Casimir Effect and Holographic Dual of Wedges

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

###### Abstract

This paper investigates the Casimir effect of a wedge and its holographic
dual. We prove that the displacement operator universally determines the wedge
Casimir effect in the smooth limit. Besides, we argue that the wedge Casimir
energy increases with the opening angle and test it with several examples.
Furthermore, we construct the holographic dual of wedges in AdS/BCFT in
general dimensions. We verify that our proposal can produce the expected
Casimir effect within smooth and singular limits. We observe that the Casimir
energy density of a wedge increases with the brane tension. Next, we discuss
the wedge contribution to holographic entanglement entropy and find it
increases with the opening angle, similar to the wedge Casimir energy.
Finally, we briefly discuss the holographic polygon in AdS3/BCFT2 and its
generalization to higher dimensions.

###### Contents

  1. 1 Introduction
  2. 2 Universal Casimir effect of wedges
    1. 2.1 Displacement operator
    2. 2.2 Universality in smooth limit
    3. 2.3 Monotonicity of wedge Casimir effect
  3. 3 Gravity dual of wedges
    1. 3.1 General theory
    2. 3.2 Exact examples
    3. 3.3 Two typical limits
  4. 4 Holographic wedge entanglement
    1. 4.1 AdS4/BCFT3
    2. 4.2 AdSd+1/BCFTd
  5. 5 Toward holographic polygons
  6. 6 Conclusions and Discussions
  7. A Wedge Casimir effect for free scalars
    1. A.1 Heat kernel method
    2. A.2 Mirror method
  8. B Poincaré AdS

##  1 Introduction

Casimir effect [1] is a novel quantum effect that originates from the changes
of zero-point energy due to the boundary or the background geometry. Casimir
effect has been measured in experiments [2, 3, 4] and has wild applications in
quantum field theory, condensed matter, and cosmology [5, 6, 7]. It is a
candidate of dark energy [8] and plays a vital role in worm-hole physics [9,
10].

We are interested in the universal nature of the Casimir effect for boundary
conformal field theories (BCFT) in flat wedge space

$\displaystyle
ds^{2}=dr^{2}+r^{2}d\theta^{2}+\sum_{\hat{i},\hat{j}=1}^{d-2}\eta_{\hat{i}\hat{j}}dy^{\hat{i}}dy^{\hat{j}},\
\ 0\leq\theta\leq\Omega,$ (1)

where $\eta_{\hat{i}\hat{j}}=\text{diag}(-1,1,...,1)$ and $\Omega$ is the
opening angle of the wedge. See [11, 12, 13] for recent studies on the BCFT in
a wedge. For simplicity, we impose the same conformal boundary conditions on
the two boundaries $\theta=0$ and $\theta=\Omega$. As a result, there is only
a geometrical defect at the corner of the wedge $r=0$. From the symmetries of
the wedge and the vanishing of the trace and divergence of the renormalized
stress tensor, we have generally [13, 14]

$\displaystyle\langle
T^{rr}\rangle=\frac{f(\theta,\Omega)-\frac{1}{(d-2)}\partial_{\theta}^{2}f(\theta,\Omega)}{r^{d}},\
\ \langle T^{\theta\theta}\rangle=\frac{-(d-1)f(\theta,\Omega)}{r^{d+2}},$ (2)
$\displaystyle\langle
T^{r\theta}\rangle=\frac{-(d-1)\partial_{\theta}f(\theta,\Omega)}{(d-2)r^{d+1}},\
\ \langle
T^{\hat{i}\hat{j}}\rangle=\frac{f(\theta,\Omega)+\frac{1}{(d-2)^{2}}\partial_{\theta}^{2}f(\theta,\Omega)}{r^{d}}\eta^{\hat{i}\hat{j}}$
(3)

where $f(\theta,\Omega)$ is a function vanishing when $\Omega=\pi$. There are
more symmetries for a ground state, where $\langle T^{r\theta}\rangle$ is
expected to vanish, and $f(\theta,\Omega)$ becomes $\theta$-independent. Thus,
it is expected for a ground state [14]

$\displaystyle\langle T^{i}_{\
j}\rangle_{\text{wedge}}=\frac{f(\Omega)}{r^{d}}\text{diag}\Big{(}1,-(d-1),1,..,1\Big{)}.$
(4)

This is indeed the case for free BCFTs [13, 14, 15] and holographic BCFTs, as
discussed in this paper. For simplicity, we have abbreviated
$f(\theta,\Omega)$ as $f(\Omega)$ above. The function $f(\Omega)$
characterizes the properties of BCFTs and has the following interesting limits

$\displaystyle f(\Omega)\to\begin{cases}\kappa_{1}/\Omega^{d},\ \ \ \ \ \ \
\text{for }\Omega\to 0,\\\ \kappa_{2}(\pi-\Omega),\ \ \text{for
}\Omega\to\pi.\end{cases}$ (5)

In the limits $r\to\infty,\Omega\to 0$ with $r\Omega=L$ fixed, the large $r$
region of a wedge approaches to a strip [12, 14]. In this way, we see that the
constant $\kappa_{1}$ is related to the Casimir effect of a flat strip with
the width $L$

$\displaystyle\langle T^{i}_{\
j}\rangle_{\text{strip}}=\frac{\kappa_{1}}{L^{d}}\text{diag}\Big{(}1,-(d-1),1,..,1\Big{)}.$
(6)

On the other hand, the physical meaning of $\kappa_{2}$ is unclear so far.
This paper shows that $\kappa_{2}$ is universally determined by the norm of
the displacement operator. The displacement operator $D(y)$ describes the
violation of the conservation of energy-stress tensors due to the boundary
[16]

$\displaystyle\nabla_{i}T^{ij}=-\delta(x)D^{j}(y),$ (7)

where $D^{j}(y)=-D(y)n^{j}$ with $n^{j}$ is the outward-pointing normal vector
on the boundary $x=0$. The Zamolodchikov norm of displacement operator is
defined as [16]

$\displaystyle\langle D(y)D(0)\rangle=\frac{C_{D}}{|y|^{2d}}.$ (8)

We prove the following universal relation between the wedge Casimir effect and
displacement operator

$\displaystyle\kappa_{2}=\frac{d\pi^{\frac{d-2}{2}}\Gamma(\frac{d}{2})}{2(d-1)\Gamma[d+2]}C_{D},$
(9)

and verify it with free BCFTs and AdS/BCFT. Besides, we argue that the Casimir
energy density of the wedge $T_{tt}=-f(\Omega)/r^{d}$ increases with the
opening angle

$\displaystyle f^{\prime}(\Omega)\leq 0,$ (10)

and test it with several examples.

On the other hand, we investigate the holographic dual of the wedge in
AdS/BCFT [17] 111Note that we focus on the AdS/BCFT correspondence for wedges,
where the wedge is on the AdS boundary. Please do not confuse it with wedge
holography [18, 19], where the wedge is in bulk.. See [18, 19, 20, 21, 22, 23,
24, 25, 26, 27] for some recent developments of AdS/BCFT and its
generalizations. Constructing the holographic dual of a BCFT with a fixed
boundary shape in flat space is a non-trivial problem 222We thank C. S. Chu
for valuable discussions on this and related problems.. Many typical solutions
in AdS/CFT are not allowed in AdS/BCFT with a specific BCFT boundary. Let us
explain the reasons. We impose the Neumann boundary condition (NBC) on the EOW
brane $Q$ [17]

$\displaystyle\Big{(}K^{ij}-(K-T)h^{ij}\Big{)}|_{Q}=0,$ (11)

and Dirichlet boundary condition on the AdS boundary $M$

$\displaystyle\delta g_{ij}|_{M}=0,$ (12)

were $K$ is the extrinsic curvature and $T=(d-1)\tanh(\rho_{*})$ is the brane
tension 333Following [17], we parameterize the brane tension as
$T=(d-1)\tanh(\rho_{*})$ in this paper. Here $\rho_{*}$ denotes the location
of EOW brane $\rho=\rho_{*}$ in the metric
$ds^{2}=d\rho^{2}+\cosh^{2}(\rho)ds^{2}_{\text{AdS}_{d}}$ [17]. We also set
the AdS radius $l=1$ for simplicity. . Note that this paper focuses on
Takayanagi’s model of AdS/BCFT [17] with NBC (11) on the brane. See [28, 29,
30, 31] for discussions on other boundary conditions. Given a bulk metric
obeying Einstein gravity in bulk $N$ and DBC (12) on the AdS boundary $M$, we
can fix the embedding function of EOW brane from the trace of NBC (11), i.e.,
$(1-d)K+dT=0$. That is because the EOW brane is a codim-1 surface; we need
only one equation to fix its location. For the less-symmetric case, the
traceless parts of NBC $K_{ij}-(K/d)h_{ij}=0$ impose additional constraints
apart from the trace. Not all given bulk solutions automatically satisfy these
additional constraints. Thus, not all solutions in AdS/CFT are solutions in
AdS/BCFT with a given BCFT geometry.

So far, we only know the gravity duals of BCFT in flat space with some
symmetrical boundaries, such as the half-space, strip, and Euclidean disk [17,
32]. There are some perturbative discussions for the less-symmetrical
boundaries with non-zero extrinsic curvatures [33, 34]. In this paper, we
construct the holographic dual of wedge space (1), providing an exact example
of AdS/BCFT with a less-symmetric boundary. We remark that the Poincaré AdS
cannot be dual to the wedge space. One can directly check that Poincaré AdS
disobeys the NBC (11) for the BCFT living in a flat wedge space. See Appendix
B. Another way to see this is that Poincaré AdS has zero holographic stress
tensors, which cannot produce the non-zero Casimir effect (2,3,4) of a wedge.
One may guess Poincaré AdS could be dual to an excited state instead of a
ground state of the wedge. This is not the case. The geometric singularity at
the wedge corner is expected to produce a divergent expectation value of
stress tensors near the corner generally, i.e., $\langle T^{i}_{\
j}\rangle\sim 1/r^{d}$ 444It is similar to the case near a curved smooth
boundary [14, 33], i.e., $\langle T^{i}_{\ j}\rangle\sim\bar{k}^{i}_{\
j}/r^{d-1}$, where $\bar{k}^{i}_{\ j}$ are the traceless-parts of extrinsic
curvatures. The divergent terms of $\langle T^{i}_{\ j}\rangle$ are
universally determined by the Weyl anomaly and are independent of the states
of BCFTs [33].. See section 2 for an example, where we prove the renormalized
stress tensor takes a universal non-zero expression (29) in the limit $r\to
0,\Omega\to\pi$ for general cases. Thus, Poincaré AdS with zero Brown-York
stress tensors cannot be dual to a wedge with non-zero $\langle T^{i}_{\
j}\rangle$. Note that we focus on the initial model of AdS/BCFT [17] instead
of the one [35, 36] with defects on the EOW branes. For the later model, the
EOW brane is singular in bulk. As a result, the Poincaré AdS could be dual to
the wedge for some parameters [35]. However, the above discussions imply these
parameters are unphysical and should be ruled out.

Let us summarize our holographic results. We construct the holographic dual of
wedges in general dimensions. Our construction produces the expected wedge
Casimir effect (4) together with the correct singular and smooth limits (5).
In particular, it obeys the universal relation (9) between the displacement
operator and Casimir effects. These are solid supports for our results. Next,
we consider the wedge contribution to the holographic entanglement entropy and
find it increases with the opening angle, similar to the wedge Casimir energy.
Finally, we discuss the holographic dual of polygons in AdS3/BCFT2 and its
possible generalizations to higher dimensions.

The paper is organized as follows. In section 2, we study the universal
feature of the Casimir effect of wedges and its relation to the displacement
operator. In section 3, we formula the holographic dual of wedges and verify
that it satisfies the universal relation (9). Section 4 investigates the wedge
contributions to holographic entanglement entropy. Section 5 discusses the
holographic duals of polygons in AdS3/BCFT2. Finally, we conclude with some
open problems in section 6.

##  2 Universal Casimir effect of wedges

This section proves the universal relation (9) between the Casimir effect and
the displacement operator and tests it by free BCFTs. Besides, we argue that
$f^{\prime}(\Omega)<0$ and verify it with several examples. This section
mainly focuses on the field theory, and we will discuss the gravity dual in
the next section.

###  2.1 Displacement operator

Let us give a quick review of the displacement operator. Variating the
effective action of BCFT with respect to the metric and the position of the
boundary, we have [16, 37]

$\displaystyle\delta
I_{\text{eff}}=\frac{1}{2}\int_{M}d^{d}x\sqrt{|g_{M}|}T^{ij}\delta g_{M\
ij}+\frac{1}{2}\int_{\partial
M}d^{d-1}y\sqrt{|\sigma|}\left(t^{ij}\delta\sigma_{ij}+2D\delta x\right)$ (13)

where $T^{ij}$, $t^{ij}$ are bulk and boundary stress tensors of BCFTs,
respectively, and $D=-D^{i}n_{i}$ is the displacement operator. For
simplicity, we focus on the flat Euclidean half space

$\displaystyle ds^{2}=dx^{2}+\sum_{a=1}^{d-1}dy^{a}dy^{a},\ \ x\geq 0,$ (14)

where we turn off the variations of derivatives of the boundary metric
$\partial_{x}^{n}h_{ij}$, such as the extrinsic curvatures.

Consider the diffeomorphism in bulk

$\displaystyle\delta_{\xi}y^{a}=0,\ \ \delta_{\xi}x^{i}=-\xi^{i},\ \
\delta_{\xi}g_{M\ ij}=2\nabla_{(i}\xi_{j)},$ (15)

we have

$\displaystyle\delta_{\xi}I_{\text{eff}}$ $\displaystyle=$
$\displaystyle-\int_{M}d^{d}x\sqrt{|g_{M}|}\xi_{j}\nabla_{i}T^{ij}$ (16)
$\displaystyle+\int_{\partial
M}d^{d-1}y\sqrt{|\sigma|}\left(T^{ij}n_{i}\xi_{j}-\hat{\xi}_{j}\hat{\nabla}_{i}t^{ij}+\xi_{j}n^{j}D\right)$
$\displaystyle=$ $\displaystyle 0,$

where $\hat{\nabla}_{i}$ is the induced covariant derivative on the boundary
and $\hat{\xi}_{i}=\xi_{l}h^{l}_{i}$ is the pull back of the bulk vector
$\xi_{i}$ into the boundary. We have used $t^{ij}n_{j}=0$ and the vanishing of
extrinsic curvatures in the above derivations. For an infinite-small $\xi$ in
bulk, we derive from (16)

$\displaystyle\nabla_{i}T^{ij}=0,\ \text{for}\ x>0.$ (17)

As for the infinite-small $\xi$ on the boundary, we get [38]

$\displaystyle\hat{\nabla}_{i}t^{ij}=T^{il}n_{i}h_{l}^{j},$ (18)
$\displaystyle D=-T^{ij}n_{i}n_{j}.$ (19)

Thus, the displacement operator $D=-T^{xx}|_{x=0}$ is given by the normal-
normal component of the stress tensor in half plane [16, 37, 38]. By using
this fact and the conformal symmetries, one can derive the one-point function
$\langle D(y)\rangle=0$ and two-point functions [16, 37] in flat half space,

$\displaystyle\langle T^{xx}(x_{1})D(y)\rangle=-\frac{dC_{D}}{(d-1)}\
\frac{1}{(x_{1}^{2}+y^{2})^{d}}\left(\frac{(x_{1}^{2}-y^{2})^{2}}{(x_{1}^{2}+y^{2})^{2}}-\frac{1}{d}\right),$
(20) $\displaystyle\langle T^{ab}(x_{1})D(y)\rangle=-\frac{dC_{D}}{(d-1)}\
\left(\frac{4x_{1}^{2}y^{a}y^{b}}{(x_{1}^{2}+y^{2})^{d+2}}-\frac{\delta^{ab}}{d(x_{1}^{2}+y^{2})^{d}}\right),$
(21) $\displaystyle\langle T^{ax}(x_{1})D(y)\rangle=-\frac{2dC_{D}}{(d-1)}\
\left(\frac{y^{a}x_{1}}{(x_{1}^{2}+y^{2})^{d+1}}-\frac{2y^{a}x_{1}^{3}}{(x_{1}^{2}+y^{2})^{d+2}}\right),$
(22)

where $T^{ij}(x_{1})=T^{ij}(x_{1},y_{1}^{a}=0)$ and $D(y)=D(x=0,y^{a})$. Note
that, in the limit $x_{1}\to 0$, (20) reduces to the two-point function of
displacement operator (8).

###  2.2 Universality in smooth limit

According to [39], one can derive the one-point function of an operator near
the deformed boundary from the two-point function of this operator with
displacement operator on the non-deformed boundary. For our case of the stress
tensor, we have

$\displaystyle\langle T_{ij}(x_{1})\rangle_{f}=\langle
T_{ij}(x_{1})\rangle_{0}-\int d^{d-1}y\langle T_{ij}(x_{1})D(y)\rangle
f(y)+O(f^{2}),$ (23)

where $\delta x=f(y)$ denotes the small deformation of the boundary, $\langle
T_{ij}(x_{1})\rangle_{0}=0$ in the half space with undeformed boundary, and we
have set $y_{1}^{a}=0$ for $T_{ij}$ and $x=0$ for $D$ for simplicity.

We are ready to derive the Casimir effect of wedges in the smooth limit
$\Omega\to\pi$. By slightly deforming the plane boundary

$\displaystyle\delta x=f(y)=\Pi(y^{1})y^{1}(\Omega-\pi),$ (24)

we get a wedge with opening angle $\Omega\to\pi$. Here $\Pi(y^{1})$ is the
step function, equal to one when $y^{1}\geq 0$ and zero otherwise.
Substituting (24) and correlation functions (20,21,22) into (23) and
performing the coordinate transformations $\rho^{2}=\sum_{a=2}^{d}y^{a}y^{a}$,
we derive

$\displaystyle\langle T_{xx}(x_{1})\rangle_{f}$ $\displaystyle=$
$\displaystyle\frac{2\pi^{\frac{d-2}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\int_{0}^{\infty}\rho^{d-3}d\rho\int_{0}^{\infty}dy^{1}y^{1}(\Omega-\pi)$
(25) $\displaystyle\times\frac{dC_{D}}{(d-1)}\
\frac{1}{(x_{1}^{2}+\rho^{2}+{y^{1}}^{2})^{d}}\left(\frac{(x_{1}^{2}-\rho^{2}-{y^{1}}^{2})^{2}}{(x_{1}^{2}+\rho^{2}+{y^{1}}^{2})^{2}}-\frac{1}{d}\right)$
$\displaystyle=$
$\displaystyle\frac{d\pi^{\frac{d-2}{2}}\Gamma\left(\frac{d}{2}\right)C_{D}}{2(d-1)\Gamma(d+2)}\frac{(\pi-\Omega)}{x_{1}^{d}}+O(\pi-\Omega)^{2},$
$\displaystyle\langle T_{y^{1}y^{1}}(x_{1})\rangle_{f}$ $\displaystyle=$
$\displaystyle\frac{2\pi^{\frac{d-2}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\int_{0}^{\infty}\rho^{d-3}d\rho\int_{0}^{\infty}dy^{1}y^{1}(\Omega-\pi)$
(26) $\displaystyle\times\frac{dC_{D}}{(d-1)}\
\left(\frac{4x_{1}^{2}y^{1}y^{1}}{(x_{1}^{2}+\rho^{2}+{y^{1}}^{2})^{d+2}}-\frac{1}{d(x_{1}^{2}+\rho^{2}+{y^{1}}^{2})^{d}}\right)$
$\displaystyle=$
$\displaystyle-(d-1)\frac{d\pi^{\frac{d-2}{2}}\Gamma\left(\frac{d}{2}\right)C_{D}}{2(d-1)\Gamma(d+2)}\frac{(\pi-\Omega)}{x_{1}^{d}}+O(\pi-\Omega)^{2},$

and for $a\neq 1$

$\displaystyle\langle T_{y^{a}y^{a}}(x_{1})\rangle_{f}$ $\displaystyle=$
$\displaystyle\frac{2\pi^{\frac{d-2}{2}}}{\Gamma\left(\frac{d-2}{2}\right)}\int_{0}^{\infty}\rho^{d-3}d\rho\int_{0}^{\infty}dy^{1}y^{1}(\Omega-\pi)$
(27) $\displaystyle\times\frac{dC_{D}}{(d-1)}\
\left(\frac{4x_{1}^{2}\rho^{2}/(d-2)}{(x_{1}^{2}+\rho^{2}+{y^{1}}^{2})^{d+2}}-\frac{1}{d(x_{1}^{2}+\rho^{2}+{y^{1}}^{2})^{d}}\right)$
$\displaystyle=$
$\displaystyle\frac{d\pi^{\frac{d-2}{2}}\Gamma\left(\frac{d}{2}\right)C_{D}}{2(d-1)\Gamma(d+2)}\frac{(\pi-\Omega)}{x_{1}^{d}}+O(\pi-\Omega)^{2}.$

Similarly, one can check that the off-diagonal elements of stress tensors
vanish, i.e., $\langle T_{xy^{a}}(x_{1})\rangle_{f}=\langle
T_{y^{a}y^{b}}(x_{1})\rangle_{f}=0$. The above results can be rewritten into a
unite form

$\displaystyle\langle
T^{i}_{j}(x_{1})\rangle_{f}=\frac{d\pi^{\frac{d-2}{2}}\Gamma\left(\frac{d}{2}\right)C_{D}}{2(d-1)\Gamma(d+2)}\frac{(\pi-\Omega)}{x_{1}^{d}}\text{diag}\Big{(}1,-(d-1),1,...,1\Big{)}+O(\pi-\Omega)^{2}.$
(28)

Comparing (28) with (4, 5) and identifying $x_{1}$ and $r$ in the smooth limit
$\Omega\to\pi$, we finally obtain the universal relation (9).

Some comments are in order.

  * •

The above discussions focus on flat space with a slightly deformed boundary.
It can be generalized to curved wedge space as long as we focus on the region
near the boundary, i.e., $r\ll 1/\sqrt{R}$, where $r$ is the distance to the
boundary and $R$ labels the curvature of curved wedge space. That is because
the spacetime is locally flat, and the two-point functions (20,21,22) still
work in the near-boundary region for local field theories. Our results imply a
universal Casimir effect near the wedge $r\sim 0$

$\displaystyle\langle T^{i}_{\
j}\rangle_{\text{wedge}}=\frac{d\pi^{\frac{d-2}{2}}\Gamma(\frac{d}{2})C_{D}}{2(d-1)\Gamma[d+2]}\frac{(\pi-\Omega)}{r^{d}}\text{diag}\Big{(}1,-(d-1),1,..,1\Big{)}+O\Big{(}\frac{1}{r^{d-1}},(\pi-\Omega)^{2}\Big{)},$
(29)

in the smooth limit $\Omega\sim\pi$ in general curved wedge space, where $r$
is proper distance to the corner of the wedge.

  * •

The above result (29) applies to a singular boundary, which reminds the
universal Casimir effect of BCFT near a smooth boundary [33, 34]

$\displaystyle\langle T_{ij}\rangle=\alpha\frac{\bar{k}_{ij}}{r^{d-1}}+...,\ \
r\sim 0,$ (30)

where $\bar{k}_{ij}$ are the traceless parts of extrinsic curvatures, $r$ is
the proper distance to the boundary, and $\alpha$ is given by the norm of
displacement operator [34]

$\displaystyle\alpha=-\frac{d\Gamma[\frac{d+1}{2}]\pi^{\frac{d-1}{2}}}{(d-1)\Gamma[d+2]}C_{D}.$
(31)

Interestingly, (29) and (30) depend on both $C_{D}$, which is related to the
B-type boundary central charge of Weyl anomaly [33, 34, 40, 41]. Since Weyl
anomaly is universal and is independent of the states of BCFTs 555The vacuum
state, thermal state, and excited states of BCFTs all have the same Weyl
anomaly in a fixed background spacetime., so does the universal Casimir
effects (29) and (30).

  * •

Let us test the universal relation (9) by free BCFTs. According to [14], we
have for 4d free BCFTs

$\displaystyle
f(\Omega)=\begin{cases}\frac{1}{1440\Omega^{2}}(\frac{\pi^{2}}{\Omega^{2}}-\frac{\Omega^{2}}{\pi^{2}}),\
\ \ \ \ \ \ \ \ \ \ \text{free scalar},\\\
\frac{1}{720\pi^{2}}(\frac{\pi^{2}}{\Omega^{2}}+11)(\frac{\pi^{2}}{\Omega^{2}}-1),\
\ \text{Maxwell theory},\end{cases}$ (32)

which yields

$\displaystyle\kappa_{2}=\begin{cases}\frac{1}{360\pi^{3}},\ \ \ \ \ \ \
\text{free scalar},\\\ \frac{1}{30\pi^{3}},\ \ \ \ \ \ \ \ \text{Maxwell
theory}.\end{cases}$ (33)

It’s worth noting that (32) holds for both Dirichlet and Robin(Neumann)
boundary conditions (BC) for free scalar and applies to both Absolute and
Relative BCs for Maxwell theory. Again, we stress that we impose the same BCs
on the two boundaries of the wedge. If we choose different BCs on the two
wedge boundaries, the discontinuous BCs induce a new singularity in addition
to the geometric singularity at the corner of the wedge. We leave this non-
trivial case to future work. The corresponding $C_{D}$ of free BCFTs can be
read off from [33, 40, 41, 42, 43]. We have

$\displaystyle C_{D}=\begin{cases}\frac{1}{2\pi^{4}},\ \ \ \ \ \ \ \text{free
scalar},\\\ \frac{6}{\pi^{4}},\ \ \ \ \ \ \ \ \text{Maxwell
theory}.\end{cases}$ (34)

The above two equations give $\kappa_{2}/C_{D}=\pi/180$, which agrees with the
universal relation (9) for $d=4$. $C_{D}$ for conformally coupled scalar in
general dimensions is given by [34]

$\displaystyle C_{D}=\frac{\Gamma[\frac{d}{2}]^{2}}{2\pi^{d}}.$ (35)

The corresponding $\kappa_{2}$ can be read off from (148,156,157,158) of
appendix A. We list $\kappa_{2}$ and $C_{D}$ in Table. 1, and verify that the
universal relation (9) is obeyed in higher dimensions too.

Table 1: $\kappa_{2}$ and $C_{D}$ for conformally coupled scalar d | $2$ | 3 | $4$ | $5$ | 6 | 7 | 8
---|---|---|---|---|---|---|---
$\kappa_{2}$ | $\frac{1}{12\pi^{2}}$ | $\frac{1}{512\pi}$ | $\frac{1}{360\pi^{3}}$ | $\frac{3}{16384\pi^{2}}$ | $\frac{1}{2100\pi^{4}}$ | $\frac{25}{524288\pi^{3}}$ | $\frac{1}{5880\pi^{5}}$
$C_{D}$ | $\frac{1}{2\pi^{2}}$ | $\frac{1}{8\pi^{2}}$ | $\frac{1}{2\pi^{4}}$ | $\frac{9}{32\pi^{4}}$ | $\frac{2}{\pi^{6}}$ | $\frac{225}{128\pi^{6}}$ | $\frac{18}{\pi^{8}}$
  * •

As we will show in the next section, the universal relation (9) also holds for
AdS/BCFT.

###  2.3 Monotonicity of wedge Casimir effect

Now we provide a simple argument that $f^{\prime}(\Omega)\leq 0$. Without loss
of generality, we focus on the range $0\leq\Omega\leq\pi$ with $f(\Omega)\geq
0$ below. The range $\pi\leq\Omega\leq 2\pi$ with $f(\Omega)\leq 0$ is
similar. In the following argument, the key point is that the energy density
$T_{tt}$ and the pressure $T_{\theta\theta}$ have the same sign. On the other
hand, the sign of $T_{tt}\sim-f(\Omega)$ is unimportant.

Suppose we adiabatically make larger the wedge by $d\Omega>0$. Since the
pressure along the $\theta$ direction is negative, we need to do positive work
to enlarge the wedge

$\displaystyle dW=-pdV\sim-
T_{\theta\theta}d\Omega\sim(d-1)\frac{f(\Omega)}{r^{d}}d\Omega\geq 0.$ (36)

As a result, the total energy of the wedge should become larger. However,
since the energy density is negative

$\displaystyle T_{tt}=-\frac{f(\Omega)}{r^{d}}\leq 0,$ (37)

the total energy $\int T_{tt}dV$ becomes smaller if the energy density remains
invariant. To have a larger energy, the energy density $T_{tt}$ must becomes
larger, which yields

$\displaystyle f^{\prime}(\Omega)\leq 0.$ (38)

Below, let us test the inequality (38). From (5), we have in the singular and
smooth limits

$\displaystyle
f^{\prime}(\Omega)\to\begin{cases}-\frac{d\kappa_{1}}{\Omega^{d+1}},\ \ \
\text{for }\Omega\to 0,\\\ -\kappa_{2},\ \ \ \ \ \ \text{for
}\Omega\to\pi,\end{cases}<0,$ (39)

which indeed obeys (38). Above, we have used $\kappa_{1}\geq 0$ that the
Casimir energy of parallel plates is negative, and $\kappa_{2}\sim C_{D}\geq
0$ that the norm of the displacement operator is positive. The inequality (38)
also holds for free 4d BCFTs

$\displaystyle
f^{\prime}(\Omega)=\begin{cases}-\frac{\pi^{2}}{360\Omega^{5}},\ \ \ \ \ \ \ \
\ \ \ \text{free scalar},\\\ -\frac{5\Omega^{2}+\pi^{2}}{180\Omega^{5}},\ \
\text{Maxwell theory},\end{cases}<0,$ (40)

and holographic BCFT with zero brane tension (73)

$\displaystyle
f^{\prime}(\Omega)=-\frac{2\pi^{2}\Omega^{d-3}\left(\frac{\sqrt{(d-2)d\Omega^{2}+\pi^{2}}-\pi}{d-2}\right)^{1-d}}{\sqrt{(d-2)d\Omega^{2}+\pi^{2}}}<0.$
(41)

Fig. 2 suggests that the inequality (38) also works for the AdS/BCFT with non-
zero brane tension.

In summary, in the smooth limit, we prove the universal relation (9) between
the displacement operator and the wedge Casimir effect. Besides, we argue that
the wedge Casimir energy density increases with the opening angle, i.e.,
$f^{\prime}(\Omega)\leq 0$. Finally, we test our results with free BCFTs. We
will also test them using AdS/BCFT in the next section.

##  3 Gravity dual of wedges

In this section, we construct the holographic dual of wedges in AdS/BCFT. We
find it is given by a cutout of the AdS soliton. We stress again that we focus
on AdS/BCFT [17] instead of wedge holography [18, 19] in this paper. We verify
that our construction produces the expected Casimir effect of wedges (4, 5).
In particular, it obeys the universal relation (9) between the Casimir effect
and the displacement operator.

###  3.1 General theory

We aim to find the holographic dual of a BCFT living in the wedge space (1).
As we have discussed in the introduction, it cannot be the Poincaré AdS since
it does not obey NBC (11) and cannot produce the correct Casimir effect (4).
Instead, we find the AdS soliton can do the job

$\displaystyle
ds^{2}=\frac{\frac{dz^{2}}{h(z)}+h(z)d\theta^{2}+\frac{dr^{2}+\sum_{\hat{i},\hat{j}=1}^{d-2}\eta_{\hat{i}\hat{j}}dy^{\hat{i}}dy^{\hat{j}}}{r^{2}}}{z^{2}},$
(42)

where $h(z)=1-z^{2}-c_{1}z^{d}$. We label the constant
$c_{1}=(1-z_{h}^{2})/z_{h}^{d}$ so that $h(z_{h})=0.$ To be absence of conical
singularity in bulk, we choose the angle period in bulk as

$\displaystyle\beta=\frac{4\pi}{|h^{\prime}(z_{h})|}=\frac{4\pi
z_{h}}{d+(2-d)z_{h}^{2}}.$ (43)

On the other hand, the range of angle is $0\leq\theta\leq\Omega<\beta$ on the
AdS boundary $z\to 0$. It should be mentioned that the metric takes the form
of a hyperbolic black hole (with time replaced by angle $\theta$), which plays
a vital role in holographic Rényi entropy [44, 45] and cone holography [20,
21].

Let us show the metric (42) has the expected features to be dual to a wedge.
On the AdS boundary $z\to 0$, the BCFT metric is conformally equivalent to the
wedge metric (1) up to a Weyl factor $1/r^{2}$. To get exactly the wedge
metric (1), One can perform a Weyl transformation on the AdS boundary,
equivalently, a coordinate transformation in bulk. The coordinate
transformations for $d=2$ are

$\displaystyle
z=\frac{4r_{1}z_{1}}{\sqrt{8\left(c_{1}+2\right)r_{1}^{2}z_{1}^{2}+c_{1}^{2}z_{1}^{4}+16r_{1}^{4}}},$
(44) $\displaystyle
r=r_{1}\left(\frac{\left(2\sqrt{c_{1}+1}+c_{1}+2\right)z_{1}^{2}+4r_{1}^{2}}{\left(-2\sqrt{c_{1}+1}+c_{1}+2\right)z_{1}^{2}+4r_{1}^{2}}\right){}^{\frac{1}{2\sqrt{c_{1}+1}}},$
(45)

which yields the bulk metric

$\displaystyle
ds^{2}=\frac{dz_{1}^{2}+\Big{(}1+f_{r}(\frac{z_{1}}{r_{1}})\Big{)}dr_{1}^{2}+r_{1}^{2}\Big{(}1+f_{\theta}(\frac{z_{1}}{r_{1}})\Big{)}d\theta^{2}}{z_{1}^{2}},$
(46)

with $f_{r}(z_{1})=\frac{1}{16}c_{1}z_{1}^{2}\left(c_{1}z_{1}^{2}+8\right)$
and
$f_{\theta}(z_{1})=\frac{1}{16}c_{1}z_{1}^{2}\left(c_{1}z_{1}^{2}-8\right)$.
Clearly, the above metric gives the expected wedge metric
$ds^{2}=dr_{1}^{2}+r_{1}^{2}d\theta^{2}$ and BCFT stress tensor on the AdS
boundary

$\displaystyle\langle T^{i}_{\
j}\rangle_{d=2}=\frac{c_{1}}{r_{1}^{2}}\text{diag}\Big{(}1,-1\Big{)}.$ (47)

For $d>2$, it is difficult to find the exact coordinate transformations.
Instead, we obtain the perturbative ones up to $O(z_{1}^{d+1})$

$\displaystyle
z=\frac{z_{1}}{\sqrt{r_{1}^{2}+z_{1}^{2}}}\Big{(}1-\frac{c_{1}z_{1}^{d}}{2dr_{1}^{d}}+O(z_{1}^{d+1})\Big{)},$
$\displaystyle r=\sqrt{r_{1}^{2}+z_{1}^{2}}\Big{(}1+O(z_{1}^{d+1})\Big{)},$
(48)

which transforms the bulk metric to be

$\displaystyle
ds^{2}=\frac{dz_{1}^{2}+\Big{(}1+\frac{c_{1}z_{1}^{d}}{dr_{1}^{d}}\Big{)}\Big{(}dr_{1}^{2}+\sum_{\hat{i},\hat{j}=1}^{d-2}\eta_{\hat{i}\hat{j}}dy^{\hat{i}}dy^{\hat{j}}\Big{)}+r_{1}^{2}\Big{(}1-\frac{c_{1}(d-1)z_{1}^{d}}{dr_{1}^{d}}\Big{)}d\theta^{2}+O(z_{1}^{d+1})}{z_{1}^{2}}.$
(49)

The above metric gives the expected wedge metric (1) and renormalized stress
tensor (4) with

$\displaystyle f(\Omega)=c_{1}.$ (50)

In the above discussions, we have used the formula of holographic stress
tensor (setting $16\pi G_{N}=1$)[46]

$\displaystyle\langle T_{ij}\rangle=dg^{(d)}_{ij},$ (51)

where $g^{(d)}_{ij}$ is defined in the Fefferman-Graham expansion of the
asymptotic AdS space

$\displaystyle
ds^{2}=\frac{dz^{2}+(g^{(0)}_{ij}+z^{2}g^{(2)}_{ij}+...+z^{d}g^{(d)}_{ij}+...)dy^{i}dy^{j}}{z^{2}}.$
(52)

In general, there are corrections to the formula (51) from Weyl anomaly for
even $d$. For our case of flat space with zero Weyl anomaly, these potential
corrections disappear. Below we mainly focus on the coordinates $(z,r)$
instead of $(z_{1},r_{1})$, since the corresponding metric (42 has an exact
expression.

Figure 1: Geometry for holographic dual of wedge in coordinates
$(z_{1},r_{1})$. The gray plane denotes the AdS boundary $M$, which is divided
by region I and region II. BCFT can be defined either on region I or region
II. Recall that the angle period is $\beta$ (43), which is not necessarily
$2\pi$. Thus, the opening angles of region I and region II can be both smaller
than $\pi$. Region I and region II share the same smooth EOW brane $Q$ (the
red surface). Since extrinsic curvatures flip signs when crossing the red
surface, from NBC (11), the brane tension differs by a minus sign for the BCFT
defined on region I and region II. Thus, once we obtain the AdS/BCFT of wedges
with negative brane tension, by considering it’s complement, we automatically
get the one with positive brane tension.

Next, we show the metric (42) obeys NBC (11) for suitable embedding function
of EOW brane

$\displaystyle z=z(\theta).$ (53)

As we have stressed in the introduction, the NBC for a fixed BCFT geometry is
not always satisfied by a given bulk metric such as Poincaré AdS. See Fig. 1
for the schematic geometry of AdS/BCFT for wedges. BCFT can be defined either
on region I or region II on the AdS boundary $M$. These two regions share the
same EOW brane $Q$ (red surface). Since extrinsic curvatures change signs when
crossing a surface, from NBC (11), the brane tension differs by a minus sign
for the BCFT defined on region I and region II. Thus, once we obtain the
AdS/BCFT of wedges with negative tension, by considering it’s complement, we
automatically get the one with positive tension. It is similar to the case of
AdS/BCFT for strips [32]. Without loss of generality, we focus on the region I
with negative brane tension below. We remark that both the opening angles of
region I and region II can be smaller than $\pi$, since the angle period
$\beta$ (43) is not necessarily $2\pi$. In fact, as we will show below, the
opening angles are the same for the case of $d=2$, i.e.,
$\Omega_{\text{I}}=\Omega_{\text{II}}=\pi/\sqrt{1+c_{1}}$.

Substituting the embedding function (53) and metric (42) into NBC (11), we get
an independent equation for the region I with a negative brane tension
$T=(d-1)\tanh(\rho_{*})$

$\displaystyle\frac{h(z(\theta))}{\sqrt{\frac{z^{\prime}(\theta)^{2}}{h(z(\theta))}+h(z(\theta))}}+\tanh\left(\rho_{*}\right)=0.$
(54)

At the turning point $z(\Omega/2)=z_{\text{max}}$, we have
$z^{\prime}(\Omega/2)=0$. Then the above equation becomes

$\displaystyle\sqrt{h(z_{\text{max}})}+\tanh\left(\rho_{*}\right)=0,$ (55)

which yields

$\displaystyle
c_{1}=f(\Omega)=z_{\max}^{-d}\left(\text{sech}^{2}\left(\rho_{*}\right)-z_{\max}^{2}\right).$
(56)

From (54) and (55), we derive

$\displaystyle
d\theta=\pm\frac{dz}{h(z)\sqrt{\frac{h(z)}{h\left(z_{\max}\right)}-1}},$ (57)

which yields the embedding function

$\displaystyle\theta_{<}(z)=\int_{0}^{z}\frac{ds}{h(s)\sqrt{\frac{h(s)}{h\left(z_{\max}\right)}-1}},\
\ \text{for}\ 0\leq\theta\leq\Omega/2.$ (58)

The case of $\Omega/2\leq\theta\leq\Omega$ can be obtained by symmetry, i.e.
$\theta_{>}(z)=\Omega-\theta_{<}(z)$. From (58), we obtain the opening angle
for region I with negative $\rho_{*}$

$\displaystyle\Omega_{\text{I}}(\rho_{*})=2\theta(z_{\max})=2\int_{0}^{z_{\text{max}}}\frac{dz}{h(z)\sqrt{\frac{h(z)}{h\left(z_{\max}\right)}-1}}.$
(59)

Recall that region II is the complement of region I. We get the opening angle
for region II with positive $\rho_{*}$

$\displaystyle\Omega_{\text{II}}(\rho_{*})=\beta-\Omega_{\text{I}}(\rho_{*}\to-\rho_{*}),$
(60)

where we have used the fact that the tensions of region I and region II differ
by a minus sign. From (56), we solve $z_{\text{max}}$ in terms of $f(\Omega)$
and then substitute it into (59). Then, (59) becomes the inverse function of
$f(\Omega)$, which can be obtained numerically generally. In fact, the
integral expressions (58,59) are good enough for many purposes. For instance,
they can draw the figures of $f(\Omega)$. See Fig. 2, which shows that
$f(\Omega)$ decreases with the tension $T=(d-1)\tanh(\rho_{*})$ and the
opening angle $\Omega$.

Figure 2: Holographic Casimir effects of 3d wedge (left) and 4d wedge (right).
Here, $\rho_{*}=-1,0,1$ denote the negative, zero, and positive brane
tensions. It shows $f(\Omega)$ approaches zero at $\Omega\to\pi$ and infinity
at $\Omega\to 0$. Besides, it suggests $f(\Omega)$ decreases with the tension
$T=(d-1)\tanh(\rho_{*})$ and the opening angle $\Omega$.

###  3.2 Exact examples

Let us study some examples, where the integrals (58,59) can be calculated
exactly.

Case I:  $d=2$

For $d=2$, we have $\beta=\frac{2\pi}{\sqrt{c_{1}+1}}$ and
$z_{\max}=\frac{\text{sech}\left(\rho_{*}\right)}{\sqrt{c_{1}+1}}$.
Integrating (57,59, 60), we obtain

$\displaystyle\theta_{<}(z)=\frac{\tan^{-1}\left(\frac{z}{\sqrt{\frac{\text{csch}^{2}\left(\rho_{*}\right)}{c_{1}+1}-z^{2}\coth^{2}\left(\rho_{*}\right)}}\right)}{\sqrt{c_{1}+1}},$
(62)
$\displaystyle\Omega_{\text{I}}=\Omega_{\text{II}}=\frac{\pi}{\sqrt{c_{1}+1}}=\frac{\beta}{2}.$

From (62), we get the characteristic function of 2d wedge

$\displaystyle f_{2d}(\Omega)=c_{1}=\frac{\pi^{2}-\Omega^{2}}{\Omega^{2}},$
(63)

which agrees with the results of field theory [47]

$\displaystyle
f_{2d}(\Omega)=\frac{c}{24\pi}\frac{\pi^{2}-\Omega^{2}}{\Omega^{2}},$ (64)

where $c=\frac{3l}{2G_{N}}=24\pi$ is the central charge of 2d BCFT. Recall
that we have set $16\pi G_{N}=1$ and $l=1$. Since a wedge can be obtained by
the conformal transformation of half space for $d=2$, the wedge Casimir effect
takes universal form (64) [47]. This is indeed the case for the holographic
result (63).

Case II:  $d=4$

We first express the bulk angle period $\beta$ and turning point $z_{\max}$ in
functions of $c_{1}$

$\displaystyle\beta=\frac{2\sqrt{2}\pi}{\sqrt{\frac{\left(4c_{1}+1\right)\left(\sqrt{4c_{1}+1}-1\right)}{c_{1}}}},\
\
z_{\max}=\frac{\sqrt{\frac{\sqrt{4c_{1}\text{sech}^{2}\left(\rho_{*}\right)+1}-1}{c_{1}}}}{\sqrt{2}}.$
(65)

From (58), we obtain the embedding function for $0\leq\theta\leq\Omega/2$

$\displaystyle\theta_{<}(z)$ $\displaystyle=$
$\displaystyle\frac{-z_{\max}\sqrt{\frac{c_{1}\left(z_{\max}^{4}-z^{4}\right)+z_{\max}^{2}-z^{2}}{c_{1}z_{\max}^{4}+z_{\max}^{2}}}}{2\sqrt{\left(4c_{1}+1\right)\left(\frac{c_{1}z^{4}+z^{2}-1}{c_{1}z_{\max}^{4}+z_{\max}^{2}-1}-1\right)}}\times$
(66)
$\displaystyle\Big{[}(\sqrt{4c_{1}+1}-1)\Pi\left(\frac{-2c_{1}z_{\max}^{2}}{\sqrt{4c_{1}+1}+1};-\sin^{-1}\left(\frac{z}{z_{\max}}\right)|\frac{-c_{1}z_{\max}^{2}}{c_{1}z_{\max}^{2}+1}\right)$
$\displaystyle+\left(\sqrt{4c_{1}+1}+1\right)\Pi\left(\frac{2c_{1}z_{\max}^{2}}{\sqrt{4c_{1}+1}-1};-\sin^{-1}\left(\frac{z}{z_{\max}}\right)|\frac{-c_{1}z_{\max}^{2}}{c_{1}z_{\max}^{2}+1}\right)\Big{]},$

and $\theta_{>}(z)=\Omega-\theta_{<}(z)$ for $\Omega/2\leq\theta\leq\Omega$,
where $\Pi$ denotes the elliptic integral of the third kind. For region I with
$\rho_{*}<0$, the opening angle is given by

$\displaystyle\Omega_{\text{I}}$ $\displaystyle=$ $\displaystyle
2\theta_{<}(z_{\max})=\frac{-\sinh\left(\rho_{*}\right)}{\sqrt{\frac{c_{1}\left(8c_{1}+2\right)}{\sqrt{4c_{1}\text{sech}^{2}\left(\rho_{*}\right)+1}-1}}}\times$
(67)
$\displaystyle\Big{[}\left(\sqrt{4c_{1}+1}-1\right)\Pi\left(\frac{1-\sqrt{4c_{1}\text{sech}^{2}\left(\rho_{*}\right)+1}}{\sqrt{4c_{1}+1}+1}|\frac{2}{\sqrt{4c_{1}\text{sech}^{2}\left(\rho_{*}\right)+1}+1}-1\right)$
$\displaystyle+\left(\sqrt{4c_{1}+1}+1\right)\Pi\left(\frac{\sqrt{4c_{1}\text{sech}^{2}\left(\rho_{*}\right)+1}-1}{\sqrt{4c_{1}+1}-1}|\frac{2}{\sqrt{4c_{1}\text{sech}^{2}\left(\rho_{*}\right)+1}+1}-1\right)\Big{]}$

where we have used (65) above. Recall we have
$\Omega_{\text{II}}=\beta-\Omega_{\text{I}}(\rho_{*}\to-\rho_{*})$ for region
II with $\rho_{*}>0$. In the tensionless limit, we derive

$\displaystyle\lim_{\rho_{*}\to 0}\Omega_{\text{I}}=\lim_{\rho_{*}\to
0}\Omega_{\text{II}}=\frac{\sqrt{2}\pi}{\sqrt{\frac{\left(4c_{1}+1\right)\left(\sqrt{4c_{1}+1}-1\right)}{c_{1}}}}=\frac{\beta}{2}.$
(68)

It means $\Omega$ is a continuous function of $\rho_{*}$, which can be
regarded as a check of our results. From (68), we solve

$\displaystyle\lim_{\rho_{*}\to 0}f(\Omega)=c_{1}$ $\displaystyle=$
$\displaystyle\frac{-8\Omega^{4}+4\pi^{2}\Omega^{2}+\pi^{3}\sqrt{8\Omega^{2}+\pi^{2}}+\pi^{4}}{32\Omega^{4}},$
(69) $\displaystyle\approx$
$\displaystyle\begin{cases}\frac{\pi^{4}}{16\Omega^{4}},\ \ \ \ \ \ \ \ \ \
\text{for }\Omega\to 0,\\\ \frac{2}{3\pi}(\pi-\Omega),\ \ \text{for
}\Omega\to\pi,\end{cases}$ (70)

which takes the expected limits (5) and gives $\lim_{\rho_{*}\to
0}\kappa_{2}=2/(3\pi)$. From [34, 33], we have $\lim_{\rho_{*}\to
0}C_{D}=120/\pi^{2}$, which agrees with the universal relation (9). This is
another support of our results. In section 3.3, we verify the universal
relation (9) for general $d$ and $\rho_{*}$.

Case III: $\rho_{*}=0$

As discussed above, the calculations can be highly simplified in the
tensionless case $\rho_{*}=0$. In such limit, we have by symmetry

$\displaystyle\Omega_{\text{I}}=\Omega_{\text{II}}=\frac{\beta}{2}=\frac{2\pi
z_{\max}}{d+(2-d)z_{\max}^{2}},$ (71)

where we have used $\lim_{\rho_{*}\to 0}z_{h}=z_{\max}$ from
$c_{1}=(1-z_{h}^{2})/z_{h}^{d}=(\text{sech}^{2}(\rho_{*})-z_{\max}^{2})/z_{\max}^{d}$.
From (56), we have

$\displaystyle\lim_{\rho_{*}\to
0}f(\Omega)=c_{1}=z_{\max}^{-d}\left(1-z_{\max}^{2}\right).$ (72)

The above two equations yield

$\displaystyle\lim_{\rho_{*}\to 0}f(\Omega)$ $\displaystyle=$
$\displaystyle\Omega^{d}\left(\frac{\sqrt{(d-2)d\Omega^{2}+\pi^{2}}-\pi}{d-2}\right)^{-d}\left(1-\frac{\left(\pi-\sqrt{(d-2)d\Omega^{2}+\pi^{2}}\right)^{2}}{(d-2)^{2}\Omega^{2}}\right)$
(73) $\displaystyle\approx$
$\displaystyle\begin{cases}d^{-d}(2\pi)^{d}\frac{1}{\Omega^{d}},\ \ \ \ \ \ \
\ \text{for }\Omega\to 0,\\\ \frac{2}{\pi(d-1)}(\pi-\Omega),\ \ \ \ \
\text{for }\Omega\to\pi,\end{cases}$ (74)

which yields $\lim_{\rho_{*}\to 0}\kappa_{2}=2/((d-1)\pi)$. From [34], we read
off $\lim_{\rho_{*}\to
0}C_{D}=2^{d+2}\pi^{-\frac{d}{2}-\frac{1}{2}}\Gamma\left(\frac{d+3}{2}\right)$,
which obeys the universal relation (9). We verify the universal relation (9)
for general $\rho_{*}$ in the next subsection.

###  3.3 Two typical limits

Now we consider two typical limits (5) of opening angles, where the integral
(59) can be performed exactly. We verify that $\kappa_{1}$ is given by the
coefficient (6) of holographic strip in the small angle limit, and
$\kappa_{2}$ obeys the universal relation (9) in the smooth limit. These are
strong supports of our results.

Limit I: $\Omega\to 0$

Performing a transformation $z=y\ z_{\max}$, we rewrite the integral (59) as

$\displaystyle\Omega_{\text{I}}=\int_{0}^{1}dy\frac{2z_{\max}}{\left(1-ay^{d}-\left(y^{2}-y^{d}\right)z_{\max}^{2}\right)\sqrt{\frac{a\left(y^{d}-1\right)+\left(y^{2}-y^{d}\right)z_{\max}^{2}}{a-1}}},$
(75)

where $a=\text{sech}^{2}\left(\rho_{*}\right)$. In the small angle limit, we
have $f(\Omega)\to\kappa_{1}/\Omega^{d}\to\infty$, which yields $z_{\max}\to
0$ from (56). In such limit, we obtain

$\displaystyle
f(\Omega)=c_{1}=\frac{a}{z_{\max}^{d}}+O(\frac{1}{z_{\max}^{d-1}}),$ (76)

and

$\displaystyle\Omega_{\text{I}}$ $\displaystyle=$
$\displaystyle\frac{2z_{\max}}{\sqrt{\frac{a}{1-a}}}\int_{0}^{1}dy\frac{1}{\sqrt{1-y^{d}}\left(1-ay^{d}\right)}+O(z_{\max}^{2}),$
(77) $\displaystyle=$
$\displaystyle\frac{2z_{\max}}{\sqrt{\frac{a}{1-a}}}\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{1}{d}\right)}{d\Gamma\left(\frac{1}{2}+\frac{1}{d}\right)}\,_{2}F_{1}\left(\frac{1}{d},1;\frac{d+2}{2d};a\right)+O(z_{\max}^{2}),$

and

$\displaystyle\Omega_{\text{II}}$ $\displaystyle=$ $\displaystyle\frac{4\pi
z_{\max}}{a^{1/d}d}-\frac{2z_{\max}}{\sqrt{\frac{a}{1-a}}}\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{1}{d}\right)}{d\Gamma\left(\frac{1}{2}+\frac{1}{d}\right)}\,_{2}F_{1}\left(\frac{1}{d},1;\frac{d+2}{2d};a\right)+O(z_{\max}^{2}).$
(78)

In the above derivations, we have used the integral formula

$\displaystyle\int_{0}^{1}dy\frac{1}{\sqrt{1-y^{d}}\left(1-ay^{d}\right)}=\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{1}{d}\right)}{d\Gamma\left(\frac{1}{2}+\frac{1}{d}\right)}\,_{2}F_{1}\left(\frac{1}{d},1;\frac{d+2}{2d};a\right),$
(79)

which can be derived from the standard integral expression of hypergeometric
function by taking $t=y^{d}$

$\,{}_{2}F_{1}\left(a,b;c;z\right)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{1}dt\
t^{b-1}(1-t)^{c-b-1}(1-tz)^{-a}.$ (80)

From (76, 77,78) and $a=\text{sech}^{2}\left(\rho_{*}\right)$, we obtain

$\displaystyle\kappa_{1}=\begin{cases}\left(\frac{2\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{1}{d}\right)\left(-\tanh\left(\rho_{*}\right)\right)\text{sech}^{\frac{2}{d}-1}\left(\rho_{*}\right)\,_{2}F_{1}\left(1,\frac{1}{d};\frac{1}{d}+\frac{1}{2};\text{sech}^{2}\left(\rho_{*}\right)\right)}{d\Gamma\left(\frac{1}{2}+\frac{1}{d}\right)}\right){}^{d},\
\ \ \ \ \text{for }\rho_{*}\leq 0,\\\
\left(\frac{4\pi}{d}-\frac{2\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{1}{d}\right)\tanh\left(\rho_{*}\right)\text{sech}^{\frac{2}{d}-1}\left(\rho_{*}\right)\,_{2}F_{1}\left(1,\frac{1}{d};\frac{1}{d}+\frac{1}{2};\text{sech}^{2}\left(\rho_{*}\right)\right)}{d\Gamma\left(\frac{1}{2}+\frac{1}{d}\right)}\right){}^{d},\
\ \text{for }\rho_{*}\geq 0.\end{cases}$ (81)

As discussed in the introduction, $\kappa_{1}$ is related to the Casimir
effects of strips. The holographic strip is studied in [32], which yields the
same $\kappa_{1}$ as above. It is a strong support to our proposal of
holographic wedges.

Limit II: $\Omega\to\pi$

Let us go on to discuss the smooth limit $\Omega\to\pi$. We have

$\displaystyle
f(\Omega)=c_{1}=2a^{\frac{1}{2}-\frac{d}{2}}\left(\sqrt{a}-z_{\max}\right)+O\left(\sqrt{a}-z_{\max}\right)^{2},$
(82)

and

$\displaystyle\Omega_{\text{I}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dy\frac{2\sqrt{\frac{1-a}{1-y^{2}}}}{1-ay^{2}}+O\left(\sqrt{a}-z_{\max}\right)^{2}$
(83) $\displaystyle+$
$\displaystyle\int_{0}^{1}dy\frac{2a\left(\frac{1-a}{a\left(1-y^{2}\right)}\right)^{3/2}\left(y^{d}-1-a\left(3y^{d+2}-2y^{d}-2y^{4}+y^{2}\right)\right)}{(1-a)\left(1-ay^{2}\right)^{2}}\left(\sqrt{a}-z_{\max}\right)$
$\displaystyle=$ $\displaystyle\pi+O\left(\sqrt{a}-z_{\max}\right)^{2}$
$\displaystyle+$
$\displaystyle\frac{2\sqrt{\pi}\Gamma\left(\frac{d+1}{2}\right)\left(a(1-a)^{-\frac{d}{2}}\,_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+2}{2};\frac{a}{a-1}\right)-\frac{d}{\sqrt{1-a}}\right)}{\sqrt{a}d\Gamma\left(\frac{d}{2}\right)}\left(\sqrt{a}-z_{\max}\right)$

and

$\displaystyle\Omega_{\text{II}}$ $\displaystyle=$
$\displaystyle\beta-\Omega_{\text{I}}=\pi+O\left(\sqrt{a}-z_{\max}\right)^{2}$
$\displaystyle+$
$\displaystyle\Big{(}2\pi(1-d)a^{\frac{1}{2}-\frac{d}{2}}-\frac{2\sqrt{\pi}\Gamma\left(\frac{d+1}{2}\right)\left(a(1-a)^{-\frac{d}{2}}\,_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+2}{2};\frac{a}{a-1}\right)-\frac{d}{\sqrt{1-a}}\right)}{\sqrt{a}d\Gamma\left(\frac{d}{2}\right)}\Big{)}\left(\sqrt{a}-z_{\max}\right).$

In the above calculations, we have used the integral formula 666One can prove
this formula by applying Mathematica together with various identities of
hypergeometric functions for general $d$. Note that Mathematica cannot
calculate directly the definite integral (85) for general $d$. The trick is
that one first performs the indefinite integration by using Mathematica.

$\displaystyle\int_{0}^{1}dy\frac{2a\left(\frac{1-a}{a\left(1-y^{2}\right)}\right)^{3/2}\left(y^{d}-1-a\left(3y^{d+2}-2y^{d}-2y^{4}+y^{2}\right)\right)}{(1-a)\left(ay^{2}-1\right)^{2}}$
(85) $\displaystyle=$
$\displaystyle\frac{2\sqrt{\pi}\Gamma\left(\frac{d+1}{2}\right)\left(a(1-a)^{-\frac{d}{2}}\,_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+2}{2};\frac{a}{a-1}\right)-\frac{d}{\sqrt{1-a}}\right)}{\sqrt{a}d\Gamma\left(\frac{d}{2}\right)}.$

From (82,83,3.3) together with $a=\text{sech}^{2}\left(\rho_{*}\right)$, we
derive

$\displaystyle\kappa_{2}=\begin{cases}\frac{-d\Gamma\left(\frac{d}{2}\right)\text{sech}^{3}\left(\rho_{*}\right)\left(-\text{csch}\left(\rho_{*}\right)\right){}^{-d}}{\sqrt{\pi}\Gamma\left(\frac{d+1}{2}\right)\left(\text{sech}^{3}\left(\rho_{*}\right)F+d\text{csch}\left(\rho_{*}\right)\left(-\tanh\left(\rho_{*}\right)\right){}^{d}\right)},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for }\rho_{*}\leq 0,\\\
\frac{d\Gamma\left(\frac{d}{2}\right)}{\pi(d-1)d\Gamma\left(\frac{d}{2}\right)-\frac{\sqrt{\pi}\Gamma\left(\frac{d+1}{2}\right)\left(d\left(\tanh\left(\rho_{*}\right)\text{sech}\left(\rho_{*}\right)\right){}^{d}-\sinh\left(\rho_{*}\right)\text{sech}^{d+3}\left(\rho_{*}\right)F\right)}{\text{sech}^{2}\left(\rho_{*}\right)\tanh^{d+1}\left(\rho_{*}\right)}},\
\text{for }\rho_{*}\geq 0,\end{cases}$ (86)

where
$F=\,_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+2}{2};-\text{csch}^{2}\left(\rho_{*}\right)\right)$.
We remark that $\kappa_{2}$ is continuous at $\rho_{*}=0$. In fact,
$\kappa_{2}$ with $\rho_{*}\geq 0$ and $\rho_{*}\leq 0$ can be derived from
the analytical continuations of each others. Take $d=4$ as an example, (86)
yields

$\displaystyle\kappa_{2}=\frac{2}{3\pi\left(1+\tanh\left(\rho_{*}\right)\right)},\
\ \ \ \text{for }d=4,$ (87)

for both $\rho_{*}\leq 0$ and $\rho_{*}\geq 0$. $C_{D}$ for AdS/BCFT can be
found in [34]. We verify that $\kappa_{2}$ (86,87) and $C_{D}$ of [34] indeed
obeys the universal relation (9), which is a non-trivial test of our results.

To summarize, we construct the holographic dual of wedges and show that it
yields the expected Casimir effects in this section. In particular, it obeys
the universal relation (9), which strongly supports our results. To end this
section, we draw the wedge Casimir effects for various 4d BCFTs in Fig. 3. It
shows that $f^{\prime}(\Omega)\leq 0$, which agrees with the discussions of
section 2.3. Besides, it implies that the holographic wedge Casimir energy
increases with the brane tension $T=(d-1)\tanh(\rho_{*})$. In particular, the
negative tension (smaller tension) tends to produce larger $f(\Omega)$ and
$f(\Omega)/C_{D}$, or equivalently, smaller Casimir energy density
$T_{tt}=-f(\Omega)/r^{d}$. It is natural that the more negative the brane
tension, the more negative the Casimir energy density.

Figure 3: Wedge Casimir effects of various 4d BCFTs. The left figure is for
$f(\Omega)$ and the right figure is for $f(\Omega)/C_{D}$. The blue, orange,
green, purple and red curves denotes the Casimir effects for holographic BCFTs
with $(\rho_{*}=-1,0,1)$, Maxwell theory and free scalar. The left figure
suggests that
$f_{\rho_{*}=-1}>f_{\rho_{*}=0}>f_{\rho_{*}=1}>f_{\text{Maxwell}}>f_{\text{scalar}}$.
The right figure implies that
$f_{\rho_{*}=-1}/C_{D}>f_{\text{scalar}}/C_{D}>f_{\rho_{*}=0}/C_{D}>f_{\text{Maxwell}}/C_{D}>f_{\rho_{*}=1}/C_{D}$.

##  4 Holographic wedge entanglement

This section studies the holographic entanglement entropy (HEE) of the
subsystem $0\leq r\leq L$ in a wedge 777This paper studies the entanglement
entropy for BCFT living in a wedge. Please distinguish the entanglement
entropy between a wedge region and its complement for CFTs. The latter case
has been studied in [48] with interesting results.. The RT surface generally
depends on the angle $\theta$, complicating the calculations. For the
tensionless case $\rho_{*}=0$, we have
$\Omega_{\text{I}}=\Omega_{\text{II}}=\beta/2$ (71) by symmetry. It means that
regions I and II, as well as their bulk duals and RT surfaces, are symmetric.
Thus, the RT surface of a wedge with opening angle $\beta/2$ is just half that
of the whole space and is $\theta$-independent. We focus on this case and
leave the tensive case for future works. For $\rho_{*}=0$, all the non-trivial
contributions to entanglement entropy come from the corner of the wedge. We
call such contribution wedge entanglement, which vanishes when the wedge
disappears $\Omega=\pi$. We find that wedge entanglement increases with the
opening angle $\Omega$, similar to the wedge Casimir energy.

We focus on the bulk metric (42) with $0\leq\theta\leq\Omega=\beta/2$ and the
following ansatz of RT surface

$\displaystyle t=\text{constant},\ r=F(z).$ (88)

Then, the area of the RT surface reads

$\displaystyle
A=\Omega\int_{\epsilon}^{z_{h}}\frac{1}{z^{d-1}F(z)^{d-3}}\sqrt{1+\frac{h(z)F^{\prime}(z)^{2}}{F(z)^{2}}}\
dz,$ (89)

where $\epsilon$ is the UV cutoff and $z_{h}$ is defined by $h(z_{h})=0$. We
have set the tangential volume $V=\int d^{d-3}y=1$ for simplicity. Taking
variations of (89) with respect to $F(z)$, we derive the equation of motion
(EOM)

$\displaystyle F^{\prime\prime}(z)$ $\displaystyle=$
$\displaystyle-\frac{(d-3)F(z)}{h(z)}-\frac{(d-4)F^{\prime}(z)^{2}}{F(z)}$
(90) $\displaystyle+$
$\displaystyle\frac{F^{\prime}(z)^{3}\left(2(d-1)h(z)-zh^{\prime}(z)\right)}{2zF(z)^{2}}+\frac{F^{\prime}(z)\left((d-1)h(z)-zh^{\prime}(z)\right)}{zh(z)}.$

###  4.1 AdS4/BCFT3

Let us first study the case $d=3$, for which we can make analytical
discussions. When $d=3$, there is an exact solution to EOM (90)

$\displaystyle r=F(z)=L,$ (91)

where $L$ denotes the scale of the subregion $0\leq r\leq L$. Substituting
(91) into the area functional (89), we get

$\displaystyle A$ $\displaystyle=$
$\displaystyle\Omega\Big{(}\frac{1}{\epsilon}-\frac{1}{z_{h}}\Big{)}$ (92)
$\displaystyle=$
$\displaystyle\Omega\left(\frac{1}{\epsilon}+\frac{\Omega}{\pi-\sqrt{3\Omega^{2}+\pi^{2}}}\right),$

where we have used (43) and $\Omega=\beta/2$ in the above calculations. The
divergent term labels the area law of entanglement entropy. We are interested
in the wedge contributions to entanglement entropy (EE), so we deduct the
plane EE with $z_{h}=1$,

$\displaystyle
A_{0}=\int_{0}^{\Omega}d\theta\Big{(}\frac{1}{\epsilon}-1\Big{)}.$ (93)

Recall that for Casimir energy of parallel plates, we subtract the vacuum
contribution between the parallel plates $\int_{0}^{L}dx...$ rather than the
whole space $\int_{-\infty}^{\infty}dx...$. Thus, above, we deduct
$\int_{0}^{\Omega}d\theta...$ instead of $\int_{0}^{\pi}d\theta...$. In this
way, we get a finite renormalized area

$\displaystyle
A_{\text{reg}}=A-A_{0}=\Omega\Big{(}1-\frac{1}{z_{h}}\Big{)}=\Omega\left(1+\frac{\Omega}{\pi-\sqrt{3\Omega^{2}+\pi^{2}}}\right).$
(94)

The renormalized EE is given by $A_{\text{reg}}/(4G_{N})$. See Fig. 4, which
shows that the renormalized wedge EE increases with the opening angle
$\Omega$. Note that the renormalized wedge EE can be negative generally. Like
the Casimir energy, nothing goes wrong for a renormalized physical quantity to
be negative. We stress that the non-renormalized wedge EE (92) is positive and
increases with $\Omega$ too.

Figure 4:  Wedge contribution to entanglement entropy for $d=3$. It shows the
renormalized wedge EE $A_{\text{reg}}/(4G_{N})$ increases with the opening
angle $\Omega$.

###  4.2 AdSd+1/BCFTd

Let us go on to study the case $d>3$. By the shooting method, we can solve EOM
(90) numerically. We show only vital points below. See [49] for more technical
details, where a similar problem (the island in AdS/dCFT) has been studied.
Here, we have different motivations.

Solving EOM (90) perturbingly around $z=z_{h}$, we get

$\displaystyle F(z)=L_{0}+b_{1}(z-z_{h})+b_{2}(z-z_{h})^{2}+...$ (95)

where

$\displaystyle b_{1}=-\frac{(d-3)z_{h}}{(d-2)z_{h}^{2}-d}L_{0},\
b_{2}=-\frac{(d-3)((d-14)d+29)z_{h}^{2}}{8\left(d-(d-2)z_{h}^{2}\right){}^{2}}L_{0}.$
(96)

We define a new function $F_{0}(z)=L_{0}+b_{1}(z-z_{h})+b_{2}(z-z_{h})^{2}$
with $b_{i}$ given by (96). Then, the initial values become

$\displaystyle F(z_{h}-\epsilon_{1})=F_{0}(z_{h}-\epsilon_{1}),\
F^{\prime}(z_{h}-\epsilon_{1})=F^{\prime}_{0}(z_{h}-\epsilon_{1}),$ (97)

where $\epsilon_{1}$ is a small cutoff for numeral calculations. For any given
$L_{0},z_{h}$, we can numerically solve EOM (90). We adjust the input $L_{0}$
to satisfy the boundary condition on the AdS boundary $z=\epsilon$

$\displaystyle F(\epsilon)=L.$ (98)

It is the so-called shooting method. Without loss of generality, we set $L=1$
below.

Similar to the above subsection, we can obtain the wedge EE by deducting the
plane EE. See Fig. 5, which shows the renormalized wedge EE increases with the
opening angle $\Omega$. We remark that the Casimir energy density of wedges
$-f(\Omega)/r^{d}$ also increases with $\Omega$ since $f^{\prime}(\Omega)<0$.
It is interesting to explore if some relations existed between the wedge
Casimir effect and wedge EE. We leave it to future works.

Figure 5:  Wedge contribution to entanglement entropy for $d=4$ (left) and
$d=5$ (right). It shows the renormalized wedge EE $A_{\text{reg}}/(4G_{N})$
increases with the opening angle $\Omega$.

##  5 Toward holographic polygons

This section generalizes the discussions to polygons. For simplicity, we focus
on AdS3/BCFT2 so that we can employ the power of conformal mappings. In two
dimensions ($d=2$), we can conformally map the upper half-plane to the
interior of a plane polygon. Unfortunately, one cannot do so for polytopes or
“polygons” 888Similar to wedges, by “polygons” in higher dimensions, we mean
the direct product of a 2-dimensional polygon and $(d-2)$-dimensional flat
space, i.e., $P^{2}\times R^{d-2}$.  in higher-dimensional flat space.
Generally, one gets a “polygon” in curved space instead of flat space for
$d>2$. We leave the non-trivial problems in higher dimensions to future works
and focus on $d=2$ in this paper. We hope it could shed some light on the
holographic constructions of polytopes and “polygons” in higher dimensions.

By making the Schwarz–Christoffel mapping,

$\displaystyle
w=h^{-1}(\xi)=\int^{\xi}\frac{c_{0}}{(s-a_{1})^{1-\alpha_{1}/\pi}(s-a_{2})^{1-\alpha_{2}/\pi}...(s-a_{n-1})^{1-\alpha_{n-1}/\pi}}ds,$
(99)

one can map the upper half-plane $\text{Im}(\xi)\geq 0$ to the interior of a
plane n-polygon, where $\alpha_{i}$ are interior angles of the polygon, and
$a_{i}$ are real numbers related to the vertices of the polygon. The above
conformal mappings correspond to the following coordinate transformations in
bulk

$\displaystyle\xi=h(w)-\frac{2z^{2}(h^{\prime})^{2}\bar{h}^{\prime\prime}}{4|h^{\prime}|^{2}+z^{2}|h^{\prime\prime}|^{2}},$
(100)
$\displaystyle\bar{\xi}=\bar{h}(\bar{w})-\frac{2z^{2}(\bar{h}^{\prime})^{2}h^{\prime\prime}}{4|h^{\prime}|^{2}+z^{2}|h^{\prime\prime}|^{2}},$
(101)
$\displaystyle\eta=\frac{4z(h^{\prime}\bar{h}^{\prime})^{3/2}}{4|h^{\prime}|^{2}+z^{2}|h^{\prime\prime}|^{2}}.$
(102)

Under these coordinate transformations, the Poincaré AdS3

$\displaystyle ds^{2}=\frac{d\eta^{2}+d\xi d\bar{\xi}}{\eta^{2}},$ (103)

becomes

$\displaystyle
ds^{2}=\frac{dz^{2}+\Big{(}1+z^{4}T(w)\bar{T}(\bar{w})\Big{)}dwd\bar{w}+z^{2}T(w)dw^{2}+z^{2}\bar{T}(\bar{w})d\bar{w}^{2}}{z^{2}},$
(104)

where $T(w)$ and $\bar{T}(\bar{w})$ are chiral and anti-chiral energy stress
tensors

$\displaystyle
T(w)=\frac{3(h^{\prime\prime})^{2}-2h^{\prime}h^{\prime\prime\prime}}{4h^{\prime
2}},\ \
\bar{T}(\bar{w})=\frac{3(\bar{h}^{\prime\prime})^{2}-2\bar{h}^{\prime}\bar{h}^{\prime\prime\prime}}{4\bar{h}^{\prime
2}}.$ (105)

Note that, according to (51), the holographic stress tensor reads
$T_{ww}=2T(w)$. Similarly, the EOW brane becomes

$\displaystyle\frac{\xi-\bar{\xi}}{2i}=-\sinh(\rho_{*})\eta,$ (106)

where $(\xi,\bar{\xi},\eta)$ are given by (100,101,102). Let us study some
examples below.

Case I: semi-infinite strip

The semi-infinite strip can be considered a special triangle limit with the
interior angles $(\pi/2,\pi/2,0)$. The corresponding Schwarz–Christoffel
mapping is

$\displaystyle
w=h^{-1}(\xi)=\int^{\xi}\frac{1}{(s-1)^{1/2}(s+1)^{1/2}}ds=\cosh^{-1}(\xi),$
(107)

where we have chosen suitable integral constant so that
$h^{-1}(1)=0,h^{-1}(-1)=i\pi,h^{-1}(\infty)=\infty$. From (107), we get
$h(w)=\cosh(w)$, which yields

$\displaystyle T(w)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(1+3\text{csch}^{2}(w)\right)$ (108)
$\displaystyle\approx$ $\displaystyle\begin{cases}\frac{3}{4w^{2}},\ \ \
\text{for }w\to 0,\\\ \frac{1}{4},\ \ \ \ \ \ \text{for
}w\to\infty.\end{cases}$

The above stress tensor has the expected limits near the corner and at
infinity. Transforming into the coordinates $(r,\theta)$ with
$w=r\exp(i\theta)$, we derive near the corner $w\to 0$

$\displaystyle T_{rr}\approx\frac{3}{r^{2}},$ (109)

which agrees with the stress tensor (63) of 2d wedge with $\Omega=\pi/2$.
Similarly, in coordinates $(x,y)$ with $w=x+iy$, we have at infinity
$w\to\infty$

$\displaystyle T_{xx}\approx 1,$ (110)

which agrees with the strip stress tensor $\kappa_{1}/L^{2}$ with $L=\pi$ and
$\kappa_{1}=\pi$ (81). We draw the EOW brane (106) in Fig. 6, which is a
smooth surface in bulk.

Figure 6: AdS3/BCFT2 for semi-infinite strip with $\rho_{*}=-1$. The EOW brane
is a smooth surface in bulk and the singularity appears only at the corner of
semi-infinite strip on the AdS boundary.

Case II: triangle

For the triangle with interior angles $\alpha,\beta$ and $\pi-\alpha-\beta$,
the conformal mapping is

$\displaystyle w$ $\displaystyle=$ $\displaystyle
h^{-1}(\xi)=\int^{\xi}\frac{c_{0}}{(s-1)^{1-\alpha/\pi}(s+1)^{1-\beta/\pi}}ds$
(111) $\displaystyle=$ $\displaystyle c_{0}\frac{\pi
2^{\frac{\beta}{\pi}-1}(\xi-1)^{\alpha/\pi}\,_{2}F_{1}\left(\frac{\alpha}{\pi},1-\frac{\beta}{\pi};\frac{\alpha+\pi}{\pi};\frac{1-\xi}{2}\right)}{\alpha},$

where we have chosen a suitable integral constant and the parameter

$\displaystyle
c_{0}=\frac{\Gamma\left(1-\frac{\beta}{\pi}\right)\Gamma\left(\frac{\alpha+\beta}{\pi}\right)}{\pi
e^{i\alpha}2^{\frac{\alpha+\beta-\pi}{\pi}}\Gamma\left(\frac{\alpha}{\pi}\right)\csc(\beta)},$
(112)

so that
$h^{-1}(1)=0,h^{-1}(-1)=1,h^{-1}(\infty)=e^{-i\alpha}\sin(\beta)\csc(\alpha+\beta)$.
Unfortunately, one cannot get the exact expression of $h(w)$ but only its
inverse function $h^{-1}(\xi)$. By using
$h^{\prime}(w)=1/(h^{-1})^{\prime}(\xi)$ and so on, we rewrite (105) as

$\displaystyle
T(w)=\frac{-3((h^{-1})^{\prime\prime})^{2}+2(h^{-1})^{\prime}(h^{-1})^{\prime\prime\prime}}{4(h^{-1})^{\prime
4}},\ \
\bar{T}(\bar{w})=\frac{-3((\bar{h}^{-1})^{\prime\prime})^{2}+2(\bar{h}^{-1})^{\prime}(\bar{h}^{-1})^{\prime\prime\prime}}{4(\bar{h}^{-1})^{\prime
4}}.$ (113)

Near the corner $\xi\to 1,w\to 0$, we derive from (111,113)

$\displaystyle
w=(\xi-1)^{\alpha/\pi}\left(\frac{2^{-\frac{\alpha}{\pi}}e^{-i\alpha}\Gamma\left(\frac{\alpha+\beta}{\pi}\right)\Gamma\left(1-\frac{\beta}{\pi}\right)\sin(\beta)}{\alpha\Gamma\left(\frac{\alpha}{\pi}\right)}+O\left((\xi-1)^{1}\right)\right),$
(114)

and

$\displaystyle T(w)$ $\displaystyle=$
$\displaystyle(\xi-1)^{-\frac{2\alpha}{\pi}}\left(\frac{2^{\frac{2\alpha}{\pi}-2}e^{2i\alpha}\left(\pi^{2}-\alpha^{2}\right)\csc^{2}(\beta)\Gamma\left(\frac{\alpha}{\pi}\right)^{2}}{\Gamma\left(\frac{\alpha+\beta}{\pi}\right)^{2}\Gamma\left(1-\frac{\beta}{\pi}\right)^{2}}+O\left((\xi-1)^{1}\right)\right)$
(115) $\displaystyle=$
$\displaystyle\frac{\pi^{2}-\alpha^{2}}{4\alpha^{2}w^{2}}+O(\frac{1}{w}),$

where we have used (114) to derive the second line of (115). Changing into the
coordinates $(r,\theta)$, we get near the corner

$\displaystyle
T_{rr}=\frac{\pi^{2}-\alpha^{2}}{\alpha^{2}r^{2}}+O(\frac{1}{r}),$ (116)

which exactly agrees with the holographic stress tensor (63) of 2d wedge. It
is a double-check of our results. From (106) and (111), we can numerically
obtain the embedding function of EOW brane. See Fig. 7 (left). The cases of
quadrangles and polygons are similar to that of triangles. One can obtain
their holographic duals following the above approach. For simplicity, we do
not repeat the calculations. See Fig. 7 (right) for an example of the
holographic square.

Figure 7: AdS3/BCFT2 for triangle (left) and square (right) with
$\rho_{*}=-1$. The interior angles of triangle are
$(\frac{\pi}{2},\frac{\pi}{4},\frac{\pi}{4})$. These figures show that the EOW
brane is a smooth surface in bulk and the singularity appears only at the
corner of the polygon on the AdS boundary.

In summary, in this section, we construct the holographic duals of 2d polygons
by applying the Schwarz-Christoffel mappings and their bulk correspondences.
We verify that the holographic stress tensors approach those of wedges near
the corners of polygons. Besides, the EOW branes dual to polygons are smooth
surfaces in bulk. It is similar to the RT surface of a corner, which is also a
smooth surface in bulk [50, 51]. From NBC (11), we see that the extrinsic
curvatures of the EOW brane are proportional to the induced metrics. Since the
induced metrics are continuous functions, so are the extrinsic curvatures. It
means that the EOW brane is smoothly embedded into the bulk. Thus, it must be
a smooth surface.

To end this section, let us comment on the constructions of holographic
“polygons” in higher dimensions. First, the holographic stress tensor should
approach that of a wedge near the corner of the “polygon”. Second, the EOW
brane is a smooth surface in bulk, and the singularity appears only at the
corner of the “polygon” on the AdS boundary. Third, due to the non-trivial
Casimir effect, the gravity dual of the “polygon” cannot be Poincaré AdS, even
not locally AdS. Recall the example of holographic wedges in section 3, where
the bulk geometry is not an AdS.

##  6 Conclusions and Discussions

This paper investigates the Casimir effect of the wedge and its holographic
dual. We are interested in the universal properties of the wedge Casimir
effect for BCFTs. We prove a universal relation between the displacement
operator and wedge Casimir effect in the smooth limit, i.e., $\Omega\to\pi$.
It should be stressed that this universal relation applies to general states
of BCFTs in general curved wedge space for $r\to 0$ and $\Omega\to\pi$. For
instance, it works for $r\ll 1/T,r\ll 1/\sqrt{R}$ for a thermal state in
curved space, where $r$ is the proper distance to the corner of the wedge, $T$
is the temperature and $R$ denotes the curvature scale. Besides, we find that
the wedge Casimir energy increases with the opening angle, i.e.,
$f^{\prime}(\Omega)\leq 0$, and verify it with several examples. Next, we
construct the holographic dual of wedges in the initial model of AdS/BCFT. We
observe that it cannot be the Poincaré AdS due to the non-trivial Casimir
effect of wedges. Instead, it is given by a cutout of the AdS soliton. We
verify that our proposal can produce the correct smooth and singular limits:
it obeys the universal relation between the displacement operator and Casimir
effect for $\Omega\to\pi$; it reduces to the holographic strip for
$r\to\infty,\Omega\to 0$ with fixed $r\Omega=L$. These are strong supports of
our results. Next, we discuss the wedge contribution to holographic
entanglement entropy. Interestingly, the renormalized wedge entanglement
entropy increases with the opening angle, similar to the wedge Casimir energy.
Finally, we study the holographic polygon in AdS3/BCFT2 and briefly discuss
its generalization to higher dimensions. The important lesson we learned is
that the EOW brane is a smooth surface in bulk due to the continuous extrinsic
curvatures determined by the Neumann boundary condition.

Let us discuss several interesting problems worthy of further exploration.

First, we notice that the wedge Casimir effect is similar to the so-called
corner entanglement [52, 53, 54] in several aspects. Since the wedge becomes a
corner for $d=3$, we take 3d CFT as an example. The entanglement entropy of 3d
CFT takes the following form in the ground state

$\displaystyle S=Bl/\epsilon-a(\Omega)\ln(l/\epsilon)+O(1),$ (117)

where $a(\Omega)$ characterizes the underlying CFT and plays a similar role of
$f(\Omega)$ of this paper. $a(\Omega)$ obeys $a(\Omega)=a(2\pi-\Omega)$
999Entanglement entropy obeys $S(A)=S(\bar{A})$ for a pure state, where
$\bar{A}$ is the complement of subsystem $A$. Thus, we have
$a(\Omega)=a(2\pi-\Omega)$ for a pure state, which is different from
$f(\Omega)$ of wedge Casimir effect., and behaves as

$\displaystyle a(\Omega)\to\begin{cases}k_{1}/\Omega,\ \ \ \ \ \ \ \ \
\text{for }\Omega\to 0,\\\ k_{2}(\pi-\Omega)^{2},\ \ \text{for
}\Omega\to\pi.\end{cases}$ (118)

In particular, $k_{2}$ obeys the universal relation

$\displaystyle k_{2}=\frac{\pi^{2}}{24}C_{T},$ (119)

where $C_{T}$ is the central charge defined by two-point functions of the
stress tensor. Remarkably, $a(\Omega)$ obeys similar limits (118) and
universal relation (119) as (5,9) of wedge Casimir effect. It implies a
mysterious relationship between entanglement entropy and the Casimir effect
may exist, which is worth further study. One possible is that they are related
by the Weyl anomaly. Thus, it is also interesting to derive the Weyl anomaly
in general wedge space.

Second, the corner entanglement obeys the inequalities [50, 55]

$\displaystyle a(\Omega)\geq 0,\ a^{\prime}(\Omega)\leq 0,\
a^{\prime\prime}(\Omega)\leq 0,\ \ \ \text{for}\ 0\leq\Omega\leq\pi,$ (120)

which can be derived from the strong subadditivity of entanglement entropy. We
argue that $f^{\prime}(\Omega)\leq 0$ by applying the virtual displacement
method in this paper. We have checked that $f^{\prime\prime}(\Omega)\geq 0$ is
obeyed by all the examples studied in this paper. Getting a strict proof of
$f^{\prime\prime}(\Omega)\geq 0$ is an interesting problem. Inspired by the
corner entanglement, if strong-subadditivity-like inequality exists for the
Casimir effect, it is also an interesting problem.

Third, there are two natural generalizations of the results of this paper. One
is the holographic dual of “polygon” and polyhedron in higher dimensions, as
mentioned in section 5. The other one is the holographic dual of a cone

$\displaystyle
ds^{2}=dr^{2}+r^{2}d\Omega_{m}^{2}-dt^{2}+\sum_{a=1}^{d-m-2}dy_{a}^{2},$ (121)

where $d\Omega_{m}^{2}$ denotes the line element of unit sphere $S^{m}$ with
$m\geq 2$.

Fourth, this paper focuses on pure Einstein gravity for simplicity. It is
interesting to generalize to the case of matter fields and higher derivative
gravity. Besides, we focus on the holographic dual of the ground state of the
wedge. Studying the gravity duals of the thermal and excited states is
interesting.

Last, this paper focuses on the case with the same boundary conditions on the
two boundaries $\theta=0,\Omega$ of the wedge. What happens if we impose
different boundary conditions? In addition to the geometric singularity, there
is a new singularity from the discontinuous boundary conditions at the corner
of the wedge. It is interesting to explore the holographic dual of this case.

## Acknowledgements

We thank Chong-Sun Chu for valuable discussions on the general problem of
constructing AdS/BCFT with fixed BCFT geometry. We are grateful to Jian-Xin Lu
for the helpful comments and discussions. This work is supported by the
National Natural Science Foundation of China (No. 12275366 and No. 11905297).

##  Appendix A Wedge Casimir effect for free scalars

###  A.1 Heat kernel method

This appendix investigates the Casimir effect of wedges in general dimensions
by applying the heat kernel method. Unfortunately, we only obtain the results
for even dimensions. We leave the discussions for odd $d$ to the next
appendix.

We are interested in the conformally coupled scalar with the action

$\displaystyle
I=\frac{1}{2}\int_{M}d^{d}x\sqrt{|g|}(\nabla_{i}\phi\nabla^{i}\phi+\xi
R\phi^{2})+\int_{\partial M}d^{d-1}y\sqrt{|h|}\xi k\phi^{2},$ (122)

where $\xi=\frac{d-2}{4(d-1)}$ and $k$ is the trace of extrinsic curvatures.
We impose the following conformally invariant boundary conditions

$\begin{split}&\text{Dirichlet BC}:\phi|_{\partial M}=0,\\\ &\text{Robin BC}:\
\ (\nabla_{n}+2\xi k)\phi|_{\partial M}=0.\end{split}$ (123)

The heat kernel satisfies the equation of motion (EOM)

$\displaystyle\partial_{s}K(s,x_{i},x^{\prime}_{i})-(\nabla^{j}\nabla_{j}-\xi
R)K(s,x_{i},x^{\prime}_{i})=0,$ (124)

and the limit

$\displaystyle\lim_{s\to
0}K(s,x_{i},x^{\prime}_{i})=\frac{\delta^{(d)}(x_{i}-x^{\prime}_{i})}{\sqrt{|g|}}.$
(125)

It also obeys the BC (123) on the boundary of the wedge. From the heat kernel,
we can obtain the Green function

$\displaystyle
G(x_{i},x^{\prime}_{i})=\int_{0}^{\infty}dsK(s,x_{i},x^{\prime}_{i}),$ (126)

and then derive the renormalized stress tensor by

$\displaystyle\langle T_{ij}\rangle=\lim_{x^{\prime}_{i}\to
x_{i}}\left[(1-2\xi)\nabla_{i}\nabla_{j^{\prime}}-2\xi\nabla_{i}\nabla_{j}+(2\xi-\frac{1}{2})g_{ij}\nabla_{l}\nabla^{l^{\prime}}+\xi(R_{ij}+\frac{4\xi-1}{2}Rg_{ij})\right]\hat{G}(x_{i},x^{\prime}_{i}),$

where $\hat{G}=G-G_{0}$ with $G_{0}$ the Green function in the free space
without boundaries.

For the flat wedge space (1) $W=R^{d-2}\times C^{2}$, the heat kernel is given
by

$\displaystyle K(s,x_{i},x^{\prime}_{i})=K_{R^{d-2}}K_{C^{2}}$ (128)

where [56]

$\displaystyle K_{R^{d-2}}=\frac{1}{(4\pi
s)^{\frac{d-2}{2}}}\exp(-\frac{\sum_{\hat{i}=1}^{d-2}(x_{\hat{i}}-x^{\prime}_{\hat{i}})^{2}}{4s})$
(129)

and for DBC and (Robin boundary condition) RBC [14]

$\displaystyle
K_{C^{2}}=\int_{0}^{\infty}dp\frac{2p}{\Omega}\exp\left(-sp^{2}\right)J_{\frac{m\pi}{\Omega}}(pr)J_{\frac{m\pi}{\Omega}}\left(pr^{\prime}\right)\begin{cases}\sum_{m=1}^{\infty}\sin\left(\frac{m\pi\theta}{\Omega}\right)\sin\left(\frac{m\pi\theta^{\prime}}{\Omega}\right),\\\
\sum_{m=0}^{\infty}\cos\left(\frac{m\pi\theta}{\Omega}\right)\cos\left(\frac{m\pi\theta^{\prime}}{\Omega}\right).\end{cases}$
(130)

Substituting (128) into (126) and applying the integral formula [14]

$\displaystyle\int_{0}^{\infty}dpp^{-\lambda}J^{2}_{v}\left(pr\right)=\frac{r^{\lambda-1}\Gamma(\lambda)\Gamma(v+\frac{1}{2}-\frac{\lambda}{2})}{2^{\lambda}\Gamma^{2}(\frac{1}{2}+\frac{\lambda}{2})\Gamma(v+\frac{1}{2}+\frac{\lambda}{2})},$
(131)

we obtain

$\displaystyle\lim_{x^{\prime}\to
x}G=\frac{\pi^{1-\frac{d}{2}}\Gamma(3-d)\Gamma\left(\frac{d}{2}+\frac{m\pi}{\Omega}-1\right)}{r^{d-2}\Omega\Gamma\left(2-\frac{d}{2}\right)\Gamma\left(-\frac{d}{2}+\frac{m\pi}{\Omega}+2\right)}\begin{cases}\sum_{m=1}^{\infty}\sin^{2}\left(\frac{m\pi\theta}{\Omega}\right)\\\
\sum_{m=0}^{\infty}\cos^{2}\left(\frac{m\pi\theta}{\Omega}\right)\end{cases},$
(132) $\displaystyle\lim_{x^{\prime}\to
x}\partial^{2}_{\hat{i}}G=-\frac{2\pi^{1-\frac{d}{2}}\Gamma(1-d)\Gamma\left(\frac{d}{2}+\frac{m\pi}{\Omega}\right)}{r^{d}\Omega\Gamma\left(1-\frac{d}{2}\right)\Gamma\left(-\frac{d}{2}+\frac{m\pi}{\Omega}+1\right)}\begin{cases}\sum_{m=1}^{\infty}\sin^{2}\left(\frac{m\pi\theta}{\Omega}\right)\\\
\sum_{m=0}^{\infty}\cos^{2}\left(\frac{m\pi\theta}{\Omega}\right)\end{cases},$
(133) $\displaystyle\lim_{x^{\prime}\to
x}\partial^{2}_{\theta}G=-\frac{\pi^{3-\frac{d}{2}}\Gamma(3-d)\Gamma\left(\frac{d}{2}+\frac{m\pi}{\Omega}-1\right)}{r^{d-2}\Omega^{3}\Gamma\left(2-\frac{d}{2}\right)\Gamma\left(-\frac{d}{2}+\frac{m\pi}{\Omega}+2\right)}\begin{cases}\sum_{m=1}^{\infty}m^{2}\sin^{2}\left(\frac{m\pi\theta}{\Omega}\right)\\\
\sum_{m=0}^{\infty}m^{2}\cos^{2}\left(\frac{m\pi\theta}{\Omega}\right)\end{cases},$
(134) $\displaystyle\lim_{x^{\prime}\to
x}\partial_{\theta}\partial_{\theta^{\prime}}G=\frac{\pi^{3-\frac{d}{2}}\Gamma(3-d)\Gamma\left(\frac{d}{2}+\frac{m\pi}{\Omega}-1\right)}{r^{d-2}\Omega^{3}\Gamma\left(2-\frac{d}{2}\right)\Gamma\left(-\frac{d}{2}+\frac{m\pi}{\Omega}+2\right)}\begin{cases}\sum_{m=0}^{\infty}m^{2}\cos^{2}\left(\frac{m\pi\theta}{\Omega}\right)\\\
\sum_{m=1}^{\infty}m^{2}\sin^{2}\left(\frac{m\pi\theta}{\Omega}\right)\end{cases}.$
(135)

From the above four equations, we obtain the relations

$\displaystyle\lim_{x^{\prime}\to
x}\partial^{2}_{\hat{i}}G=\lim_{x^{\prime}\to
x}\frac{\frac{(d-2)^{2}}{4}G+\partial^{2}_{\theta}G}{r^{2}(1-d)},$ (136)

and

$\displaystyle\lim_{x^{\prime}\to
x}\partial_{\theta}\partial_{\theta^{\prime}}\begin{cases}G_{D}\\\
G_{R}\end{cases}=-\lim_{x^{\prime}\to
x}\partial^{2}_{\theta}\begin{cases}G_{R}\\\ G_{D}\end{cases},$ (137)

where $D$ and $R$ denotes DBC and RBC, respectively. So we can transform the
derivatives of renormalized stress tensor (A.1) into the derivatives of
$\theta$ due to the symmetries of wedges [14]. From (4, A.1) together with
(134,136,137), we get the characteristic function

$\displaystyle f(\Omega)$ $\displaystyle=$ $\displaystyle\lim_{x^{\prime}\to
x}\frac{r^{d-2}}{2(d-1)}\partial^{2}_{\theta}(G_{D}+G_{R}-2G_{0})$ (138)
$\displaystyle=$
$\displaystyle\sum_{m=1}^{\infty}\frac{2^{-d}\pi^{\frac{5}{2}-\frac{d}{2}}m^{2}\Gamma\left(\frac{1}{2}-\frac{d}{2}\right)\Gamma\left(\frac{d}{2}+\frac{m\pi}{\Omega}-1\right)}{\Omega^{3}\Gamma\left(-\frac{d}{2}+\frac{m\pi}{\Omega}+2\right)}-(\Omega\to\pi).$

The above sum can be calculated directly for even $d$. For instance, we have
for $d=2$

$\displaystyle f_{2d}(\Omega)=\sum_{m=1}^{\infty}-\frac{\pi
m}{2\Omega^{2}}-(\Omega\to\pi)=-\frac{\pi\zeta(-1)}{2\Omega^{2}}-(\Omega\to\pi)=\frac{\frac{\pi^{2}}{\Omega^{2}}-1}{24\pi},$
(139)

which takes the expected form. For $d=4$, we get

$\displaystyle
f_{4d}(\Omega)=\sum_{m=1}^{\infty}\frac{\pi^{2}m^{3}}{12\Omega^{4}}-(\Omega\to\pi)=\frac{\pi^{2}\zeta(-3)}{12\Omega^{4}}-(\Omega\to\pi)=\frac{\frac{\pi^{4}}{\Omega^{4}}-1}{1440\pi^{2}},$
(140)

which agrees with the result of [14]. Similarly, we obtain

$\displaystyle
f_{6d}(\Omega)=\frac{\pi\zeta(-3)}{120\Omega^{4}}-\frac{\pi^{3}\zeta(-5)}{120\Omega^{6}}-(\Omega\to\pi)=\frac{-31\Omega^{6}+21\pi^{4}\Omega^{2}+10\pi^{6}}{302400\pi^{3}\Omega^{6}},$
(141)

and

$\displaystyle
f_{8d}(\Omega)=\frac{\pi^{4}\zeta(-7)}{1680\alpha^{8}}-\frac{\pi^{2}\zeta(-5)}{336\alpha^{6}}+\frac{\zeta(-3)}{420\alpha^{4}}-(\Omega\to\pi)=\frac{-289\Omega^{8}+168\pi^{4}\Omega^{4}+100\pi^{6}\Omega^{2}+21\pi^{8}}{8467200\pi^{4}\Omega^{8}}.$
(142)

Unfortunately, we do not derive results for odd $d$. We leave this problem to
the next appendix.

###  A.2 Mirror method

In this appendix, we derive the wedge Casimir effect for the open angle
$\Omega=\pi/m$ with an integer $m$, and then analytically extend the results
to arbitrary open angle $\Omega$. By apply the mirror method, we obtain the
Green functions for DBC and RBC

$\displaystyle
G_{D}=\frac{\Gamma(\frac{d}{2}-1)}{4\pi^{\frac{d}{2}}}\sum_{i=1}^{2m}\frac{(-1)^{i+1}}{(r^{2}+r^{\prime
2}-2rr^{\prime}\cos(\theta-\theta_{i}))^{\frac{d-2}{2}}},$ (143)
$\displaystyle
G_{R}=\frac{\Gamma(\frac{d}{2}-1)}{4\pi^{\frac{d}{2}}}\sum_{i=1}^{2m}\frac{1}{(r^{2}+r^{\prime
2}-2rr^{\prime}\cos(\theta-\theta_{i}))^{\frac{d-2}{2}}},$ (144)

where $\theta_{i}=(-1)^{i+1}\theta^{\prime}+\frac{2\pi}{m}[\frac{i}{2}]$, and
$[\ ]$ denotes the integer part. The background Green function in free space
is given by

$\displaystyle
G_{0}=\frac{\Gamma(\frac{d}{2}-1)}{4\pi^{\frac{d}{2}}}\frac{1}{(r^{2}+r^{\prime
2}-2rr^{\prime}\cos(\theta-\theta^{\prime}))^{\frac{d-2}{2}}}.$ (145)

From (138), we have

$\displaystyle f(\Omega)$ $\displaystyle=$ $\displaystyle\lim_{r^{\prime}\to
r,\theta^{\prime}\to\theta}\frac{r^{d-2}}{2(d-1)}\partial_{\theta}^{2}(G_{D}+G_{R}-2G_{0})$
(146) $\displaystyle=$
$\displaystyle\frac{2^{-d-1}\pi^{-\frac{d}{2}}\Gamma\left(\frac{d}{2}\right)}{d-1}\sum_{j=1}^{m-1}\csc^{d}\left(\frac{\pi
j}{m}\right)\left(d-1-(d-2)\sin^{2}\left(\frac{\pi j}{m}\right)\right).$

For even $d$, the above finite sum can be calculated by Mathematica. For
instance, we have

$\displaystyle f(\frac{\pi}{m})=\begin{cases}\frac{m^{2}-1}{24\pi},\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for d=2},\\\
\frac{m^{4}-1}{1440\pi^{2}},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\text{for d=4},\\\ \frac{10m^{6}+21m^{4}-31}{302400\pi^{3}},\ \ \ \ \ \ \ \ \
\ \ \ \text{for d=6},\\\
\frac{21m^{8}+100m^{6}+168m^{4}-289}{8467200\pi^{4}},\ \text{for
d=8}.\end{cases}$ (147)

Performing the analytical continuation $\frac{\pi}{m}\to\Omega$, we get

$\displaystyle
f(\Omega)=\begin{cases}\frac{\frac{\pi^{2}}{\Omega^{2}}-1}{24\pi},\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for d=2},\\\
\frac{\frac{\pi^{4}}{\Omega^{4}}-1}{1440\pi^{2}},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \text{for d=4},\\\
\frac{\frac{10\pi^{6}}{\Omega^{6}}+\frac{21\pi^{4}}{\Omega^{4}}-31}{302400\pi^{3}},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \text{for d=6},\\\
\frac{\frac{21\pi^{8}}{\Omega^{8}}+\frac{100\pi^{6}}{\Omega^{6}}+\frac{168\pi^{4}}{\Omega^{4}}-289}{8467200\pi^{4}},\
\ \ \ \text{for d=8},\end{cases}$ (148)

which agree with the results of appendix A.1

The cases of odd $d$ are more subtle. Take $d=3$ as an example, we need to
calculate the sum

$\displaystyle
f_{3d}(\frac{\pi}{m})=\sum_{j=1}^{m-1}\left(\frac{\csc^{3}\left(\frac{\pi
j}{m}\right)}{32\pi}-\frac{\csc\left(\frac{\pi j}{m}\right)}{64\pi}\right).$
(149)

Unfortunately, the sum formula for $\sum_{j=1}^{m-1}\csc^{2n+1}\left(\frac{\pi
j}{m}\right)$ ($n,m$ integers) is unknown. The main idea is to replace the sum
with an integral so that it is easier to perform the analytical continuation.
We have

$\displaystyle\csc(\pi s)=\int_{0}^{\infty}\frac{x^{s-1}}{\pi(x+1)}\,dx,$
(150)

which yields

$\displaystyle\sum_{j=1}^{m-1}\csc(\frac{\pi
j}{m})=\int_{0}^{\infty}\frac{x-x^{1/m}}{\pi
x(x+1)\left(x^{1/m}-1\right)}\,dx.$ (151)

Taking two derivatives of (150) with respect to $s$, we get

$\displaystyle\partial_{s}^{2}\csc(\pi s)=2\pi^{2}\csc^{3}(\pi
s)-\pi^{2}\csc(\pi
s)=\int_{0}^{\infty}\frac{x^{s-1}\log^{2}(x)}{\pi(x+1)}\,dx.$ (152)

From (150,152), we solve

$\displaystyle\csc^{3}(\pi
s)=\int_{0}^{\infty}\frac{x^{s-1}\left(\log^{2}(x)+\pi^{2}\right)}{2\pi^{3}(x+1)}\,dx,$
(153)

which gives

$\displaystyle\sum_{j=1}^{m-1}\csc^{3}(\frac{\pi
j}{m})=\int_{0}^{\infty}\frac{\left(x-x^{1/m}\right)\left(\log^{2}(x)+\pi^{2}\right)}{2\pi^{3}x(x+1)\left(x^{1/m}-1\right)}\,dx.$
(154)

Substituting (151,154) into (149) and making analytical the continuation
$m\to\pi/\Omega$, we obtain

$\displaystyle f_{3d}(\Omega)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{\left(x-x^{\Omega/\pi}\right)\log^{2}(x)}{64\pi^{4}x(x+1)\left(x^{\Omega/\pi}-1\right)}\,dx$
(155) $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{\left(x-x^{\Omega/\pi}\right)\log^{2}(x)}{32\pi^{4}x(x+1)\left(x^{\Omega/\pi}-1\right)}\,dx.$

where we have used $\int_{0}^{1}...dx=\int_{1}^{\infty}...dx$ for the first
line of integral. The above equation has the correct smooth limit
$f_{3d}(\pi)=0$, which is test of the result. At the next order, we have

$\displaystyle f_{3d}(\Omega)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{(\pi-\Omega)\log^{3}(x)}{32\left(\pi^{5}\left(x^{2}-1\right)\right)}\,dx+O(\Omega-\pi)^{2}$
(156) $\displaystyle=$
$\displaystyle\frac{\pi-\Omega}{512\pi}+O(\Omega-\pi)^{2},$

which yields $\kappa_{2}=1/(512\pi)$. Recall that we have $C_{D}=1/(8\pi^{2})$
from (35). One can check that $\kappa_{2}$ and $C_{D}$ obeys the universal
relation (9), which is another support of our results. Following the above
approach, we can derive $f(\Omega)$ for higher odd $d$ one by one. We list the
results below.

$\displaystyle f_{5d}(\Omega)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{\left(x-x^{\Omega/\pi}\right)\log^{2}(x)\left(\log^{2}(x)+\pi^{2}\right)}{1024\pi^{7}x(x+1)\left(x^{\Omega/\pi}-1\right)}\,dx$
(157) $\displaystyle=$
$\displaystyle\frac{3(\pi-\Omega)}{16384\pi^{2}}+O(\Omega-\pi)^{2}.$
$\displaystyle f_{7d}(\Omega)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{\left(x-x^{\Omega/\pi}\right)\log^{2}(x)\left(\log^{2}(x)+\pi^{2}\right)\left(\log^{2}(x)+9\pi^{2}\right)}{49152\pi^{10}x(x+1)\left(x^{\Omega/\pi}-1\right)}\,dx$
(158) $\displaystyle=$
$\displaystyle\frac{25(\pi-\Omega)}{524288\pi^{3}}+O(\Omega-\pi)^{2}.$

We verify that the above $\kappa_{2}$ obeys the universal relation (9) for
$d=5,7$ too. See discussions above Table. 1.

##  Appendix B Poincaré AdS

In this section, we show that Poincaré AdS space cannot be the dual of a
wedge. For simplicity, we focus on Poincaré AdS3

$\displaystyle ds^{2}=\frac{dz^{2}+dr^{2}+r^{2}d\theta^{2}}{z^{2}},\ \
0\leq\theta\leq\Omega.$ (159)

From scaling symmetry of AdS and the fact that there is no other scale in the
problem, we take the following ansatz of the EOW brane

$\displaystyle z=rF(\theta),$ (160)

where $F(\theta)$ obeys the boundary conditions

$\displaystyle F(0)=F^{\prime}(\frac{\Omega}{2})=0.$ (161)

We remark that (160) is inspired from the RT surface of wedge [50, 51].
Similar to section 3, we focus on region I with negative tension below. By
considering its complement, we automatically get region II with positive
tension. See Fig. 1. Substituting (160) into NBC (11), we get one independent
equation

$\displaystyle\frac{1}{\sqrt{F^{\prime}(\theta)^{2}+F(\theta)^{2}+1}}+\tanh\left(\rho_{*}\right)=0,$
(162)

which can be solved as

$\displaystyle
F(\theta)=\pm\text{csch}\left(\rho_{*}\right)\sin(\theta+c_{0}).$ (163)

Imposing the boundary conditions (161), we obtain a constraint for the opening
angle of wedge

$\displaystyle\Omega=\pi,$ (164)

which means the wedge becomes a plane. In other words, Poincaré AdS3 is not
dual to a wedge but a half space. The above discussions can be generalized to
higher dimensions straightforwardly. To end this section, we mention that the
metric (46) of section 3 is dual to a wedge. The embedding function of EOW
brane can be read off from (44,62).

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