}\right)\,.$ (5.15)

The areal law term of the entanglement entropy for $A\cup Is$ reads

$\displaystyle S^{(Q)}_{A\cup
Is}\,\approx\,\frac{\gamma}{\epsilon^{d-2}}\int_{-\frac{\pi}{2}+\delta^{\prime}}^{\frac{\pi}{2}-\delta^{\prime}}d\theta\int_{0}^{\pi}\prod_{i=1}^{d-3}d\phi_{i}\sqrt{\det(h_{d-2})}\,+\,S_{0}\,,$
(5.16)

where the regularization parameter $\delta^{\prime}$ is defined by
$l\cos\delta^{\prime}=\delta$.

Using the decomposition of the determinant, the determinant of the induced
metric is given by

$\displaystyle\det(h_{d-2})\,$
$\displaystyle=\,\left(\frac{\delta}{\cos\theta}\right)^{2(d-2)}\,\det\Big{(}h_{d-2}^{(0)}+vv^{T}\Big{)}$
$\displaystyle=\,\left(\frac{\delta}{\cos\theta}\right)^{2(d-2)}\,\det\big{(}h_{d-2}^{(0)}\big{)}\det\big{(}I_{d-2}+v^{T}(h_{d-2}^{(0)})^{-1}v\big{)}\,,$
(5.17)

where $I_{d-2}$ is the $d-2$ dimensional identity matrix and we used
Sylvester’s determinant theorem (or the matrix determinant lemma) for the
second line. Since the first determinant $\det\big{(}h_{d-2}^{(0)}\big{)}$
does not depend on the surface profile $r=r(\theta,\phi_{i})$, we need to
consider to minimize the second determinant
$\det\big{(}I_{d-2}+v^{T}(h_{d-2}^{(0)})^{-1}v\big{)}$. This is archived by
the vanishing vector $v=0$, so that
$\det\big{(}I_{d-2}+v^{T}(h_{d-2}^{(0)})^{-1}v\big{)}=1$. Therefore, the
minimal surface is semi-circle with the constant radius $r=l$. Hence in this
case, the entanglement entropy is given by

$\displaystyle S_{A}=S^{(Q)}_{A\cup Is}\,$
$\displaystyle=\,\frac{\gamma\delta^{d-2}}{\epsilon^{d-2}}\int_{-\frac{\pi}{2}+\delta^{\prime}}^{\frac{\pi}{2}-\delta^{\prime}}d\theta\,\frac{(\sin\theta)^{d-3}}{(\cos\theta)^{d-2}}\int_{0}^{\pi}\prod_{i=1}^{d-3}d\phi_{i}\big{(}\sin\phi_{i}\big{)}^{d-3-i}\,+\,S_{0}$
$\displaystyle\approx\,\frac{\gamma\delta
S_{d-3}l^{d-3}}{(d-3)\epsilon^{d-2}}\,+\,S_{0}\,,$ (5.18)

where $S_{d-3}=2\pi^{(d-2)/2}/\Gamma((d-2)/2)$. The above result of the
$\theta$ integral assumes $d>3$ and the case of $d=3$ must be treated
separately.

For $d=3$, we actually obtain

$\displaystyle S_{A}\simeq
2\gamma\cdot\frac{\delta}{\epsilon}\log\frac{2l}{\delta}+S_{0},$ (5.19)

where we have $S_{0}=\gamma\frac{\pi l}{\epsilon}$.

We also would like to note an important fact. We initially assumed the
subsystem $A$ to be the outside of a round half ball $B$ specified by (5.2).
However, the above calculation based on the Island formula only uses the
location of the boundary $\partial A$ at the interface between the CFT and the
gravity i.e. we have $x=\pm l$ at $w=0$. Thus we obtain the same result (5.19)
for any shape of the subsystem $A$ as long as it ends on $|x|=\pm l$ at $w=0$.

###  5.2 Entanglement entropy in BCFT from AdS/BCFT

Next we would like to compare the previous result based on the Island formula
with the result obtained from a holographic BCFT, using the AdS/BCFT. In the
AdS/BCFT, the holographic entanglement entropy $S_{A}$ [106, 107, 108], is
given by

$\displaystyle
S_{A}=\mbox{Ext}_{\Gamma_{A}}\left[\frac{\mbox{Area}(\Gamma_{A})}{4G_{N}}\right].$
(5.20)

with an extra rule [5, 6] that the surface $\gamma_{A}$ is a $d-1$ dimensional
surface whose boundary is given by $\partial A\cup\gamma_{Is}$, where
$\gamma_{Is}$ is a $d-2$ dimensional surface on the EOW brane $Q$. In the end,
$\gamma_{Is}$ will be identified with the boundary of the Island i.e.
$\partial Is$. Refer to Fig. 7 for a sketch. Note that for static backgrounds,
which we focus in this section, $\Gamma_{A}$ becomes a minimal surface on a
constant time slice.

Figure 7: Calculation of holographic entanglement entropy in AdS/BCFT

We choose the coordinates of the AdSd+1 to be

$\displaystyle
ds_{d+1}^{2}\,=\,d\rho^{2}+\frac{\cosh^{2}\rho}{y^{2}}\left(-dt^{2}+dy^{2}+\sum_{i=1}^{d-2}dx_{i}^{2}\right)\,.$
(5.21)

When $A$ is a semi-disk with radius $l$ as before, the minimal surface is
given by a half sphere even in the presence of EOW brane as we explain in
appendix B. We employ the spherical coordinates by $y=l\cos\theta$ and (5.6)
for $x_{i}$’s such that the minimal surface is given by $r=l$. Then, the
induced metric on the two minimal surface in this AdSd+1 is given by

$\displaystyle
ds_{d-1}^{2}\,=\,d\rho^{2}+\frac{\cosh^{2}\rho}{\cos^{2}\theta}\,d\Omega_{d-2}^{2}\,.$
(5.22)

In the previous sections, we placed the EOW brane at $\rho=\rho_{*}$ with
$\rho_{*}<0$, but for this section, it’s more convenient to denote the brane
location by $\rho=-\rho_{*}$ with $\rho_{*}>0$, so the sign of $\rho_{*}$ here
is different from the one used in the previous sections. Then, the holographic
EE is computed as

$\displaystyle S_{A}\,$
$\displaystyle=\,\frac{1}{4G_{N}}\int_{-\rho_{*}}^{\rho_{\infty}}d\rho\,(\cosh\rho)^{d-2}\int_{-\frac{\pi}{2}+\frac{\epsilon}{l}\cosh\rho}^{\frac{\pi}{2}-\frac{\epsilon}{l}\cosh\rho}d\theta\,\frac{(\sin\theta)^{d-3}}{(\cos\theta)^{d-2}}\int_{0}^{\pi}\prod_{i=1}^{d-3}d\phi_{i}\big{(}\sin\phi_{i}\big{)}^{d-3-i}$
$\displaystyle\approx\,S^{half}_{A}\,+\,\frac{S_{d-3}l^{d-3}L^{d-1}}{4(d-3)G_{N}\epsilon^{d-3}}\,\sinh\rho_{*}\,.$
(5.23)

Again the $\theta$ integral assumes $d>3$ and the case of $d=3$ must be
treated separately.

For $d=3$, we actually obtain

$\displaystyle S_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2G_{N}}\int^{\rho_{\infty}}_{-\rho_{*}}d\rho\cosh\rho\int^{l}_{\epsilon\cosh\rho}dy\frac{l}{y\sqrt{l^{2}-y^{2}}}$
(5.24) $\displaystyle\simeq$
$\displaystyle\frac{L^{2}}{2G^{(4)}_{N}}\sinh\rho_{*}\log\frac{2l}{\epsilon}+S_{0}.$

Comparing (5.23) with (5.18), the corresponding dictionary for AdSd+1/BCFTd is
given by

$\displaystyle\frac{\gamma\delta}{\epsilon}\,=\,\frac{L^{d-1}}{4G_{N}^{(d+1)}}\,\sinh\rho_{*}\,.$
(5.25)

Note that since we expect $\gamma\sim\frac{L^{d-1}}{2G^{(d+1)}_{N}}$, which is
a sort of the central charge, we expect
$\sinh\rho_{*}\sim\frac{\delta}{\epsilon}$. This is indeed what is expected in
the AdS/BCFT as we have

$\displaystyle\frac{w}{z}\left(=\frac{\delta}{\epsilon}\right)=\sinh\rho_{*},$
(5.26)

on the EOW brane. In this way, we find a nice agreement of entanglement
entropy between the Island and the holographic BCFT calculation.

Notice that this analysis of entanglement entropy in a holographic BCFT is
correct for any value of tension $\sigma$ or angle $\rho_{*}$ of the EOW
brane. However, when the tension takes a generic value, we need to take into
account the subleading terms other than the area law term in the Island
calculation of the dual system, via the Island/BCFT correspondence. This is
clear in our AdS/BCFT analysis presented above because the BCFT contribution
to the holographic entanglement entropy comes from the region deep in the
bulk. When we deform the shape of the subsystem $A$ with keeping the location
at the interface $w=0$, the extremal surface $\Gamma_{A}$ will clearly be
deformed accordingly. Nevertheless, if we assume the tension of EOW brane is
very large ($\sigma\gg 1$ or $\theta\simeq\pi$) such that it sits very close
to the asymptotically AdS boundary, then it is obvious that the leading
contribution to the holographic entanglement entropy is still given by (5.24)
even if we deform the shape of subsystem $A$. This agrees with our observation
in the Island calculation mentioned in the final part of the previous
subsection.

##  6 One point functions in higher dimensional AdS/BCFT

In general, one-point functions in BCFT are non-vanishing [1]. For a scalar
primary operator $\mathcal{O}$ with a conformal dimension
$\Delta_{\mathcal{O}}$, the one-point function has a form

$\displaystyle\big{\langle}\mathcal{O}(w)\big{\rangle}\,=\,\frac{\mathcal{N}_{\mathcal{O}}}{|w|^{\Delta_{\mathcal{O}}}}\,,$
(6.1)

where $|w|$ is the distance from the boundary with a numerical normalization
factor $\mathcal{N}_{\mathcal{O}}$. In this section, we will see that in order
to reproduce this non-vanishing one-point function, the dual bulk scalar field
must couple to the bulk gravity with a non-trivial expectation value. This was
pointed out in [40] in the case of AdS${}_{3}/$BCFT2 (see also earlier work
[6, 109]). Below we will extend this to higher dimensional cases.

###  6.1 The model

We study the Einstein-dilaton theory defined by the action

$\displaystyle I\,=\,-\frac{1}{16\pi G}\int
d^{d+1}x\sqrt{g}\Big{[}R-g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-U(\phi)\Big{]}\,,$
(6.2)

where $R$ is the $d+1$-dimensional Ricci scalar and $U(\phi)$ is a generic
potential of the dilaton $\phi$. This model and the solutions we are looking
for can be thought as a generalization of the Janus solutions [110, 111]. (See
also [40]).

We can obtain a class of setups of AdSd+1/BCFTd by placing the EOW brane $Q$
at $\rho=\rho_{*}$, such that the bulk gravity extends in the region
$\rho_{*}<\rho<\infty$. For the scalar field $\phi$, we assume the linear
interaction on the brane

$\displaystyle I_{\textrm{bdy}}\,=\,\frac{a_{0}}{8\pi
G}\int_{Q}d^{d}x\sqrt{h}\,\phi\,,$ (6.3)

where $a_{0}$ is a coupling constant and $h_{ij}$ is the induced metric on
$Q$. Combined with the bulk action, the variation of $\phi$ leads to the
Neumann-like boundary condition at $Q$:

$\displaystyle\partial_{\rho}\phi\big{|}_{\rho=\rho_{*}}\,=\,a_{0}\,.$ (6.4)

The Neumann boundary condition of the gravity is now written as

$\displaystyle
K_{ij}-h_{ij}K\,=\,-\left(\frac{\sigma}{2}+a_{0}\phi\right)h_{ij}\,.$ (6.5)

From the bulk action, the Einstein equation and the Klein-Gordon equation are
given by

$\displaystyle
R_{\mu\nu}\,-\,\frac{U(\phi)}{d-1}\,g_{\mu\nu}\,=\,\partial_{\mu}\phi\partial_{\nu}\phi\,,$
(6.6) $\displaystyle
2\partial_{\mu}\big{(}\sqrt{g}g^{\mu\nu}\partial_{\nu}\phi\big{)}\,=\,\sqrt{g}\partial_{\phi}U(\phi)\,.$
(6.7)

For the background solution, we consider the Janus type ansatz 888When there
is no dilaton field, the warp factor $f(\rho)$ coincides with
$\mathfrak{a}^{2}(\rho)=\cosh^{2}(\rho)$ defined in (3.8). However in this
section, we consider non-trivial backreaction of the dilaton $\phi$, so it
differs from $\mathfrak{a}^{2}(\rho)$ and we instead use $f(\rho)$ which is
the standard notation for the Janus solutions.

$\displaystyle ds_{d+1}^{2}\,$
$\displaystyle=\,d\rho^{2}+f(\rho)ds_{AdS_{d}}^{2}\,,$ (6.8)
$\displaystyle\phi\,$ $\displaystyle=\,\phi(\rho)\,.$ (6.9)

Given this ansatz, the Einstein equation and the Klein-Gordon equation are
reduced to

$\displaystyle
2(d-1)ff^{\prime\prime}\,+\,(d-1)(d-2)(f^{\prime})^{2}\,+\,4(d-1)^{2}f\,+\,4Uf^{2}\,=\,0\,,$
(6.10) $\displaystyle
d(d-1)\big{(}2ff^{\prime\prime}-(f^{\prime})^{2}\big{)}\,+\,4Uf^{2}\,+\,4(d-1)f^{2}(\phi^{\prime})^{2}\,=\,0\,,$
(6.11) $\displaystyle
2f\phi^{\prime\prime}\,+\,d\,f^{\prime}\phi^{\prime}-f\partial_{\phi}U\,=\,0\,,$
(6.12)

where the prime denotes a derivative with respect to $\rho$. In fact, the two
equations (6.10), (6.11) coming from the Einstein equation are not independent
to each other provided the Klein-Gordon equation (6.12) and they can be
combined into

$\displaystyle
d(d-1)(f^{\prime})^{2}\,+\,4d(d-1)f\,+\,4Uf^{2}\,-\,4f^{2}(\phi^{\prime})^{2}\,=\,0.$
(6.13)

Therefore, we need to solve the Klein-Gordon equation (6.12) and the Einstein
equation (6.13) simultaneously.

###  6.2 Free massless case

For free massless case, we have $U(\phi)=2\Lambda=-d(d-1)$. For this case, the
Einstein equation and the Klein-Gordon equation are

$\displaystyle
d(d-1)(f^{\prime})^{2}\,+\,4d(d-1)f\,-\,4d(d-1)f^{2}\,-\,4f^{2}(\phi^{\prime})^{2}\,=\,0,$
(6.14) $\displaystyle
2f\phi^{\prime\prime}\,+\,d\,f^{\prime}\phi^{\prime}\,=\,0\,.$ (6.15)

Since the Klein-Gordon equation is a total derivative, it is solved by

$\displaystyle\phi^{\prime}\,=\,\frac{p_{0}}{f^{d/2}}\,,$ (6.16)

where $p_{0}$ is an integration constant and this constant is fixed by the
parameter $a_{0}$ in (6.4) by $p_{0}=a_{0}f^{d/2}(\rho_{*})$. Substituting
this into the Einstein equation, we find

$\displaystyle
d(d-1)(f^{\prime})^{2}\,+\,4d(d-1)f\,-\,4d(d-1)f^{2}\,-\,4c^{2}f^{2-d}\,=\,0.$
(6.17)

Since this is a first order differential equation, we need only one boundary
condition which we choose

$\displaystyle
f(\rho)\,\to\,\frac{1}{4}\,e^{2\rho}\,+\,\cdots\qquad\textrm{for}\ \
\rho\to\infty\,.$ (6.18)

This equation is still difficult to find analytical solutions, except for
$d=2$ case [110]. However, numerically we can easily find solutions and we
plotted some numerical solutions for $d=3$ and $d=4$ in Fig. 8. As in [40], we
defined

$\displaystyle u\,=\,\tanh\rho\,,$ (6.19)

which map the coordinate into $-1\leq u\leq 1$ and

$\displaystyle
f(u)\,=\,\frac{4g(u)}{1-u^{2}}\,,\qquad\phi(u)\,=\,\left(\frac{1-u^{2}}{4}\right)^{\frac{\Delta}{2}-1}\chi(u)\,.$
(6.20)

These solutions are singular solutions in the sense that we see a naked
singularity at $u=u_{s}$ where $f(u_{s})=0$ and $\phi(u_{s})$ diverges. Such a
naked singularity must be prohibited for the usual AdS/CFT without an EOW
brane. However, in the context of the AdS/BCFT setup, this is not a problem
[40] but just means that we have to place the EOW brane before this
singularity ($u_{s}<u_{*}=\tanh\rho_{*}$).

Figure 8: Massless solutions for $d=3$ (left) and $d=4$ (right).

###  6.3 Free massive case

For free massive case, we have $U(\phi)=-d(d-1)+m^{2}\phi^{2}$. For this case,
the Einstein equation and the Klein-Gordon equation are

$\displaystyle
d(d-1)(f^{\prime})^{2}\,+\,4d(d-1)f\,-\,4d(d-1)f^{2}\,+\,4m^{2}f^{2}\phi^{2}\,-\,4f^{2}(\phi^{\prime})^{2}\,=\,0,$
(6.21) $\displaystyle
2f\phi^{\prime\prime}\,+\,d\,f^{\prime}\phi^{\prime}\,-\,2m^{2}f\phi\,=\,0\,.$
(6.22)

The boundary conditions we impose are

$\displaystyle f(\rho)\,$
$\displaystyle\to\,\frac{1}{4}\,e^{2\rho}\,+\,\frac{1}{2}\,+\,\cdots\,,$
(6.23) $\displaystyle\phi(\rho)\,$
$\displaystyle\to\,\alpha\,e^{-\Delta\rho}\,+\,\cdots\,,$ (6.24)

where we parametrized the mass by $m^{2}=\Delta(\Delta-d)$. In this note, we
focus on the masses above the BF bound $m^{2}\geq-(d/2)^{2}$, this implies for
the dimension we have $\Delta\geq d/2$. In terms of $g$ and $\chi$ defined in
(6.20), the boundary conditions are written as

$\displaystyle
g(u)\,\to\,\frac{1}{4}\,+\,\cdots\,,\qquad\chi(u)\,\to\,\alpha\left(\frac{1-u^{2}}{4}\right)\,+\,\cdots\,.$
(6.25)

We plotted numerical singular solutions for $d=3$ in Fig. 9 and $d=4$ in Fig.
10. In Fig. 11, we also showed some non-singular solutions for $d=3$.

Figure 9: Massive singular solutions for $d=3$ with $\alpha=4.0$.

Figure 10: Massive singular solutions for $d=4$ with $\alpha=7.0$.

Figure 11: Massive non-singular solutions for $d=3$ with $\alpha=$4.2756,
4.0026, 3.6125, 3.1150, 2.5210 for $\Delta=$2.2, 2.4, 2.6, 2.8, 3.0,
respectively.

###  6.4 Boundary one-point function

If we take the Poincare coordinates for AdSd+1

$\displaystyle
ds^{2}\,=\,\frac{d\tau^{2}+dz^{2}+dw^{2}+\sum_{i=0}^{d-2}dx_{i}^{2}}{z^{2}}\,,$
(6.26)

the bulk scalar behaves near the AdS boundary $z\to 0$ as

$\displaystyle\phi(\tau,z,w,x_{i})\,=\,J(\tau,w,x_{i})\,z^{d-\Delta}\,+\,\alpha(\tau,w,x_{i})z^{\Delta}\,+\,\cdots\,,$
(6.27)

where $J$ is the source for the dual operator $\mathcal{O}$ and $\alpha$
corresponds to the expectation value as

$\displaystyle\big{\langle}\mathcal{O}(\tau,w,x_{i})\big{\rangle}\,=\,\alpha(\tau,w,x_{i})\,.$
(6.28)

Using the coordinate change $z=w/\sinh\rho$, we can move from the hyperbolic
slicing coordinates to the Poincare coordinates. In particular, near the
boundary we have

$\displaystyle e^{-\rho}\,\simeq\,\frac{z}{2w}\,.$ (6.29)

Therefore, given the boundary condition of the scalar field as in (6.24),

$\displaystyle\big{\langle}\mathcal{O}(\tau,w,x_{i})\big{\rangle}\,=\,\frac{\alpha}{(2w)^{\Delta}}\,.$
(6.30)

This agrees with the form of the expected one-point function from the BCFT
side.

For the free massless case, with the solution (6.16) and the boundary
behaviour of $f$ (6.18), the scalar field behaves near the boundary as

$\displaystyle\phi\,\simeq\,-\frac{p_{0}}{d}\,(2e^{-\rho})^{d}\,+\,\cdots\,.$
(6.31)

Since for the massless case we have $\Delta=d$, the one-point function is
obtained as

$\displaystyle\big{\langle}\mathcal{O}(\tau,w,x_{i})\big{\rangle}\,=\,-\frac{p_{0}}{d}\frac{1}{(w)^{d}}\,.$
(6.32)

This again agrees with the form of the expected one-point function from the
BCFT side.

##  7 Conclusions and Discussions

In this paper, we have studied several dynamical aspects of holographic
boundary conformal field theories (BCFTs), by employing a gravity dual
construction known as the AdS/BCFT correspondence. A key feature of AdS/BCFT
is the presence of end of the world-brane (EOW brane), which stretches from
the boundary of the BCFT towards the AdS bulk. When we excite a holographic
BCFT, the EOW brane starts to vibrate in addition to bulk metric excitations
in the gravity dual. The EOW brane is characterized by the value of its
tension $\sigma$. In this way, we can analyze the dynamical property of
holographic BCFT via the gravity dual in a way similar to the one employed in
earlier studies of brane-world models.

First we noted that the gravity dual of a two dimensional BCFT, i.e.
AdS${}_{3}/$BCFT2 is special in that there are no propagating modes in the
bulk of the pure AdS3 gravity. Owing to this property, we can find analytical
solutions to the gravity duals of an arbitrary excited states in a holographic
BCFT, whose stress energy tensor satisfies the boundary condition of complete
reflection. However, the main aim of this paper is to investigate dynamical
properties of AdS/BCFT in higher dimensions.

In higher dimensional AdS${}_{d+1}/$BCFTd ($d\geq 3$), the bulk graviton mode
is propagating and its dynamics becomes much richer. In this paper we
performed a complete analysis of metric perturbations in the presence of EOW
brane. We found that they are organized into the helicity 0, 1 and 2 mode. In
particular, the lightest mode of helicity 0 can be identified with the brane
bending mode. We can calculate the holographic stress energy tensor from these
solutions to the metric perturbations. The brane bending mode provides a
diagonal component of stress energy tensor which shows an universal behavior
such that it approaches a constant value at the boundary of BCFT, being
independent from the value of brane tension. On the other hand, all other
modes of helicity 0, 1 and 2 are sensitive to the value of tension and the
corresponding stress energy tensors are proportional to characteristic powers
of the distance $w$ between the point and the boundary. These powers depend on
the tension and interestingly, they become integer valued when the tension
vanishes. We confirmed that the boundary condition of stress energy tensor
$T_{iw}|_{w=0}=0$, which is expected for any BCFTs, is indeed satisfied for
any values of tension. It will be an important future problem to understand
the above behaviors of stress energy tensor, predicted by the gravity dual
analysis, from field theoretic calculations in BCFTs.

We also examined solutions of the scalar field perturbations in the AdS/BCFT
setup. This is dual to the scalar operator excitation in a holographic BCFT.
This analysis again confirms that the bulk excitation propagates to the EOW
brane and is completely reflected back, as expected from the dual BCFT. We
construct an explicit numerical solution which shows this behavior. It would
be an interesting future problem to study both scalar field and metric
perturbations at the same time and construct a gravity dual of realistic black
hole evaporation in higher dimensions, where we expect that the dynamics of
brane world plays a crucial role.

Then we analyzed another important aspect of AdS/BCFT, namely this has the
third description in terms of a CFT coupled to gravity along the boundary.
This is expected from the brane-world holography, so called double holography.
If we focus on the duality in lower dimensional (i.e. $d$ dimensional)
descriptions, this leads to the equivalence that a $d$ dimensional BCFT is
equivalent to a $d$ dimensional CFT on a half plane, coupled to $d$
dimensional gravity on AdSd, which is called Island/BCFT correspondence. In
this paper, we provided an evidence which supports this duality by calculating
the entanglement entropy for a semi-disk subsystem. The upshot is that the
entanglement entropy in a $d$ dimensional holographic BCFT agrees with that in
the theory defined by a $d$ dimensional CFT coupled to a $d$ dimensional
gravity, where the latter gravitational theory is assumed to be an induced
gravity. It will be intriguing to generalize this analysis to more general
subsystems.

Finally, we present a gravity dual computation of one point functions in
holographic BCFTs, where the scalar field solution was given numerically in
the presence of a non-trivial boundary condition on EOW brane. This reproduces
the expected form of one-point functions in BCFTs. An interesting future
problem is to consider a string theory embedding of the AdS/BCFT and calculate
the one point functions both from string theory and the dual BCFT.

## Acknowledgements

We are grateful to Shan-Ming Ruan and Zixia Wei for useful comments on the
draft of this paper. We also thank Yu-ki Suzuki and Seiji Terashima for
informing us of their independent work [88] when we were writing up this
draft. This work is supported by MEXT KAKENHI Grant-in-Aid for Transformative
Research Areas (A) through the “Extreme Universe” collaboration: Grant Number
21H05182, 21H05187 and 21H05189. KI and TS is supported by JSPS Grants-in-Aid
for Scientific Research (A) 17H01091. KI is also supported by JSPS Grants-in-
Aid for Scientific Research (B) JP20H01902. TS is also supported by JSPS
Grants-in-Aid for Scientific Research (C) JP21K03551. KS and TT are supported
by the Simons Foundation through the “It from Qubit” collaboration. TT is also
supported by Inamori Research Institute for Science, World Premier
International Research Center Initiative (WPI Initiative) from the Japan
Ministry of Education, Culture, Sports, Science and Technology (MEXT), and
JSPS Grant-in-Aid for Scientific Research (A) No. 21H04469. NT is supported in
part by JSPS Grants-in-Aid for Scientific Research (C) 18K03623.

##  Appendix A A class of explicit metric perturbations at $d=3$

Here, we show a class of explicit solutions for the perturbative Einstein
equation under the following metric ansatz for $d=3$:

$\displaystyle
ds^{2}=d\rho^{2}+\hat{H}_{\mu\nu}(\rho,\tau,x,y)dx^{\mu}dx^{\mu},$ (A.1)

where we assume the following form

$\displaystyle\hat{H}_{\mu\nu}(\rho,\tau,x,y)=\delta_{\mu\nu}\frac{\cosh^{2}\rho}{y^{2}}+y\cdot\hat{h}_{\mu\nu}(\rho,\tau,x)+O(y^{2}).$
(A.2)

This form is motivated by picking up perturbations which lead to a constant
value of holographic stress energy tensor. We work with the Euclidean
signature. As opposed to the section 3, we will not impose the TT-gauge
condition. Instead we require that the location of EOW brane is at
$\rho=\rho_{*}$ even after we take into account the gravitational
backreaction.

###  A.1 Perturbative solutions to Einstein equation

By solving the Einstein equation perturbatively, we find the solutions as
follows

$\displaystyle\hat{h}_{\tau\tau}=-\frac{1}{2}B\cosh^{2}\rho+\frac{U_{1}}{2}\sinh\rho\cosh\rho+\frac{U_{2}}{2}\cosh\rho\left(1+2\sinh\rho\cdot\arctan\left[\tanh\frac{\rho}{2}\right]\right)$
$\displaystyle\ \ \ \ \ \
-V_{1}\tanh\rho(\sinh^{2}\rho+3)-\frac{V_{2}}{\cosh\rho},$
$\displaystyle\hat{h}_{xx}=-\frac{1}{2}B\cosh^{2}\rho+\frac{U_{1}}{2}\sinh\rho\cosh\rho+\frac{U_{2}}{2}\cosh\rho\left(1+2\sinh\rho\cdot\arctan\left[\tanh\frac{\rho}{2}\right]\right)$
$\displaystyle\ \ \ \ \ \
+V_{1}\tanh\rho(\sinh^{2}\rho+3)+\frac{V_{2}}{\cosh\rho},$
$\displaystyle\hat{h}_{yy}=\frac{3}{2}B\cosh^{2}\rho-U_{1}\sinh\rho\cosh\rho-
U_{2}\cosh\rho\left(1+2\sinh\rho\cdot\arctan\left[\tanh\frac{\rho}{2}\right]\right),$
$\displaystyle\hat{h}_{\tau
x}=A_{1}\tanh\rho(\sinh^{2}\rho+3)+\frac{A_{2}}{\cosh\rho},$
$\displaystyle\hat{h}_{\tau
y}=Q_{1}\cosh^{2}\rho+Q_{2}\cosh^{2}\rho\left(2\arctan\left[\tanh\frac{\rho}{2}\right]+\frac{\sinh\rho}{\cosh\rho^{2}}\right),$
$\displaystyle\hat{h}_{xy}=R_{1}\cosh^{2}\rho+R_{2}\cosh^{2}\rho\left(2\arctan\left[\tanh\frac{\rho}{2}\right]+\frac{\sinh\rho}{\cosh\rho^{2}}\right),$
(A.3)

where $A_{1},A_{2},B,U_{1},U_{2},V_{1},V_{2},P_{1},P_{2},Q_{1},Q_{2}$ are
arbitrary functions of $\tau$ and $x$. Here we imposed the Einstein equation
up to $O(y^{3})$, $O(y^{2})$ and $O(y)$ for $(\rho,\rho)$, $(\rho,\mu)$ and
$(\mu,\nu)$ components, respectively. It is also useful to note that we can
rewrite

$\displaystyle\arctan\left[\tanh\frac{\rho}{2}\right]=\frac{i}{2}\cdot\log\left[\frac{e^{\rho/2}+ie^{-\rho/2}}{e^{\rho/2}-ie^{-\rho/2}}\right]+\frac{\pi}{4}.$
(A.4)

###  A.2 Comparison with AdS4 black brane solution

Consider the static black brane solution (AdS4 Schwarzschild)

$\displaystyle
ds^{2}=\frac{f(\hat{z})}{\hat{z}^{2}}dw^{2}+\frac{d\hat{z}^{2}}{f(\hat{z})\hat{z}^{2}}+\frac{d\tau^{2}+dx^{2}}{\hat{z}^{2}},$
(A.5)

where $f(\hat{z})=1-3\alpha\hat{z}^{3}$. Via the coordinate transformation

$\displaystyle z=\hat{z}+\frac{\alpha}{2}\hat{z}^{4}+\cdots,$ (A.6)

we can write the metric in the Graham-Fefferman form

$\displaystyle ds^{2}\simeq\frac{dz^{2}+(1-2\alpha z^{3})dw^{2}+(1+\alpha
z^{3})d\tau^{2}+(1+\alpha z^{3})dx^{2}}{z^{2}}.$ (A.7)

More generally, at the first order of perturbation, the solution to the
Einstein equation looks like

$\displaystyle ds^{2}\simeq\frac{dz^{2}+(1+\alpha z^{3})d\tau^{2}+(1+\beta
z^{3})dx^{2}+(1+\gamma z^{3})dw^{2}}{z^{2}},$ $\displaystyle\mbox{with the
condition:}\ \alpha+\beta+\gamma=0.$ (A.8)

Note that the above condition is equivalent to the traceless condition of
holographic stress energy tensor.

Now by applying the coordinate transformation (3.2) or equally

$\displaystyle y=\sqrt{z^{2}+w^{2}},\ \ \
\cosh\rho=\frac{\sqrt{w^{2}+z^{2}}}{z},$ (A.9)

we obtain

$\displaystyle ds^{2}$ $\displaystyle\simeq$ $\displaystyle
d\rho^{2}+\cosh^{2}\rho\left[\frac{d\tau^{2}+dx^{2}+dy^{2}}{y^{2}}\right]+\frac{\alpha
y}{\cosh\rho}d\tau^{2}+\frac{\beta y}{\cosh\rho}dx^{2}$ (A.10) $\displaystyle\
+\frac{\gamma y}{\cosh\rho}\left(\tanh\rho
dy+\frac{yd\rho}{\cosh^{2}\rho}\right)^{2}$ $\displaystyle=$
$\displaystyle\left(1+\frac{\gamma
y^{3}}{\cosh^{5}\rho}\right)d\rho^{2}+2\frac{\gamma
y^{2}\sinh\rho}{\cosh^{4}\rho}dyd\rho+\left(\frac{\cosh^{2}\rho}{y^{2}}+\gamma
y\frac{\sinh^{2}\rho}{\cosh^{3}\rho}\right)dy^{2}$ $\displaystyle\
+\left(\frac{\cosh^{2}\rho}{y^{2}}+\frac{\alpha
y}{\cosh\rho}\right)d\tau^{2}+\left(\frac{\cosh^{2}\rho}{y^{2}}+\frac{\beta
y}{\cosh\rho}\right)dx^{2}.$

To make this metric into that of a Gaussian normal coordinate, we further
perform the coordinate transformation into the new coordinate $\tilde{\rho}$
and $\tilde{y}$:

$\displaystyle\rho=\tilde{\rho}+\eta(\tilde{\rho},\tilde{y}),\ \ \
y=\tilde{y}+\xi(\tilde{\rho},\tilde{y}).$ (A.11)

We require

$\displaystyle
g_{\tilde{\rho}\tilde{\rho}}=1+\frac{\gamma\tilde{y}^{3}}{\cosh^{5}\tilde{\rho}}+2\partial_{\tilde{\rho}}\eta\equiv
1,$ $\displaystyle
g_{\tilde{\rho}\tilde{y}}=\gamma\tilde{y}^{2}\frac{\sinh\tilde{\rho}}{\cosh^{4}\tilde{\rho}}+\partial_{\tilde{y}}\eta+\frac{\cosh^{2}\tilde{\rho}}{\tilde{y}^{2}}\partial_{\tilde{\rho}}\xi\equiv
0.$ (A.12)

These are solved as follows:

$\displaystyle\eta=-\frac{\gamma}{2}\tilde{y}^{3}\left(\frac{3}{4}\arctan\left[\tanh\frac{\tilde{\rho}}{2}\right]+\frac{3}{8}\frac{\sinh\tilde{\rho}}{\cosh^{2}\tilde{\rho}}+\frac{1}{4}\frac{\sinh\tilde{\rho}}{\cosh^{4}\tilde{\rho}}\right),$
$\displaystyle\xi=\gamma\tilde{y}^{4}\left(\frac{9}{16}\frac{1}{\cosh\tilde{\rho}}-\frac{3}{16}\frac{1}{\cosh^{3}\tilde{\rho}}+\frac{1}{8}\frac{1}{\cosh^{5}\tilde{\rho}}-\frac{9}{8}\tanh\rho\cdot\arctan\left[\tanh\frac{\tilde{\rho}}{2}\right]\right).$

This leads to the solution in the Gaussian normal coordinate:

$\displaystyle ds^{2}$ $\displaystyle\simeq$ $\displaystyle
d\tilde{\rho}^{2}+\cosh^{2}\tilde{\rho}\left[\frac{d\tau^{2}+dx^{2}+d\tilde{y}^{2}}{\tilde{y}^{2}}\right]+3\gamma\tilde{y}\cosh\tilde{\rho}\left(1+2\arctan\left[\tanh\frac{\tilde{\rho}}{2}\right]\sinh\tilde{\rho}\right)d\tilde{y}^{2}$
(A.14)
$\displaystyle+\tilde{y}\left(-\frac{3}{2}\gamma\cosh\tilde{\rho}+\left(\alpha+\frac{\gamma}{2}\right)\frac{1}{\cosh\tilde{\rho}}-\frac{3\gamma}{2}\arctan\left[\tanh\frac{\tilde{\rho}}{2}\right]\sinh
2\tilde{\rho}\right)d\tau^{2}$
$\displaystyle+\tilde{y}\left(-\frac{3}{2}\gamma\cosh\tilde{\rho}+\left(\beta+\frac{\gamma}{2}\right)\frac{1}{\cosh\tilde{\rho}}-\frac{3\gamma}{2}\arctan\left[\tanh\frac{\tilde{\rho}}{2}\right]\sinh
2\tilde{\rho}\right)dx^{2}.$

Due to the constraint $\alpha+\beta+\gamma=0$, there are two independent
solutions: one is $(\alpha,\beta,\gamma)=(-2V_{2},2V_{2},0)$ and the other is
$(\alpha,\beta,\gamma)=(U_{2}/6,U_{2}/6,-U_{2}/3)$. Each of them coincide with
those in the perturbative solutions in (A.3). Moreover, the solutions to
(A.12) have two ambiguities which allow us to shift
$\eta\to\eta+C_{1}\tilde{y}^{3}$ and $\xi\to\xi+C_{2}\tilde{y}^{4}$. They
correspond to $U_{1}$ and $B$ modes in (A.3), respectively. In the same way,
we can add a perturbation

$\displaystyle ds^{2}\to ds^{2}+\mbox{(const.)}\cdot zdtdx.$ (A.15)

This corresponds to the mode $A_{2}$ in (A.3), In this way, these static AdS
deformations can explain the modes $(U_{2},V_{2},A_{2})$ and the redundant
modes $(U_{1},B)$.

###  A.3 Full solutions with boundary condition imposed and relation to the TT
gauge result

Now we impose the Neumann boundary condition (A.16) to find perturbative
solutions in the AdS/BCFT. We can asssume that the EOW brane is situated at
$\rho=\rho_{*}$. Then the Neumann boundary condition leads to:

$\displaystyle\frac{1}{2}\frac{\partial}{\partial\rho}\hat{h}_{\mu\nu}=\tanh\rho_{*}\cdot\hat{h}_{\mu\nu}.$
(A.16)

First note that the modes $B$, $Q_{1}$ and $R_{1}$ automatically satisfy this
condition (A.16), though they are non-normalizble in the $\rho\to\infty$
region. This means that we cannot turn on either $B$, $Q_{1}$, $R_{2}$,
$Q_{2}$ or $R_{2}$. Then we can also find that for any $\rho_{*}$, a suitable
linear combination of $U_{1}$ and $U_{2}$ can satisfy the normalizability and
the boundary condition. Indeed, as we have seen in (A.14), it is clear that
the modes $(U_{1},U_{2})$ are normalizable, which corresponds to the $a=0$
mode (3.102) in the TT gauge analysis.

On the other hand, since $A_{1}$ and $V_{1}$ are non-normalizable and thus are
not available in our setup of AdS/BCFT, $V_{2}$ and $A_{2}$ can satisfy the
boundary condition (A.16) only when $\rho_{*}=0$ i.e. the vanishing tension
case. This is consistent with the speciality of the vanishing tension case we
found in subsection 3.6.

If we also take into account non-normalizable modes (i,e, which gets larger
than $O(e^{-\rho})$ in $\rho\to\infty$ limit), which are not dynamical in
AdS/BCFT, we have the complete matching between our analysis here and that in
the TT gauge result in section 3 as follows (note that Legendre $P$ function
gives non-normalizable mode near the AdS boundary):

$\displaystyle(i)\ \lambda_{\rho}=-1\ (a=0):$ $\displaystyle\sqrt{\cosh\rho}\
P^{3/2}_{-1/2}(\tanh\rho)\propto\sinh\rho\cosh\rho,\ \to U_{1}$
$\displaystyle\sqrt{\cosh\rho}\
P^{-3/2}_{-1/2}(\tanh\rho)\propto\left[\cosh\rho(1+2\sinh\rho\arctan\left(\tanh\frac{\rho}{2}\right)-\frac{\pi}{2}\sinh\rho\cosh\rho\right],\
\to(U_{1},U_{2})$ $\displaystyle(ii)\ \lambda_{\rho}=0\ (a=1):$
$\displaystyle\sqrt{\cosh\rho}\ P^{3/2}_{1/2}(\tanh\rho)\propto\cosh^{2}\rho,\
\to Q_{1},R_{1}$ $\displaystyle\sqrt{\cosh\rho}\
P^{-3/2}_{1/2}(\tanh\rho)\propto\cosh^{2}\rho\left[2\arctan\left(\tanh\frac{\rho}{2}\right)+\frac{\sinh\rho}{\cosh^{2}\rho}-\frac{\pi}{2}\right],\
\to(Q_{1},Q_{2}),(R_{1},R_{2})$ $\displaystyle(iii)\ \lambda_{\rho}=3\ (a=2):$
$\displaystyle\sqrt{\cosh\rho}\
P^{3/2}_{3/2}(\tanh\rho)\propto\tanh\rho(\sinh^{2}\rho+3),\ \to A_{1},V_{1}$
$\displaystyle\sqrt{\cosh\rho}\
P^{-3/2}_{3/2}(\tanh\rho)\propto\frac{1}{\cosh\rho}.\ \to A_{2},V_{2}.$ (A.17)

In the above, physical modes are $(U_{1},U_{2})$, $A_{2}$ and $V_{2}$. Other
modes are either non-normalizable or violating the boundary condition. The
mode (ii) $\lambda_{\rho}=0$ is not allowed due to the condition (3.60). Note
also that $B$ mode does not satisfy the traceless gauge condition
$h^{\mu}_{\mu}=0$. These perfectly match with the physical modes in the TT
gauge, discussed in subsection 3.6.

##  Appendix B Minimal surface in AdSd+1

In this appendix, we study the condimension-2 minimal surface in AdSd+1 with
the metric (5.21)

$\displaystyle
ds_{d+1}^{2}\,=\,d\rho^{2}+\frac{\cosh^{2}\rho}{y^{2}}\left(-dt^{2}+dy^{2}+\sum_{i=1}^{d-2}dx_{i}^{2}\right)\,,$
(B.1)

with a condition that the boundary of the surface in the $\rho\to\infty$ limit
is given by

$\displaystyle y^{2}\,+\,\sum_{i=1}^{d-2}x_{i}^{2}\,=\,l^{2}\,,\qquad
y\,\geq\,0\,.$ (B.2)

The determination of the minimal surface in this case is very much similar to
the one we discussed in section 5.1 for AdSd space.

First, we again move to the spherical coordinates by

$\displaystyle y\,$ $\displaystyle=\,r\cos\theta\,,$ (B.3) $\displaystyle
x_{i}\,$
$\displaystyle=\,r\sin\theta\cos\phi_{i}\prod_{n=1}^{i-1}\sin\phi_{n}\,,\qquad(i=1,\cdots,d-3)$
(B.4) $\displaystyle x_{d-2}\,$
$\displaystyle=\,r\sin\theta\prod_{n=1}^{d-3}\sin\phi_{n}\,,$ (B.5)

which brings the metric into

$\displaystyle
ds_{d+1}^{2}\,=\,d\rho^{2}+\frac{\cosh^{2}\rho}{r^{2}\cos^{2}\theta}\big{(}-dt^{2}+dr^{2}+r^{2}d\Omega_{d-2}^{2}\big{)}\,.$
(B.6)

We specify the surface by

$\displaystyle r\,=\,r(\rho,\theta,\phi_{1},\cdots,\phi_{d-3})\,,$ (B.7)

so that the induced metric is found as

$\displaystyle ds_{d-1}^{2}\,$
$\displaystyle=\,d\rho^{2}+\frac{\cosh^{2}\rho}{\cos^{2}\theta}\,d\Omega_{d-2}^{2}$
$\displaystyle\qquad\,+\,\frac{\cosh^{2}\rho}{\cos^{2}\theta}\left(\frac{\partial\log
r}{\partial\rho}d\rho+\frac{\partial\log
r}{\partial\theta}d\theta+\sum_{i=1}^{d-3}\frac{\partial\log
r}{\partial\phi_{i}}d\phi_{i}\right)^{2}\,.$ (B.8)

As before, we decompose this induced metric into two contributions as one from
the first line and the other from the second line of (B.8)

$\displaystyle\big{(}h_{d-1}\big{)}_{\mu\nu}\,=\,\big{(}h_{d-1}^{(0)}\big{)}_{\mu\nu}\,+\,\big{(}\tilde{h}_{d-1}\big{)}_{\mu\nu}\,.$
(B.9)

Again the $\big{(}\tilde{h}_{d-1}\big{)}_{\mu\nu}$ metric is given by a tensor
product of a vector

$\displaystyle\big{(}\tilde{h}_{d-1}\big{)}_{\mu\nu}\,=\,v_{\mu}\,(v^{T})_{\nu}\,,$
(B.10) $\displaystyle
v^{T}_{\mu}\,=\,\frac{\cosh\rho}{\cos\theta}\left(\frac{\partial\log
r}{\partial\rho}\,,\,\frac{\partial\log
r}{\partial\theta}\,,\,\frac{\partial\log
r}{\partial\phi_{1}}\,,\,\cdots\,,\,\frac{\partial\log
r}{\partial\phi_{d-3}}\right)\,,$ (B.11)

so that the determinant is decomposed as

$\displaystyle\det(h_{d-1})\,$
$\displaystyle=\,\det\Big{(}h_{d-1}^{(0)}+vv^{T}\Big{)}$
$\displaystyle=\,\det\big{(}h_{d-1}^{(0)}\big{)}\det\big{(}I_{d-1}+v^{T}(h_{d-1}^{(0)})^{-1}v\big{)}\,,$
(B.12)

Minimization of this determinant requires $v=0$, therefore, the minimal
surface is given by $y=l$.

## References

  * [1] J. L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B 240 (1984) 514–532.
  * [2] J. L. Cardy, Boundary conformal field theory, hep-th/0411189.
  * [3] D. M. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys. B 406 (1993) 655–680, [hep-th/9302068].
  * [4] D. M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522–576, [cond-mat/9505127].
  * [5] T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602, [arXiv:1105.5165].
  * [6] M. Fujita, T. Takayanagi, and E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043, [arXiv:1108.5152].
  * [7] M. Nozaki, T. Takayanagi, and T. Ugajin, Central Charges for BCFTs and Holography, JHEP 06 (2012) 066, [arXiv:1205.1573].
  * [8] A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, JHEP 06 (2001) 063, [hep-th/0105132].
  * [9] A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149, [arXiv:1908.10996].
  * [10] G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002, [arXiv:1905.08255].
  * [11] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063, [arXiv:1905.08762].
  * [12] L. Randall and R. Sundrum, A Large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370–3373, [hep-ph/9905221].
  * [13] L. Randall and R. Sundrum, An Alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690–4693, [hep-th/9906064].
  * [14] S. S. Gubser, AdS / CFT and gravity, Phys. Rev. D 63 (2001) 084017, [hep-th/9912001].
  * [15] A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008, [hep-th/0011156].
  * [16] S. B. Giddings, E. Katz, and L. Randall, Linearized gravity in brane backgrounds, JHEP 03 (2000) 023, [hep-th/0002091].
  * [17] T. Shiromizu and D. Ida, Anti-de Sitter no hair, AdS / CFT and the brane world, Phys. Rev. D 64 (2001) 044015, [hep-th/0102035].
  * [18] T. Shiromizu, T. Torii, and D. Ida, Brane world and holography, JHEP 03 (2002) 007, [hep-th/0105256].
  * [19] S. Nojiri, S. D. Odintsov, and S. Zerbini, Quantum (in)stability of dilatonic AdS backgrounds and holographic renormalization group with gravity, Phys. Rev. D 62 (2000) 064006, [hep-th/0001192].
  * [20] S. Nojiri and S. D. Odintsov, Brane world inflation induced by quantum effects, Phys. Lett. B 484 (2000) 119–123, [hep-th/0004097].
  * [21] S. W. Hawking, T. Hertog, and H. S. Reall, Brane new world, Phys. Rev. D 62 (2000) 043501, [hep-th/0003052].
  * [22] K. Koyama and J. Soda, Strongly coupled CFT in FRW universe from AdS / CFT correspondence, JHEP 05 (2001) 027, [hep-th/0101164].
  * [23] S. Kanno and J. Soda, Brane world effective action at low-energies and AdS / CFT, Phys. Rev. D 66 (2002) 043526, [hep-th/0205188].
  * [24] A. Almheiri, R. Mahajan, and J. E. Santos, Entanglement islands in higher dimensions, SciPost Phys. 9 (2020), no. 1 001, [arXiv:1911.09666].
  * [25] M. Rozali, J. Sully, M. Van Raamsdonk, C. Waddell, and D. Wakeham, Information radiation in BCFT models of black holes, JHEP 05 (2020) 004, [arXiv:1910.12836].
  * [26] H. Z. Chen, Z. Fisher, J. Hernandez, R. C. Myers, and S.-M. Ruan, Information Flow in Black Hole Evaporation, JHEP 03 (2020) 152, [arXiv:1911.03402].
  * [27] V. Balasubramanian, A. Kar, O. Parrikar, G. Sárosi, and T. Ugajin, Geometric secret sharing in a model of Hawking radiation, JHEP 01 (2021) 177, [arXiv:2003.05448].
  * [28] H. Geng and A. Karch, Massive islands, JHEP 09 (2020) 121, [arXiv:2006.02438].
  * [29] H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes, and J. Sandor, Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane, JHEP 10 (2020) 166, [arXiv:2006.04851].
  * [30] H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes, and J. Sandor, Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane, JHEP 12 (2020) 025, [arXiv:2010.00018].
  * [31] R. Bousso and E. Wildenhain, Gravity/ensemble duality, Phys. Rev. D 102 (2020), no. 6 066005, [arXiv:2006.16289].
  * [32] H. Z. Chen, Z. Fisher, J. Hernandez, R. C. Myers, and S.-M. Ruan, Evaporating Black Holes Coupled to a Thermal Bath, JHEP 01 (2021) 065, [arXiv:2007.11658].
  * [33] Y. Chen, V. Gorbenko, and J. Maldacena, Bra-ket wormholes in gravitationally prepared states, JHEP 02 (2021) 009, [arXiv:2007.16091].
  * [34] I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi, and Z. Wei, Entanglement Entropy in a Holographic Moving Mirror and the Page Curve, Phys. Rev. Lett. 126 (2021), no. 6 061604, [arXiv:2011.12005].
  * [35] M. Miyaji, Island for gravitationally prepared state and pseudo entanglement wedge, JHEP 12 (2021) 013, [arXiv:2109.03830].
  * [36] I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi, and Z. Wei, Holographic moving mirrors, Class. Quant. Grav. 38 (2021), no. 22 224001, [arXiv:2106.11179].
  * [37] H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas, and S. Shashi, Entanglement Phase Structure of a Holographic BCFT in a Black Hole Background, arXiv:2112.09132.
  * [38] A. Bhattacharya, A. Bhattacharyya, P. Nandy, and A. K. Patra, Bath deformations, islands, and holographic complexity, Phys. Rev. D 105 (2022), no. 6 066019, [arXiv:2112.06967].
  * [39] Q.-L. Hu, D. Li, R.-X. Miao, and Y.-Q. Zeng, AdS/BCFT and Island for curvature-squared gravity, arXiv:2202.03304.
  * [40] K. Suzuki and T. Takayanagi, BCFT and Islands in Two Dimensions, arXiv:2202.08462.
  * [41] T. Anous, M. Meineri, P. Pelliconi, and J. Sonner, Sailing past the End of the World and discovering the Island, arXiv:2202.11718.
  * [42] T. Kawamoto, T. Mori, Y.-k. Suzuki, T. Takayanagi, and T. Ugajin, Holographic Local Operator Quenches in BCFTs, arXiv:2203.03851.
  * [43] L. Bianchi, S. De Angelis, and M. Meineri, Radiation, entanglement and islands from a boundary local quench, arXiv:2203.10103.
  * [44] I. Akal, T. Kawamoto, S.-M. Ruan, T. Takayanagi, and Z. Wei, On the Page curve under final state projection, arXiv:2112.08433.
  * [45] I. Akal, T. Kawamoto, S.-M. Ruan, T. Takayanagi, and Z. Wei, Zoo of holographic moving mirrors, arXiv:2205.02663.
  * [46] M. Fujita, M. Kaminski, and A. Karch, SL(2,Z) Action on AdS/BCFT and Hall Conductivities, JHEP 07 (2012) 150, [arXiv:1204.0012].
  * [47] T. Ugajin, Two dimensional quantum quenches and holography, arXiv:1311.2562.
  * [48] J. Erdmenger, M. Flory, C. Hoyos, M.-N. Newrzella, A. O’Bannon, and J. Wu, Holographic impurities and Kondo effect, Fortsch. Phys. 64 (2016) 322–329, [arXiv:1511.09362].
  * [49] D. Seminara, J. Sisti, and E. Tonni, Corner contributions to holographic entanglement entropy in AdS4/BCFT3, JHEP 11 (2017) 076, [arXiv:1708.05080].
  * [50] D. Seminara, J. Sisti, and E. Tonni, Holographic entanglement entropy in AdS4/BCFT3 and the Willmore functional, JHEP 08 (2018) 164, [arXiv:1805.11551].
  * [51] Y. Hikida, Y. Kusuki, and T. Takayanagi, Eigenstate thermalization hypothesis and modular invariance of two-dimensional conformal field theories, Phys. Rev. D 98 (2018), no. 2 026003, [arXiv:1804.09658].
  * [52] T. Shimaji, T. Takayanagi, and Z. Wei, Holographic Quantum Circuits from Splitting/Joining Local Quenches, JHEP 03 (2019) 165, [arXiv:1812.01176].
  * [53] P. Caputa, T. Numasawa, T. Shimaji, T. Takayanagi, and Z. Wei, Double Local Quenches in 2D CFTs and Gravitational Force, JHEP 09 (2019) 018, [arXiv:1905.08265].
  * [54] M. Mezei and J. Virrueta, Exploring the Membrane Theory of Entanglement Dynamics, JHEP 02 (2020) 013, [arXiv:1912.11024].
  * [55] W. Reeves, M. Rozali, P. Simidzija, J. Sully, C. Waddell, and D. Wakeham, Looking for (and not finding) a bulk brane, JHEP 12 (2021) 002, [arXiv:2108.10345].
  * [56] Y. Kusuki, Analytic bootstrap in 2D boundary conformal field theory: towards braneworld holography, JHEP 03 (2022) 161, [arXiv:2112.10984].
  * [57] T. Numasawa and I. Tsiares, Universal Dynamics of Heavy Operators in Boundary CFT2, arXiv:2202.01633.
  * [58] M. Gutperle and J. Samani, Holographic RG-flows and Boundary CFTs, Phys. Rev. D 86 (2012) 106007, [arXiv:1207.7325].
  * [59] J. Estes, K. Jensen, A. O’Bannon, E. Tsatis, and T. Wrase, On Holographic Defect Entropy, JHEP 05 (2014) 084, [arXiv:1403.6475].
  * [60] N. Kobayashi, T. Nishioka, Y. Sato, and K. Watanabe, Towards a $C$-theorem in defect CFT, JHEP 01 (2019) 039, [arXiv:1810.06995].
  * [61] Y. Sato, Boundary entropy under ambient RG flow in the AdS/BCFT model, Phys. Rev. D 101 (2020), no. 12 126004, [arXiv:2004.04929].
  * [62] S. Chapman, D. Ge, and G. Policastro, Holographic Complexity for Defects Distinguishes Action from Volume, JHEP 05 (2019) 049, [arXiv:1811.12549].
  * [63] Y. Sato and K. Watanabe, Does Boundary Distinguish Complexities?, JHEP 11 (2019) 132, [arXiv:1908.11094].
  * [64] P. Braccia, A. L. Cotrone, and E. Tonni, Complexity in the presence of a boundary, JHEP 02 (2020) 051, [arXiv:1910.03489].
  * [65] Y. Sato, Complexity in a moving mirror model, arXiv:2108.04637.
  * [66] J. Hernandez, R. C. Myers, and S.-M. Ruan, Quantum extremal islands made easy. Part III. Complexity on the brane, JHEP 02 (2021) 173, [arXiv:2010.16398].
  * [67] S. Collier, D. Mazac, and Y. Wang, Bootstrapping Boundaries and Branes, arXiv:2112.00750.