11institutetext: Center for Theoretical Physics and Department of Physics,
University of California, Berkeley, California 94720, U.S.A.

# Magnetic Braneworlds: Cosmology and Wormholes

Stefano Antonini 1 Luis Gabriel C. Bariuan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We construct 4D flat Big Bang-Big Crunch cosmologies and Anti-de Sitter (AdS)
planar eternally traversable wormholes using braneworlds embedded in
asymptotically AdS5 spacetimes. The background geometries are the AdS5
magnetic black brane and the magnetically charged AdS5 soliton, respectively.
The two setups arise from different analytic continuations of the same saddle
of the gravitational Euclidean path integral, in which the braneworld takes
the form of a Maldacena-Maoz Euclidean wormhole. We show the existence of a
holographic dual description of this setup in terms of a microscopic Euclidean
boundary conformal field theory (BCFT) on a strip. By analyzing the BCFT
Euclidean path integral, we show that the braneworld cosmology is encoded in a
pure excited state of a CFT dual to a black brane microstate, whereas the
braneworld wormhole is encoded in the ground state of the BCFT. The latter
confines in the IR, and we study its confining properties using holography. We
also comment on the properties of bulk reconstruction in the two Lorentzian
pictures and their relationship via double analytic continuation. This work
can be interpreted as an explicit, doubly-holographic realization of the
relationship between cosmology, traversable wormholes, and confinement in
holography, first proposed in arXiv:2102.05057, arXiv:2203.11220.

##  1 Introduction

How to describe cosmology remains one of the most important open questions in
holography. The holographic duality tHooft:1993dmi ; Susskind:1994vu  is a
powerful framework to describe quantum gravity and it is best understood in
the context of the Anti-de Sitter/Conformal Field Theory (AdS/CFT)
correspondence Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc ;
Aharony:1999ti . The structure of the AdS/CFT correspondence seems prima facie
to rely on the features of asymptotically AdS spacetimes, which have a
negative cosmological constant $\Lambda<0$—which asymptotically is the only
relevant source in the Einsten’s equations—and have a timelike conformal
boundary where the dual theory is defined. These are all specific features
distinguishing AdS from cosmological models of our universe, which typically
do not have any asymptotic boundary, are filled with energy density (almost)
homogeneously and isotropically, and, at least in the simplest possible models
in accordance with observations, have a positive cosmological constant
$\Lambda>0$ Weinberg:2008zzc ; Baumann:2022mni .

However, there is evidence that holography is an intrinsic feature of quantum
gravity, extending well beyond the AdS realm. In fact, the idea that gravity
stores information holographically can be dated back to the work of Bekenstein
Bekenstein:1972tm ; Bekenstein:1973ur ; Bekenstein:1974ax  and Hawking
Hawking:1974rv ; Hawking:1975vcx , and was understood before tHooft:1993dmi ;
Susskind:1994vu  its explicit realization in AdS/CFT. The holographic
description of different backgrounds, mainly de Sitter Alishahiha:2004md ;
Coleman:2021nor ; Batra:2024kjl ; Susskind:2021omt ; Shaghoulian:2022fop ;
Araujo-Regado:2022gvw ; Chandrasekaran:2022cip ; Witten:2023xze ;
Cotler:2023eza ; Cotler:2024xzz 111See also McFadden:2009fg  for a different
approach to the holographic description of $\Lambda>0$ cosmology. and
asymptotically flat Strominger:2017zoo ; Pasterski:2021raf , have been under
investigation for the last two decades. In particular, de Sitter holography,
unlike AdS/CFT, has the advantage of naturally describing a universe
undergoing an accelerated expansion.222Although our universe is currently
undergoing an accelerated expansion and possibly contains a positive
cosmological constant (which would imply the acceleration lasts forever and
our universe approaches de Sitter in the far future, modulo the conjectured
metastability of de Sitter leading to vacuum decay to a neighboring AdS vacuum
in the string theory landscape), it is not exactly de Sitter. A similar
reasoning applies to the inflationary era, during which the universe was
nearly de Sitter. Therefore, de Sitter holography should be regarded as one
step forward from AdS/CFT towards a holographic description of $\Lambda>0$
cosmology, rather than the final goal. 333More recently, using general quantum
information-theoretic properties of holography Ryu2006a ; Ryu2006b ;
Hubeny:2007xt ; Engelhardt:2014gca ; Swingle:2009bg ; VanRaamsdonk:2010pw ,
steps towards a more generic holographic duality independent of the global
properties of the spacetime were taken Bousso:2022hlz ; Bousso:2023sya .
Although these approaches are promising and could lead to breakthroughs in the
holographic description of cosmology in the next few years, they are not yet
as well understood as AdS/CFT and their application to a concrete description
of a cosmological universe seems premature.

An alternative line of research Maldacena:2004rf ; McInnes:2004nx ;
Cooper:2018cmb ; Antonini:2019qkt ; VanRaamsdonk:2020tlr ;
VanRaamsdonk:2021qgv ; Antonini:2022blk ; Antonini:2022ptt ; Antonini:2022fna
; Sahu:2023fbx ; Antonini:2023hdh 444See also Usatyuk:2024mzs  for a lower-
dimensional construction. is aimed at describing $\Lambda<0$ cosmology in
holography. Broadly speaking, the reason for this endeavor is that quantum
gravity with a negative cosmological constant is better understood thanks to
AdS/CFT and we can hope to obtain a well-controlled holographic description of
the cosmological universe, as we will see. The resulting cosmology will
typically be a Big Bang-Big Crunch universe which, after an expanding phase,
starts recollapsing due to the effect of the negative cosmological constant.
This is clearly unlike the $\Lambda>0$ $\Lambda$CDM standard model of
cosmology, which is the simplest model matching current observational data.
However, other mechanisms could lead to phases of accelerated expansion
Peebles:1987ek ; Ratra:1987rm , with the resulting models able to in principle
match current cosmological observations Antonini:2022ptt ; Antonini:2022fna ;
VanRaamsdonk:2023ion .

Regardless of the phenomenological viability of these models, the holographic
description of $\Lambda<0$ cosmologies has great theoretical value. Having a
negative cosmological constant does not immediately imply a simple, ordinary
AdS/CFT description. In fact, these $\Lambda<0$ cosmological universes in
general are not asymptotically AdS. Nonetheless, as we will see, a holographic
description using AdS/CFT can be obtained with some effort. Therefore, these
constructions provide us with a more “ordinary” holographic setting where to
learn general lessons about how spacetimes without an asymptotic AdS boundary
are encoded in a dual microscopic quantum mechanical theory. Hopefully, these
lessons can then be applied to the description of phenomenologically relevant
cosmological universes.

Motivated by these ideas, in this paper we study (3+1)-dimensional,
$\Lambda<0$, symmetric Big Bang-Big Crunch braneworld cosmologies, which we
build by embedding a codimension-1 end-of-the-world (ETW) brane in a
(4+1)-dimensional magnetic black brane (MBB) background DHoker:2009mmn . This
setup is associated with a dominant Euclidean saddle of the gravitational path
integral and has a well-defined holographic dual description in terms of an
excited pure state of a microscopic 4D CFT. Interestingly, the Euclidean
saddle also admits a different analytic continuation to a (4+1)-dimensional
magnetically charged AdS5 soliton cut off by a static ETW brane. In this case,
the braneworld physics555Here and in the rest of the paper, when we talk about
“braneworld physics” or “braneworld picture” we mean the effective 4D
description of physics on the ETW brane. is given by a planar eternally
traversable wormhole connecting two distinct asymptotic AdS4 boundaries.

The existence of this alternative analytic continuation is of its own interest
both from a bulk and from a boundary dual viewpoint. Our braneworld
construction is, to the best of our knowledge, the first construction of
higher dimensional eternally traversable wormholes connecting two different
asymptotic AdS4 boundaries.666Other eternally traversable wormhole
constructions include lower dimensional examples Gao:2016bin ;
Maldacena:2018lmt , 4D wormholes connecting two regions of the same
asymptotically flat spacetime Maldacena:2018gjk ; Fu:2019vco , and 4D
braneworld wormholes connecting two different regions of the same asymptotic
AdS4 boundary Maldacena:2020sxe . See Section 5.5 for a discussion of the
relationship between our construction and those presented in Maldacena:2018gjk
; Maldacena:2020sxe . We also remark that the costruction of our wormhole
involves embedding a braneworld in a higher dimensional spacetime, and is
different in this sense from the constructions in Gao:2016bin ;
Maldacena:2018gjk ; Maldacena:2018lmt ; Fu:2019vco . However, an effective 4D
description of the braneworld wormhole not involving the 5D spacetime does
exist, see Sections 1.1 and 3.3. These wormholes admit a dual description in
terms of the vacuum state of a microscopic theory which exhibits a mass gap
and IR confinement, as we will discuss. Additionally, our construction
realizes in an explicit and controlled setup the ideas advanced in
VanRaamsdonk:2021qgv ; Antonini:2022blk  connecting cosmology, traversable
wormholes, and confining properties of a holographic microscopic theory.
Before summarizing the main results of our paper, we will review these ideas
and explain their connection to our construction.

###  1.1 Relation to previous work and motivation

First of all, let us review the ideas presented in VanRaamsdonk:2021qgv ;
Antonini:2022blk , which are one basic motivation for the present work.

The realization that certain time symmetric $\Lambda<0$ Big Bang-Big Crunch
cosmologies are obtained by analytic continuation of reflection-symmetric
Euclidean wormholes connecting two AdS asymptotic boundaries dates back to the
work of Maldacena and Maoz Maldacena:2004rf . The metric of these Euclidean
wormholes generically takes the form

$ds^{2}=dw^{2}+a^{2}(w)ds^{2}_{\Sigma_{d}}$ (1)

where $ds^{2}_{\Sigma_{d}}$ is the metric on the $d$-dimensional manifold
$\Sigma_{d}$ and $a(w)$ is symmetric $a(w)=a(-w)$ and diverges exponentially
for $w=\pm\infty$. This behavior is typical of asymptotically AdS spacetimes,
as it becomes clear by recasting the metric in the conformal form (which will
be useful in the rest of the paper)

$ds^{2}=a^{2}(\tau)\left(d\tau^{2}+ds^{2}_{\Sigma_{d}}\right)$ (2)

in which $a(\tau)\approx 1/(\tau_{\infty}\pm\tau)$ for
$\tau\to\mp\tau_{\infty}$, with $\tau_{\infty}$ a finite number.

These wormholes analytically continue under $w=i\lambda$ to Friedmann-
Lemaître-Robertson-Walker (FLRW) metrics for homogeneous and isotropic
cosmological universes. The cosmological scale factor is simply given by
$a(\lambda)$ after substituting $w\to i\lambda$ in the wormhole scale factor.
$a(\lambda)$ is guaranteed to be real by reflection symmetry and it goes to
zero at the Big Bang and Big Crunch for $\lambda=\pm\lambda_{0}$. The simplest
explicit example is given by a flat FLRW universe sourced by a negative
cosmological constant and a radiation term, giving rise to a scale factor
$a(\lambda)=A\sqrt{\cos(B\lambda)}$, where $A,B$ are constants simply related
to $\Lambda$ and the radiation density parameter. Analytically continuing to
Euclidean signature, the scale factor becomes $a(w)=A\sqrt{\cosh(Bw)}$, which
indeed diverges exponentially at the two asymptotic AdS boundaries
$w\to\pm\infty$.

Figure 1: Recap of all the constructions discussed in this paper. The yellow
slices labeled by $A$ are the reflection-symmetric slices associated with the
wormhole picture, whereas the red slices labeled by $B$ are the reflection-
symmetric slices associated with the cosmology picture. In the first line we
have the microscopic, UV complete description, with boundaries indicated by a
purple line. From left to right: the vacuum state of the BCFT on
$\mathbb{R}^{2,1}\times I$, the Euclidean BCFT on $\mathbb{R}^{3}\times I$,
and the excited state of the CFT on $\mathbb{R}^{3,1}$. In the second line we
have the singly holographic/braneworld description. From left to right: the
eternally traversable wormhole (in purple) with boundaries coupled by non-
gravitational DOF (in blue), the Maldacena-Maoz Euclidean wormhole with
boundaries coupled by non-gravitational DOF, and the Big Bang-Big Crunch
cosmology entangled to the bath on $\mathbb{R}^{3,1}$. In the third line we
have the doubly holographic description. The black dot indicates $r=r_{H}$.
From left to right: the magnetically charged AdS5 soliton spacetime cut off by
the ETW brane (in purple), the 5D Euclidean saddle cut off by the ETW brane,
and the double-sided magnetic black brane cut off by the ETW brane.

In VanRaamsdonk:2020tlr ; VanRaamsdonk:2021qgv ; Antonini:2022blk ;
Antonini:2022ptt  it was proposed that Maldacena-Maoz-like wormholes in
$d+1=4$ dimensions are saddles of a Euclidean gravitational path integral dual
to the path integral of a microscopic Euclidean theory given by two 3D
holographic CFTs on $\Sigma_{3}$ coupled via a relevant interaction to 4D, in
general non-holographic auxiliary degrees of freedom (DOF) on
$\Sigma_{3}\times I$. $I$ here is a finite interval connecting the two 3D
CFTs. See the top line, center panel of Figure 1. The holographic 3D theories
are taken to have many more degrees of freedom than the 4D auxiliary theory,
such that an effective 4D holographic description (the Euclidean wormhole)
exists. Following VanRaamsdonk:2020tlr ; VanRaamsdonk:2021qgv ;
Antonini:2022blk ; Antonini:2022ptt , let us focus our attention on the case
$\Sigma_{3}=\mathbb{R}^{3}$. Given this microscopic description, the bulk
effective field theory (EFT) is given by a 4D Maldacena-Maoz-like Euclidean
wormhole—which can be understood to be dual to the two 3D holographic
CFTs—with the two asymptotic boundaries coupled by auxiliary, non
gravitational DOF on $\mathbb{R}^{3}\times I$, which account for the non-
holographic microscopic 4D DOF.777This is an intuitive, simplified description
of the relationship between microscopic and bulk EFT DOF. In fact, the
wormhole physics does not encode only the holographic 3D microscopic boundary
DOF, but also the coupling of these with the 4D microscopic DOF. In
particular, matter fields associated with both 3D and 4D microscopic DOF live
on the wormhole background and are coupled by appropriate boundary conditions
to the non-gravitational bath Antonini:2022blk . 888This can be interpreted in
analogy with the (1+1)-dimensional holographic models of black hole
evaporation studied in Almheiri:2019psf ; Almheiri:2019qdq . In that case, the
microscopic theory was given by a holographic (0+1)-dimensional theory coupled
to a non-holographic (1+1)-dimensional CFT. In the bulk setup, a
(1+1)-dimensional black hole replaces the holographic DOF and is coupled to a
non-gravitational bath at the asymptotic AdS boundary. See the center line,
center panel of Figure 1. Notice that the presence of these auxiliary non-
gravitational DOF coupling the two boundaries is fundamental to avoid a
factorization puzzle Maldacena:2004rf ; VanRaamsdonk:2020tlr .

In this Euclidean bulk EFT, the reflection-symmetric slice $w=0$ where the
initial state of the corresponding Lorentzian cosmology is prepared also
intersect the non-gravitational DOF (the “bath”). The Lorentzian description
therefore includes a Big Bang-Big Crunch cosmology and a non-gravitational
bath on $\mathbb{R}^{3,1}$, with bulk fields in the cosmology entangled to
bulk fields in the bath, see the center line, right panel of Figure 1. This
has interesting implications for the encoding of the cosmological universe in
the dual microscopic theory and is related to the construction of cosmological
islands Hartman:2020khs ; Bousso:2022gth ; Antonini:2022blk , see Section
4.3.1.

What is the state of the microscopic theory dual to the Lorentzian cosmology
(plus the non-gravitational bath)? From the associated slicing of the
microscopic Euclidean path integral, we see that the interval $I$ is taken to
be the Euclidean time direction. This Euclidean path integral therefore
prepares an excited pure state of the 4D non-holographic DOF only on
$\mathbb{R}^{3,1}$, see top line, right panel of Figure 1. Note that the 3D
holographic theories are not present at all in the Lorentzian description, but
rather they are in the Euclidean past and future, representing boundary
conditions for the Euclidean path integral preparing the microscopic state.

It was observed in VanRaamsdonk:2020tlr ; VanRaamsdonk:2021qgv ;
Antonini:2022blk ; Antonini:2022ptt  that, given the translational symmetry in
the $\mathbb{R}^{3}$ directions, the bulk Euclidean saddle admits a different
analytic continuation also yielding a real Lorentzian theory. This is obtained
by identifying one of the $\mathbb{R}^{3}$ directions, let us label it by $z$,
with the Euclidean time. The resulting bulk effective field theory is static
and it is given by a planar eternally traversable wormhole (namely the
Lorentzian version of a Maldacena-Maoz-like wormhole), with the two boundaries
coupled by non-gravitational auxiliary DOF, see the center line, left panel of
Figure 1. Once again, the presence of these additional DOF is fundamental,
because a traversable wormhole can only exist if the holographic theories
living on the two boundaries are coupled to each other Antonini:2022blk ;
Maldacena:2018lmt .

The microscopic dual description of the traversable wormhole is given by the
vacuum state999The $z$ direction, which is chosen to be the Euclidean time in
this picture, is non-compact. Therefore, the microscopic path integral
naturally prepares the vacuum state of the theory. of the 3D-4D-3D theory on
$\mathbb{R}^{2,1}\times I$, see top-left panel of Figure 1. Note that in the
infrared (IR), namely at scales much larger than the interval $I$, this
3D-4D-3D theory on $\mathbb{R}^{2,1}\times I$ flows to a 3D theory on
$\mathbb{R}^{2,1}$. Because the center of the traversable wormhole is at a
finite proper distance away from any other bulk point in the wormhole, it acts
as a cut off in the emergent holographic direction. In the spirit of the
holographic renormalization group (RG) flow deBoer:1999tgo , this suggests the
existence of a IR energy cutoff in the dual microscopic theory, below which
physics is trivial. This fact was interpreted in VanRaamsdonk:2020tlr ;
VanRaamsdonk:2021qgv ; Antonini:2022blk ; Antonini:2022ptt  as evidence that
the 3D and 4D theories and their coupling must be such that the IR theory on
$\mathbb{R}^{2,1}$ confines and possesses a mass gap (hence the choice of
relevant coupling between the 3D CFTs and the 4D DOF). A similar argument for
confinement in holography was first introduced by Witten Witten:1998zw  (see
Section 2.4).

The existence of these two possible slicings of the same Euclidean path
integral, leading to two different states of two different microscopic
theories and, holographically, to two different bulk setups, was dubbed
“slicing duality” in Antonini:2022blk . It was argued VanRaamsdonk:2021qgv ;
Antonini:2022blk ; Antonini:2022opp  that observables in the two Lorentzian
bulk setups are related to each other by double analytic continuation, and
that the slicing duality could be used to reconstruct observables in the
cosmology from vacuum observables in the confining theory dual to the
Lorentzian wormhole. Because direct reconstruction of cosmological observables
from the dual highly excited pure state is understood to be very complex, the
slicing duality has the potential to provide a new, much simpler dictionary in
this context. We will come back to these points in Sections 4.3.1 and 5.4.

#### Double holography and braneworlds

In order to build an explicit realization of these ideas in a concrete
setting, it is useful to consider the special case in which the 4D microscopic
auxiliary degrees of freedom that couple the two 3D CFTs are also holographic.
Let us focus initially on the Euclidean setup. The microscopic theory can then
be understood to be a 4D holographic boundary conformal field theory (BCFT)
Cardy:1989ir ; Affleck:1991tk ; Cardy:2004hm  on $\mathbb{R}^{3}\times I$.
According to the AdS/BCFT prescription Takayanagi:2011zk ; Fujita:2011fp , the
bulk description is now given by a 5D Euclidean bulk geometry cut off by a
constant tension ETW brane,101010Depending on the setup, there could be
different saddles of the 5D gravitational path integral, with some of them
having two disconnected ETW branes each one intersecting the boundary at one
of the two endpoints of the interval Cooper:2018cmb . In our setup, this
alternative phase does not exist, and the gravitational ensemble is dominated
by the connected phase of our interest (see Appendix C). which intuitively is
an extension into the bulk of the boundary of the BCFT. See the bottom center
panel of Figure 1. The tension parameter $T$ is related to the boundary
entropy of the Cardy state $\ket{B}$ in the dual BCFT Takayanagi:2011zk ;
Fujita:2011fp , and is therefore uniquely determined by the specific choice of
conformal boundary conditions at the endpoints of the interval $I$.

Under specific conditions that we will explain in Section 3.3, an effective 4D
description of gravity localized on the brane via the Karch-Randall-Sundrum
mechanism Randall:1999ee ; Randall:1999vf ; Karch:2000ct  is possible. When
this is the case, there exists an effective 4D description of physics on the
ETW brane (the “braneworld”) without referring to the higher dimensional 5D
bulk setup. Once again, the braneworld will be coupled to non-gravitational
auxiliary DOF accounting for the physics of the 5D bulk.111111Similarly to
what we explained in Footnote 7, this is just an intuitive description. In
reality, the physics of the 5D bulk affects both the auxiliary DOF and the 4D
gravitational effective theory on the ETW brane. This braneworld description
takes the form of the 4D Euclidean Maldacena-Maoz wormhole reviewed above. The
5D setup with ETW branes can then be seen as a doubly holographic realization
of the ideas introduced in VanRaamsdonk:2020tlr ; VanRaamsdonk:2021qgv ;
Antonini:2022blk ; Antonini:2022ptt , in a similar way as doubly holographic
setups for black hole evaporation were built in Almheiri:2019hni . The
advantage is that this doubly holographic realization can be built explicitly
in setups with a well-understood holographic dual description.

The first construction of this AdS/BCFT setup was given in Cooper:2018cmb
(see also Kourkoulou:2017zaj ; Almheiri:2018ijj  for earlier lower-dimensional
realizations of similar constructions), where the 5D bulk spacetime was taken
to be a Euclidean AdS-Schwarzschild black hole cutoff by an ETW brane, see
Figure 2. The analytic continuation corresponding to the cosmology picture
yields a Lorentzian double-sided black hole with one of the two asymptotic
boundaries completely cut off by an ETW brane, similar to the bottom-right
panel of Figure 1. This is a single-sided black hole microstate, with the
corresponding pure excited state of the dual microscopic theory living on the
only remaining asymptotic boundary (the left one in Figure 1). An observer
comoving with the brane experiences the expansion and contraction of a Big
Bang-Big Crunch FLRW universe.

Figure 2: Left: Euclidean black hole cut off by a codimension-1 ETW brane with
trajectory in the $r-\tau$ directions. Right: in the AdS-Schwarzschild case
Cooper:2018cmb  the brane overlaps itself for large tension (which is needed
for gravity localization), signaling an ill-defined bulk setup and no
holographic dual description in this regime. In our magnetic setup, gravity
localization can be achieved without the brane overlapping.

Notice that, in the planar limit, the same Euclidean saddle admits also an
alternative analytic continuation yielding a real Lorentzian spacetime: the
static AdS5 soliton spacetime Horowitz:1998ha ; Witten:1998zw  cut off by a
static ETW brane, see bottom-left panel of Figure 1. The braneworld
description is now that one of an eternal traversable wormhole connecting two
asymptotic AdS4 boundaries, realizing the Lorentzian wormhole picture of the
singly holographic description reviewed above VanRaamsdonk:2021qgv ;
Antonini:2022blk ; Antonini:2022ptt . AdS5 solitons are understood to be dual
to negative energy ground states of the dual holographic theory on
$\mathbb{R}^{2,1}\times S^{1}$ Horowitz:1998ha , which is expected to confine
in the IR at length scales much larger than the size of the $S^{1}$
Witten:1998zw . In our case, the soliton cut off by the ETW brane is then dual
to the ground state of the BCFT, which is also expected to confine at length
scales much larger than the interval $I$.

One important issue with the AdS-Schwarzschild construction of Cooper:2018cmb
is that gravity localization (and therefore an effective 4D braneworld
description) cannot be achieved in a well-defined bulk setup with a well-
defined dual description. In fact, if we try to increase the tension $T$ of
the brane enough to obtain gravity localization121212As we will discuss in
Section 3.3, increasing the tension allows us to “push” the brane far into the
asymptotic AdS region, which is necessary for gravity localization to hold.
From a microscopic point of view, this corresponds to increasing the number of
boundary degrees of freedom of the BCFT., the brane overlaps itself in the
Euclidean setup, completely cutting off the asymptotic boundary, see Figure 2.
Therefore, we lose control over the 5D bulk EFT and no well-defined
microscopic dual description exists. This problem was addressed in
Antonini:2019qkt  by replacing the AdS-Schwarzschild spacetime by a near-
extremal AdS-Reissner-Nordström (AdS-RN) black hole. In this case, it was
shown that the tension can be taken arbitrarily close to the critical value
$T_{crit}=1/L$ (where $L$ is the 5D AdS radius), and the brane does not
overlap if the black hole charge is close enough to its extremal value.
However, in the AdS-RN setup there is no corresponding well-defined soliton
solution because, under double analytic continuation, the electric field in
AdS-RN becomes an imaginary magnetic field in the soliton.

In this paper, we consider a magnetically charged setup which allows for
gravity localization on the brane while retaining a well-defined dual theory.
Furthermore, both analytic continuations (to a Lorentzian magnetic black brane
(MBB) and to a magnetically charged AdS5 soliton) of the Euclidean saddle are
real and well-defined, giving rise to a braneworld cosmology and a braneworld
traversable wormhole, respectively. Our setup is therefore a first explicit
and complete doubly-holographic realization of the ideas advanced in
VanRaamsdonk:2021qgv ; Antonini:2022blk ; Antonini:2022ptt . It realizes in a
concrete setting the embedding of a $\Lambda<0$ cosmological universe in a
holographic setup, it relates this to the physics of traversable wormholes in
AdS, and allows us to study explicitly the features of the dual holographic
theory.

###  1.2 Summary of results and outline

Let us briefly summarize the main constructions and results of this paper:

  * •

We start by studying the (4+1)-dimensional MBB Lorentzian spacetime first
constructed in DHoker:2009mmn  (obtained numerically) and we observe that it
arises from analytic continuation of a Euclidean saddle of the gravitational
path integral. We also observe the existence of an alternative analytic
continuation of the same Euclidean saddle yielding a static, magnetically
charged AdS5 soliton spacetime. Unlike the AdS-RN case, this soliton solution
is well-defined and has a real magnetic field. This is thanks to the breaking
of rotational symmetry due to the presence of a magnetic field in 5D. The
drawback of this symmetry breaking is that the spacetime metric cannot be
obtained analytically.

  * •

The holographic theory on $\mathbb{R}^{2,1}\times S^{1}$ dual to the magnetic
soliton spacetime is confining on scales much larger than the $S^{1}$.
Following Witten:1998zw , we compute the discrete spectrum of massive
particles (glueballs Csaki:1998qr ) by means of a bulk calculation and show
the existence of a mass gap.

  * •

Embedding an ETW brane in the Euclidean 5D spacetime, for a sufficiently large
magnetic field the brane tension can be increased enough to allow for gravity
localization on the brane without the latter overlapping with itself. This
guarantees a well-defined 5D bulk solution and dual theory (given by a 4D
BCFT), and a well-defined 4D effective braneworld description on the brane.
The Euclidean braneworld takes the form of a Maldacena-Maoz wormhole.

  * •

One analytic continuation of this Euclidean saddle leads to the double-sided
MBB with one boundary completely cut off by an ETW brane. This MBB microstate
is dual to an excited state of a 4D CFT living on the only remaining
asymptotic boundary (which is $\mathbb{R}^{3,1}$). The 4D braneworld
description is given by a Big Bang-Big Crunch homogeneous but anisotropic
(Bianchi I Taub:1950ez ; Ellis:1968vb ; Montani:2009hju ) cosmology. In the
region where a good 4D braneworld description exists, the anisotropies are
very small and the braneworld cosmology approximately takes the form of a flat
FLRW universe. In the braneworld cosmology setting, gravity localization only
holds for (a large) part of the brane trajectory around the time-symmetric
point.

  * •

At early times (but where gravity localization still holds and an effective 4D
braneworld description still exists), the braneworld cosmology undergoes a
phase of accelerated expansion, likely driven by a negative (from the point of
view of the braneworld) energy density proportional to the magnetic field. The
expansion then decelerates and eventually stops when the negative cosmological
constant $\Lambda_{4}<0$ takes over. The recollapse is time-symmetric.

  * •

A different analytic continuation of the same Euclidean saddle leads to the
magnetically charged AdS5 soliton with part of the boundary cut off by an ETW
brane. This is dual to the ground state of a BCFT on $\mathbb{R}^{2,1}\times
I$, which confines on scales much larger than the size of the interval $I$.
The 4D braneworld description is given by a nearly Poincaré symmetric
eternally traversable wormhole connecting two distinct AdS4 boundaries. In the
braneworld wormhole setting, gravity localization holds everywhere on the
brane. The confined spectrum of the dual theory can be computed, following
Antonini:2022opp , directly from the 4D braneworld effective theory.

  * •

The Euclidean saddle giving rise to the results above, which contains a
connected ETW brane, is the only known saddle contributing to the
gravitational ensemble. The existence of a natural alternative Euclidean
saddle containing two disconnected ETW branes is ruled out in Appendix C.

The paper is organized as follows. In Section 2 we study in depth the
properties of the AdS5 magnetic black brane (MBB) solution, its double
analytic continuation given by the magnetically charged AdS5 soliton, and
their holographic dual description. Many interesting features of these
constructions are discussed, and a derivation of the mass spectrum of
confining particles (glueballs) of the theory dual to the soliton is given.
Although knowledge of these properties and results will deepen the
understanding of the following sections, the reader eager to learn the main
upshot related to braneworld cosmology and braneworld wormholes can find the
MBB and AdS5 soliton background metrics in equations (10) and (15), with the
metric functions obtained numerically plotted in Figure 4. This knowledge
should be enough for a surface-level understanding of Sections 3, 4, and 5. In
Section 3 we embed an ETW brane in the Euclidean geometry and show that well-
defined solutions supporting gravity localization and admitting a holographic
dual description can be obtained. We then show that the braneworld description
is given by a Maldacena-Maoz-like wormhole. In Section 4 we study the
braneworld cosmology embedded in the Lorentzian MBB solution and its dual
description, and comment on the properties of bulk reconstruction in this
setting. In Section 5 we study the Lorentzian traversable braneworld wormhole
embedded in the magnetically charged AdS5 soliton, study the confining
properties of the dual theory, comment on the slicing duality, and explain the
relationship of our solution with other constructions of 4D eternally
traversable wormholes. Finally, in Section 6 we give our final remarks. The
appendices contain technical details related to the main sections of the
paper. In particular, Appendix C shows that a naively possible alternative
saddle of the gravitational Euclidean path integral does not exist in our
setup. The Euclidean saddle examined in the main text is therefore the only
(known) saddle of the Euclidean path integral and it dominates the
gravitational ensemble.

In order to facilitate the reading, we include at the beginning of each
section a quick overview of the most relevant results and the organization of
that section.

#### Glossary

To avoid confusion, we recap here the different descriptions depending on how
many layers of holography and which analytic continuation we consider
(depicted in Figure 1), and we specify the terms we will use in the rest of
the paper to refer to each one of them.

  * •

We refer to the microscopic UV complete description depicted in the top row of
Figure 1 interchangeably as: microscopic description/theory/DOF, 4D BCFT. This
is what in the double holography literature Almheiri:2019hni  is typically
called the “fundamental description”.

  * •

We refer to the intermediate 4D bulk EFT depicted in the central row of Figure
1 given by the wormhole/cosmology and non-gravitational auxiliary DOF
interchangeably as: effective 4D EFT/description, singly holographic setup,
braneworld description.

  * •

We refer to the 5D bulk setup including an ETW brane depicted in the bottom
row of Figure 1 interchangeably as: 5D bulk theory/description, doubly
holographic setup, higher dimensional setup.

Depending on which description we are analyzing (microscopic, singly, or
doubly holographic, respectively), we refer to the Euclidean and the two
Lorentzian pictures as:

  * •

Left column of Figure 1, generically referred to as “wormhole picture”. Top to
bottom: vacuum state of the confining theory on $\mathbb{R}^{2,1}\times I$;
AdS4 eternally traversable wormhole or braneworld wormhole; magnetically
charged AdS5 soliton cut off by ETW brane.

  * •

Center column of Figure 1, generically referred to as “Euclidean picture”. Top
to bottom: Euclidean BCFT on $\mathbb{R}^{3}\times I$; Maldacena-
Maoz/Euclidean wormhole; 5D Euclidean bulk geometry.

  * •

Right column of Figure 1, generically referred to as “cosmology picture”. Top
to bottom: highly excited pure state of CFT on $\mathbb{R}^{3,1}$; braneworld
cosmology; MBB cut off by ETW brane.

##  2 $\text{AdS}_{5}$ Magnetic Black Brane and soliton

In this section we introduce the asymptotically AdS5 background spacetimes of
our interest: the AdS5 magnetic black brane (MBB), the magnetically charged
AdS5 soliton, and the Euclidean saddle they arise from. The MBB is a solution
of Einstein-Maxwell equations in five dimensions with a negative cosmological
constant $\Lambda<0$ and a constant magnetic field $B$ tangent to the
asymptotic boundary. First introduced by D’Hoker and Kraus DHoker:2009mmn
(see also DHoker:2009ixq ; Martinez-y-Romero:2017awl ; Avila:2018sqf ), it is
a one-parameter family of solutions at finite temperature $\mathcal{T}$ with a
spatially flat asymptotic boundary, parametrized by the dimensionless ratio
$\mathcal{T}/\sqrt{B}$.

In order to get acquainted with the properties of planar solutions of
Einstein-Maxwell equations, we first consider in Section 2.1 the planar limit
of the electrically charged AdS-Reissner-Nordström (AdS-RN) black hole,
pointing out the similarities and differences with the MBB solution of our
interest. Working out the AdS-RN case, on which we have full analytic control,
is particularly useful to guide our intuition when treating the MBB solution,
which can only be found numerically. We then numerically construct the MBB
solution in Section 2.2. We use a different choice of coordinates with respect
to DHoker:2009mmn , which will turn out to be favorable to study ETW branes
embedded in the MBB background. In Section 2.3 we consider a double analytic
continuation of the MBB solution yielding a magnetically charged AdS5 soliton
spacetime and comment on its properties and role in the context of AdS/CFT.
Both the MBB and the soliton arise from analytic continuation of the same
Euclidean saddle, which we describe. Finally, in Section 2.4 we comment on the
microscopic dual description of the MBB and the soliton and, following
Witten:1998zw , provide evidence for the existence of a mass gap and a
spectrum of confined particles in the holographic theory dual to the magnetic
soliton.

###  2.1 Planar limit of AdS-Reissner-Nordström

As a warm up, let us consider the planar limit of the electrically charged
AdS-RN black hole in five dimensions. This can be obtained by generalizing the
procedure employed in Witten:1998zw  to derive the planar limit of the AdS-
Schwarzschild black hole. The metric of the 5D AdS-RN black hole is given by

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{3}^{2}$ (3)

where

$f(r)=1+\frac{r^{2}}{L^{2}}-\frac{2M}{r^{2}}+\frac{Q^{2}}{r^{4}},$ (4)

$M$ is the mass parameter (proportional to the ADM mass), and $Q$ is the
charge parameter. This metric has two horizons at $r=r_{\pm}$ and its inverse
temperature, given by $\beta=4\pi/f^{\prime}(r_{+})$, diverges in the extremal
limit for which $r_{-}=r_{+}$. The electromagnetic field strength has
$F_{tr}=-F_{rt}\neq 0$, with all other components vanishing.

First, we can rescale the radial coordinate as
$r=(2M/L^{2})^{\frac{1}{4}}\rho$ and the time coordinate as
$t=(L^{2}/2M)^{\frac{1}{4}}\tilde{t}$, obtaining the metric

$ds^{2}=-\tilde{f}(\rho)d\tilde{t}^{2}+\frac{d\rho^{2}}{\tilde{f}(\rho)}+\rho^{2}\sqrt{\frac{2M}{L^{2}}}d\Omega_{3}^{2},$
(5)

where we defined

$\tilde{f}(\rho)=\sqrt{\frac{L^{2}}{2M}}f(\rho)=\sqrt{\frac{L^{2}}{2M}}+\left(\frac{\rho^{2}}{L^{2}}-\frac{L^{2}}{\rho^{2}}+\frac{\alpha
L^{4}}{\rho^{4}}\right),\quad\quad\alpha\equiv\frac{Q^{2}}{L(2M)^{\frac{3}{2}}}.$
(6)

We can then take the limit of infinite mass and infinite charge, keeping the
ratio $\alpha$ fixed. In this limit, we can drop the first term on the right-
hand side of equation (6). From (5) we also see that the size of the $S^{3}$
diverges as $M\to\infty$ yielding a 3-plane,
$\sqrt{(2M/L^{2})}d\Omega_{3}^{2}\to dx^{2}+dy^{2}+dz^{2}$. We are therefore
left with the planar AdS-RN metric

$ds^{2}=-\left(\frac{\rho^{2}}{L^{2}}-\frac{L^{2}}{\rho^{2}}+\frac{\alpha
L^{4}}{\rho^{4}}\right)dt^{2}+\frac{d\rho^{2}}{\left(\frac{\rho^{2}}{L^{2}}-\frac{L^{2}}{\rho^{2}}+\frac{\alpha
L^{4}}{\rho^{4}}\right)}+\rho^{2}(dx^{2}+dy^{2}+dz^{2}),$ (7)

which in the zero charge limit $\alpha=0$ reduces to the planar AdS-
Schwarzschild solution Witten:1998zw .

A few comments are in order. First, notice that taking the planar limit we
reduced a two-parameter ($M,Q$) family of solutions to a one-parameter
($\alpha$) family of solutions. This can be readily understood in terms of the
conformal symmetry of the dual boundary theory. Initially, there are three
length scales in the boundary theory131313The dual boundary theory has a
finite chemical potential $\mu$, which is proportional to the charge parameter
$Q$ introduced above, see e.g Guo:2015swu ; Andrade:2013rra .: the inverse
temperature $\beta$ (i.e. the periodicity of the Euclidean time circle), the
size of the spatial $S^{3}$, and the length scale associated with the chemical
potential $l_{\mu}\propto\mu^{-1}$. Since conformal invariance guarantees that
only adimensional parameters are physically meaningful, these three length
scales can be combined into two physical parameters, e.g. the ratios $\beta/R$
and $l_{\mu}/R$ with $R$ the radius of the spatial $S^{3}$. In the planar
limit $R\to\infty$ we are clearly left with only two length scales, $\beta$
and $l_{\mu}$, and therefore with a single dimensionless parameter given by
their ratio, which can be expressed in terms of bulk parameters as a
combination of $Q$, $M$, and $L$, namely the $\alpha$ parameter of equation
(6). This fact remains true for the magnetic black branes we will study in the
next subsections, which are indeed a one-parameter family of solutions as we
will discuss.

Second, the planar solution still admits a well-defined extremal limit for
which $l_{\mu}/\beta=0$. By setting
$\tilde{f}(\rho)=\tilde{f}^{\prime}(\rho)=0$ after taking the planar limit,
the extremal value of the free parameter can be found to be
$\alpha_{e}=2/(3\sqrt{3})$. This is contrast with the MBB solution which, as
we will see, is not well defined in the strictly extremal limit.

Finally, the Euclidean geometry associated with the planar AdS-RN black hole
admits a different analytic continuation yielding a real Lorentzian geometry,
where one of the planar directions (e.g. $z$) becomes timelike. Notice that
the $\tau=it$ direction is now compact and it pinches off at $r=r_{+}$. This
static cigar geometry is an AdS5 soliton spacetime. AdS5 solitons are
understood to be dual to negative energy ground states of CFTs living on
manifolds with one compact direction Horowitz:1998ha . In this case, the
(3+1)-dimensional dual theory would live on a spatial $\mathbb{R}^{2}\times
S^{1}$ manifold. As we will explain in Section 2.4, similar theories flow in
the IR to (2+1)-dimensional theories on spatial $\mathbb{R}^{2}$ which are
expected to be confining Witten:1998zw . However, in the electrically charged
setup studied in this subsection, this alternative analytic continuation does
not yield a well-defined Lorentzian theory. In fact, it is straightforward to
show that the electric field $F_{tr}$ becomes an imaginary magnetic field in
the bulk theory, and, correspondingly, the chemical potential becomes
imaginary in the dual boundary theory.141414One possible way to make sense of
this theory is to Wick rotate also the charge $Q$, see e.g. Anabalon:2021tua .
Note that in this case the Euclidean saddles associated to the two Lorentzian
spacetimes are different, because the size of the $S^{1}$ at infinity (i.e.
the periodicity $\beta$) changes under analytic continuation of $Q$. The
physical motivation and implications of this procedure are not entirely clear
and we will not explore this possibility in the present paper.

On the other hand, in the magnetically charged black branes studied in the
next subsections, the soliton solution and the corresponding dual confining
theory will be well-defined.

###  2.2 AdS5 magnetic black brane

With the properties of the planar AdS-RN solution in mind, let us now
construct the MBB solution of our interest following DHoker:2009mmn . The
Lorentzian Einstein-Maxwell action in five dimensions is given by

$S=\frac{1}{16\pi G}\int d^{5}x\sqrt{-\bar{g}}(R-2\Lambda-
F_{\mu\nu}F^{\mu\nu})+S_{\partial}$ (8)

where $G$ is the 5D Newton’s constant, $\Lambda=-6/L^{2}$ is the negative
cosmological constant,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\nu}$ is the
electromagnetic field strength (with $A_{\mu}$ the electromagnetic potential),
and $S_{\partial}$ includes boundary terms such as the Gibbons-Hawking-York
term, the necessary electromagnetic boundary terms Hawking:1995ap , and other
counterterms DHoker:2009ixq ; Ammon:2017ded .151515A Chern-Simons term can
also be added, and our action then becomes the action for the bosonic sector
of minimal gauged supergravity in 5D DHoker:2009mmn . We do not include this
term here, which would not give a contribution to our solution. We denoted the
determinant of the 5D metric by $\bar{g}$ to distinguish it from the metric
function $g$ appearing in the ansatz (10). With these conventions, the
Einstein-Maxwell equations are161616These equations need to be supplemented by
the Bianchi identity.

$\displaystyle R_{\mu\nu}-\frac{1}{2}(R-2\Lambda)\bar{g}_{\mu\nu}=2T_{\mu\nu}$
(9) $\displaystyle\nabla_{\mu}F^{\mu\nu}=0$

where
$T_{\mu\nu}=\bar{g}^{\rho\sigma}F_{\mu\rho}F_{\nu\sigma}-\frac{1}{4}\bar{g}_{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}$.
We are interested in finding an asymptotically AdS, planar solution sourced by
a constant magnetic field $B$ tangent to the planar directions, which we label
by $x,y,z$ in analogy with (7). We can then simply take
$F_{xy}=-F_{yx}=B$,171717A simple choice of potential yielding this field
strength is $A=Bxdy$. with all other components of the field strength
vanishing, and plug it into the Einstein-Maxwell equations. Notice that a
magnetic field breaks rotational symmetry in five dimensions by distinguishing
the $x-y$ plane from the perpendicular $z$ direction. The most general ansatz
for the spacetime metric is therefore

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+g(r)(dx^{2}+dy^{2})+h(r)dz^{2}$ (10)

with the AdS asymptotics requiring $f(r)\sim g(r)\sim h(r)\sim r^{2}$ as
$r\to\infty$. The planar directions $x,y,z$ can be taken to be compactified
into a torus $\mathbb{T}^{3}$, but we will be interested in the non-compact
case $\mathbb{R}^{3}$. The absence of rotational symmetry is the reason why an
analytic solution of Einstein’s equations is hard (if possible at all) to
find.

By defining $g(r)\equiv\exp(2V(r))$, $h(r)\equiv\exp(2W(r))$, Einstein’s
equations simplify significantly, taking the form

$\displaystyle
f^{\prime\prime}+f^{\prime}(2V^{\prime}+W^{\prime})=8+\frac{4}{3}B^{2}e^{-4V}$
(11) $\displaystyle
2V^{\prime\prime}+W^{\prime\prime}+2(V^{\prime})^{2}+(W^{\prime})^{2}=0$
$\displaystyle
f(V^{\prime\prime}-W^{\prime\prime})+[f^{\prime}+f(2V^{\prime}+W^{\prime})](V^{\prime}-W^{\prime})=-2B^{2}e^{-4V}$
$\displaystyle
f^{\prime}(2V^{\prime}+W^{\prime})+2fV^{\prime}(V^{\prime}+2W^{\prime})=12-2B^{2}e^{-4V}$

where a prime denotes a derivative with respect to $r$. Notice that,
naturally, $f(r)=g(r)=h(r)=r^{2}$ is a solution for $r\to\infty$. Moreover, an
exact solution of Einstein’s equations is given by the product of the BTZ
black hole and $\mathbb{R}^{2}$, namely $f(r)=3(r^{2}-r_{H}^{2})$,
$g(r)=B/\sqrt{3}$, $h(r)=3r^{2}$ DHoker:2009mmn , where $r_{H}$ is the black
hole horizon radius. This is not asymptotically AdS5, but it suggests that the
solution of our interest, i.e. the MBB, has a horizon and it interpolates
between a near-horizon region given by BTZ$\times\mathbb{R}^{2}$ and the AdS5
asymptotics.181818This suggests that the near-horizon dynamics is captured by
an emergent (1+1)-dimensional CFT in the $t-z$ directions DHoker:2009mmn .
Although we could not construct this solution analytically, it is
straightforward to find numerically.

The boundary conditions to impose to find a numerical solution are (see
Appendix A for a detailed derivation, we set here and in the following $L=1$
and shift the $r$ coordinated such that $r_{H}=1$)

$f(1)=V(1)=W(1)=0,\quad f^{\prime}(1)=\frac{8}{3}(3-b^{2}),\quad
V^{\prime}(1)=\frac{1}{2},\quad W^{\prime}(1)=\frac{6+b^{2}}{4(3-b^{2})}.$
(12)

With these boundary conditions (which are completely general), $g$ and $h$
behave asymptotically as $g(r)\sim v(b)r^{2}$, $h(r)\sim w(b)r^{2}$, with
$v(b)$ and $w(b)$ constants depending only on the value of the magnetic field
parameter $b$. After obtaining our solution, we will therefore need to
numerically find the value of $v(b),w(b)$ and appropriately rescale the $x$,
$y$ coordinates to obtain the correct asymptotically AdS5 solution. The
physical magnetic field is then given by $B=b/v(b)$. We remark that the
difference between our boundary conditions in (12) and those used in
DHoker:2009mmn  is non-physical and only corresponds to a different choice of
coordinates. Our choice is more convenient to numerically study gravity
localization on ETW branes embedded in this background. See Appendix A for
details.

We can now solve the three dynamical equations in the system (11) with the
boundary conditions (12).191919For the sake of numerical implementation, the
boundary conditions (12) must be perturbed slightly away from the black hole
horizon. Similar to the AdS-RN case studied in Section 2.1, this defines a
one-parameter family of solutions parametrized by $b$ in our numerical
solution. The only dimensionless physical parameter is given by

$\alpha_{B}\equiv\frac{\mathcal{T}}{\sqrt{B}}=\frac{2(3-b^{2})\sqrt{v(b)}}{3\pi\sqrt{b}}.$
(13)

Smooth solutions can be found in the range $0\leq|b|<\sqrt{3}$.202020Without
loss of generality, we will focus on $b\geq 0$ in the rest of the paper. The
solutions for $b<0$, which differ only by the direction of the magnetic field,
are identical. For $b=0$, two solutions are simply Poincaré-AdS5 and the
planar limit of AdS-Schwarzschild, with the latter dominating the
gravitational thermodynamic ensemble, because the planar limit is equivalent
to a large temperature limit Witten:1998zw .212121This is readily understood
because the ratio of the size of the thermal $S^{1}$ to the size of the
spatial $S^{3}$ at the boundary vanishes in the planar limit. It is also made
manifest by the divergence of the $\alpha_{B}$ parameter defined in equation
(13). On the other hand, because $\alpha_{B}\to 0$ for $b\to\sqrt{3}$ (see
Figure 3), it is tempting to interpret $b=\sqrt{3}$ as an extremal solution,
similar to the AdS-RN case. However, from (12) we also see that
$W^{\prime}(1)\to\infty$ in the same limit, signaling that no smooth extremal
solution exists and the MBB solution simply ceases to exist in the
$b\to\sqrt{3}$ limit.222222Our choice of coordinates makes this breakdown of
the MBB solution for $b\to\sqrt{3}$ analytically manifest in the blowup of the
boundary condition for $W^{\prime}$. With different coordinate choices the
breakdown is not immediately manifest, but nonetheless numerical solutions
with the correct AdS5 asymptotics can again only be found for $b<\sqrt{3}$
DHoker:2009mmn . Notice also that $v(b)$ and $w(b)$ blow up for $b\to\sqrt{3}$
(with the combination $(3-b^{2})\sqrt{v(b)}\to 0$, guaranteeing $\alpha_{B}\to
0$), which is another sign of the breakdown of the MBB solution. Another
indication of the difference between the $b\to\sqrt{3}$ limit and the extremal
limit of the AdS-RN solution is that the entropy density in the MBB case
vanishes in the $b\to\sqrt{3}$ limit DHoker:2009mmn , whereas the extremal
AdS-RN solution has a finite entropy. It was shown in DHoker:2009mmn  that at
exactly zero-temperature a different solution exists, interpolating between
Poincaré-AdS${}_{3}\times\mathbb{R}^{2}$ deep into the bulk and AdS5
asymptotically. This solution exhibits Poincaré invariance in the $t-z$ plane,
i.e. $f(r)=h(r).$232323The absence of a Poincaré symmetry in the $x,y$
directions is due to the presence of the magnetic field, which breaks
rotational invariance in the planar directions. Similar to the planar AdS-
Schwarzschild case, in this limit there are no dimensionless parameters and
the solution is therefore unique, independent of the value of the magnetic
field. More details of this zero temperature solution and why it does not
contribute to the Euclidean path integral in our braneworld setup are reported
in Appendix C. We refer to DHoker:2009mmn  for details of the numerical
construction of this solution.

Figure 3: Dimensionless parameter $\alpha_{B}$ (defined in equation (13)) as a
function of the magnetic field parameter $b$. $\alpha_{B}$ diverges for $b\to
0$, signaling a large temperature phase dominated by the planar limit of AdS-
Schwarzschild, and it vanishes for $b\to\sqrt{3}$, where the MBB solution
ceases to exist. At exactly zero temperature, the gravitational thermodynamic
ensemble is dominated by a different (unique for all values of $b$) solution
interpolating between AdS${}_{3}\times\mathbb{R}^{2}$ deep into the bulk and
AdS5 asymptotically DHoker:2009mmn .

From now on, we will focus on the range $0<b<\sqrt{3}$ where the MBB solution
is well-defined. As an exemplificatory numerical solution, we report in Figure
4 the metric functions $f(r)$, $g(r)$, $h(r)$ for $b=1$ (with $g$ and $h$
appropriately rescaled by $v(b)$ and $w(b)$, respectively). It is worth
mentioning that our numerical results show $f(r)$ and $h(r)$ becoming
indistinguishable for any value of $r$ in the $b\to\sqrt{3}$ limit. This is in
accordance with the expectation that we should recover Poincaré invariance in
the $t-z$ plane in the zero temperature limit.

Figure 4: Metric functions $f(r)$, $g(r)$, $h(r)$ as a function of the radial
coordinate for $b=1$ (i.e. $\alpha_{B}=0.239494$). In the
BTZ$\times\mathbb{R}^{2}$ near-horizon region (see the inset) the three
functions are distinct, but they become indistinguishable for larger values of
$r$, as expected for an asymptotically AdS5 solution.

The BTZ$\times\mathbb{R}^{2}$ structure of the near-horizon region of the MBB
suggests that the causal structure of the MBB is similar to AdS-Schwarzschild,
namely presenting a single horizon and a spacelike singularity. This
expectations is confirmed by the extension of our numerical solution inside
the black hole horizon (see also Avila:2018sqf ; HosseiniMansoori:2018gdu ).
The maximally extended Penrose diagram for the MBB is then depicted on the
right of Figure 5, where we also depict one of the non-compact directions
transverse to the $r-t$ directions.

###  2.3 Magnetically charged AdS5 soliton

Let us now consider the Euclidean metric associated with the MBB metric (10):

$ds^{2}=f(r)d\tau^{2}+\frac{dr^{2}}{f(r)}+g(r)(dx^{2}+dy^{2})+h(r)dz^{2}.$
(14)

The Euclidean time $\tau$ is periodic with periodicity
$\beta=4\pi/f^{\prime}(1)$. The Euclidean geometry is then given, as usual for
Euclidean black holes, by a cigar geometry in the $\tau-r$ plane with the
$\tau$ circle pinching off at $r=1$, see Figure 5. The transverse directions
$x,y,z$ are all non-compact. This Euclidean geometry is the dominant saddle of
the Euclidean gravitational path integral.242424It is the only known saddle
given our boundary conditions for $\beta<\infty$. If other saddles exist, it
is also expected to be the dominant one because, as we shall see, it gives
rise to the AdS5 soliton Lorentzian solution, which is understood to be dual
to the ground state of the dual CFT Horowitz:1998ha . For $\beta=\infty$ there
exists a different saddle DHoker:2009mmn , but this clearly does not
contribute to the finite $\beta$ thermodynamic ensemble.

Figure 5: Center: Euclidean saddle associated with the MBB and magnetic AdS5
soliton solutions. The black dot indicates $r=1$, where the $\tau$ circle
pinches off. We are suppressing two non-compact directions ($x$ and $y$).
Right: Identifying $\tau$ with the Euclidean time, the semiclassical
gravitational Euclidean path integral prepares the initial state for bulk
fields in the Lorentzian maximally extended MBB solution on the
$\tau=0,\pm\beta/2$ slice represented by the red dotted line, which is left
invariant by the analytic continuation. The causal structure of the MBB is
similar to AdS-Schwarzschild. Left: Identifying $z$ with the Euclidean time,
the initial state for Lorentzian time evolution is prepared on the $z=0$
slice, identified by the yellow dotted line. The corresponding Lorentzian
geometry is the static AdS5 soliton (15). The black dot indicates $r=1$, where
the $\tau$ circle pinches off.

This Euclidean saddle has multiple reflection symmetries, see Figure 5. The
first one is the one around the $\tau=0,\pm\beta/2$ slice (depicted in red).
The others are around any slice identified by a fixed value of one of the non-
compact directions, e.g. $x=0$, or $y=0$, or $z=0$ (depicted in yellow). By
identifying any of these coordinates ($\tau$, $x$, $y$, or $z$) with the
Euclidean time, the semiclassical gravitational Euclidean path integral in the
saddle point approximation can be interpreted as preparing the initial state
for bulk fields on any such reflection symmetric slices. These initial states
can then be evolved in real time yielding real Lorentzian geometries, which
can be easily obtained by analytically continuing the corresponding coordinate
in the Euclidean metric (14).

Choosing $\tau=it$ to be the Euclidean time leads to the maximally extended
Lorentzian MBB studied in the previous subsection, with Penrose diagram given
on the right of Figure 5 and a real magnetic field $F=Bdx\wedge dy$. Choosing
$x=i\chi$ or $y=i\upsilon$ to be the Euclidean time yields a real Lorentzian
geometry, but an imaginary electric field, e.g. $F=iBd\chi\wedge dy$. This is
a problem similar to that seen for the double analytic continuation of the
planar AdS-RN case discussed at the end of Section 2.1, and the resulting
Lorentzian theory is not well-defined. Interestingly, choosing $z=i\zeta$ to
be the Euclidean time yields a real geometry

$ds^{2}=f(r)d\tau^{2}+\frac{dr^{2}}{f(r)}+g(r)(dx^{2}+dy^{2})-h(r)d\zeta^{2}$
(15)

and a real magnetic field $F=Bdx\wedge dy$ (which is left invariant by both
this and the MBB analytic continuations). This is the magnetically charged
AdS5 soliton. The existence of this well-defined double analytic continuation
of the MBB solution is a consequence of the breaking of rotational symmetry
introduced by the magnetic field in five dimensions.

Note that each time slice of the spacetime (15) is given by a cigar geometry
in the $\tau-r$ plane times two non-compact directions (see Figure 6), and the
spacetime is static. This comes as no surprise, because the soliton should be
regarded as a ground state solution of the semiclassical Einstein’s equations,
and representing the ground state of the dual holographic CFT. In fact, the
$z$ direction is non-compact and a Euclidean path integral performed over a
manifold with a non-compact Euclidean time direction prepares the vacuum
state.

Figure 6: Time slice of the magnetically charged AdS5 soliton. Only the $\tau$
and $r$ directions are represented, with the non compact $x$, $y$ directions
suppressed. The $S^{1}$ in the $\tau$ direction pinches off in the bulk at
$r=1$, a sign of the presence of a mass gap and a confined spectrum of massive
particles in the dual theory Witten:1998zw , see Section 2.4.

The asymptotic AdS boundary of the soliton spacetime has spatial topology
$S^{1}\times\mathbb{R}^{2}$, with the $S^{1}$ in the $\tau$ direction. Note
that the $S^{1}$ pinches off in the bulk at $r=1$, where $f(1)=0$. This
feature is associated with a mass gap and IR confinement in the dual theory,
as we will explore in the next subsection Witten:1998zw . AdS soliton
spacetimes similar to (15) have been studied and shown to have negative
energy.252525The zero energy configuration is taken here to be pure AdS with
one direction periodically identified Horowitz:1998ha . There is evidence that
they represent the negative energy ground state of the dual theory with one
compact $S^{1}$ spatial direction. In the dual theory, the negative energy is
interpreted as a Casimir energy arising from the supersymmetry-breaking
antiperiodic boundary conditions for fermions on the $S^{1}$ Horowitz:1998ha .

Before turning to the properties of the holographic dual description of the
MBB and of the soliton, we would like to point out that these two solutions
are associated with the same saddle of the gravitational Euclidean path
integral. Therefore, although the two geometries and the state of quantum
fields on top of them (when they are present) are radically different,
observables in the two setups are expected to be related by double analytic
continuation. In particular, observables in the MBB on the codimension-2 slice
$t=z=0$, which is left invariant by the double analytic continuation, are
equal to the corresponding observables in the soliton spacetime on the
codimension-2 slice $\tau=\zeta=0$. As we have pointed out in Section 1, this
“slicing duality” has interesting implications for the holographic description
of cosmology (see also Antonini:2022blk ; Antonini:2022ptt ; Antonini:2022opp
and Section 5).

###  2.4 The dual description: $\mathcal{N}=4$ SYM and confinement

Now that we have explored in some detail the properties of the bulk spacetimes
of our interest, we can describe their holographic dual description. In
general, the boundary value of a bulk gauge field acts as an external source
turned on for a global $U(1)$ conserved current in the dual theory. In
DHoker:2009mmn  this conserved current $J^{\alpha}$ was understood to be
associated with a $U(1)$ subgroup in the Cartan subalgebra of the $SU(4)$
R-symmetry of $\mathcal{N}=4$ Super-Yang-Mills (SYM):

$S=S_{SYM}+\int d^{4}xJ^{\alpha}A_{\alpha}^{ext},$ (16)

where $A_{\alpha}^{ext}$ is the boundary value of the bulk gauge field.
Evidence for the duality between the MBB solution and $\mathcal{N}=4$ SYM with
such an external magnetic field was provided in DHoker:2009mmn ;
DHoker:2009ixq ; Ammon:2017ded . Notice that the coupling of the external
source to the $U(1)$ subgroup of the R-symmetry partially breaks supersymmetry
DHoker:2016ncv .

In the following we will explore some generic features of the boundary theory
which will apply to any holographic theory dual to our bulk setup, regardless
of whether it takes the exact form described above. For simplicity, we will
refer to it as $\mathcal{N}=4$ SYMB (with the subscript $B$ indicating the
presence of the external magnetic field), but our results are completely
general.

Let us start by noticing that the Euclidean boundary theory lives on a
manifold $S^{1}\times\mathbb{R}^{3}$. The analytic continuation corresponding
to the MBB solution is the one identifying the $S^{1}$ as the Euclidean time
direction. It is then immediate to conclude that the state dual to the MBB and
prepared by the boundary Euclidean path integral is the thermofield double
(TFD) state of two copies of $\mathcal{N}=4$ SYMB on $\mathbb{R}^{3,1}$ (see
Figure 7, right panel):262626With an abuse of notation, we are using here the
familiar discrete sum definition of the TFD, even though $\mathcal{N}=4$ SYMB
on spatial $\mathbb{R}^{3}$ has a continuous energy spectrum and the sum
should be replaced by an integral.

$\ket{TFD}=\frac{1}{\sqrt{Z(\beta\sqrt{B})}}\sum_{n}e^{-\frac{\beta
E_{n}}{2}}\ket{E_{n},B}_{L}\ket{E_{n},B}_{R},$ (17)

where we restricted the sum to a fixed magnetic charge subsector and we
emphasized, as we have pointed out in Sections 2.1 and 2.2, that the finite
temperature partition function only depends on the dimensionless quantity
$\beta\sqrt{B}$, as dictated by conformal symmetry.

Figure 7: Holographic theory dual to our 5D bulk geometries. This could for
example be $\mathcal{N}=4$ SYMB. Center: the Euclidean theory lives on
$S^{1}\times\mathbb{R}^{3}$ (two non-compact directions are suppressed).
Right: the Lorentzian theory dual to the MBB is given by two copies of
$\mathcal{N}=4$ SYMB on $\mathbb{R}^{3,1}$ in the thermofield double state.
Left: the Lorentzian theory dual to the magnetic AdS5 soliton is given by
$\mathcal{N}=4$ SYMB on $S^{1}\times\mathbb{R}^{2,1}$ in the vacuum state.
This theory flows in the IR to a (2+1)-dimensional confining theory on
$\mathbb{R}^{2,1}$.

On the other hand, the analytic continuation corresponding to the magnetic
AdS5 soliton is the one where one of the non-compact directions (the one
breaking rotational symmetry, which we labeled by $z$) is identified with the
Euclidean time. The state dual to the AdS5 soliton is then the negative energy
ground state of $\mathcal{N}=4$ SYMB on $S^{1}\times\mathbb{R}^{2,1}$, see
left panel of Figure 7. The negative energy arises from the antiperiodic
boundary conditions for fermions around the $S^{1}$ Witten:1998zw ;
Horowitz:1998ha . This theory flows in the IR (on length scales much larger
than the size of the $S^{1}$) to a (2+1)-dimensional theory on
$\mathbb{R}^{2,1}$. This lower-dimensional theory is essentially a pure
$SU(N)$ gauge theory272727This is because the long distance dynamics in the
$\mathbb{R}^{2,1}$ directions of $\mathcal{N}=4$ SYM on
$S^{1}\times\mathbb{R}^{2,1}$ is that one of (2+1)-dimensional $SU(N)$ gauge
fields, because all other fields become massive from the (2+1)-dimensional
point of view due to the boundary conditions along the $S^{1}$ Witten:1998zw .
and is expected to display a mass gap and a discrete mass spectrum of
confining particles Witten:1998zw ; VanRaamsdonk:2021qgv .282828On
intermediate length scales much larger than the magnetic field scale
$1/\sqrt{B}$ but smaller than the size of the $S^{1}$, a 2D CFT emerges in the
$\tau-\zeta$ directions DHoker:2009mmn ; Maldacena:2020sxe . On length scales
much larger than the size of the $S^{1}$ this conformal structure is also lost
and the remaining theory on $\mathbb{R}^{2,1}$ is confining.

From the bulk spacetime point of view, the gapped nature (in the
(2+1)-dimensional sense explained above) of the dual theory is manifested in
the $S^{1}$ pinching off in the bulk at $r=1$. In fact, this feature acts as a
radial cutoff at small $r$ in the bulk at a finite proper distance from any
point in the bulk spacetime. By means of the holographic renormalization group
deBoer:1999tgo , we can identify this bulk radial cutoff with a IR cutoff in
the dual boundary theory. This intuitively implies that, in the boundary
theory, physics is trivial at arbitrarily long wavelengths. In other words,
the dual theory is not a CFT in the IR, but rather a confining theory with a
mass gap.292929This is different from the pure AdS spacetime with spatially
$S^{3}$ boundary, where the center of AdS $r=0$ is also a finite proper
distance away from any bulk point. In that case, the IR cutoff is due to the
compact nature of the $S^{3}$. In our case, the dual theory lives on a non-
compact manifold, and the IR cutoff cannot be interpreted in terms of a
geometric feature of the boundary manifold. Our case should be compared
instead to the Poincaré-AdS case, in which the boundary is spatially
$\mathbb{R}^{3}$ and $r=0$ is infinitely far away in proper distance from any
bulk point, corresponding to no IR cutoff in the dual boundary theory. We can
make this argument precise using a bulk calculation similar to the one carried
out in Witten:1998zw ; Csaki:1998qr . The goal is to show that bulk fields on
the background (15) give rise to a mass gap and a discrete tower of massive
particles (glueballs Csaki:1998qr ) in the dual IR theory on
$\mathbb{R}^{2,1}$.303030Another way to investigate confinement is by
computing the expectation values of spatial Wilson loops and check whether
they satisfy an area law Witten:1998zw . We will not pursue this direction in
the present paper.

####  2.4.1 Mass gap and confinement in the dual theory

Let us focus our attention on a bulk scalar field $\phi$ with mass $\mu$
satisfying the Klein-Gordon equation
$(\nabla^{\mu}\nabla_{\mu}-\mu^{2})\phi=0$, which can be conveniently
rewritten as

$\left[\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\right)-\mu^{2}\right]\phi=0.$
(18)

Exploiting the translational symmetry in the $\zeta$, $x$, $y$, $\tau$
directions, we can decompose the scalar field as

$\phi=\psi(r)e^{-i\omega\zeta+ik_{x}x+ik_{y}y+ik_{\tau}\tau},$ (19)

where $k_{\tau}=2\pi n/\beta$ due to the periodicity
$\beta=4\pi/f^{\prime}(1)$ of the $\tau$ coordinate. Using the metric (15),
the Klein-Gordon equation (18) takes the form

$\mathcal{D}_{r}\psi(r)=\left(\frac{k_{\tau}^{2}}{f(r)}+\frac{k_{x}^{2}+k_{y}^{2}}{g(r)}-\frac{\omega^{2}}{h(r)}\right)\psi(r),$
(20)

where the differential operator $\mathcal{D}_{r}$ is given by313131Notice that
by defining $\bar{\psi}(r)=\sqrt{f(r)g(r)\sqrt{h(r)}}\psi(r)$ we could
eliminate the linear derivative term in equation (21) at the expense of
introducing a complicated potential term $V(r)$. However, we find that keeping
the differential operator in the form (21) simplifies our numerical analysis.
323232For the numerical analysis it is convenient to rewrite the operator (21)
in terms of the $V,W$ functions appearing in the Einstein’s equations (11).

$\mathcal{D}_{r}=f(r)\partial_{r}^{2}+\left[f^{\prime}(r)+\frac{f(r)g^{\prime}(r)}{g(r)}+\frac{f(r)h^{\prime}(r)}{2h(r)}\right]\partial_{r}-\mu^{2}.$
(21)

When solving the Klein-Gordon equation (18) we must impose smoothness at the
tip of the cigar, i.e. $\psi^{\prime}(1)=0$, as well as require a normalizable
solution at $r=\infty$. Asymptotically, where $f(r)\sim g(r)\sim h(r)\sim
r^{2}$, there are two linearly independent solutions, one behaving as
$r^{\mu}$ and the other one behaving as $r^{-4-\mu}$. We must therefore pick
the latter (for which clearly $\psi(\infty)=\psi^{\prime}(\infty)=0$). For any
given value of $k_{\tau},k_{x},k_{y}$ this set of boundary conditions can only
be satisfied by a discrete set of solutions associated to a discrete set of
values of $\omega$. In particular, let us label the $\omega$ eigenvalues
associated with $k_{\tau}=k_{x}=k_{y}=0$ by $\omega=M_{n}$. The first five
$M_{n}$ eigenvalues found numerically for $b=1$ and a scalar field of mass
$\mu=1$ are

$M_{n}=\\{5.10,8.81,12.39,15.92,19.44\\}.$ (22)

Notice that the eigenvalues are strictly positive, $M_{n}>0$.

Let us now think of the bulk fields as (2+1)-dimensional fields living on the
spatial plane identified by $x,y$. The (2+1)-dimensional dispersion relation
arising from our analysis above of the scalar field takes the form

$\omega_{bulk}=\sqrt{M_{n}^{2}+F(k_{\tau})+G(k_{x}^{2}+k_{y}^{2})}$ (23)

with $F(0)=G(0)=0$. Notice that $F$ and $G$333333$G$ only depends on the
combination $k_{x}^{2}+k_{y}^{2}$ because of rotational symmetry in the $x-y$
plane. are complicated functions in general, and in particular
$F(k_{\tau})\neq k_{\tau}^{2}$ and $G(k_{x}^{2}+k_{y}^{2})\neq
k_{x}^{2}+k_{y}^{2}$ because the $\zeta-\tau$ and $\zeta-x-y$ Poincaré
invariance is broken in the bulk, as it is clear from the soliton metric
(15).343434We thank Gary Horowitz for a useful discussion on this point. In
this (2+1)-dimensional sense, a scalar field on our AdS5 soliton spacetime
gives therefore rise to discrete towers of massive particles with mass
$\sqrt{M_{n}^{2}+F(k_{\tau})}$.

As we approach the asymptotic boundary $r\to\infty$, Poincaré symmetry is
restored because $f(r)\sim g(r)\sim h(r)\sim r^{2}$. Therefore, in the
boundary theory the dispersion relation reduces to

$\omega_{\partial}=\sqrt{M_{n}^{2}+k_{\tau}^{2}+k_{x}^{2}+k_{y}^{2}}.$ (24)

In terms of the IR boundary theory on $\mathbb{R}^{2,1}$, this is associated
to towers of massive particles with mass
$m_{n}=\sqrt{M_{n}^{2}+k_{\tau}^{2}}$. The strict positivity of the $M_{n}$
eigenvalues (see equation (22)) guarantees the existence of a mass gap in the
dual IR theory on $\mathbb{R}^{2,1}$.

We emphasize that this spectrum of massive particles is not simply the
discrete spectrum one would obtain by Kaluza-Klein (KK) reduction of a theory
on $S^{1}\times\mathbb{R}^{2,1}$ to a theory on $\mathbb{R}^{2,1}$. In that
case, the mass spectrum would simply be given by $m_{n}^{KK}=k_{\tau}=2\pi
n/\beta$, and the KK zero mode implies there would be no mass gap. Our mass
spectrum differs from the KK one due to the presence of the $M_{n}$
eigenvalues, which arise from the presence of the radial cutoff in the bulk,
i.e. the IR cutoff in the boundary theory. The infinite, discrete tower of
$M_{n}$’s should be therefore interpreted as a confining spectrum, i.e. as the
masses of glueballs Csaki:1998qr . Notice that for each $M_{n}$ a KK tower of
massive particles exists associated with different discrete values of
$k_{\tau}$. The dispersion relation (24) therefore really describes a tower of
towers of massive particles, with one tower associated to confinement and the
other one to KK reduction.

Figure 8: Mass gap $M_{0}$ as a function of the magnetic field parameter $b$
for a scalar field of mass $\mu=1$. As $b\to\sqrt{3}$ the gap closes,
signaling deconfinement. The gap between the different $M_{n}$ eigenvalues
also vanishes in this limit.

Finally, we would like to remark that the mass gap reduces to zero and the
$M_{n}$ eigenvalues become continuous in the limit $b\to\sqrt{3}$, see Figure
8.353535We remind that this limit cannot be taken strictly, as we have
explained in Section 2.2. This result has a simple interpretation both in the
bulk and in the boundary. From the bulk point of view, the radial cutoff at
$r=1$ (i.e. the tip of the cigar) becomes infinitely far away in proper radial
distance from any bulk point. This is because $f^{\prime}(1)\to 0$ in this
limit, implying that $f(r)$ develops a double pole at $r=1$ which causes the
proper distance $l_{r}\propto\int_{1}^{r^{*}}dr/\sqrt{f(r)}$ to diverge
logarithmically. Therefore the radial cutoff responsible for the confining
spectrum is removed, the gap closes $M_{0}\to 0$, and the spacing between
different $M_{n}$’s also vanishes. Analogously, from the boundary theory
perspective, the corresponding IR cutoff determining the confining scale is
taken to zero. In fact, as we have explained in Section 2.2, the
$b\to\sqrt{3}$ limit corresponds to the $\beta\to\infty$ limit (in the MBB
picture this is the zero temperature limit). In this limit, the dual theory
becomes $\mathcal{N}=4$ SYMB on $\mathbb{R}^{3,1}$ in its zero energy vacuum
state, which does not flow in the IR to a theory on $\mathbb{R}^{2,1}$ and is
known to not be confining, see e.g. Witten:1998zw .

Now that we have analyzed in full detail the MBB solution, the magnetically
charged AdS5 soliton, and the properties of the holographic dual theories, we
are ready to consider ETW branes embedded in these spacetimes, which will
allow us to study braneworld cosmologies, braneworld traversable wormholes,
and their holographic descriptions.

##  3 Euclidean braneworld analysis

In the previous section, we constructed the asymptotically AdS5 MBB background
spacetime and the associated Euclidean saddle (14), whose metric we report
here:

$ds^{2}=f(r)d\tau^{2}+\frac{1}{f(r)}dr^{2}+g(r)(dx^{2}+dy^{2})+h(r)dz^{2},$
(25)

where $\tau$ is periodically identified with period $\beta$. We will take the
$\tau$ coordinate in the range $\tau\in[-\beta/2,\beta/2]$. As we explained in
the previous section, this Euclidean solution is dual to $\mathcal{N}=4$ SYMB
with an external magnetic field on $\mathbb{R}^{3}\times S^{1}$.

Following Cooper:2018cmb ; Antonini:2019qkt , let us now place a dynamical
codimension-1 End-of-the-World (ETW) brane with a constant tension $T$ to
excise a portion of the asymptotic region of this Euclidean spacetime. Neumann
boundary conditions for bulk fields, including the metric,363636Technically,
the boundary condition for the metric is Robin, see Footnote 39. are imposed
at the brane, allowing it to dynamically evolve in the bulk. We are interested
in a brane whose trajectory lies in the $\tau-r$ directions and cuts off a
portion of the asymptotic boundary identified by an interval in the $\tau$
direction, see Figure 9. The brane extends infinitely in the $x,y,z$
coordinates.

Figure 9: Brane trajectory in the $\tau-r$ direction. The region shaded in
purple is cut off by the ETW brane. The transverse $x$, $y$, and $z$
directions are suppressed.

In the Euclidean picture, the brane evolves from the asymptotic boundary
$r\to\infty$ to a minimum radius $r_{0}$, then expands back to $r\to\infty$.
We are interested in brane trajectories preserving the reflection symmetry of
the Euclidean saddle around the $\tau=0,\pm\beta/2$ slice.373737By
construction the reflection symmetry around $z=0$ is also preserved. We will
therefore take the minimum radius $r_{0}$ of the trajectory to occur for
$\tau=\pm\beta/2$, and study symmetric trajectories around this point. We
label by $\tau_{\infty}$ the amount of $\tau$ coordinate employed by the brane
to reach the asymptotic boundary from $r_{0}$. The trajectory therefore
excises a portion of the asymptotic boundary of size $2\tau_{\infty}$, leaving
us with a strip-shaped boundary of width

$l\equiv 2\tau_{0}=\beta-2\tau_{\infty},$ (26)

where we defined the boundary width parameter $\tau_{0}$.383838In
Cooper:2018cmb ; Antonini:2019qkt  $\tau_{0}$ was labeled “preparation time”,
because $\tau$ was identified with the Euclidean time and $\tau_{0}$
corresponds to the amount of Euclidean time evolution applied to a Cardy state
$\ket{B}$ to prepare the state dual to the double-sided black hole (or MBB in
our case) with the second asymptotic region cut off by the ETW brane, see
Section 4.3. Here we avoid this terminology, because we will also be
interested in a different analytic continuation in which $z$ is identified
with the Euclidean time and $2\tau_{0}$ is simply the width of the boundary
strip. See Figure 9. Clearly, the only dimensionless parameter in the boundary
theory is now $\tau_{0}\sqrt{B}$, where the boundary width parameter
$\tau_{0}$ took the place of $\beta$, the length of the $S^{1}$ at the
boundary.

Notice that the AdS boundary is now a manifold with a boundary and it is
topologically a strip $\mathbb{R}^{3}\times I$, where $I=[-\tau_{0},\tau_{0}]$
is an interval in the $\tau$ direction. The boundary theory dual to this bulk
setup is therefore a Euclidean BCFT Cardy:1989ir ; Affleck:1991tk ;
Cardy:2004hm , as customary in the context of the AdS/BCFT correspondence
Takayanagi:2011zk ; Fujita:2011fp . In particular, following DHoker:2009ixq
and Section 2.4, the Euclidean dual theory is given by $\mathcal{N}=4$ SYMB on
$\mathbb{R}^{3}\times I$, with appropriate conformal boundary conditions
imposed at the boundaries of the interval. The choice of conformal boundary
conditions determines the tension of the brane in the bulk setup
Takayanagi:2011zk ; Fujita:2011fp . In order for this dual theory to exist,
the brane must not excise all of the asymptotic boundary, i.e. it must not
overlap with itself. In other words, the boundary width parameter $\tau_{0}$
must be positive.

On the other hand, since our goal is to describe braneworld cosmologies and
braneworld wormholes, namely effective (3+1)-dimensional theories localized on
the ETW brane, it is crucial that an effective (3+1)-dimensional description
of gravity localized on the brane exists. This can be achieved in our setup
via the Karch-Randall-Sundrum gravity localization mechanismRandall:1999ee ;
Randall:1999vf ; Karch:2000ct . Gravity localization will hold only locally
and for portions of the brane trajectory lying deep into the asymptotic AdS
region Antonini:2019qkt ; Clarkson:2005mg . Since the minimum radius $r_{0}$
becomes a maximum radius in the braneworld cosmology setting studied in
Section 4, effective 4D gravity for part of the evolution of the braneworld
cosmology is possible only if $r_{0}$ is large enough to be deep into the
asymptotic AdS region (i.e. the metric (25) approximately reduces to Euclidean
AdS5 for $r\sim r_{0}$).

From this discussion we understand that two conditions are necessary for our
setup to realize braneworld cosmologies and braneworld wormholes with well-
defined holographic dual descriptions:

  1. 1.

$\tau_{0}>0$ to ensure that a well-defined dual description exists;

  2. 2.

$r_{0}$ lies deep into the AdS region to guarantee Karch-Randall gravity
localization works for at least a portion of the evolution of the braneworld.

It was shown in Cooper:2018cmb  that satisfying these two conditions
simultaneously in a Euclidean AdS-Schwarzschild setup is impossible.
Contrarily, the AdS-RN setup of Antonini:2019qkt  achieved this goal by
considering a near-extremal black hole and a near-critical brane tension $T\to
1/L$. The drawback is that there is no well-defined solitonic solution
associated with the AdS-RN Euclidean saddle as we discussed in Section 2.1,
and so no braneworld wormhole can be defined.

In this section we will study the embedding of an ETW brane in the Euclidean
background (25) and show that, similar to the AdS-RN setup, a regime of
parameters exist (namely, $b\lesssim\sqrt{3}$, $T\lesssim 1/L$) for which both
conditions can be satisfied simultaneously. In this case, as we have
discussed, a solitonic solution also exists, which will allow us to build a
braneworld traversable wormhole in Section 5. In particular, in Sections 3.1
and 3.2 we analyze the brane trajectory and numerically obtain $r_{0}$ and
$\tau_{0}$ across different values of the magnetic field parameter $b$ and
tension $T$. In Section 3.3, we study at what values of the radial coordinate
$r$ the metric (25) approximately takes the form of a pure AdS metric (i.e. at
what values of $r$ we are deep within the AdS asymptotic region) for various
values of $b$. We then show that a range of parameters $b,T$ exists such that
$r_{0}$ is well within such region while keeping $\tau_{0}>0$. In Section 3.4
we discuss the 4D effective braneworld description of the Euclidean setup,
which is given by a Maldacena-Maoz-like wormhole with the two AdS4 asymptotic
boundaries coupled by non-gravitational auxiliary degrees of freedom.

We remark that, by analogy with Cooper:2018cmb , one could expect the
existence of an alternative saddle built by cutting off the zero temperature
solution of DHoker:2009mmn  with two disconnected ETW branes. As we show in
Appendix C, this alternative phase does not exist in our setup, and the
Euclidean saddle studied in the present section is the only known saddle
contributing to the Euclidean path integral. It therefore dominates the
gravitational ensemble.

###  3.1 Brane Trajectory

The Euclidean ETW brane action is given by

$S_{ETW}=-\frac{1}{8\pi G}\int_{ETW}d^{4}x\sqrt{h}(K-3T)+S_{ETW}^{em}$ (27)

where $h$ is the determinant of the induced metric on the brane, $K$ is the
trace of the extrinsic curvature, and $S_{ETW}^{em}$ is a necessary
electromagnetic term needed for the well-posedness of the variational problem
Hawking:1995ap ; Antonini:2019qkt , which will not affect the brane’s
equations of motion.

We can describe the motion of the brane by varying the full action of the bulk
(equation (8)) and brane and imposing Robin boundary conditions, 393939In most
of the literature (e.g. Takayanagi:2011zk ; Fujita:2011fp ; Cooper:2018cmb ;
Antonini:2019qkt ), it is common to regard this boundary condition as
“Neumann” rather than “Robin”. However, in principle, the boundary condition
imposed is between the induced metric and its derivative along the brane.
Thus, it is more precise to characterize this condition as the “Robin”
boundary condition which we adopt in the rest of the work.

$K_{ab}-Kh_{ab}=-3Th_{ab}$ (28)

where $K_{ab}$ is the extrinsic curvature and $K$ its trace. The tension can
take values $-1<T<1$ for a $\Lambda<0$ brane and we will focus on the positive
case $T>0$ (we remind that $L=1$). We relegate the explicit derivation of the
brane equation to Appendix B. Parametrizing the brane trajectory by $\tau$,
the final result we obtain in terms of the $f,U,V$ metric functions defined in
Section 2 is:

$\frac{dr}{d\tau}=\pm\frac{f\sqrt{4fV^{\prime}W^{\prime}+4fV^{\prime
2}+fW^{\prime 2}-9T^{2}}}{3T}$ (29)

where a prime denotes a derivative with respect to $r$. The $+$ and $-$ sign
denotes the contracting (top half of Figure 9) and expanding (bottom half of
Figure 9) phase, respectively.

We define the minimum radius $r_{0}$ (reached at the reflection-symmetric
slice $\tau=\pm\beta/2$) of the brane’s evolution as the largest $r$ such that
$dr/d\tau=0$ in equation (29) is satisfied.

More explicitly,

$r_{0}=\max{\\{r|4f(r)V^{\prime}(r)W^{\prime}(r)+4f(r)V^{\prime}(r)^{2}+f(r)W^{\prime}(r)^{2}-9T^{2}=0\\}}.$
(30)

Figure 10: The minimum radius $r_{0}$ of the ETW brane as a function of the
brane tension $T$ across different values of the magnetic field parameter $b$.
For any fixed value of $b$, $r_{0}$ diverges for $T\to T_{crit}=1$.

We numerically evaluate the minimum radius $r_{0}$ across different values of
the magnetic field parameter $b$ and tension $T$ and plot the results in
Figure 10. As expected, $r_{0}$ increases as $T$ increases for any value of
$b$, and for $T=0$ we have $r_{0}=r_{H}=1$, i.e. the brane passes through the
tip of the cigar and we are left with exactly half of the Euclidean geometry.
In principle, as $T$ is taken to criticality $T_{\text{crit}}=1$, $r_{0}$
approaches infinity for any fixed value of $b$, which corresponds to the brane
approaching the asymptotic boundary.

Ultimately, the numerical evaluation of our Euclidean MBB scenario shows that,
given high numerical precision, we can attain a larger value of $r_{0}$ by
simply tuning the tension closer to criticality. This will play an important
role when considering the range of parameters that allow gravity localization
to be satisfied in the Euclidean, cosmology, and wormhole pictures.

###  3.2 Boundary width parameter

An essential property we need to consider for a well-defined state is the
positivity of the boundary width parameter $\tau_{0}$, defined in equation
(26). Only if $\tau_{0}>0$ the brane does not overlap and a portion of the
asymptotic boundary where to define the dual BCFT is retained. This condition
generically competes with the large tension condition seen above needed to
support gravity localization, as demonstrated in both the AdS-Schwarzschild
and the AdS-RN scenarios Cooper:2018cmb ; Antonini:2019qkt . More
specifically, a tension too close to criticality $T_{\text{crit}}=1$ leads to
a vanishing and eventually negative boundary width parameter, signaling an
ill-defined solution with an overlapping brane cutting off the entire
asymptotic boundary.

$\tau_{0}$ can be computed in our bulk solution by numerically evaluating the
interval $2\tau_{\infty}$ in $\tau$ coordinate taken by the brane to undergo
its trajectory. This can then be subtracted from the $\tau$ periodicity
$\beta=4\pi/f^{\prime}(r)|_{r=r_{H}}=3\pi/(2(3-b^{2}))$ to find twice the
boundary width parameter. Thus, $\tau_{0}$ has the explicit form

$\tau_{0}=\frac{3\pi}{4(3-b^{2})}-\int_{r_{0}}^{\infty}dr\frac{3T}{f\sqrt{4fV^{\prime}W^{\prime}+4fV^{\prime
2}+fW^{\prime 2}-9T^{2}}}.$ (31)

(a)

(b)

Figure 11: (a) The boundary width parameter $\tau_{0}$ as a function of the
brane tension $T$ across different magnetic fields $b$. For any fixed $b$,
$\tau_{0}$ becomes negative if $T$ is tuned too close to criticality. The
inset shows how the value of $T$ for which $\tau_{0}$ becomes negative
increases as $b$ is tuned closer to $\sqrt{3}$. (b) $\tau_{0}$ as a function
of the magnetic field $b$ evaluated across different brane tensions $T$. For
any fixed $T$, $\tau_{0}$ can be made positive by tuning $b$ sufficiently
close to $\sqrt{3}$.

We numerically evaluate $\tau_{0}$ and plot the results in Figure 11. There
are several features to notice. As the tension approaches criticality $T\to
T_{\text{crit}}$, $\tau_{0}$ vanishes and eventually becomes negative for any
value of $b$. However, for the purposes of constructing a valid solution that
provides a large enough $r_{0}$ such that gravity localization is possible, we
only need to take $T$ close to criticality but strictly smaller than
$T_{\text{crit}}$. The second feature to notice is that taking $b\to\sqrt{3}$
while keeping $T$ fixed increases $\tau_{0}$. In particular, $\tau_{0}$ can
always be made positive for any value of $T$ by tuning $b$ close enough to
$\sqrt{3}$. Intuitively, this is because taking $b\to\sqrt{3}$ corresponds to
taking $\beta\to\infty$ (see Section 2), whereas $\tau_{\infty}$ remains
finite in this limit for fixed $T$. This suggests that to allow for solutions
which can support gravity localization we should tune $b$ close to $\sqrt{3}$
to provide a larger range of valid brane tensions with $\tau_{0}>0$. This
behavior is completely analogous to the AdS-RN case studied in
Antonini:2019qkt .

We remark that within our family of solutions parameterized by $b$ and $T$,
there exist values of the parameters that allow for a positive boundary width
parameter and locally localized gravity, as we will see in the next
subsection.

###  3.3 Gravity Localization

The second property we need to examine is whether observers living on the ETW
brane see 4-dimensional gravity. Gravity localization on a brane was initially
explored by Karch:2000ct ; Randall:1999ee ; Randall:1999vf . To review,
Randall and Sundrum Randall:1999ee ; Randall:1999vf  demonstrated that a 4D
Minkowski brane embedded in a warped AdS5 geometry supports 4-dimensional
gravity: a stable 4D massless graviton exists on the brane. A massive tower of
Kaluza-Klein (KK) modes is also present, and the 4D graviton is the zero mode.
Karch and Randall Karch:2000ct  extended this result to AdS4 branes embedded
in AdS5, showing that in this case the 4D graviton acquires a small mass,
leading to localized gravity on shorter length scales. In the case of more
general asymptotically AdS5 spacetimes, local gravity localization can still
be achieved as long as the brane sits in a region which is nearly pure AdS5.
For instance, for a black hole background the brane needs to sit far from the
black hole horizon and deep into the AdS asymptotic region Clarkson:2005mg ;
Antonini:2019qkt . In this case the 4D graviton is also massive and it has a
finite lifetime, i.e. it is a quasi-bound mode. However, if we only consider
relatively short spatial and time scales, gravity is effectively localized on
the brane and the graviton is nearly massless. Thus, 4D observers living on
the brane would experience 4D gravity.

Therefore, the question we need to answer is: in the Lorentzian braneworld
cosmology and braneworld wormhole pictures studied in Sections 4 and 5, can
the brane sit, at least for part of its trajectory, in the asymptotic AdS
region of the respective Lorentzian spacetimes (the MBB and the soliton)? The
answer to this question resides in the value of $r_{0}$ studied in Section
3.1. If $r_{0}$ is large enough to be in a region where the spacetime looks
nearly AdS5, then the Lorentzian brane trajectories will support local gravity
localization. In the braneworld cosmology picture, as we will see, $r_{0}$
becomes a maximum radius and the brane collapses into the MBB horizon.
Therefore, gravity localization is achieved only for part of the trajectory
around the time-symmetric slice, when the brane is far away from the horizon.
On the other hand, $r_{0}$ remains a minimum radius in the braneworld wormhole
picture, implying that gravity localization will hold for the whole trajectory
of the brane.

As we have seen, $r_{0}$ can be made arbitrarily large by tuning the tension
close to criticality.404040This is in line with the idea that a 4D effective
description of physics on the brane exists whenever the number of DOF
associated with the boundary of the microscopic BCFT is large compared to the
number of DOF associated with the bulk of the BCFT Karch:2000ct ;
VanRaamsdonk:2021qgv ; Antonini:2022blk , and therefore the boundary entropy
is large. The difficulty resides in also retaining a positive value of
$\tau_{0}$, warranty of a well-defined solution and dual description. In the
AdS-Schwarzschild setup of Cooper:2018cmb , even in the planar limit, valid
solutions that allow for gravity localization do not exist because, increasing
$T$, $\tau_{0}$ becomes negative well before $r_{0}$ reaches the asymptotic
AdS region. In the AdS-RN setup, Antonini:2019qkt  explicitly demonstrated
gravity localization is possible for values of the electric charge parameter
sufficiently close to extremality. Because, as we have seen in the previous
subsection, $\tau_{0}$ can be made positive for any value of $T$ by tuning
$b\to\sqrt{3}$, the same result should be attainable in our magnetically
charged setup.

To verify whether our setup supports valid ($\tau_{0}>0$) solutions with
gravity localization, we will now check for which value $r=\bar{r}$ of the
radial coordinate the metric (25) is approximately pure AdS5 given a certain
value of the parameter $b$, and then make sure that, for the same value of
$b$, it is possible to choose a value of $T$ such that $r_{0}>\bar{r}$ and
$\tau_{0}>0$.414141Note that we cannot simply find the asymptotic AdS region
by numerically checking the range of $r$ for which $f(r)\approx g(r)\approx
h(r)\approx r^{2}$. This is because we are always allowed (and in fact made
use of in Section 2) a shift in the radial coordinate $r\to r+c$, which
implies $f(r)\approx(r+c)^{2}$ might hold in the asymptotic AdS region.
Although eventually $f(r)\approx r^{2}$ for large enough $r$, for large values
of $c$ we could be deep into the AdS asymptotic region with $f(r)\neq r^{2}$.

Beginning with the metric (25), we perform a coordinate transformation such
that $\rho^{2}=f(r)$, which recasts the metric into the form

$ds^{2}=\rho^{2}d\tau^{2}+\frac{4d\rho^{2}}{(f^{\prime}(r)|_{r=r(\rho)})^{2}}+g(r(\rho))(dx^{2}+dy^{2})+h(r(\rho))dz^{2}.$
(32)

As we have seen in Section 2, $f(r)\approx g(r)\approx h(r)$ for large enough
$r>\bar{r}$. Assuming we are in this range of values of $r$, the metric
reduces to

$ds^{2}\approx\rho^{2}(d\tau^{2}+dx^{2}+dy^{2}+dz^{2})+\frac{4d\rho^{2}}{(f^{\prime}(r)|_{r=r(\rho)})^{2}}$
(33)

Finally, if $f^{\prime}(r)|_{r=r(\rho)}=2\rho$, then the metric takes the form
of Euclidean AdS5:

$ds^{2}\approx\rho^{2}(d\tau^{2}+dx^{2}+dy^{2}+dz^{2})+\frac{d\rho^{2}}{\rho^{2}}.$
(34)

The range of $r$ associated with the asymptotic AdS region is therefore
$r>\bar{r}$, where $\bar{r}$ is the value of $r$ for which (1) $f(r)\approx
g(r)\approx h(r)$ starts to be satisfied and (2)
$f^{\prime}(r)|_{r=r(\rho)}\approx 2\rho$ starts to hold. The numerical plots
in Figure 12 show that, for an appropriate choice of $b$ and $T$ (we chose
here $b=\sqrt{3}-0.001$ and $T=0.9999$), these conditions are satisfied for
values of $r$ considerably smaller than $r_{0}$ while $\tau_{0}>0$ holds.
Therefore, at least part of the brane trajectory sits in the asymptotic AdS
region in all Lorentzian pictures. The results of Antonini:2019qkt  then imply
that local gravity localization can be achieved. We leave a more thorough
study of gravity localization and the 4D quasi-bound mode in our setup to
future work.

Figure 12: Numerical evaluation of the two conditions defining the asymptotic
AdS region for $b=\sqrt{3}-0.001$ and $T=0.9999$. The (red) dashed line
denotes $\rho_{0}=\rho(r_{0})$, namely the value of $\rho$ corresponding to
the minimum radius of the brane. Notice that for these values of the
parameters, $\tau_{0}>0$ also holds. The (black) dashed line denotes
$\bar{\rho}$ such that for all $\rho<\bar{\rho}$ at least one of the
conditions for gravity localization is violated by more than $1\%$. The brane
minimum radius is clearly deep into the asymptotic AdS region, implying that
our solutions support gravity localization.

###  3.4 Maldacena-Maoz wormhole on the brane

Before moving on to the study of Lorentzian continuations of our Euclidean
setup, we would like to emphasize the connection between our Euclidean
braneworld solution and the setup discussed in VanRaamsdonk:2021qgv ;
Antonini:2022blk ; Antonini:2022ptt  and reviewed in Section 1. In particular,
the gravity localization analysis of the previous subsection suggests that an
effective 4D bulk description on the Euclidean brane should exist, which does
not rely on the 5D Euclidean geometry. We can therefore “remove one layer of
holography”. This should be understood in complete analogy with the
relationship between singly and doubly holographic models of black hole
evaporation Almheiri:2019hni .

The metric induced on the brane takes the form

$ds^{2}=\left(f(\tau)+\frac{(r^{\prime}(\tau))^{2}}{f(\tau)}\right)d\tau^{2}+g(\tau)(dx^{2}+dy^{2})+h(\tau)dz^{2}$
(35)

where we defined $f(\tau)\equiv f(r(\tau))$ (and similarly for $g$ and $h$),
with $r(\tau)$ describing the brane trajectory. In the asymptotically AdS
region, where the brane sits in the solutions supporting gravity localization,
this reduces to the nearly isotropic form424242In regions where gravity
localization is possible, the $(r^{\prime}(\tau))^{2}/f(\tau)$ can be
neglected. We verified this numerically for the examples we displayed.
Moreover, it can be seen analytically by examining equation (29).

$ds^{2}\approx f(\tau)(d\tau^{2}+dx^{2}+dy^{2}+dz^{2}).$ (36)

$f(\tau)$ has a minimum at $\tau=0$ and two double poles at
$\tau=\pm\tau_{\infty}$,434343It is easy to show that $f(\tau)$ has double
poles by solving equation (29) in the asymptotic region where $f(r)\approx
g(r)\approx h(r)\approx r^{2}$, see Section 5.1. where the brane meets the
asymptotic boundary, see Figure 13. The geometry of the brane is therefore
that of a Euclidean wormhole connecting two distinct asymptotically AdS4
boundaries, corresponding to the two 3D boundaries of the dual 4D BCFT. Note
that in the convention used so far, the turnaround point of the brane was at
$\tau=\pm\beta/2$, but for the braneworld wormhole description we choose to
set $\tau=0$ there for simplicity of notation.

Figure 13: Wormhole conformal factor $f(\tau)$ for the (Euclidean and
Lorentzian) wormhole solution given $b=\sqrt{3}-0.001$ and $T=0.9999$.
$f(\tau)$ has double poles at the two asymptotic AdS4 boundaries at
$\tau=\pm\tau_{\infty}$. In the doubly holographic setup, this is where the
ETW brane meets the asymptotic AdS5 boundary.

This wormhole is similar to those studied by Maldacena and Maoz
Maldacena:2004rf , with the metric taking the same form (2). Notice that in
our setup the two boundaries connected by the wormhole are also coupled by
non-gravitational auxiliary degrees of freedom living on $\mathbb{R}^{3}\times
I$, see center line, center panel of Figure 1. These can be understood to
account for the coupling of the brane to the 5D bulk physics, and are a low-
energy effective version of the bulk444444By bulk here we mean the region of
the microscopic 4D BCFT away from its boundaries. degrees of freedom of the
microscopic 4D holographic BCFT.454545These auxiliary non-gravitational
degrees of freedom are the analog of the bath degrees of freedom in the singly
holographic description of the black hole evaporation setting Almheiri:2019hni
. This is in fact exactly the setup considered in VanRaamsdonk:2021qgv ;
Antonini:2022blk ; Antonini:2022ptt , of which our higher dimensional
braneworld setup can be interpreted as a doubly holographic realization.

##  4 Lorentzian Braneworld Cosmology

In the previous section, we embedded an ETW brane in the Euclidean geometry
that excises a portion of the asymptotic boundary. As we have discussed, our
Euclidean solution including the ETW brane possesses two reflection
symmetries, and therefore admits two well-defined analytic continuations. In
this section, we consider the MBB analytic continuation, i.e. we identify
$\tau$ with the Euclidean time direction and obtain the metric (10). This
leads to a double-sided MBB bulk spacetime with an ETW brane cutting off the
second asymptotic boundary, see Figure 14. This is therefore a one-sided black
brane microstate, with the dual holographic theory living on the only
asymptotic AdS boundary (the left one in Figure 14).

From the point of view of an observer comoving with the brane, the expansion
and contraction of the brane looks like the evolution of a Big Bang-Big Crunch
cosmology. As we will see, this is a special case of a Bianchi type I universe
Taub:1950ez ; Ellis:1968vb ; Montani:2009hju . As we have examined in Section
3.3, there exist values of $b$ and $T$ for which gravity is locally localized
on the brane for part of its trajectory. In particular, gravity is effectively
4D on the brane when the brane is deep into the AdS5 asymptotic region, but
not when it sits near or inside the MBB horizon. Therefore, the effective 4D
description in terms of a cosmology is only sensible for part of the evolution
of the crunching universe, namely around the time-symmetric point. As we will
see, this includes a large portion of the cosmological evolution in cosmic
time. Notice that the past and future singularities of the braneworld
cosmology correspond to the past and future singularities of the higher
dimensional MBB. However, as we have explained, in the regions near the
singularities there is no effective 4D cosmological description.

Figure 14: Left: MBB spacetime with an ETW brane (purple) cutting off the
right asymptotic boundary. Two non-compact directions ($x$ and $y$) are
suppressed. Right: The effective 4D description on the brane is that of a Big
Bang-Big Crunch Bianchi I cosmology. The time-symmetric slice is denoted by
the red dashed line. Gravity is effectively 4D for a large portion of the
cosmological evolution around the time-symmetric point.

In Section 4.1, we derive the Bianchi I metric of the braneworld cosmology and
the equation of motion for the cosmological evolution. The cosmic scale
factors can be numerically evaluated in the region outside the MBB horizon.
Because this contains the region where gravity localization holds and the
effective 4D description exists, this is enough for the study of our
braneworld cosmology. In Section 4.2, we study the behavior of the scale
factors of this cosmology, observe the presence of a phase of accelerated
expansion, and comment on the possible source of this accelerated expansion.
Finally, in Section 4.3 we will comment on the holographic dual description of
this bulk setup, which is given by a specific pure excited state of a theory
on $\mathbb{R}^{3,1}$, and on the properties of bulk reconstruction, pointing
out the connection with entanglement islands in cosmology Hartman:2020khs ;
Bousso:2022gth .

###  4.1 Big Bang/Big Crunch Cosmology

To describe cosmological evolution, we derive the brane equation of motion in
Lorentzian signature. The first step is to analytically continue $\tau=it$ and
define the brane proper time as

$d\lambda=\sqrt{f(r(t))-\frac{1}{f(r(t))}\left(\frac{dr}{dt}\right)^{2}}dt$
(37)

where $r(t)$ denotes the brane trajectory. The induced metric on the brane
then reads

$ds^{2}_{ETW}=-d\lambda^{2}+g(\lambda)(dx^{2}+dy^{2})+h(\lambda)dz^{2}$ (38)

where $g(\lambda)\equiv g(r(\lambda))$, and similarly for $h$. This is the
metric of a homogeneous but anisotropic cosmology with scale factors
$g(\lambda)$ and $h(\lambda)$, namely a special case of a Bianchi I universe
Taub:1950ez ; Ellis:1968vb ; Montani:2009hju  in which two directions ($x$ and
$y$) share the same scale factor. Here $\lambda$ plays the role of the cosmic
time in the braneworld cosmology. Although generically $g(\lambda)\neq
h(\lambda)$ and the cosmology is anisotropic, we will see that it remains
relatively isotropic $g(\lambda)\approx h(\lambda)$ for a large portion of its
evolution around the turning point, where the brane is the furthest from the
MBB horizon. In fact, $g\approx h$ was one of the necessary conditions for
gravity localization to hold, see Section 3.3. This implies that whenever
gravity is localized on the brane and therefore an effective 4D cosmological
description exists, the 4D universe is nearly-isotropic and approximately
takes the form of a flat Big Bang-Big Crunch FLRW universe.

Notice that the analytic continuation $\tau=it$ has the effect of flipping the
sign of the terms under the square root in the brane equation of motion (29).
This implies that the minimum radius $r_{0}$ of the brane in Euclidean
signature becomes a maximum radius in the Lorentzian picture. Combining the
analytic continuation of equation (29) with the definition of the comoving
time $\eqref{eqn:dlambda}$, the brane equation of motion in Lorentzian
signature can be cast in the form

$\left(\frac{dr}{d\lambda}\right)^{2}=\frac{9T^{2}}{4V^{\prime}W^{\prime}+4V^{\prime
2}+W^{\prime 2}}-f.$ (39)

We can integrate this equation with initial condition $r(\lambda=0)=r_{0}$ to
find the brane trajectory $r(\lambda)$ as a function of the brane comoving
time. We can then evaluate the scale factors $g(\lambda)$ and $h(\lambda)$
numerically for $r>1$. We display the results in Figure 15, where we selected
$b=\sqrt{3}-0.001$ and $T=0.9999$. We remark that this choice of parameters
satisfies the conditions needed for gravity localization around the turning
point of the brane evolution. Evidently, the cosmology is nearly isotropic for
a large part of its trajectory. Additionally, gravity localization holds and
an effective 4D cosmological description exists for a large fraction of the
total brane proper time outside the horizon.

To quantify the portion of the trajectory for which gravity localization holds
(and the cosmology is nearly isotropic), we can evaluate the total proper time
as the brane moves from the horizon $r_{H}=1$ to $r_{0}$ and eventually back
to $r_{H}=1$:

$\lambda_{\text{tot}}=2\int_{1}^{r_{0}}\sqrt{\frac{4V^{\prime}W^{\prime}+4V^{\prime
2}+W^{\prime 2}}{9T^{2}-4fV^{\prime}W^{\prime}-4fV^{\prime 2}-fW^{\prime
2}}}dr$ (40)

where the factor of two accounts for the time symmetry of the brane evolution.
We can then evaluate the total proper time for which the conditions for
gravity localization hold by simply substituting the lower extreme of the
integral (40) with $\bar{r}$ defined in Section 3.3. By choosing $\bar{r}$
such that the conditions for gravity localization are satisfied to more than
$1\%$ accuracy, for our choice of parameters $b=\sqrt{3}-0.001$, $T=0.9999$ we
obtain that gravity localization holds for $85.8\%$ of the total brane proper
time of the trajectory outside the horizon.

A 4D effective cosmological description on the brane is therefore well-defined
for a large portion of the braneworld cosmology evolution. We leave a more
accurate quantification of gravity localization to future work.

(a)

(b)

Figure 15: (a) The scale factors $g(\lambda)$ and $h(\lambda)$ as a function
of the brane comoving time $\lambda$ (i.e. the cosmic time). We chose
$b=\sqrt{3}-0.001$ and $T=0.9999$ for the free parameters. Note that for a
large portion of the evolution, $g(\lambda)$ and $h(\lambda)$ are practically
equal to each other. Thus, the cosmology is approximately isotropic. (b)
Detail of the scale factors at early times for the expanding phase of the
cosmological evolution. At early times, where the cosmology becomes more
anisotropic, we also lose gravity localization and a 4D effective description
of the cosmology ceases to make sense. The (black) dashed lines denote
$\lambda=\pm\bar{\lambda}\equiv\pm|\lambda(\bar{r})|$ such that for all
$|\lambda|>\bar{\lambda}$ at least one of the conditions for gravity
localization is violated by more than $1\%$.

###  4.2 Accelerating cosmology at early times

Examining Figure 15 reveals that a phase of accelerated cosmic expansion takes
place near the beginning of the braneworld cosmology
evolution.464646Naturally, this corresponds to a phase of decelerated
contraction near the end of the braneworld cosmology evolution, because the
full evolution is time-symmetric. In particular, notice that gravity
localization still holds during part of this accelerated phase. After the
accelerated phase, the universe undergoes a decelerated expansion and
eventually it starts to symmetrically recollapse when the effective negative
4D cosmological constant, given by (we restore here the AdS radius $L$)

$\Lambda_{4}=T^{2}-\frac{1}{L^{2}}<0,$ (41)

becomes dominant.

In order to quantify the acceleration, we can evaluate the cosmological
deceleration parameters $q_{g}$ and $q_{h}$ defined as

$q_{g}=-\frac{\ddot{g}g}{\dot{g}^{2}},\quad\quad
q_{h}=-\frac{\ddot{h}h}{\dot{h}^{2}}$ (42)

where the derivatives are taken with respect to the cosmic (i.e. comoving)
time $\lambda$. In Figure 16, we plot the results for $q_{g}$ and $q_{h}$ for
$b=\sqrt{3}-0.001$ and $T=0.9999$. We remark that $q_{g},q_{h}<0$ (signaling
an accelerated expansion) for an interval of time where gravity localization
still holds and therefore the 4D effective description of the cosmology makes
sense. Therefore, our braneworld cosmology undergoes a phase of accelerated
expansion.

Figure 16: The deceleration parameters $q_{g,h}$ for $g(\lambda)$, and
$h(\lambda)$ evaluated in the accelerated expansion phase. The (black) dashed
line denotes the cosmic time $\lambda$ where gravity localization starts to be
possible (i.e. the conditions for gravity localization are satisfied to better
than $1\%$ accuracy). Notice that $q<0$, i.e. the cosmological expansion is
accelerated, in a region where gravity localization likely holds and 4D
braneworld cosmology is well-defined.

What drives this accelerated expansion? In Cooper:2018cmb ; Antonini:2019qkt ,
where the form of the 5D bulk metric is known analytically, the Friedmann
equation governing the scale factor evolution was obtained explicitly,
allowing an identification of the energy densities driving the cosmological
evolution. For the AdS-RN case, for instance, the Friedmann equation read

$\left(\frac{\dot{a}}{a}\right)^{2}=\Lambda_{4}-\frac{1}{a^{2}}+\frac{2\mu}{a^{4}}-\frac{Q^{2}}{a^{6}},$
(43)

where $a$ is the scale factor and we can identify a (negative) cosmological
constant $\Lambda_{4}=T^{2}-1/L^{2}$, a curvature term, a radiation term
proportional to mass of the AdS-RN black hole, and a repulsive term
proportional to the black hole charge (known in the cosmology literature as
stiff matter with negative energy density Chavanis:2014lra ). The last term,
which arises from the repulsive nature of the AdS-RN timelike singularity and
is a negative energy source from the point of view of the braneworld
cosmology, is responsible for an acceleration at early times. This eventually
leads to a Big Bounce, but this accelerated phase takes place in a region
where gravity localization does not hold, and the AdS-RN solution itself is
unstable (namely inside the Cauchy horizon) Antonini:2019qkt .

The absence of an analytic form for the MBB metric and therefore for the scale
factors’ equations of motion prevents us from identifying what kind of energy
density sources the accelerated expansion in our setup. By analogy with the
AdS-RN case, it seems plausible to assume that, from the 4D cosmology
viewpoint, such an energy density is negative and is proportional to the
magnetic field. This is also consistent with the species driving the
acceleration being dominant at early times, see Appendix D. We leave a more
thorough investigation of this possibility to future work. In our numerical
setting, all we can do is constrain the equation of state of the species
driving the accelerated expansion. We report this analysis in Appendix D.

###  4.3 Holographic dual description

We can now turn to the question of what is the microscopic state dual to this
MBB bulk spacetime cut off by the ETW brane. First of all, notice that the
dual theory lives on the $\mathbb{R}^{3,1}$ asymptotic boundary of the MBB
microstate. In fact, the microscopic state is defined by slicing the
microscopic Euclidean path integral in the middle of the interval $I$, see the
top line, right panel of Figure 1. The boundary of the Euclidean BCFT is in
the Euclidean past. The conformal boundary conditions imposed there should be
seen as initial conditions for the Euclidean path integral preparing the
microscopic state, which is given by

$\ket{\psi}_{B}=e^{-\tau_{0}H}\ket{B}$ (44)

where $\ket{B}$ is the Cardy state defined at the boundary of the BCFT at
$\tau=-\tau_{0}$, and $\tau_{0}$ plays here the role of the preparation time.
The state $\ket{\psi}_{B}$ is therefore an excited pure state of the dual
theory ($\mathcal{N}=4$ SYMB) on $\mathbb{R}^{3,1}$, obtained by Euclidean
time evolution of a boundary state. It can be also understood to arise from
the TFD (17) dual to the double-sided MBB spacetime after a projective
measurement of the entire right boundary is performed in an appropriate basis
Kourkoulou:2017zaj ; Cooper:2018cmb ; Antonini:2022sfm ; Antonini:2023aza .
This construction is analogous to the one described in Cooper:2018cmb ;
Antonini:2019qkt  and can be interpreted as a higher dimensional realization
of the Kourkoulou-Maldacena setup Kourkoulou:2017zaj , see also
Almheiri:2018ijj ; Antonini:2021xar .

####  4.3.1 Hartman-Maldacena surfaces and cosmological islands

We remark that probing the cosmological physics from the dual microscopic
theory is difficult, because the brane sits behind the MBB horizon. In fact,
the braneworld cosmology is inside a Python’s lunch Brown:2019rox ;
Engelhardt:2021mue ; Engelhardt:2021qjs , with the non-minimal quantum
extremal surface being the bifurcate horizon and the minimal one the empty
set. Thus, for a CFT observer living on the left asymptotic boundary,
reconstructing observables within the ETW brane is exponentially complex.

Some known observables are nonetheless expected to be able to probe the
cosmology. One example is the entanglement entropy of large subregions of the
dual theory. In fact, for large enough subregions the Ryu-Takayanagi surface
computing the holographic entanglement entropy transitions from a connected to
a disconnected phase, extending behind the horizon and ending on the ETW
brane, see Figure 17. These surfaces, studied in Cooper:2018cmb , are similar
to Hartman-Maldacena surfaces Hartman:2013qma . In this case, the entanglement
entropy should display a time evolution matching the brane’s evolution in the
bulk. The scale factor behavior is therefore encoded in the entanglement
entropy of these large subregions.

Interestingly, these Hartman-Maldacena-like surfaces ending on the brane are
strictly related to the existence of entanglement islands in cosmology
Hartman:2020khs ; Bousso:2022gth . To understand that, we should realize that
the 4D (singly holographic) effective description of this Lorentzian setup
involves the Big Bang-Big Crunch cosmology and a non gravitational bath on
$\mathbb{R}^{3,1}$ disconnected from but entangled with the cosmology
VanRaamsdonk:2021qgv ; Antonini:2022blk ; Antonini:2022ptt , as explained in
Section 1, see the center line, right panel of Figure 1. This bath once again
accounts for the physics of the 5D bulk and is a low-energy EFT version of the
microscopic dual theory on $\mathbb{R}^{3,1}$. This is exactly the setup in
which cosmological islands arise. In particular, for a flat cosmology such as
ours, it was shown in Hartman:2020khs  that a contribution coming from an
island in the cosmological universe is present when computing the holographic
entanglement entropy of large regions of the microscopic theory.

In our 5D setup, which can be seen as a doubly-holographic realization of the
construction in Hartman:2020khs , the island is identified by the region
comprised between the two intersection points of the Hartman-Maldacena-like
surface with the brane, see Figure 17. This is completely analogous to what
happens in doubly holographic models of black hole evaporation
Almheiri:2019hni , where islands can be understood in terms of classical RT
surfaces ending on an ETW brane. We leave a thorough study of the quantitative
agreement between the doubly holographic calculation and the cosmological
island calculation in our setup to future study.

Figure 17: Left: in the effective (singly-holographic) 4D theory, the
entanglement entropy of a large region $R$ of the microscopic theory receives
contributions from the same region $R$ in the non-gravitational bath and from
an entanglement island $I$ in the cosmology. Both $R$ and $I$ are depicted in
light yellow. Right: in our 5D (doubly-holographic) setup, the RT surface
associated with subregions of the microscopic theory has a connected phase (in
brown) and a disconnected phase given by a Hartman-Maldacena-like surface
ending on the brane (in green). The latter becomes dominant for large $R$. The
endpoints on the brane of this surface identify the boundary of the island
$I$.

Although it is possible to reconstruct some cosmological observables such as
the scale factor using entanglement entropy or holographic complexity
Cooper:2018cmb , these boundary quantities are very complex observables from a
microscopic theory point of view. Additionally, probing other cosmological
observables—such as correlation functions of fields in the cosmology—from the
dual theory is also an exponentially complex task, and the dictionary is
unknown and expected to be state-dependent Papadodimas:2012aq ;
Papadodimas:2013jku ; Akers:2021fut ; Akers:2022qdl . However, the slicing
duality introduced in Section 1 could provide a solution to this issue. In
fact, one can relate braneworld cosmological observables to observables in a
traversable braneworld wormhole, which will be the subject of the next
section. Those observables can then be reconstructed from the confining theory
dual to the wormhole using an ordinary holographic dictionary such as HKLL.

##  5 Eternally Traversable Lorentzian Wormhole on the Brane

Figure 18: Left: AdS5 soliton geometry cut off by an ETW brane (in purple).
The black dot denotes $r=1$, where the compact direction $\tau$ pinches off.
The orange dashed line denotes the time reflection symmetric slice on which
the bulk (vacuum) state is prepared by the Euclidean path integral. Right: a
time-slice of the Lorentzian eternally traversable wormhole on the brane,
connecting two asymptotic AdS4 boundaries.

In this section, we focus on the other possible analytic continuation of our
Euclidean saddle, where the $z$ coordinate is identified with the Euclidean
time. This yields the magnetically charged AdS5 soliton (15) with part of the
asymptotic boundary cut off by the ETW brane, see Figure 18. Notice that in
this setup the brane is static, since both $r$ and $\tau$ are spatial
directions. Its trajectory is exactly the same as in the Euclidean analysis of
Section 3. The only difference is that the planar $z$ direction is now
analytically continued and plays the role of the Lorentzian time $\zeta=-iz$.

The geometry induced on the brane is that one of a traversable planar wormhole
connecting two distinct AdS4 boundaries.474747See Maldacena:2020sxe  for a
different construction of braneworld traversable wormholes starting from the
zero temperature 5D solution constructed in DHoker:2009mmn . Notice that the
authors of Maldacena:2020sxe  take a different approach, embedding the ETW
brane in a Lorentzian geometry analogous to our MBB rather than to the soliton
solution. See Section 5.5 for additional comments on this point. In fact, it
is a Lorentzian version of the Maldacena-Maoz-like wormhole seen in Section
3.4. Given that the induced geometry on the brane, just like the 5D geometry,
is static, and given that the proper distance between any two bulk points in
this wormhole is finite, we can conclude that this is an eternally traversable
wormhole.

We remark that this 4D effective description of the brane as a wormhole is
valid for the whole brane trajectory. In fact, we have seen that for an
appropriate choice of the $b,T$ parameters (e.g. the ones employed in the
numerical analysis of the previous section), the radius of the brane at the
turnaround point $r_{0}$ lies in a region where gravity can be localized on
the brane. For radii $r>r_{0}$ gravity localization only works better because
the brane is farther into the asymptotic AdS region, approaching the setup of
Karch:2000ct . Since in this Lorentzian soliton picture, just like in the
Euclidean case, $r_{0}$ is the minimum radius of the brane, a 4D local
effective description of gravity on the brane exists everywhere, and the
braneworld traversable wormhole description is sensible everywhere. We also
emphasize that in the 4D effective description, the two boundaries of the
wormhole are also coupled by 4D non-gravitational auxiliary degrees of freedom
on $\mathbb{R}^{2,1}\times I$, just like in the Euclidean case discussed in
Section 3.4. This is something to be expected: a traversable wormhole
connecting two AdS boundaries can only exist if the two asymptotic boundaries
(where the holographic theories live) are coupled Maldacena:2018lmt . To the
best of our knowledge, our braneworld construction is the first example of a
higher dimensional eternally traversable wormholes connecting two distinct
asymptotic AdS4 boundaries.

Another interesting feature is that, due to the slicing duality of the
Euclidean saddle, cosmological observables in the braneworld cosmology picture
of the previous section can be analytically continued to observables in the
braneworld wormhole of this section. The absence of a horizon in the soliton
spacetime (and, equivalently, in the 4D effective description of the brane as
a traversable wormhole) implies that reconstruction of these observables is
simple, and can be performed using the tools needed for causal wedge
reconstruction, such as the HKLL dictionary. This simplifies greatly the
reconstruction of braneworld cosmology observables with respect to a direct
reconstruction in the dual theory described in Section 4.3.

In Section 5.1 we derive the traversable wormhole geometry by examining the
induced metric on the brane and show via a numerical evaluation that the
wormhole connects two AdS asymptotic boundaries. In Section 5.2 we explore
traversability of the wormhole by evaluating the amount of boundary time
needed for a light ray shot from one boundary to reach the other one and
showing that it is finite. We also find that in the limit $b\to\sqrt{3}$
(corresponding to the $\beta\to\infty$ limit), the wormhole becomes non-
traversable. We comment how this result allows us to reconcile our solution
with known “no-go” arguments for Poincaré-symmetric traversable wormholes
Freivogel:2019lej . In Section 5.3, we discuss the dual theory, which is given
by the vacuum state of a BCFT on $\mathbb{R}^{2,1}\times I$, and demonstrate
that it is confining by calculating the mass gap using the 4D braneworld
wormhole description via the procedure described in Antonini:2022opp . We
compare and find qualitative agreement with the behavior of the mass gap
obtained in the 5D soliton calculation in Section 2.4. In Section 5.4 we
comment about bulk reconstruction and the slicing duality between the
braneworld cosmology and the braneworld wormhole pictures. Finally, in Section
5.5 we explain the relationship between our braneworld wormholes and other
previously constructed eternally traversable wormhole solutions in four
dimensions Maldacena:2018gjk ; Maldacena:2020sxe .

###  5.1 Braneworld Wormhole Solution

The metric induced on the brane embedded in the magnetically charged AdS5
soliton (15) is given by

$ds^{2}=\left(f(\tau)+\frac{(r^{\prime}(\tau))^{2}}{f(\tau)}\right)d\tau^{2}+g(\tau)(dx^{2}+dy^{2})-h(\tau)d\zeta^{2}$
(45)

and it can be obtained by simply analytically continuing the Euclidean induced
metric (35).

As we have seen in Section 3.4, in regions where gravity localization is
possible, we can recast the metric into an approximately conformally flat form

$ds^{2}\approx f(\tau)(d\tau^{2}+dx^{2}+dy^{2}-d\zeta^{2}).$ (46)

A numerical evaluation of $f(\tau)$ for $b=\sqrt{3}-0.001$ and $T=0.9999$ was
reported in Figure 13. $f(\tau)$ has a positive minimum (given by $f(r_{0})$)
which we set to be at $\tau=0$,484848Like for the Euclidean Maldacena-Maoz
wormhole, we choose to set $\tau=0$ at the turnaround point in the 4D
braneworld wormhole description, as opposed to $\tau=\pm\beta/2$ in the doubly
holographic description. and it diverges for finite values of $\tau$, which we
label $\tau=\pm\tau_{\infty}$. Naturally, this divergence corresponds to the
intersection between the brane and the asymptotic boundary of the AdS5 soliton
spacetime, and $\tau_{\infty}$ is the amount of coordinate $\tau$ elapsed
during the brane trajectory from the symmetric point to the boundary, first
defined at the beginning of Section 3.

By inspecting the brane equation of motion (29) it is easy to conclude that,
near the asymptotic boundary where $f(r)\approx g(r)\approx h(r)\approx
r^{2}$, we have $\tau(r)\approx\pm(\tau_{\infty}-1/r)$. Therefore, near
$\tau=\mp\tau_{\infty}$ the conformal factor $f(\tau)$ in the metric (46)
behaves as

$f(\tau)\approx\frac{-3/\Lambda_{4}}{(\tau_{\infty}\pm\tau)^{2}}$ (47)

where the effective 4D cosmological constant on the brane is
$\Lambda_{4}=T^{2}-1/L^{2}<0$ (where we temporarily restored the AdS radius
$L$). Therefore, in the 4D effective theory on the brane, the two regions
around $\tau=\pm\tau_{\infty}$ are two (planar) asymptotically AdS4 regions
connected by a planar eternally traversable wormhole. The $\tau$ coordinate is
along the wormhole.

The traversability of the wormhole is guaranteed by the positive minimum of
the wormhole conformal factor $f(\tau)$. It can be quantitatively confirmed by
a simple numerical check that the proper distance along the $\tau$ direction

$\Delta\ell=\int_{\tau_{1}}^{\tau_{2}}\sqrt{f(\tau)+\frac{(r^{\prime}(\tau))^{2}}{f(\tau)}}d\tau$
(48)

between any two bulk points, with
$-\tau_{\infty}<\tau_{1},\tau_{2}<\tau_{\infty}$, is finite for $b<\sqrt{3}$,
$T<1$. Because the wormhole geometry (45) is static, this guarantees
traversability. Clearly, timelike geodesics cannot reach the asymptotic
boundaries in a finite proper time, but can connect bulk points near the two
asymptotic boundaries. On the other hand, as we will see in the next
subsection, light rays can travel from one boundary to the other in a finite
amount of coordinate time $\zeta$, which is identified with the boundary time.

We remark that, in the 4D effective description, the two boundaries connected
by the traversable wormhole are also coupled by auxiliary non-gravitational
degrees of freedom on $\mathbb{R}^{2,1}\times I$, see the center line, left
panel of Figure 1. This “bath” accounts for the 5D DOF we integrated out and
is a low-energy effective description of the 4D DOF in the microscopic BCFT,
whereas the microscopic 3D boundary DOF are represented holographically by the
wormhole geometry.494949We remind that this is a simplified description of the
role of microscopic DOF in the 4D effective description, see Footnote 7. This
coupling between the two asymptotic boundaries is what allows the wormhole to
be traversable. From an effective 4D bulk point of view, it guarantees the
presence of negative energy supporting the wormhole (see Section 5.5 for
further comments); from a microscopic point of view, it shows that the 3D
holographic DOF living on the asymptotic AdS4 boundaries are coupled, and the
information transfer possible through the wormhole is possible also
microscopically.

###  5.2 Traversability and no-go arguments

In the previous subsection we showed the existence of a 4D, planar, nearly-
Poincaré-symmetric, eternally traversable wormhole on the brane. It was argued
in Freivogel:2019lej  that, in more than two spacetime dimensions, it is
generically not possible to build semiclassical eternally traversable
wormholes that satisfy Poincaré invariance in the boundary directions. There
are two reasons why our construction is not in tension with the results of
Freivogel:2019lej . The first one is that the 4D effective description on the
brane is only approximate and it involves massive gravity (although the mass
of the brane-localized graviton is small), as we have already explained. The
second one is that Poincaré invariance is also not exact: the wormhole is
slightly anisotropic, see equations (45)-(46).