Communicating Multiverses in Holographic
de-Sitter Braneworld

Gopal Yadav111email: gopalyadav<EMAIL_ADDRESS>
Chennai Mathematical Institute,
H1 SIPCOT IT Park, Siruseri 603103, India

In this paper, we construct the wedge holography for the de-Sitter space a
bulk theory. First, we discuss a more general mathematical construction of
wedge holography in parallel with wedge holography construction for the AdS
bulk, and then we construct the wedge holography in the extended static patch.
In the first case, we prove that one can construct wedge holography for the
de-Sitter bulk for the static patch as well as global coordinate metric on
end-of-the-world branes. Extended static patch wedge holography is constructed
by joining the two copies of double holography in DS/dS correspondence. We
find that wedge holography with extended static patch metric leads to the
emergence of communicating universes. Further, we propose a model which
provides the theoretical evidence of communicating multiverses.

One day there may well be proof of multiple universes…and in that universe,
Zayn is still in one direction. Stephen Hawking

###### Contents

  1. 1 Introduction
  2. 2 Wedge holography for the de-Sitter bulk: general description
    1. 2.1 The setup
    2. 2.2 Wedge holography in various dimensions
      1. 2.2.1 $d=3$
      2. 2.2.2 $d=4$
      3. 2.2.3 $d=5$
      4. 2.2.4 $d=6$
    3. 2.3 Pictorial representation of wedge holography for the de-Sitter bulk
  3. 3 Wedge holography in extended static patch
  4. 4 Communicating universes in extended static patch wedge holography?
  5. 5 Communicating Multiverses in de-Sitter wedge holography
    1. 5.1 Multiverse
    2. 5.2 Communicating Multiverses
  6. 6 Discussion

##  1 Introduction

It has always been fascinating to question whether our universe is the only
universe, whether there are other universes in nature, or whether many
universes exist in a multiverse. Is our universe part of a multiverse? Some
questions related to Multiverse have been explored in [5, 2, 1, 4, 3]. There
is also progress done in this direction within braneworld by the use of wedge
holography [6, 7] where the bulk theory is anti de-Sitter spacetime which has
negative cosmological constant. In this work, we explore this type of question
from the perspective of wedge holography where the bulk is de-Sitter space
with positive cosmological constant.

Holography has played an important role in understanding various phenomena in
many branches of physics. Originally, it was started as a duality between
${\cal N}$=4 supersymmetric Yang-Mills theory and type IIB string theory on
$AdS_{5}\times S^{5}$ background [8]. We are able to decode the information
about boundary theory from the bulk theory using holography and vice-versa.
There are many generalizations of holography, e.g., dS/CFT correspondence [9,
10] which is a duality between de-Sitter space in bulk and Euclidean CFT
living at the future/past boundary of de-Sitter space. This is a co-dimension
one holography. In this paper, we discuss co-dimension two holography in the
context of de-Sitter space. There is also so-called DS/dS correspondence [11]
which describes duality between $d$-dimensional de-Sitter space and two CFTs
which are coupled to each other and ($d-1$)-dimensional de-Sitter gravity.
Further, see [12, 13, 14] for the application of holography to thermal QCD
from a top-down approach, [15, 16] for condensed matter physics, [17, 18, 19]
where holography has been used in cosmology, [20, 21, 22] for the discussion
on multi-boundary wormholes. There is also a new version of holography in the
context of AdS/CFT where time is emergent dimension known as “Cauchy slice
holography” [23]222There are many applications of holography, and 25 years of
the same was celebrated at ICTP-SAIFR (https://www.ictp-
saifr.org/holography25/)..

As an application of holography, we are able to resolve the information
paradox [24]. For this purpose, there are three proposals: island proposal
[25, 26], double holography [27, 28, 29], and wedge holography [30, 31]. In
double holography, we couple the black hole living on an end-of-the-world
brane with an external CFT bath living at the conformal boundary of the AdS
bulk. This external bath acts as a sink to collect the Hawking radiation.
Using double holography, the Page curve [32] has been obtained by computing
the areas of Hartman-Maldacena [33] and island surfaces. Further, efforts have
been made to collect the Hawking radiation in a gravitating bath, and the
setup is obtained by introducing two Karch-Randall branes in which one brane
acts as a black hole, whereas another brane acts as a bath. The setup with two
gravitating branes is known as wedge holography [30, 31].

Wedge holography is a co-dimension two holography, which can be understood as
follows. First, we localize the $d$-dimensional bulk Einstein gravity on
$(d-1)$-dimensional Karch-Randall branes, which is called as braneworld
holography [34, 35] and then the gravity living on Karch-Randall branes is
dual to the CFT living at the $(d-2)$-dimensional defect because of AdS/CFT
correspondence [8]. Hence, $d$-dimensional Einstein gravity living in the
wedge formed by Karch-Randall branes is dual to $(d-2)$-dimensional defect
CFT. For the AdS bulk, wedge holography has been explored in detail in [36,
40, 43, 42, 47, 46, 45, 41, 44, 37, 38, 39] and for flat spacetime bulk, see
[48]. An interesting application of wedge holography is that it describes
multiverse [6, 7] but in the AdS bulk spacetime. The multiverse is obtained by
$2n$ Karch-Randall branes, which are joined at the common defect. Since there
is Einstein gravity on each branch, therefore, we obtain a Multiverse made up
of $2n$ universes [6].

Wedge holography has been successfully constructed in the AdS and flat
spacetime bulk. The motivation of the paper is to construct the wedge
holography for the de-Sitter bulk. We will see that it is possible to
construct wedge holography even for the de-Sitter bulk. This setup provides
interesting results. It gives a hint for dSd/CFTd-2 correspondence, the
existence of communicating universes, multiverse, and communicating
multiverses within braneworld.

The rest of the paper is organized as follows. We start with the construction
of wedge holography for the de-Sitter bulk in section 2 via three subsections
2.1, 2.2 and 2.3. In subsection 2.1, we describe the mathematical description
of de-Sitter wedge holography; in subsection 2.2, we prove that we have
consistent wedge holographic realization for the de-Sitter bulk, and in
subsection 2.3, we describe de-Sitter wedge holography pictorially. We
construct the wedge holography in an extended static patch in section 3 and
use this model to discuss whether parallel universes exist or not in section
4. In section 5, we discuss the existence of multiverse (subsection 5.1) and
communicating multiverses (subsection 5.2) and then discuss our results in
section 6.

##  2 Wedge holography for the de-Sitter bulk: general description

In this section, we construct wedge holography for the general de-Sitter bulk
and prove that the metric on the end-of-the-world branes333At many place we
will use “EOW branes” instead of “end-of-the-world branes”. can be static
patch and global coordinate. We consider dimensionality of de-Sitter bulk as
$d$-dimensions whereas dimensionality of end-of-the-world branes [located at
$\omega=\omega_{1}$ and $\omega=\omega_{2}$] as $(d-1)$-dimensions. The
discussion in this section will be general, similar to wedge holography
construction for the AdS bulk.

###  2.1 The setup

The bulk gravitational action with positive cosmological constant in
$d$-dimensions including the gravitational action on the branes is given as
[49]

$\displaystyle S=\frac{1}{16\pi G_{N}^{(d)}}\int d^{d}x\sqrt{-g}{\left({\cal
R}-2\Lambda\right)}-\frac{1}{8\pi G_{N}^{(d)}}\int
d^{d-1}x\sqrt{-\hat{g}_{\beta}}\left({\cal K}_{\beta}-T_{\beta}\right),$ (1)

where $\beta=1,2$ and $\Lambda=\frac{(d-1)(d-2)}{2}$. $G_{N}^{(d)}$, ${\cal
R}$ and $\Lambda$ are the Newton constant, Ricci scalar and cosmological
constant in $d$-dimensions; ${\cal K}_{\beta}$ and $T_{\beta}$ are the trace
of extrinsic curvature and tensions of end-of-the-world branes. Einstein
equation associated with the bulk metric is given by

$\displaystyle R_{\mu\nu}-\frac{R}{2}g_{\mu\nu}{\bf+}\Lambda g_{\mu\nu}=0.$
(2)

The above equation has the following solution [11]

$\displaystyle ds^{2}=d\omega^{2}+e^{2A(\omega)}\hat{g}_{ij}dy^{i}dy^{j},$ (3)

where $\hat{g}_{ij}$ corresponds to the metric of dSd-1 space. Depending upon
the warp factor, the bulk metric (3) can be either de-Sitter or anti de-
Sitter444Bulk action:  $\displaystyle S=-\frac{1}{16\pi G_{N}^{(d)}}\int
d^{d}x\sqrt{-g}{\left({\cal R}-2\Lambda\right)}-\frac{1}{8\pi G_{N}^{(d)}}\int
d^{d-1}x\sqrt{-\hat{g}_{\beta}}\left({\cal K}_{\beta}-T_{\beta}\right),$ (4)
with $\Lambda=-\frac{(d-1)(d-2)}{2}$. as follows [11]

$\displaystyle{\bf dS_{d}}:\ \ \ \ \ \ \ \ \
e^{2A(\omega)}=\sin^{2}\left(\omega/R\right),$ $\displaystyle{\bf AdS_{d}}:\ \
\ \ \ \ \ e^{2A(\omega)}=\sinh^{2}\left(\omega/R\right),$ (5)

where $R$ is the length scale associated with corresponding background. Since
we are interested in constructing wedge holography for the de-Sitter bulk, we
will consider the de-Sitter case here. Let us write the explicit metric for
the de-Sitter bulk as

$\displaystyle ds_{\rm general\
bulk}^{2}=d\omega^{2}+\sin^{2}\left(\omega/R\right)\hat{g}_{ij}^{\beta}dy^{i}dy^{j},$
$\displaystyle ds_{\rm static\
patch}^{2}=d\omega^{2}+\sin^{2}\left(\omega/R\right)\left[-(1-r^{2})dt^{2}+\frac{dr^{2}}{1-r^{2}}+r^{2}d\Omega_{d-3}^{2}\right],$
$\displaystyle ds_{\rm global\ de-
Sitter}^{2}=d\omega^{2}+\sin^{2}\left(\omega/R\right)\left[-d\tau^{2}+\cosh^{2}\left(\tau\right)d\Omega_{d-2}^{2}\right],$
(6)

where $t\in(-\infty,\infty)$, $r\in(0,1)$555We set $l=1$ and the horizon is
located at $r=1$., $\tau\in(-\infty,\infty)$, and
$d\Omega_{d-2}^{2}=d\theta_{1}^{2}+\sin^{2}\theta_{1}d\theta_{2}^{2}+......+\sin^{2}\theta_{1}\sin^{2}\theta_{2}......\sin^{2}\theta_{d-3}d\theta_{d-2}^{2}$.

The Neumann boundary condition (NBC) satisfied by the bulk metric (3) at the
location of end-of-the-world branes is given as

$\displaystyle{\cal K}_{ij}^{\beta}-({\cal
K}^{\beta}-T^{\beta})\hat{g}_{ij}^{\beta}=0,$ (7)

where ${\cal
K}_{ij}^{\beta}=\frac{1}{2}\partial_{\omega}{\hat{g}_{ij}^{\beta}}|_{\omega=\omega_{1,2}}$
and ${\cal K}^{\beta}=\hat{g}^{ij,\beta}{\cal K}_{ij,\beta}$.

It has been discussed in [11] that the de-Sitter bulk (2.1) is dual to two
CFTs on $dS_{d-1}$ which are coupled to each other and $(d-1)$-dimensional de-
Sitter gravity at $\omega=\frac{\pi R}{2}$. The warp factor is maximum at
$\omega=\frac{\pi R}{2}$ and it is zero at the horizon, $\omega=0,\pi R$. For
the discussion in this section, we will consider localization of
$d$-dimensional de-Sitter gravity on $(d-1)$-dimensional EOW brane [50].

Having discussed the general structure of $d$-dimensional de-Sitter space, let
us see how we can construct wedge holography in de-Sitter space. Let us
consider two EOW branes located at $\omega=\omega_{1}$ and $\omega=\omega_{2}$
which are basically constant $\omega$ slicing of the bulk metric (2.1). It is
to be noted that the story here is different from the AdS bulk because wedge
holography construction of de-Sitter bulk is a little bit difficult. We will
discuss this pictorially in subsection 2.3 using the idea of [51]. For now, we
are considering EOW branes at some arbitrary locations $\omega_{1,2}$. The
wedge holography is obtained by joining two $(d-1)$-dimensional EOW branes
(say $Y_{1}$ and $Y_{2}$) with de-Sitter metric at the $(d-2)$-dimensional
defect in $d$-dimensional de-Sitter bulk (see Fig. 1).

Similar to AdS and flat bulk spacetime, we expect that de-Sitter wedge
holography has the following three descriptions:

  * •

Boundary description: $(d-2)$-dimensional defect CFT living at the corner of
the wedge formed by two EOW branes.

  * •

Intermediate description: the gravitating branes ($Y_{1}$ and $Y_{2}$) with
metric $dS_{d-1}$ are interacting with each other via the transparent boundary
conditions at the defect.

  * •

Bulk description: CFT living on the $(d-2)$-dimensional defect is dual to
$d$-dimensional de-Sitter gravity living in the wedge formed by gravitating
EOW branes with metric dSd-1.

When we take de-Sitter slicing of (3) then we expect that defect CFT should be
non-unitary because of dS/CFT correspondence [9, 10]. Let us make statement
about the wedge holography [30, 31] in the context of de-Sitter bulk as
follows:

Classical gravity in $d$-dimensional de-Sitter space
$\equiv$ (Quantum) gravity on two $(d-1)$-dimensional EOW branes with metric
dSd-1
$\equiv$ CFT living at the $(d-2)$-dimensional defect.

The first and the second lines are related via braneworld holography in de-
Sitter space [50] whereas the second and third lines exist because of dS/CFT
correspondence [9, 10]. Hence, classical gravity in $d$-dimensional de-Sitter
space is dual to $(d-2)$-dimensional defect CFT living at the corner of the
wedge, which is the signature of dSd/CFTd-2 correspondence.

To get a consistent wedge holographic realization of the de-Sitter space, we
need to check the following conditions:

  1. 1.

The bulk metric (2.1) should satisfy Einstein’s equation with a positive
cosmological constant (2).

  2. 2.

The bulk metric (2.1) must satisfy Neumann boundary conditions (7) on the
branes, i.e. at $\omega=\omega_{1,2}$ where $\omega_{1,2}$ are some constant
values of $\omega$.

  3. 3.

The induced metric $\hat{g}_{ij}^{\beta}$ on the branes should be the solution
of Einstein’s equation with a positive cosmological constant in
$(d-1)$-dimensions:

$\displaystyle
R_{ij}^{\beta}-\frac{1}{2}\hat{g}_{ij}^{\beta}R[\hat{g}_{ij}^{\beta}]+\frac{(d-2)(d-3)}{2}\hat{g}_{ij}^{\beta}=0.$
(8)

One can obtain the gravitational action on the branes from the substitution of
(2.1) into the bulk action (1), using the Neumann boundary condition (7) and
integration over $\omega$ and the general form is given as

$\displaystyle S_{\rm brane}=\lambda_{\beta}^{\rm dS}\int
d^{d-1}x\left(R[\hat{g}_{ij}^{\beta}]-2\Lambda_{\rm brane}^{\rm dS}\right),$
(9)

where $\Lambda_{\rm brane}^{\rm dS}=\frac{(d-2)(d-3)}{2}$,
$\lambda_{\beta}^{\rm dS}=\frac{1}{16\pi G_{N}^{(d-1),\beta}}$. Here, we show
that condition 2 is satisfied, provided EOW branes have some tensions. Let us
see this in more detail. For the bulk metric (2.1), extrinsic curvature and
the trace of the same on EOW branes are obtained as

$\displaystyle{\cal
K}_{ij}^{\beta}=\frac{1}{2}\partial_{\omega}{\hat{g}_{ij}}|_{\omega=\omega_{1,2}}=\frac{1}{R}\cot\left(\frac{\omega_{1,2}}{R}\right)\hat{g}_{ij}|_{\omega=\omega_{1,2}},$
$\displaystyle{\cal K}^{\beta}=\hat{g}^{ij,\beta}{\cal
K}_{ij}^{\beta}=\frac{d-1}{R}\cot\left(\frac{\omega_{1,2}}{R}\right).$ (10)

Hence, using the Neumann boundary condition (7) and the information given in
(2.1), we see that NBC for the bulk metric (2.1) is satisfied when tensions of
the branes are given by

$\displaystyle
T^{\beta}=\frac{d-2}{R}\cot\left(\frac{\omega_{1,2}}{R}\right).$ (11)

Hence at $\omega_{1,2}=\frac{\pi R}{2}$, we get tensionless branes as
discussed in [49].

###  2.2 Wedge holography in various dimensions

In this subsection, we construct wedge holography for various dimensional de-
Sitter bulk. For simplicity, we work with $R=1$ unit.

####  2.2.1 $d=3$

For three-dimensional de-Sitter space, the bulk metric is given as

$\displaystyle
ds^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\hat{g}_{ij}^{\beta}dy^{i}dy^{j},$
$\displaystyle ds_{\rm static\
patch}^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\left[-(1-r^{2})dt^{2}+\frac{dr^{2}}{1-r^{2}}\right],$
$\displaystyle ds_{\rm global\ de-
Sitter}^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\left[-d\tau^{2}+\cosh^{2}\left(\tau\right)d\Omega_{1}^{2}\right],$
(12)

where $d\Omega_{1}^{2}=d\theta_{1}^{2}$. For this case, the Ricci scalar is
$R=6$, and the Ricci tensors for the static patch and global de-Sitter are
summarized below.

  * •

Ricci tensors for static patch:

$\displaystyle R_{11}=2\left(r^{2}-1\right)\sin^{2}(w);\ \
R_{22}=-\frac{2\sin^{2}(\omega)}{r^{2}-1};\ \ R_{33}=2.$ (13)

  * •

Ricci tensors for global coordinate:

$\displaystyle R_{11}=-2\sin^{2}(\omega);\ \
R_{22}=2\cosh^{2}(\tau)\sin^{2}(\omega);\ \ R_{33}=2.$ (14)

The cosmological constant for three-dimensional de-Sitter space is
$\Lambda=1$, and hence bulk EOM is given as

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+g_{\mu\nu}=0,$ (15)

where $\mu,\nu=1,2,3$. One can check that bulk EOM666EOM is the short form of
equation of motion. (15) is satisfied for the Ricci tensors given in equations
(13) and (14) and the Ricci scalar obtained above. Hence, condition 1 is
satisfied for bulk dS3. Therefore, we get dS2 slicing of the bulk metric
(2.2.1) in the static patch and global coordinate both. Further, the tensions
of the end-of-the-world branes and trace of the extrinsic curvature for dS3
bulk are given as

$\displaystyle T^{\beta}=\cot\left(\omega_{1,2}\right);\ \ \ {\cal
K}^{\beta}=2\cot\left(\omega_{1,2}\right).$ (16)

####  2.2.2 $d=4$

The metric for the four-dimensional de-Sitter space is given by

$\displaystyle
ds^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\hat{g}_{ij}^{\beta}dy^{i}dy^{j},$
$\displaystyle ds_{\rm static\
patch}^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\left[-(1-r^{2})dt^{2}+\frac{dr^{2}}{1-r^{2}}+r^{2}d\Omega_{1}^{2}\right],$
$\displaystyle ds_{\rm global\ de-
Sitter}^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\left[-d\tau^{2}+\cosh^{2}\left(\tau\right)d\Omega_{2}^{2}\right],$
(17)

where $d\Omega_{2}^{2}=d\theta_{1}^{2}+\sin^{2}\theta_{1}d\theta_{2}^{2}$ and
the Ricci tensors for the static patch and global de-Sitter are given as
below.

  * •

Static patch:

$\displaystyle\hskip-14.45377ptR_{11}=3\left(r^{2}-1\right)\sin^{2}(\omega);\
\ R_{22}=-\frac{3\sin^{2}(\omega)}{r^{2}-1};\ \
R_{33}=3r^{2}\sin^{2}(\omega);\ \ R_{44}=3.$ (18)

  * •

Global coordinate:

$\displaystyle\hskip-36.135ptR_{11}=-3\sin^{2}(\omega);\ \
R_{22}=3\cosh^{2}(\tau)\sin^{2}(\omega);\ \
R_{33}=3\sin^{2}\left(\theta_{1}\right)\cosh^{2}(\tau)\sin^{2}(\omega);\ \
R_{44}=3.$

The Ricci scalar for dS4 metric (2.2.2) is $R=12$ and the cosmological
constant is, $\Lambda=3$ and hence the EOM for the bulk metric associated with
dS4 space is

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+3g_{\mu\nu}=0,$ (20)

where $\mu,\nu=1,2,3,4$. We see that (20) is satisfied for the Ricci tensors
given in (18) and (• ‣ 2.2.2) and the Ricci scalar as mentioned above which
proves condition 1 of wedge holography for the dS4 bulk. For the dS4 bulk,
tensions of the end-of-the-world branes and trace of the extrinsic curvature
are

$\displaystyle T^{\beta}=2\cot\left(\omega_{1,2}\right);\ \ \ {\cal
K}^{\beta}=3\cot\left(\omega_{1,2}\right).$ (21)

####  2.2.3 $d=5$

The five-dimensional bulk de-Sitter metric is written as

$\displaystyle
ds^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\hat{g}_{ij}^{\beta}dy^{i}dy^{j},$
$\displaystyle ds_{\rm static\
patch}^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\left[-(1-r^{2})dt^{2}+\frac{dr^{2}}{1-r^{2}}+r^{2}d\Omega_{2}^{2}\right],$
$\displaystyle ds_{\rm global\ de-
Sitter}^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\left[-d\tau^{2}+\cosh^{2}\left(\tau\right)d\Omega_{3}^{2}\right],$
(22)

where
$d\Omega_{3}^{2}=d\theta_{1}^{2}+\sin^{2}\theta_{1}d\theta_{2}^{2}+\sin^{2}\theta_{1}\sin^{2}\theta_{2}d\theta_{3}^{2}$.
The Ricci tensors are given below.

  * •

Static patch:

$\displaystyle R_{11}=4\left(r^{2}-1\right)\sin^{2}(\omega);\ \
R_{22}=-\frac{4\sin^{2}(\omega)}{r^{2}-1};\ \ R_{33}=4r^{2}\sin^{2}(\omega);$
$\displaystyle R_{44}=4r^{2}\sin^{2}\left(\theta_{1}\right)\sin^{2}(\omega);\
\ R_{55}=4.$ (23)

  * •

Global coordinate:

$\displaystyle R_{11}=-4\sin^{2}(\omega);\ \
R_{22}=4\cosh^{2}(\tau)\sin^{2}(\omega);\ \
R_{33}=4\sin^{2}\left(\theta_{1}\right)\cosh^{2}(\tau)\sin^{2}(\omega);$
$\displaystyle
R_{44}=4\sin^{2}\left(\theta_{1}\right)\sin^{2}\left(\theta_{2}\right)\cosh^{2}(\tau)\sin^{2}(\omega);\
\ R_{55}=4.$

Ricci scalar for dS5 bulk is $R=20$, and the cosmological constant for the
same is $\Lambda=6$. Hence bulk EOM is given as

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+6g_{\mu\nu}=0,$ (25)

where $\mu,\nu=1,2,3,4,5$. Again we can check that bulk EOM (25) is satisfied
for the Ricci tensors and Ricci scalar mentioned above for the dS5 bulk for
the static patch as well as global coordinate metric on EOW branes and hence
condition 1 is satisfied.

####  2.2.4 $d=6$

For the six-dimensional de-Sitter space, the bulk metric is given as

$\displaystyle
ds^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\hat{g}_{ij}^{\beta}dy^{i}dy^{j},$
$\displaystyle ds_{\rm static\
patch}^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\left[-(1-r^{2})dt^{2}+\frac{dr^{2}}{1-r^{2}}+r^{2}d\Omega_{3}^{2}\right],$
$\displaystyle ds_{\rm global\ de-
Sitter}^{2}=d\omega^{2}+\sin^{2}\left(\omega\right)\left[-d\tau^{2}+\cosh^{2}\left(\tau\right)d\Omega_{4}^{2}\right],$
(26)

where
$d\Omega_{4}^{2}=d\theta_{1}^{2}+\sin^{2}\theta_{1}d\theta_{2}^{2}+\sin^{2}\theta_{1}\sin^{2}\theta_{2}d\theta_{3}^{2}+\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin^{2}\theta_{3}d\theta_{4}^{2}$
and the Ricci tensors are written below.

  * •

Static patch:

$\displaystyle R_{11}=5\left(r^{2}-1\right)\sin^{2}(\omega);\ \
R_{22}=-\frac{5\sin^{2}(\omega)}{r^{2}-1};\ \ R_{33}=5r^{2}\sin^{2}(\omega);$
$\displaystyle R_{44}=5r^{2}\sin^{2}\left(\theta_{1}\right)\sin^{2}(\omega);\
\
R_{55}=5r^{2}\sin^{2}\left(\theta_{1}\right)\sin^{2}\left(\theta_{2}\right)\sin^{2}(\omega);\
\ R_{66}=5.$ (27)

  * •

Global coordinate:

$\displaystyle R_{11}=-5\sin^{2}(\omega);\ \
R_{22}=5\cosh^{2}(\tau)\sin^{2}(\omega);\ \
R_{33}=5\sin^{2}\left(\theta_{1}\right)\cosh^{2}(\tau)\sin^{2}(\omega);$
$\displaystyle
R_{44}=5\sin^{2}\left(\theta_{1}\right)\sin^{2}\left(\theta_{2}\right)\cosh^{2}(\tau)\sin^{2}(\omega);$
$\displaystyle
R_{55}=5\sin^{2}\left(\theta_{1}\right)\sin^{2}\left(\theta_{2}\right)\sin^{2}\left(\theta_{3}\right)\cosh^{2}(\tau)\sin^{2}(\omega);\
\ R_{66}=5.$ (28)

Ricci scalar for dS6 is $R=30$, and the cosmological constant for dS6 is
$\Lambda=10$, which leads to the following EOM for the bulk metric (2.2.4)

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+10g_{\mu\nu}=0,$ (29)

where $\mu,\nu=1,2,3,4,5,6$. Similar to other cases discussed earlier, it is
easy to check that the bulk metric (2.2.4) satisfies (29).

To summarize, we proved that one could construct wedge holography for the de-
Sitter bulk, and we have done this in $d=3,4,5,6$, and in general, it is valid
for arbitrary $d$-dimensional de-Sitter space. Now, we are done with the
mathematical description of wedge holography in de-Sitter space, and hence in
subsection 2.3, we will discuss the pictorial realization of the same and
other related issues.

###  2.3 Pictorial representation of wedge holography for the de-Sitter bulk

We can construct the wedge holography from a single de-Sitter bulk by
considering end-of-the-world branes at constant $\omega$ slices as done in
[36, 52]. For example, we have to consider $Y_{1}$ at $\omega=\omega_{1}$ and
$Y_{2}$ at $\omega=\omega_{2}$ as shown in Fig. 1 where we have shown the
constant $t$ slice of de-Sitter space.

Figure 1: Pictorial representation of wedge holography in a single de-Sitter
space. Green and red curves denote EOW branes at $\omega=\omega_{1}$ ($Y_{1}$)
and $\omega=\omega_{2}$ ($Y_{2}$). Blue dots are $(d-2)$-dimensional defects.

Wedge holography by joining two copies of double holography in de-Sitter
space: Here, we discuss the idea that will play an important role in
constructing extended static patch wedge holography. The wedge holography in
de-Sitter space can also be described pictorially, as shown in Fig. 2.

Figure 2: Pictorial representation of wedge holography in de-Sitter space.
Green and red curves denote EOW branes at $\omega=\omega_{1}$ ($Y_{1}$) and
$\omega=\omega_{2}$ ($Y_{2}$). We can glue (a) and (b) [51] which results in
(c) and (d).

We take the spatial slice of de-Sitter space and take a constant $\omega$
slice at $\omega=\omega_{1}$; this is analogous to chopping off the geometry
beyond $\omega=\omega_{1}$ (description (a) in Fig. 2). We call this slice as
$Y_{1}$. Similarly, we take one more copy of the setup above and chop off the
geometry at $\omega=\omega_{2}$; we name this constant $\omega$ slice as
$Y_{2}$ (description (b) in Fig. 2). Now we join (a) and (b) using [51].
Conditions for gluing $Y_{1}$ (with metric $\hat{g}_{ij}^{(1)}$, tension
$T^{(1)}$) and $Y_{2}$ (with metric $\hat{g}_{ij}^{(2)}$, tension $T^{(2)}$)
are Israel junction conditions777This condition has to be satisfied whenever
we join two spacetimes along the two branes.

$\displaystyle\hat{g}_{ij}^{(1)}=\hat{g}_{ij}^{(2)};\ \ \ \
[K_{ij}]-[K]\hat{g}_{ij}=-(T^{(1)}+T^{(2)})\hat{g}_{ij},$

where $[K_{ij}]=K_{ij}^{(1)}+K_{ij}^{(2)}$. In the end, $Y_{1}$ and $Y_{2}$
will merge and result in a new brane, say “$Y$”, and we see that we joined two
de-Sitter spaces and obtained a new de-Sitter space.

Therefore, if we compare the above construction of the wedge holography in de-
Sitter space with the usual construction in AdS bulk, then we see that we can
see a wedge holography-like structure as long as $Y_{1}$ and $Y_{2}$ are
separated. Whenever two EOW branes $Y_{1}$ and $Y_{2}$ merge into a single
brane $Y$, we obtain a single de-Sitter space instead of wedge holography.

In the next section, we will construct wedge holography for the de-Sitter bulk
for the extended static patch, including two Randall-Sundrum branes [53, 54].
In that case, wedge holography is constructed by two Randall-Sundrum branes
$Q_{1}$ and $Q_{2}$, similar to the wedge holography in AdS bulk.

##  3 Wedge holography in extended static patch

Authors in [49] constructed double holography in an extended static patch in
the context of DS/dS correspondence. The bulk metric of ($d+1$)-dimensional
de-Sitter space is

$\displaystyle
ds_{d+1}^{2}=d\omega^{2}+\sin^{2}(\omega)ds_{d}^{2}=d\omega^{2}+\sin^{2}(\omega)\left(-\cos^{2}\beta
dt^{2}+d\beta^{2}+\sin^{2}\beta d\Omega_{d-2}^{2}\right),$ (30)

where $\beta\in(0,\pi)$ for $d\geq 3$ and
$d\Omega_{d-2}^{2}=d\chi^{2}+\sin^{2}\chi d\Omega_{d-3}^{2}$ such that
$0\leq\chi\leq\pi$ for $d\geq 4$. The idea to construct double holography is
that we need to consider two d-dimensional de-Sitter branes, dS${}_{d}^{1}$
and dS${}_{d}^{2}$ and treat dS${}_{d}^{1}$ brane as Randall-Sundrum brane $Q$
(described by $\chi=\pi/2$) slice and take the $\mathbb{Z}_{2}$ quotient of
dS${}_{d}^{2}$, i.e., dS${}_{d}^{2}/\mathbb{Z}_{2}$ which removes half of de-
Sitter space of dS${}_{d}^{2}$ and its spatial slice looks like hemisphere. In
[49], dS${}_{d}^{2}/\mathbb{Z}_{2}$ is described by $\omega=\pi/2$ slice and
is treated as a bath analogous to double holography in AdS backgrounds. In de-
Sitter double holography, $d$-dimensional Randall-Sundrum brane $Q$ interact
with dS${}_{d}^{2}/\mathbb{Z}_{2}$ via $(d-1)$-dimensional defect where
authors considered transparent boundary conditions.

Since $\omega=\pi/2$ slice is the UV part of the geometry [11] therefore, we
can treat dS${}_{d}^{2}/\mathbb{Z}_{2}$ as the UV brane. By following [51], we
can glue two copies of the double holography described above, and this is
described pictorially in Fig. 3.

Figure 3: Realization of wedge holography in DS/dS correspondence. Green and
red curves denote UV branes $Y_{1}$ and $Y_{2}$ at $\omega_{1,2}=\pi/2$. (a)
and (b) are joined along the branes $Y_{1}$ and $Y_{2}$ which results in the
final picture (d).

Mathematical descriptions of gluing two copies of double holography are
discussed below.

  * •

Consider the bulk metric as

$\displaystyle ds_{d+1}^{2}=d\omega_{1}^{2}+\sin^{2}(\omega_{1})dS_{d}^{2}$
$\displaystyle\hskip
28.90755pt=d\omega_{1}^{2}+\sin^{2}(\omega_{1})\left(-\cos^{2}\beta
dt^{2}+d\beta^{2}+\sin^{2}\beta\left(d\chi_{1}^{2}+\sin^{2}{\chi_{1}}d\Omega_{d-3}^{2}\right)\right).$
(31)

In geometry (• ‣ 3), we consider two slicing, $\omega_{1}=\pi/2$
[dS${}_{d}^{2}/\mathbb{Z}_{2}$, we denote this as $Y_{1}$] and $\chi=\pi/2$
[Randall-Sundrum brane, $Q_{1}$ which is dS${}_{d}^{1}$]. $Q_{1}$ interact
with $Y_{1}$ via the transparent boundary conditions at the
$(d-1)$-dimensional defect.

  * •

Let us take the bulk metric as

$\displaystyle ds_{d+1}^{2}=d\omega_{2}^{2}+\sin^{2}(\omega_{2})dS_{d}^{2}$
$\displaystyle\hskip
28.90755pt=d\omega_{2}^{2}+\sin^{2}(\omega_{2})\left(-\cos^{2}\beta
dt^{2}+d\beta^{2}+\sin^{2}\beta\left(d\chi_{2}^{2}+\sin^{2}{\chi_{2}}d\Omega_{d-3}^{2}\right)\right).$
(32)

In the bulk (• ‣ 3), $\omega_{2}=\pi/2$ slice is dS${}_{d}^{4}/\mathbb{Z}_{2}$
($Y_{2}$) and $\chi=\pi/2$ slice is dS${}_{d}^{3}$ (Randall-Sundrum brane,
$Q_{2}$). The interaction between $Q_{2}$ and $Y_{2}$ is happening due to the
transparent boundary conditions at the $(d-1)$-dimensional defect.

In double holography, the space available is between the Randall-Sundrum brane
$Q$ and UV brane $Y$. In this case, wedge holography consists of two
$d$-dimensional Randall-Sundram branes $Q_{1}$ and $Q_{2}$, which are joined
at the $(d-1)$-dimensional defect. In the language of [51], we have a de-
Sitter space bounded between $Q_{1}$ and $Q_{2}$.

Therefore, wedge holography in the extended static patch is constructed by two
Randall-Sundrum branes $Q_{1}$ and $Q_{2}$ (description (d) in Fig. 3). We
have ($d+1$)-dimensional de-Sitter gravity in the bulk, which is localized on
the $d$-dimensional Randall-Sundrum branes $Q_{1}$ and $Q_{2}$. Randall-
Sundrum branes $Q_{1}$ and $Q_{2}$ form a wedge, and Randall-Sundrum branes
are joined at the ($d-1$)-dimensional defect (blue dots in description (d) of
Fig. 3).

##  4 Communicating universes in extended static patch wedge holography?

The setup described in section 3 can describe the communicating universes
similar to [7]888We can also call this kind of setup as “parallel universe”
because all the universes are parallel to each other [2]. All de-sitter branes
are not joined at the common defect. See [1] for the classification of
parallel universes.. We consider “$n$” copies of the wedge holography
discussed in section 3 to describe communicating universes. For three copies,
we have Randall-Sundrum branes as $Q_{1,2,3,4,5,6}$ and UV branes as
$Y_{1,2,3,4,5,6}$ . Communicating universes are obtained by joining $Y_{1}$
with $Y_{2}$, $Q_{1}$ with $Q_{4}$, $Y_{3}$ with $Y_{4}$, $Q_{3}$ with $Q_{6}$
and so on. This is represented in Fig. 4.

Figure 4: Communicating universes in de-Sitter wedge holography.

If we follow the above idea and join the various branes following [51], then
Fig. 4 end up at the end of gluing as shown in Fig. 5 where we have replaced
“sphere” with “box” only for convenience so that we can draw the cartoon
picture easily. The notations are, e.g., $Q_{14}=Q_{1}+Q_{4}$,
$Y_{12}=Y_{1}+Y_{2}$, $Q_{36}=Q_{3}+Q_{6}$, $Y_{34}=Y_{3}+Y_{4}$ and so on; in
general $Q_{ij}=Q_{i}+Q_{j}$ and $Y_{ij}=Y_{i}+Y_{j}$ where $i$ and $j$ are
the labels of the branes.

Figure 5: Structure of the communicating universes in de-Sitter wedge
holography. We have represented the gluing of $Y_{i}$’s brane
($Y_{ij}=Y_{i}+Y_{j}$) in yellow color whereas the gluing of Randall-Sundrum
branes, $Q_{i}$’s ($Q_{ij}=Q_{i}+Q_{j}$) by red color.

Let us understand the physical meaning of Fig. 4 and 5. In the beginning, we
take two copies of double holography in DS/dS correspondence [49], and we
obtain the geometrical structure as the one copy of wedge holography as shown
in Fig. 4 with Randall-Sundrum branes $Q_{1,2}$ and UV branes $Y_{1,2}$. When
we join $Y_{1}$ and $Y_{2}$ using [51], then we obtain the first box as shown
in the upper part of Fig. 5 with Randall-Sundrum branes $Q_{1}$ and $Q_{2}$
and one UV brane $Y_{12}$. This describes a single de-Sitter space with cut-
off by the Randall-Sundrum branes $Q_{1}$ and $Q_{2}$. We can treat this kind
of box as a building block (say $P_{1}$) and then join many copies of this
building block as shown in the upper part of Fig. 5, e.g., $P_{1}$ with
$P_{2}$, $P_{2}$ with $P_{3}$, $P_{3}$ with $P_{4}$ and so on. $P_{i}$’s are
like a single de-Sitter space with two Randall-Sundrum branes. As long as
$P_{i}$s are separated, this model will provide the structure of parallel
universes. Whenever we join all the $P_{i}$’s with each other, then we obtain
the lower part of Fig. 5, which describes a single de-Sitter space with two
Randall-Sundrum branes at the end of de-Sitter space. Hence, the existence of
a parallel universe in extended static patch holography depends upon whether
we join all copies of extended static patch holography or not.

##  5 Communicating Multiverses in de-Sitter wedge holography

This section discusses the existence of multiverse and communicating
multiverses in de-Sitter wedge holography via subsections 5.1 and 5.2.

###  5.1 Multiverse

We can construct a multiverse for a single de-Sitter space as shown in Fig. 6.

Figure 6: Multiverse in de-Sitter wedge holography. $Y_{1,2,...,n-1,n}$ are
the constant $\omega$ slices in a single de-Sitter space.

We obtain the Multiverse model in a single de-Sitter space by taking the
constant $\omega$ slices of a single de-Sitter bulk (2.1). As we discussed
earlier, we have Einstein gravity with positive cosmological constant on
constant $\omega$ slices, and hence we have a Multiverse made up of “$n$”
universes with geometry on each of them as dSd-1, and all these universes are
embedded in a single de-Sitter bulk with geometry dSd. This kind of model for
the AdS bulk was discussed in [6]. Here, we are able to construct the
Multiverse model in de-Sitter space which is more realistic as we are living
in a universe with positive cosmological constant. One can check that the bulk
metric (2.1) satisfies the Neumann boundary condition (7) at
$\omega=\omega_{1,2,....,n-1,n}$ provided tensions of the end-of-the-world
branes should be given as follows

$\displaystyle
T^{\beta}=\frac{d-2}{R}\cot\left(\frac{\omega_{1,2,....,n-1,n}}{R}\right),$
(33)

where $\beta=1,2,....,n-1,n$. The interesting point about this setup is that
all the universes ($Y_{1}$, $Y_{2}$,….,$Y_{n-1}$, and $Y_{n}$) are joined at
the single defect (blue dot in Fig. 6) and hence there is the possibility of
exchange of information among each other.

###  5.2 Communicating Multiverses

Given that Multiverse exists in de-Sitter wedge holography as discussed in
subsection 5.1. Now, we discuss the possibility of the existence of
communicating multiverses. It is possible by joining many copies of the
Multiverse as shown in Fig. 7.

Figure 7: Communicating multiverses in de-Sitter wedge holography. For
simplicity, we replaced the sphere with a rectangle in the lower part of this
figure.

To do so, we join $n$th brane of one copy with $(n-1)$th brane of the second
copy, $n$th brane of the second copy with $(n-1)$th brane of the third copy,
and so on; this is represented pictorially in Fig. 7. More precisely, the left
outermost brane of the first copy will join with the right outermost brane of
the second copy; the left outermost brane of the second copy will join with
the rightmost outer brane of the third copy, and so on. We have done it for
three copies in Fig. 7.

In the process of constructing communicating multiverses, we have to lose some
of the universes. For example, if we join two multiverses, then we have to
lose two universes: one from the left copy($Y_{n}$) and one copy from the
other copy ($Z_{n-1}$). These two branes will form a new brane, which is
represented by a yellow line in the lower part of Fig. 7. Similarly, if we
join Multiverse made up of $Z_{1,2,...,n-1,n}$-branes with the Multiverse made
up of $X_{1,2,....,n-1,n}$ branes, then the combination of $Z_{n}$ brane and
$X_{n-1}$ brane will form a new brane and the same is represented by dark
purple in the lower part of Fig. 7.

To have a consistent model of communicating multiverses, we have to stop the
process of joining branes from the consecutive multiverses; otherwise, all
left branes of one copy will join with the all right branes of the right copy,
and at the end, we have a single universe as obtained in section 4. To be more
specific in the context of Fig. 7, all the left branes of multiverse
“$Y$”(which consist of universes $Y_{1,2,...,n-1,n}$) will join with all the
right branes of multiverse “$Z$” consists of universes $Z_{1,2,...,n-1,n}$;
all the left branes of multiverse “$Z$” with universes $Z_{1,2,...,n-1,n}$
will join with all the right branes of multiverse “$X$” with universes
$X_{1,2,...,n-1,n}$ and so on. In the end, we obtain a single de-Sitter space
cut-off by two end-of-the-world branes.

##  6 Discussion

In this paper, we have constructed the wedge holography for the de-Sitter
space and discussed its application to multiverse models. First, we did this
for general de-Sitter background using the idea similar to AdS bulk in section
2 where we used the localization of de-Sitter gravity [50] on the brane. A
clue to the dSd/CFTd-2 duality is provided by this. In this case, it is
possible to obtain a multiverse-like structure by taking many constant
$\omega$ slices of single de-Sitter space, see Fig. 6. We see the existence of
communicating multiverses [Fig. 7] by gluing many multiverses with each other
following the concept of [51]. It must be noted that communicating multiverses
exist as long as all left branes of one multiverse are not joined with the all
right branes of the other multiverse. Otherwise, we obtain a single de-Sitter
space bounded by two end-of-the-world branes.

In section 3, we constructed wedge holography in the extended static patch.
This was done using the idea of double holography in DS/dS correspondence
[49]. We can construct wedge holography in this case by gluing two copies of a
doubly holographic setup by following [51]. We can have four branes in
extended static patch wedge holography if we keep separating the UV branes
$Y_{1}$ and $Y_{2}$. If we join them, we obtain wedge holography with two
Randall-Sundrum branes $Q_{1}$ and $Q_{2}$, see Fig. 3. It is nice to see that
one can describe parallel universes from the perspective of wedge holography.
This can be achieved by taking “$n$” copies of wedge holography and then
gluing them together in parallel. Again, parallel universes exist as long as
“$n$” copies are not completely joined, i.e., we can have disconnected
parallel universes. If we join the left brane of one copy with the right brane
of the other, we obtain a single de-Sitter space bounded by two Randall-
Sundrum branes (Fig. 5).

Based on the theoretical construction of Multiverse discussed in subsection
5.1, we can say that we are living in a universe that is part of the
Multiverse because this model is constructed in de-Sitter space, which has a
positive cosmological constant. Many universes exist (where different species
may be living) within the same Multiverse, but we are not able to travel among
other universes because of our limitations. Suppose we are living in a
universe and we want to travel in a different universe, then we have to cross
the defect that is common to both universes. In this paper, we have not
discussed this issue, but we hope to explore it in our future work. We have
made a qualitative statement on this issue in [6] for the AdS bulk spacetime.

In subsection 5.2, in the process of constructing communicating multiverses.
There is a possibility that we can have disconnected multiverses. In this
scenario, there can’t be any communication between any universe of one
Multiverse and any universe of another multiverse. However, to make the
communication between two multiverses, we need to have connected multiverses
where one universe (left outermost) of one Multiverse will get combined with
one universe(right outermost) of the other Multiverse. In this case, there
could be possibility of communication between different multiverses.

It would be interesting if one could construct this kind of model from top-
down holography similar to [55] where the authors discussed the existence of
binary black holes from top-down triple holography999See [58, 57, 56] for the
top-down construction of double holography and [59] for the realization of de-
Sitter vacua in string theory.. So far we have discussed the existence of
parallel universes, multiverses, and communicating multiverses for the
identical branes, i.e., we have multiple copies of the same universe. As we
discussed in [6] that issue of mismatched branes [60] was present even in
wedge holography which prevented us to contruct the wedge holographic
realization of Schwarzschild de-Sitter black hole. If one could resolve the
issue of mismatched branes from the perspective of [51] then one may obtain
the Page curve of Schwarzschild de-Sitter black hole [62, 61, 63] from wedge
holography. It would be nice to include the grey-body factors [64] and see the
affect of the same on information exchange between different universes or in
usual wedge holography where Hawking radiation is exchanged between
gravitating branes. Further, one can use our model to discuss the information
paradox in de-Sitter space by considering one brane as the source of Gibbons-
Hawking radiation [65] and another as a bath for collecting the Gibbons-
Hawking radiation.

## Acknowledgements

The author would like to thank the Isaac Newton Institute for Mathematical
Sciences, Cambridge, for support and hospitality during the programme Bridges
between holographic quantum information and quantum gravity. I would like to
thank Prof. Aron Wall for inspiring conversations during the aforementioned
programme. I am extremely grateful to the Physics department at Swansea
University for the wonderful hospitality during my visit, especially to Prof.
Timothy J. Hollowood for hosting me and fruitful discussions. The idea about
this work originated in the event Holography@25 at ICTP-SAIFR during the
conversation with Prof. Juan Maldacena and then further discussion with Prof.
Timothy J. Hollowood. I would like to thank them and all the participants,
speakers, and organizers of these two events. I would also like to thank Prof.
Tadashi Takayanagi for the helpful clarification. This work is partially
supported by a grant to CMI from the Infosys Foundation and by EPSRC grant
EP/R014604/1. I would also like to thank the Physics group at CMI for
providing a vibrant research environment.

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