Effective Action, Spectrum and First Law of Wedge Holography

Peng-Ju Hu and Rong-Xin Miao 111Email<EMAIL_ADDRESS>
School of Physics and Astronomy, Sun Yat-Sen University, 2 Daxue Road, Zhuhai
519082, China

###### Abstract

In this paper, we study the effective action, the mass spectrum and the first
law of entanglement entropy for a novel doubly holographic model called wedge
holography. We work out the effective action of quantum gravity on the branes.
In the perturbative formulation, it is given by an infinite sum of Pauli-Fierz
actions. In the non-perturbative formulation, the effective action is composed
of a higher derivative gravity and a matter action. Usually, a higher
derivative gravity can be renormalizable but suffers the ghost problem. For
our case, since the effective theory on the brane is equivalent to Einstein
gravity in the bulk, it must be ghost-free. We notice that the matter action
plays an important role in eliminating the ghost. We also provide evidences
that the higher derivative gravity on the brane is equivalent to a ghost-free
multi-gravity. Besides, we prove that the effective action yields the correct
Weyl anomaly. Interestingly, although the effective action on the brane is an
infinite tower of higher derivative gravity, the holographic Weyl anomaly is
exactly the same as that of Einstein gravity. We also analyze the mass
spectrum of wedge holography. Remarkably, there is always a massless mode of
gravitons on the end-of-the-world branes in wedge holography. This happens
because one imposes Neumann boundary condition on both branes. On the other
hand, the massless mode disappears if one imposes Dirichlet boundary condition
on one of the branes as in brane world theory and AdS/BCFT. Finally, we verify
the first law of entanglement entropy for wedge holography. Interestingly, the
massive fluctuations are irrelevant to the first order perturbation of the
holographic entanglement entropy. Thus, in many aspects, the effective theory
on the brane behaves like massless Einstein gravity.

###### Contents

  1. 1 Introduction
  2. 2 Effective action on the brane
    1. 2.1 Higher derivative gravity on the brane
    2. 2.2 Mechanism to eliminate ghosts
  3. 3 Holographic Weyl anomaly
  4. 4 Vector on the brane
    1. 4.1 Mass spectrum
    2. 4.2 Perturbative action
  5. 5 Gravity on the brane
    1. 5.1 Mass spectrum
    2. 5.2 Perturbative action
  6. 6 First law of entanglement entropy
    1. 6.1 The massive fluctuations
    2. 6.2 The massless fluctuations
  7. 7 Conclusions and Discussions
  8. A Multi-gravity on the brane
    1. A.1 Multi-gravity with four metrics
    2. A.2 Multi-gravity with six metrics

##  1 Introduction

Recently, a novel codimension two holography called wedge holography is
proposed [1], which conjectures that the following dualities hold

$\displaystyle\text{Classical gravity on wedge}\ W_{d+1}$
$\displaystyle\simeq$ $\displaystyle\text{(Quantum) gravity on two branes}\
(Q_{1}\cup Q_{2})$ $\displaystyle\simeq$ $\displaystyle\text{CFT}_{d-1}\
\text{on corner of wedge}\ \Sigma,$

where $W_{d+1}$ is the $d+1$ dimensional wedge space, which is bounded by two
end-of-the-world branes, i.e., $\partial W=Q_{1}\cup Q_{2}$, and $\Sigma$ is
the corner of the wedge. See Fig.1 (left) for the geometry. See also [2, 3, 4,
5, 6, 7] for some related works. Wedge holography is a generalization of the
AdS/CFT correspondence [8, 9, 10], and is closely related to brane world
holography [11, 12, 13], AdS/BCFT [14, 15, 16, 17, 18, 19, 20] and the doubly
holographic model [21, 22, 23]. See [24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] for recent works on double
holography and island. As shown in Fig.1 (right), wedge holography can be
obtained from AdS/BCFT by taking the zero-volume limit $M\to 0$. In this
limit, the bulk mode of CFTs on $M$ disappears and only the edge mode on the
boundary $\Sigma$ survives. Thus wedge holography can be regarded as a
holographic dual of the edge mode on the boundary (codim-1 defect) [4].
Generalizing wedge holography to codim-n defects, [4] proposes the so-called
cone holography, which can be derived from suitable limit of holographic
defect CFT [46, 47, 48].

The gravitational action of wedge holography is given by [1]

$\displaystyle I_{W}=\frac{1}{16\pi
G_{N}}\int_{W}\sqrt{|g|}(R-2\Lambda)+\frac{1}{8\pi G_{N}}\int_{Q_{1}\cup
Q_{2}}\sqrt{|h|}(K-T),$ (1)

where $W$ denotes $d+1$ dimensional wedge space, $Q_{1}$ and $Q_{2}$ denote
two end-of-the-world branes, $K$ is the trace of the extrinsic curvature, and
$T$ is the tension of the brane. Following [14], [1] proposes to impose
Neumann boundary condition (NBC) on the end-of-the-world branes

$\displaystyle\text{NBC}:\ \left(K^{ij}-(K-T)h^{ij}\right)|_{Q}=0.$ (2)

Actually, the Dirichlet boundary condition (DBC) [17]

$\text{DBC}:\ \delta h_{ij}|_{Q}=0,$ (3)

and the conformal boundary condition (CBC) [20]

$\displaystyle K=\frac{d}{d-1}T,$ (4a) $\displaystyle\delta
h_{ij}|_{Q}=2\sigma(y)h_{ij}|_{Q},$ (4b)

with $\sigma(y)$ a conformal factor also work well. See also [49, 50, 51, 52,
53, 54] for discussions on various boundary conditions of gravity. For
simplicity, we mainly focus on NBC in this paper. It is found that wedge
holography can yield the expected free energy, Weyl anomaly,
entanglement/Rényi entropy, two point functions and so on [1, 3]. In
particular, it obeys the holographic c-theorem [3]. For one novel class of
solutions, it is proved that wedge holography is equivalent to AdS/CFT with
Einstein gravity [3]. These are all strong supports for wedge holography.

The previous works [1, 3] mainly focus on a special class of solutions. In
this paper, we discuss the general solutions and gain more understanding of
wedge holography. We work out the effective action of quantum gravity on the
end-of-the-world branes with NBC. In the perturbative formulation, the
effective action is given by an infinite sum of Pauli-Fierz actions of massive
gravity. In the non-perturbative formulation, it is given by a higher
derivative gravity plus a matter action. Usually, a higher derivative gravity
suffers the problem of ghost. However, since the effective action on the brane
is equivalent to Einstein gravity in the bulk, it must be ghost-free. We
discuss the mechanism to eliminate the ghost and find that the matter action
plays an important role. We argue that the higher derivative gravity on the
brane is equivalent to a ghost-free multi-gravity. Besides, we prove that the
effective action can produce the correct Weyl anomaly. Interestingly, although
the effective action on the brane is an infinite tower of higher derivative
gravity, the holographic Weyl anomaly is exactly the same as that of Einstein
gravity. We also study the mass spectrum of wedge holography. Interestingly,
we find that there is a massless mode on the end-of-the-world branes if we
impose NBC on both of the two branes. On the other hand, the massless mode
disappears if one imposes DBC/CBC on one of the branes as in brane world
holography and AdS/BCFT. Finally, we verify the first law of entanglement
entropy and provide more supports for wedge holography.

Figure 1: (left) Geometry of wedge holography; (right) Wedge holography from
AdS/BCFT.

The paper is organized as follows. In section 2, we investigate the effective
action of gravity and discuss the mechanism to eliminate the ghost. In section
3, we derive the holographic Weyl anomaly from the effective action of gravity
on the brane. In section 4, we study the mass spectrum and perturbative
effective action of vectors on the end-of-the-world branes. In section 5, we
generalize the discussions to the gravitons on the end-of-the-world branes. In
section 6, we discuss the first law of entanglement entropy for wedge
holography. Finally, we conclude with some open problems in section 6.

Note added: It should be mentioned that, after this work has been finished,
there appears an interesting paper [55] in arXiv, which also discusses the
massless graviton on the brane.

##  2 Effective action on the brane

In this section, we investigate the effective action of wedge holography. For
simplicity, we focus on NBC. We leave the study of DBC/CBC to future works. We
consider only gravity in this section. The generalization to the vector and
the scalar is straightforward, but a little complicated.

###  2.1 Higher derivative gravity on the brane

We apply the method of [30], which is initially developed for the case of
large brane tension $T=(d-1)\tanh(\rho)\to(d-1)$, where the end-of-the-world
brane approaches to the AdS boundary. In fact, the method of [30] can be
generalized to arbitrary brane tension. See [56] for discussions of the case
$d=3$.

To warm up, let us first consider the case of large brane tension, i.e.,
$\rho\to\infty$ ($T\to d-1$), which means that the brane is set at infinity
and the geometry of wedge holography becomes that of AdS/CFT. In this limit,
the bulk gravitational action $I_{W}$ (1) is divergent. To get a finite
action, one can perform the holographic renormalization by adding the
following counterterms on the brane [57, 58]

$\displaystyle I_{c}=\frac{-1}{16\pi
G_{N}}\int_{Q}\sqrt{|h|}\Big{[}2(d-1)-2T+\frac{1}{d-2}\mathcal{R}+\frac{1}{(d-4)(d-2)^{2}}\left(\mathcal{R}^{ij}\mathcal{R}_{ij}-\frac{d}{4(d-1)}\mathcal{R}^{2}\right)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\mathcal{L}_{\mathcal{R}3}+...\Big{]},$ (5)

where $Q$ denotes $Q_{1}\cup Q_{2}$, $T=(d-1)\tanh(\rho)$ 111We add T in the
counterterm eq.(2.1) in order to compensate the brane action $-\frac{1}{8\pi
G_{N}}\int_{Q}\sqrt{|h|}T$ in the bulk action eq.(1). In the standard
holographic renormalization, we have $T=0$ due to the absence of brane
action., $\mathcal{R}_{ij}$ denotes the intrinsic curvature on the brane and
$\mathcal{L}_{\mathcal{R}3}$ is given by [59]

$\displaystyle\mathcal{L}_{\mathcal{R}3}=\frac{-2}{(d-2)^{3}(d-4)(d-6)}\Big{(}\frac{3d+2}{4(d-1)}\mathcal{R}\mathcal{R}_{ij}\mathcal{R}^{ij}-\frac{d(d+2)}{16(d-1)^{2}}\mathcal{R}^{3}-2\mathcal{R}^{ij}\mathcal{R}_{ikjl}\mathcal{R}^{kl}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \
+\frac{d}{4(d-1)}\mathcal{R}\Box\mathcal{R}-\mathcal{R}^{ij}\Box\mathcal{R}_{ij}\Big{)}.$
(6)

Now the renormalized action (the effective action of CFTs)

$\displaystyle I_{\text{CFT}}=I_{W}+I_{c},$ (7)

becomes finite. From eq.(7), we read off the action of wedge holography

$\displaystyle I_{W}=I_{\text{CFT}}-I_{c},$ (8)

which is composed of a matter part $I_{\text{CFT}}$ and a gravitational part
$(-I_{c})$.

Now set the brane at a finite place $r=\pm\rho$ instead of at infinity
$r=\pm\infty$. Then $I_{W}$ becomes finite. One can always separate $I_{W}$
into a matter part and a gravitational part. The natural conjecture is that
the gravitational part is still given by $(-I_{c})$ eq.(2.1) even if the brane
is located at a finite position. One support for this proposal is that
$(-I_{c})$ eq.(2.1) yields the correct leading terms of entanglement entropy
[30]. Let us explain more on this point. In double holography, one has two
methods to calculate the entanglement entropy. One method is by using
holographic entanglement entropy of higher derivative gravity $(-I_{c})$
eq.(2.1) on the brane [60, 61, 62], the other way is by applying TR formula of
Einstein gravity in the bulk [63]. It turns out that these two methods yield
the same entanglement entropy for large but finite brane tension [30]. Another
support for this proposal is that, as we will show in sect.3, $(-I_{c})$
eq.(2.1) yields the correct Weyl anomaly for wedge holography.

Note that $I_{c}$ eq.(2.1) includes infinite higher derivative terms for the
finite tension $T<(d-1)$. Note also that eq.(2.1) works well only for odd $d$.
For even $d$, one needs to correct $I_{c}$ eq.(2.1). See [30] for some
examples. For simplicity, we focus on the case of odd $d$ in this section.

To end this subsection, let us give a further evidence for the proposal
eq.(8). We verify that eq.(8) yields the correct effective action for the
following type of solutions

$\displaystyle ds^{2}=dr^{2}+\cosh^{2}(r)\bar{h}_{ij}dy^{i}dy^{i},$ (9)

where $-\rho\leq r\leq\rho$, and $\bar{h}_{ij}$ obeys the vacuum Einstein
equations on the brane

$\displaystyle R_{\bar{h}\
ij}-\frac{R_{\bar{h}}+(d-1)(d-2)}{2}\bar{h}_{ij}=0.$ (10)

Here $R_{\bar{h}\ ij}$ is the curvature with respect to $\bar{h}_{ij}$. The
effective action of wedge holography for the solution eq.(9) is given by [3]

$\displaystyle I_{\text{W Ein}}=\frac{1}{16\pi
G_{N}}\int_{0}^{\rho}\cosh^{d-2}(r)dr\int_{Q}dy^{d}\sqrt{|\bar{h}|}\Big{(}R_{\bar{h}}+(d-1)(d-2)\Big{)},$
(11)

where we focus on half of the wedge space, i.e, $0\leq r\leq\rho$, for
simplicity. Rewriting the above action in form of the induced metric
$h_{ij}=\cosh^{2}(\rho)\bar{h}_{ij}$ on the brane, we get

$\displaystyle I_{\text{W Ein}}=\frac{1}{16\pi
G_{N}}\int_{0}^{\rho}\frac{\cosh^{d-2}(r)}{\cosh^{d-2}(\rho)}dr\int_{Q}dy^{d}\sqrt{|h|}\Big{(}\mathcal{R}+\frac{(d-1)(d-2)}{\cosh^{2}(\rho)}\Big{)}.$
(12)

For the class of solution eq.(9), we have

$\displaystyle\mathcal{R}_{ij}=-(d-1)\text{sech}^{2}(\rho)h_{ij},\ \
\mathcal{R}=-d(d-1)\text{sech}^{2}(\rho).$ (13)

By applying the above formula, eq.(12) can be further simplified as

$\displaystyle I_{\text{W Ein}}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G_{N}}\int_{Q}dy^{d}\sqrt{|h|}(d-1)\text{sech}^{d}(\rho)B_{\text{sech}^{2}(\rho)}\left(1-\frac{d}{2},\frac{1}{2}\right)$
(14) $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G_{N}}\int_{Q}dy^{d}\sqrt{|h|}\Big{(}-\frac{2(d-1)\epsilon^{2}}{d-2}+\frac{(1-d)\epsilon^{4}}{d-4}-\frac{3(d-1)\epsilon^{6}}{4(d-6)}+O\left(\epsilon^{8}\right)\Big{)},$

for odd $d$, where $B_{a}(b,c)$ denotes the beta function and we have
reparameterized $\rho$ by

$\displaystyle\epsilon=\text{sech}(\rho).$ (15)

Note that the $\mathcal{R}^{n}$ terms in the counterterm eq.(2.1) are of
orders $O(\epsilon^{2n})$. Substituting eqs.(13,15) into the effective action
eq.(8), we derive

$\displaystyle I_{W}=I_{\text{CFT}}+\frac{1}{16\pi
G_{N}}\int_{Q}dy^{d}\sqrt{|h|}\Big{(}-\frac{2(d-1)\epsilon^{2}}{d-2}+\frac{(1-d)\epsilon^{4}}{d-4}-\frac{3(d-1)\epsilon^{6}}{4(d-6)}+O\left(\epsilon^{8}\right)\Big{)},$
(16)

which agrees with eq.(14) provided that $I_{\text{CFT}}=0$. The vanishing of
the CFT action has a natural explanation in odd dimensions. In the limit
$\rho\to\infty(\epsilon\to 0)$, the geometry of wedge holography becomes that
of AdS/CFT. We can check by direct calculations that $I_{\text{CFT}}$ eq.(7)
indeed vanishes for the class of solutions eq.(9) when $d$ is odd. It is
natural that $I_{\text{CFT}}$ remains zero as we adiabatically pull the brane
from infinity to a finite place. $I_{\text{CFT}}=0$ means that eq.(12) is the
vacuum solution to AdS/CFT, and the complete expressions of the counterterms
$(-I_{c})$ eq.(2.1) is given by eq.(12) for the solutions eq.(9). Let us go on
to discuss the case of finite tension. For simplicity, we assume that the
geometry on the brane is an AdS. Recall that the vacuum energy of CFTs
vanishes in odd AdS space, since there is no Weyl anomaly in odd dimensions
222One the other hand, the vacuum energy is non-zero for CFTs in even AdS
space, due to the non-trivial Weyl anomaly. . As a result, it is natural that
the effective action of CFTs also vanishes for odd $d$.

Now we finish the verification that $I_{W}$ eq.(8) gives the correct effective
action of wedge holography for the class of solutions eq.(9). As a by-product,
we notice that, $I_{\text{CFT}}=0$ and the complete expression of counterterms
$(-I_{c})$ eq.(2.1) is given by eq.(12) for the solutions eq.(9).

###  2.2 Mechanism to eliminate ghosts

As we have shown in the above subsection, the effective action on the end-of-
the-world brane is an infinite tower of higher derivative gravity. Usually, a
higher derivative gravity suffers the ghost problem. However, since the
effective action is obtained from Einstein gravity in the bulk, it must be
ghost-free. In this subsection, we discuss the mechanism to eliminate the
ghost and find that the matter action $I_{\text{CFT}}$ plays an important
role. We also argue that higher derivative gravity on the brane is equivalent
to a ghost-free multi-gravity.

Let us first give a quick review of the ghost problem of higher derivative
gravity. Take curvature squared gravity as an example. In general, it includes
a scalar mode, a massless graviton and a massive graviton. Consider the linear
perturbation equation around a flat-space background

$\displaystyle\Box\left(\Box-m^{2}\right)\delta h_{ij}=0,$ (17)

where $\Box$ is the D’Alembert operator and the metric perturbation $\delta
h_{ij}$ obeys $D_{i}\delta h^{ij}=0$ and $h^{ij}\delta h^{ij}=0$. From
eq.(17), we read off the propagator

$\displaystyle D(p)\sim\Big{(}\frac{1}{p^{2}}-\frac{1}{p^{2}+m^{2}}\Big{)},$
(18)

where we have ignored the indexes and the tensor structures. From eq.(18), it
is clear that the massless graviton and the massive graviton cannot both have
the correct sign. In other words, one of them is a ghost. Our case is similar.
The linear perturbation equation of the higher derivative gravity on the brane
takes the form

$\displaystyle\Pi_{n=0}^{\infty}(\Box+\frac{2}{L_{\text{eff}}^{2}}-m_{n}^{2})\delta
h_{ij}=0,$ (19)

where $m_{0}=0$ and $m_{n}$ denotes the mass of the nth mode. Without loss of
generality, we set $m_{0}^{2}<m_{1}^{2}<m_{2}^{2}<...$. For the large AdS
radius $L_{\text{eff}}=\cosh(\rho)\to\infty$, eq.(19) yields the following
propagator

$\displaystyle D(p)\sim\sum_{i=0}^{\infty}\Big{(}\Pi_{j\neq
i}\frac{1}{m_{j}^{2}-m_{i}^{2}}\Big{)}\frac{1}{p^{2}+m_{i}^{2}}$
$\displaystyle\ \ \ \ \ \ \
\sim\Big{(}\frac{a^{2}_{0}}{p^{2}}-\frac{a_{1}^{2}}{p^{2}+m_{1}^{2}}+\frac{a_{2}^{2}}{p^{2}+m_{2}^{2}}-\frac{a_{3}^{2}}{p^{2}+m_{3}^{2}}+...\Big{)},$
(20)

where $a_{i}^{2}$ are some positive constants related to the mass of
gravitons. The above propagator implies that half of the massive modes are
ghosts. However, this contradicts with the fact the effective action on the
brane must be ghost-free, since it is obtained from Einstein gravity in the
bulk.

Let us discuss the resolution to the above puzzle. To start, we want to
mention an enlightening example. It is found in [64] that, by eliminating one
of the metric, the ghost-free bimetric theory [65, 66]

$\displaystyle S[h,f]=m_{h}^{d-2}\int
dy^{d}\left(\sqrt{|h|}\mathcal{R}(h)+\alpha^{d-2}\sqrt{|f|}\mathcal{R}(f)-2m^{2}\sqrt{|h|}\sum_{n=0}^{d}\beta_{n}e_{n}(S)\right),$
(21)

is equivalent to the higher derivative gravity

$\displaystyle S^{HD}=m_{h}^{d-2}\int
dy^{d}\sqrt{|h|}\left(\Lambda+c_{R}\mathcal{R}(h)-\frac{c_{RR}}{m^{2}}\left(\mathcal{R}^{ij}\mathcal{R}_{ij}-\frac{d}{4(d-1)}\mathcal{R}^{2}\right)+O(\frac{\mathcal{R}^{3}}{m^{4}})\right),$
(22)

where $\alpha=m_{f}/m_{h}$ is the ratio of the Planck masses, $\beta_{n}$ are
dimensionless free parameters, $m^{2}$ sets the mass scale of the massive
mode, $\Lambda,c_{R},c_{RR}$ are some unimportant constants depending on the
parameters of bimetric gravity, $S^{i}_{j}=(\sqrt{h^{-1}f})^{i}_{j}$ and
$e_{n}(S)$ is given by

$\displaystyle
e_{n}(S)=-\frac{1}{n}\sum_{k=1}^{n}(-1)^{k}\text{Tr}(S^{k})e_{n-k}(S).$ (23)

Due to the equivalence to the ghost-free bimetric theory, the higher
derivative gravity eq.(22) must be ghost-free too. However, as we have
reviewed above, the curvature squared gravity includes a ghost. [64] argues
that the ghost problem is an artifact of the truncation at
$O(\mathcal{R}^{2})$. If one considers the complete action with infinite
higher derivative terms, the ghost can be removed. This is indeed the case in
a toy model of higher derivative scalar theory with suitable couplings to the
source [64]. It is found that the source structure plays an important role in
eliminating the ghost in the higher derivative theory [64].

Now turn to our case. Remarkably, the curvature squared term of eq.(22) takes
exactly the same form as that of the higher derivative gravity $I_{W}$
eqs.(2.1,8) on the brane. Following the approach of [64], we find that, by
choosing suitable parameters of the bimetric gravity, the $O(\mathcal{R}^{3})$
term eq.(2.1) of $I_{W}$ can also be recovered [67]. This strongly implies
that the higher derivative gravity on the brane is actually ghost-free. It
should be mentioned that, although the $O(\mathcal{R}^{2})$,
$O(\mathcal{R}^{3})$ terms are the same, the effective theory on the brane
cannot be equivalent to the bimetric gravity eq.(21). That is because they
have different degrees of freedom. There are infinite massive modes on the
brane, while there are only a massless mode and a massive mode in bimetric
gravity. The inconsistency of degrees of freedom can be naturally resolved by
considering multi-gravity with infinite metrics [68, 69]

$\displaystyle S_{N}=\lim_{N\to\infty}\frac{M^{d-2}}{2}\sum_{n=1}^{N}\int
dy^{d}\sqrt{|h_{n}|}\left(\mathcal{R}_{n}+\frac{m_{N}^{2}}{2}\sum_{m=0}^{d}\alpha_{m}^{(n)}\mathcal{L}_{m}\left(\mathcal{K}(h_{n},h_{n+1})\right)\right),$
(24)

where $h_{n}$ denotes the nth metric, $\mathcal{R}_{n}$ is the Ricci scalar of
$h_{n}$ and $\mathcal{L}_{m}\left(\mathcal{K}(h_{n},h_{n+1})\right)$ is the
interaction between neighboring metrics. Naturally, $h_{1}$ and $h_{N}$ are
the metrics on the two branes. Following the approach of [64], we eliminate
the other metrics and obtain two higher derivative gravity on the two branes

$\displaystyle S_{N}^{HD}$ $\displaystyle=\frac{M^{d-2}}{2}\int
dy^{d}\sqrt{|h_{1}|}\left(\Lambda_{1}+c_{R}\mathcal{R}_{1}-\frac{c_{RR}}{m_{N}^{2}}\left(\mathcal{R}_{1}^{\;ij}\mathcal{R}_{1\
ij}-\frac{d}{4(d-1)}\mathcal{R}_{1}^{2}\right)+O(\frac{\mathcal{R}_{1}^{3}}{m_{N}^{4}})\right)$
$\displaystyle+\frac{M^{d-2}}{2}\int
dy^{d}\sqrt{|h_{N}|}\left(\Lambda_{N}+c_{R}\mathcal{R}_{N}-\frac{c_{RR}}{m_{N}^{2}}\left(\mathcal{R}_{N}^{\;ij}\mathcal{R}_{N\
ij}-\frac{d}{4(d-1)}\mathcal{R}_{N}^{2}\right)+O(\frac{\mathcal{R}_{N}^{3}}{m_{N}^{4}})\right)$
$\displaystyle+\frac{M^{d-2}}{2}\int dy^{d}\mathcal{L}_{int}(h_{1},h_{N}),$
(25)

where $\mathcal{L}_{int}(h_{1},h_{N})$ denotes the interaction of
$(h_{1},h_{N})$. See the appendix for the derivations. Remarkably, the
$O(\mathcal{R}^{2})$ term takes exactly the same form as that of the effective
action $I_{W}$ eqs.(2.1,8). This is a strong evidence that the effective
theory on the brane is equivalent to a ghost-free multi-gravity. We leave a
careful study of this equivalence to future work [67].

To end this section, we provide an alternative mechanism to remove the ghost
in higher derivative gravity. To have a ghost-free unitary theory, we require
that the Euclidean effective action satisfies the following condition

$\displaystyle e^{-I_{W}}=e^{-I_{\text{CFT}}+I_{c}}\leq 1.$ (26)

Due to the ghost, the exponent of higher derivative action $e^{I_{c}}$ could
be larger than one. Thus, the matter action $I_{\text{CFT}}$ plays an
important role in defining a unitary theory on the brane. Indeed,
$I_{\text{CFT}}$ is large enough to make $e^{-I_{W}}$ less than one. That is
because the effective action $I_{W}$ on the brane is obtained from the
Einstein gravity in the bulk. As a result, it must satisfy the above condition

$\displaystyle e^{-I_{W}}=e^{-I_{\text{Ein bulk}}}\leq 1.$ (27)

This teaches us an interesting mechanism to eliminate the ghost. By adding
suitable matter fields to the higher derivative gravity, it is possible to
construct a ghost-free total theory. As we have discussed above, wedge
holography provides such an example. However, the effective action of wedge
holography includes infinite higher derivative terms, thus is too complicated.
Is it possible to construct a ghost-free total theory with finite higher
derivative terms? We hope this interesting problem could be addressed in
futures.

##  3 Holographic Weyl anomaly

In this section, we derive the holographic Weyl anomaly from the effective
action eq.(8) of wedge holography and show that it agrees with the result of
[3]. This can be regarded as a test of the effective action eq.(8).

According to [70], the Weyl anomaly [71] can be obtained from the UV
logarithmic divergent term of the gravitational action. Since $I_{\text{CFT}}$
is finite, it is expected that it does not contribute to the UV logarithmic
divergent term. Thus, we focus on the gravitational part $(-I_{c})$ of the
effective action eq.(8). We assume that the spacetime on the brane is
asymptotically AdS

$\displaystyle
ds_{Q}^{2}=h_{ij}dy^{i}dy^{j}=\cosh^{2}(\rho)\frac{dz^{2}+\sigma_{ab}dy^{a}dy^{b}}{z^{2}},$
(28)

where $y^{i}=(z,y^{a})$,
$\sigma_{ab}=\sigma^{(0)}_{ab}+z^{2}\sigma^{(1)}_{ab}+...+z^{d-1}(\sigma^{(\frac{d-1}{2})}_{ab}+\lambda^{(\frac{d-1}{2})}_{ab}\ln
z)$. By analyzing the asymptotical symmetry of AdS, [72] obtains a universal
relation for general higher derivative gravity

$\displaystyle\sigma^{(0)ab}\sigma^{(1)}_{ab}=-\frac{R_{\Sigma}}{2},\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for}\ d=3$ (29)
$\displaystyle\sigma^{(1)}_{ab}=\frac{-1}{d-3}(R_{\Sigma\
ab}-\frac{R_{\Sigma}}{2(d-2)}\sigma^{(0)}_{ab}),\ \ \text{for}\ d>3$ (30)

where $R_{\Sigma\ ab}$ is the curvature with respect to $\sigma^{(0)}_{ab}$.

We use the background-field method developed in [73] to investigate the
holographic Weyl anomaly. This method is quite useful for the study of Weyl
anomaly [73], correlation functions [74], entanglement/Rényi entropy [75, 76,
77] of higher derivative gravity. Expanding $I_{W}$ eq.(8) in terms of
background curvature $\bar{\mathcal{R}}$ defined below

$\displaystyle\mathcal{R}=\bar{\mathcal{R}}-d(d-1)\text{sech}^{2}(\rho),$ (31)
$\displaystyle\mathcal{R}_{ij}=\bar{\mathcal{R}}_{ij}-(d-1)\text{sech}^{2}(\rho)h_{ij},$
(32)
$\displaystyle\mathcal{R}_{ijkl}=\bar{\mathcal{R}}_{ijkl}-\text{sech}^{2}(\rho)(h_{ik}h_{jl}-h_{il}h_{jk}),$
(33)

we get

$\displaystyle I_{W}=I_{\text{CFT}}+I_{\text{W Ein}}+\frac{1}{16\pi
G_{N}}\int_{Q}\sqrt{|h|}\Big{[}\frac{\left(\bar{\mathcal{R}}^{ij}\bar{\mathcal{R}}_{ij}-\frac{d}{4(d-1)}\bar{\mathcal{R}}^{2}\right)}{(d-4)(d-2)^{2}}+\mathcal{L}_{\mathcal{R}3}(\bar{\mathcal{R}})+O(\bar{\mathcal{R}}^{4})\Big{]},$
(34)

where $I_{\text{W Ein}}$ eq.(12) is the action of Einstein gravity with an
effective Newton’s constant and AdS radius

$\displaystyle\frac{1}{G_{N\text{
eff}}}=\frac{1}{G_{N}}\int_{0}^{\rho}\frac{\cosh^{d-2}(r)}{\cosh^{d-2}(\rho)}dr,\
\ \ \ L_{\text{eff}}=\cosh(\rho).$ (35)

Recall that for the class of solutions eq.(48), we have $I_{\text{CFT}}=0$ and
$\bar{\mathcal{R}}_{ij}=\bar{\mathcal{R}}=\mathcal{L}_{\mathcal{R}3}(\bar{\mathcal{R}})=O(\bar{\mathcal{R}}^{4})=0$.
As a result, we get $I_{W}=I_{\text{W Ein}}$ for solutions eq.(48), which
agrees with the discussions of sect.2.1. It should be stressed that, in
general, $I_{W}$ and $I_{\text{W Ein}}$ are different.

We are interested of only the Weyl anomaly for 2d, 4d and 6d CFTs ($d=3,5,7$).
According to [73], we have for general solutions

$\displaystyle\sqrt{|h|}\left(\bar{\mathcal{R}}^{ij}\bar{\mathcal{R}}_{ij}-\frac{d}{4(d-1)}\bar{\mathcal{R}}^{2}\right)\sim
O(z^{8-d}),$ (36)
$\displaystyle\sqrt{|h|}\mathcal{L}_{\mathcal{R}3}(\bar{\mathcal{R}})\sim\sqrt{|h|}O(\bar{\mathcal{R}}^{4})\sim
O(z^{8-d}).$ (37)

Thus, the $O(\bar{\mathcal{R}}^{2}),O(\bar{\mathcal{R}}^{3})$ and
$O(\bar{\mathcal{R}}^{4})$ terms of eq.(34) are irrelevant to the UV
logarithmic divergent term of the effective action for $d<9$. Recall that
$I_{\text{CFT}}$ is irrelevant to the UV logarithmic divergent term too. Thus
only the Einstein action $I_{\text{W Ein}}$ of eq.(34) contributes to the
holographic Weyl anomaly. As a result, the holographic Weyl anomaly of wedge
holography $\text{AdSW}_{d+1}/\text{CFT}_{d-1}$ is exactly the same as that of
$\text{AdS}_{d}/\text{CFT}_{d-1}$ with Einstein gravity, provided that the
Newton’s constant and AdS radius are given by the effective ones eq.(35).

For the convenience of readers, we list the holographic Weyl anomaly for 2d,
4d and 6d CFTs below

$\displaystyle\mathcal{A}_{2d}=\int_{\Sigma}dx^{2}\sqrt{|\sigma|}\frac{c_{2d}}{24\pi}R_{\Sigma},$
(40)
$\displaystyle\mathcal{A}_{4d}=\int_{\Sigma}dx^{4}\sqrt{|\sigma|}[\frac{c}{16\pi^{2}}C_{\Sigma}^{ijkl}C_{\Sigma\
ijkl}-\frac{a}{16\pi^{2}}(R_{\Sigma}{}^{ijkl}R_{\Sigma}{}_{ijkl}-4R_{\Sigma}{}^{ij}R_{\Sigma}{}_{ij}+R_{\Sigma}^{2})],$
$\displaystyle\mathcal{A}_{6d}=\int_{\Sigma}dx^{6}\sqrt{|\sigma|}[\sum_{n=1}^{3}B_{n}I_{n}+2A\
E_{6}],$

where $C_{\Sigma\ ijkl}$ is the Weyl tensor on $\Sigma$, $I_{n}$ are the Weyl
invariant terms constructed from curvatures and their covariant derivatives,

$\displaystyle I_{1}=C_{\Sigma\ kijl}C^{\Sigma\ imnj}C_{\Sigma\ m\ \ n}^{\ \ \
\ kl},\ \ I_{2}=C_{\Sigma\ ij}^{\ \ \ \ kl}C_{\Sigma\ kl}^{\ \ \ \
mn}C_{\Sigma\ mn}^{\ \ \ \ \ \ ij},$ (41) $\displaystyle I_{3}=C_{\Sigma\
iklm}(\Box_{\Sigma}\delta^{i}_{j}+4R_{\Sigma}{}^{i}_{j}-\frac{6}{5}R_{\Sigma}\delta^{i}_{j})C^{\Sigma\
jklm},$ (42)

and $E_{2p}$ is the Euler density defined by

$\displaystyle
E_{2p}(R_{\Sigma})=\frac{1}{(8\pi)^{p}\Gamma(p+1)}\delta^{i_{1}i_{2}...i_{2p-1}i_{2p}}_{j_{1}j_{2}...j_{2p-1}j_{2p}}R_{\Sigma}{}^{j_{1}j_{2}}_{\
\ \ \ i_{1}i_{2}}...R_{\Sigma}{}^{j_{2p-1}j_{2p}}_{\ \ \ \ \ \ \ \
i_{2p-1}i_{2p}}.$ (43)

For wedge holography, the central charges of above Weyl anomaly are given by

$\displaystyle
c_{2d}=\frac{3}{2G_{N}}\int_{0}^{\rho}\cosh(r)dr=\frac{3}{2G_{N}}\sinh(\rho),$
(44) $\displaystyle
a=c=\frac{\pi}{8G_{N}}\int_{0}^{\rho}\cosh^{3}(r)dr=\frac{\pi}{96G_{N}}(9\sinh(\rho)+\sinh(3\rho)),$
(45) $\displaystyle A=\frac{\pi^{2}}{16G_{N}}\int_{0}^{\rho}\cosh^{4}(r)dr,\
B_{1}=-\frac{1}{256\pi G_{N}}\int_{0}^{\rho}\cosh^{4}(r)dr,$ (46)
$\displaystyle B_{2}=-\frac{1}{1024\pi G_{N}}\int_{0}^{\rho}\cosh^{4}(r)dr,\
B_{3}=\frac{1}{3072\pi G_{N}}\int_{0}^{\rho}\cosh^{4}(r)dr.$ (47)

Recall that we focus on half of the wedge space $0\leq r\leq\rho$ in above
discussions. For the whole wedge space, we should double the above central
charges. Note also that although the metrics on the two branes $Q_{1}$ and
$Q_{2}$ are different generally, the induced metrics $\sigma_{ab}$ on the
corner $\Sigma$ are the same.

As a summary, we obtain the holographic Weyl anomaly for wedge holography with
general solutions in this section. Remarkably, although the effective action
on the brane are infinite towers of higher derivative gravity, the holographic
Weyl anomaly is exactly the same as that of Einstein gravity. It should be
mentioned that the holographic Weyl anomaly for 2d and 4d CFTs eqs.(40,40)
have been derived in [3] for the special class of solution eq.(48). Here we
re-derive the same holographic Weyl anomaly for the most general solutions in
wedge holography. Since Weyl anomaly is independent of the states of CFTs
333CFTs in vacuum state and thermal state have the same central charges and
thus the same Weyl anomaly., the holographic Weyl anomaly should be irrelevant
to the bulk solutions. As we have shown in this section, this is indeed the
case. This self-consistency can be regarded as a test of wedge holography and
the effective action eq.(8).

##  4 Vector on the brane

In this section, we study the mass spectrum and the perturbative effective
action of vectors on the end-of-the-world brane. The case of gravity is
similar.

###  4.1 Mass spectrum

Figure 2: Geometry of wedge holography associated with the coordinate system
of eqs.(48,49).

Let us first discuss the mass spectrum. For simplicity, we focus on the probe
limit, where the bulk metric and the embedding function of $Q$ are given by

$\displaystyle ds^{2}=dr^{2}+\cosh^{2}(r)\bar{h}^{(0)}_{ij}(y)dy^{i}dy^{j},$
(48) $\displaystyle Q:\ r=\pm\rho.$ (49)

See Fig.2 for the geometry, where the branes are located at $r=\pm\rho$, and
$\bar{h}^{(0)}_{ij}(y)$ is the AdS metric with the unit radius $L=1$ on the
branes. We take the following ansatz of Maxwell’s fields in the bulk

$\displaystyle\mathcal{A}_{r}=0,\ \mathcal{A}_{i}=S(r)A_{i}(y).$ (50)

Substituting eq.(50) into Maxwell’s equations
$\nabla_{\mu}\mathcal{F^{\mu\nu}}=0$ and separating variables, we obtain

$\displaystyle\bar{D}_{i}F^{ij}-m^{2}_{v}A^{j}=0,$ (51)
$\displaystyle\cosh^{2}(r)S^{\prime\prime}(r)+(d-2)\sinh(r)\cosh(r)S^{\prime}(r)+m_{v}^{2}S(r)=0,$
(52)

where $F=dA$, $\bar{D}_{i}$ are the covariant derivatives with respect to
$\bar{h}^{(0)}_{ij}(y)$, and $m_{v}$ denotes the mass of vectors. Solving
eq.(52), we get

$\displaystyle
S(r)=\text{sech}^{\frac{d-2}{2}}(r)\left(c_{1}P_{\lambda_{v}}^{\frac{d-2}{2}}(\tanh
r)+c_{2}Q_{\lambda_{v}}^{\frac{d-2}{2}}(\tanh r)\right),$ (53)

where $P_{\lambda_{v}}^{\frac{d-2}{2}}$ and $Q_{\lambda_{v}}^{\frac{d-2}{2}}$
are the Legendre polynomials, $c_{1}$ and $c_{2}$ are integral constants and
$\lambda_{v}$ is given by

$\displaystyle\lambda_{v}=\frac{1}{2}\left(\sqrt{(d-3)^{2}+4m_{v}^{2}}-1\right).$
(54)

From eq.(54), we derive the expected Breitenlohner-Freedman (BF) bound of
massive vectors in $\text{AdS}_{d}$

$\displaystyle m_{v}^{2}\geq-(\frac{d-3}{2})^{2}.$ (55)

One can impose either the absolute BC or the relative BC for vectors on the
end-of-the-world branes

$\begin{split}&\text{absolute BC}:\mathcal{F}_{ni}|_{Q}=0,\\\ &\text{relative
BC}:{}^{*}\mathcal{F}_{ni}|_{Q}=0,\end{split}$ (56)

where $n$ denotes the normal direction, $i$ denotes the tangent direction,
$\mathcal{F}=d\mathcal{A}$ is the bulk field strength and ${}^{*}\mathcal{F}$
is the Hodge dual of $\mathcal{F}$. For our ansatz eqs.(48,49,50), the
absolute BC and relative BC become Neumann BC (NBC) and Dirichlet BC (DBC),
respectively

$\displaystyle\text{NBC}:S^{\prime}(\pm\rho)=0,$ (57)
$\displaystyle\text{DBC}:S(\pm\rho)=0.$ (58)

The dynamical field on the brane is the induced vector and its conjugate
momentum for NBC and DBC, respectively

$\begin{cases}\delta\mathcal{A}_{i}=\ S(\pm\rho)A_{i},&\mbox{for absolute
BC/NBC},\\\ \delta\mathcal{F}_{ni}=\pm S^{\prime}(\pm\rho)A_{i},&\mbox{for
relative BC/DBC}.\end{cases}$ (59)

Now we are ready to study the mass spectrum of vectors on the end-of-the-world
branes. Let us first study the case of NBC. For the following purpose, we
denote $S(r)$ by $S(r)=c_{1}\bar{P}(r)+c_{2}\bar{Q}(r)$ where
$\bar{P}(r)=\text{sech}^{\frac{d-2}{2}}(r)P_{\lambda_{v}}^{\frac{d-2}{2}}(\tanh
r)$ and
$\bar{Q}(r)=\text{sech}^{\frac{d-2}{2}}(r)Q_{\lambda_{v}}^{\frac{d-2}{2}}(\tanh
r)$. Imposing NBC eq.(57), we get

$\displaystyle M\cdot
c=\left(\begin{matrix}\bar{P}^{\prime}(\rho)&\bar{Q}^{\prime}(\rho)\\\
\bar{P}^{\prime}(-\rho)&\bar{Q}^{\prime}(-\rho)\end{matrix}\right)\cdot\left(\begin{matrix}c_{1}\\\
c_{2}\end{matrix}\right)=0.$ (60)

To have non-trivial solutions of $c_{1}$ and $c_{2}$, we must have

$\displaystyle|M|=\bar{P}^{\prime}(\rho)\bar{Q}^{\prime}(-\rho)-\bar{Q}^{\prime}(\rho)\bar{P}^{\prime}(-\rho)=0,$
(61)

which gives a constraint for the mass of the vector

$\displaystyle\text{NBC}:\ m_{v}^{2}\
\text{sech}^{d}(\rho)\left(P_{\lambda_{v}}^{\frac{d}{2}-2}(x)Q_{\lambda_{v}}^{\frac{d}{2}-2}(-x)-P_{\lambda_{v}}^{\frac{d}{2}-2}(-x)Q_{\lambda_{v}}^{\frac{d}{2}-2}(x)\right)=0,$
(62)

where $x=\tanh(\rho)$ and $\lambda_{v}$ is a function of the mass eq.(54).

It is clear that the massless mode with $m_{v}=0$ is a solution to eq.(62) for
wedge holography with NBC. On the other hand, there is no massless mode for
the usual brane world theory 444For simplicity, we focus on one side of the
brane in brane world theory, where the junction condition is equivalent to
NBC. and AdS/BCFT defined in the region $-\rho\leq r\leq\infty$ with NBC on
the brane $S^{\prime}(-\rho)=0$ and DBC $S(\infty)=0$ on the AdS boundary. Let
us explain more on this point. Solving eq.(52) with $m_{v}=0$, we get

$\displaystyle S(r)=c_{1}+c_{2}\int_{0}^{r}\text{sech}^{d-2}(s)ds.$ (63)

Imposing NBC $S^{\prime}(\pm\rho)=0$ for wedge holography, we derive

$\displaystyle S(r)=c_{1},$ (64)

which yields non-zero induced vector
$\mathcal{A}_{i}=S(\pm\rho)A_{i}=c_{1}A_{i}$ on the brane. On the other hand,
the BCs $S^{\prime}(-\rho)=0$ and $S(\infty)=0$ for brane world theory and
AdS/BCFT yield $S(r)=0$. As a result, there is no massless mode in the usual
brane world holography and AdS/BCFT. We take the existence of massless modes
as an advantage of the wedge holography with NBC, since there are Maxwell’s
fields on the end-of-the-world branes. Furthermore, the massless mode can be
normalized since wedge holography is defined in a finite region $-\rho\leq
r\leq\rho$. However, since $S(r)$ eq.(64) is a constant in the bulk, the
massless vector is not located perfectly on the brane. By perfect
localization, we mean that the wave function $S(r)$ peaks on the brane only
and decays when it goes far from the brane. The massless mode is a critical
case: the wave function $S(r)$ neither increases nor decreases but keeps a
constant when it goes far from the brane. The localization looks better in the
coordinate that the EOM eq.(51) takes the form of schrödinger equation. See
sect. 3.1 for more discussions.

Similar to [20], there is no solution to (62) for $m^{2}<-(d-3)^{2}/4$ outside
the Breitenlohner-Freedman (BF) bound. Naively, the negative value of $m^{2}$
within the BF bound

$m_{v}^{2}=\begin{cases}-\frac{1}{4}((d-3)^{2}-1),&\ \text{even $d$},\\\
-\frac{1}{4}(d-3)^{2},&\ \text{odd $d$},\end{cases}$ (65)

is a solution to eq.(62). However, according to [20], for these special
values, (53) no longer gives the general solution since the Legendre functions
$P^{n}_{0}(x),Q^{n+\frac{1}{2}}_{-\frac{1}{2}}(x)$ vanish identically for
integer $n$. In this case, the general solution is given by

$S(r)=\begin{cases}\text{sech}^{\frac{d-2}{2}}(r)\left(c_{1}e^{\frac{d-2}{2}r}+c_{2}e^{-\frac{d-2}{2}r}\right),&\
\text{even $d$},\\\
\text{sech}^{\frac{d-2}{2}}(r)\left(c_{1}P^{\frac{d-2}{2}}_{-\frac{1}{2}}(\tanh
r)+c_{2}P^{-\frac{d-2}{2}}_{-\frac{1}{2}}(\tanh r)\right),&\ \text{odd
$d$}.\end{cases}$ (66)

For $\rho>0$, the above solutions cannot satisfy NBC $S^{\prime}(\pm\rho)=0$,
so we should rule out (65) from the spectrum. As a result, we have
$m_{v}^{2}\geq 0$ for the mass spectrum of vectors with NBC.

Let us draw some figures to illustrate the spectrum. See Fig.3, where the
intersections of the curves and $m_{v}^{2}$-axis denote the mass squares. As
shown in Fig.3, the massless mode is indeed a solution. Besides, the larger
the tension $T=(d-1)\tanh(\rho)$ is, the continuous the mass spectrum is.

Figure 3: Mass spectrum for vectors with NBC. The left figure is for $d=3$,
and the right figure is for $d=4$, $|M|$ is given by eq.(62) and the roots of
$|M|=0$ denote the mass squares. The massless mode is always a solution to
NBC. Besides, the larger the tension $T=(d-1)\tanh(\rho)$ is, the continuous
the mass spectrum is.

To get more understandings of the spectrum, let us study two limits. For the
large $\rho\to\infty$ ($x\to 1$), eq.(62) can be approximately by

$0\sim\text{sech}^{\frac{d}{2}+2}(\rho)\begin{cases}\sin(\lambda_{v}\pi)\,&\
\text{even d},\\\ \cos(\lambda_{v}\pi),&\ \text{odd d},\end{cases}$ (67)

which has the roots

$m_{v}^{2}\approx k(k+d-3),\,\ \ \ \ \text{for large }\rho,$ (68)

where $k\geq 0$ are integers. In the contrary limit of small $\rho$,
$-\rho\leq r\leq\rho$ is also small. Then eq.(52) becomes approximately by

$\displaystyle S^{\prime\prime}(r)+m_{v}^{2}S(r)=0,$ (69)

which can be solved as

$\displaystyle S(r)=c_{1}\cos(|m_{v}|r)+c_{2}\sin(|m_{v}|r).$ (70)

Imposing NBC eq.(57) and following the above approach, we get

$\displaystyle m_{v}^{2}\sin(2|m_{v}|\rho)=0,$ (71)

which yields the mass spectrum

$m_{v}^{2}\approx\frac{k^{2}\pi^{2}}{4\rho^{2}},\,\ \ \ \ \text{for small
}\rho.$ (72)

As a summary, the mass spectrum of vectors with NBC is non-negative, i.e.,
$m_{v}^{2}\geq 0$. In particular, it contains a massless mode, which is quite
different from the brane world holography and AdS/BCFT. For the large $\rho$,
the mass spectrum is given by eq.(68). While for small $\rho$, the mass
spectrum is approximately by eq.(72), which is independent of the dimensions
of spacetime.

Let us go on to study the mass spectrum of DBC eq.(58). Since the approach is
quite similar to that of NBC eq.(57), we do not repeat the calculations but
just list main results below. The constraint of the mass spectrum with DBC is
given by

$\displaystyle\text{DBC}:\
\text{sech}^{d-2}(\rho)\left(P_{\lambda_{v}}^{\frac{d}{2}-1}(x)Q_{\lambda_{v}}^{\frac{d}{2}-1}(-x)-P_{\lambda_{v}}^{\frac{d}{2}-1}(-x)Q_{\lambda_{v}}^{\frac{d}{2}-1}(x)\right)=0,$
(73)

where $x=\tanh(\rho)$. The mass spectrum of vectors with DBC includes only
positive $m_{v}^{2}>0$. Unlike NBC, the massless mode is no longer a solution
to DBC. One can easily check that the solution eq.(63) with $m_{v}=0$ cannot
satisfy the DBC $S(\pm\rho)=0$ unless $S(r)=0$. In the large and small $\rho$
limit, except a massless mode, the mass spectrum of DBC is exactly the same as
that of NBC

$\text{DBC}:\ m_{v}^{2}\approx\begin{cases}l(l+d-3),&\ \text{for large
$\rho$},\\\ \frac{l^{2}\pi^{2}}{4\rho^{2}},&\ \text{for small
$\rho$},\end{cases}$ (74)

where $l>0$ is a positive integer. See Fig. 4 for the mass spectrum of vectors
with DBC.

Figure 4: Mass spectrum for vectors with DBC. The left figure is for $d=3$,
and the right figure is for $d=4$, $|M|$ is given by eq.(73) and the roots of
$|M|=0$ denote the mass squares. The small circle at origin means that the
massless mode is removed from the mass spectrum for DBC. The larger the
tension $T=(d-1)\tanh(\rho)$ is, the continuous the mass spectrum is.

###  4.2 Perturbative action

Let us go on to study the perturbative action of vectors on the end-of-the-
world branes. We expand the bulk vectors in infinite powers of Kaluza-Klein
(KK) modes

$\displaystyle\mathcal{A}_{r}=0,\
\mathcal{A}_{i}=\sum_{m_{v}}S^{(m_{v})}(r)A^{(m_{v})}_{i}(y),$ (75)

where $\sum_{m_{v}}$ denotes the sum over the mass spectrum,
$A^{(m_{v})}_{i}(y)$ and $S^{(m_{v})}(r)$ obey EOM eqs.(51,52) and the
orthogonal condition

$\displaystyle\int_{-\rho}^{\rho}\cosh(r)^{d-4}S^{(m_{v})}(r)S^{(m^{\prime}_{v})}(r)dr=\delta^{m_{v},m^{\prime}_{v}}.$
(76)

Substituting eq.(75) into the bulk action

$\displaystyle
I=-\frac{1}{4}\int_{N}d^{d+1}x\sqrt{|g|}\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu},$
(77)

we get

$\displaystyle
I=-\frac{1}{4}\sum_{m_{v},m^{\prime}_{v}}\int_{-\rho}^{\rho}dr\int_{Q}d^{d}y\sqrt{|\bar{h}^{(0)}|}\Big{[}2\cosh(r)^{d-2}S^{\prime(m_{v})}(r)S^{\prime(m^{\prime}_{v})}(r)A^{(m_{v})}_{i}A^{(m^{\prime}_{v})i}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\
+\cosh(r)^{d-4}S^{(m_{v})}S^{(m^{\prime}_{v})}F^{(m_{v})}_{ij}F^{(m^{\prime}_{v})ij}\Big{]},$
(78)

where $A^{i}$ and $F^{ij}$ are raised by $\bar{h}^{(0)ij}$. Integrating by
parts and imposing either NBC $S^{\prime}(\pm\rho)=0$ or DBC $S(\pm\rho)=0$,
we derive

$\displaystyle
I=-\frac{1}{4}\sum_{m_{v},m^{\prime}_{v}}\int_{-\rho}^{\rho}dr\int_{Q}d^{d}y\sqrt{|\bar{h}^{(0)}|}\Big{[}-\frac{d}{dr}\left(2\cosh(r)^{d-2}S^{\prime(m_{v})}(r)\right)S^{(m^{\prime}_{v})}(r)A^{(m_{v})}_{i}A^{(m^{\prime}_{v})i}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\
+\cosh(r)^{d-4}S^{(m_{v})}S^{(m^{\prime}_{v})}F^{(m_{v})}_{ij}F^{(m^{\prime}_{v})ij}\Big{]}.$
(79)

By using EOM eq.(52) and orthogonal condition eq.(76), the above action can be
simplified as

$\displaystyle
I=-\frac{1}{4}\sum_{m_{v}}\int_{-\rho}^{\rho}dr\cosh(r)^{d-4}S^{(m_{v})}(r)^{2}\int_{Q}d^{d}y\sqrt{|\bar{h}^{(0)}|}\Big{[}2m_{v}^{2}A^{(m_{v})}_{i}A^{(m_{v})i}+F^{(m_{v})}_{ij}F^{(m_{v})ij}\Big{]}$
$\displaystyle\ \ \
=-\frac{1}{4}\sum_{m_{v}}\int_{Q}d^{d}y\sqrt{|\bar{h}^{(0)}|}\Big{[}2m_{v}^{2}A^{(m_{v})}_{i}A^{(m_{v})i}+F^{(m_{v})}_{ij}F^{(m_{v})ij}\Big{]},$
(80)

which is the correct action of massive vectors.

##  5 Gravity on the brane

In this section, we investigate the mass spectrum and the perturbative
effective action of gravitons on the end-of-the-world branes. Interestingly,
the mass spectrum of gravitons in $d$ dimensions is the same as that of
vectors in $(d+2)$ dimensions. In particular, there is a massless mode when
one imposes NBC on both branes.

###  5.1 Mass spectrum

Let us first discuss the mass spectrum of gravitons on the end-of-the-world
branes. We choose the following ansatz of the perturbation metric and the
embedding function of $Q$

$\displaystyle ds^{2}=dr^{2}+\cosh^{2}(r)\left(\bar{h}^{(0)}_{ij}(y)+\epsilon
H(r)\bar{h}^{(1)}_{ij}(y)\right)dy^{i}dy^{j}+O(\epsilon^{2}),$ (81)
$\displaystyle Q:\ r=\pm\rho+O(\epsilon^{2}),$ (82)

where $\bar{h}^{(0)}_{ij}(y)$ is the AdS metric with a unit radius and
$\bar{h}^{(1)}_{ij}(y)$ denotes the perturbation. In terms of bulk metric
perturbations, we have

$\displaystyle\delta g_{r\mu}=0,\ \delta
g_{ij}=\cosh^{2}(r)H(r)\bar{h}^{(1)}_{ij}(y).$ (83)

Imposing the transverse traceless gauge

$\displaystyle\nabla^{\mu}\delta g_{\mu\nu}=0,\ \ \ g^{\mu\nu}\delta
g_{\mu\nu}=0,$ (84)

we get

$\displaystyle\bar{D}^{i}\bar{h}^{(1)}_{ij}=0,\ \ \
\bar{h}^{(0)ij}\bar{h}^{(1)}_{ij}=0,$ (85)

where $\nabla_{\mu}$ and $\bar{D}_{i}$ are the covariant derivatives with
respect to $g_{\mu\nu}$ and $\bar{h}^{(0)}_{ij}$, respectively. Note that the
gauge $\bar{h}^{(0)ij}\bar{h}^{(1)}_{ij}=0$ eliminates the scalar mode in the
metric fluctuation. Since there is only a scalar degree of freedom for the
metric in two dimensions, the gauge eq.(85) removes all the degree of freedom
for the 2-dimensional metric. As a result, the discussions of this section
works only for $d\geq 3$.

One can impose either NBC [14], DBC [17] or CBC [20] on the end-of-the-world
branes

$\displaystyle\text{NBC}:\ \ \ \ \ \ \ \ H^{\prime}(\pm\rho)=0,$ (86)
$\displaystyle\text{DBC/CBC}:H(\pm\rho)=0,$ (87)

where CBC specifies the conformal geometry of the boundary and the trace of
the extrinsic curvature. Note that DBC and CBC are the same at the linear
order of perturbations, they are different at higher orders generally [20].
For DBC/CBC, we have the freedom to choose the induced metric on the brane.
For simplicity, we choose the same induced metric, the same background metric
eq.(48) in the bulk and the same location of branes eq.(82) as those of NBC.
Remarkably, the mass spectrum is independent of the choice of the induced
metric $\bar{h}^{(0)}_{ij}(y)$, as long as it satisfies Einstein equations
eq.(10) on the brane. That is because, for any $\bar{h}^{(0)}_{ij}(y)$ obeying
eq.(10), the boundary conditions eqs.(86,87) and the equation of motion of
$H(r)$ eq.(90) are the same. As a result, the mass spectrum determined by the
boundary condition and the equation of motion of $H(r)$ is independent of the
choice of the induced metric $\bar{h}^{(0)}_{ij}(y)$, as long as it obeys
eq.(10).

In the gauge eq.(84), Einstein equations become

$\displaystyle\left(\nabla_{\alpha}\nabla^{\alpha}+2\right)\delta
g_{\mu\nu}=0.$ (88)

Substituting eq.(83) together with eq.(85) into eq.(88) and separating
variables, we obtain

$\displaystyle\left(\bar{D}_{k}\bar{D}^{k}+2-m_{g}^{2}\right)\bar{h}^{(1)}_{ij}(y)=0,$
(90)
$\displaystyle\cosh^{2}(r)H^{\prime\prime}(r)+d\sinh(r)\cosh(r)H^{\prime}(r)+m_{g}^{2}H(r)=0,$

where $m_{g}$ denotes the mass of gravitons. Note that we have assumed that
$\bar{h}^{(0)}_{ij}(y)$ is an AdS metric in the above derivations. For general
$\bar{h}^{(0)}_{ij}(y)$ obeying eq.(10), eq.(90) is unchanged but eq.(90)
becomes

$\displaystyle\left(\bar{D}_{k}\bar{D}^{k}+2-m_{g}^{2}\right)\bar{h}^{(1)}_{ij}(y)+2C^{(0)}_{imjn}\bar{h}^{(1)}{}^{mn}(y)=0,$
(91)

where $C^{(0)}_{imjn}$ is the Weyl tensor defined by $\bar{h}^{(0)}_{ij}(y)$.

Solving eq.(90), we get

$\displaystyle
H(r)=\text{sech}^{\frac{d}{2}}(r)\left(c_{1}P_{\lambda_{g}}^{\frac{d}{2}}(\tanh
r)+c_{2}Q_{\lambda_{g}}^{\frac{d}{2}}(\tanh r)\right),$ (92)

where $P_{\lambda_{g}}^{\frac{d}{2}}$ and $Q_{\lambda_{g}}^{\frac{d}{2}}$ are
the Legendre polynomials, $c_{1}$ and $c_{2}$ are integral constants and
$\lambda_{g}$ is given by

$\displaystyle\lambda_{g}=\frac{1}{2}\left(\sqrt{(d-1)^{2}+4m_{g}^{2}}-1\right),$
(93)

which yields the correct BH bound of massive gravity in $\text{AdS}_{d}$

$\displaystyle m_{g}^{2}\geq-(\frac{d-1}{2})^{2}.$ (94)

Recall that we have set the AdS radius $L=1$.

Remarkably, the EOM of $H(r)$ eq.(90) in $d$ dimensions is exactly the same as
that of $S(r)$ eq.(52) in $(d+2)$ dimensions. Furthermore, $H(r)$ and $S(r)$
obey the same boundary conditions eqs.(57,58,86,87). As a result, the mass
spectrum of gravitons in $d$ dimensions is the same as the spectrum of vectors
in $(d+2)$ dimensions. Thus there is no need to repeat the calculations again.
For the convenience of readers, we list the key characteristics of the
gravitational spectrum below.

1. The gravitational spectrum obeys the following constraint

$\displaystyle\text{NBC}:\ m_{g}^{2}\
\text{sech}^{d+2}(\rho)\left(P_{\lambda_{g}}^{\frac{d}{2}-1}(x)Q_{\lambda_{g}}^{\frac{d}{2}-1}(-x)-P_{\lambda_{g}}^{\frac{d}{2}-1}(-x)Q_{\lambda_{g}}^{\frac{d}{2}-1}(x)\right)=0,$
(95) $\displaystyle\text{DBC/CBC}:\
\text{sech}^{d}(\rho)\left(P_{\lambda_{g}}^{\frac{d}{2}}(x)Q_{\lambda_{g}}^{\frac{d}{2}}(-x)-P_{\lambda_{g}}^{\frac{d}{2}}(-x)Q_{\lambda_{g}}^{\frac{d}{2}}(x)\right)=0,$
(96)

where $x=\tanh(\rho)$ and $\lambda_{g}=\lambda_{v}(d\to d+2)$ is given by
eq.(93).

2. We have $m_{g}^{2}\geq 0$ for NBC and $m_{g}^{2}>0$ for DBC/CBC. In particular, there is a massless mode when one imposes NBC on both branes. This is consistent with the result of [3], which finds that, for one class of solutions to NBC, the effective gravity on the branes is Einstein gravity. On the other hand, there is no massless mode in the brane-world holography [11, 12, 13] unless the brane tension approaches the critical value $T=(d-1)$.

3. Let us make some comments on the localization for the massless mode on the brane. By localization, we mean that the wave function peaks on the brane only and decays when it goes far from the brane. In the geodesic coordinate $r$, the wave function $H(r)$ is a constant for the massless mode. Thus, it is not located perfectly on the brane. Instead, it is a critical case. In the coordinate $w$ defined below, the localization looks better. Performing the transformations

$\displaystyle dw=\frac{dr}{\cosh(r)},\ \Psi(w)=\cosh^{\frac{d-1}{2}}(r)H(r),$
(97)

we rewrite eq.(90) into the form of schrödinger equation

$\displaystyle-\Psi^{\prime\prime}(w)+V(w)\Psi(w)=m_{g}^{2}\Psi(w),$ (98)

where the “volcano potential” is given by

$\displaystyle
V(w)=\frac{1}{4}(d-1)\Big{(}(d+1)\sec^{2}(w)-d+1\Big{)}-(d-1)\sinh(\rho)\delta(w\pm
w_{0}),$ (99)

where $w_{0}=\sin^{-1}(\text{tanh}(\rho))$ and the two branes are located at
$w=\pm w_{0}$. Due to the “volcano potential”, the massless mode tends to be
located on the branes at low energies. Indeed, as shown in Fig. 5, the wave
function $\Psi(w)$ peaks on the brane only and decays when it goes far from
the brane. However, $\Psi(w)$ does not vanish in the middle of the wedge space
$w=0$. Thus, the massless mode is not perfectly located on the brane.

Figure 5: (Left) wave function of massless graviton, which shows that the
massless graviton is located on the branes at $w=\pm w_{0}$; (Right) “volcano
potential”, where the negative delta function potential on the brane is
ignored. We have $d=4$ and $w_{0}\approx 0.87$ ($\rho=1$) for above figures.

4. In the large and small $\rho$ limit, except a massless mode, the mass spectrum of NBC is exactly the same as that of DBC/CBC

$\text{NBC/DBC/CBC}:\ m_{g}^{2}\approx\begin{cases}k(k+d-1),&\ \text{for large
$\rho$},\\\ \frac{k^{2}\pi^{2}}{4\rho^{2}},&\ \text{for small
$\rho$},\end{cases}$ (100)

where $k$ is an integer, $k\geq 0$ for NBC and $k>0$ for DBC/CBC. Recall that
$m^{2}_{g}$ is defined in eq.(90) with respect to $\bar{h}_{ij}^{(1)}$ instead
of the induced metric $\cosh^{2}(\rho)\bar{h}_{ij}^{(1)}$. For the induced
metric, the mass is defined by

$\displaystyle\left(\Box+\frac{2}{\cosh^{2}(\rho)}-M_{g}^{2}\right)\bar{h}^{(1)}_{ij}(y)=0,$
(101)

where $\Box=\bar{D}^{i}\bar{D}_{i}/\cosh^{2}(\rho)$ is the D’Alembert operator
with respect to $\cosh^{2}(\rho)\bar{h}_{ij}^{(1)}$. Comparing eq.(90) with
eq.(101), we read off

$\displaystyle M_{g}^{2}=\frac{m_{g}^{2}}{\cosh^{2}(\rho)}.$ (102)

For the large $\rho$, the mass spectrum of $M_{g}^{2}$ becomes almost
continuous. While for the small $\rho$, the massive modes are frozen at low
energy due to the infinite masses and only the massless mode is excited. As a
result, the effective theory of wedge holography at low energy is Einstein
gravity.

5. Let us draw some figures to illustrate the spectrum. See Fig.6 and Fig.7 for NBC and DBC respectively, where the intersections of the curves and $m_{g}^{2}$-axis denote the mass squares. We notice that the massless mode is indeed a solution to NBC. For both NBC and DNC, the larger the tension $T=(d-1)\tanh(\rho)$ is, the continuous the mass spectrum is.

Figure 6: Mass spectrum for gravitons with NBC. The left figure is for $d=3$,
and the right figure is for $d=4$, $|M|$ is given by eq.(95) and the roots of
$|M|=0$ denote the mass squares. The massless mode is always a solution to
NBC. Besides, the larger the tension $T=(d-1)\tanh(\rho)$ is, the continuous
the mass spectrum is.

Figure 7: Mass spectrum for gravitons with DBC. The left figure is for $d=3$,
and the right figure is for $d=4$, $|M|$ is given by eq.(96) and the roots of
$|M|=0$ denote the mass squares. The small circle at origin means that the
massless mode is removed from the mass spectrum for DBC. The larger the
tension $T=(d-1)\tanh(\rho)$ is, the continuous the mass spectrum is.

###  5.2 Perturbative action

Let us go on to discuss the perturbative action of gravitons on the end-of-
the-world branes. Similarly, we expand the metric perturbations eq.(83) in
powers of KK modes

$\displaystyle\delta g_{r\mu}=0,\ \ \delta
g_{ij}=\cosh(r)^{2}\sum_{m_{g}}H^{(m_{g})}(r)\bar{h}^{(m_{g})}_{ij}(y),$ (103)

where $\sum_{m_{g}}$ denotes the sum over the gravitational mass spectrum,
$\bar{h}^{(m_{g})}_{ij}(y)$ and $H^{(m_{g})}(r)$ satisfy EOM eqs.(90,90) and
the orthogonal condition

$\displaystyle\int_{-\rho}^{\rho}\cosh(r)^{d-2}H^{(m_{g})}(r)H^{(m^{\prime}_{g})}(r)dr=\delta^{m_{g},m^{\prime}_{g}}.$
(104)

The Einstein-Hilbert action expanded to second order of the metric
perturbation is given by

$\displaystyle
I_{EH}=\int_{W}d^{d+1}x\sqrt{|g|}\Big{[}-\frac{1}{4}\nabla_{\alpha}\bar{H}_{\mu\nu}\nabla^{\alpha}\bar{H}^{\mu\nu}+\frac{1}{2}\nabla_{\alpha}\bar{H}_{\mu\nu}\nabla^{\nu}\bar{H}^{\mu\alpha}-\frac{d}{2}\bar{H}_{\mu\nu}\bar{H}^{\mu\nu}\Big{]}$
$\displaystyle\ \ \ \ \ \ \ +\int_{\partial
W}d^{d}y\sqrt{|h|}n^{\mu}\bar{H}^{\alpha\beta}(\nabla_{\mu}\bar{H}_{\alpha\beta}-\nabla_{\alpha}\bar{H}_{\beta\mu}),$
(105)

where we have set $16\pi G_{N}=1$, $\bar{H}_{\mu\nu}=\delta g_{\mu\nu}$ denote
the metric perturbation and we have used the gauge eq.(84) for simplicity. The
second order of perturbations of the Gibbons-Hawking term plus the brane
action are

$\displaystyle I_{GH}=2\int_{\partial W}d^{d}y\sqrt{|h|}(K-T)=\int_{\partial
W}d^{d}y\sqrt{|h|}\tanh(\rho)(-\frac{1}{2}\bar{H}_{\alpha\beta}\bar{H}^{\alpha\beta}),$
(106)

where we have used $K-T=\tanh(\rho)$, which applies to NBC and CBC at all
orders, to DBC at the linear order. For the metric ansatz eq.(81) and the
gauge eq.(84), we have $\nabla_{r}\bar{H}_{ij}|_{\partial W}=0$ and
$\nabla_{i}\bar{H}_{jr}=-\tanh(r)\bar{H}_{ij}$, which yield

$\displaystyle
n^{\mu}\bar{H}^{\alpha\beta}\nabla_{\mu}\bar{H}_{\alpha\beta}|_{\partial
W}=0,\ \
n^{\mu}\bar{H}^{\alpha\beta}\nabla_{\alpha}\bar{H}_{\beta\mu}|_{\partial
W}=-\tanh(\rho)\bar{H}_{\alpha\beta}\bar{H}^{\alpha\beta}.$ (107)

From eqs.(5.2,106,107), we obtain the total action

$\displaystyle
I=\int_{W}d^{d+1}x\sqrt{|g|}\Big{[}-\frac{1}{4}\nabla_{\alpha}\bar{H}_{\mu\nu}\nabla^{\alpha}\bar{H}^{\mu\nu}+\frac{1}{2}\nabla_{\alpha}\bar{H}_{\mu\nu}\nabla^{\nu}\bar{H}^{\mu\alpha}-\frac{d}{2}\bar{H}_{\mu\nu}\bar{H}^{\mu\nu}\Big{]}$
$\displaystyle\ \ \ \ \ +\frac{1}{2}\int_{\partial
W}d^{d}y\sqrt{|h|}\tanh(\rho)\bar{H}_{\alpha\beta}\bar{H}^{\alpha\beta}.$
(108)

Substituting eq.(103) into eq.(5.2) and following the approach of vectors, we
finally derive

$\displaystyle
I=\sum_{m_{g}}\int_{Q}d^{d}y\sqrt{|\bar{h}^{(0)}|}\Big{[}-\frac{1}{4}\bar{D}_{k}\bar{h}^{(m_{g})}_{ij}\bar{D}^{k}\bar{h}^{(m_{g})}{}^{ij}+\frac{1}{2}\bar{D}_{k}\bar{h}^{(m_{g})}_{ij}\bar{D}^{i}\bar{h}^{(m_{g})}{}^{jk}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\frac{d-1}{2}\bar{h}^{(m_{g})}_{ij}\bar{h}^{(m_{g})}{}^{ij}-\frac{1}{4}m_{g}^{2}\bar{h}^{(m_{g})}_{ij}\bar{h}^{(m_{g})}{}^{ij}\Big{]},$
(109)

which is an infinite sum of the Fierz-Pauli action in $\text{AdS}_{d}$ with
the gauge eq.(85). To derive eq.(5.2), we find the following formulas are
useful

$\displaystyle\nabla_{r}\bar{H}_{ij}=\cosh^{2}(r)\sum_{m_{g}}\bar{h}^{(m_{g})}_{ij}(y)\frac{d}{dr}H^{(m_{g})}(r),$
(110)
$\displaystyle\nabla_{i}\bar{H}_{jr}=-\cosh(r)\sinh(r)\sum_{m_{g}}\bar{h}^{(m_{g})}_{ij}(y)H^{(m_{g})}(r),$
(111)
$\displaystyle\nabla_{i}\bar{H}_{jk}=\cosh^{2}(r)\sum_{m_{g}}\bar{D}_{i}\bar{h}^{(m_{g})}_{jk}(y)H^{(m_{g})}(r).$
(112)

Note that the induced metric on the brane is
$\cosh^{2}(\rho)H(\pm\rho)\bar{h}^{(m_{g})}_{ij}$ instead of
$\bar{h}^{(m_{g})}_{ij}$. Taking into account this fact and the normalization
eq.(104), from eq.(5.2) we can recover the effective Newton’s constant of [3].

Note that action eq.(5.2) has the correct sign of kinetic term, thus it is
ghost-free. This is consistent with the discussions of sect. 2.2, which shows
that the higher derivative effective action on the brane is ghost-free.

##  6 First law of entanglement entropy

In this section, we prove that the first law of entanglement entropy is
satisfied for wedge holography. Since the holographic formula of entanglement
entropy is still un-known for wedge holography with CBC/DBC [20] 555 Recall
that, for CBC/DBC, the dynamical gravity on the brane is the extrinsic
curvature instead of the induced metric. So far we know little about this kind
of gravity. What we know is that, the holographic entanglement entropy for
CBC/DBC must be different from that of NBC, since they have different spectrum
and effective action on the brane. Two natural questions are as follows.
First, is the holographic entanglement entropy in the bulk still given by RT
surface (extremal surface) with respect to the metric? Second, if one could
impose CBC/DBC for gravity, can one impose CBC/DBC for the RT surface, that
the endpoint of RT surface is fixed on the brane? See [40] for some
discussions on the RT surface with DBC. We leave a careful study of these
problems to future works., we focus on NBC in this section. Interestingly, we
find that the massive fluctuations do not change the holographic entanglement
entropy. For simplicity, we focus on the first order perturbation including
the massless and massive modes of gravity in this section.

To start, let us give a brief review of the relative entropy and the first law
of entanglement entropy. The relative entropy measures the fundamental
distance between two states in the same Hilbert space

$\displaystyle
S(\rho_{1}|\rho_{0})=\text{tr}(\rho_{1}\ln\rho_{1})-\text{tr}(\rho_{1}\ln\rho_{0}).$
(113)

It is non-negative, i.e., $S(\rho_{1}|\rho_{0})\geq 0$, and vanishes if and
only if the states are equal. The relative entropy can be re-expressed as

$\displaystyle S(\rho_{1}|\rho_{0})=\Delta\langle H\rangle-\Delta S,$ (114)

with

$\displaystyle\Delta\langle
H\rangle=\text{tr}(\rho_{1}H)-\text{tr}(\rho_{0}H),\ \ \Delta
S=S(\rho_{1})-S(\rho_{0}),$ (115)

where $H$ is the modular Hamiltonian defined by
$\rho_{0}=e^{-H}/\text{tr}(e^{-H})$, and $S(\rho)=-\text{tr}(\rho\ln\rho)$ is
the entanglement entropy. In general, the positivity of the relative entropy
requires

$\displaystyle\Delta\langle H\rangle\geq\Delta S,$ (116)

which is a quantum generalization of the Bekenstein bound. At the first order
of perturbations, the above inequality is saturated [78]

$\displaystyle\delta\langle H\rangle=\delta S.$ (117)

This is the so-called first law of entanglement entropy. Interestingly, by
applying the first law of entanglement entropy eq.(117), one can derive the
linearized Einstein Equations in the bulk [79].

In the followings, we focus on the first order perturbations and verify that
the first law of entanglement entropy is obeyed by wedge holography.

###  6.1 The massive fluctuations

Let us first discuss the massive fluctuations. We take the ansatz of the
metric (81)

$\displaystyle
ds^{2}=dr^{2}+\textrm{cosh}^{2}(r)\Big{(}\frac{dz^{2}+\eta_{ab}dy^{a}dy^{b}}{z^{2}}+\epsilon\sum_{m_{g}}H^{(m_{g})}(r)\bar{h}_{ab}^{(m_{g})}(y)dy^{a}dy^{b}\Big{)},$
(118)

where $y^{i}=(z,y^{a}),y^{a}=(t,x^{A})=(t,x^{1},...,x^{d-2})$, the index $a$
runs from 0 to $(d-2)$ and $\sum_{m_{g}}$ denotes the sum over the spectrum.

Consider the entanglement entropy of $\text{CFT}_{d-1}$ on a $(d-2)$
dimensional round disk $\sum_{A=1}^{d-2}(x^{A})^{2}\leq R_{0}^{2}$, where
$R_{0}$ is the radius of the disk. According to [63], it can be calculated by
the area of the RT surface in the bulk

$\displaystyle S=\frac{\text{Area}(\gamma)}{4G_{N}},$ (119)

where $\gamma$ is the minimal surface anchored at the entangling surface of
the disk, i.e., $\partial\gamma=\sum_{A=1}^{d-2}(x^{A})^{2}=R_{0}^{2}$. For
the background metric eq.(118) with $\epsilon=0$, the RT surface in the bulk
is given by [1]

$\displaystyle\text{RT surface}\ \gamma:\ z^{2}+r_{0}^{2}=R_{0}^{2},\
t=\text{constant}.$ (120)

where $r_{0}^{2}=\sum_{A=1}^{d-2}(x^{A})^{2}$. Since the RT surface is a
minimal surface, the first order perturbation does not change its location. As
a result, the RT surface for the perturbative metric eq.(118) is still given
by eq.(120). From eq.(118) and eq.(120), we read off the induced metric on the
RT surface

$\displaystyle
ds_{\gamma}^{2}=dr^{2}+\frac{\textrm{cosh}^{2}(r)}{z^{2}}\gamma_{AB}dx^{A}dx^{B},$
(121)

where

$\displaystyle\gamma_{AB}=\delta_{AB}+\partial_{A}z\partial_{B}z+\epsilon\
z^{2}\sum_{m_{g}}H^{(m_{g})}(r)\bar{h}_{AB}^{(m_{g})}(y).$ (122)

By applying the RT formula eq.(119), we get the holographic entanglement
entropy of a disk

$\displaystyle S=\frac{1}{4G_{N}}\int_{-\rho}^{\rho}dr\int_{r_{0}\leq
R_{0}}d^{d-2}x{}\frac{\cosh^{d-2}(r)}{z^{d-2}}\sqrt{|\gamma_{AB}|},$ (123)

which yields the first order variation of entanglement entropy

$\displaystyle\delta
S=\sum_{m_{g}}\frac{\epsilon}{8G_{N}}\int_{-\rho}^{\rho}dr\cosh^{d-2}(r)H^{(m_{g})}(r)\int_{r_{0}\leq
R_{0}}d^{d-2}x{}f^{(m_{g})}(x^{A}),$ (124)

with

$\displaystyle
f^{(m_{g})}(x^{A})=\frac{1}{z^{d-4}}\left(\delta^{CD}\bar{h}_{CD}^{(m_{g})}\sqrt{1+\delta^{AB}\partial_{A}z\partial_{B}z}-\frac{\bar{h}^{(m_{g})CD}\partial_{C}z\partial_{D}z}{\sqrt{1+\delta^{AB}\partial_{A}z\partial_{B}z}}\right).$
(125)

Here $\bar{h}^{(m_{g})CD}$ are raised by $\delta^{CD}$ and $z$ obeys eq.(120).
In the above derivations, we have used the formula
$|g_{AB}+\partial_{A}z\partial_{B}z|=|g_{AB}|(1+g^{AB}\partial_{A}z\partial_{B}z)$
[78].

Recall the orthogonal condition (104) and the fact that $H^{(0)}(r)$ is a
constant for the massless mode. Immediately, we get

$\displaystyle\int_{-\rho}^{\rho}dr\cosh^{d-2}(r)H^{(m_{g})}(r)\sim\int_{-\rho}^{\rho}dr\cosh^{d-2}(r)H^{(m_{g})}(r)H^{(0)}(r)=0,$
(126)

for the massive modes $m_{g}>0$. As a result, the first order variation of
entanglement entropy eq.(124) vanishes for the massive fluctuations.

The first order variation of the expectation value of the modular Hamiltonian
$\delta\langle H\rangle$ for a spherical entangling surface is given by [78]

$\displaystyle\delta\langle H\rangle=\frac{\pi}{R_{0}}\int_{r_{0}\leq
R_{0}}d^{d-2}x\ {}z^{2}{}\delta\langle T_{00}\rangle,$ (127)

where $T_{00}$ is the stress tensor of $\text{CFT}_{d-1}$. Applying
$\text{AdS}_{d}/\text{CFT}_{d-1}$ on the brane, we get the holographic stress
tensor [58]

$\displaystyle\langle T_{ab}\rangle=\frac{d-1}{16\pi
G_{N}^{(d)}}h_{ab}^{(d-1)},$ (128)

where $G_{N}^{(d)}$ is the effective Newton’s constant, and $h_{ab}^{(d-1)}$
is defined in the Fefferman-Graham (FG) expansion on the brane

$\displaystyle
ds_{Q}^{2}=\frac{dz^{2}+(\eta_{ab}+z^{d-1}h_{ab}^{(d-1)}+...)dy^{a}dy^{b}}{z^{2}}.$
(129)

Note that, for the massive modes, the FG expansion behaves as

$\displaystyle
ds_{Q}^{2}=\frac{dz^{2}+(z^{d-1-\Delta}h_{ab}^{(d-1-\Delta)}+z^{\Delta}h_{ab}^{(\Delta)}+...)dy^{a}dy^{b}}{z^{2}},$
(130)

where $\Delta=\frac{d-1}{2}+\sqrt{\frac{(d-1)^{2}}{4}+m_{g}^{2}}$ is the
conformal dimension, which can be derived from EOM eq.(90). Comparing eq.(129)
with eq.(130), we get $\langle T_{ab}\rangle\sim h_{ab}^{(d-1)}=0$ for the
massive modes. As a result, the first order variation of the expectation value
of the modular Hamiltonian eq.(127) vanishes too. Now we have verified the
first law of entanglement entropy for the massive fluctuations

$\displaystyle\delta S=\delta\langle H\rangle=0,\ \ \text{for}\ m_{g}^{2}>0.$
(131)

###  6.2 The massless fluctuations

Let us go on to discuss the massless fluctuations, where
$\bar{h}_{ab}^{(m_{g}=0)}(y^{i})$ of eq.(118) takes the form,

$\displaystyle\bar{h}_{ab}^{(m_{g}=0)}(y^{i})=z^{d-3}h_{ab}^{(d-1)}(y^{a})=\frac{16\pi
G_{N}^{(d)}}{d-1}z^{d-3}\langle T_{ab}(y^{a})\rangle,$ (132)

and the effective Newton’s constant is given by [3]

$\displaystyle\frac{1}{G_{N}^{(d)}}=\frac{1}{G_{N}}\int_{-\rho}^{\rho}dr\cosh^{d-2}(r).$
(133)

Note that we have set $H^{(m_{g}=0)}(r)=1$ for simplicity. Combining equations
eq.(120,124,132), we get the first order perturbation of entanglement entropy

$\displaystyle\delta S=\frac{2\pi
R_{0}S_{d-4}}{d-1}\int_{0}^{R_{0}}dr_{0}r_{0}^{d-3}\int_{0}^{\pi}d\theta\textrm{sin}^{d-4}\theta\left(\langle
T_{\;\;\;A}^{A}(y)\rangle-\langle
T_{AB}(y)\rangle\frac{x^{A}x^{B}}{R_{0}^{2}}\right),$ (134)

where $S_{d-4}$ is the volume of $(d-4)$ dimensional unit sphere. On the other
hand, the first order perturbation of the expectation value of the modular
Hamiltonian is eq.(127)

$\displaystyle\delta\langle H\rangle$
$\displaystyle=\frac{\pi}{R_{0}}\int_{r_{0}\leq R_{0}}d^{d-2}x\ z^{2}\langle
T_{00}(y)\rangle$ $\displaystyle=\frac{\pi
S_{d-4}}{R_{0}}\int_{0}^{R_{0}}dr_{0}r_{0}^{d-3}\int_{0}^{\pi}d\theta\textrm{sin}^{d-4}\theta(R_{0}^{2}-r_{0}^{2})\langle
T_{00}(y)\rangle.$ (135)

It is convenient to express $\delta S,\delta\langle H\rangle$ eqs.(134,6.2) in
the Fourier expansion

$\displaystyle T_{ab}(y)=\int d^{d-1}pe^{-ip\cdot y}\hat{T}_{ab}(p).$ (136)

Without loss of generality, we drop the integral $\int d^{d-1}pe^{ip^{0}t}$
below. Following [78], we set the spatial direction of momentum in direction
$x^{1}$. The conservation and tracelessness of $\langle T_{ab}\rangle$ yield
the following useful formulas [78]

$\displaystyle\langle\hat{T}_{\;\;\;A}^{A}\rangle=\langle\hat{T}_{00}\rangle,\;\;\;$
$\displaystyle\langle\hat{T}_{10}\rangle=-\frac{p^{0}}{p^{1}}\langle\hat{T}_{00}\rangle,\;\;\;\langle\hat{T}_{11}\rangle=\left(\frac{p^{0}}{p^{1}}\right)^{2}\langle\hat{T}_{00}\rangle,$
$\displaystyle\langle\hat{T}_{\;\;\;A}^{A}(p)\rangle-\langle\hat{T}_{AB}(p)\rangle\frac{x^{A}x^{B}}{R_{0}^{2}}$
$\displaystyle=\langle\hat{T}_{00}\rangle\left(1-(\frac{p^{0}x^{1}}{p^{1}R_{0}})^{2}-\frac{(1-(\frac{p^{0}}{p^{1}})^{2})(r_{0}^{2}-(x^{1})^{2})}{(d-3)R_{0}^{2}}\right)$
$\displaystyle=\langle\hat{T}_{00}\rangle\left(\frac{R_{0}^{2}-r_{0}^{2}\textrm{cos}^{2}(\theta)}{R_{0}^{2}}+\mathcal{O}(p^{2})\right),$
(137)

where $p^{0}=\sqrt{\text{(}p^{1})^{2}-p^{2}}$.

Substituting eqs.(136,6.2) into eqs.(134,6.2), we finally derive

$\displaystyle\delta S$ $\displaystyle=\frac{2\pi
R_{0}S_{d-4}}{d-1}\langle\hat{T}_{00}\rangle\int_{0}^{R_{0}}dr_{0}r_{0}^{d-3}\int_{0}^{\pi}d\theta\textrm{sin}^{d-4}\theta
e^{-ip^{1}r_{0}\textrm{cos($\theta$)}}\left(\frac{R_{0}^{2}-r_{0}^{2}\textrm{cos}^{2}(\theta)}{R_{0}^{2}}\right)$
$\displaystyle=\frac{\pi^{\frac{3}{2}}R_{0}^{d-1}S_{d-4}\Gamma(\frac{d-3}{2})}{2\Gamma(\frac{d+2}{2})}{}_{0}F_{1}\left(\frac{d+2}{2};-\frac{1}{4}(p^{1}R_{0})^{2}\right)\langle\hat{T}_{00}\rangle,$
(138)

and

$\displaystyle\delta\langle H\rangle$ $\displaystyle=\frac{\pi
S_{d-4}}{R_{0}}\langle\hat{T}_{00}\rangle\int_{0}^{R_{0}}dr_{0}r_{0}^{d-3}\int_{0}^{\pi}d\theta\textrm{sin}^{d-4}\theta
e^{-ip^{1}r_{0}\textrm{cos($\theta$)}}(R_{0}^{2}-r_{0}^{2})$
$\displaystyle=\frac{\pi^{\frac{3}{2}}R_{0}^{d-1}S_{d-4}\Gamma(\frac{d-3}{2})}{2\Gamma(\frac{d+2}{2})}{}_{0}F_{1}\left(\frac{d+2}{2};-\frac{1}{4}(p^{1}R_{0})^{2}\right)\langle\hat{T}_{00}\rangle,$
(139)

where ${}_{0}F_{1}\left(a;z\right)$ is the hypergeometric function. As
expected, eqs.(6.2,6.2) yield the correct first law of entanglement entropy
$\delta S=\delta\langle H\rangle$. It is a strong support for wedge
holography.

To end this section, let us make some comments. First, for the massive
fluctuations, the first law of entanglement entropy eq.(131) holds for more
general background instead of only AdS. That is because the EOM eq.(90), the
spectrum and the orthogonal condition eq.(104) of $H(r)$ are independent of
the choices of the background metric $h^{(0)}_{ij}$, as long as it obeys the
Einstein equations on the brane. As a result, $\delta S$ eq.(124) always
includes a vanishing pre-factor
$\int_{-\rho}^{\rho}dr\cosh^{d-2}(r)H^{(m_{g})}(r)=0$ for the massive
fluctuations. Similarly, we always have $\delta\langle H\rangle=0$ for the
massive fluctuations, since the massive modes are irrelevant to the
holographic stress tensor $\langle T_{ab}\rangle$ with the conformal dimension
$(d-1)$. Thus, we can prove the first law of entanglement entropy $\delta
S=\delta\langle H\rangle=0$ for the massive fluctuations around more general
background. Second, it is straightforward to generalize the discussions of the
massless modes to higher order perturbations. Recall that the effective theory
of the massless modes is Einstein gravity eq.(11) on the brane [3]. According
to [78], for Einstein gravity, we have $\delta S\leq\delta\langle H\rangle$ at
the second order perturbations. Third, inspired by [79], it is interesting to
turn the logic around, and to derive the Einstein equations in the $(d+1)$
dimensional wedge space from the first law of entanglement entropy of CFTs on
the $(d-1)$ dimensional corner of the wedge.

##  7 Conclusions and Discussions

In this paper, we have investigated the effective action, the spectrum and the
first law of entanglement entropy for wedge holography. We work out the
effective action on the brane. In the perturbative formulation, it is given by
an infinite sum of Pauli-Fierz actions. In the non-perturbative formulation,
the effective action is composed of a higher derivative gravity and a matter
action. Usually, a higher derivative gravity suffers the problem of ghost. Due
to the equivalence to Einstein gravity in the bulk, the effective action on
the brane must be ghost-free. We find that the matter action plays an
important role in eliminating the ghost. Besides, we provide evidences that
the higher derivative gravity on the brane is equivalent to a ghost-free
multi-gravity [68, 69]. We show that the higher derivative gravity gives the
correct Weyl anomaly of wedge holography, which is a support to both wedge
holography and the non-perturbative effective action on the brane. We also
study the mass spectrum for various boundary conditions. We find that the
spectrum is discrete and non-negative for all kinds of boundary conditions. In
particular, there is a massless mode if one imposes NBC on both branes. On the
other hand, the massless mode disappears if one imposes CBC/DBC on one or two
of the branes. Finally, we verify that the first law of entanglement entropy
is satisfied for wedge holography. Interestingly, the massive fluctuations are
irrelevant to the first order perturbations of entanglement entropy. This is
also a strong support for wedge holography.

In this paper, we mainly focus on NBC. It is interesting to discuss the
effective action and the first law of entanglement entropy for CBC/DBC [17,
20]. It is expected that, similar to the case of NBC, the non-perturbative
effective action for CBC/DBC is also equivalent to a higher derivative gravity
or a multi-gravity on the brane. The only difference are the parameters of the
theory, since, as we have shown in sect.3, the perturbative effective action
is given by an infinite sum of Pauli-Fierz massive gravity with different mass
spectrums for different BCs. It is also interesting to study the holographic
entanglement entropy for gravity with CBC/DBC. Recall that the dynamical
gravity on the brane is the extrinsic curvature instead of the induced metric.
Is the holographic entanglement entropy still given by the area of a minimal
surface with respect to the metric in the bulk? Which boundary condition
should one choose for the RT surface ending on the brane with CBC/DBC?
Finally, inspired by [79], it is interesting to derive the Einstein equations
in the $(d+1)$ dimensional wedge space from the first law of entanglement
entropy on the $(d-1)$ dimensional corner of the wedge. We leave a careful
study of these interesting problems to future works.

## Acknowledgements

We thank Chong-Sun Chu and Jie Ren for valuable discussions. This work is
supported by the National Natural Science Foundation of China (Grant
No.11905297) and Guangdong Basic and Applied Basic Research Foundation
(No.2020A1515010900).

##  Appendix A Multi-gravity on the brane

By applying the “deconstruction” method [69], one can obtain the so-called
multi-gravity from Einstein gravity in the bulk. The main idea is to make
discretization of Einstein gravity and replace the extra dimension $r$ by a
series of sites $r_{b}$ ($1\leq b\leq N$). See Fig. 8 for an example. The
Einstein gravity in the bulk can be written as

$\displaystyle I=\int_{-\rho}^{\rho}dr\int
dy^{d}\sqrt{|h|}\Big{(}\mathcal{R}[h]+K^{2}-K_{ij}K^{ij}-2\Lambda\Big{)},$
(140)

where we have set $16\pi G_{N}=1$ for simplicity, $\mathcal{R}[h]$ is the
intrinsic Ricci scalar on the constant $r$ surface,
$K_{ij}=\frac{1}{2}\partial_{r}g_{ij}$ is the extrinsic curvature and
$\Lambda$ is the cosmological constant. By discretization, the extrinsic
curvature becomes

$\displaystyle K_{i}^{a}=e^{ja}K_{ij}\backsimeq m_{N}(e_{n+1}\
{}^{a}_{i}-e_{n}\ {}^{a}_{i}),$ $\displaystyle
K^{i}_{j}[h_{n},h_{n+1}]=-m_{N}\left(\delta^{i}_{j}-\left(\sqrt{h_{n}^{-1}h_{n+1}}\right)^{i}_{j}\right),$
(141)

where $e_{ia}$ is the vielbein and $m_{N}=N/(2\rho)\backsimeq\partial_{r}$
denotes the discrete derivative. The integration over the whole of the extra
dimension can be replaced by summing over all the sites after discretization.
In this way, we get the multi-gravity eq.(2.2) [68, 69]

$\displaystyle I$ $\displaystyle=$ $\displaystyle\sum_{n=1}^{N}\int
dy^{d}\sqrt{|h_{n}|}\left(\mathcal{R}[h_{n}]+K^{2}[h_{n},h_{n+1}]-K_{ij}[h_{n},h_{n+1}]K^{ij}[h_{n},h_{n+1}]-2\Lambda\right)$
(142) $\displaystyle=$ $\displaystyle\sum_{n=1}^{N}\int
dy^{d}\sqrt{|h_{n}|}\left(\mathcal{R}_{n}+\frac{m_{N}^{2}}{2}\sum_{m=0}^{d}\alpha_{m}^{(n)}\mathcal{L}_{m}\left(K(h_{n},h_{n+1})\right)\right),$

where $\alpha_{m}^{(n)}$ are some coefficients,
$\mathcal{L}_{m}[Q]=\epsilon\epsilon Q^{m}$ denote the interaction between
neighboring metrics and $\epsilon$ is the Levi-Civita symbol. For instance, we
have
$\mathcal{L}_{2}[Q]=\epsilon^{ijkl}\epsilon_{i_{1}j_{1}kl}Q^{i_{1}}_{i}Q^{j_{1}}_{j}$
in four dimensions. By construction, the multi-gravity eq.(142) is ghost-free
[68, 69].

Taking the limit $N\to\infty$, the multi-gravity eq.(142) reduces to Einstein
gravity eq.(140) in the bulk. Since the multi-gravity eq.(142) with
$N\to\infty$ and the effective higher derivative gravity eq.(8) on the two
branes are both obtained from the bulk Einstein gravity, they must be
equivalent. Following the approach of [64], by eliminating the metrics
$h_{2},h_{3},...,h_{N-1}$ of the multi-gravity with $N\to\infty$, we should
recover the higher derivative gravity eq.(8) on the branes. We leave a careful
study of this problem to future work [67]. In this appendix, we consider
finite $N$ for simplicity. We recover the correct form of $O(\mathcal{R}^{2})$
terms in the higher derivative gravity eq.(8). This is a strong evidence for
the equivalence between the higher derivative gravity on the brane and the
ghost-free multi-gravity. For simplicity, we focus on the symmetric
deconstruction of the bulk gravity, where the action is invariant under the
transformations $(b)\leftrightarrow(N+1-b)$, so that we only need to consider
half of the wedge space $-\rho\leq r\leq 0$ below.

Figure 8: Schematic diagram of deconstruction. The main idea is to make
discretization of Einstein gravity and replace the extra dimension $r$ by a
series of sites $r_{b}$ ($1\leq b\leq N$). In this way, one can derive ghost-
free multi-gravity from the Einstein gravity in the bulk.

###  A.1 Multi-gravity with four metrics

Let us first discuss the multi-gravity with four metrics $h_{b}$ $(1\leq b\leq
N=4)$. The action includes three parts

$I_{4}(h_{b})=\int
d^{d}y\left(\mathcal{L}_{12}+\mathcal{L}_{43}+\mathcal{L}_{int\;23}\right),$
(143)

where $\mathcal{L}_{12}$ and $\mathcal{L}_{43}$ are the bi-metric Lagrangian
density for $(h_{1},h_{2})$ and $(h_{4},h_{3})$ respectively,
$\mathcal{L}_{int\;23}$ denotes the interaction of $(h_{2},h_{3})$

$\displaystyle\mathcal{L}_{12}=m_{(1)}^{d-2}\left(\sqrt{|h_{1}|}\mathcal{R}_{1}+\gamma_{(2)}^{d-2}\sqrt{|h_{2}|}\mathcal{R}_{2}-2m^{2}\sqrt{|h_{1}|}\sum_{n=0}^{d}\beta_{n}^{(1)}e_{n}(S_{(1)})\right),$
(144)
$\displaystyle\mathcal{L}_{43}=m_{(4)}^{d-2}\left(\sqrt{|h_{4}|}\mathcal{R}_{4}+\gamma_{(3)}^{d-2}\sqrt{|h_{3}|}\mathcal{R}_{3}-2m^{2}\sqrt{|h_{4}|}\sum_{n=0}^{d}\beta_{n}^{(4)}e_{n}(\bar{S}_{(4)})\right),$
(145)
$\displaystyle\mathcal{L}_{int\;23}=-m^{2}\sum_{n=0}^{d}\left(m_{(1)}^{d-2}\beta_{n}^{(2)}\sqrt{|h_{2}|}e_{n}(S_{(2)})+m_{(4)}^{d-2}\beta_{n}^{(3)}\sqrt{|h_{3}|}e_{n}(\bar{S}_{(3)})\right).$
(146)

Let us recall some notations. $\gamma_{(2)}=m_{(2)}/m_{(1)}$ and
$\gamma_{(3)}=m_{(3)}/m_{(4)}$ are the ratio of the Planck masses,
$\beta_{n}^{(b)}$ are dimensionless free parameters, $\mathcal{R}_{b}$ is the
Ricci scalar of $h_{b}$, $S_{(b)}=\sqrt{h_{b}^{-1}h_{b+1}}$ and
$\bar{S}_{(b)}=\sqrt{h_{b}^{-1}h_{b-1}}$. Note that $\mathcal{L}_{12}$ and
$\mathcal{L}_{43}$ are symmetric under the interchanges
$\left\\{h_{b}\leftrightarrow h_{N+1-b},(b)\leftrightarrow(N+1-b)\right\\}$
because of the symmetric deconstruction of the bulk gravity. The equations of
motion (EOM) of $h_{1}$ and $h_{2}$ are given by [64, 81]

$\displaystyle\mathcal{P}_{1\;j}^{\;i}-\mathcal{P}_{1}\delta_{j}^{i}+m^{2}\sum_{n=0}^{d-1}(-1)^{n}\beta_{n}^{(1)}\mathcal{\mathbb{Y}}_{(n)\nu}^{\mu}(S_{(1)})=0,$
(147)
$\displaystyle\mathcal{P}_{2\;j}^{\;i}-\mathcal{P}_{2}\delta_{j}^{i}+\frac{m^{2}}{\gamma_{(2)}^{d-2}}\sum_{n=0}^{d-1}(-1)^{n}\left[\beta_{n}^{(2)}\mathcal{\mathbb{Y}}_{(n)\nu}^{\mu}(S_{(2)})+\beta_{d-n}^{(1)}\mathcal{\mathbb{Y}}_{(n)\nu}^{\mu}(S_{(1)}^{-1})\right]=0,$
(148)

where
$\mathcal{P}_{b\;ij}=\mathcal{R}_{b\;ij}-\frac{\mathcal{R}_{b}}{2(d-1)}h_{b\;ij}$,
$\mathcal{\mathbb{Y}}_{(n)\nu}^{\mu}(S_{(b)})=\sum_{k=0}^{n}(-1)^{k}e_{k}(S_{(b)})(S_{(b)}^{n-k})_{\;j}^{i}$
.

We aim to express $h_{2}$ in functions of $h_{1}$. To do so, we first solve
$S_{(1)\;j}^{\;i}=(\sqrt{h_{1}^{-1}h_{2}})^{i}_{j}$ perturbatively. The
general ansatz of $S_{(1)\;j}^{\;i}$ is given by

$\displaystyle S_{(1)\;j}^{\;i}=$ $\displaystyle
a^{(1)}\delta_{j}^{i}+\frac{1}{m^{2}}\left(b_{1}^{(1)}\mathcal{P}_{1\;j}^{\;i}+b_{2}^{(1)}\mathcal{P}_{1}\delta_{j}^{i}\right)$
$\displaystyle+\frac{1}{m^{4}}\left(c_{1}^{(1)}\mathcal{P}_{1\;k}^{\;i}\mathcal{P}_{1\;j}^{\;k}+c_{2}^{(1)}\mathcal{P}_{1}\mathcal{P}_{1\;j}^{\;i}+c_{3}^{(1)}\mathcal{P}_{1\;kl}\mathcal{P}_{1}^{\;kl}\delta_{j}^{i}+c_{4}^{(1)}\mathcal{P}_{1}^{\;2}\delta_{j}^{i}\right)+\mathcal{O}(m^{-6}).$
(149)

where $a^{(1)},b^{(1)}$ and $c^{(1)}$ are the coefficients to be determined
below. Substituting eq.(A.1) into eq.(147), we get

$\displaystyle\mathcal{P}_{1\;j}^{\;i}-\mathcal{P}_{1}\delta_{j}^{i}+m^{2}\sum_{n=0}^{d-1}\beta_{n}^{(1)}a^{(1)n}\sum_{k=0}^{n}\left(\sum_{r=0}^{n-k}\sum_{m=0}^{k}(-1)^{n+k}C_{n-k}^{r}C_{d-m}^{k-m}(M_{(1)}^{r})_{\;j}^{i}e_{m}(M_{(1)})\right)=0,$
(150)

where we have replaced $S_{(1)}$ by $M_{(1)}$ defined below

$\displaystyle
M_{(b)\;j}^{\;i}=\frac{1}{a^{(b)}}S_{(b)\;j}^{\;i}-\delta_{j}^{i},$ (151)
$\displaystyle(S_{(b)}^{n-k})_{\;j}^{i}=a^{(b)n-k}\sum_{r=0}^{n-k}(M_{(b)}^{r})_{\;j}^{i},\
\ \ e_{k}(S_{(b)})=a^{(b)k}\sum_{m=0}^{k}C_{d-m}^{k-m}e_{m}(M_{(b)}).$ (152)

From eq.(A.1) and eq.(151), we have

$\displaystyle M_{(1)\;j}^{\;i}$
$\displaystyle=\frac{1}{m^{2}}\frac{1}{a^{(1)}}\left(b_{1}^{(1)}\mathcal{P}_{1\;j}^{\;i}+b_{2}^{(1)}\mathcal{P}_{1}\delta_{j}^{i}\right)$
$\displaystyle+\frac{1}{m^{4}}\frac{1}{a^{(1)}}\left(c_{1}^{(1)}\mathcal{P}_{1\;k}^{\;i}\mathcal{P}_{1\;j}^{\;k}+c_{2}^{(1)}\mathcal{P}_{1}\mathcal{P}_{1\;j}^{\;i}+c_{3}^{(1)}\mathcal{P}_{1\;kl}\mathcal{P}_{1}^{\;kl}\delta_{j}^{i}+c_{4}^{(1)}\mathcal{P}_{1}^{\;2}\delta_{j}^{i}\right)+\mathcal{O}(m^{-6}).$
(153)

It’s convenient to use the following notations and formulas

$\displaystyle
s_{k}^{(b)}=\sum_{n=k}^{d-1}C_{d-k-1}^{n-k}\beta_{n}^{(b)}a^{(b)n},$ (154)
$\displaystyle\sum_{k=0}^{n}(-1)^{k+n}C_{n-k}^{r}C_{d-m}^{k-m}=(-1)^{r}C_{d-1-(r+m)}^{n-(r+m)}.$
(155)

Combining eqs.(150,154,155), we obtain

$\displaystyle\mathcal{P}_{1\;j}^{\;i}-\mathcal{P}_{1}\delta_{j}^{i}$
$\displaystyle+s_{0}^{(1)}m^{2}\delta_{j}^{i}+s_{1}^{(1)}m^{2}\left[-M_{(1)\;j}^{\;i}+\delta_{j}^{i}e_{1}(M_{(1)})\right]$
$\displaystyle+s_{2}^{(1)}m^{2}\left[(M_{(1)}^{2})_{\;j}^{i}-M_{(1)\;j}^{\;i}e_{1}(M_{(1)})+\delta_{j}^{i}e_{2}(M_{(1)})\right]+\mathcal{O}(M_{(1)}^{3})=0.$
(156)

Substituting eq.(A.1) into the above equation and solving it order by order in
$O(1/m^{2})$, we get

$\displaystyle\mathcal{O}(m^{2}),$ $\displaystyle s_{0}^{(1)}=0,$
$\displaystyle\mathcal{O}(m^{0}),$ $\displaystyle
b_{1}^{(1)}=\frac{a^{(1)}}{s_{1}^{(1)}},\;\;\;b_{2}^{(1)}=0,$
$\displaystyle\mathcal{O}(m^{-2}),$ $\displaystyle
c_{1}^{(1)}=-c_{2}^{(1)}=\frac{a^{(1)}s_{2}^{(1)}}{s_{1}^{(1)3}},\;\;\;c_{3}^{(1)}=-c_{4}^{(1)}=-\frac{a^{(1)}s_{2}^{(1)}}{2(d-1)s_{1}^{(1)3}}.$
(157)

Substituting eq.(A.1) into eq.(A.1), we finally obtain $S_{(1)}$

$\displaystyle
S_{(1)\;j}^{\;i}=a^{(1)}\delta_{j}^{i}+\frac{a^{(1)}}{s_{1}^{(1)}m^{2}}\mathcal{P}_{1\;j}^{\;i}+\frac{a^{(1)}s_{2}^{(1)}}{s_{1}^{(1)3}m^{4}}\left(\mathcal{P}_{1\;k}^{i}\mathcal{P}_{1\;j}^{k}-\mathcal{P}_{1}\mathcal{P}_{1\;j}^{i}+\frac{e_{2}(\mathcal{P}_{1})}{d-1}\delta_{j}^{i}\right)+\mathcal{O}(m^{-6}).$
(158)

and express $h_{2}$ in terms of $h_{1}$

$\displaystyle h_{2\;ij}$ $\displaystyle=h_{1\;ik}(S_{(1)}^{\;2})_{\;j}^{k}$
$\displaystyle=a^{(1)2}\left[h_{1\;ij}+\frac{2\mathcal{P}_{1\;ij}}{s_{1}^{(1)}m^{2}}+\frac{2s_{2}^{(1)}}{s_{1}^{(1)3}m^{4}}\left((1+\frac{s_{1}^{(1)}}{2s_{2}^{(1)}})\mathcal{P}_{1\;ik}\mathcal{P}_{1\;j}^{k}-\mathcal{P}_{1}\mathcal{P}_{1\;ij}+\frac{e_{2}(\mathcal{P}_{1})}{d-1}h_{1\;ij}\right)\right]$
$\displaystyle\ \ +\mathcal{O}(m^{-6}).$ (159)

Following the same approach, we can express $h_{3}$ in terms of $h_{4}$. From
$h_{2}(h_{1})$ eq.(A.1) and $h_{3}(h_{4})$, the interaction potential
$\mathcal{L}_{int\;23}(h_{2},h_{3})$ can be converted to
$\mathcal{L}_{int\;14}(h_{1},h_{4})$. We find the following relations are
useful in the calculations

$\displaystyle\sum_{n=0}^{d}\beta_{n}^{(b)}e_{n}(S_{(b)})=\sum_{n=0}^{d}\alpha_{n}^{(b)}e_{n}(M_{(b)}),\
\ \ $
$\displaystyle\alpha_{n}^{(b)}=\sum_{k=n}^{d}C_{k-n}^{d-n}a^{(b)k}\beta_{k}^{(b)}.$
(160)

Finally, combining eqs.(143,144,145,158,A.1,160), we obtain the higher
derivative gravity with respect to two metrics $h_{1}$ and $h_{4}$

$\displaystyle I_{4}^{HD}(h_{1},h_{4})$
$\displaystyle=m_{(1)}^{d-2}\int_{Q_{1}}d^{d}y\sqrt{|h_{1}|}\left[\hat{\Lambda}^{(1)}+\hat{c}_{\mathcal{R}}^{(1)}\mathcal{R}_{1}+\frac{\hat{c}_{\mathcal{R}\mathcal{R}}^{(1)}}{m^{2}}\left(\mathcal{R}_{1}^{\;ij}\mathcal{R}_{1\;ij}-\frac{d}{4(d-1)}\mathcal{R}_{1}^{\;2}\right)\right]$
$\displaystyle+m_{(4)}^{d-2}\int_{Q_{2}}d^{d}y\sqrt{|h_{4}|}\left[\hat{\Lambda}^{(4)}+\hat{c}_{\mathcal{R}}^{(4)}\mathcal{R}_{4}+\frac{\hat{c}_{\mathcal{R}\mathcal{R}}^{(4)}}{m^{2}}\left(\mathcal{R}_{4}^{\;ij}\mathcal{R}_{4\;ij}-\frac{d}{4(d-1)}\mathcal{R}_{4}^{\;2}\right)\right]$
$\displaystyle+\mathcal{O}(\mathcal{R}_{1}^{\;3})+\mathcal{O}(\mathcal{R}_{4}^{\;3})+\mathcal{L}_{int\;14}(h_{1},h_{4}),$
(161)

where the parameters are given by

$\displaystyle\hat{\Lambda}^{(1)}$ $\displaystyle=-2m^{2}\alpha_{0}^{(1)},\ \
\
\hat{c}_{\mathcal{R}}^{(1)}=1+(\gamma_{(2)}a^{(1)})^{d-2}-\frac{\alpha_{1}^{(1)}(d-2)}{s_{1}^{(1)}(d-1)},$
$\displaystyle\hat{c}_{\mathcal{R}\mathcal{R}}^{(1)}$
$\displaystyle=-\frac{1}{s_{1}^{(1)2}}\left(2s_{1}^{(1)}(\gamma_{(2)}a^{(1)})^{d-2}-\alpha_{2}^{(1)}+\frac{\alpha_{1}^{(1)}s_{2}^{(1)}(d-2)}{s_{1}^{(1)}(d-1)}\right),$
(162)

and
$\hat{\Lambda}^{(4)},\hat{c}_{\mathcal{R}}^{(4)},\hat{c}_{\mathcal{RR}}^{(4)}$
can be obtained from eq.(A.1) by replacing indexes $(b)$ by $(5-b)$.

Remarkably, the higher derivative gravity eq.(A.1) reduced from the multi-
gravity takes similar forms as the effective action eq.(8) on the brane, where
the interaction $\mathcal{L}_{int\;14}(h_{1},h_{4})$ can be naturally
explained as the CFT action $I_{\text{CFT}}$.

###  A.2 Multi-gravity with six metrics

Let us now discuss the multi-gravity with six metrics $h_{b}$ $(1\leq b\leq
N=6)$

$I_{6}(h_{b})=\int
d^{d}y\left(\mathcal{L}_{123}+\mathcal{L}_{654}+\mathcal{L}_{int\;34}\right),$
(163)

where $\mathcal{L}_{123}$ and $\mathcal{L}_{654}$ are the Lagrangian density
of multi-gravity with three metrics for $(h_{1},h_{2},h_{3})$ and
$(h_{6},h_{5},h_{4})$ respectively, $\mathcal{L}_{int\;34}$ denotes the
interaction of $(h_{3},h_{4})$

$\displaystyle\mathcal{L}_{123}=\mathcal{L}_{12}+\mathcal{L}_{3}=\mathcal{L}_{12}+m_{(1)}^{d-2}\left(\gamma_{(3)}^{d-2}\sqrt{|h_{3}|}\mathcal{R}_{3}-2m^{2}\sqrt{|h_{2}|}\sum_{n=0}^{d}\beta_{n}^{(2)}e_{n}(S_{(2)})\right),$
(164)
$\displaystyle\mathcal{L}_{654}=\mathcal{L}_{65}+\mathcal{L}_{4}=\mathcal{L}_{65}+m_{(6)}^{d-2}\left(\gamma_{(4)}^{d-2}\sqrt{|h_{4}|}\mathcal{R}_{4}-2m^{2}\sqrt{|h_{5}|}\sum_{n=0}^{d}\beta_{n}^{(5)}e_{n}(\bar{S}_{(5)})\right),$
(165)
$\displaystyle\mathcal{L}_{int\;34}=-m^{2}\sum_{n=0}^{d}\left(m_{(1)}^{d-2}\beta_{n}^{(3)}\sqrt{|h_{3}|}e_{n}(S_{(3)})+m_{(6)}^{d-2}\beta_{n}^{(4)}\sqrt{|h_{4}|}e_{n}(\bar{S}_{(4)})\right),$
(166)

where $\mathcal{L}_{12}$ is given by eq.(144), $\mathcal{L}_{65}$ can be
obtained from eq.(145) by replacing indexes (4,3) with (6,5). Since we have
already rewritten $\mathcal{L}_{12}$ in functions of $h_{1}$ in eq.(A.1), we
focus on $\mathcal{L}_{3}$ below.

In order to express $h_{3}$ in functions of $h_{1}$, we first solve
$S_{(2)\;j}^{\;i}=(\sqrt{h_{2}^{-1}h_{3}})^{i}_{j}$ perturbatively. The
general ansatz of $S_{(2)\;j}^{\;i}$ is given by

$\displaystyle S_{(2)\;j}^{\;i}=$ $\displaystyle\
a^{(2)}\delta_{j}^{i}+\frac{1}{m^{2}}\left(b_{1}^{(2)}\mathcal{P}_{1\;j}^{\;i}+b_{2}^{(2)}\mathcal{P}_{2}\delta_{j}^{i}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{m^{4}}\left(c_{1}^{(2)}\mathcal{P}_{2\;k}^{\;i}\mathcal{P}_{2\;j}^{\;k}+c_{2}^{(2)}\mathcal{P}_{2}\mathcal{P}_{2\;j}^{\;i}+c_{3}^{(2)}\mathcal{P}_{2\;kl}\mathcal{P}_{2}^{\;kl}\delta_{j}^{i}+c_{4}^{(2)}\mathcal{P}_{2}^{\;2}\delta_{j}^{i}\right)+\mathcal{O}(m^{-6}),$
(167)

where $a^{(2)},b^{(2)}$ and $c^{(2)}$ are the coefficients to be determined
below. Substituting (A.2) into eq.(148), we get

$\displaystyle\mathcal{P}_{2\;j}^{\;i}-$
$\displaystyle\mathcal{P}_{2}\delta_{j}^{i}+\text{$\frac{m^{2}}{\gamma_{2}^{d-2}}$}\sum_{n=0}^{d-1}\beta_{n}^{(2)}a^{(2)n}\sum_{k=0}^{n}\left(\sum_{r=0}^{n-k}\sum_{m=0}^{k}(-1)^{n+k}C_{n-k}^{r}C_{d-m}^{k-m}(M_{(2)}^{r})_{\;j}^{i}e_{m}(M_{(2)})\right)$
$\displaystyle+$
$\displaystyle\frac{m^{2}}{\gamma_{2}^{d-2}}\sum_{n=0}^{d-1}\beta_{d-n}^{(1)}a^{(1)-n}\sum_{k=0}^{n}\left(\sum_{r=0}^{n-k}\sum_{m=0}^{k}(-1)^{n+k}C_{n-k}^{r}C_{d-m}^{k-m}(\hat{M}_{(1)}^{r})_{\;j}^{i}e_{m}(\hat{M}_{(1)})\right)=0,$
(168)

where $M_{(2)}$ is given by eqs.(151,A.2) and we have replaced $S_{(1)}^{-1}$
by $\hat{M}_{(1)}$ defined below

$\displaystyle\hat{M}_{(b)\;j}^{\;i}=a^{(b)}(S_{(b)}^{-1})_{\;j}^{i}-\delta_{j}^{i},$
(169)
$\displaystyle\left((S_{(b)}^{-1})^{n-k}\right)_{\;j}^{i}=a^{(b)k-n}\sum_{r=0}^{n-k}(\hat{M}_{(b)}^{r})_{\;j}^{i},\
\ \
e_{k}(S_{(b)}^{-1})=a^{(b)-k}\sum_{m=0}^{k}C_{d-m}^{k-m}e_{m}(\hat{M}_{(b)}).$
(170)

From eq.(158) and eq.(169), we have

$\hat{M}_{(1)\;j}^{\;i}=-\frac{\mathcal{P}_{1\;j}^{\;i}}{s_{1}^{(1)}m^{2}}+\frac{1}{s_{1}^{(1)3}m^{4}}\left(s_{2}^{(1)}\mathcal{P}_{1}\mathcal{P}_{1\;j}^{i}+(s_{1}^{(1)}-s_{2}^{(1)})\mathcal{P}_{1\;k}^{i}\mathcal{P}_{1\;j}^{k}-s_{2}^{(1)}\frac{e_{2}(\mathcal{P}_{1})}{d-1}\delta_{j}^{i}\right)+\mathcal{O}(m^{-6}).\\\
$ (171)

Combining eqs.(154,155,A.2), we obtain

$\displaystyle\mathcal{P}_{2\;j}^{\;i}-$
$\displaystyle\mathcal{P}_{2}\delta_{j}^{i}+\frac{m^{2}}{\gamma_{2}^{d-2}}\left[s_{0}^{(2)}\delta_{j}^{i}+s_{1}^{(2)}\left(-M_{(2)\;j}^{\;i}+\delta_{j}^{i}e_{1}(M_{(2)})\right)\right]$
$\displaystyle+$
$\displaystyle\frac{m^{2}}{\gamma_{2}^{d-2}}\left[s_{2}^{(2)}\left((M_{(2)}^{2})_{\;j}^{i}-M_{(2)\;j}^{\;i}e_{1}(M_{(2)})+\delta_{j}^{i}e_{2}(M_{(2)})\right)\right]$
$\displaystyle+$
$\displaystyle\frac{m^{2}}{\gamma_{2}^{d-2}}\left[q_{2}^{(1)}\left((\hat{M}_{(1)}^{2})_{\;j}^{i}-\hat{M}_{(1)\;j}^{\;i}e_{1}(\hat{M}_{(1)})+\delta_{j}^{i}e_{2}(\hat{M}_{(1)})\right)\right]$
$\displaystyle+$
$\displaystyle\frac{m^{2}}{\gamma_{2}^{d-2}}\left[q_{0}^{(1)}\delta_{j}^{i}+q_{1}^{(1)}\left(-\hat{M}_{(1)\;j}^{\;i}+\delta_{j}^{i}e_{1}(\hat{M}_{(1)})\right)\right]+\mathcal{O}(\hat{M}_{(1)}^{3},M_{(2)}^{3})=0,$
(172)

where $q_{k}^{(b)}=\sum_{n=k}^{d-1}C_{d-k-1}^{n-k}\beta_{d-n}^{(b)}a^{(b)-n}$.
Note that $\mathcal{P}_{2}$ in eq.(A.2) can be converted to $\mathcal{P}_{1}$
by eq.(A.1)

$\displaystyle\mathcal{P}_{2\;j}^{\;i}-\mathcal{P}_{2}\delta_{j}^{i}=$
$\displaystyle\frac{1}{a^{(1)2}}\left[\mathcal{P}_{1\;j}^{\;i}-\mathcal{P}_{1}\delta_{j}^{i}+\frac{1}{s_{1}^{(1)}m^{2}}\left((\mathcal{P}_{1}^{\;kl}\mathcal{P}_{1\;kl}+\frac{\mathcal{P}_{1}^{\;2}}{d-2})\delta_{j}^{i}-2(\mathcal{P}_{1\;k}^{i}\mathcal{P}_{1\;j}^{k}+\frac{\mathcal{P}_{1}\mathcal{P}_{1\;j}^{i}}{d-2})\right)\right]$
$\displaystyle+\mathcal{O}(m^{-4}),$ (173)

where we have ignored the total derivative terms of $\mathcal{P}_{1}$, which
have no contribution to the gravitational action at order $O(\mathcal{R}^{2})$
666At order $O(\mathcal{R}^{2})$, the only possible terms including
derivatives are $\Box\mathcal{R}$ and $\nabla_{i}\nabla_{j}\mathcal{R}^{ij}$,
which are total derivative terms and can be dropped in the action..
Substituting eqs.(171,A.2) and $M_{(2)}$ into eq.(A.2) and solving it order by
order in $\mathcal{O}(1/m^{2})$, we obtain the coefficients of
$S_{(2)\;j}^{\;i}$

$\displaystyle\mathcal{O}$ $\displaystyle(m^{2}),\ \
s_{0}^{(2)}=-q_{0}^{(1)},$ $\displaystyle\mathcal{O}$ $\displaystyle(m^{0}),\
\
b_{1}^{(2)}=\frac{a^{(2)}}{s_{1}^{(2)}}\left(\frac{q_{1}^{(1)}}{s_{1}^{(1)}}+\frac{\gamma_{(2)}^{d-2}}{a^{(1)2}}\right),\;\;\;b_{2}^{(2)}=0,$
$\displaystyle\mathcal{O}$ $\displaystyle(m^{-2}),\ \
c_{1}^{(2)}=-2(d-1)c-\frac{a^{(2)}q_{1}^{(1)}}{s_{1}^{(1)2}s_{1}^{(2)}}-\frac{2a^{(2)}\gamma_{(2)}^{d-2}}{a^{(1)2}s_{1}^{(1)}s_{1}^{(2)}},\;\;\;c_{3}^{(2)}=c+\frac{a^{(2)}\gamma_{(2)}^{d-2}}{a^{(1)2}(d-1)s_{1}^{(1)}s_{1}^{(2)}},$
$\displaystyle
c_{2}^{(2)}=2(d-1)c-\frac{2a^{(2)}\gamma_{(2)}^{d-2}}{(d-2)a^{(1)2}s_{1}^{(1)}s_{1}^{(2)}},\;\;\;c_{4}^{(2)}=-c+\frac{a^{(2)}\gamma_{(2)}^{d-2}}{a^{(1)2}(d-1)(d-2)s_{1}^{(1)}s_{1}^{(2)}},$
(174)

where the parameter $c$ is

$\displaystyle c=$
$\displaystyle-\frac{a^{(2)}q_{2}^{(1)}}{2(d-1)s_{1}^{(1)2}s_{1}^{(2)}}-\frac{a^{(2)}q_{1}^{(1)}s_{2}^{(1)}}{2(d-1)s_{1}^{(1)3}s_{1}^{(2)}}-\frac{a^{(2)}q_{1}^{(1)2}s_{2}^{(2)}}{2(d-1)s_{1}^{(1)2}s_{1}^{(2)3}}$
$\displaystyle-\frac{a^{(2)}q_{1}^{(1)}s_{2}^{(2)}\gamma_{(2)}^{d-2}}{(d-1)a^{(1)2}s_{1}^{(1)}s_{1}^{(2)3}}-\frac{a^{(2)}s_{2}^{(2)}\gamma_{(2)}^{2(d-2)}}{2(d-1)a^{(1)4}s_{1}^{(2)3}}.$
(175)

Substituting eqs.(A.2,A.2) into eq.(A.2), we get $S_{(2)}$

$\displaystyle S_{(2)\;j}^{\;i}=$ $\displaystyle
a^{(2)}\delta_{j}^{i}+\frac{a^{(2)}}{s_{1}^{(1)}m^{2}}(\frac{q_{1}^{(1)}}{s_{1}^{(1)}}+\frac{\gamma_{(2)}^{d-2}}{a^{(1)2}})\mathcal{P}_{1\;j}^{\;i}$
$\displaystyle+$
$\displaystyle\frac{1}{m^{4}}\left(c_{1}^{(2)}\mathcal{P}_{1\;k}^{i}\mathcal{P}_{1\;j}^{k}+c_{2}^{(2)}\mathcal{P}_{1}\mathcal{P}_{1\;j}^{i}+c_{3}^{(2)}\mathcal{P}_{1\;kl}\mathcal{P}_{1}^{\;kl}\delta_{j}^{i}+c_{4}^{(2)}\mathcal{P}_{1}^{\;2}\delta_{j}^{i}\right)+\mathcal{O}(m^{-6})\text{,}$
(176)

where the parameters $c^{(b)}_{n}$ are given by eq.(A.2). Combining equations
eqs.(A.1,A.2), we express the metric $h_{3}$ in terms of the metric $h_{1}$

$\displaystyle h_{3\;ij}=$ $\displaystyle h_{2\;ik}(S_{(2)}^{\;2})_{\;j}^{k}$
$\displaystyle=$ $\displaystyle
a^{(1)2}a^{(2)2}\left[h_{1\;ij}+\frac{2(1+\frac{b_{1}^{(2)}s_{1}^{(1)}}{a^{(2)}})}{s_{1}^{(1)}m^{2}}\mathcal{P}_{1\;ij}+\text{$\frac{2\mathcal{P}_{1}\mathcal{P}_{1\;ij}}{s_{1}^{(1)3}m^{4}}$}(\frac{c_{2}^{(2)}s_{1}^{(1)3}}{a^{(2)}}-s_{2}^{(1)})\right]$
$\displaystyle+$ $\displaystyle
a^{(1)2}a^{(2)2}\cdot\frac{\mathcal{P}_{1\;ik}\mathcal{P}_{1\;j}^{k}}{a^{(2)2}s_{1}^{(1)3}m^{4}}\left(b_{1}^{(2)2}s_{1}^{(1)3}+2a^{(2)}s_{1}^{(1)2}(2b_{1}^{(2)}+c_{1}^{(2)}s_{1}^{(1)})+a^{(2)2}(s_{1}^{(1)}+2s_{2}^{(1)})\right)$
$\displaystyle+$ $\displaystyle
a^{(1)2}a^{(2)2}\cdot\frac{2s_{2}^{(1)}h_{1\;ij}}{s_{1}^{(1)3}m^{4}}\left(\frac{e_{2}(\mathcal{P}_{1})}{d-1}+\frac{s_{1}^{(1)3}}{a^{(2)}s_{2}^{(1)}}(c_{3}^{(2)}\mathcal{P}_{1\;kl}\mathcal{P}_{1}^{\;kl}+c_{4}^{(2)}\mathcal{P}_{1}^{\;2})\right).$
(177)

Finally, combining eqs.(160,163,164,A.2,A.2,A.2), we obtain the higher
derivative gravity with respect to two metrics $h_{1}$ and $h_{6}$

$\displaystyle I_{6}^{HD}(h_{1},h_{6})$
$\displaystyle=m_{(1)}^{d-2}\int_{Q_{1}}d^{d}y\sqrt{|h_{1}|}\left[\Lambda^{(1)}+c_{\mathcal{R}}^{(1)}\mathcal{R}_{1}+\frac{c_{\mathcal{R}\mathcal{R}}^{(1)}}{m^{2}}\left(\mathcal{R}_{1}^{\;ij}\mathcal{R}_{1\;ij}-\frac{d}{4(d-1)}\mathcal{R}_{1}^{\;2}\right)\right]$
$\displaystyle+m_{(6)}^{d-2}\int_{Q_{2}}d^{d}y\sqrt{|h_{6}|}\left[\Lambda^{(6)}+c_{\mathcal{R}}^{(6)}\mathcal{R}_{6}+\frac{c_{\mathcal{R}\mathcal{R}}^{(6)}}{m^{2}}\left(\mathcal{R}_{6}^{\;ij}\mathcal{R}_{6\;ij}-\frac{d}{4(d-1)}\mathcal{R}_{6}^{\;2}\right)\right]$
$\displaystyle+\mathcal{L}_{int\;16}(h_{1},h_{6})+\mathcal{O}(\mathcal{R}_{1}^{\;3})+\mathcal{O}(\mathcal{R}_{4}^{\;3}),$
(178)

where the parameters are given by

$\displaystyle\Lambda^{(1)}$
$\displaystyle=-2m^{2}(\alpha_{0}^{(1)}+a^{(1)d}\alpha_{0}^{(2)}),\;\;\;c_{\mathcal{R}\mathcal{R}}^{(1)}=\hat{c}_{\mathcal{RR}}^{(1)}+f_{1}+f_{2}+f_{3},$
$\displaystyle c_{\mathcal{R}}^{(1)}$
$\displaystyle=\hat{c}_{\mathcal{R}}^{(1)}+a^{(1)d}\left[\frac{\text{(}a^{(2)}\gamma_{(3)})^{d-2}}{a^{(1)2}}-\frac{(d-2)(a^{(2)}\alpha_{0}^{(2)}+b_{1}^{(2)}s_{1}^{(1)}\alpha_{1}^{(2)})}{(d-1)s_{1}^{(1)}a^{(2)}}\right],$
(179)

and

$\displaystyle f_{1}=$
$\displaystyle\frac{(a^{(1)}\gamma_{(2)})^{d-2}\left(\alpha_{2}^{(2)}\gamma_{(2)}^{d-2}-2s_{1}^{(2)}(a^{(2)}\gamma_{(3)})^{d-2}\right)}{a^{(1)2}s_{1}^{(2)2}},$
$\displaystyle f_{2}=$
$\displaystyle\frac{2\left(q_{1}^{(1)}\alpha_{2}^{(2)}+\frac{d-2}{d-1}s_{1}^{(2)}\alpha_{1}^{(2)}\right)(a^{(1)}\gamma_{(2)})^{d-2}-2s_{1}^{(2)}(q_{1}^{(1)}+s_{1}^{(2)})(a^{(1)}a^{(2)}\gamma_{(3)})^{d-2}}{s_{1}^{(1)}s_{1}^{(2)2}},$
$\displaystyle f_{3}=$
$\displaystyle\frac{a^{(1)d}}{a^{(2)}s_{1}^{(1)3}s_{1}^{(2)2}}\Bigg{[}a^{(2)}q_{1}^{(1)2}s_{1}^{(1)}\alpha_{2}^{(2)}+a^{(2)}s_{1}^{(2)2}\alpha_{0}^{(2)}(s_{1}^{(1)}-\frac{d-2}{d-1}s_{2}^{(1)})$
$\displaystyle+2s_{1}^{(1)}s_{1}^{(2)}\alpha_{1}^{(2)}\left(a^{(2)}q_{1}^{(1)}+(d-2)cs_{1}^{(1)2}s_{1}^{(2)}\right)\Bigg{]}.$
(180)

Similarly, $\Lambda^{(6)},c_{\mathcal{R}}^{(6)}$ and $c_{\mathcal{RR}}^{(6)}$
can be obtained from eq.(A.2) by replacing indexes $(b)$ with $(N+1-b)$.
Remarkably, the multi-gravity with eight metrics ($N=8$) eq.(A.2) yields the
same curvature squared terms as the effective action eq.(8) on the brane. It
is expected that the results can be generalized to arbitrary $N$. This is a
strong evidence for the equivalence between ghost-free multi-gravity and the
higher derivative gravity eq.(8) on the brane.

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