Aspects of Dynamical Cobordism in AdS/CFT

Jesús Huertas, Angel M. Uranga
Instituto de Física Teórica IFT-UAM/CSIC,
C/ Nicolás Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

The cobordism conjecture implies that consistent theories of Quantum Gravity
must admit the introduction of boundaries. We study the dynamical realization
of the cobordism conjecture in type IIB in AdS${}_{5}\times{\bf{S}}^{5}$,
using the existing gravity duals of 4d ${\cal{N}}=4$ SYM with Gaiotto-Witten
superconformal boundary conditions (near-horizon limits of D3-branes ending on
NS5- and D5-branes). We show that these configurations are, from the 5d
perspective, dynamical cobordism solutions which start from an asymptotic AdS5
vacuum and evolve until they hit an end of the world (ETW) brane with AdS4
worldvolume. The latter displays localization of gravity, and provide a
completion of the Karch-Randall (KR) AdS branes, in which the backreaction of
running scalars replace the KR cusp in the warp factor with a smooth bump. The
dynamical scalars are either in the $SO(6)$ invariant AdS5 bulk sector (e.g.
describing the ${\bf{S}}^{5}$ size and its shrinking at the cobordism
boundary) or brane localized (e.g. the $SO(6)\to SO(3)\times SO(3)$ squashing
due to boundary conditions). We introduce a novel double scaling limit which
zooms into the ETW brane and makes localization of gravity manifest, and which
shows a tantalizing relation with wedge holography. We extend the picture to
AdS5 theories with less (super)symmetry, via orbifolds and S-folds, leading to
dynamical cobordisms for gravity duals of 4d theories with ${\cal{N}}=2$ and
${\cal{N}}=3$ supersymmetry.

###### Contents

  1. 1 Introduction
  2. 2 Overview of ETW branes for AdS${}_{5}\times{\bf{S}}^{5}$
    1. 2.1 Bagpipe geometries
    2. 2.2 Towards one AdS${}_{5}\times{\bf{S}}^{5}$ asymptotic region
    3. 2.3 Boundaries in 4d ${\cal{N}}=4$ SYM
  3. 3 Dynamical cobordism for AdS${}_{5}\times{\bf{S}}^{5}$
    1. 3.1 The bagpipe geometries are dynamical cobordisms
    2. 3.2 A 5d effective theory description
  4. 4 Zoom into the ETW brane: A double scaling limit
    1. 4.1 A scaling symmetry
    2. 4.2 Double scale zooming into the ETW configuration
    3. 4.3 The limit revisited: closing off AdS${}_{5}\times{\bf{S}}^{5}$
    4. 4.4 Reduction in the double scaled limit
  5. 5 ETW branes in gravity duals of 4d ${\cal{N}}=2$ orbifold theories
    1. 5.1 The 4d ${\cal{N}}=2$ quiver gauge theory and its gravity dual
    2. 5.2 The ETW orbifold configurations
    3. 5.3 Boundaries from Brane Box construction
  6. 6 ETW branes in gravity duals of ${\cal{N}}=3$ S-fold theories
    1. 6.1 The 4d ${\cal{N}}=3$ theories on D3-branes
    2. 6.2 The ETW configurations in the gravity duals
  7. 7 Conclusions

##  1 Introduction

The cobordism conjecture McNamara:2019rup  implies that consistent theories of
quantum gravity, in particular string theory, should admit the introduction of
boundaries, configurations (possibly dressed with localized objects to absorb
charges/fluxes) ending spacetime. We will refer to such configurations End of
the World (ETW) branes. At the topological level, this has led to non-trivial
results regarding anomalies and the existence of (possibly non-supersymmetric)
extended states in string theory GarciaEtxebarria:2020xsr ; Ooguri:2020sua ;
Montero:2020icj ; Dierigl:2020lai ; Hamada:2021bbz ; Blumenhagen:2021nmi ;
Andriot:2022mri ; Dierigl:2022reg ; Debray:2023yrs . At the level of physical
realizations, it has led to the notion of dynamical cobordisms Buratti:2021yia
; Buratti:2021fiv ; Angius:2022aeq ; Blumenhagen:2022mqw ; Blumenhagen:2023abk
(see also Dudas:2000ff ; Blumenhagen:2000dc ; Dudas:2002dg ; Dudas:2004nd ;
Hellerman:2006nx ; Hellerman:2006ff ; Hellerman:2007fc  for related early work
and Basile:2018irz ; Antonelli:2019nar ; Mininno:2020sdb ; Basile:2020xwi ;
Basile:2021mkd ; Mourad:2022loy ; Angius:2022mgh ; Angius:2023xtu  for other
related recent developments), spacetime dependent solutions developing real
codimension-1 singularities ending spacetime, at which certain scalars run off
to infinite distance in field space Buratti:2021fiv . Hence, dynamical
cobordisms are useful probes of infinite distance limits in field space and
the swampland distance conjecture Ooguri:2006in  (and in fact relate to other
probes, such as small black holes Hamada:2021yxy ; Angius:2022aeq ;
Angius:2023xtu  or 4d EFT strings Lanza:2020qmt ; Lanza:2021qsu ;
Marchesano:2022avb .

It is natural to explore the cobordism conjecture in the arguably best
understood vacua in string theory, those arising in AdS/CFT holography, such
as type IIB on AdS${}_{5}\times{\bf{S}}^{5}$ with $N$ units of RR 5-form flux.
The simplest proposal for a boundary configuration for the latter would seem
to simply correspond to shrinking the ${\bf{S}}^{5}$ at some value of the
radial coordinate in a Minkowski slicing of AdS5, including $N$ explicit
D3-branes to remove the flux; however, the backreaction of this large number
of D3-branes would reconstruct a non-compact AdS5 throat, preventing an actual
end of spacetime.

This puzzle can however be overcome by the introduction of a different class
of boundaries, familiar in the setup known as AdS/BCFT or double holography
Karch:2000gx ; Takayanagi:2011zk . The key idea is to consider the boundary of
an AdSd+1 spacetime111Our convention is to use the term ETW boundary, or just
boundary, for the cobordism one, and holographic boundary for the conformal
boundary at infinity. to be localized at a codimension-1 AdSd slice. These so
called Karch-Randall (KR) branes Karch:2000ct ; Karch:2001cw ; Karch:2000gx
have the property of localizing gravity on their worldvolume, and reaching off
to the holographic boundary, so that they define a boundary in the dual CFTd.
Double holography is the statement that the gravitational theory on the AdSd
worldvolume of the ETW brane is the gravity dual of a $(d-1)$-dimensional
boundary CFT (BCFT) of the $d$-dimensional CFT holographically dual to the
AdSd+1 gravitational bulk. This kind of AdS ETW branes in AdS3/CFT2 also play
a prominent role in the recent explanation of the black hole Page curve and
quantum islands Almheiri:2019hni ; Almheiri:2019psy ; Chen:2020uac ;
Chen:2020hmv ; Geng:2020fxl ; Geng:2021mic ; Uhlemann:2021nhu ;
Demulder:2022aij .

Despite their importance, most explorations of double holography and quantum
islands involve a bottom-up simplified modeling of ETW branes as sharp
boundaries, with no microscopic description. However, for
AdS${}_{5}\times{\bf{S}}^{5}$ there is a class of 10d supergravity solutions,
including D5- and NS5-branes, which provide such description for boundaries
preserving an $SO(3)\times SO(3)$ subgroup of the $SO(6)$ symmetry
DHoker:2007zhm ; DHoker:2007hhe ; Aharony:2011yc ; Assel:2011xz ;
Bachas:2017rch ; Bachas:2018zmb  (see also Raamsdonk:2020tin ;
VanRaamsdonk:2021duo ; Demulder:2022aij ; Karch:2022rvr  for recent
applications, and DHoker:2008lup ; DHoker:2008rje ; DHoker:2009lky ;
DHoker:2009wlx  for related results on AdS${}_{4}\times{\bf{S}}^{7}$ and
AdS${}_{7}\times{\bf{S}}^{4}$); these configurations are dual to 4d
${\cal{N}}=4$ SYM with a boundary, with the $OSp(4|4)$ superconformal symmetry
preserving boundary conditions studied in Gaiotto:2008sa ; Gaiotto:2008ak .
The 5-branes source NSNS and RR 3-form fluxes, whose Chern-Simons couplings
remove the 5-form flux and allow the ${\bf{S}}^{5}$ to shrink (see
Buratti:2021yia  for a similar phenomenon in the Klebanov-Strassler solution
Klebanov:2000hb ) with no non-compact throats arising from the backreaction.

In this paper we discuss aspects of this class of solutions, with emphasis in
extracting their interpretation from the perspective of the 5d theory, and
their relation with the bottom-up approach. We show that the 5d description
takes the form of a dynamical cobordism with fields running along the
coordinate transverse to the AdS4 slicing. The description includes the 5d
metric and the dilaton, but a prominent role is played by the scalar breathing
mode of the ${\bf{S}}^{5}$. In particular, it is necessary to define the
shrinking of the ${\bf{S}}^{5}$ at the boundary, in a way dictated by the
local ETW brane description in Angius:2022aeq . In addition, it smooths out
the cusp of the naive 5d KR metric into a bump, which nevertheless retains the
feature of localizing gravity. Our analysis allows to systematically include
further KK modes, and enrich the picture. For instance, they encode the
breaking $SO(6)\to SO(3)\times SO(3)$ via scalars localized on the ETW brane.
We also introduce a double scaling limit of the solutions (interestingly
related to configurations in Assel:2011xz ; Bachas:2017rch  and to wedge
holography Akal:2020wfl ; VanRaamsdonk:2021duo ), which allows to isolate the
ETW brane dynamics and make some of its properties manifest. We carry out the
discussion mainly for AdS${}_{5}\times{\bf{S}}^{5}$, but subsequently discuss
S-fold and orbifold quotients preserving the structure of the 10d solution,
hence providing a description for ETW branes in AdS5 spaces in cases with
lower supersymmetry, gravity duals of 4d ${\cal{N}}=3$ and ${\cal{N}}=2$ field
theories with boundaries.

This paper is organized as follows. In Section 2 we review the 10d solutions
describing AdS${}_{5}\times{\bf{S}}^{5}$ ending on ETW brane configurations
and their interpretation as gravity duals of 4d ${\cal{N}}=4$ SYM with
boundaries defined by NS5- and D5-branes. Section 2.1 describes the general
class of solutions, section 2.2 particularizes to the ETW branes of
AdS${}_{5}\times{\bf{S}}^{5}$, and section 2.3 discusses the holographic
relation. In Section 3 we interpret them as dynamical cobordisms. Section 3.1
describes the solutions as dynamical cobordisms of several
AdS${}_{5}\times{\bf{S}}^{5}$ asymptotic regions, or cobordisms to nothing for
a single one. In section 3.2 we provide a 5d interpretation of the solutions
after reduction on the ${\bf{S}}^{5}$, including the 5d metric and
localization of gravity, and bulk and localized scalars. Section 4 discusses a
double scaling limit isolating the dynamics of the ETW brane. In section 4.1
we present a scaling symmetry of the solutions, and section 4.2 describes the
scaling limit, for which we provide an alternative interpretation in section
4.3; section 4.4 deals with the effective theory reduction of the resulting
solutions. In Section 5 we extend the discussion to gravity duals of 4d
${\cal{N}}=2$ orbifold field theories. Section 5.1 introduces the field
theories and their gravity duals. In section 5.2 we present the orbifold ETW
configurations, which we recast in section 5.3 in terms of T-dual brane box
configurations. Section 6 discusses ETW configurations in gravity duals of 4d
S-fold field theories. In section 6.1 we introduce the field theories, and in
section 6.2 we describe their gravity duals and the corresponding solutions
for ETW configurations. In Section 7 we offer some final remarks.

Note added: As this paper was being preparing for submission we noticed
DeLuca:2023kjj , which deals with gravity localization in this and other
setups.

##  2 Overview of ETW branes for AdS${}_{5}\times{\bf{S}}^{5}$

In this section we review the configurations ending
AdS${}_{5}\times{\bf{S}}^{5}$ on ETW configurations along an AdS4 slice, on
which the ${\bf{S}}^{5}$ is able to contract while suitable NSNS and RR 3-form
fluxes manage to remove the $N$ units of RR 5-form flux on it. The ETW
configuration reaches off to the holographic boundary, hence is
holographically dual to a boundary of the 4d ${\cal{N}}=4$ SYM theory. The
geometries, and their holographic interpretation in BCFT, have been studied in
DHoker:2007zhm ; DHoker:2007hhe ; Aharony:2011yc ; Assel:2011xz ;
Bachas:2017rch ; Bachas:2018zmb , see also Raamsdonk:2020tin ;
VanRaamsdonk:2021duo ; Demulder:2022aij  for recent applications. We merely
outline the results useful for our purposes, and refer the reader for further
details. For concreteness, we follow the notation and conventions in
Aharony:2011yc .

###  2.1 Bagpipe geometries

The symmetry of the solutions is the 3d ${\cal{N}}=4$ superconformal group
$OSp(4|4)$, whose bosonic symmetry is $SO(2,3)\times SO(3)\times SO(3)$. They
have the structure

$\displaystyle{\rm
AdS}_{4}\times{\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}\times\Sigma\,,$ (2.1)

where $\Sigma$ is an oriented Riemann surface, over which the AdS4 and the two
${\bf{S}}^{2}$’s vary. The ansatz for the Einstein frame 10d metric is

$\displaystyle
ds^{2}=f_{4}^{2}ds^{2}_{AdS_{4}}+f_{1}^{2}ds^{2}_{{\bf{S}}_{1}^{2}}+f_{2}^{2}ds^{2}_{{\bf{S}}_{2}^{2}}+ds^{2}_{\Sigma}\,.$
(2.2)

Here $f_{1}$, $f_{2}$, $f_{3}$ are functions of a complex coordinate $w$ of
$\Sigma$, in terms of which

$\displaystyle ds^{2}_{\Sigma}=4\rho^{2}|dw|^{2}\,,$ (2.3)

for some real function $\rho$. There are also non-trivial backgrounds for the
NSNS and RR 2-forms and the RR 4-form, which we will mostly skip in our
discussing, refering the reader to the references for details.

There are closed expressions for the different functions in the above metric,
describing the BPS solution. As an intermediate step, we define the real
functions

$\displaystyle
W\equiv\partial_{w}h_{1}\partial_{\bar{w}}h_{2}+\partial_{w}h_{2}\partial_{\bar{w}}h_{1}\;\;,\;\;N_{1}\equiv
2h_{1}h_{2}|\partial_{w}h_{1}|^{2}-h_{1}^{2}W\;\;,\;\;N_{2}\equiv
2h_{1}h_{2}|\partial_{w}h_{2}|^{2}-h_{2}^{2}W\,.$ (2.4)

The dilaton is given by

$\displaystyle e^{2\Phi}=\frac{N_{2}}{N_{1}}$ (2.5)

and the functions are given by

$\displaystyle\rho^{2}=e^{-\frac{1}{2}\Phi}\frac{\sqrt{N_{2}|W|}}{h_{1}h_{2}}\;,\;f_{1}^{2}=2e^{\frac{1}{2}\Phi}h_{1}^{2}\sqrt{\frac{|W|}{N_{1}}}\;,\;f_{2}^{2}=2e^{-\frac{1}{2}\Phi}h_{2}^{2}\sqrt{\frac{|W|}{N_{2}}}\;,\;f_{4}^{2}=2e^{-\frac{1}{2}\Phi}\sqrt{\frac{N_{2}}{|W|}}$
(2.6)

The functions $h_{1}$, $h_{2}$ are defined using an auxiliary genus $g$ hyper-
elliptic Riemann surface, parametrized by a coordinate $u$ in the lower half-
plane ${\rm Im}\,u\leq 0$. This coordinate will be related to the variable $w$
above later on, so we denote the Riemann surface by $\Sigma$. It is defined by
the hyper-elliptic equation

$\displaystyle
s^{2}(u)=(u-e_{1})\prod_{k=1}^{g}(u-e_{2k})(u-e_{2k+1})\quad{\rm
with}\,e_{i}\in{\bf{R}}\,.$ (2.7)

Note that we have fixed the $SL(2,{\bf{R}})$ symmetry to locate the branch
point $e_{2g+2}$ at infinity. The holomorphic differentials $\partial h_{1}$,
$\partial h_{2}$ on this surface (i.e. 1-forms with branch points at $e_{i}$)
are defined as

$\displaystyle\partial
h_{1}=-i\frac{P(u)}{Q_{1}(u)}{s^{3}(u)}du\quad,\quad\partial
h_{2}=-\frac{P(u)}{Q_{2}(u)}{s^{3}(u)}du$ (2.8)

where $P$, $Q_{1}$, $Q_{2}$ are real polynomials of degrees $2g$, $g+1$,
$g+1$, respectively, and the following structure of zeros

$\displaystyle P(u)=\prod_{a=1}^{g}(u-u_{a})(u-{\bar{u}}_{a})\quad,\quad
Q_{1}(u)=\prod_{b=1}^{g+1}(u-\alpha_{b})\quad,\quad
Q_{2}(u)=\prod_{b=1}^{g+1}(u-\beta_{b})$ (2.9)

where ${\rm Im}(u_{a})\leq 0$. Regularity of the solution requires the
following ordering of the zeroes

$\displaystyle\alpha_{g+1}\leq e_{2g+1}\leq\beta_{g+1}\leq
e_{2g}\leq\ldots\leq\alpha_{b}\leq e_{2b-1}\leq\beta_{b}\leq\ldots\leq
e_{2}\leq\alpha_{1}\leq e_{1}\leq\beta_{1}\quad\quad$ (2.10)

namely, alternating $\alpha$’s and $\beta$’s, with branch points $e_{i}$
separating them. The parameters $u_{a}$ can be determined in terms of these
and are hence not independent quantities. The generic solution where the
ordering (2.10) is obeyed with strict inequalities corresponds to regular
solutions; when some of the values coincide, the solution develops interesting
singularities, see later.

The distribution of branch points on the real line ${\rm Im}(u)=0$ encodes
interesting properties of the topology of the solution, see Figure 1. The
fibration of ${\bf{S}}_{1}^{2}$ and ${\bf{S}}_{2}^{2}$ over this real line has
the following structure: the ${\bf{S}}_{1}^{2}$ shrinks to zero size on top of
the segments $(e_{2p},e_{2p-1})$, for $p=1,\ldots,g+1$ (and
$e_{2g+2}=-\infty$), while the ${\bf{S}}_{2}^{2}$ shrinks to zero size on top
of the segments $(e_{2p-1},e_{2p-2})$, for $p=1,\ldots,g+1$ (and
$e_{0}\equiv+\infty$). This implies that the boundary of the Riemann surface
is not a boundary of the full geometry, as the closing off of the two-spheres
seals it off.

Figure 1: Fibration of ${\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$ over the
$u$-plane, indicating the degenerations of fibres over the real line, and the
kinds of non-trivial cycles in the geometry.

Hence, the fibration of ${\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$ over a
segment in the lower half-plane, with its two endpoints on this real line and
separated by one brach point, has different two-spheres shrinking at its
endpoints, so it has the topology of an ${\bf{S}}^{5}$. Actually, at these
points the 10d geometry develops a semi-infinite spike, corresponding to an
AdS${}_{5}\times{\bf{S}}^{5}$ region. The values of the RR 5-form flux $N$ and
$g_{s}$ will be discussed later on.

These geometries were dubbed ‘bagpipes’ in Bachas:2018zmb , as they morally
correspond to AdS${}_{4}\times{\bf{X}}_{6}$ (the bag), with ${\bf{X}}_{6}$ a
compact 6d manifold, with AdS${}_{5}\times{\bf{S}}^{5}$ asymptotic regions
sticking out of it (the pipes), see Figure 2.

Figure 2: The bagpipe geometries.

On the other hand, fibering ${\bf{S}}_{1}^{2}$ and ${\bf{S}}_{2}^{2}$ over a
segment in the lower half-plane, with its two endpoints on this real line and
separated by two branch points, has the same two-sphere collapsing at its
endpoints, so it has the topology of ${\bf{S}}^{2}\times{\bf{S}}^{3}$. The
${\bf{S}}^{3}$ acquires a sharp physical meaning if one considers the limit in
which the two branch points coincide (and so does the corresponding $\alpha$
or $\beta$ between them in the ordering (2.10). If we have the collapse of a
subset $(e,\alpha,e)$, the resulting geometry locally corresponds to a
NS5-brane solution; on the other hand, the collapse of a subset $(e,\beta,e)$,
the resulting geometry locally corresponds to a D5-brane solution. The
interpretation of these objects will become clear in the holographic
description in section 2.3.

One may also consider mixed collapses $(\alpha,e,\beta)$ or
$(\beta,e,\alpha)$. Interestingly, the effect is the vanishing of the value of
the RR 5-form flux in the would-be local AdS${}_{5}\times{\bf{S}}^{5}$ spike,
meaning that the ${\bf{S}}^{5}$ is actually contractible and the spike is
closed off into a smooth point of the geometry.

###  2.2 Towards one AdS${}_{5}\times{\bf{S}}^{5}$ asymptotic region

Note that the collapse of branch points leads to an effective reduction of the
genus of the Riemann surface, hence of the number of asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$ regions. In the following we focus on the case
of at most two asymptotic AdS${}_{5}\times{\bf{S}}^{5}$ regions. This arises
when the ordering (2.10) collapses as

$\displaystyle\alpha\;\underbrace{(e\;\beta\;e)\;\alpha\;(e\;\beta\;e)\;\alpha\;(e\;\beta\;e)}_{m}\;\alpha\;e\;\beta\;\underbrace{(e\;\alpha\;e)\;\beta\;(e\;\alpha\;e)\;\beta\;(e\;\alpha\;e)}_{n}\;\beta\,,$
(2.11)

with $n+m=g$. This corresponds to a configuration with two asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$ regions, around the isolated branch point $e$ in
the middle, and that at infinity. In addition, there are $n$ NS5-brane
positions and $m$ D5-brane positions. In the following we focus on these
configurations. Let us introduce the quantities $k_{a}^{2}$, $l_{b}^{2}$ with
$a=1,\ldots,n$ and $b=1,\ldots,m$ to define the positions of the NS5- and
D5-branes, respectively, so that the ordering is

$\displaystyle\alpha_{g+1}<-l_{m}^{2}<\alpha_{g}<\cdots<\alpha_{n+2}<-l_{1}^{2}<\alpha_{n+1}<0<\beta_{n+1}<k_{n}^{2}<\cdots<\beta_{2}<k_{1}^{2}<\beta_{1}\,,$
(2.12)

where, without loss of generality, we choose to locate the middle branch point
at $u=0$.

The interpretation of these geometries is the following. For large $|u|$, the
geometry asymptotes to AdS${}_{5}\times{\bf{S}}^{5}$ with some RR 5-form flux.
For intermediate values of $|u|$, there is a non-trivial warping, describing
some transition region. Finally, the solution near the origin $u=0$ describes
AdS${}_{5}\times{\bf{S}}^{5}$ with some (possibly different) RR 5-form flux.
The solution thus describes an effective domain wall configuration connecting
two AdS${}_{5}\times{\bf{S}}^{5}$, described by a ‘fat brane’ with AdS4
symmetry. Hence it corresponds to a string theory version of the Karch-Randall
branes Karch:2000ct  (already anticipated in Karch:2001cw ).

We focus on the case in which, in a subsequent step, we also collapse of the
middle cluster by setting $\alpha_{n+1}=\beta_{n+1}=0$, namely (2.12) turns
into

$\displaystyle\alpha_{g+1}<-l_{m}^{2}<\alpha_{g}<\cdots<\alpha_{n+2}<-l_{1}^{2}<0<k_{n}^{2}<\cdots<\beta_{2}<k_{1}^{2}<\beta_{1}\,,$
(2.13)

As explained above, this corresponds to closing off the asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$ spike at $u=0$, leaving only that at
infinity222The inquisitive reader may wonder if this last asymptotic region
can be closed off as well. This is in fact the punchline of section 4..

It is useful to introduce the coordinate change $u=w^{2}$. This means that $w$
parametrizes the second quadrant in the complex plane, and that the boundary
of the Riemann surfaces contains two semi-infinite lines: the negative real
axis (along with the NS5-brane positions are located) and the positive
imaginary axis (along which the D5-brane positions are located). The closed-
off asymptotic AdS${}_{5}\times{\bf{S}}^{5}$ sits at $w=0$, while the only
left-over asymptotic AdS${}_{5}\times{\bf{S}}^{5}$ is at infinity. This is
shown in Figure 3.

Figure 3: Fibration of ${\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$ over the
$w$-plane for one asymptotic AdS${}_{5}\times{\bf{S}}^{5}$ region. The arcs
correspond to slices with AdS${}_{4}\times{\bf{S}}^{5}$ geometry. We indicate
the location of NS5- and D5-branes in the ETW region and the asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$. (Inspired in Fig. 4 of Raamsdonk:2020tin ).

The resulting structure of the functions $h_{1}$, $h_{2}$ is

$\displaystyle h_{1}=\;4{\rm
Im}(w)+2\sum_{b=1}^{m}{\tilde{d}}_{b}\log\left(\frac{|w+il_{b}|^{2}}{|w-il_{b}|^{2}}\right)$
$\displaystyle h_{2}=-4{\rm
Re}(w)-2\sum_{a=1}^{n}d_{a}\log\left(\frac{|w+k_{a}|^{2}}{|w-k_{a}|^{2}}\right)\,,$
(2.14)

where

$\displaystyle d_{a}\equiv\frac{(\beta_{a}-k_{a}^{2})}{2k_{a}}\prod_{c\neq
a}^{n}\frac{(k_{a}^{2}-\beta_{c})}{(k_{a}^{2}-k_{c}^{2})}\quad,\quad{\tilde{d}}_{b}\equiv\frac{(-\alpha_{b+n+1-l_{b}^{2}}-l_{b}^{2})}{2l_{b}}\prod_{c\neq
b}^{m}\frac{(l_{b}^{2}+\alpha_{c+n+1})}{(l_{b}^{2}-l_{c}^{2})}\,.$ (2.15)

We note that the parameters $d_{a}$, ${\tilde{d}}_{b}$ are positive. Moreover,
the computation of the NSNS and RR 3-form fluxes on the corresponding
${\bf{S}}^{3}$’s shows that they are related to the numbers $n_{a}$, $m_{b}$
of NS5- and D5-branes in the corresponding location, hence obey a quantization
condition

$\displaystyle n_{a}\equiv 32\pi^{2}d_{a}\in{\bf{Z}}\quad,\quad m_{b}\equiv
32\pi^{2}{\tilde{d}}_{b}\in{\bf{Z}}\,.$ (2.16)

There is also a RR 5-form flux on the corresponding
${\bf{S}}^{2}\times{\bf{S}}^{3}$’s, which implies some induced D3-brane charge
on the corresponding 5-branes, given by the quantities

$\displaystyle K_{a}\equiv 32\pi k_{a}\in{\bf{Z}}\quad,\quad L_{b}\equiv 32\pi
l_{b}\in{\bf{Z}}\,.$ (2.17)

The total amount of 5-form flux, which also pierces the 5-sphere in the
asymptotic AdS${}_{5}\times{\bf{S}}^{5}$, is thus

$\displaystyle N=\sum_{a}n_{a}K_{a}+\sum_{b}m_{b}L_{b}$ (2.18)

The fact that one recovers an asymptotic AdS${}_{5}\times{\bf{S}}^{5}$ with RR
5-form flux $N$ given by (2.18) can be easily checked by performing the
asymptotic expantion of (2.14). We have

$\displaystyle
h_{1}=4\Big{(}r+\frac{2}{r}\sum_{b}{\tilde{d}}_{b}l_{b}\Big{)}\sin\varphi\quad,\quad
h_{2}=-4\Big{(}r+\frac{2}{r}\sum_{a}d_{a}k_{a}\Big{)}\cos\varphi$ (2.19)

Using (2.4), we have constant dilaton, and a 10d metric (2.2) given by

$\displaystyle
ds^{2}=8\sqrt{2}R\Big{(}\frac{dr^{2}}{r^{2}}+\frac{r^{2}}{2R^{2}}ds^{2}_{AdS_{4}}+\big{(}d\varphi^{2}+\sin^{2}\varphi\,ds_{{\bf{S}}_{1}^{2}}^{2}+\cos^{2}\varphi\,ds_{{\bf{S}}_{2}^{2}}^{2}\big{)}\Big{)}$
(2.20)

The first two terms describe an AdS5, while those in brackets correspond to an
${\bf{S}}^{5}$. Their radius $R$ is related to (2.18) by the usual holographic
relation $R^{4}\sim N^{2}$.

The parameters of the ETW configuration will receive a natural interpretation
in terms of the holographic dual, in the next section. We note that, although
the parameters $k_{a}$, $l_{b}$, $d_{a}$, ${\tilde{b}}_{b}$ are subject to the
quantization conditions (2.16), (2.17), we treat them as continuous in the
supergravity approximation if needed.

The interpretation of these geometries is the following. For large $r=|w|$,
the geometry asymptotes to AdS${}_{5}\times{\bf{S}}^{5}$ with $N$ units of RR
5-form flux. For smaller values of $|w|$, there is a non-trivial warping, and
the solution ends at the origin $w=0$. The solution thus describes an
effective end of the world (ETW) configuration for
AdS${}_{5}\times{\bf{S}}^{5}$, described by a ‘fat brane’ with AdS4 symmetry.
Hence it corresponds to a string theory version of the Karch-Randall branes
Karch:2000ct .

A key property of these ETW branes is that they extend off to the holographic
boundary of the AdS5 geometry, which in fact, engulfs the holographic boundary
of the AdS4. This in fact sets the stage for the (double) holographic
interpretation in the next section. A picture of the geometry displaying the
ETW brane reaching to the holographic boundary is shown in Figure 4

Figure 4: The geometry shown as a piece of Poincaré
AdS${}_{5}\times{\bf{S}}^{5}$ ending on the ETW brane. The colored radial
lines correspond to the arcs of Figure 3 (inspired in Fig. 4 of
Raamsdonk:2020tin ).

###  2.3 Boundaries in 4d ${\cal{N}}=4$ SYM

In this section we review the holographic description of the solutions in
section 2.2.

As already mentioned, the ETW configurations in AdS5 in the previous section
reach out to the holographic boundary. Hence, they define the gravity dual of
4d ${\cal{N}}=4$ $SU(N)$ SYM in a semi-infinite 4d spacetime, namely with a 3d
boundary defect. The boundary defect is itself a 3d ${\cal{N}}=4$ CFT, and the
configuration ending the AdS${}_{5}\times{\bf{S}}^{5}$ provides its 4d gravity
dual, in the picture known as double holography Karch:2000gx ;
Takayanagi:2011zk  (see Karch:2022rvr  for a recent discussion in this
context). The boundaries preserve half of the super(conformal) symmetry, and
are of the kind considered in Gaiotto:2008ak , which can be described using
Hanany-Witten brane configurations Hanany:1996ie . In the following we review
the basic ingredients needed for our discussion.

Consider $N$ D3-branes along the directions 0123 and located at the same
position in the other directions. We can define a boundary for this theory by
letting the D3-branes end on suitable configurations of NS5- and D5-branes
localized in the direction 3, and spanning 012, and some additional
directions. In order to preserve the maximum supersymmetry, the NS5-branes
span the directions 012 456 (and are located in the same position as the
D3-branes in 789), while the D5-branes span the directions 012 789 (and are
located in the same position as the D3-branes in 456). We consider all
5-branes to be located on the same position in the direction 3, so that they
define a boundary for the whole 4d $SU(N)$ theory.333We note that the 5-branes
are taken to the same position in 3 starting from a setup where they maintain
a specific ordering, which encodes the specifics of the boundary
configuration. This has its reflection in the ordering of 5-brane poles
supergravity solution, but we will skip its discussion in the holographic
field theory dual context, referring the reader to Gaiotto:2008ak  for
details.

As studied in Gaiotto:2008ak , the boundary conditions of D3-branes ending on
a 5-brane (NS5- and D5-branes admit a similar description, due to S-duality)
are associated to a representation of $SU(2)$, which describes the commutation
relations of the three (matrix-valued) complex scalars of the 4d SYM, namely
$[X^{i},X^{j}]\sim\epsilon^{ijk}X^{k}$. Hence, if we consider a configuration
with $N_{5}$ NS5-branes and $N_{5}^{\prime}$ D5-branes, we need to partition
each of these quantities in a number of copies of the different irreducible
representations of $SU(2)$. In other words

$\displaystyle{\cal R}_{n}=\oplus_{a=1}^{n}n_{a}{\cal
R}_{K_{a}}\quad,\quad{\cal R}_{m}=\oplus_{b=1}^{m}m_{b}{\cal R}_{L_{b}}\,,$
(2.21)

where ${\cal R}_{K_{a}}$, ${\cal R}_{L_{b}}$ correspond to the irreducible
representations of $SU(2)$ of dimensions $K_{a}$, $L_{b}$, respectively (here
clearly $K_{a}\neq K_{a^{\prime}}$ for $a\neq a^{\prime}$, and similarly for
the $L_{b}$’s).

In other words, we split the NS5-branes into $n$ ‘stacks’, with multiplicity
$n_{a}$, $a=1\ldots,n$, with each NS5-brane in the $a^{th}$ defining boundary
conditions associated to the irreducible representation of $SU(2)$ of
dimension $K_{a}$; namely, there are $K_{a}$ D3-branes ending on each of the
$n_{a}$ NS5-branes in the $a^{th}$ stack. And similarly for the D5-branes.
Clearly, to define a boundary for the 4d ${\cal{N}}=4$ $SU(N)$, all the
D3-branes must end on some 5-brane, hence

$\displaystyle N=\sum_{a}n_{a}K_{a}+\sum m_{b}L_{b}\,,$ (2.22)

which matches (2.18). The double holographic interpretation is clear. The
boundary configuration is a Hanany-Witten Hanany:1996ie  set of D3-branes
suspended among NS5- and D5-branes, describing a 3d ${\cal{N}}=4$ CFT, in the
particular limit of all NS5- and D5-branes coincident in the direction 3
(strong coupling limit of the theory at the origin of its (completely) Higgsed
phase). Hence, the AdS4 ETW brane ending the AdS5 ($\times{\bf{S}}^{5}$)
spacetime is just the gravitational dual of this 3d CFT (see Assel:2011xz ;
Assel:2012cj  for this particular class of AdS4/CFT3 duals). The relation of
the parameters of the 4d CFT and its 3d boundary CFT determine the Poincaré
angle formed by the holographic boundary and the ETW boundary, as discussed in
detail in Raamsdonk:2020tin ; VanRaamsdonk:2021duo . Note that, the latter
reference considered the regimes where this angle is large; in contrast, we
impose no constraint on the parameters and consider general configurations.

Note that a similar description applies to the gravity solutions with two
asymptotic AdS${}_{5}\times{\bf{S}}^{5}$ regions (Janus configurations), even
with different numbers $N_{1}$, $N_{2}$ of units of RR 5-form flux, in section
2.2. Consider without loss of generality that $N_{1}>N_{2}$, and denote
$N=N_{1}-N_{2}$. The holographic description corresponds to $N_{2}$ infinite
D3-branes, intersected by the set of NS5- and D5-branes, and $N$ semi-infinite
D3-branes on top; the latter end on the 5-brane configuration with a similar
split into $SU(2)$ irreducible representations.

##  3 Dynamical cobordism for AdS${}_{5}\times{\bf{S}}^{5}$

###  3.1 The bagpipe geometries are dynamical cobordisms

The bagpipe geometries in the previous section allow to connect
AdS${}_{5}\times{\bf{S}}^{5}$ compactifications in a cobordism, which is in
fact realized in terms of a fully supersymmetric configuration.

An important point in these statements is that the configurations may in
general include regions corresponding to NS5- or D5-branes, as discussed in
section 2.1. One may fear that that they lead to additional non-compact spikes
which spoil the actual cobordism between AdS asymptotic spaces, basically as
in the problematic cobordism with D3-branes mentioned in the introduction.
However, it is easy to check that there is no such problem, and that the local
geometries including the backreaction of the 5-branes and the induced D3-brane
charge are actually compact. Indeed, it was shown in Aharony:2011yc  that the
10d metric around a 5-brane location (taken as ${\tilde{r}}=0$) has the form

$\displaystyle
ds^{2}=f_{4}^{2}\big{[}ds_{AdS_{4}}^{2}+ds_{{\bf{S}}^{2}_{2}}^{2}\big{]}+4{\tilde{r}}^{2}\rho^{2}\Big{[}\,d\psi^{2}+\sin^{2}\psi\,ds_{{\bf{S}}_{1}^{2}}^{2}+\frac{1}{{\tilde{r}}^{2}}d{\tilde{r}}^{2}\,\Big{]}\,,$
(3.1)

with

$\displaystyle\rho^{2}\sim{\tilde{r}}^{-\frac{3}{2}}|\log{\tilde{r}}|^{-\frac{1}{4}}\quad,\quad
f_{4}^{2}\sim{\tilde{r}}^{\frac{1}{2}}|\log{\tilde{r}}|^{\frac{3}{4}}\,.$
(3.2)

The local geometry is AdS${}_{4}\times{\bf{S}}^{2}\times{\bf{S}}^{3}$ fibered
over the radial direction ${\tilde{r}}$, which ends at a singularity at
${\tilde{r}}=0$, at finite distance in spacetime. The singularity is simply
due to the presence of an explicit source in the configuration, but it leads
to no non-compactness.

Hence the solutions describe interpolating solutions describing cobordisms
among AdS${}_{5}\times{\bf{S}}^{5}$ regions. It is clear that being able to
connect arbitrary numbers of AdS${}_{5}\times{\bf{S}}^{5}$ pipes actually
implies that the cobordism class of a single AdS${}_{5}\times{\bf{S}}^{5}$ is
trivial. This is in perfect agreement with the swampland cobordism
conjecture444This picture is amusingly reminiscent of the (admittedly non-
supersymmetric) solutions in Hellerman:2010dv ..

In particular, this must imply that any two AdS${}_{5}\times{\bf{S}}^{5}$
regions can be connected by some domain wall. This indeed corresponds to the
configurations with two asymptotic AdS${}_{5}\times{\bf{S}}^{5}$ regions in
section 2.2. Similarly, triviality in cobordism must imply that a single
AdS${}_{5}\times{\bf{S}}^{5}$ admits a boundary ending spacetime; this is
again indeed achieved by the geometry with a single asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$ in section 2.2.

The configurations with a single AdS${}_{5}\times{\bf{S}}^{5}$ spacetime
ending on an ETW configuration, when regarded from the 5d perspective, has the
structure of a dynamical cobordism, in the sense of Buratti:2021yia ;
Buratti:2021fiv ; Angius:2022aeq ; Blumenhagen:2022mqw ; Blumenhagen:2023abk .
Indeed, the solution corresponding to (2.14) describes, for large
$r\equiv|w|\gg\alpha_{g+1},\beta_{1}$, an asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$. As $|w|$ decreases towards the region
$r\simeq\alpha_{g+1},\beta_{1}$, there is a non-trivial running of the
solution as we move along the AdS4 slices of the foliation. In particular,
considering the ${\bf{S}}^{5}$ obtained by fibering the
${\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$ over the segment $w=re^{i\varphi}$,
$\pi/s\leq\varphi\leq\pi$, its geometry varies along $r$ (it squashes while
preserving the $SO(3)^{2}$ symmetry of the ${\bf{S}}^{2}$’s); similarly, the
dilaton also runs with $r$. The discussion of this running solution, from the
perspective of the 5d theory, is described in section 3.2. Finally, for $r\ll
l_{1}^{2},k_{n}^{2}$, the ${\bf{S}}^{5}$ becomes essentially round again, and
shrinks to zero size, closing off the geometry and ending spacetime. In
section 3.2.4 we discuss this small $r$ behaviour in term of the local
dynamical cobordism of Angius:2022aeq .

In the next section we expand of this interpretation of the solutions as
dynamical cobordisms of the 5d theory resulting after reduction of type IIB
theory on the ${\bf{S}}^{5}$ coordinates. Note that the interpretation of
these solutions from the lower-dimensional perspective aligns with the
viewpoint implicit in holography. Hence, we expect our approach to relate to
the bottom up descriptions in the study of AdS/BCFT and quantum islands.

###  3.2 A 5d effective theory description

The actual description of the above solutions as dynamical cobordisms requires
describing them as 5d running solutions ending in a 4d ETW configuration. In
this section we develop such 5d effective description.

We recall that, since already in AdS${}_{5}\times{\bf{S}}^{5}$ there is no
scale separation (in agreement with the AdS distance conjecture Lust:2019zwm
), this 5d theory should not be regarded as an effective field theory in the
Wilsonian sense, but rather as an ${\bf{S}}^{5}$ truncation. Namely, we
perform a reduction along the ${\bf{S}}^{5}$ keeping a finite set of modes
which capture the essential dynamics of the full configuration. As is well
known, for AdS${}_{5}\times{\bf{S}}^{5}$ this corresponds to a consistent
truncation, and the dynamics of the $SO(6)$ invariant sector has little mixing
with the KK modes, despite the lack of mass hierarchy in the KK tower. For the
configuration with ETW branes, the lower symmetry implies a more significant
role of the KK modes involved in the breaking $SO(6)\to SO(3)\times SO(3)$. We
will first describe the physics of the $SO(6)$ invariant truncation, and show
that it is the key player in the ending of spacetime; we subsequently discuss
the symmetry breaking modes, and show that they manifest as ETW brane
worldvolume modes. For a recent discussion of scale separation in this and
other contexts of gravity localization, see DeLuca:2023kjj .

####  3.2.1 The ${\bf{S}}^{5}$ reduction of the metric

We start the discussion of the $SO(6)$ invariant sector, and show it suffices
to capture the ending of the AdS5 spacetime on a smoothed out version of an
AdS4 Karch-Randall brane. The ETW configuration is to be described as a
running solution of a 5d theory including the 5d metric, the IIB complex
dilaton (although we restrict the discussion to vanishing axion), and a
breathing mode of a round ${\bf{S}}^{5}$. As explained, in the next section we
enrich the system by the inclusion of higher KK modes, describing the
squashing $SO(6)\to SO(3)^{2}$ of the ${\bf{S}}^{5}$.

Consider the 10d ETW solutions in section 2.2, whose the 10d metric (2.2) we
repeat for convenience

$\displaystyle
ds^{2}=f_{4}^{2}ds_{AdS_{4}}^{2}+f_{1}^{2}ds_{{\bf{S}}^{2}_{1}}^{2}+f_{2}^{2}ds_{{\bf{S}}_{2}^{2}}^{2}+4\rho^{2}(dr^{2}+r^{2}d\varphi^{2})$
(3.3)

where have have used polar coordinates for the Riemann surface $\Sigma$.
Recall that the ${\bf{S}}^{5}$ is obtained by fibering the
${\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$ over the interval
$\pi/2\leq\varphi\leq\pi$ at fixed value of the radial coordiante $r$

Hence, we carry out this reduction in two steps. First, we restrict to the
$SO(3)\times SO(3)$ invariant sector by performing the reduction on
${\bf{S}}^{2}_{1}\times{\bf{S}}^{2}_{2}$, by using the general ansatz

$\displaystyle
ds^{2}=e^{-\frac{1}{2}(\sigma_{1}+\sigma_{2})}ds_{6}^{2}+e^{\frac{1}{\sqrt{2}}(a\sigma_{1}-b\sigma_{2})}ds_{{\bf{S}}^{2}_{1}}^{2}+e^{\frac{1}{\sqrt{2}}(a\sigma_{2}-b\sigma_{1})}ds_{{\bf{S}}_{2}^{2}}^{2}\,,$
(3.4)

where $a,b$ will be fixed shortly. The resulting 6d metric is

$\displaystyle
ds_{6}^{2}=\sqrt{f_{1}^{2}f_{2}^{2}}\,[\,f_{4}^{2}ds_{AdS_{4}}^{2}+4\rho^{2}(dr^{2}+r^{2}d\varphi^{2})\,]\,.$
(3.5)

We also have two 6d scalars $\sigma_{1}$, $\sigma_{2}$ with profiles

$\displaystyle\sigma_{1}=\log(f_{1}^{a}f_{2}^{b})\quad,\quad\sigma_{2}=\log(f_{1}^{b}f_{2}^{a})\;,\;{\rm
with}\,a=1+\frac{1}{\sqrt{2}}\;,\;b=1-\frac{1}{\sqrt{2}}\,,$ (3.6)

where the parameters $a,b$ are fixed so that these scalars have canonical
kinetic term in the resulting 6d action, although we will not need it.

Using (2.4), (2.5), (2.6), the metric components can be recast as

$\displaystyle\sqrt{f_{1}^{2}f_{2}^{2}}\,f_{4}^{2}=4h_{1}h_{2}\quad,\quad\sqrt{f_{1}^{2}f_{2}^{2}}\,\rho^{2}=2|W|$
(3.7)

Using the harmonic function (2.14), they are

$\begin{split}\sqrt{f_{1}^{2}f_{2}^{2}}\,f_{4}^{2}=&-32r^{2}\sin
2\varphi-32r\cos\varphi\sum^{m}_{b=1}\tilde{d}_{b}\log\left(\frac{r^{2}+l_{b}^{2}+2rl_{b}\sin\varphi}{r^{2}+l_{b}^{2}-2rl_{b}\sin\varphi}\right)\\\
&-32r\sin\varphi\sum^{n}_{a=1}d_{a}\log\left(\frac{r^{2}+k_{a}^{2}+2rk_{a}\cos\varphi}{r^{2}+k_{a}^{2}-2rk_{a}\cos\varphi}\right)\\\
&-16\sum^{n}_{a=1}\sum^{m}_{b=1}d_{a}\tilde{d}_{b}\log\left(\frac{r^{2}+k_{a}^{2}+2rk_{a}\cos\varphi}{r^{2}+k_{a}^{2}-2rk_{a}\cos\varphi}\right)\log\left(\frac{r^{2}+l_{b}^{2}+2rl_{b}\sin\varphi}{r^{2}+l_{b}^{2}-2rl_{b}\sin\varphi}\right)\end{split}$
(3.8) $\begin{split}\sqrt{f_{1}^{2}f_{2}^{2}}\,\rho^{2}=&-32r^{2}\sin
2\varphi\left[\sum^{n}_{a=1}\frac{d_{a}k_{a}}{(r^{4}+k_{a}^{4}-2r^{2}k_{a}^{2}\cos
2\varphi)}+\sum^{m}_{b=1}\frac{\tilde{d}_{b}l_{b}}{(r^{4}+l_{b}^{4}+2r^{2}l_{b}^{2}\cos
2\varphi)}+\right.\\\
&+2\left.\sum^{n}_{a=1}\sum^{m}_{b=1}\frac{d_{a}\tilde{d}_{b}k_{a}l_{b}(k_{a}^{2}+l_{b}^{2})}{(r^{4}+k_{a}^{4}-2r^{2}k_{a}^{2}\cos
2\varphi)(r^{4}+l_{b}^{4}+2r^{2}l_{b}^{2}\cos 2\varphi)}\right]\end{split}$
(3.9)

For large $r$ the 6d metric behaves as

$\displaystyle ds^{2}=-64R^{2}\sin
2\varphi\bigg{(}\,\frac{dr^{2}}{r^{2}}+\frac{r^{2}}{2R^{2}}ds_{AdS_{4}}^{2}+d\varphi^{2}\bigg{)}\,.$
(3.10)

This clearly includes an AdS5 factor, and an ${\bf{S}}^{1}$ left over from
${\bf{S}}^{5}$ upon integration over ${\bf{S}}^{2}\times{\bf{S}}^{2}$, of
radius

$\displaystyle
R^{2}=\sum^{n}_{a=1}(\beta_{a}-k_{a}^{2})-\sum^{m}_{b=1}(\alpha_{b+n+1}+l_{b}^{2})$
(3.11)

We recall that this relates to the number of D3 branes $N$ by the usual
AdS/CFT dictionary.

The interpretation of the $\varphi$-dependent prefactor in (3.10) is simply
the zero mode $f_{0}$ in the KK tower of the 5d graviton. Fixing the
normalization $\int_{\pi/2}^{\pi}f_{0}^{2}=1$, we have

$f_{0}(\varphi)=-\frac{2}{\sqrt{\pi}}\sin 2\varphi\,,$ (3.12)

where the minus sign is introduced for later convenience. This allows to
define the ${\bf{S}}^{1}$ truncation onto the $SO(6)$ invariant sector. Since
KK modes form an orthonormal basis, the 5d avatar of the full 6d metric can be
extracted as

$\displaystyle\hat{g}_{\mu\nu}=\int_{\pi/2}^{\pi}f_{0}(\varphi)g_{\mu\nu}(r,\varphi)d\varphi$
(3.13)

This gives

$\displaystyle
ds_{5}^{2}=\hat{f}_{4}^{2}ds_{AdS_{4}}^{2}+4\hat{\rho}^{2}\,dr^{2}\,.$ (3.14)

with

$\begin{split}\hat{f}_{4}^{2}=&16\sqrt{\pi}r^{2}+32\sqrt{\pi}r\sum^{m}_{b=1}\tilde{d}_{b}\left(B_{b}-\frac{B_{b}^{3}}{3}\right)+32\sqrt{\pi}r\sum^{n}_{a=1}d_{a}\left(A_{a}-\frac{A_{a}^{3}}{3}\right)+\\\
&+64\sqrt{\pi}\sum^{n}_{a=1}\sum^{m}_{b=1}d_{a}\tilde{d}_{b}\left[A_{a}B_{b}+\frac{A_{a}^{2}+B_{b}^{2}}{2A_{a}^{2}B_{b}^{2}}\left(A_{a}B_{b}-(1+A_{a}^{2}B_{b}^{2})\tan^{-1}(A_{a}B_{b})\right)\right]\end{split}$
(3.15)

$\hat{\rho}^{2}=16\sqrt{\pi}\left[\sum^{n}_{a=1}\frac{d_{a}}{k_{a}}A_{a}^{2}+\sum^{m}_{b=1}\frac{\tilde{d}_{b}}{l_{b}}B_{b}^{2}+2\sum^{n}_{a=1}\sum^{m}_{b=1}d_{a}\tilde{d}_{b}k_{a}l_{b}\frac{A_{a}^{2}+B_{b}^{2}}{r^{4}+k_{a}^{2}l_{b}^{2}}\right]$
(3.16)

where

$\displaystyle A_{a}=\left\\{\begin{array}[]{cc}r/k_{a},&\ \ r\leq k_{a}\\\
k_{a}/r,&\ \ r>k_{a}\\\ \end{array}\right.\quad,\quad
B_{b}=\left\\{\begin{array}[]{cc}r/l_{b},&\ \ r\leq l_{b}\\\ l_{b}/r,&\ \
r>l_{b}\end{array}\right.\,.$ (3.21)

In order to compare with KR brane, it is convenient to change coordinates to
gather all the non-trivial structure of the metric in a unique warp factor $A$
in front of the AdS4 piece. We choose the parametrization

$\displaystyle
ds_{5d}^{2}=e^{2A(x)}ds_{AdS_{4}}^{2}+\frac{128R^{2}}{\sum^{n}_{a=1}k_{a}^{2}+\sum^{m}_{b=1}l_{b}^{2}}dx^{2}\,,$
(3.22)

Some illustrative plots of the quantities $\hat{f}_{4}^{2}$ and $A$, and their
implication in the KR localization of gravity are discussed in section 3.2.3.

####  3.2.2 The single 5-brane example

In order to get some insight we now consider the particular case of a single
stack of NS5-branes are the position $k$, the resulting factors for the 6d
metric (3.5) are

$\displaystyle\sqrt{f_{1}^{2}f_{2}^{2}}\,f_{4}^{2}=-32r^{2}\sin
2\varphi-16\,r\sin\varphi\,\frac{\beta-k^{2}}{k}\log\left(\frac{r^{2}+k^{2}+2rk\cos\varphi}{r^{2}+k^{2}-2rk\cos\varphi}\right)$
(3.23)

$\displaystyle\sqrt{f_{1}^{2}f_{2}^{2}}\,\rho^{2}=\frac{-16\,r^{2}(\beta-k^{2})\sin
2\varphi}{r^{4}+k^{4}-2r^{2}k^{2}\cos 2\varphi}$ (3.24)

In this case the radius is given by

$\displaystyle R^{2}=\beta-k^{2}$ (3.25)

For the components of the 5d metric (3.14) we have

$\hat{f}_{4}^{2}=16\sqrt{\pi}\,r^{2}+16\sqrt{\pi}\,r\,\frac{\beta-k^{2}}{k}\left(A-\frac{A^{3}}{3}\right)$
(3.26)

$\hat{\rho}^{2}=8\sqrt{\pi}\,\frac{\beta-k^{2}}{k^{2}}A^{2}$ (3.27)

with $A=\left\\{\begin{array}[]{cc}r/k,&\ \ r\leq k\\\ k/r,&\ \ r>k\\\
\end{array}\right.$.

The metric is continuous at $r=k$, and so are its first derivatives. The
curvature is finite but displays a discontinuity

$\displaystyle\Delta
R=R_{r=k^{+}}-R_{r=k^{-}}=\frac{-\beta}{2\sqrt{\pi}\,k^{2}(\beta-k^{2})}$
(3.28)

It is easy to check that there is no localized stress-energy tensor at the
discontinuity. The jump in the curvature is associated to a jump in the matter
stress-energy tensor, ultimately related to the change in the fluxes across
the 5-brane location.

It is interesting to express the 5d metric in the parametrization of
Karch:2000ct . Hence, we recast (3.14) as:

$\displaystyle
ds_{5d}^{2}=e^{2A(x)}ds_{AdS_{4}}^{2}+128\frac{\beta-k^{2}}{k^{2}}dx^{2}\,,$
(3.29)

where the prefactor of the $dx^{2}$ term is introduced to simplify the
expression for $A(x)$. Here $x$ relates to the previous $r$ as:

$\frac{r^{2}}{k^{2}}=\left\\{\begin{array}[]{cc}\frac{4x}{\pi^{1/4}k},&\,r\leq
k\\\ \exp\left(\frac{4x}{\pi^{1/4}k}-1\right),&\,r>k\\\ \end{array}\right.$
(3.30)

and there is an analytical, yet admittedly not simple, expression for $A(x)$:

$e^{2A(x)}=16\sqrt{\pi}k^{2}\left\\{\begin{array}[]{cc}\frac{4x}{\pi^{1/4}k}\left(\frac{\beta}{k^{2}}-\frac{\beta-k^{2}}{3k^{2}}\frac{4x}{\pi^{1/4}k}\right)&\,,\;x\leq\frac{\pi^{1/4}k}{4}\\\
\exp\left(\frac{4x}{\pi^{1/4}k}-1\right)-\frac{\beta-k^{2}}{3k^{2}}\exp\left(1-\frac{4x}{\pi^{1/4}k}\right)+\frac{\beta-k^{2}}{k^{2}}&\,,\;x>\frac{\pi^{1/4}k}{4}\\\
\end{array}\right.$ (3.31)

####  3.2.3 Localization of gravity

In order to get some intuition about the solutions, we provide plots for the
warp factors $\hat{f}_{4}^{2}$, $\hat{\rho}^{2}$ and $A(x)$, in a few
illustrative examples. The configurations display localized gravity when the
shape contains a bump, since it corresponds to a smoothed version of the cusp
in the KR branes in Karch:2000ct . The smoothing is because our 5d theory is
not one of pure gravity, but contains scalar degrees of freedom, to be
discussed in the next section. We also note that the 5d metric becomes
singular at $r=0$, reflecting the fact that the ${\bf{S}}^{5}$ shrinks at that
point, as we discuss in more detail in section 3.2.4.

We start considering one single 5-brane. Due to S-duality, we can focus
without loss of generality on the case of a single NS5-brane, located at
$r=k$. Figure 5 shows two illustrative examples of the KR warp factor $A$ in
(3.31) as a function of the spacetime distance $\Delta$ (proportional to $x$).

Figure 5: The warp factors $\hat{f}_{4}^{2}$ and $A$ for a single NS5-brane
for different values of its parameters. The curve is monotonic and exhibits no
bump, hence no localization of gravity.

The main features are that, as mentioned, the metric and first derivative are
continuous across $r=k$. The solution asymptotes to AdS5 at $r\to\infty$ and
develops a singularity ($A\to-\infty$) at the origin. But the most specific
feature of this case is the fact that it is monotonous and contains no bump.
Hence it does not lead to localization of gravity. It is easy to check that
this hold for different choices of the parameters, in particular of the
relative values of $\beta$ and $k^{2}$, as illustrated in Figure 5. For
$\beta\gg k^{2}$, the curve flattens out, but develop no bump.

It is easy to show that in order to have a bump in the warp factor $A$ and
hence localization of gravity, one needs to introduce both kinds of 5-branes
simultaneously. We illustrate this in Figure 6 in examples with one NS5- and
one D5-brane (taking $\alpha=-\beta$, $k=l$, similar results hold also for
unequal values, see later). The curves have the same asymptotic features as
discussed above, but generically have a bump, leading to a smooth version of
the cusp in the bottom-up description of KR branes.

Let us note the interesting pattern upon changing the relative size of $\beta$
and $k^{2}$, see Figure 6. For small value of $\beta-k^{2}$ there is no bump
even if we have both kinds of 5-branes in the system. On the other hand, as
the value of $\beta-k^{2}$ gets larger, the bump gets stronger, and more
decoupled from the asymptotic AdS5. This will be easily understood in Section
4.

Figure 6: Warp factors $\hat{f}_{4}^{2}$ and $A$ in examples with one NS5- and
one D5-branes, with equal parameters, for fixed $\beta$ and different choices
of $k$. There is a gravity localizing bump, which gets stronger as $k^{2}$
becomes smaller.

The above results continue to hold even if we allow the 5-branes to have
different parameters, see Figure 7 for an illustrative example.

Figure 7: The bump in the warp factors $\hat{f}_{4}^{4}$ and $A$ in an
illustrative example with two 5-branes with different parameters.

In conclusion, our 5d reduction of the metric reproduces a smooth version of
the warp factor of KR branes, in particular with a bump localizing gravity.
The smoothing of the cusp of KR branes is due to the presence of additional
scalar fields in the 5d theory. They are also responsible for the singularity
at the origin, associated to the shrinking of the ${\bf{S}}^{5}$, and hence
encoded in the dynamics of the its volume modulus, to which we turn next.

####  3.2.4 The 5d scalars

#### The $SO(6)$ invariant sector

There are several scalars that to be included in the 5d description. In this
section we focus on the $SO(6)$ invariant sector, namely the dilaton and the
${\bf{S}}^{5}$ breathing mode.

The 5d profile of the dilaton is obtained by the reduction of its 10d solution
on ${\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$, which is trivial, followed by a
truncation to the zero mode in the reduction on the ${\bf{S}}^{1}$
parametrized by $\varphi$. The zero mode is read out from the asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$ region, and is simply a constant function. Hence
the last ${\bf{S}}^{1}$ reduction is simply an average in $\varphi$ of the
dilaton profile.

One interesting observation is that, due to S-duality, the effects of NS5- and
D5-branes on the dilaton have compensating effects. Hence, in order to better
appreciate the generic behaviour of the dilaton it is useful to consider
configurations in which both kinds of 5-branes have different parameters. In
Figure 8 we show an illustrative example with one NS5- and one D5-brane. The
dilaton is asymptotically constant and gets perturbed in the vicinity of the
5-branes, which bend its slope in opposite ways for NS- and D5-branes. The
dilaton behaves as approximately piecewise linear function. This is however
not an exact behaviour, as can be seen in more generic situations, see for
instance Figure 9.

Figure 8: The dilaton for a configuration of one NS5- and one D5-brane with
different parameters, chose as in Figure 7. The dilaton behaves as
approximately piecewise linear function in the ETW brane region.

Figure 9: The dilaton for a fairly general configuration of NS5- and
D5-branes, for parameters $k_{1}=4$, $k_{2}=3$, $k_{3}=0.8$, $\beta_{1}=22$,
$\beta_{2}=12$, $\beta_{3}=1.1$, $l_{1}=1.4$, $l_{2}=3.5$, $\alpha_{5}=-4$
(and finally $\alpha_{6}=-17$ in the first figure, and $\alpha_{6}=-24$ in the
second).

We now consider the ${\bf{S}}^{5}$ breathing mode. In the asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$ region, this can be characterized by using the
6d scalars $\sigma_{1}$, $\sigma_{2}$ describing the breathing modes of the
${\bf{S}}^{2}$’s. In particular one may use any function of them, invariant
under the $1\leftrightarrow 2$ exchange. Different choices can differ in
detailed behaviour in the mid region, but this ambiguity is a mere reflection
of the non-trivial mixing of KK modes due to the breaking of symmetries at the
core of the ETW brane. In the following we proceed with a particular simple
choice, warning the reader that the resulting picture is necessarily only
reliable at the qualitative level.

We consider the two scalars $\phi_{1}$, $\phi_{2}$ corresponding to the
coefficients of the $ds_{{\bf{S}}_{1}^{2}}$, $ds_{{\bf{S}}_{2}^{2}}$ elements
in the 10d metric (2.2). Hence their profiles in the solution are given by

$\phi_{1}\equiv f_{1}^{2}\ ,\ \ \ \ \phi_{2}\equiv f_{2}^{2}$ (3.32)

Note that at large $r$ the asymptotic metric (2.20) implies that the
(normalized) zero modes are

$\phi_{1}^{(0)}(\varphi)=\frac{4}{\sqrt{3\pi}}\sin^{2}\varphi\quad,\quad\phi_{2}^{(0)}(\varphi)=\frac{4}{\sqrt{3\pi}}\cos^{2}\varphi$
(3.33)

The scalar $\phi_{+}$ parametrizing the size of the ${\bf{S}}^{5}$ can thus be
taken to correspond to the projection of $\phi_{1}+\phi_{2}$ in the solution
along the zero mode, namely

$\phi_{+}=\frac{4}{\sqrt{3\pi}}\int_{\pi/2}^{\pi}\left(\sin^{2}\varphi\,f_{1}^{2}+\cos^{2}\varphi\,f_{2}^{2}\right)\,.$
(3.34)

This cannot be solved analytically, but can be computed numerically. Some
illustrative examples of the behaviour of $\phi_{1}$, $\phi_{2}$ and
$\phi_{+}$ are shown in Figure 10, for one NS5-brane and for one NS5- and one
D5-brane. In the latter case, we have chosen different parameters for the two
5-branes, so as to see their different effect in $\phi_{1}$ and $\phi_{2}$.

Figure 10: The ${\bf{S}}^{5}$ breathing mode in two illustrative examples of a
single NS5-brane, and one NS5- and one D5-brane with different parameters
(chosen as in Figure 7).

As expected, the ${\bf{S}}^{5}$ breathing mode goes to a constant value in the
asymptotic AdS5 region, and it goes to zero at the origin, where the
${\bf{S}}^{5}$ shrinks and spacetime ends. It is easy to show that the scalar
traverses an infinite distance in field space traversed in the finite
spacetime distance to the origin; this leads to the metric singularity at the
origin mentioned in the previous section. In fact, the behaviour of the 5d
solution near the origin falls in the local description of dynamical
cobordisms in Angius:2022aeq , and in particular leads to the scaling
relations of the spacetime distance $\Delta$, the field space distance ${\cal
D}$ and the spacetime curvature scalar $R$, as follows.

Near $r=0$ the dilaton is essentially constant, there is no RR 5-form flux,
and the ${\bf{S}}^{5}$ metric is round, the local description of the dynamical
cobordism is just an ${\bf{R}}^{6}$ split as ${\bf{S}}^{5}$ slices shrinking
as $r\to 0$. Indeed expanding (2.14) for small $r$ we have $h_{1}=4y$,
$h_{2}=-4x$, which lead to the 10d metric

$\displaystyle
ds^{2}=F_{1}\Big{(}F_{2}\,ds^{2}_{AdS_{4}}+dr^{2}+r^{2}\big{(}d\varphi^{2}+\sin^{2}\varphi\,ds_{{\bf{S}}_{1}^{2}}^{2}+\cos^{2}\varphi\,ds_{{\bf{S}}_{2}^{2}}^{2}\big{)}\Big{)}\,,$
(3.35)

where $F_{1}$, $F_{2}$ are constants involving the brane parameters. We see
that the ${\bf{S}}^{5}$ is round, so that we recover a simpler reduction to
get the 5d Einstein metric with a breathing mode

$\displaystyle
ds^{2}=e^{\sqrt{\frac{5}{6}}\omega}ds_{5}^{2}+e^{-\sqrt{\frac{3}{10}}\omega}ds_{{\bf{S}}^{5}}^{2}\,,$
(3.36)

such that the 5d metric is in the Einstein frame and the scalar is canonically
normalized. The spacetime distance to the singularity is

$\displaystyle\Delta=\int g_{rr}^{1/2}dr\propto\int r^{5/3}dr\sim r^{8/3}$
(3.37)

The spacetime scalar curvature near $r=0$ is given by

$\displaystyle R=-20F_{1}^{-8/3}r^{-16/3}\quad\Rightarrow R\sim\Delta^{-2}$
(3.38)

On the other hand, the distance traversed in field space555One may worry that
we are measuring the distance using the field variable $\omega$, whereas above
we have used $\phi_{+}$ to describe the breathing mode. The result is however
unchanged since near $r=0$ they are related by a field redefinition. is

$\displaystyle{\cal D}=\int
g_{\omega\omega}^{1/2}d\omega=\omega\,\sim-\sqrt{\frac{10}{3}}\log
r^{2}\sim\,-\sqrt{\frac{15}{8}}\log\Delta$ (3.39)

This reproduces the scaling laws in Angius:2022aeq  for a critical exponent
$\delta=2\sqrt{\frac{8}{15}}$. Hence, the breathing mode attains infinite
distance in field space at finite spacetime distance, in agreement for the
criteria of a dynamical cobordism to nothing in Buratti:2021fiv ;
Angius:2022aeq .

It is interesting to notice that the local description in Angius:2022aeq
captures only the collapse of the ${\bf{S}}^{5}$, but not other features of
the ETW brane, such as the KR bump. This shows that effects such as
localization of gravity requires a description beyond that in Angius:2022aeq .
We will however show in Section 4 that it is possible to construct a local
description rich enough to account for all features of the ETW brane,
including localized gravity.

The behaviour of the ${\bf{S}}^{5}$ breathing mode thus confirms that the 10d
solutions describing ETW branes for AdS${}_{5}\times{\bf{S}}^{5}$ correspond
to dynamical cobordisms from the 5d perspective. Note however that, although
the behaviour near the origin is a local dynamical cobordism of the kind in
Angius:2022aeq , the whole ETW configurations involves additional structure,
at a scale fixed by $\beta_{1}$, $\alpha_{g+1}$, necessary to remove the RR
5-form flux, and ultimately leading to localization of gravity. This is a
novel feature compared with earlier realizations of dynamical cobordisms.

Another interesting feature is that, in the case with both NS5- and D5-branes
the ${\bf{S}}^{5}$ describes a bump with a maximum around the ETW brane
region. It is easy to check numerically, see Figure 11, that the bump
strengthens when $\beta\gg k^{2}$. This signals the growing of an almost
independent compact geometry in the ETW region, with respect to which the
asymptotic AdS5 geometry is suppressed. This will receive an intuitive
explanation in Section 4.

Figure 11: The ${\bf{S}}^{5}$ breathing mode for one NS5- and one D5-brane,
for the choices of parameters in Figure 6. As $k$ decreases, for fixed
$\beta$, the size of the ETW region increases.

#### The $SO(3)\times SO(3)$ squashing

The above discussion leads to a picture of a boundary described by a
configuration corresponding to a KR-like brane in AdS5, dressed with a running
scalar dealing with the evolution and shrinking of the ${\bf{S}}^{5}$, albeit
in an $SO(6)$ invariant fashion. One may argue that this description misses
important information about the boundary conditions, in particular the fact
that they preserve only $SO(3)\times SO(3)$. It should be clear from our
discussion above that it is easy to systematically include extra degrees of
freedom, corresponding to higher KK modes in the reduction in the
${\bf{S}}^{1}$ parametrized by $\varphi$. A general feature is that, since the
${\bf{S}}^{5}$ tends to the round metric both at large $r$ and at small
$r\simeq 0$, those 5d scalars will have a non-trivial profile only around the
bump in the 5d metric; namely, they correspond to worldvolume modes of the
smoothed KR brane (see Kanda:2023zse  for KR branes with worldvolume scalars).

In particular, a simple $SO(6)\to SO(3)\times SO(3)$ breaking mode $\phi_{-}$
is obtained from the reduction of the 6d combination
$\phi_{-}=\phi_{1}-\phi_{2}$ along the zero modes (3.33), namely the
antisymmetric combination of (3.34). Again, this integration is not amenable
to analytic treatment, but can be studied numerically. The resulting profiles
in some illustrative examples are shown in Figures 12.

Figure 12: The $SO(6)\to SO(3)\times SO(3)$ squashing mode, for the choice of
5-branes and parameters of Figure 10.

As announced, the mode $\phi_{-}$ is supported on the bump and goes to zero at
large and small distance. It is interesting to notice that the scalar
associated to the corresponding 5-brane (namely, $\phi_{1}$ for D5-branes and
$\phi_{2}$ for NS5-branes) suffers a discontinuity in crossing the 5-brane
location. This implies that the discontinuities in $\phi_{-}$ have opposite
sign for NS5- and D5-branes, providing a 5d criterion to distinguish them.

As a last remark, we would like to emphasize that, besides the kind of KK
modes of the metric discussed above, there are additional scalars arising from
the reduction of the 2-forms, but we skip their discussion (see Aharony:2011yc
for a discussion of the lowest such mode). In fact, there is actually a tower
of $SO(6)\to SO(3)\times SO(3)$ breaking modes localized on the ETW branes,
with no mass hierarchy among them. This is actually expected, as these modes
in AdS4 are the gravity duals of the chiral operators in the holographic BCFT.
In fact, this latter language may provide the best approach to characterizing
the ETW brane worldvolume dynamics Assel:2011xz , as we also discuss in the
next Section.

##  4 Zoom into the ETW brane: A double scaling limit

In this section we present a, to our knowledge new, scaling limit of the 10d
solutions in section 2.2, which will be useful to isolate the physics of the
ETW brane configuration ending the AdS5 spacetime. It allows for a simple
intuitive understanding of some key properties of the ETW branes, noticed in
earlier sections. Interestingly, the limit ends up being equivalent to closing
off the last AdS${}_{5}\times{\bf{S}}^{5}$ asymptotic region in the solutions
in section 2.2, and is closely related to solutions considered in Assel:2011xz
, and setups of wedge holography Akal:2020wfl ; VanRaamsdonk:2021duo .

###  4.1 A scaling symmetry

Let us start by considering an invariance property of the 10d solutions in
section 2.2, focusing on the case of one asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$ region. Consider the scaling

$\displaystyle w\to\lambda w\quad,\quad k_{a}\to\lambda k_{a}\quad,\quad
l_{b}\to\lambda
l_{b}\quad,\quad\beta_{a}\to\lambda^{2}\beta_{a}\quad,\quad\alpha_{b}\to\lambda^{2}\alpha_{b}\,,$
(4.1)

which preserves the ordering (2.13). From (2.15) and (2.14), we have the
scaling

$\displaystyle d_{a}\to\lambda
d_{a}\quad,\quad{\tilde{d}}_{b}\to\lambda{\tilde{d}}_{a}\quad,\quad
h_{1}\to\lambda h_{1}\quad,\quad h_{2}\to\lambda h_{2}\,.$ (4.2)

Then from (2.4) and (2.5), we get

$\displaystyle W\to W\quad,\quad N_{1}\to\lambda^{2}N_{1}\quad,\quad
N_{2}\to\lambda^{2}N_{2}\quad e^{2\Phi}\to e^{2\Phi}\,,$ (4.3)

and the scalings of the functions (2.6) are

$\displaystyle\rho^{2}\to\lambda^{-1}\rho^{2}\quad,\quad f_{1}^{2}\to\lambda
f_{1}^{2}\quad,\quad f_{2}^{2}\to\lambda f_{2}^{2}\quad,\quad
f_{4}^{2}\to\lambda f_{4}^{2}$ (4.4)

Hence, using also (2.3), the 10d metric (2.2) experiences an overall rescaling

$\displaystyle ds^{2}\to\lambda ds^{2}\,.$ (4.5)

The metric is thus invariant up to an overall constant rescaling.

###  4.2 Double scale zooming into the ETW configuration

Our motivation in this work is to better understand the structure of the ETW
configuration ending spacetime. For that purpose, we now consider a double
scaling limit which zooms into the ETW configuration. This will be an
interesting approach to isolate its properties, and to provide a direct 10d
description of the KR AdS4 branes.

A naive limit to focus on the region ending spacetime would be to just zoom
onto $w=0$. However, this is only a smooth point in the 10d geometry, which
misses all the information about the 5-brane configurations, hence leads to a
rather trivial limit. Therefore we consider a more interesting double scaling
limit

$\displaystyle w\to\lambda w\quad,\quad k_{a}\to\lambda k_{a}\quad,\quad
l_{b}\to\lambda l_{b}\quad,\quad{\rm as}\;\lambda\to 0\,,$ (4.6)

which sends all the 5-branes towards the origin as we zoom onto it (and them).
In order to preserve the ordering (2.13), we also take

$\displaystyle\beta_{a}\to\lambda^{2}\beta_{a}\;\;{\rm for}\;a\neq
1\quad;\quad\alpha_{b}\to\lambda^{2}\alpha_{b}\;\;{\rm for}\;b\neq g+1\,.$
(4.7)

Namely, we send all points in the ordering to the origin, except for the
extreme ones $\beta_{1}$, $\alpha_{g+1}$. We emphasize that this scaling is
possible only if the configuration contains both NS5- and D5-branes
simultaneously. In the absence of D5-branes (i.e. no $l_{b}$’s), the sequence
(2.13) contains no $\alpha$’s (equivalently, the only one in (2.12) had
already been sent to $0$ in closing off the AdS${}_{5}\times{\bf{S}}^{5}$
region at the origin, so it cannot kept finite in the scaling limit). We will
come back to the need of both kinds of 5-branes later on.

The reason to keep $\beta_{1}$, $\alpha_{g+1}$ finite is that, as explained
above, they set the effective size of the fat ETW brane. Note also that this
is the crucial difference with respect to the scaling (4.1) above, and will
lead to a non-trivial change in the configuration.

In particular, the leading terms in (2.15) are

$\displaystyle d_{1}=\lambda^{-1}\frac{\beta_{1}}{2k_{1}}\prod_{c\neq
1}^{n}\frac{(k_{1}^{2}-\beta_{c})}{(k_{1}^{2}-k_{c}^{2})}\quad,\quad
d_{a}=-\lambda^{-1}\frac{\beta_{1}(\beta_{a}-k_{a}^{2})}{2k_{a}(k_{a}^{2}-k_{1}^{2})}\prod_{c\neq
a,1}^{n}\frac{k_{a}^{2}-\beta_{c}}{k_{a}^{2}-k_{c}^{2}}\quad\,\;a\neq
1\quad\quad\quad\quad\quad\quad\quad$ (4.8)
$\displaystyle\\!\\!{\tilde{d}}_{g-n}\\!=\\!-\lambda^{-1}\frac{\alpha_{g+1}}{2l_{g-n}}\prod_{c\neq
g-n}^{m}\\!\\!\\!\\!\frac{(l_{g-n}^{2}+\alpha_{c+n+1})}{(l_{g-n}^{2}-l_{c}^{2})},\;{\tilde{d}}_{b}\\!=\\!\lambda^{-1}\frac{\alpha_{g+1}(-\alpha_{b+n+a}-l_{b}^{2})}{2l_{b}(l_{b}^{2}-l_{g-n}^{2})}\\!\\!\\!\prod_{c\neq
b,g-n}^{n}\\!\\!\\!\\!\frac{(l_{b}^{2}-\alpha_{c+n+1})}{(l_{b}^{2}-l_{c}^{2})}\;,\quad\quad\quad\quad$
$\displaystyle b\neq g-n\,.\quad\quad\quad\quad$

Since they scale as $\lambda^{-1}$, the first term in (2.14) is subleading and
drops out, leading to

$\displaystyle
h_{1}=\;2\sum_{b=1}^{m}{\tilde{d}}_{b}\log\left(\frac{|w+il_{b}|^{2}}{|w-il_{b}|^{2}}\right)\quad,\quad
h_{2}=-2\sum_{a=1}^{n}d_{a}\log\left(\frac{|w+k_{a}|^{2}}{|w-k_{a}|^{2}}\right)\,,$
(4.9)

They also scale as $\lambda^{-1}$, and using (2.4) we get that $W$ scales as
$\lambda^{-4}$, and $N_{1}$, $N_{2}$ scale as $\lambda^{-6}$. From (2.5), the
dilaton has no scaling with $\lambda$, and from (2.6), we have that $\rho^{2}$
scales as $\lambda^{-3}$, while $f_{1}^{2}$, $f_{2}^{2}$ and $f_{4}^{4}$ all
scale as $\lambda^{-1}$. Note then that the 10d metric (2.2) scales as
$\lambda^{-1}$. Forgetting this overall factor, we have a metric of the kind
considered in section 2.2, with the modified $h$-functions in (4.9).

Let us discuss some key properties of the resulting geometry: The asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$ region at $r\to\infty$ disappears.
Correspondingly, the asymptotic 5-form flux effectively disappears. The
simplest way to show this is that the point $r\to\infty$ is actually at finite
distance in the 10d Einstein metric, and in fact corresponds to a smooth point
in the geometry. In other words, the 6d geometry ${\bf{X}}_{6}$ transverse to
AdS4 is actually compact: the configuration is a compactification
AdS${}_{4}\times{\bf{X}}_{6}$.

This may seem surprising, but actually matches the intuition that the scaling
limit encodes the dynamics of the ETW brane configuration. In particular, its
compactness implies the existence of a localized graviton, one of the main
properties of our ETW branes. Actually the localized graviton is a massive one
in the full geometry before the scaling limit, due to its coupling to the bulk
AdS5; this is a subleading $\lambda^{-1}$ effect in our description and
disappears in the limit. The fact that the massive graviton can become
massless in a continuous fashion is related to the discussion in Karch:2001jb
.

There is also an extra bonus: the fact that the scaling limit (and its
graviton zero mode) exist only if both NS5- and D5-are present explains the
observation in section 3.2.1 that the KR bump localizing gravity in the 5d
metric appears only when both kinds of 5-branes are present. Note also that an
additional requirement for localized gravity in section 3.2.1 was the
hierarchy between $\alpha,\beta$ and the 5-brane locations $k,l$; this is
automatically built in in the scaling limit, since the former are kept finite
while the latter are sent to 0.

The fact that the solution has only 5-brane charge and no 5-form flux would
seem to suggest that it is just the near horizon solution of a stack of
5-branes. However, our solution maintains all the information about the
$k_{a}$, $l_{b}$, $d_{a}$, ${\tilde{d}}_{b}$, hence all the information about
the boundary conditions of the holographic CFT. The limit thus isolates the
relevant information about the ETW brane in the gravity side.

In fact, 10d geometries defined by logarithmic functions $h_{1},h_{2}$ as in
(4.9) were considered in Assel:2011xz ; VanRaamsdonk:2021duo  to define the
gravitational dual of 3d ${\cal{N}}=4$ theories. In our context, this
AdS4/CFT3 duality is just the holography of the gravitational theory on the
ETW brane worldvolume and the BCFT of the initial 4d CFT.

###  4.3 The limit revisited: closing off AdS${}_{5}\times{\bf{S}}^{5}$

We now provide an alternative description for the limit (4.6), (4.7). Recall
the invariance at the beginning of this section i.e. the rescaling (4.1) of
$w$ and all the special points $\alpha_{b}$, $\beta_{a}$, $k_{a}^{2}$,
$l_{b}^{2}$. Using this invariance, the above limit of sending to zero $w$ and
all the special points, except $\alpha_{g+1}$, $\beta_{1}$, is equivalent to
keeping $w$ and all special points fixed, except for $\alpha_{g+1}$,
$\beta_{1}$, which are sent to infinity. This can be seen from (4.8), by
reinterpreting the prefactor $\lambda^{-1}$ as a scaling associated to
$\alpha_{g+1}$, $\beta_{1}$, while the other quantities in the expressions
remain constant. Note that, as above, the limit exists only if both NS5- and
D5-branes are present in the configuration, otherwise at least one of the
parameters $\alpha_{g+1}$, $\beta_{1}$ is missing, and we cannot send it to
infinity.

In this description, many of the features observed in the previous section
have a simple interpretation. Recalling the description in section 2.1,
asymptotic AdS${}_{5}\times{\bf{S}}^{5}$ regions arise ‘bare’ branch points
$e_{i}$, and they are closed off when the adjacent $\alpha$ and $\beta$
collapse onto it. In this description, the single asymptotic
AdS${}_{5}\times{\bf{S}}^{5}$ region in the solution in section 2.2 is
associated to the branch point $e_{2g+2}$, which is located at infinity. The
above limit of sending $\beta_{1}$ and $\alpha_{g+1}$ to infinity simply
corresponds to collapsing them onto $e_{2g+2}$, and therefore to closing off
the AdS${}_{5}\times{\bf{S}}^{5}$ region.

Since this was the only non-compact spike of the initial geometry, the
resulting space is compact. It describes the gravitational dual of the BCFT
defined by the 5-brane configuration. Since it corresponds to a
compactification, it has a graviton zero mode; this nicely corresponds to the
graviton localized by the ETW brane, matching the expected behaviour for AdS4
KR branes. Again, the coupling of the ETW brane to the to the asymptotic AdS5
region, and the corresponding mass for the localized graviton, are switched
off in the limit, leaving the graviton massless. Note also that we recover the
connection between having both NS5- and D5-branes simultaneously (and the
hierarchy of the parameters $\alpha_{g+1},\beta_{2}$ and $k_{a},l_{b}$) and
the appearance of localized gravity, observed in section 3.2.1, as in the
previous section.

###  4.4 Reduction in the double scaled limit

We have noticed some relations between the localization of gravity in the 5d
effective action approach in section 3.2.1 and in the scaling limit. This
motivates performing a 5d reduction of the 10d solution obtained after the
scaling limit, namely using the functions (4.9). This is very analogous to the
procedure in section 3.2, hence we simply sketch some general features, in a
few plots analogous to those in section 3.2. Note that, due to the compactness
of the internal space, the plots have finite extent in the radial direction.

Figure 13 provides the warp factors of the metric in the scaling limit, for
the choices of brane configurations and parameters of Figure 6. The axes scale
with $\lambda$ as dictated by the KR metric (3.22). It is clear that the
solution after the scaling provides a very good description of the ETW brane
region of the full solution in Figure 6. Note that $\hat{f}_{4}^{2}$ goes to
zero (and $A\to-\infty$) at the endpoints of the finite extent of the
geometry, signalling the closing off of the asymptotic AdS5 region.

Figure 13: Warp factors in the ETW brane for one NS5- and one D5-brane, for a
choice of parameters related to those in Figure 6.

The behaviour of the dilaton is shown in two prototypical cases in Figure 14.
Since there is no AdS5 asymptotic regions, there is an undetermined constant
(the average of the dilaton in the ETW volume) which we have fixed to some
arbitrary value.

Figure 14: Dilaton in the ETW brane for one NS5- and one D5-brane, for the
choice of parameters of Figure 8 and Figure 9-right.

One can also study the scalars $\phi_{+}$, describing the evolution of the
${\bf{S}}^{5}$ size, and $\phi_{-}$, describing its squashing. In Figure 15 we
show them in an illustrative example. Note that the ETW geometry has
${\bf{S}}^{6}$ topology, with the 5-branes introducing deviations from the
round metric.

Figure 15: The scalars $\phi_{\pm}$ for one NS5- and one D5-brane, for the
choice of parameters of Figure 13.

In the above discussion we have worked in the 5d perspective and studied how
different quantities evolve in the direction $r$. Since the latter is compact,
it also makes sense to perform a dimensional reduction on it and describe the
dynamics on AdS4. This is actually in the spirit of AdS$4$/CFT3 already
considered in Assel:2011xz , so we will not develop it further in this work.

##  5 ETW branes in gravity duals of 4d ${\cal{N}}=2$ orbifold theories

In holography, understanding a gauge/gravity dual pair leads to many other
examples with reduced (super)symmetry by the procedure of
orbifolding/orientifolding. In this and the following sections we will
describe quotients which are compatible with structure of the BCFT, and which
hence provide ETW configurations for AdS5 geometries with orbifolded internal
spaces.

In this section we consider D3-branes at perturbative orbifold singularities
${\bf{C}}^{2}/{\bf{Z}}_{k}$, projecting the 4d ${\cal{N}}=4$ SYM theory down
to a 4d ${\cal{N}}=2$ quiver gauge theory Douglas:1996sw , whose gravity duals
are AdS${}_{5}\times{\bf{S}}^{5}/{\bf{Z}}_{k}$ geometries. We will show that
the configurations of NS5- and D5-branes of the parent theory are invariant
under the orbifold action an descend to the orbifold quotient. Hence, they
define BCFTs for the 4d ${\cal{N}}=2$ quiver gauge theories, and their gravity
dual are suitable orbifolds of the ETW configurations in section 2.2.

###  5.1 The 4d ${\cal{N}}=2$ quiver gauge theory and its gravity dual

Consider a set of D3-branes located at the origin in a
${\bf{C}}^{2}/{\bf{Z}}_{k}$ singularity666One may consider
${\bf{C}}^{2}/\Gamma$ singularities, with $\Gamma$ other subgroups of $SU(2)$
(i.e. $D_{k}$, $E_{6,7,8}$). We leave this development for future work.. For
concreteness, we let the D3-branes span the direction 0123, and let the
generator $\Theta$ of ${\bf{Z}}_{k}$ act on the complex coordinates
$z_{1}=x^{4}+ix^{5}$, $z_{2}=x^{7}+ix^{8}$ as

$\displaystyle\Theta:(z_{1},z_{2})\;\rightarrow\;(e^{\frac{2\pi
i}{k}z_{1}},e^{-\frac{2\pi i}{k}z_{2}})\,.$ (5.1)

One must also specify the action of $\Theta$ on the Chan-Paton indices of the
D3-branes. Considering a stack of $Nk$ D3-branes in the covering space, the
orbifold Chan-Paton action is a $U(Nk)$ matrix $\gamma_{\Theta,3}$ of order
$k$, in other words, a representation of dimension $Nk$ of ${\bf{Z}}_{k}$. In
order to obtain 4d ${\cal{N}}=2$ SCFT’s Kachru:1998ys ; Lawrence:1998ja ;
Hanany:1998ru ; Hanany:1998it , we choose $N$ copies of the regular
representation

$\displaystyle\gamma_{\Theta,3}={\bf{1}}_{N}\otimes(\,1,e^{2\pi
i/k},\ldots,e^{2\pi i\,(k-1)/k}\,)\,.$ (5.2)

The theory on the worldvolume of the D3-branes has gauge group777 The $U(1)$’s
are massive due to BF couplings on the D3-brane worldvolumes; this is dual to
the brane bending phenomenon in Witten:1997sc . $\prod_{i}SU(N)_{i}$ and
hypermultiplets in the bifundamental representation
$(~{}\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}~{}_{i},~{}\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}~{}_{i+1})$.
This is usually represented as a circular quiver with $k$ nodes, and links
joining neighbouring nodes.

As discussed in Kachru:1998ys ; Lawrence:1998ja ; Hanany:1998it , the gravity
dual of these theories is given by type IIB on
AdS${}_{5}\times{\bf{S}}^{5}/{\bf{Z}}_{k}$. The action of ${\bf{Z}}_{k}$ is as
follows: Introduce and extra complex coordinate $z_{3}=x^{6}+ix^{9}$, and
describe the ${\bf{S}}^{5}$ as

$\displaystyle|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}=R^{2}\,.$ (5.3)

The action of ${\bf{Z}}_{k}$ on ${\bf{S}}^{5}$ leaves $z_{3}$ invariant and
acts on $z_{1},z_{2}$ as in (5.1). There is an ${\bf{S}}^{1}$ of
${\bf{C}}^{2}/{\bf{Z}}_{k}$ fixed points corresponding to $z_{1}=z_{2}=0$,
$|z_{3}|=R$. This leads to a set of fields localized at such singularities,
see Hanany:1998it ; Gukov:1998kk  for their holographic dictionary.

###  5.2 The ETW orbifold configurations

In the brane construction it is clear that we can introduce NS5- and D5-brane
boundary defects in the parent ${\cal{N}}=4$ theory which survive the orbifold
projection and hence yield boundary defects in the daughter ${\cal{N}}=2$
theories. In fact, as in Section 2.3, we introduce NS5-branes along 012456
(hence, spanning $z_{1}$, localized at the origin in $z_{2}$, and along the
real axis in $z_{3}$) and D5-branes along 012789 (hence, at the origin in
$z_{1}$, spanning $z_{2}$, and along the imaginary axis in $z_{3}$).

The action of ${\bf{Z}}_{k}$ on D5-branes is easily discussed, in close
analogy with D3-branes (the action on NS5-branes888Orbifolds of brane
constructions with NS5-branes and ${\bf{C}}^{2}/{\bf{Z}}_{k}$ orbifolds were
considered in a closely related setup with D4-branes in Lykken:1997gy , which
are a T-dual realization of brane boxes Hanany:1997tb ; Hanany:1998ru ;
Hanany:1998it . should also be similar, due to S-duality of the underlying
${\cal{N}}=2$ theory). Skipping a detailed derivation, it suffices to say that
the orbifold action on 5-brane gauge degrees of freedom splits them into $k$
stacks, which are also organized into a cyclic quiver. The ${\bf{Z}}_{k}$
quantum symmetry of the orbifold theory (simultaneous ${\bf{Z}}_{k}$ rotations
of the cyclic quivers of D3- and 5-branes), implies that the $i^{th}$
fractional D3-brane stack in (5.2) can end on the $i^{th}$ stack of 5-branes.
We also note that, since a complete ${\bf{Z}}_{k}$ orbit of fractional
D3-branes (a regular D3-brane) behaves as a D3-brane in the covering space, it
inherits the interpretation of boundary conditions of the parent 4d
${\cal{N}}=4$ theories, namely the quantities $K_{a}$, $L_{b}$. Hence, they
are naturally encoded in the dual geometries, which we consider next.

The gravity duals of the 4d ${\cal{N}}=2$ theories coupled to the BCFT are
just an orbifold of those considered in section 2.2. Recall that we have a
warped AdS${}_{4}\times{\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$ varying over
the Riemann surface $\Sigma$ parametrized by a quadrant in the $w$-plane. The
fibration preserves an $SO(3)\times SO(3)$ symmetry, corresponding to the
rotation of the 3-planes 456 and 789 in the brane picture. This means that the
${\bf{Z}}_{k}$ action (5.1) is in a subgroup of $SO(3)_{1}\times SO(3)_{2}$.
Let us introduce angular coordinates in the 3-planes 456 and 789 to
parametrize the ${\bf{S}}^{2}$’s as follows,

$\displaystyle x_{4}=R_{1}\cos\theta_{1}\sin\varphi_{1}\quad,\quad$
$\displaystyle x_{7}=R_{2}\cos\theta_{2}\sin\varphi_{2}$ (5.4) $\displaystyle
x_{5}=R_{1}\cos\theta_{1}\cos\varphi_{1}\quad,\quad$ $\displaystyle
x_{8}=R_{2}\cos\theta_{2}\cos\varphi_{2}$ $\displaystyle
x_{6}=R_{1}\sin\theta_{1}\quad,\quad$ $\displaystyle
x_{9}=R_{2}\sin\theta_{2}$

The generator $\Theta$ of ${\bf{Z}}_{k}$ acts as

$\displaystyle\Theta:\;\varphi_{1}\rightarrow e^{2\pi
i/k}\varphi_{1}\quad,\quad\varphi_{2}\rightarrow e^{-2\pi i/k}\varphi_{2}\,,$
(5.5)

leaving $\theta_{1},\theta_{2}$ invariant, as well as $w$ on the base
$\Sigma$.

At a generic position in $w$, the above action on
${\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$ has 4 fixed points, locally of the
form ${\bf{C}}^{2}/{\bf{Z}}_{k}$, with coordinates at the poles of the
spheres. Hence, one might think that there are four copies of the
corresponding twisted sector fields, propagating over $w$. This replication of
fields would disagree with those in the
AdS${}_{5}\times{\bf{S}}^{5}/{\bf{Z}}_{k}$ dual; it would also be puzzling
because $w$ parametrizes just a quadrant $\Sigma$, so the geometry on which
these twisted fields propagate would have boundaries. Both problems are
related, and are happily solved at once if we notice that the boundaries in
$\Sigma$ correspond to points where some ${\bf{S}}^{2}$ shrinks to zero, which
leads to a reduced number of fixed points. In other words, the fixed point set
of the ${\bf{Z}}_{k}$ action is a four-sheeted cover of $\Sigma$, with
different sheets joining at the boundaries of $\Sigma$. One can check that the
complete fixed set is topologically a complex plane (an orientable non-compact
Riemann surface with no handles and no boundaries) as follows:

Consider the polar coordinates $w=re^{i\varphi}$ and consider the
${\bf{S}}^{5}$ given by the fibration of
${\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$ over $\varphi$. At a generic point in
$\varphi$, there are four fixed points of the ${\bf{Z}}_{k}$ action on
${\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$, which we denote $P_{ij}$, with
$i,j=1,2$ labeling the two poles on each ${\bf{S}}^{2}$. However, at one of
the endpoints of $\varphi$ one of the 2-spheres e.g. ${\bf{S}}_{1}^{2}$
shrinks to zero and the four points collapse pairwise into two,
$P_{11}=P_{21}$, $P_{12}=P_{22}$; at the other endpoint of $\varphi$,
${\bf{S}}_{2}^{2}$ shrinks to zero size and the four points collapse to two in
a different pattern $P_{11}=P_{12}$, $P_{21}=P_{22}$. It is easy to see that
in the whole fibration over $\varphi$, the 4-sheeted fixed point set describes
a single ${\bf{S}}^{1}$ with no boundary. Adding now the radial coordinate
builds a real cone over this ${\bf{S}}^{1}$, showing the complete fixed point
set is a complex plane, as announced. Incidentally, note that the above
${\bf{S}}^{1}\subset{\bf{S}}^{5}$ is precisely the geometry of the fixed point
set in the AdS${}_{5}\times{\bf{S}}^{5}/{\bf{Z}}_{k}$ gravity dual of the bulk
${\cal{N}}=2$ theory; this is as should be, since the 10d ETW geometry
reproduces precisely such gravity dual in the large $r$ regime.

The fact that the fixed point set has no non-trivial topology in the ETW brane
region suggests that there are no localized modes of the bulk twisted sector
fields in the KR brane for these AdS5 geometries. Hence, they are adequately
described by the effective action description in section 3.2, by simply
reinterpreting $\phi_{+}$ as the breathing mode of
${\bf{S}}^{5}/{\bf{Z}}_{k}$.

###  5.3 Boundaries from Brane Box construction

It is known that 4d ${\cal{N}}=2$ quiver theories admit a brane construction
in terms of a configuration of D4-branes spanning 0123 and suspended in a
direction 6 between sets of NS5 branes spanning 012345 Witten:1997sc . Linear
quiver theories are obtained by having non-compact direction 6, while cyclic
quivers are obtained when the direction 6 parametrizes an ${\bf{S}}^{1}$. The
latter correspond to the theories of D3-branes at ${\bf{C}}^{2}/{\bf{Z}}_{k}$
singularities, and both setups are in fact related Karch:1998yv  by a
T-duality along 6, which turns the suspended D4- into fractional D3-branes,
and NS5-branes into the orbifold geometry.

In this section we show that the T-duality continues to hold in the presence
of the 5-brane configurations which define boundary conditions for the 4d
${\cal{N}}=2$ theories on D3-branes at ${\bf{C}}^{2}/{\bf{Z}}_{k}$
singularities. In other words, we describe how to introduce sets of 5-branes
in the above Hanany-Witten NS5/D4-brane configurations so as to introduce
boundary conditions.

Consider the following sets of branes

D4 | 0 | 1 | 2 | 3 | $\times$ | $\times$ | 6 | $\times$ | $\times$ | $\times$
---|---|---|---|---|---|---|---|---|---|---
NS5 | 0 | 1 | 2 | 3 | 4 | 5 | $\times$ | $\times$ | $\times$ | $\times$
NS5’ | 0 | 1 | 2 | $\times$ | 4 | $\times$ | 6 | 7 | $\times$ | $\times$

where a cross indicates the brane is localized in that direction, and
NS5’-branes are primed to emphasize their different orientation with respect
to NS5-branes.

It is easy to check that the system preserves 4 supercharges. As above, the
D4-branes are suspended in 6 between different NS5-branes, and we make the
direction 6 periodic. In addition, we introduce a single stack of NS5’-branes
and we specify that the D4-brane end on the NS5’-branes, so that they are
semi-infinite in the direction 3, see Figure 16. The 4d theory, arising from
the NS5- and D4-brane configurations, is 4d ${\cal{N}}=2$, but the
introduction of NS5’- branes breaks the 4d Poincaré invariance and breaks the
supersymmetry by 1/2. This is a confirmation of the symmetry preserved by the
boundary conditions we have discussed in the previous sections.

Figure 16: Configuration of D4-branes bounded by NS5- and NS5’-branes. The
direction 6 is compact by the identification of the dashed lines. The
direction 3 is part of the dimensions of the 4d ${\cal{N}}=2$ quiver gauge
theory with $k=4$ defined by the D4-branes suspended among the NS5-branes in
the direction 6; the NS5’-branes define the boundary in the direction 3.

Note that this is a D4-brane version of the D5-brane brane box configurations
introduced in Hanany:1997tb , with the NS’ brane bounding semi-infinite boxes,
filled with D4-branes on one side and empty on the other, to reproduce the
bounded dimension of the 4d gauge theory. We also note that the T-duality
along 6, relating the NS5/D4-systems to D3-branes at
${\bf{C}}^{2}/{\bf{Z}}_{k}$, also maps the NS5’-branes into the same kind of
NS5-branes in the previous section.

We may also consider the brane box version of the D5-branes defining
boundaries in the picture of D3-branes at ${\bf{C}}^{2}/{\bf{Z}}_{k}$. They
are easily identified as D6-branes with the following orientation

D6’ | 0 | 1 | 2 | $\times$ | $\times$ | 5 | 6 | $\times$ | 8 | 9
---|---|---|---|---|---|---|---|---|---|---

(where they are primed in analogy with the NS5’-branes). These can be
introduced in the earlier configuration of NS5/D4-branes and NS5’-branes,
still preserving 4 supersymmetries. Since the D6’-branes are localized in the
direction 3, they admit D4-branes ending on them and hence define boundary
conditions

Hence, the set of NS5’- and D6’-branes provide a set of boundary conditions,
specified by the partitioning of the set of $N$ gauge D4-branes (for each
quiver node) into sets ending on different boundary branes. As in the previous
discussions, the interpretation of the quantities $K_{a}$, $L_{b}$ remains as
describing the numbers of (regular) D3-branes ending on a given 5-brane with
boundary conditions associated to some $SU(2)$ representation, in the parent
theory.

##  6 ETW branes in gravity duals of ${\cal{N}}=3$ S-fold theories

In this section we extend the logic of the previous section, generalizing to
quotients including non-trivial duality relations. We focus on the particular
case of S-folds Garcia-Etxebarria:2015wns .

###  6.1 The 4d ${\cal{N}}=3$ theories on D3-branes

Let us go back to the configuration of D3-branes in flat space and perform an
orbifold quotient of the kind considered in Garcia-Etxebarria:2015wns .
Namely, we combine coordinates as

$\displaystyle z_{1}=x_{4}+ix_{7}\quad,\quad z_{2}=x_{5}+ix_{8}\quad,\quad
z_{3}=x_{6}+ix_{9}\,,$ (6.1)

and we also introduce the coordinate $z_{4}$ for the F-theory ${\bf{T}}^{2}$,
and mod by

$\displaystyle\Theta:z_{k}\to e^{2\pi iv_{k}}z_{k}\,,$ (6.2)

with

$\displaystyle
v=\big{(}\,\frac{1}{k},\frac{1}{k},\frac{1}{k},\frac{1}{k}\,\big{)}\,.$ (6.3)

This is the action in Garcia-Etxebarria:2015wns , up to an irrelevant change
in the choice of complex structure. The allowed values are $k=2,3,4,6$, so as
to have a consistent crystallographic action on the F-theory ${\bf{T}}^{2}$.

We are interested in considering S-fold actions of this kind leaving invariant
the configuration of NS5- and D5-branes providing boundary conditions for the
D3-branes as in section 2.3. The only possible choices are $k=2,4$. The simple
choice is $k=2$, which as explained in Garcia-Etxebarria:2015wns  corresponds
to introducing an O3-plane on top of the D3-branes, and actually preserves 4d
${\cal{N}}=4$. The O3-plane maps the NS5-branes to themselves and the
D5-branes to themselves, so their structure of numbers of stacks and
multiplicities are independent. Also, the string coupling is arbitrary, as the
action of $\Theta$ on $\tau$ is trivial. The gravity dual will be discussed in
the next section.

The only genuinely ${\cal{N}}=3$ case hence corresponds to $k=4$, generated by

$\displaystyle\Theta:(z_{1},z_{2},z_{3},z_{4})\to(iz_{1},iz_{2},iz_{3},iz_{4})\,.$
(6.4)

In terms of the cartesian coordinates we have

$\displaystyle\Theta:$
$\displaystyle(x_{4},x_{5},x_{6})\to(x_{7},x_{8},x_{9})$ (6.5)
$\displaystyle(x_{7},x_{8},x_{9})\to(-x_{4},-x_{5},-x_{6})$

and a simultaneous $SL(2,{\bf{Z}})$ action by the monodromy
$\begin{pmatrix}0&1\cr-1&0\end{pmatrix}$, which acts on $\tau$ as
$\tau\to-1/\tau$, so it is the $S$ generator of $SL(2,{\bf{Z}})$. Hence, in
this $k=4$ case, the NS5-branes are mapped to the D5-branes and viceversa999We
note that the element $\Theta^{2}$ in the S-fold group corresponds to the
above O3-plane ${\bf{Z}}_{2}$ action. So acting twice with $\Theta$ brings
each 5-brane to itself, up to this orientifold action., due to the
simultaneous rotation of the 3-planes they span and the $SL(2,{\bf{Z}})$
action on them. This implies that the structure of numbers of stacks and
multiplicities of must be identical for NS5- and D5-branes. Also, the solution
necessarily contains regimes of strong coupling, as in the parent S-fold
theory.

###  6.2 The ETW configurations in the gravity duals

In this section we sketch some properties of the ETW configurations
gravitational dual to the above BCFT configurations for the above 4d
${\cal{N}}=3$ S-fold theories.

#### The $k=2$ case

We start with the $k=2$ case. In the absence of boundaries, the gravity dual
of the D3-brane theory is AdS${}_{5}\times{\bf RP}_{5}$ Witten:1998xy , where
${\bf RP}_{5}$ is the quotient of ${\bf{S}}^{5}$ by the orientifold action
$\Omega(-1)^{F_{L}}{\cal R}$, with ${\cal R}$ sending any point to its
antipodes in ${\bf{S}}^{5}$. Writing the ${\bf{S}}^{5}$ in the ${\bf{R}}^{6}$
transverse to the D3-branes as

$\displaystyle
x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}+x_{9}^{2}=R^{2}$ (6.6)

the action is simply inherited from ${\cal R}:x^{i}\to-x^{i}$. Note that,
since the origin $x_{i}=0$ is the only fixed point, the ${\bf{Z}}_{2}$ is
acting freely on the ${\bf{S}}^{5}$ and there are no singularities.

In the presence of 5-branes defining boundaries, the ETW configuration is
obtained from the ETW configuration of section 2.2 by quotienting by the
orientifold action $\Omega(-1)^{F_{L}}{\cal R}$. The geometric action ${\cal
R}$ acts on the ${\bf{S}}^{2}\times{\bf{S}}^{2}$ fibration over $\Sigma$ as
follows: it leaves $w$ invariant, while it acts on each ${\bf{S}}^{2}$ sending
any point to its antipode in ${\bf{S}}^{2}$. This just follows from the fact
that the ${\bf{S}}^{2}$’s are given by the angular piece of the
${\bf{R}}^{3}$’s, on which ${\cal R}$ acts by flipping the 3 coordinates. It
is easy to see that this agrees with the action of ${\cal R}$ in the
asymptotic AdS${}_{5}\times{\bf{S}}^{5}$ region.

An interesting point is that, although the ${\bf{Z}}_{2}$ quotient has no
fixed points on the ${\bf{S}}^{5}$, there is a fixed point at $w=0$, because
the ${\bf{S}}^{5}$ collapses (and so do the ${\bf{S}}^{2}$’s). In the F-theory
description, this is locally described by a ${\bf{C}}^{4}/{\bf{Z}}_{2}$
singularity. This is a familiar example of terminal singularity, i.e. one not
admitting a crepant resolution (i.e. there is no blow-up preserving the
Calabi-Yau property). This nicely agrees with the fact that, in the
perturbative description, the singularity corresponds to an O3-plane at $w=0$,
and O3-planes carry no localized degrees of freedom.

#### The $k=4$ case

Consider now the $k=4$ case. In the absence of boundaries for the 4d
${\cal{N}}=3$ S-fold theory, the gravity dual is given by an F-theory
fibration over AdS${}_{5}\times{\bf{S}}^{5}/{\bf{Z}}_{4}$. The action of the
${\bf{Z}}_{4}$ is obtained from (6.5) by simply cutting out the ${\bf{S}}^{5}$
shell in ${\bf{R}}^{6}$, c.f. (6.6). Again, since the origin $x_{i}=0$ is the
only fixed point under (6.5), the ${\bf{Z}}_{4}$ is acting freely on the
${\bf{S}}^{5}$, and there are no fixed points.

As explained, in the presence of 5-branes defining boundaries, the ETW
configuration is obtained by starting with a parent ETW configuration of
section 2.2 with symmetric distributions of NS- and D5-brane poles. Namely, we
take $n=m$, and symmetric poles $k_{a}=l_{a}$ and coefficients
$d_{a}={\tilde{d}}_{a}$. Then, also using $w=-x+iy$ in (2.14), we have

$\displaystyle
h_{1}=\;4y+2\sum_{a=1}^{n}d_{a}\log\left(\frac{x^{2}+(y+k_{a})^{2}}{x^{2}+(y-k_{a})^{2}}\right)$
$\displaystyle
h_{2}=4x+2\sum_{a=1}^{n}d_{a}\log\left(\frac{(x+k_{a})^{2}+y^{2}}{(x-k_{a})^{2}+y^{2}}\right)\,,$
(6.7)

We then have to perform the ${\bf{Z}}_{4}$ quotient, making sure we recover
the right behaviour in the asymptotic AdS${}_{5}\times{\bf{S}}^{5}$ region,
namely (6.5) (plus the $SL(2,{\bf{Z}})$ monodromy). Let us split (6.6) in two
equations, to display more explicitly the phase $\varphi$ of the base
coordinate $w=re^{i\varphi}$

$\displaystyle x_{4}^{2}+x_{5}^{2}+x_{6}^{2}=R^{2}\cos^{2}\varphi\equiv
R_{1}^{2}(\varphi)$ $\displaystyle
x_{7}^{2}+x_{8}^{2}+x_{9}^{2}=R^{2}\sin^{2}\varphi\equiv R_{2}^{2}(\varphi)$
(6.8)

The action (6.5) of $\Theta$ requires exchanging these two equations, namely

$\displaystyle\Theta:\varphi\to\frac{3\pi}{2}-\varphi$ (6.9)

so that the values $\varphi=\pi/2$ and $\varphi=\pi$ are exchanged, and so are
the NS5- and D5-brane poles. This corresponds to the exchange
$x\leftrightarrow y$ in (6.7), clearly a symmetry. By propagating the exchange
$h_{1}\leftrightarrow h_{2}$ through (2.4), (2.5), we see that it corresponds
to $\phi\to-\phi$, which agrees with the $SL(2,{\bf{Z}})$ action.

Although the action of $\Theta$ on $\Sigma$ is just a ${\bf{Z}}_{2}$ flip, we
should not forget that there is also the action on the ${\bf{S}}^{2}$’s. To
determine this, let us introduce the coordinates $\theta_{1}$, $\varphi_{1}$
and $\theta_{2}$, $\varphi_{2}$ as in (5.4). Noticing that the
${\bf{S}}^{2}$’s are the angular manifolds in the 3-planes 456 and 789, the
action (6.5) corresponds to

$\displaystyle\Theta:$ $\displaystyle\theta_{1}\to\theta_{2}\quad,\quad$
$\displaystyle\theta_{2}\to-\theta_{1}$ (6.10)
$\displaystyle\varphi_{1}\to\varphi_{2}\quad,\quad$
$\displaystyle\varphi_{2}\to\varphi_{1}+\pi$

This exchanges the two ${\bf{S}}^{2}$, but with a ${\bf{Z}}_{4}$ action: a
point of ${\bf{S}}_{1}^{2}$ goes to the corresponding point in
${\bf{S}}_{2}^{2}$, and a point of ${\bf{S}}_{2}^{2}$ goes the antipodes of
the corresponding point in ${\bf{S}}_{1}^{2}$. This also shows that there are
no fixed points of this action on ${\bf{S}}_{1}^{2}\times{\bf{S}}_{2}^{2}$.
The action of $\Theta$ on the ETW configuration is thus the combination of
(6.9) and (6.10).

An interesting feature, similar to the $k=2$ case, is that, although there are
no fixed points in the asymptotic AdS${}_{5}\times{\bf{S}}^{5}/{\bf{Z}}_{4}$
case, there is a fixed point in the 10d ETW configuration at the origin $w=0$
in $\Sigma$, where the ${\bf{S}}^{2}$’s shrink to zero size and (6.10) is not
freely acting.

The singular point is locally given by F-theory on
${\bf{C}}^{4}/{\bf{Z}}_{4}$, so it is strongly coupled and hence non-trivial
to analyze. We are not aware of a discussion of this singularity in the
F-theory literature. Anyway, in analogy with the ${\bf{Z}}_{2}$ case in the
previous section, the ${\bf{C}}^{4}/{\bf{Z}}_{4}$ with orbifold vector
$v=(1/4,1/4,1/4,1/4)$ is terminal Morrison:1984 . This also agrees with the
absence of massless twisted sector states of the type IIA perturbative
orbifold Font:2004et , which relates to the F-theory setup via dualities. In
brief, the S-fold fixed point seems to have no localized degrees of freedom,
in agreement with its interpretation as a non-perturbative generalization of
O3-planes. Hence we expect no additional modes localized on the ETW brane from
the appearance of this singular point.

The absence of localized twisted modes in the KR brane for these AdS5
geometries implies that they are adequately described by the effective action
description in section 3.2, by simply reinterpreting $\phi_{+}$ as the
breathing mode of the quotiented ${\bf{S}}^{5}$.

##  7 Conclusions

In this work we have studied the 10d $SO(3)\times SO(3)$ preserving solutions
defining the gravity dual of 4d ${\cal{N}}=4$ SYM with boundary defined by a
3d ${\cal{N}}=4$ BCFT (and quotientes thereof), emphasizing their
interpretation as cobordisms to nothing for the corresponding
AdS${}_{5}\times{\bf{S}}^{5}$ geometries. We have provided a description in
terms of dynamical cobordisms in the 5d theory after reduction on the
${\bf{S}}^{5}$, and shown that they describe a generalization of the AdS4 KR
branes. Our 5d solutions display a smoothing out of the KR cusp into a bump,
which still leads to localization of gravity. The smoothing out is due to the
presence of additional scalars, which encode the varying internal geometry; in
particular, there is a bulk $SO(6)$ invariant breathing mode which describes
the shrinking of ${\bf{S}}^{5}$ and the end of spacetime, and there are
localized modes which encode the $SO(6)\to SO(3)\times SO(3)$ breaking due to
the boundary conditions. We have clarified the conditions of the 10 solution
under which such gravity localization arises, and we have introduced a new
scaling limit of the 10d solution which isolates the properties of the ETW
brane. Finally, we have extended the discussion to several orbifold and S-fold
quotients, whose ETW branes are inherited from those of the parent theory; we
have shown that, despite the presence of genuinely new singular points in the
quotient, they lead to no new localized degrees of freedom.

There are several interesting directions to be pursued:

$\bullet$ It would be interesting to study gravity duals of 4d ${\cal{N}}=1$
theories with boundaries. The naive extension of our construction for 4d
${\cal{N}}=2$ theories, namely using 4d ${\cal{N}}=1$ theories on D3-branes at
${\bf{C}}^{3}/{\bf{Z}}_{k}$ singularities Kachru:1998ys ; Lawrence:1998ja ;
Ibanez:1998xn ; Hanany:1998it ; Aldazabal:2000sa , do not work: the orbifold
does not preserve the geometry of the NS5- and D5-branes of the parent theory.
This difficulty is linked to the fact that the 4d ${\cal{N}}=1$ orbifold
theories are generically chiral, and so to the difficulties to introduce
boundaries in chiral theories.

$\bullet$ Even in the 4d ${\cal{N}}=4$ theory, it is possible to introduce
boundaries preserving less (super)symmetry, by simply using NS5- and D5-branes
with different orientations. A simple proposal, inspired by type IIA brane
setups Elitzur:1997fh ; Barbon:1997zu ; Elitzur:1997fh  is to use NS5 branes
along 012456 and 012678, and D5-branes along 012789 and 012457. There is no
known 10d solution for the corresponding gravity dual, but it may be possible
to prescribe an approximate 5d effective theory, based on the our same $SO(6)$
invariant sector, but a richer set of ETW brane localized modes to account for
the further breaking of symmetries.

$\bullet$ It would be interesting to exploit the tools of AdS/BCFT to gain
further understanding of the dynamics of the ETW localized modes and their
interplay with AdS5 bulk modes.

$\bullet$ Finally, it would be interesting to extend the present analysis to
other holographic setups, such as AdS${}_{4}\times{\bf{S}}^{7}$ and
AdS${}_{7}\times{\bf{S}}^{4}$, possibly based on the solutions discussed in
DHoker:2008lup ; DHoker:2008rje ; DHoker:2009lky ; DHoker:2009wlx .

We hope to come back to these and other questions in the future.

## Acknowledgments

We are pleased to thank Matilda Delgado, Miguel Montero and Irene Valenzuela
for useful discussions originating this project, and Roberta Angius, José
Calderón-Infante, Anamaria Font, Iñaki Garcia-Etxebarria, Luis E. Ibáñez and
Fernando Marchesano for useful discussions. This work is supported through the
grants CEX2020-001007-S and PID2021-123017NB-I00, funded by
MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe. The work by
J. H. is supported by the FPU grant FPU20/01495 from the Spanish Ministry of
Education and Universities.

## References

  * (1) J. McNamara and C. Vafa, Cobordism Classes and the Swampland, arXiv:1909.10355.
  * (2) I. n. García Etxebarria, M. Montero, K. Sousa, and I. Valenzuela, Nothing is certain in string compactifications, JHEP 12 (2020) 032, [arXiv:2005.06494].
  * (3) H. Ooguri and T. Takayanagi, Cobordism Conjecture in AdS, arXiv:2006.13953.
  * (4) M. Montero and C. Vafa, Cobordism Conjecture, Anomalies, and the String Lamppost Principle, JHEP 01 (2021) 063, [arXiv:2008.11729].
  * (5) M. Dierigl and J. J. Heckman, Swampland cobordism conjecture and non-Abelian duality groups, Phys. Rev. D 103 (2021), no. 6 066006, [arXiv:2012.00013].
  * (6) Y. Hamada and C. Vafa, 8d supergravity, reconstruction of internal geometry and the Swampland, JHEP 06 (2021) 178, [arXiv:2104.05724].
  * (7) R. Blumenhagen and N. Cribiori, Open-Closed Correspondence of K-theory and Cobordism, arXiv:2112.07678.
  * (8) D. Andriot, N. Carqueville, and N. Cribiori, Looking for structure in the cobordism conjecture, arXiv:2204.00021.
  * (9) M. Dierigl, J. J. Heckman, M. Montero, and E. Torres, IIB string theory explored: Reflection 7-branes, Phys. Rev. D 107 (2023), no. 8 086015, [arXiv:2212.05077].
  * (10) A. Debray, M. Dierigl, J. J. Heckman, and M. Montero, The Chronicles of IIBordia: Dualities, Bordisms, and the Swampland, arXiv:2302.00007.
  * (11) G. Buratti, M. Delgado, and A. M. Uranga, Dynamical tadpoles, stringy cobordism, and the SM from spontaneous compactification, JHEP 06 (2021) 170, [arXiv:2104.02091].
  * (12) G. Buratti, J. Calderón-Infante, M. Delgado, and A. M. Uranga, Dynamical Cobordism and Swampland Distance Conjectures, JHEP 10 (2021) 037, [arXiv:2107.09098].
  * (13) R. Angius, J. Calderón-Infante, M. Delgado, J. Huertas, and A. M. Uranga, At the end of the world: Local Dynamical Cobordism, JHEP 06 (2022) 142, [arXiv:2203.11240].
  * (14) R. Blumenhagen, N. Cribiori, C. Kneissl, and A. Makridou, Dynamical Cobordism of a Domain Wall and its Companion Defect 7-brane, arXiv:2205.09782.
  * (15) R. Blumenhagen, C. Kneissl, and C. Wang, Dynamical Cobordism Conjecture: Solutions for End-of-the-World Branes, arXiv:2303.03423.
  * (16) E. Dudas and J. Mourad, Brane solutions in strings with broken supersymmetry and dilaton tadpoles, Phys. Lett. B 486 (2000) 172–178, [hep-th/0004165].
  * (17) R. Blumenhagen and A. Font, Dilaton tadpoles, warped geometries and large extra dimensions for nonsupersymmetric strings, Nucl. Phys. B 599 (2001) 241–254, [hep-th/0011269].
  * (18) E. Dudas, J. Mourad, and C. Timirgaziu, Time and space dependent backgrounds from nonsupersymmetric strings, Nucl. Phys. B 660 (2003) 3–24, [hep-th/0209176].
  * (19) E. Dudas, G. Pradisi, M. Nicolosi, and A. Sagnotti, On tadpoles and vacuum redefinitions in string theory, Nucl. Phys. B 708 (2005) 3–44, [hep-th/0410101].
  * (20) S. Hellerman and I. Swanson, Cosmological solutions of supercritical string theory, Phys. Rev. D 77 (2008) 126011, [hep-th/0611317].
  * (21) S. Hellerman and I. Swanson, Dimension-changing exact solutions of string theory, JHEP 09 (2007) 096, [hep-th/0612051].
  * (22) S. Hellerman and I. Swanson, Charting the landscape of supercritical string theory, Phys. Rev. Lett. 99 (2007) 171601, [arXiv:0705.0980].
  * (23) I. Basile, J. Mourad, and A. Sagnotti, On Classical Stability with Broken Supersymmetry, JHEP 01 (2019) 174, [arXiv:1811.11448].
  * (24) R. Antonelli and I. Basile, Brane annihilation in non-supersymmetric strings, JHEP 11 (2019) 021, [arXiv:1908.04352].
  * (25) A. Mininno and A. M. Uranga, Dynamical Tadpoles and Weak Gravity Constraints, arXiv:2011.00051.
  * (26) I. Basile, On String Vacua without Supersymmetry: brane dynamics, bubbles and holography.  PhD thesis, Scuola normale superiore di Pisa, Pisa, Scuola Normale Superiore, 2020.  arXiv:2010.00628.
  * (27) I. Basile, Supersymmetry breaking, brane dynamics and Swampland conjectures, JHEP 10 (2021) 080, [arXiv:2106.04574].
  * (28) J. Mourad and A. Sagnotti, A 4D IIB Flux Vacuum and Supersymmetry Breaking. I. Fermionic Spectrum, arXiv:2206.03340.
  * (29) R. Angius, M. Delgado, and A. M. Uranga, Dynamical Cobordism and the beginning of time: supercritical strings and tachyon condensation, JHEP 08 (2022) 285, [arXiv:2207.13108].
  * (30) R. Angius, J. Huertas, and A. M. Uranga, Small Black Hole Explosions, arXiv:2303.15903.
  * (31) H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B766 (2007) 21–33, [hep-th/0605264].
  * (32) Y. Hamada, M. Montero, C. Vafa, and I. Valenzuela, Finiteness and the Swampland, arXiv:2111.00015.
  * (33) S. Lanza, F. Marchesano, L. Martucci, and I. Valenzuela, Swampland Conjectures for Strings and Membranes, arXiv:2006.15154.
  * (34) S. Lanza, F. Marchesano, L. Martucci, and I. Valenzuela, The EFT stringy viewpoint on large distances, arXiv:2104.05726.
  * (35) F. Marchesano and M. Wiesner, 4d strings at strong coupling, arXiv:2202.10466.
  * (36) A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, JHEP 06 (2001) 063, [hep-th/0105132].
  * (37) T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602, [arXiv:1105.5165].
  * (38) A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008, [hep-th/0011156].
  * (39) A. Karch and L. Randall, Localized gravity in string theory, Phys. Rev. Lett. 87 (2001) 061601, [hep-th/0105108].
  * (40) A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149, [arXiv:1908.10996].
  * (41) A. Almheiri, R. Mahajan, and J. E. Santos, Entanglement islands in higher dimensions, SciPost Phys. 9 (2020), no. 1 001, [arXiv:1911.09666].
  * (42) H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes, and J. Sandor, Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane, JHEP 10 (2020) 166, [arXiv:2006.04851].
  * (43) H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes, and J. Sandor, Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane, JHEP 12 (2020) 025, [arXiv:2010.00018].
  * (44) H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas, and S. Shashi, Information Transfer with a Gravitating Bath, SciPost Phys. 10 (2021), no. 5 103, [arXiv:2012.04671].
  * (45) H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas, and S. Shashi, Entanglement phase structure of a holographic BCFT in a black hole background, JHEP 05 (2022) 153, [arXiv:2112.09132].
  * (46) C. F. Uhlemann, Islands and Page curves in 4d from Type IIB, JHEP 08 (2021) 104, [arXiv:2105.00008].
  * (47) S. Demulder, A. Gnecchi, I. Lavdas, and D. Lust, Islands and light gravitons in type IIB string theory, JHEP 02 (2023) 016, [arXiv:2204.03669].
  * (48) E. D’Hoker, J. Estes, and M. Gutperle, Exact half-BPS Type IIB interface solutions. I. Local solution and supersymmetric Janus, JHEP 06 (2007) 021, [arXiv:0705.0022].
  * (49) E. D’Hoker, J. Estes, and M. Gutperle, Exact half-BPS Type IIB interface solutions. II. Flux solutions and multi-Janus, JHEP 06 (2007) 022, [arXiv:0705.0024].
  * (50) O. Aharony, L. Berdichevsky, M. Berkooz, and I. Shamir, Near-horizon solutions for D3-branes ending on 5-branes, Phys. Rev. D 84 (2011) 126003, [arXiv:1106.1870].
  * (51) B. Assel, C. Bachas, J. Estes, and J. Gomis, Holographic Duals of D=3 N=4 Superconformal Field Theories, JHEP 08 (2011) 087, [arXiv:1106.4253].
  * (52) C. Bachas and I. Lavdas, Quantum Gates to other Universes, Fortsch. Phys. 66 (2018), no. 2 1700096, [arXiv:1711.11372].
  * (53) C. Bachas and I. Lavdas, Massive Anti-de Sitter Gravity from String Theory, JHEP 11 (2018) 003, [arXiv:1807.00591].
  * (54) M. V. Raamsdonk and C. Waddell, Holographic and localization calculations of boundary F for $\mathcal{N}$ = 4 SUSY Yang-Mills theory, JHEP 02 (2021) 222, [arXiv:2010.14520].
  * (55) M. Van Raamsdonk and C. Waddell, Finding AdS${}^{5}\times S^{5}$ in 2+1 dimensional SCFT physics, JHEP 11 (2021) 145, [arXiv:2109.04479].
  * (56) A. Karch, H. Sun, and C. F. Uhlemann, Double holography in string theory, JHEP 10 (2022) 012, [arXiv:2206.11292].
  * (57) E. D’Hoker, J. Estes, M. Gutperle, and D. Krym, Exact Half-BPS Flux Solutions in M-theory. I: Local Solutions, JHEP 08 (2008) 028, [arXiv:0806.0605].
  * (58) E. D’Hoker, J. Estes, M. Gutperle, and D. Krym, Exact Half-BPS Flux Solutions in M-theory II: Global solutions asymptotic to AdS(7) x S**4, JHEP 12 (2008) 044, [arXiv:0810.4647].
  * (59) E. D’Hoker, J. Estes, M. Gutperle, and D. Krym, Janus solutions in M-theory, JHEP 06 (2009) 018, [arXiv:0904.3313].
  * (60) E. D’Hoker, J. Estes, M. Gutperle, and D. Krym, Exact Half-BPS Flux Solutions in M-theory III: Existence and rigidity of global solutions asymptotic to AdS(4) x S**7, JHEP 09 (2009) 067, [arXiv:0906.0596].
  * (61) D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789–855, [arXiv:0804.2902].
  * (62) D. Gaiotto and E. Witten, S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009), no. 3 721–896, [arXiv:0807.3720].
  * (63) I. R. Klebanov and M. J. Strassler, Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities, JHEP 08 (2000) 052, [hep-th/0007191].
  * (64) I. Akal, Y. Kusuki, T. Takayanagi, and Z. Wei, Codimension two holography for wedges, Phys. Rev. D 102 (2020), no. 12 126007, [arXiv:2007.06800].
  * (65) G. B. De Luca, N. De Ponti, A. Mondino, and A. Tomasiello, Harmonic functions and gravity localization, arXiv:2306.05456.
  * (66) A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nucl. Phys. B492 (1997) 152–190, [hep-th/9611230].
  * (67) B. Assel, C. Bachas, J. Estes, and J. Gomis, IIB Duals of D=3 N=4 Circular Quivers, JHEP 12 (2012) 044, [arXiv:1210.2590].
  * (68) S. Hellerman and M. Kleban, Dynamical Cobordisms in General Relativity and String Theory, JHEP 02 (2011) 022, [arXiv:1009.3277].
  * (69) D. Lüst, E. Palti, and C. Vafa, AdS and the Swampland, arXiv:1906.05225.
  * (70) H. Kanda, M. Sato, Y.-k. Suzuki, T. Takayanagi, and Z. Wei, AdS/BCFT with brane-localized scalar field, JHEP 03 (2023) 105, [arXiv:2302.03895].
  * (71) A. Karch, E. Katz, and L. Randall, Absence of a VVDZ discontinuity in AdS(AdS), JHEP 12 (2001) 016, [hep-th/0106261].
  * (72) M. R. Douglas and G. W. Moore, D-branes, quivers, and ALE instantons, hep-th/9603167.
  * (73) S. Kachru and E. Silverstein, 4-D conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998) 4855–4858, [hep-th/9802183].
  * (74) A. E. Lawrence, N. Nekrasov, and C. Vafa, On conformal field theories in four-dimensions, Nucl. Phys. B 533 (1998) 199–209, [hep-th/9803015].
  * (75) A. Hanany, M. J. Strassler, and A. M. Uranga, Finite theories and marginal operators on the brane, JHEP 06 (1998) 011, [hep-th/9803086].
  * (76) A. Hanany and A. M. Uranga, Brane boxes and branes on singularities, JHEP 05 (1998) 013, [hep-th/9805139].
  * (77) E. Witten, Solutions of four-dimensional field theories via M theory, Nucl. Phys. B500 (1997) 3–42, [hep-th/9703166]. [,452(1997)].
  * (78) S. Gukov, Comments on N=2 AdS orbifolds, Phys. Lett. B 439 (1998) 23–28, [hep-th/9806180].
  * (79) J. D. Lykken, E. Poppitz, and S. P. Trivedi, Chiral gauge theories from D-branes, Phys. Lett. B 416 (1998) 286–294, [hep-th/9708134].
  * (80) A. Hanany and A. Zaffaroni, On the realization of chiral four-dimensional gauge theories using branes, JHEP 05 (1998) 001, [hep-th/9801134].
  * (81) A. Karch, D. Lust, and D. J. Smith, Equivalence of geometric engineering and Hanany-Witten via fractional branes, Nucl. Phys. B 533 (1998) 348–372, [hep-th/9803232].
  * (82) I. n. García-Etxebarria and D. Regalado, $\mathcal{N}=3$ four dimensional field theories, JHEP 03 (2016) 083, [arXiv:1512.06434].
  * (83) E. Witten, Baryons and branes in anti-de Sitter space, JHEP 07 (1998) 006, [hep-th/9805112].
  * (84) D. R. Morrison and G. Stevens, Terminal Quotient Singularities in Dimensions Three and Four, Proc. Amer. Math. Soc. 90 (1984) 15\.
  * (85) A. Font and J. A. Lopez, Strings on eight-orbifolds, Nucl. Phys. B 703 (2004) 177–198, [hep-th/0405151].
  * (86) L. E. Ibanez, A Chiral D = 4, N=1 string vacuum with a finite low-energy effective field theory, JHEP 07 (1998) 002, [hep-th/9802103].
  * (87) G. Aldazabal, L. E. Ibanez, F. Quevedo, and A. M. Uranga, D-branes at singularities: A Bottom up approach to the string embedding of the standard model, JHEP 08 (2000) 002, [hep-th/0005067].
  * (88) S. Elitzur, A. Giveon, and D. Kutasov, Branes and N=1 duality in string theory, Phys. Lett. B 400 (1997) 269–274, [hep-th/9702014].
  * (89) J. L. F. Barbon, Rotated branes and N=1 duality, Phys. Lett. B 402 (1997) 59–63, [hep-th/9703051].