10in(165mm,15mm) 11 October 2023

Cobordism and Bubbles of Anything

in the String Landscape

Bjoern Friedrich1, Arthur Hebecker1, Johannes Walcher1,2

1Institute for Theoretical Physics, Heidelberg University,

Philosophenweg 19, 69120 Heidelberg, Germany

2Institute for Mathematics, Heidelberg University,

Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

E-Mail<EMAIL_ADDRESS><EMAIL_ADDRESS>heidelberg.de,

<EMAIL_ADDRESS>
###### Abstract

We study bubble of nothing decays and their reverse processes, the creation of
vacua through ‘bubbles of something’, in models of the Universe based on
string theory. From the four-dimensional perspective, the corresponding
gravitational instantons contain an end-of-the-world (ETW) boundary or brane,
realized by the internal manifold shrinking to zero size. The existence of
such ETW branes is predicted by the Cobordism Conjecture. We develop the 4d
EFT description of such boundaries at three levels: First, by generalizing the
Witten bubble through an additional defect. Second, by replacing the compact
$S^{1}$ with a Calabi-Yau orientifold and allowing it to shrink and disappear
through a postulated defect. Third, by explicitly constructing an ETW brane
for type IIB Calabi-Yau orientifolds with O3/O7 planes through an appropriate
additional O5 orientifolding. Our 4d EFT formalism allows us to compute the
decay/creation rates for bubbles of anything depending on two parameters: The
size of the relevant defect and its tension a.k.a. the induced (generalized)
deficit angle.

###### Contents

  1. 1 Introduction
  2. 2 Bubbles of anything – technical preliminaries
    1. 2.1 The Witten bubble
    2. 2.2 Bubbles of anything from the four-dimensional point of view
  3. 3 Generalized Witten bubbles
    1. 3.1 Witten bubble with defect
    2. 3.2 The decay rate
    3. 3.3 Generality of the method developed
    4. 3.4 Bubbles of nothing in Calabi-Yau compactifications
    5. 3.5 Bubbles of something
    6. 3.6 Comparison to other results
  4. 4 Explicit realization in type IIB string theory
    1. 4.1 The ETW brane
    2. 4.2 Creation/decay rates
  5. 5 Bubbles of anything – further remarks
    1. 5.1 Nothing as AdS with infinitely negative vacuum energy
    2. 5.2 Refining the decay/creation rate estimates
      1. 5.2.1 Non-trivial potentials
      2. 5.2.2 Fluxes
      3. 5.2.3 Other fields
      4. 5.2.4 Higher-order corrections
  6. 6 Conclusions
  7. A Dimensional reduction
  8. B Bubble of nothing decays in general dimensions
  9. C Supersymmetry and the ETW brane
  10. D AdS bubble transitions in the limit of small AdS length

##  1 Introduction

In the presence of multiple vacua, for example in Calabi-Yau compactifications
of string theory, the decay of one vacuum into another is in general a common
phenomenon. Many such processes admit an effective semi-classical description
involving scalar fields in potentials with multiple critical points [1, 2].
Witten [3] discovered that, in addition, for some vacua arising from
compactifications of a higher-dimensional theory, a decay process to ‘nothing’
also exists, whose effective field theory description is much less obvious.
Specifically in the case of an $S^{1}$ compactification from 5d to 4d, the
relevant instanton is characterized by the $S^{1}$ shrinking to zero size over
the locus of an $S^{3}$, cf. l.h.s. of Fig. 1. The inside of the $S^{3}$ is
empty while far away from that region the geometry approaches the vacuum.

Figure 1: Illustration of the bubble of nothing (left) and the bubble of
something (right) instanton. In both cases, the four-dimensional space
$\mathcal{M}$ has a spherical boundary which coincides with the locus where
the internal manifold, shown above $\mathcal{M}$, shrinks to zero size.
Figure 2: Physical processes corresponding to bubble of nothing (left) and
bubble of something (right). The dashed line denotes the locus of analytic
continuation from euclidean half instanton to the Lorentzian, expanding
bubble. The solid black line is the boundary of 4d space.

In the corresponding Lorentzian spacetime, a hole of nothing emerges as a
quantum effect and then starts growing, eating up all space, as depicted on
the l.h.s. of Fig. 2. This process and the corresponding instanton is known as
a ‘bubble of nothing’. Further studies of such decay processes considered
models with stabilized extra dimensions as well as geometries where an
originally present supersymmetry is broken by compactification on spheres [4,
5, 6, 7, 8, 9, 10, 11, 12] or in string-theoretic toy-models [13, 14, 15, 16,
17, 18, 19].

The reverse process of a decay to nothing is vacuum creation from nothing with
an instanton of the type illustrated on the r.h.s. of Fig. 1. The geometry is
that of a disk with an $S^{1}$ over every point. At the boundary of the disk,
the $S^{1}$ shrinks to zero size. As the counterpart to the ‘bubble of
nothing’, we refer to the creation process as a ‘bubble of something’111In
[20], the term ‘bubbles from nothing’ is used.. Again, these processes are
different from the most familiar vacuum creation proposals of [21, 22, 23].
Crucially, from the 4d perspective, the instanton contains a boundary and the
cosmological constant of the new vacuum need not be positive. Furthermore, the
created universe is open in the sense that its spatial slices are negatively
curved. The idea of a bubble of something was first introduced in [24, 25] and
then connected to extra dimensions in [26]. Further research on this topic
includes [27, 28, 7, 20, 29].

Figure 3: A bordism between $S^{1}$ and nothing, including a defect (drawn as
a red disk).

The Witten bubble of nothing can not be naively generalized to supersymmetric
configurations since only spinors with anti-periodic boundary conditions on
the $S^{1}$ can be defined on the cigar geometry of Figure 1. The reason is
that a circle with periodic boundary conditions for the spinor is not in the
trivial class of the spin cobordism group $\Omega_{1}^{Spin}=\mathbb{Z}_{2}$.
As a consequence, the supersymmetric vacuum does not decay to nothing via the
smooth bubble of nothing [3]. More recently, however, it has been argued that
this argument for stability is not quite correct in a fundamental theory of
quantum gravity. Namely, according to the cobordism conjecture [30] any
internal manifold of a string compactification must be cobordant to nothing.
In other words, there can not be a topological obstruction to letting
spacetime end by the shrinking of the compact space to zero size.
Consequently, defects ensuring the triviality of compact spaces in cobordisms
are predicted [30, 31, 32, 33]. A bordism with defect is illustrated in Fig.
3. From the 4d point of view, the conjecture implies that all string
compactifications possess end-of-the-world (ETW) branes, but it remains a
question of dynamics whether bubbles of nothing or something (bubbles of
anything for short) arise. A key dynamical ingredient is the tension of the
defect. In particular, one expects that in the case of supersymmetric
compactifications the relevant defects are always heavy enough to ensure
stability. Other applications of the cobordism conjecture can be found in [34,
35, 36, 37, 38, 39, 40, 41].

In what follows, we will develop a 4d effective field theory approach to
vacuum decay/creation via bubbles of anything, and use it extensively. Let us
illustrate this by discussing the Witten bubble, allowing for a defect as
depicted in Fig. 3. In this case (which of course is just a toy model), the
defect has co-dimension 2 in the 5-dimensional spacetime, and its tension is
equivalent to a deficit angle $\theta$ of the shrinking circle. The relevant
4d Lagrangian consists of the Einstein-Hilbert term and the kinetic term for a
scalar $\phi$, characterising the radius of the $S^{1}$. In addition, there is
a boundary action including the Gibbons-Hawking-York term and the tension of
our ETW brane. Our analysis of this setting is related to[42, 10, 11]. It goes
beyond these references by allowing for a defect. In addition, we differ in
the treatment of the total derivative and boundary terms: We remove the former
by integration, obtaining a standard 4d EFT with a completely local
description of the ETW brane.

Boundary conditions at the brane fix the scalar field to a certain value,
corresponding from the 5-dimensional perspective to a characteristic finite
size, $\eta$, of the defect. The deficit angle $\theta$ is microscopically
determined by the tension of the defect. In the 4d EFT, it is identified with
the gradient with which $\phi$ approaches the ETW brane. We will show that
$\eta$ and $\theta$ are the key input needed for a simple 4d derivation of the
instanton solution corresponding to the Witten bubble with defect and its
action, which is related to the decay/creation rate in the usual way.

It can be shown that in the 5d model it is mathematically consistent to view
the defect as being point-like, i.e. to perform the limit $\eta\to 0$. In this
case, $\phi$ runs to infinity at the ETW brane. The original Witten bubble
without a defect can now be obtained by setting $\theta=0$ and performing the
above limit $\eta\to 0$. However, even in this case it is useful to keep
$\eta$ finite since it now plays the role of a regulator making a pure 4d
Einstein-frame description possible.

As remarked above, defects of finite size and tension become intrinsic for
string theory models such as Calabi-Yau orientifold compactifications with
flux. Apart from that, the discussion of the two previous paragraphs carries
over almost word by word. The CY is assumed to shrink at the ETW boundary. A
defect of size $\eta$, which in contrast to the Witten case is now
unavoidable, hides the appearing singularity. An appropriately generalized
deficit angle $\theta$ characterizes the impact of the defect on the nearby
geometry, and contributes to the tension of the ETW brane. While instanton
solution and action can be given rather explicitly, the limit $\eta\to 0$ can
not be taken. We expect that topology can fluctuate freely at string scale, so
$\eta\sim 1$ in string units is plausible, but clearly also much larger ETW
defects can exist.

Having thus developed an effective method for computing bubble-of-anything
creation/decay rates, we turn to the construction of an explicit ETW brane for
type IIB CY orientifold compactifications with D3/D7 branes. This accounts for
many vacua of the best understood part of the string landscape [43, 44, 45,
46]. The defect corresponding to our proposed ETW brane is an O5/D5 system
wrapping a three-cycle of the CY. It can be understood as the mirror dual of a
type-IIA orientifold model where, rather obviously, the world can end on an
O8/D8 system wrapping the entire mirror manifold and filling three non-compact
dimensions. We provide leading order estimates for the transition rates for
such vacua and compare the decay rates to traditional decay mechanisms [1, 47,
48, 49].

We emphasize that in distinction to the previous discussion based on a rather
generic Calabi-Yau cobordism, the explicit orientifold construction does not
require the compact space to become small. This illustrates that in controlled
situations large defects of size $\eta=R_{\it KK}\gg 1$ are possible. On the
other hand, being supersymmetric, the O5/D5 system will lead to a flat ETW
brane and the associated transition rate will vanish at face value. However,
including SUSY breaking and other corrections, it is also possible that our
explicit ETW brane is driven to the previously discussed regime of small
$\eta$.

The paper is organized as follows: In Sec. 2, we set the stage by reviewing
the Witten bubble and its 4d description. In Sec. 3, we present the Witten
bubble of nothing with defect and characterize it in 4d language. Then we use
the 4d intuition gained to analyze more general bubbles of anything, including
CY compactifications of string theory. In Sec. 4, we explicitly construct an
ETW brane for vacua arising from compactifying type IIB string theory on a CY
orientifold. We use Sec. 5 to defend the bubble of something against
fundamental criticism raised in [7] and to discuss the potential for making
our analysis more precise.

##  2 Bubbles of anything – technical preliminaries

In this section, we first review the Witten bubble of nothing and then discuss
the general setup for analyzing bubbles of anything from the four-dimensional
perspective.

###  2.1 The Witten bubble

Witten [3] has discovered that the Kaluza-Klein (KK) geometry
$\mathbb{M}_{4}\times S^{1}$ exhibits a peculiar instability. The relevant
gravitational instanton is precisely the five-dimensional Schwarzschild black
hole geometry, with metric

$\displaystyle
ds^{2}=\left(1-\frac{R_{KK}^{2}}{\tilde{r}^{2}}\right)^{-1}d\tilde{r}^{2}+\tilde{r}^{2}d\Omega_{3}^{2}+\left(1-\frac{R_{KK}^{2}}{\tilde{r}^{2}}\right)R_{KK}^{2}dy^{2}\,,\quad
y\sim y+2\pi\,,$ (1)

satisfying Einstein’s equations. Here, we have chosen to name the radial
coordinate $\tilde{r}$ instead of $r$ to reserve the latter for later use. At
$\tilde{r}\gg R_{KK}$, this geometry approaches the KK vacuum
$\mathbb{M}_{4}\times S^{1}$ with circle radius $R_{KK}$. Crucially, the
geometry is defined only for $\tilde{r}\geq R_{KK}$ and at $\tilde{r}=R_{KK}$
the internal circle shrinks to zero size (cf. the l.h. side of Fig. 1). From
the four-dimensional point of view, this locus represents the end of the
world. Despite the vanishing radius of the $S^{1}$, the five-dimensional
curvature remains small as long as $R_{KK}$ is large in 5d Planck units,
ensuring the reliability of a semiclassical analysis.

The Lorentzian geometry of the bubble of nothing is obtained by an analytic
continuation of (1): One first writes

$\displaystyle d\Omega_{3}^{2}=d\psi^{2}+\sin^{2}\psi d\Omega_{2}^{2}$ (2)

and then lets $\psi\to\frac{\pi}{2}+it$, where $t\in(0,\infty)$. This step is
analogous to the analytic continuation employed in Coleman-de Luccia
instantons [1]. From the 4d perspective, a spherical hole of nothing with
radius $R_{KK}$ nucleates at $t=0$, starts accelerating and rapidly approaches
the speed of light. This Lorentzian geometry corresponds to the region above
the dashed line on the l.h. side of Fig. 2.

The nucleation probability per time and volume of such a bubble can be
estimates as

$\displaystyle\Gamma\sim\exp(-B)\,,\qquad B=S_{instanton}-S_{vacuum}\,.$ (3)

Here $S_{instanton}$ and $S_{vacuum}$ are defined by evaluating the Einstein-
Hilbert action supplemented by the Gibbons-Hawking-York term,222We set the 5d
Planck mass $M_{5}=1/8\pi G_{5}$ to unity. Moreover, our sign convention is
such that $\mathcal{K}_{5}=\nabla_{a}n^{a}$, with $n=n^{a}\partial_{a}$ being
the outward pointing normal vector of the boundary hypersurface. We adopt this
convention throughout the paper.

$\displaystyle
S=-\frac{1}{2}\int\sqrt{g}\mathcal{R}_{5}-\int_{\partial\mathcal{M}}\sqrt{h}\mathcal{K}_{5}\,,$
(4)

on the instanton geometry (1) and in the vacuum. In the above,
$\mathcal{R}_{5}$ and $\mathcal{K}_{5}$ denote the 5d Ricci- and extrinsic
curvature scalar respectively. One explicitly finds[3]

$\displaystyle B=2\pi^{3}R_{KK}^{3}\,.$ (5)

###  2.2 Bubbles of anything from the four-dimensional point of view

We want to find a 4d description of the Witten bubble and then study more
general bubbles of nothing as well as bubbles of something from a 4d point of
view. Our primary interest will eventually be in bubbles related to Calabi-Yau
(CY) compactifications of string theory. We start by considering
compactifications from $d=4+n$ to 4 dimensions on an $n$-dimensional compact
manifold. To obtain the simplest possible estimate for rates of bubble-of-
anything transitions, we restrict attention to the volume modulus. This is
also motivated by the fact that in many string models the volume modulus has
the lowest mass. We neglect all other fields in the following. As shown in
Appendix A, the $d$-dimensional-Einstein-Hilbert action with metric ansatz

$\displaystyle ds^{2}=e^{2\alpha\phi}ds_{4}^{2}+e^{2\beta\phi}ds_{n}^{2}$ (6)

reduces to a 4d theory of gravity with Einstein-frame metric $ds_{4}^{2}$ and
a canonically normalized volume modulus field $\phi$ if the constants
$\alpha,\beta$ are chosen as

$\displaystyle\alpha^{2}=\frac{n}{2(n+2)}\,,\qquad\beta=-2\frac{\alpha}{n}\,.$
(7)

Here $\phi$ is measured in 4d Planck units. In the following, we choose
$\alpha<0$ and, correspondingly, $\beta>0$. In Eq. (6), $ds_{n}^{2}$ describes
the geometry of the internal manifold with the convention that its volume is
fixed to unity in $d$-dimensional Planck units. For later convenience, we
define the quantity $R$ as

$\displaystyle 2\pi R=e^{\beta\phi}\,,$ (8)

which may be thought of as the radius of the internal manifold, also measured
in $d$-dimensional Planck units. Clearly, the case $n=1$ is appropriate for
the Witten bubble.

Motivated by general arguments [50] and similarly to [10, 11], we make an
$O(4)$-symmetric ansatz for the instanton geometry:

$\displaystyle
ds_{4}^{2}=dr^{2}+f(r)^{2}d\Omega_{3}^{2}\quad,\qquad\phi=\phi(r)\,.$ (9)

For completeness, we also record the corresponding $d$-dimensional metric:

$\displaystyle
ds^{2}=e^{2\alpha\phi(r)}\left(dr^{2}+f(r)^{2}d\Omega_{3}^{2}\right)+e^{2\beta\phi(r)}ds_{n}^{2}\,.$
(10)

The KK radius is

$\displaystyle R_{KK}=\frac{e^{\beta\phi_{0}}}{2\pi}\,,$ (11)

with $\phi_{0}$ the value of the scalar $\phi$ when the geometry approaches
the vacuum. The equations of motion, i.e. the 4d Einstein equations and the
field equation for $\phi$ take the form

$\displaystyle\phi^{\prime\prime}$
$\displaystyle+3\frac{f^{\prime}}{f}\phi^{\prime}=\frac{dV(\phi)}{d\phi}\,,$
(12) $\displaystyle f^{\prime 2}$
$\displaystyle=1+\frac{f^{2}}{3}\left(\frac{\phi^{\prime
2}}{2}-V(\phi)\right)\,,$ (13)

where primes denote derivatives with respect to $r$. A potential $V(\phi)$ can
arise for example from internal curvature or fluxes.

A flat-space bubble of nothing instanton is a solution to the field equations
where $\phi=$const. and $f^{\prime}(r)\to 1$ at $r\to\infty$ and where
$\phi\to-\infty$ at $r=0$. This implies that we chose $r$ to vanish at the
bubble wall, which has radius $f(r=0)$.

For a bubble of something instanton [26], we may choose $r=0$ to be at the
center of the bubble (see r.h.s. of Fig. 1), where $\phi$ takes some finite
value and $f=0$. The boundary is then at some finite $r>0$, where
$\phi\to-\infty$ and the value of $f$ characterizes the radius of the bubble.

Solutions to (12),(13) involve three integration constants. Two of them are
determined by boundary conditions for $\phi$ and $\phi^{\prime}$ at the bubble
wall. The final parameter is then determined by requiring $\phi^{\prime}$ to
vanish at $r=0$ in the bubble of something case and at $r\to\infty$ in the
bubble of nothing case.333 To be more precise, in the bubble of nothing case
with vanishing potential, $V(\phi)\equiv 0$, all solutions share the feature
that $\phi^{\prime}$ vanishes at $r\to\infty$. In this case, the 3rd
integration constant can be used to set the asymptotic value of $\phi$, fixing
$R_{KK}=\exp(\beta\phi_{0})$.

We now specialize our discussion to bubble of nothing solutions and focus on
internal manifolds without scalar curvature or fluxes, e.g. fluxless CY or
torus compactifications, such that the potential vanishes identically. These
are merely simplifying assumptions and we discuss their severity in Section 5.
A more detailed discussion of bubbles of something is given in Section 3.5.
With the assumptions stated above, we can work with analytic solutions.
Equation (12) can be integrated to give

$\displaystyle f^{3}\phi^{\prime}=C=const.$ (14)

Substituting the result (14) into (13) leads to

$\displaystyle f^{\prime 2}=1+\frac{C^{2}}{6f^{4}}\,,$ (15)

which can be used to obtain [10]444Here we have chosen the positive root of
(15) such that $f$ increases with growing $r$.

$\displaystyle\phi(f)=\phi_{0}-\frac{\sqrt{6}}{2}\text{arcsinh}\left(\frac{C}{\sqrt{6}f^{2}}\right)\,.$
(16)

For $C>0$, we find that $\phi$ shrinks when approaching the bubble core.
Deriving the profiles $\phi(r)$ and $f(r)$ analytically is more complicated,
see [10, 11]. However, it is clear from (13) that $f$ is a monotonically
growing function of $r$, such that knowing the profile $\phi(f)$ should be
sufficient for practical calculations. In the limit of $f\to\infty$, $\phi$
approaches $\phi_{0}$ such that the definition of $\phi_{0}$ in (16) is
compatible with Equation (11). Further, $f^{\prime}$ approaches unity and
spacetime becomes asymptotically Minkowski, as desired. In the limit $f\to 0$,
$\phi$ approaches $-\infty$ such that the internal $S^{1}$ shrinks to zero
size. As we will shortly see, $C$ can be fixed by looking at the boundary
conditions on $\phi^{\prime}$ at the bubble surface. The third integration
constant is fixed implicitly by choosing the point where $\phi$ diverges to be
at $r=0$.

From (16) one immediately sees that $f=0$ at the bubble wall, where
$\phi=-\infty$. This means that the bubble has vanishing radius in 4d Planck
units. While this may sound confusing, it is simply a consequence of the fact
that, at the wall, the compact space shrinks and hence the 4d Planck length
diverges. This technical problem of the 4d Einstein frame analysis is overcome
when, further down, we regulate the 4d solution by cutting it off when the
compact space reaches minimal size $\eta$. If appropriate, $\eta$ may be
identified with the size of an ETW defect.

##  3 Generalized Witten bubbles

We now turn to one of our central points: The development of an entirely four-
dimensional description of bubble of nothing/something instantons related to
the Witten bubble. In Sec. 3.1, we study the Witten bubble of nothing
including a defect at the bubble wall. In Sec. 3.2, we develop a method to
effectively compute decay rates for such bubbles, which will also apply to
even more general bubble geometries to be discussed below. Section 3.3
discusses a generalizion of the Witten bubble with defect to higher
dimensions. In Sec. 3.4, bubbles of nothing in 10d Calabi-Yau (CY)
compactifications in string theory are analyzed under the assumption that the
instanton geometry is similar to the Witten bubble with defect. Finally, in
Sec. 3.5, a more detailed discussion of bubbles of something is given.

###  3.1 Witten bubble with defect

We use the analytic solution from the end of Sec. 2.2, focussing on the case
$d=5$ where (7)

$\displaystyle\alpha=-\sqrt{\frac{1}{6}}\,,\qquad\beta=\sqrt{\frac{2}{3}}\,.$
(17)

The integration constant $\phi_{0}$ is straightforwardly determined by
demanding a specific KK radius at $f\to\infty$, cf. (11). The integration
constant $C$ remains to be fixed, which is done by analyzing the boundary
conditions near the bubble wall. For the geometry (10) with
$ds_{n}^{2}=dy^{2}\,,y\sim y+1$ to be completely regular, one has to demand
that the $S^{1}$ shrink to zero size without creating a deficit angle. We can
however be more general and consider the class of bubbles of nothing where an
extra defect is needed to close off the geometry, cf. Fig. 3. In this case, a
deficit angle generically arises. For related discussions of defects see [15,
8, 4, 5, 19].

Since we work semiclassically, when we approach the defect we can only trust
the geometry until the internal circle has shrunk to Planck size. If the
defect is larger than Planck size, we have to stop even earlier. Let us denote
the minimal value of $R$ up to which we trust our analysis by $\eta$, which
for the moment we assume to be an $\order{1}$ number. In other words, the
geometry stays trustworthy until a minimal $f=f_{\eta}$, defined by

$\displaystyle\eta=R(f_{\eta})\,.$ (18)

Assuming $R_{KK}\gg\eta$ and using (8),(16), we find

$\displaystyle
f_{\eta}=\left(\frac{2C}{\sqrt{6}}\right)^{1/2}\left(\frac{\eta}{R_{KK}}\right)^{1/2}\,.$
(19)

As already noted above our defect, being a 3d object in 5d, creates a deficit
angle in the geometry. This is entirely analogous to a point mass in three or
a cosmic string in four dimensions. We can thus relate the defect tension
$T_{def}$ to the deficit angle and hence to the local geometry in the standard
way [51, 23],

$\displaystyle T_{def}=\theta\,\,\,,\qquad
1-\frac{\theta}{2\pi}=\left.\frac{dR}{dx}\right|_{x=0}\,.$ (20)

Here we use 5d Planck units and $x$ measures the proper distance from the
defect. It is related to $r$ by

$\displaystyle dx=e^{\alpha\phi(r)}dr\,,$ (21)

such that one finds

$\displaystyle
1-\frac{\theta}{2\pi}=\left.\frac{\beta}{2\pi}\phi^{\prime}e^{(\beta-\alpha)\phi}\right|_{x=0}=\left.\frac{\beta}{2\pi}\frac{C}{f^{3}}e^{(\beta-\alpha)\phi}\right|_{f=f_{\eta}}=\sqrt{\frac{4\pi}{3}}\frac{C}{f_{\eta}^{3}}\eta^{3/2}\,.$
(22)

Here, we have used the solution (14), the definition of $\eta$ (18) and (17).

After substituting (19) into (22), one arrives at

$\displaystyle C=\sqrt{6}\pi
R_{KK}^{3}\left(1-\frac{\theta}{2\pi}\right)^{-2}\,.$ (23)

Interestingly, $C$ does not depend on $\eta$ which reflects the fact that,
even though a defect is present, the limit $\eta\to 0$ may be taken. In other
words, it is mathematically (though maybe not physically) consistent to take
the defect to be point-like in the two transverse directions. We will see
below that this is no longer true in higher-dimensional models.

We have now fully determined the solutions corresponding to the Witten bubble
with defect from the four-dimensional perspective by demanding the three
boundary conditions (11),(18),(22). By inserting (23) into (16), one obtains
the instanton solution depending only on the input parameters $R_{KK}$ and
$\theta$ and where the limit $\eta\to 0$ has implicitly been taken. The Witten
bubble is recovered for the special case $\theta=0$. The critical bubble
radius is

$\displaystyle\rho_{0}=\left.e^{\alpha\phi}f\right|_{f=f_{\eta}}=R_{KK}^{-1/2}\left(\frac{1}{\pi\sqrt{6}}\right)^{1/2}\sqrt{C}=\frac{R_{KK}}{1-\frac{\theta}{2\pi}}\,.$
(24)

We observe that $\rho_{0}$ matches the Witten result for $\theta=0$ and
becomes larger when turning on a deficit angle. This agrees with our physics
intuition that we expect an additional defect to result in a larger critical
bubble and a smaller decay rate.

###  3.2 The decay rate

We are now ready to calculate decay rates for bubbles of nothing with defects.
Although we have the Witten bubble with defect in mind as our primary example,
our analysis will turn out to be valid for more general bubbles of nothing,
such as the 10d string theory models we are ultimately after. We follow the
strategy explained at the end of Sec. 2.1, computing in particular the decay
exponent using (3). Crucially, we want to work with 4d effective actions for
the instanton and the vacuum contribution. Both take the form

$\displaystyle
S=\int_{\mathcal{M}}\sqrt{g}\left(-\frac{1}{2}\mathcal{R}_{4}+\frac{1}{2}(\partial\phi)^{2}+V(\phi)\right)-\int_{\partial\mathcal{M}}\sqrt{h}\left(\mathcal{K}_{4}-T_{4}\right)\,.$
(25)

Here, $\mathcal{R}_{4}$ is the 4d Ricci scalar, $\mathcal{K}_{4}$ is the
extrinsic curvature, $\partial\mathcal{M}$ is the boundary of the four-
dimensional spacetime $\mathcal{M}$ with induced metric $h$. For both
$S_{instanton}$ and $S_{vacuum}$, there is a boundary contribution at
infinity, without brane tension $T_{4}$. These contributions will cancel. At
the center, the vacuum obviously has no boundary while $S_{instanton}$
receives a contribution with both non-zero $\mathcal{K}_{4}$ and the tension
$T_{4}$ of an effective 4d ETW brane sitting at the bubble wall, i.e. at the
spherical boundary of $\mathcal{M}$ on the l.h.s. of Fig. 1. The subscript $4$
in $T_{4}$ indicates the dimension of space which is bounded by the ETW brane
and should not be confused with the tension of a $4$-brane in string theory.
In Eq. (25), the 4d Planck mass is set to unity. It is hence important to
distinguish between four- and $d$-dimensional quantities, which are measured
in the respective Planck units.

The form of the action (25) differs from the one used in [42, 10, 11] by the
absence of a term proportional to $\square\phi$ coming from dimensional
reduction of $\mathcal{R}_{d}$. We have replaced this total derivative term by
appropriate boundary terms: One at infinity and one at the inner spherical
boundary. At the former boundary, the term coming from $\square\phi$ is
cancelled by a contribution deriving from the 5d GHY term. The latter boundary
is the ETW boundary defined by $f=f_{\eta}$, where also the ETW brane with
tension $T_{4}$ sits. Here, the boundary term coming from $\square\phi$ is
absorbed in our effective description in terms of $\mathcal{K}_{4}$ and
$T_{4}$. We feel that such a description, based on (25) without total
derivative terms but viewing the bubble surface as an additional boundary, is
more appropriate for our subsequent stringy applications with physical defects
at the bubble wall.

In order to compute the on-shell action of the instanton, we first note that
the trace of the 4d Einstein equations determines

$\displaystyle\frac{1}{2}\mathcal{R}_{4}=\frac{1}{2}\phi^{\prime
2}+2V(\phi)\,,$ (26)

and hence

$\displaystyle S_{\text{on-shell}}=-2\pi^{2}\int
dr\,f^{3}V(\phi)-\int_{\partial\mathcal{M}}\sqrt{h}\left(\mathcal{K}_{4}-T_{4}\right)\,.$
(27)

In the absence of a potential, the bulk contribution simply vanishes and the
entire contribution comes from boundary terms.

For a Minkowski bubble of nothing geometry, there are two boundaries: One
located at asymptotic infinity $f\to\infty$ and one near the center of the
bubble at finite $f=f_{\eta}$. We calculate the value of the extrinsic
curvature scalar for the boundary at infinity using (13)

$\displaystyle\mathcal{K}_{4}=3\frac{f^{\prime}}{f}=\frac{3}{f}\sqrt{1+\frac{C^{2}}{6f^{4}}}\simeq\frac{3}{f}+\frac{C^{2}}{4f^{5}}\,,$
(28)

where in the last step, we have expanded the previous expression to subleading
order in the limit $f\to\infty$. The contribution from this boundary to the
decay exponent $B$ (see (3)), can be computed by noting that the vacuum
extrinsic curvature scalar is given by

$\displaystyle\mathcal{K}_{4,vacuum}=\frac{3}{f}\,,$ (29)

such that the contribution to $B$, i.e. the difference in the actions, can be
calculated

$\displaystyle
B\supset\int_{f\to\infty}f^{3}\left(-\mathcal{K}_{4}+\mathcal{K}_{4,vacuum}\right)=6\pi^{2}f^{3}\left.\left(-\frac{3}{f}-\frac{C^{2}}{4f^{5}}+\frac{3}{f}\right)\right|_{f\to\infty}=0\,.$
(30)

Hence, the entire contribution to $B$ comes from the ETW boundary at the
bubble wall. As dictated by the Israel junction conditions, an ETW boundary
with non-vanishing extrinsic curvature requires a brane with a certain
tension. The (analogue of the) Israel conditions relate the extrinsic
curvature $\mathcal{K}_{4,ab}$ to the matter energy momentum tensor $S_{ab}$
located at the end of the world [52, 53]

$\displaystyle\mathcal{K}_{4,ab}=S_{ab}-\frac{1}{2}h_{ab}h^{cd}S_{cd}\,.$ (31)

An object of pure tension $T_{4}$ has an energy momentum tensor
$S_{ab}=-T_{4}h_{ab}$ and, due to the $O(4)$ symmetry of the problem, we lose
no information by taking the trace of (31)

$\displaystyle\mathcal{K}_{4}=-\frac{1}{2}h^{ab}S_{ab}=\frac{3}{2}T_{4}\,.$
(32)

As a consequence of (32), the decay exponent takes the form555For the bubble
of nothing solution, where the action of the vacuum has to be subtracted, we
note that the vacuum action does not have a boundary at finite $f=0$. However,
if one considers $f=0$ to be a boundary of flat space one quickly notices that
the extrinsic curvature reads $\mathcal{K}_{4,vacuum}=-3/f$ and the boundary
contribution becomes $6\pi^{2}f^{3}/f|_{f=0}=0$.

$\displaystyle
B=2\pi^{2}f_{\eta}^{3}\left(-\mathcal{K}_{4}+T_{4}\right)=-2\pi^{2}f_{\eta}^{3}\frac{T_{4}}{2}=-2\pi^{2}f_{\eta}^{3}\frac{\mathcal{K}_{4}}{3}\,.$
(33)

Finally, for the ETW boundary at $f_{\eta}$, the extrinsic curvature scalar
takes the value

$\displaystyle\mathcal{K}_{4}=-3\left.\frac{f^{\prime}}{f}\right|_{f=f_{\eta}}=-\frac{3}{f_{\eta}}\sqrt{1+\frac{C^{2}}{6f_{\eta}^{4}}}\,.$
(34)

By comparing with (32), we see that a Minkowski bubble of nothing requires a
negative tension $T_{4}$, as anticipated before.

We can now compute the decay rate for the Witten bubble with defect. Using
(32), (34), (23) in the limit $\eta\ll R_{KK}$, the tension and extrinsic
curvature are calculated as

$\displaystyle
T_{4}=\frac{2}{3}\mathcal{K}_{4}=-\frac{2C}{\sqrt{6}f_{\eta}^{3}}=-\left(1-\frac{\theta}{2\pi}\right)\frac{\eta^{-\frac{3}{2}}}{\sqrt{2\pi}}\,.$
(35)

The decay exponent then follows from substituting (19),(35) into (33)

$\displaystyle
B=\frac{2\pi^{3}R_{KK}^{3}}{\left(1-\frac{\theta}{2\pi}\right)^{2}}=\frac{\pi^{2}M_{P}^{2}R_{KK}^{2}}{\left(1-\frac{\theta}{2\pi}\right)^{2}}\,,$
(36)

where we have used the definition of the 4d Planck mass $M_{P}^{2}=2\pi
R_{KK}$. We observe that the Witten bubble decay rate (5) is recovered for
$\theta=0$. Furthermore, the decay rate decreases for increasing deficit angle
(and hence increasing defect tension) and the transition becomes forbidden for
$\theta=2\pi$. By comparing with (24), we see that in this limit, the bubble
radius grows infinitely large.

Let us collect some further remarks on the above calculation. From (35), we
see that the Witten bubble solution has, from the four-dimensional
perspective, a non-vanishing extrinsic curvature contribution to the action at
the bubble surface, which for a low-energy observer signals the presence of a
boundary ETW brane. Hence, the description using $\mathcal{K}_{4}$ and $T_{4}$
is justified. In the presence of more complicated defects as discussed below,
it is likely that even the $d$-dimensional theory is described by a boundary
action at the locus of the defect, which then naturally descends to a 4d
boundary action as in (25). Importantly, the action of the form (25) also
allows us to reliably compute decay rates for more general ETW branes, as will
be discussed below.

From (35), we see that $T_{4}$ diverges in the limit $\eta\to 0$. This is
result from $T_{4}$ being measured in units of the 4d Planck mass, which in
the limit of $\eta\to 0$ locally vanishes as $\sim\eta^{1/2}$. Hence, the
diverging $T_{4}$ is a peculiar looking artefact of our 4d description which
however will turn out to be useful in the upcoming sections.

The additional defect was introduced in (20) as an object of the 5d theory
wrapping the $S^{3}$ at fixed $f$. Its tension in 4d Einstein frame can be
obtained by looking at the action

$\displaystyle S_{def}=-T_{def}\int d^{3}x\sqrt{g_{5}|_{P.B.}}=-T_{def}\int
d^{3}xe^{3\alpha\phi}f^{3}=-T_{def}e^{-\frac{3}{2}\beta\phi}\int
d^{3}x\sqrt{g_{4}|_{P.B.}}\,,$ (37)

where we have used (85). Hence, the tension in 4d Einstein frame is given by
$T_{def}(2\pi R)^{-3/2}$. Remembering that $T_{def}=\theta$, Eq. (35) becomes
intuitive: The defect tension simply adds to the negative tension arising from
the shrinking extra dimension and brings the total tension closer to zero,
making the decay less likely to occur.

The positive energy theorem [54, 55, 56] applies to KK compactifications which
asymptote to flat Minkowski space times constant KK manifold. If there exists
an asymptotically covariantly constant spinor and the weak energy condition
holds666There are multiple versions of the positive energy theorem with
slightly different assumptions. For our purposes it is appropriate to use
[55]., the ADM mass is non-negative. Furthermore, the mass only vanishes if
the asymptotic vacuum situation extends to the complete geometry. Since energy
is conserved, this implies that the vacuum is stable. Based on arguments of
this type and using the fact that the energy is the square of the
supercharges, one expects supersymmetric string vacua to be stable [57]. In
our context, we thus expect $T_{def}$ to be large enough to forbid a decay if
the relevant vacuum is supersymmetric. Once supersymmetry is broken and the
weak energy condition is violated, decays become possible[18].

###  3.3 Generality of the method developed

Our method for computing the decay rate allows us to go beyond the Witten
bubble with defect and to treat more complicated instanton geometries, e.g.
with more compact dimensions. Using the general metric ansatz (10) with
appropriate values for $\alpha$ and $\beta$ (7), the equations of motion
(12),(13) continue to hold. If the potential $V(\phi)$ vanishes, also the
solution (14),(16) remains valid. Of course, our ability to let the compact
space shrink to zero size in such a way that the higher-dimensional geometry
remains smooth is special to $S^{1}$ and higher spheres. If the
compactification space is generic, the locus where its volume vanishes will
represent a singularity. Motivated by the Cobordism Conjecture [30], we may
expect that corresponding transitions to nothing are allowed in quantum
gravity. Put differently, one may expect that spacetime topology can fluctuate
at small scales and that hence our compact space, once it has become small,
can disappear. We should then be able to cut out a region with sufficiently
small compact space and to replace it by an effective ‘ETW defect’. The
relevant region, i.e. the thickness of the ETW defect, may be Planck size or
it may be larger. This is to a certain extent a matter of definition of our
effective theory. Of course, for our approximation to be valid, the thickness
should be much smaller than the critical bubble radius.

This class of instantons, where only the volume of the internal manifold
changes until an ETW defect is reached, is conceptually similar to the Witten
bubble with defect and might hence be viewed as a generalized Witten bubble.
Many known bubble of nothing solutions fall in this category and may hence be
described by our general methods, see e.g. [18, 7, 10]. The power of our 4d
description and the generality of the ansatz (25) allow us to compute the
decay rate for a general such instanton, i.e. equations (33),(34) remain
valid.

A natural limit to consider is $\eta\ll R_{KK}$. In this case one finds from
(33),(34) that

$\displaystyle B=\frac{2\pi^{2}}{\sqrt{6}}|C|\,.$ (38)

Like for the Witten bubble, the integration constant $C$ depends on $R_{KK}$
and on the boundary conditions at the ETW defect. We will work this out below
in examples.

Another natural class of solutions has $\eta\approx R_{KK}$. In this case,
$1\gg C^{2}/f_{\eta}^{4}$, such that $f^{\prime}\simeq 1$,
$\phi^{\prime}\simeq 0$ on the entire instanton. As a result, the ETW brane
cuts out a sphere from flat Minkowski space. Moreover, the ETW defect is
maximally thick and essentially identical with the ETW brane. We will see an
explicit example in what follows. The general formulas without any restriction
on $\eta$ are shown in App. B.

We note that, in principle, one could even have exotic bubble-of-nothing or
something solutions in which the compact radius grows as one approaches the
ETW brane. One may then have $\eta\gg R_{KK}$. Although such boundary
conditions are physically unintuitive, the method how to compute the decay
rate remains unchanged.

Another interesting observation to be made is that the extrinsic curvature
(34) and hence the decay rate is symmetric under $C\to-C$. From (16), one
observes that this change in sign changes the solution from a shrinking to a
growing internal manifold when approaching the bubble surface. The Witten
bubble with defect is a solution with shrinking $S^{1}$, i.e. $C>0$, but due
to $T$-duality, it might not be entirely surprising that an analogous solution
with $C<0$ also exist. Of course, the resulting geometry is somewhat peculiar,
because the extra dimension grows large towards the bubble center and it is
unclear how the geometry closes. Under T-duality, the codimension-two defect
assumed to be present gets mapped to a codimension-one brane wrapping the
internal $S^{1}$. This brane can act as an ETW brane, i.e. fulfilling the
Israel junction conditions, such that the geometry can smoothly end. However,
for theories in $d>5$, the T-duality interpretation is not always given. It
would be interesting to study the $C\to-C$ symmetry in more detail.

###  3.4 Bubbles of nothing in Calabi-Yau compactifications

In this section, we finally discuss bubble of nothing instanton solutions and
the corresponding decay rates in string theory models arising from
compactifiction on CY-orientifolds, possibly with fluxes. We will use the
intuition from the Witten bubble with defect and the general procedure how to
calculate decay rates from above. Fluxes and non-perturbative effects
generally present in $\mathcal{N}=1$ or $\mathcal{N}=0$ models induce a
potential for $\phi$ which is particularly important at small volumes. We
assume that it is a valid approximation to treat the effects of this potential
as being part of the effective ETW defect and hence use (16) to find the decay
rate. This is justified as long as the thickness of the domain wall remains
small compared to the critical bubble radius.

Since we now work in $d=10$, $\alpha$ and $\beta$ take values according to
(7):

$\displaystyle\alpha=-\sqrt{\frac{3}{8}}\,,\qquad\beta=\frac{1}{2\sqrt{6}}\,.$
(39)

Clearly, in the present case, we always require a ‘cobordism transition’, i.e.
a region of topology change and hence a non-trivial ETW defect. Thus, we can
trust the geometry defined by (16) only up to the point where the CY radius
has changed to some critical size $\eta$. From the condition

$\displaystyle\eta=R(f_{\eta})\,,$ (40)

one then finds the relation

$\displaystyle\frac{C}{f_{\eta}^{2}}=\frac{\sqrt{6}}{2}\left[\left(\frac{R_{KK}}{\eta}\right)^{4}-\left(\frac{\eta}{R_{KK}}\right)^{4}\right]\,.$
(41)

As for the Witten bubble with defect, we expect that $C$ should be fixed by
the boundary conditions at the ETW defect. While this ETW defect is an
unfamiliar object, it is natural to characterize its effect on the surrounding
geometry by a ‘deficit angle’ $\theta$ defined in analogy to (20):

$\displaystyle
1-\frac{\theta}{2\pi}=\left.\frac{dR}{dx}\right|_{x=0}=\left.\frac{\beta}{2\pi}\frac{C}{f^{3}}e^{(\beta-\alpha)\phi}\right|_{f=f_{\eta}}=\frac{1}{4\pi\sqrt{6}}\frac{C}{f_{\eta}^{3}}(2\pi\eta)^{4}\,.$
(42)

For the remainder of this section, we assume $\eta\ll R_{KK}$ and hence
$C>0,\;\theta<2\pi$, which is the parameter range corresponding to a small ETW
defect, as motivated above. Hence, we ignore the second term in (41). General
equations, including the cases $\eta\approx R_{KK}$ and $\eta>R_{KK}$, are
provided in App. B. Solving (41) and (42) for $C$ gives

$\displaystyle
C=\frac{\sqrt{6}(2\pi)^{6}}{32\left(1-\frac{\theta}{2\pi}\right)^{2}}\frac{R_{KK}^{12}}{\eta^{4}}\,.$
(43)

This fully determines the instanton solution. Unlike the 5d case, $C$ now
explicitly depends on $\eta$ and diverges as $\eta\to 0$. Thus, the limit of a
truly point-like ETW defect can not be taken. To find the decay rate, we go on
to compute the 4d extrinsic curvature of the boundary at $f=f_{\eta}$ (see
(34)):

$\displaystyle\mathcal{K}_{4}=-6\left|1-\frac{\theta}{2\pi}\right|(2\pi)^{-3}\eta^{-4}\,.$
(44)

Note that $\mathcal{K}_{4}$ does not depend on $R_{KK}$, which in Appendix B
is shown to hold generally in the limit $\eta\ll R_{KK}$. This is consistent
with the expectation that the choice of $\eta$ and $\theta$ should completely
fix the local geometry near the boundary. By the Israel conditions, the
extrinsic curvature also determines the total tension of the ETW brane in the
4d EFT, cf. (32).

Next, we compute the critical bubble radius:

$\displaystyle\rho_{0}=e^{\alpha\phi(f_{\eta})}f_{\eta}=\frac{R_{KK}}{4\left|1-\frac{\theta}{2\pi}\right|}\left(\frac{R_{KK}}{\eta}\right)^{3}\,.$
(45)

It diverges in the limit $\eta\to 0$, showing again that it is not possible to
remove the ETW defect. In comparison to the Witten bubble radius (24),
$\rho_{0}$ is much larger due to the appearance of the factor
$(R_{KK}/\eta)^{3}\gg 1$. The decay exponent straightforwardly follows from
(38)

$\displaystyle
B=\frac{(2\pi)^{8}R_{KK}^{8}}{64\left(1-\frac{\theta}{2\pi}\right)^{2}}\left(\frac{R_{KK}}{\eta}\right)^{4}=\frac{\pi^{2}M_{P}^{2}R_{KK}^{2}}{16\left(1-\frac{\theta}{2\pi}\right)^{2}}\left(\frac{R_{KK}}{\eta}\right)^{4}\,.$
(46)

It is again enhanced relative to the Witten bubble case (36) by powers of
$R_{KK}/\eta$.

A peculiar feature of the CY bubble is the fact that when approaching the
bubble wall from infinity, the transversal three sphere radius
$\rho=e^{\alpha\phi}f$ first shrinks as expected, but at some point starts
growing again. This can be seen by looking at the profile $\rho(f)$

$\displaystyle\rho(f)=R_{KK}^{-3}\left(\frac{C}{\sqrt{6}f^{2}}+\sqrt{\left(\frac{C}{\sqrt{6}f^{2}}\right)^{2}+1}\right)^{\frac{3}{4}}f\,.$
(47)

For large values of $f$, $\rho(f)\approx R_{KK}^{-3}f$ and hence $\rho$ grows
with growing $f$. For small $f$ on the other hand,

$\displaystyle\rho(f)\approx
R_{KK}^{-3}\left(\frac{2C}{\sqrt{6}}\right)^{\frac{3}{4}}\frac{1}{\sqrt{f}}\,,$
(48)

and we see that $\rho$ decreases with growing $f$. Such a behavior has already
been observed in 6d models [8]. In Appendix B, it is shown that this peculiar
behavior always occurs for $d>5$.

Clearly, this analysis could have been carried out with $C<0$ and from (38)
one can see that the decay rate is symmetric under $C\to-C$. In contrast to
the case of $S^{1}$ compactifications, it is not immediately clear how to
relate these two types of solutions using T-duality or mirror symmetry. It
would be interesting to study this further, presumably invoking Calabi-Yau
moduli other than the volume.

###  3.5 Bubbles of something

The 4d geometry corresponding to the bubble of something instanton is the disc
on the r.h.s. of Fig. 1, with $f=0$ at the center and $f=f_{\eta}$ at the
boundary. To find the instanton solutions, we again need to consider the field
equations (12),(13) subject to the general boundary condition
$\phi^{\prime}(f=0)=0$ as well as the boundary conditions on $\phi$ and
$\phi^{\prime}$ at the bubble wall. For instanton solutions in concrete 6d
models, see [20].

Given a solution, one can calculate $-S_{instanton}$. However, we can not
apply (3) to find a rate since there is no parent vacuum. Instead, creation
rates are expected to obey

$\displaystyle\Gamma\sim\exp(-B)=\exp(-S_{instanton})\,,$ (49)

with a single $\Gamma$ not being meaningful. Physical predictions follow, for
example, from ratios of the form [58, 59, 60](see also [61] for a recent
proposal)

$\displaystyle\Gamma_{i}/\Gamma_{j}\,,$ (50)

with $i,j$ labelling vacua of the theory.

We first observe that, in general, bubbles of something do not exist for
vanishing potential. This follows from (14), which shows that the vanishing of
$f$ at the center, together with smoothness of $\phi$, implies $C=0$. This
special solution corresponds to a flat disc with $\phi=\phi_{0}$.

However, it is not difficult to show that bubble of something solutions do
exist for nontrivial potentials. They can approximate the flat-
disc/constant-$\phi$ solution above as follows: In the central part of the
disc, $\phi$ is essentially constant, sitting near a Minkowski-minimum of the
potential. Near the boundary, $\phi$ starts moving away from that minimum and
$f$ starts deviating from $f=r$, such that eventually the boundary conditions
associated with the ETW defect are met at $f=f_{\eta}$. The boundary
conditions are given by (40) and an equation for the deficit angle similar to
(42). However, we have to remember that $x$ measures the distance from the ETW
brane which, in comparison to (21), is now related to $r$ via

$\displaystyle dx=-e^{\alpha\phi}dr\,.$ (51)

As a result, the second boundary condition becomes

$\displaystyle
1-\frac{\theta}{2\pi}=\left.\frac{dR}{dx}\right|_{x=0}=-\left.\frac{\beta}{2\pi}\phi^{\prime}e^{(\beta-\alpha)\phi}\right|_{f=f_{\eta}}=-\frac{1}{4\pi\sqrt{6}}\phi^{\prime}(2\pi\eta)^{4}\,.$
(52)

Hence, a solution with deficit angle $\theta<2\pi$ now corresponds to
$\phi^{\prime}<0$ at the ETW brane, unlike in bubble of nothing models where
$\phi^{\prime}>0$ (i.e. $C>0$) was a necessary condition, cf. (42).

The instanton action can be evaluated using our general formula (27) for on-
shell actions. It receives both bulk and boundary contributions. The boundary
term contribution to $S_{instanton}$ can be computed using

$\displaystyle\frac{3}{2}T_{4}=\mathcal{K}_{4}=+\left.\frac{3f^{\prime}}{f}\right|_{f=f_{\eta}}=+\frac{3}{f_{\eta}}\sqrt{1+\frac{f_{\eta}^{2}}{3}\left(\frac{\phi^{\prime
2}}{2}-V(\phi)\right)_{f=f_{\eta}}}\geq 0\,.$ (53)

Note that, as anticipated, a bubble of something requires a positive-tension
ETW brane.

Given a solution, also the bulk term in (27) can in principle be evaluated
straightforwardly. However, in this study we do not discuss any details of the
potential and we can hence only give explicit results if this term is
negligible due to appropriate assumptions:

We can provide a creation rate estimate assuming that $\phi$ is stabilized
such that $V\simeq 0$ for a large part of the instanton. Consequently, $\phi$
moves significantly only close to the ETW boundary. Using the equations of
motion, one can show that the bulk of the instanton is essentially flat space,
i.e. $\phi^{\prime}\simeq 0,f^{\prime}\simeq 1$. In case of the critical
bubble radius being large in comparison to the region where $\phi$ moves
significantly, we may absorb these effects into the ETW brane tension and
hence obtain an effective description of a bubble of something instanton where
an ETW brane with tension $T_{4}$ bounds flat space, similar to [28]. The
assumptions stated above are most easily fulfilled for small values of $T_{4}$
or a potential stabilizing $\phi$ strongly. Of course, by absorbing the
unknown effects of the potential near the boundary into $T_{4}$, we sacrifice
the calculability of $T_{4}$ from $\eta$ and $\theta$.

The extrinsic curvature at $f=f_{\eta}$, defined as the region where $\phi$
begins rolling and hence where the ETW brane is located, is given by

$\displaystyle\mathcal{K}_{4}\simeq\frac{3}{f_{\eta}}\,.$ (54)

The ETW brane tension, now also capturing the effects of the potential and the
rolling of $\phi$ near the ETW boundary, is then related to $f_{\eta}$ by (32)

$\displaystyle T_{4}=\frac{2}{3}\mathcal{K}_{4}=\frac{2}{f_{\eta}}.$ (55)

Disregarding, as discussed, the bulk contribution and reinstating $M_{P}$ for
clarity one finds

$\displaystyle
B=-2\pi^{2}f_{\eta}^{3}\frac{M_{P}^{2}\mathcal{K}_{4}}{3}=-2\pi^{2}f_{\eta}^{2}=-8\pi^{2}\frac{M_{P}^{6}}{T_{4}^{2}}\,.$
(56)

The rates governed by this exponent have to be viewed as competing with the
famous proposals for vacuum creation [21, 22, 62] which result in closed
universes and require the initial value of the scalar potential to be
positive. By contrast, the bubbles of something governed by (56) give rise to
open universes with no restriction on the vacuum energy, as long as an ETW
brane with sufficient tension exists.

Our Coleman-De Luccia based perspective on (56) implies that the quantity in
the exponent is eventually positive, such that no tunneling-suppression
arises. This is then analogous to the Hartle-Hawking [21] rather than the
Linde/Vilenkin [22, 62] prediction. However, in light of the long-lasting
debate concerning these proposals, one should obviously ask whether a Linde-
Vilenkin-like exponential suppression by $B$ from (56) may also be argued for.

Recently, the creation of universes from nothing according to [21, 22, 62] was
analyzed based on the Lorentzian path integral [63, 64, 65, 66, 67], where it
was found that neither proposal in the original form is free of issues. An
important role is played by potentially unsuppressed field fluctuations.
Clearly, it would be desirable to study the bubble of something from a
Lorentzian perspective to validate the creation rate and to analyze the
behavior of field fluctuations.

###  3.6 Comparison to other results

Our results can be tested by comparing with [18], where a bubble of nothing
for a $T^{3}$ compactification in Einstein-Dilaton-Gauss-Bonnet gravity was
constructed. This construction also allows for a string-theoretic embedding.
What we referred to as ETW defect can be identified with the ‘inner bubble
region’ of [18]. Naively, we would compare their results with our calculations
for $d=7$. However, the presence of the dilaton in [18] modifies the theory
such that the resulting equations of motion are analogous to the $d=5$ theory
without a dilaton. Hence, one should compare with equations derived in
Sections 3.1,3.2. Using results from [18], the deficit angle can be calculated
to be 777Here, we use the profile of the volume modulus (which is identical to
the dilaton profile) at the surface where the outer bubble-region matches the
inner-bubble region. More precisely, we refer to Eq. (5.87) in [18].

$\displaystyle
1-\frac{\theta_{T^{3}}}{2\pi}=\frac{12\pi\lambda}{(2\pi\eta)^{2}}\,,$ (57)

with $\lambda$ denoting the Gauss-Bonnet coupling and $(2\pi\eta)^{3}$ being
the volume of the $T^{3}$ at the locus of the ETW defect. The critical bubble
radius $\rho_{0,T^{3}}$ as well as the decay exponent $B_{T^{3}}$ (assuming
$\eta\ll R_{KK}$ where analytical results were obtained in [18]) are

$\displaystyle\rho_{0,T^{3}}=\frac{(2\pi\eta)^{2}R_{KK}}{12\pi\lambda}\,\,\,,\qquad
B_{T^{3}}=\pi^{2}M_{P}^{2}\rho_{0}^{2}\,.$ (58)

Using (57), one finds

$\displaystyle\rho_{0,T^{3}}=\frac{R_{KK}}{1-\frac{\theta_{T^{3}}}{2\pi}}\,\,\,,\qquad
B_{T^{3}}=\frac{\pi^{2}M_{P}^{2}R_{KK}^{2}}{\left(1-\frac{\theta_{T^{3}}}{2\pi}\right)^{2}}\,,$
(59)

in perfect agreement with our previous results (24),(36).

##  4 Explicit realization in type IIB string theory

ETW branes for Calabi-Yau or Calabi-Yau orientifold compactifications, as
discussed in the previous section, could be responsible for bubble of nothing
decays or bubble of something creation processes for many vacua of the string
landscape. In the present section, we explicitly construct an ETW brane for
type IIB compactifications on CY orientifolds with O3 planes. We will comment
on the relation to the shrinking-CY analysis of Sec. 3.4 at the end of Sec.
4.1.

###  4.1 The ETW brane

The main idea is most easily explained by starting with type IIA theory on a
CY manifold $Y_{A}$. The domain wall arises by considering the geometry
$\mathbb{R}^{4}\times Y_{A}$, with $\mathbb{R}^{4}$ parameterized by
$(X^{0},\cdots,X^{3})$, and applying a standard orientifold projection
$X^{3}\to-X^{3}$. As a result, one obtains an O8 plane located at $X^{3}=0$
which also plays the role of an ETW brane.888 In the present section, we use
Lorentzian geometry with a time-like ETW brane. This makes it simpler to
discuss SUSY and SUSY breaking. We assume that we are allowed to return to
Euclidean instanton solutions as usual, by analytic continuation in the 4d
EFT.

Applying mirror symmetry and recalling its Strominger-Yau-Zaslow (SYZ)
interpretation as T-duality of the $T^{3}$ fibre of the CY [68], it is clear
that we obtain type IIB string theory on $\mathbb{R}^{4}\times Y_{B}$ with a
set of O5 planes wrapping a three-cycle of $Y_{B}$ and filling out three
directions of $\mathbb{R}^{4}$. Clearly, this slightly more complicated ETW
brane may also be constructed by directly applying an appropriate orientifold
projection to $\mathbb{R}^{4}\times Y_{B}$ on the type IIB side. We place one
D5 brane on top of each O5 plane to cancel tadpoles locally.

We can go beyond plain $\mathcal{N}=2$ in the bulk by combining the
orientifold projection above with a holomorphic orientifold projection acting
only on $Y_{B}$ and introducing (for simplicity) O3 planes only. Now, away
from the ETW brane, we have a type IIB ${\cal N}=1$ orientifold model of the
type underlying the best understood part of the landscape. At the locus of the
ETW brane, the simultaneous existence the O5/D5 and O3/D3 systems might break
SUSY completely. However, as is argued in Appendix C, this configuration
preserves two supercharges leading to $\mathcal{N}=1$ SUSY on the 3d
worldvolume of the domain wall.

The above construction can be conveniently illustrated in the following toy-
model: Consider the space $T^{6}/\mathbb{Z}_{2}$, with the $T^{6}$
parameterized by three complex coordinates

$\displaystyle{Z^{i}=U^{i}+iV^{i}}\,,\quad U^{i}\sim U^{i}+2\pi\,,\quad
V^{i}\sim V^{i}+2\pi\,,\quad i\in\\{1,2,3\\}\,.$ (60)

The $\mathbb{Z}_{2}$ generator acts as
$(Z^{i}\to-Z^{i},\;(-1)^{F_{L}}\Omega)$, with $F_{L}$ the left-moving fermion
number and $\Omega$ the worldsheet parity. This orientifold action produces
$64$ O3 planes located at $U^{i}=0$ or $U^{i}=\pi$ and $V^{i}=0$ or
$V^{i}=\pi$ for $i=1,2,3$ and filling the non-compact directions. In addition,
we introduce an O5 plane generator acting by $(X^{3}\to-X^{3}$,
$Z^{i}\to\overline{Z}^{i}+i\pi,\;\Omega)$. The fixed points of this action,
which determine the positions of the O5 planes, are given by $V^{i}=\pm\pi/2$.
The full orientifold group is $\mathbb{Z_{2}}\times\mathbb{Z}_{2}$ containing
the identity, the two aforementioned generators and their product acting by
$X^{3}\to-X^{3}$, $Z^{i}\to-\overline{Z}^{i}-i\pi$ without orientation
reversal (since $\Omega^{2}=1$). Modding out this product generator does not
induce additional singularities since the map $V^{i}\to V^{i}-\pi$ on a torus
is fixed-point-free. Table 1 summarizes the generators and Table 2 the
locations of the O-planes. Note that the O3 and O5 planes do not intersect due
to their separation along $V^{i}$. We can explicitly check that this
configuration does not break SUSY completely. To show this, we first note that
O$p$ planes preserve the same supercharges as D$p$ branes, such that we can
equivalently think about the SUSY breaking of D3 and D5 branes. From Table 2,
we can directly read off that an open string stretching between those branes
is defined by four Dirichlet-Neumann boundary conditions which is known to
preserve $1/4$ of the $32$ supercharges.

| $X^{0}$ | $X^{1}$ | $X^{2}$ | $X^{3}$ | $U^{1}$ | $V^{1}$ | $U^{2}$ | $V^{2}$ | $U^{3}$ | $V^{3}$ |
---|---|---|---|---|---|---|---|---|---|---|---
$g_{1}$ | $X^{0}$ | $X^{1}$ | $X^{2}$ | $X^{3}$ | $-U^{1}$ | $-V^{1}$ | $-U^{2}$ | $-V^{2}$ | $-U^{3}$ | $-V^{3}$ | $\Omega(-1)^{F_{L}}$
$g_{2}$ | $X^{0}$ | $X^{1}$ | $X^{2}$ | $-X^{3}$ | $U^{1}$ | $-V^{1}+\pi$ | $U^{2}$ | $-V^{2}+\pi$ | $U^{3}$ | $-V^{3}+\pi$ | $\Omega$
$g_{1}\cdot g_{2}$ | $X^{0}$ | $X^{1}$ | $X^{2}$ | $-X^{3}$ | $-U^{1}$ | $V^{1}-\pi$ | $-U^{2}$ | $V^{2}-\pi$ | $-U^{3}$ | $V^{3}-\pi$ | $(-1)^{F_{L}}$
Table 1: Action of the two orientifold generators (of O3 and O5 planes) and of their product. | $X^{0}$ | $X^{1}$ | $X^{2}$ | $X^{3}$ | $U^{1}$ | $V^{1}$ | $U^{2}$ | $V^{2}$ | $U^{3}$ | $V^{3}$
---|---|---|---|---|---|---|---|---|---|---
O3 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\crossproduct$ | $\crossproduct$ | $\crossproduct$ | $\crossproduct$ | $\crossproduct$ | $\crossproduct$
O5 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\crossproduct$ | $\checkmark$ | $\crossproduct$ | $\checkmark$ | $\crossproduct$ | $\checkmark$ | $\crossproduct$
Table 2: Summary of dimensions filled by O3/O5 planes (indicated with a
$\checkmark$).

We expect our $T^{6}/\mathbb{Z}_{2}$ toy-model to generalize to proper CY
orientifolds with O3 planes, as described at the beginning of this section. In
particular, the O5 planes in type IIB are related by mirror symmetry to the O8
plane on the IIA side, which arises from the reflection $X^{3}\to-X^{3}$.
Viewing $Y_{B}$ in terms of the SYZ fibration $T^{3}\to Y_{B}\to B_{3}$, with
$B_{3}$ denoting the base, it is clear that the O5 planes are compatible with
the symplectic structure of $Y_{B}$ (i.e. they wrap $B_{3}$). We cancel their
D5 tadpole locally by placing one D5 brane on top of every O5 plane. One might
by concerned that the O5/D5 system induces a D1 tadpole since the Chern-Simons
brane action contains a term

$\displaystyle S_{C.S.}\supset\int_{\rm O5/D5}C_{2}\wedge p_{1}(R)\,.$ (61)

However, since the O5/D5 systems wraps a three-cycle of $Y_{B}$ and the first
Pontryagin class $p_{1}(R)$ is a four-form, the above integral vanishes.
Hence, our domain wall does not introduce any uncancelled tadpoles. Following
this line of thought, we are lead to refine our initial statements about the
O8/D8 system on the mirror-dual IIA side. Indeed, we expect the O8/D8 system
to carry a non-trivial gauge bundle. To see this, note that this system wraps
the entire CY. The coupling analogous to (61),

$\displaystyle S_{C.S.}\supset\int_{\rm O8/D8}C_{5}\wedge p_{1}(R)\,,$ (62)

then induces a D4 charge, which we may cancel by an appropriately wrapped
D4-brane. This brane can be taken to be resolved in the D8-stack, inducing a
non-trivial gauge bundle and thereby ensuring that the IIA side is tadpole-
free as well. For completeness, we recall that the IIA side in addition has an
O6/D6 system and the IIB side a mirror-dual O3/D3 system, such that the non-
empty side of the ETW-brane is ${\cal N}=1$.

Let us comment in more detail on the dynamics of our effective ETW brane. At
tree level this brane, viewed as an object of the 4d EFT, has vanishing
tension: $T_{4}=0$. This is clear since we use CY geometry and cancel the
tadpoles of all O5 planes involved locally. The O3 plane charge can of course
only be cancelled globally, which implies non-vanishing $F_{5}$ field strength
and some warping but does not destabilize the compact space away from the ETW
brane or the brane itself. This is also consistent with the fact mentioned
above (cf. App. C for details) that the ETW brane preserves $\mathcal{N}=1$
SUSY in 3d.

Even in the absence of fluxes, non-perturbative effects will in general induce
a non-zero superpotential, leading to an $\mathcal{N}=1$ AdS vacuum or to a
runaway to infinite volume in the bulk. Since our domain wall preserves some
supercharges, one might expect that it also receives corresponding
corrections, giving it a tension of exactly the right magnitude to keep it
flat [69, 70, 71]. A flat domain wall with critical tension $T_{4}=\pm
T_{c}=\pm 2/\ell_{AdS}$ allows for a critical bubble of something (positive
sign) or a critical bubble of nothing (negative sign) with infinite radius.
Thus, neither of the two corresponding tunneling processes can dynamically
arise.

We note, however, that the above relies on an analysis performed for domain
walls in a 4d theory with smoothly varying superpotential[69, 70, 71]. It is
not clear to us whether the conclusion concerning the critical tension can
really be carried over to our case of a (generically singular) ETW brane. In
particular, we feel that we can not firmly exclude the possibility of
supersymmetric but non-stationary ETW branes with $AdS_{3}$ worldvolume
geometry. For instance, it was shown in [72] that 5-dimensional supergravity
does admit supersymmetric domain walls with “arbitrary” tension. In 4d, one
might expect that the ordinary BPS bound for field theory domain walls will
carry over to ETW branes, although this is not immediately compatible with the
following holographic argument: Find an AdS3 supergravity theory containing a
large-$N$ CFT in the matter sector. This CFT can be holographically
interpreted as an additional spatial dimension, implying the existence of an
AdS4 ‘bulk space’. The AdS3 we started from now plays the role of the boundary
of our 4d theory. Being AdS, this boundary is non-stationary and corresponds
to a ‘detuned’ ETW brane.

In any event, even if such ‘detuned’ ETW branes should be relevant in our
setting, both types of of bubble-of-anything processes are still excluded
since the tension lies in the range $-T_{c}\leq T_{4}\leq T_{c}$. Thus, for
bubbles of either type to arise, SUSY must be broken in the bulk or by the ETW
brane. We will now discuss both options.

First, let us focus on bulk SUSY breaking. This may arise if our ${\cal N}=1$
vacuum is modified by further effects, such as fluxes and/or anti-D3 brane
uplifts etc., leading to a metastable SUSY breaking minimum. If fluxes are
present, they would first have to be removed by an appropriate domain wall
consisting of wrapped D5/NS5 branes, which are extended in parallel to our
O5/D5 ETW brane. We will from now on view both together as our effective ETW
brane.

The bulk SUSY breaking effects will clearly modify both the vacuum energy and
the tension of the domain wall. Most naively, one expects that SUSY breaking
increases the decay probability, i.e. that bubbles of nothing become possible
similar to [18]. However, especially if the tension has been positive before
SUSY breaking, it is also plausible that a bubble of something arises. We
expect the correction to $T_{4}$ to be set by a combination of the KK scale
and the SUSY-breaking scale.

If, instead of the supersymmetric ETW brane constructed above, we manage to
find an ETW brane that does not preserve any supercharges, we expect the
tension to be larger such that bubbles of something may become allowed. For
example, one may imagine that the D5 branes which are part of our ETW
construction are wrapped on cycles that are distinct from the special
lagrangian 3-cycles of the O5-planes. Whether this is possible while still
cancelling the O5-tadpole and ensuring a sufficiently metastable or long-lived
ETW brane is an open question. It appears likely that the small-$\eta$ ETW
defects speculated to exist in Sec. 3 are SUSY-breaking and hence provide an
example of the SUSY-breaking ETW branes just discussed. Generally, since non-
supersymmetric defects have been predicted by the cobordism conjecture [30,
32, 33], there likely exist ETW branes suitable for bubble of something
creation. However, it is a general challenge to explicitly construct
metastable objects with large tension and it could be a similarly hard task as
finding metastable de Sitter vacua in string theory.999For example, one
expects a 9d non-SUSY positive-tension ETW brane for the 10d Minkowski type-
IIB vacuum to exist. Its tension presumably depends on the dilaton,
dynamically driving the latter to infinity. It would be interesting to study
whether bubble of something processes are possible in settings of this type.

For simplicity, we so far considered CY orientifolds with O3 planes only.
However, nothing prevents us from also allowing for O7 planes wrapped on
holomorphic cycles on the CY. The resulting 4d vacua still have
$\mathcal{N}=1$ SUSY and, as argued in App. C, the O5/D5 ETW brane still
preserves two supercharges. However, the O7 and the O5 plane now generically
intersect.

Note that the $T^{6}$ example above also allows for the introduction of O7
planes: Consider a compactification of type IIB string theory on
$T^{2}/\mathbb{Z}_{2}\times T^{4}$ containing four O7 planes. Using again the
parametrization (60) with $\mathbb{Z}_{2}$ action
$(Z^{1}\to-Z^{1},\;(-1)^{F_{L}}\Omega)$, the O7 planes are located at points
where $U^{1},V^{1}$ take values $0$ or $\pi$. Introducing again an O5 plane
generator compatible with the symplectic structure of the torus,
$(X^{3}\to-X^{3},Z^{i}\to\overline{Z}^{i}+i\pi,\;\Omega)$, one sees that the
O5 planes are located at $U^{i}=\pm\pi/2,\,V^{i}=\pm\pi/2$. Hence, the O5s and
O7s do not intersect in this geometry.

However, it is questionable whether this property can also arise in proper CY
compactifications. Here, in analogy to the torus example, the O7 planes are
compatible with the holomorphic structure of the CY and the O5 planes with its
symplectic structure. The non-trivial fibration of the SYZ-torus over the base
$B_{3}$ may make it unavoidable for the O5s and O7s to intersect.

Finally, we want to comment on the relation to the possibly more general type
of CY-based ETW brane of Sec. 3.4. There, the CY was assumed to shrink to some
radius $\eta$ before the 4d world terminates in a suitable defect. By
contrast, in the present section we analysed an ETW brane which terminated the
4d theory without the need of a previous shrinking of the CY. Once SUSY is
broken (either in the bullk or on the brane) this solution has corrections and
it is conceivable that, when these are included, our present solution is
dynamically driven to a solution of the type discussed in Sec. 3.4. Put
differently, we may think of the presently discussed ETW brane as of a special
case where $\eta\approx R_{KK}$. Corrections may drive this value of $\eta$ to
other, e.g. $\order{1}$, values.

###  4.2 Creation/decay rates

We now want to estimate the creation/decay rates of bubbles of anything
involving the ETW brane constructed above. We assume $\eta\approx R_{KK}$, as
discussed before and rely on our calculations for Minkowski vacua from Sec. 3.
These remain approximately valid for de Sitter (dS) or AdS models as long as
$\ell_{dS/AdS}\gg M_{P}^{2}/|T_{4}|$, which defines the class of vacua we are
considering in this section. Clearly, vacua with cosmological constant tuned
close to zero most easily fulfill this condition which is why we focus on
vacua of this type.

Let us now comment on the scales setting the magnitude of $T_{4}$. The non-
vanishing tension is a combination of two effects: The O3/D3 or O7/D7 system
present in the bulk leading to $\mathcal{N}=1$ SUSY in 4d and the additional
SUSY breaking effects either in the bulk or on the boundary. Before SUSY
breaking, the tension is generically of order $|T_{4}|\sim T_{c}\simeq
M_{P}^{2}/\ell_{AdS}$. We want to express this in terms of the physical KK
radius $R_{KK}/M_{10}$, with $M_{10}$ denoting the 10d Planck mass. Hence we
introduce a parameter $\epsilon$ measuring the separation of scales between
the KK- and the AdS curvature length

$\displaystyle R_{KK}/M_{10}=\epsilon\ell_{AdS}\,.$ (63)

We are at this point agnostic concerning possible constraints on realizing
$\epsilon\ll 1$[73, 74, 75].

Eventually, the tension before SUSY breaking can be expressed as

$\displaystyle|T_{4}|\sim\epsilon\frac{M_{P}^{2}}{R_{KK}/M_{10}}\sim\epsilon\frac{M_{P}^{3}}{R_{KK}^{4}}\,.$
(64)

We expect that corrections from the uplifting mechanism will not invalidate
this estimate. Thus we will continue to work with the ETW brane tension (64),
with both signs for $T_{4}$ being allowed.

##### Bubbles of something:

As discussed before, a tension $T_{4}>0$ suitable for a bubble of something
can arise due to SUSY breaking either in the bulk or on the ETW brane. Since
we assumed $\eta\approx R_{KK}$, the volume modulus stays approximately
constant away from the ETW brane and we may use our estimate (56) for the
creation rate:

$\displaystyle
B=-\frac{8\pi^{2}M_{P}^{6}}{T_{4}^{2}}\sim-\frac{R_{KK}^{8}}{\epsilon^{2}}\,.$
(65)

If the vacuum under consideration is dS, it would be interesting to compare
the above result with those of the Hartle-Hawking no-boundary and the
Linde/Vilenkin tunneling proposals. As discussed before in Section 3.5, it is
conceptually unclear how the different creation rates should be compared to
each other, but we can nevertheless make a few observations:

The creation exponents for the no-boundary and the tunneling proposals are
given by

$\displaystyle B=\mp 8\pi^{2}M_{P}^{2}\ell_{dS}^{2}\,,$ (66)

with the upper sign corresponding to the Hartle-Hawking and the lower sign to
the Linde/Vilenkin process. Clearly, the sign of $B$ in (65) matches that of
Hartle-Hawking, which is a straightforward consequence of our euclidean
approach. It would be interesting to investigate whether a Linde/Vilenkin-type
analysis of the bubble-of-something creation process can be implemented and
shown to produce a positive exponent.

Assuming that (65) may be directly compared to the Hartle-Hawking version of
(66), one can use our initial assumption $\ell_{dS}\gg M_{P}^{2}/|T_{4}|$ to
conclude that the Hartle-Hawking process is more likely to occur than a
bubble-of-something process. It is straightforward to check this remains true
after a more general analysis, relaxing the above assumption and including the
effects of dS geometry.

##### Bubbles of nothing:

SUSY breaking in the bulk can lead to the existence of a bubble of nothing
with $T_{4}<0$. The scales setting the tension are already discussed above. To
estimate the decay rate, we approximate the instanton geometry away from the
ETW brane by flat space, in analogy to Sec. 3.5. The decay exponent follows
from a calculation similar to (54)-(56), but its sign is reversed due to the
extrinsic curvature now being negative:

$\displaystyle
B=\frac{8\pi^{2}M_{P}^{6}}{T_{4}^{2}}\sim\frac{R_{KK}^{8}}{\epsilon^{2}}\,.$
(67)

The rate following from (67) should be compared to the rates of other decay
channels, such as tunneling to decompactification, flux transitions or the
Kachru-Pearson-Verlinde (KPV) [47] process. For traditional dS constructions
in string theory [44, 45, 46], the decay exponents of tunneling to
decompactification and flux transitions are, up to a few orders of
magnitude101010For example, in KKLT models the tunneling exponent can be
suppressed by factors of the volume., close to [44, 76, 49]

$\displaystyle B\sim M_{P}^{2}\ell_{dS}^{2}\,,$ (68)

with $\ell_{dS}$ being the de Sitter length. The above becomes more exact the
closer the dS vacuum energy approaches zero. The value of $B$ in (68) is
extremely large, of the order of magnitude needed to saturate the de Sitter
lifetime. We thus expect bubble of nothing decays to occur faster. This also
follows explicitly since, by comparing to (67), we immediately see that if
$\ell_{dS}^{2}T_{4}^{2}>M_{P}^{4}$ bubble of nothing decays are faster. But
this condition was already assumed to hold at the beginning of the section,
such that it is automatically satisfied for all vacua we are considering at
the moment.

If the vacuum contains a Klebanov-Strassler throat [77] with $\overline{D3}$
branes as an uplifting mechanism, the KPV decay process [47] might occur.
Generically, this decay happens in the field-theoretic regime and the decay
exponent is given by [47, 48]

$\displaystyle
B_{KPV}=\frac{27\,(0.932)^{6}\,M^{6}\,g_{s}}{2048\pi\,(N_{\overline{D3}})^{3}}\,.$
(69)

Here, $M$ denotes the flux number on the A-cycle, $N_{\overline{D3}}$ the
number of anti-D3 branes and $g_{s}$ is the string coupling. We see that this
$B$ is not parametrically large but only enhanced by a largish integer $M$
needed to avoid a classical instability. The KPV estimate $g_{s}M^{2}\gtrsim
12$ is under debate[78, 79, 80]. This instability issue endangers the
practicality of the KPV uplift, see e.g. [81, 82, 83, 84]. In particular, it
has been argued that practical applications must consider KPV with NS5-brane
curvature corrections, which change the potential barrier significantly [85,
86]. Remembering that we estimated $|T_{4}|$ to be parametrically bounded from
above by $\epsilon R_{KK}^{6}$, one may expect our $B\sim
R_{KK}^{8}/\epsilon^{2}$ to be larger than that of KPV. However, this story is
still evolving.

Finally, there remains the possibility of the existence of exotic domain walls
with $|\theta|\gg 1$, which can lead to a significant enhancement of the decay
rate to nothing.

##  5 Bubbles of anything – further remarks

In this section, we collect further comments on bubbles of anything and
discuss how the creation/decay rates can be computed more precisely in the
future.

###  5.1 Nothing as AdS with infinitely negative vacuum energy

In [7], it was suggested that ‘nothing’ can be modelled as an AdS space in the
limit of infinitely negative vacuum energy or, equivalently, infinitely small
AdS length $\ell_{AdS}$. Indeed, a ball of AdS of radius $R\gg\ell_{AdS}$ has
most of its negative energy located near the boundary. Hence a brane
separating de Sitter space and an AdS region (with $\ell_{dS}\gg
R\gg\ell_{AdS}$) may be seen as an effective ETW brane. Its tension receives a
correction from the ‘thin’, negative-energy AdS region on one side of the
brane. As a result, tunneling between de Sitter and AdS may be a good model
for bubble of anything processes. Observing that up-tunneling from AdS to dS
is forbidden, one may then claim that bubbles of something are impossible. We
want to argue against this conclusion.

To be more precise (cf. App. D for supporting calculations111111 For
simplicity, our calculations are performed only in the $\ell_{dS}\to\infty$
limit, which is completely sufficient for the qualitative purposes of the
present section. ), consider first tunneling from dS to AdS. As explained
above, the instanton is a large sphere with a small ball of AdS glued in. When
evaluating the action, one notices that the AdS on the inner side of the
spherical, fundamental brane with tension $T$ contributes an amount
$-2/\ell_{AdS}$ to the total tension $T_{eff}=T-2/\ell_{AdS}$ of the resulting
effective brane. One then takes the limit $\ell_{AdS}\to 0$, $T\to\infty$
while keeping $T_{eff}$ fixed, ensuring that the critical bubble radius stays
finite. Given this renormalization procedure, one finds that the Brown-Dahlen
model [7] perfectly describes the corresponding bubble of nothing.

The suggestion of [7] is that the exponent for up-tunneling from AdS to dS
should be calculated using the same instanton, but this time subtracting not a
full dS sphere but rather an infinite AdS as background. This clearly leads to
a divergence and hence to a vanishing tunneling rate. However, the divergence
comes from the infinite AdS volume and hence our initial intuition that AdS is
a good model for ‘nothing’ because only a thin slice of it contributes to the
action is not borne out.

We prefer a different model in which a small ball of dS is glued into infinite
AdS, with the subtracted background being AdS as before. Now the divergence
from the AdS bulk cancels between instanton and background and only a thin
slice of the background AdS on the inner side of the domain wall of radius $R$
contributes. This time the effective ETW brane tension reads
$T_{eff}=T+2/\ell_{AdS}$, with the fundamental tension $T$ necessarily being
negative. The reader who feels that such negative-tension branes should not be
considered may at this point simply conclude that there is no good AdS-based
model for bubbles of something. However, if one is agnostic concerning the
possibility of $T<0$ and simply carries on the analysis, one finds that a
‘renormalization’ as above can be carried out: Taking $\ell_{AdS}$ to zero,
$T$ to $-\infty$ and keeping $T_{eff}$ positive and finite, one recovers all
naive bubble-of-something results.

Our conclusion is that, allowing for a fundamental negative-tension brane as
part of an appropriate limiting procedure, one may recover a finite bubble-of-
something creation rate from a Brown-Dahlen-type AdS model. On the contrary,
excluding such negative-tension branes one simply finds no satisfactory model.
Either way, we do not see how this can be turned into an argument against the
fundamental possibility of bubbles of something.

A different way of relating deep AdS and ‘nothing’ is discussed in [29].

###  5.2 Refining the decay/creation rate estimates

Our calculations from Secs. 3 and 4 provide only leading-order estimates for
the decay/creation rates. While we have to leave a more precise treatment to
future work, we want to at least comment on the necessary improvements.

####  5.2.1 Non-trivial potentials

In the present paper, we have focused on compactifications to Minkowski space
with vanishing potential for the volume modulus. Clearly, in any realistic
compactification, a stabilizing potential must exist. Nevertheless, the
general strategy for finding bubble of nothing/something solutions stays
similar to what is discussed in Secs. 2 and 3. However, it now becomes a non-
trivial task to solve the equations of motion (12), (13), subject to the
boundary conditions (40), (42). The calculation of the extrinsic curvature,
its relation to the tension $T_{4}$ (32) and its contribution to the tunneling
action are unchanged. In addition to the boundary contribution to $B$ (3),
there will now also be a bulk term. For both the instanton and the vacuum
solution, this contribution to the action reads (27)

$\displaystyle S_{bulk}=-2\pi^{2}\int f^{3}V(\phi)dr\,.$ (70)

This has to be evaluated using the the profile $\phi(r)$ obtained by solving
the equations of motion. For some analytic and numerical estimates in 5d
bubble of nothing models see [10, 11, 12].

Allowing for non-trivial potentials includes the generalization to situations
where the decaying/created vacuum is dS or AdS. The resulting modifications
are straightforward. Moreover, if $\ell_{dS/AdS}$ is large compared to the
critical bubble radius, we expect only small corrections relative to the
results derived in the previous sections. If the considered vacuum is dS, one
can not distinguish between a bubble of nothing and a bubble of something
instanton. Any solution can be interpreted in both ways, which is a result of
euclidean dS space being compact. This is analogous to Coleman-de Luccia
instantons for dS $\to$ dS tunneling where the same instanton can be viewed as
an up- or down-tunneling event and only the difference in the vacuum
contribution to the decay exponent (3) leads to a suppressed up-tunneling
rate.

In models where $\phi$ stays approximately constant away from an ETW brane
with tension $T_{4}$, cf. Sec. 3.5, we can directly calculate the dS and AdS
decay/creation exponents.

##### de Sitter

For the dS case, similar calculations in a slightly different setting have
appeared in [28]. The ETW brane sits at some radius $f_{\eta}\leq\ell_{dS}$ of
the euclidean dS sphere. The extrinsic curvature and its relation to the
tension reads

$\displaystyle\frac{3}{2}T_{4}=M_{P}^{2}\mathcal{K}_{4}=\pm\frac{3M_{P}^{2}}{f_{\eta}}\sqrt{1-\frac{f_{\eta}^{2}}{\ell_{dS}^{2}}}\,,$
(71)

where the positive sign corresponds to a dS patch of size less than a
hemisphere and the negative sign to a dS patch of size larger than a
hemisphere. The instanton action and thus the creation exponent is given by

$\displaystyle
S_{instanton}=4\pi^{2}M_{P}^{2}\ell_{dS}^{2}\left(\sqrt{\frac{T_{4}^{2}\ell_{dS}^{2}}{T_{4}^{2}\ell_{dS}^{2}+4M_{P}^{4}}}-1\right)$
(72)

if the tension is positive and

$\displaystyle
S_{instanton}=-8\pi^{2}M_{P}^{2}\ell_{dS}^{2}-4\pi^{2}M_{P}^{2}\ell_{dS}^{2}\left(\sqrt{\frac{T_{4}^{2}\ell_{dS}^{2}}{T_{4}^{2}\ell_{dS}^{2}+4M_{P}^{4}}}-1\right)$
(73)

if the tension is negative. To obtain the decay exponent, one needs to
subtract the vacuum action $S_{vaccum}=-8\pi^{2}M_{P}^{2}\ell_{dS}^{2}$ from
$S_{instanton}$.

##### Anti de Sitter

Here, similar to Minkowski space, a bubble of nothing instanton is a non-
compact patch of AdS with a hole of radius $f_{\eta}$ meanwhile a bubble of
something instanton is a compact patch of AdS of radius $f_{\eta}$. The former
requires a negative ETW brane tension and the latter a positive one. The
relation between the extrinsic curvature and the tension reads

$\displaystyle\frac{3}{2}T_{4}=M_{P}^{2}\mathcal{K}_{4}=\pm\frac{3M_{P}^{2}}{f_{\eta}}\sqrt{1+\frac{f_{\eta}^{2}}{\ell_{AdS}^{2}}}\,,$
(74)

with the upper/lower sign appropriate for a bubble of something/nothing. One
immediately sees that $|T_{4}|>2M_{P}^{2}/\ell_{AdS}$ is a necessary
requirement to have any bubble. The creation/decay exponents are then given by

$\displaystyle B_{creation}$
$\displaystyle=-4\pi^{2}M_{P}^{2}\ell_{AdS}^{2}\left(\sqrt{\frac{T_{4}^{2}\ell_{AdS}^{2}}{T_{4}^{2}\ell_{AdS}^{2}-4M_{P}^{4}}}-1\right)\,,$
(75) $\displaystyle B_{decay}$
$\displaystyle=4\pi^{2}M_{P}^{2}\ell_{AdS}^{2}\left(\sqrt{\frac{T_{4}^{2}\ell_{AdS}^{2}}{T_{4}^{2}\ell_{AdS}^{2}-4M_{P}^{4}}}-1\right)\,.$
(76)

Using the above results, one can verify that holding fix a tension $T_{4}$,
bubbles of nothing/something become more probable if the vacuum energy
increases/decreases.

####  5.2.2 Fluxes

If the CY compactification involves fluxes, one generally requires charged ETW
branes. Such ETW branes may be constructed by noting that brane flux
transitions are possible [47], realized by wrapped branes forming a domain
wall in the non-compact directions. Bubble of nothing decays can then occur as
a ‘double process’, where the removal of the flux by a brane is followed by a
decay to nothing along the lines of Secs. 3 and 4. For a decay rate estimate
in toy models, see [12]. We may think of our desired charged domain wall as
being realized if the flux-removing brane is moved on top of the ETW brane. As
already noted in Sec. 3.4, our formalism of Sec. 3 still applies with
appropriate values of $\theta$ and $\eta$, which of course first have to be
derived.

####  5.2.3 Other fields

In an exhaustive treatment of bubbles of anything, all fields, such as moduli
fields, gauge fields and the dilaton have to be taken into account. If those
fields are only weakly stabilized, they can contribute significantly to the
decay rate, as can be seen from [18]. Here, in the outermost layer, two fields
are dynamical, namely the volume modulus and the dilaton. Both fields obtain
the same profile such that the equations of motion change to a form that
resemble the ones from the Witten bubble of nothing analyzed in Sec. 3.1. In
the scenario of [18], the presence of the dilaton increases the decay rate
when comparing to the $d=7$ estimate obtained in App. B. One can also imagine
situations where additional fields lead to a decrease of the decay rate.

####  5.2.4 Higher-order corrections

The supergravity action obtained as the low-energy limit of string theory
receives corrections at higher orders in the $\alpha^{\prime}$ expansion. Such
corrections typically become important at small distance scales which in
principle can induce additional effects in the region where the internal
manifold has shrunk to small sizes. It would be interesting to investigate the
consequences of the higher-order corrections on the instanton geometry and the
decay/creation rates.

##  6 Conclusions

Our main results in the present paper concern the theoretical development and
implementation of ‘bubble of anything’ processes in the string theory
landscape, making them potentially relevant to cosmology.

First, we formulated a 4d EFT description for Witten’s bubble of nothing based
on $S^{1}$ compactifications. In doing so, we also allowed for a defect with
variable tension, sitting at the point where the $S^{1}$ disappears.
Crucially, this defect has non-zero size $\eta$ and induces a deficit angle
$\theta$. This deficit angle measures the tension of the defect and
contributes to the effective 4d tension $T_{4}$ of the boundary. In such a
more general setting, both bubble of nothing and bubble of something
instantons exist. The latter require a non-trivial potential for the scalar
governing the $S^{1}$ radius as well as a positive effective ETW brane tension
$T_{4}$.

We have demonstrated that the results above may be generalized to Calabi-Yau
orientifold compactifications, again with the assumption that the compact
manifold shrinks to zero size at the boundary of spacetime. Since the
microscopic structure of Calabi-Yau (CY) bordisms to nothing is generally
unknown, the introduction of an appropriate defect of size $\eta$ and with a
(suitably generalized) deficit angle $\theta$ is in this case mandatory. While
it may be natural to assume the defect size to be of higher-dimensional Planck
or string scale, it can in principle also be much larger. We provide general
formulae for the actions of Calabi-Yau bubbles of anything of this type, which
turn out to be parametrically larger than for the Witten bubble. These actions
allow for estimates of the decay or creation rates of corresponding 4d vacua.

While we feel that it is well-motivated by the Cobordism Conjecture, the ad-
hoc introduction of a defect realizing the bordism of a Calabi-Yau orientifold
to nothing is certainly unsatisfactory. We therefore continued our analysis by
also providing an explicit ETW domain wall for type IIB Calabi-Yau orientifold
models with O3/O7 planes. Our construction employs an O5 orientifolding of the
type IIB compactification, involving a reflection in one of the non-compact
spatial dimensions. The O5/D5 system wraps the bubble surface and a three
cycle of the CY. The resulting ETW brane preserves two supercharges, i.e.
$\mathcal{N}=1$ supersymmetry in 3d. Naively, we then expect such an ETW brane
to be either stationary or contracting, allowing neither for a bubble of
nothing nor of something. However, once we include moduli stabilization and
SUSY breaking in the bulk, as well as (possibly) additional SUSY breaking on
the boundary, this will certainly change. One may generically expect a
combination of the KK- and SUSY breaking scale to set the brane tension. In
case a bubble of nothing is allowed, one can then compare our decay rates
(with decay exponent (67)) with the rates of traditional vacuum decay
processes, e.g. the decay to decompactification, flux transitions or the
Kachru-Pearson-Verlinde (KPV) decay process. It turns out that only the field-
theoretic decay of KPV is likely to be faster.

For bubbles of something, the nucleation exponent is given in (72), (73) and a
naive estimate for our setting in (65). Bubbles of something may be viewed as
an alternative to the Hartle-Hawking no-boundary creation process. The form of
the nucleation exponents differs significantly between these two vacuum
creation processes: In one case, the key input are the ETW brane tension and
4d vacuum energy, in the other case it is only the 4d vacuum energy. One may
ask whether the relative importance of the two creation processes for the
landscape can be quantified by directly comparing the respective nucleation
exponents.

It would be desirable to analyse the effects of the dilaton, fluxes, vacuum
energy and non-perturbative effects in more detail. We also leave a careful
study of SUSY-breaking and supersymmetric corrections to the domain wall
tension to future work. Other open questions are whether bubbles of something
can be interpreted using the Linde-Vilenkin approach to vacuum creation[22,
62] and whether they can be studied in a Lorentzian approach[87].

### Acknowledgements

We are very grateful for useful discussions with Giuseppe Dibitetto, Luca
Martucci, Jacob McNamara, Miguel Montero, Simon Schreyer, Irene Valenzuela,
Gerben Venken and Timo Weigand. This work was supported by Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES
Excellence Cluster).

##  Appendix A Dimensional reduction

In the following, we present the dimensional reduction from $d$-dimensions to
four dimensions starting from the metric ansatz (10)

$\displaystyle\begin{split}ds^{2}&=e^{2\alpha\phi(r)}\left(dr^{2}+f(r)^{2}d\Omega_{3}^{2}\right)+e^{2\beta\phi(r)}ds_{n}^{2}\\\
&=e^{2\alpha\phi}\left(ds_{4}^{2}+e^{2(\beta-\alpha)\phi}ds_{n}^{2}\right)=e^{2\alpha\phi}d\tilde{s}^{2}\,.\end{split}$
(77)

We set $M_{d}=1$ and normalize $ds_{n}^{2}$ (and hence the compact-space
metric) by

$\displaystyle\int\sqrt{g_{n}}d^{n}x=1\,,$ (78)

such that the 4d Planck mass is unity as well.

##### Einstein-Hilbert action:

When dimensionally reducing the $d$-dimensional Einstein-Hilbert action, it is
convenient to employ the metric $d\tilde{s}^{2}$ introduced in (77). Using the
transformation properties of the Ricci scalar under Weyl rescalings, the
action takes the form

$\displaystyle S$
$\displaystyle=-\frac{1}{2}\int\sqrt{g_{n}}\sqrt{g_{4}}e^{d\alpha\phi}e^{n(\beta-\alpha)\phi}e^{-2\alpha\phi}\left(\tilde{\mathcal{R}}-2(d-1)\alpha\tilde{\square}\phi-(d-2)(d-1)\alpha^{2}(\partial\phi)^{2}\right)$
(79)
$\displaystyle=-\frac{1}{2}\int\sqrt{g_{4}}e^{[2\alpha+n\beta]\phi}\left(\tilde{\mathcal{R}}-2(d-1)\alpha\square_{4}\phi-\left[(d-2)(d-1)\alpha^{2}+2(d-1)n(\beta-\alpha)\alpha\right](\partial\phi)^{2}\right)\,,$
(80)

where $\tilde{\square}_{4}$ is the Laplace operator associated with
$d\tilde{s}_{4}^{2}$ and $\square_{4}$ is the Laplace operator associated with
$ds_{4}^{2}$. Further, we have used

$\displaystyle\tilde{\square}\phi=\frac{1}{f^{3}e^{n(\beta-\alpha)\phi}}\partial_{r}(f^{3}e^{n(\beta-\alpha)\phi}\phi^{\prime})=\frac{1}{f^{3}}\partial_{r}(f^{3}\phi^{\prime})+n(\beta-\alpha)(\partial\phi)^{2}=\square_{4}\phi+n(\beta-\alpha)(\partial\phi)^{2}\,.$
(81)

We proceed by calculating $\tilde{\mathcal{R}}$, the Ricci scalar of the
metric $d\tilde{s}^{2}$. To do this, we first write the metric in the form

$\displaystyle
d\tilde{s}^{2}=e^{2(\beta-\alpha)\phi}\left(dz^{2}+g(r)^{2}d\Omega_{3}^{2}+ds_{n}^{2}\right)\,,$
(82)

where we defined $e^{(\beta-\alpha)\phi}dz=dr$ and
$g(r)e^{(\beta-\alpha)\phi(r)}=f(r)$. Using twice the transformation
properties of the Ricci scalar under conformal transformations one arrives at
the result

$\displaystyle\tilde{\mathcal{R}}=\mathcal{R}_{4}-2n(\beta-\alpha)\square_{4}\phi-n(n+1)(\beta-\alpha)^{2}(\partial\phi)^{2}+\mathcal{R}_{n}e^{-2(\beta-\alpha)\phi}\,,$
(83)

with $\mathcal{R}_{4}$ denoting the Ricci scalar of the metric $ds_{4}^{2}$
and $\mathcal{R}_{n}$ the Ricci scalar of the metric $ds_{n}^{2}$. For
completeness, we note that

$\displaystyle\mathcal{R}_{4}=6\left(\frac{1}{f^{2}}-\frac{f^{\prime
2}}{f^{2}}-\frac{f^{\prime\prime}}{f}\right)\,.$ (84)

From (80) one observes that setting

$\displaystyle\beta=-2\alpha/n=-2\alpha/(d-4)$ (85)

leads to a 4d Einstein frame action (recall that $d=n+4$)

$\displaystyle
S=\int\sqrt{g_{4}}\left(-\frac{1}{2}\mathcal{R}_{4}+\alpha\square_{4}\phi+\frac{2(n+2)}{n}\alpha^{2}\frac{(\partial\phi)^{2}}{2}-\frac{\mathcal{R}_{n}}{2}e^{\left(\frac{4}{n}+2\right)\alpha\phi}\right)\,.$
(86)

Finally, one can choose $\alpha$ such that the scalar field is canonically
normalized:

$\displaystyle\alpha^{2}=\frac{n}{2(n+2)}\,.$ (87)

##### Gibbons-Hawking-York boundary term:

We continue with the reduction of the Gibbons-Hawking-York boundary term from
$d$ to four dimensions, assuming that the boundary is a surface defined by
$r=const.$ We can directly calculate

$\displaystyle
S_{GHY}=-\int_{\Sigma}\sqrt{h}\mathcal{K}_{d}=-\int_{\Sigma}\sqrt{g_{n}}e^{3\alpha\phi}e^{n\beta\phi}f^{3}\mathcal{K}_{d}\,,$
(88)

with $h$ the induced metric on the $(d-1)$-dimensional boundary. To calculate
the $d$-dimensional extrinsic curvature $\mathcal{K}_{d}$, we note that the
outward pointing normal vector is $n=e^{-\alpha\phi}\partial_{r}$. We further
use $3\alpha+n\beta=\alpha$ from (85), resulting in

$\displaystyle\mathcal{K}_{d}=\frac{1}{e^{(4\alpha+n\beta)\phi}f^{3}}\partial_{r}\left(e^{(3\alpha+n\beta)\phi}f^{3}\right)=e^{-\alpha\phi}\left(\alpha\phi^{\prime}+3\frac{f^{\prime}}{f}\right)\,,$
(89)

and hence

$\displaystyle-\int_{\Sigma}\sqrt{h}\mathcal{K}_{d}=-\int
f^{3}\alpha\phi^{\prime}-\int f^{3}\mathcal{K}_{4}\,.$ (90)

One now observes that the term $\alpha\square_{4}\phi$ in (86) is a total
derivative and contributes to the action as

$\displaystyle\left.\alpha f^{3}\phi^{\prime}\right|_{r=0}^{r=\infty}\,.$ (91)

Hence, for Minkowski compactifications, the first term on the r.h.s. of (90)
precisely cancels the contribution from the $r=\infty$ term of (91).

Since, from the 4d perspective, the bubble wall (for bubbles of anything) is
another boundary of spacetime, it appears logical to also include a
corresponding GHY term in the four-dimensional action. In addition, the bubble
wall has an effective 4d tension, such that we supplement our total action
with a term

$\displaystyle-\int_{\partial\Sigma}\sqrt{g_{4}}\,({\cal
K}_{4}-T_{4})=-2\pi^{3}f^{3}\left(\mathcal{K}_{4}-T_{4}\right)$ (92)

on the boundary corresponding to the bubble wall. At the same time, we remove
the term $\alpha\square_{4}\phi$ from the bulk action since we take its
remaining effect at the bubble wall from (91) into account in the new boundary
term (92).

##  Appendix B Bubble of nothing decays in general dimensions

In this section, we collect the relevant equations to analyze bubble of
nothing decays in general dimensions.

For general $d$, Equations (14) and (16) are still valid, just the values of
$\alpha$ and $\beta$ change according to (7). Demanding that the geometry ends
when the extra dimensions reach critical size $\eta$ results in the defining
equation

$\displaystyle 2\pi\eta=2\pi R(f_{\eta})=e^{\beta\phi(f_{\eta})}\,$ (93)

for $f_{\eta}$. Equation (93) simplifies to

$\displaystyle\frac{2C}{\sqrt{6}f_{\eta}^{2}}=\left(\frac{R_{KK}}{\eta}\right)^{\frac{2}{\sqrt{6}\beta}}-\left(\frac{R_{KK}}{\eta}\right)^{-\frac{2}{\sqrt{6}\beta}}\,.$
(94)

Unless $\eta$ is extremely close to $R_{KK}$, we can neglect either the first
or the second term in the equation above. In the main text, $\eta/R_{KK}<1$
was assumed to improve readability of formulae. In the following, we will
present the general equations.

Introducing the (analogue of) the deficit angle $\theta$ as in (42)

$\displaystyle
1-\frac{\theta}{2\pi}=\left.\frac{dR}{dr}e^{-\alpha\phi}\right|_{r=0}=\frac{\beta}{2\pi}\frac{C}{f_{\eta}^{3}}(2\pi\eta)^{1+\frac{n}{2}}$
(95)

and solving for $C$ by using (94) results in

$\displaystyle
C=\frac{\beta^{2}}{(2\pi)^{2}}\frac{3\sqrt{6}}{4\left(1-\frac{\theta}{2\pi}\right)^{2}}(2\pi\eta)^{n+2}\left[\left(\frac{R_{KK}}{\eta}\right)^{\frac{2}{\sqrt{6}\beta}}-\left(\frac{R_{KK}}{\eta}\right)^{-\frac{2}{\sqrt{6}\beta}}\right]^{3}\,.$
(96)

The extrinsic curvature in four dimensions can now be calculated to be

$\displaystyle\mathcal{K}_{4}=-\sqrt{6}\frac{2\pi\left|1-\frac{\theta}{2\pi}\right|}{\beta}(2\pi\eta)^{-1-\frac{n}{2}}\sqrt{\left[\left(\frac{R_{KK}}{\eta}\right)^{\frac{2}{\sqrt{6}\beta}}-\left(\frac{R_{KK}}{\eta}\right)^{-\frac{2}{\sqrt{6}\beta}}\right]^{-2}+\frac{1}{4}}\,.$
(97)

Interestingly, $\mathcal{K}_{4}$ does not depend on $R_{KK}$, unless
$\eta\approx R_{KK}$. This provides evidence to the claim that bubbles of
anything are local quantum effects.

The critical bubble radius $\rho_{0}$ is computed to be

$\displaystyle\rho_{0}=\sqrt{\frac{3}{2}}\frac{\beta}{\left|1-\frac{\theta}{2\pi}\right|}\eta\left|\left(\frac{R_{KK}}{\eta}\right)^{\frac{2}{\sqrt{6}\beta}}-\left(\frac{R_{KK}}{\eta}\right)^{-\frac{2}{\sqrt{6}\beta}}\right|\,.$
(98)

Finally, the decay exponent is given by

$\displaystyle\begin{split}B&=-2\pi^{2}f_{\eta}^{3}\frac{\mathcal{K}_{4}}{3}\\\
&=\frac{3}{4}\frac{\beta^{2}}{\left(1-\frac{\theta}{2\pi}\right)^{2}}(2\pi\eta)^{n+2}\sqrt{\left[\left(\frac{R_{KK}}{\eta}\right)^{\frac{2}{\sqrt{6}\beta}}-\left(\frac{R_{KK}}{\eta}\right)^{-\frac{2}{\sqrt{6}\beta}}\right]^{4}+\frac{1}{4}\left[\left(\frac{R_{KK}}{\eta}\right)^{\frac{2}{\sqrt{6}\beta}}-\left(\frac{R_{KK}}{\eta}\right)^{-\frac{2}{\sqrt{6}\beta}}\right]^{6}}\,.\end{split}$
(99)

Inspecting the profile of the radius $\rho(f)$ of the transversal $S^{3}$

$\displaystyle\rho(f)=R_{KK}^{-\frac{n}{2}}\left(\frac{C}{\sqrt{6}f^{2}}+\sqrt{\left(\frac{C}{\sqrt{6}f^{2}}\right)^{2}+1}\right)^{-\alpha\frac{\sqrt{6}}{2}}f$
(100)

shows that for large $f$

$\displaystyle\rho(f)\approx R_{KK}^{-\frac{n}{2}}f\,,$ (101)

and hence $\rho$ grows with growing $f$. On the other hand, if
$f_{\eta}^{2}\ll C$, there will be the regime where

$\displaystyle\rho(f)\approx\begin{cases}R_{KK}^{-\frac{n}{2}}\left(\frac{2C}{\sqrt{6}}\right)^{-\alpha\sqrt{6}/2}f^{1+\alpha\sqrt{6}}&C>0\,,\\\
R_{KK}^{-\frac{n}{2}}\left(\frac{\sqrt{6}}{2C}\right)^{-\alpha\sqrt{6}/2}f^{1-\alpha\sqrt{6}}&C<0\,.\end{cases}$
(102)

It now depends on the value of $\alpha$ whether $\rho$ grows or shrinks with
growing $f$. We first note that $\alpha<0$ for all values of $d$, such that
$\rho$ grows with growing $f$ if $C<0$. Let us now analyze the case $C>0$: If,
$d=5$, $\alpha=-\frac{1}{\sqrt{6}}$ and hence $\rho$ stays approximately
constant. Since $\alpha<-1/\sqrt{6}$ for $d>5$, one observes that $\rho$
shrinks with growing $f$ near the bubble wall if $C>0$. Such a peculiar
feature has already been observed in [20].

##  Appendix C Supersymmetry and the ETW brane

In this Appendix, we show that a spacetime filling O3/D3 system together with
a O5/D5 domain wall wrapped on a special Lagrangian three-cycle of a Calabi-
Yau manifold preserves two supercharges. We mostly follow the ideas of [88,
89]. We further argue that this situation also holds for O5/D5, O7/D7 models.
Since Op planes preserve the same supercharges as Dp branes, we can
equivalently think of D-brane systems only. The two SUSY generators of type II
theories will be denoted by $\epsilon,\tilde{\epsilon}$. Then, the condition
that a brane preserves supersymmetry is given by

$\displaystyle\tilde{\epsilon}=\Gamma_{\parallel}^{Dp}\epsilon\,,\qquad\Gamma_{\parallel}^{Dp}=\frac{1}{\sqrt{g|_{Dp}}}\prod_{i}\Gamma_{i}\,,$
(103)

where the product is over all gamma matrices in directions parallel to the Dp
brane. After compactifying on a Calabi-Yau, the SUSY generators for type IIB
string theory can be parametrized as

$\displaystyle\epsilon$
$\displaystyle=\zeta_{+}\otimes\eta_{+}+\zeta_{-}\otimes\eta_{-}\,,$ (104)
$\displaystyle\tilde{\epsilon}$
$\displaystyle=\tilde{\zeta}_{+}\otimes\eta_{+}+\tilde{\zeta}_{-}\otimes\eta_{-}\,.$
(105)

Here, $\zeta_{+},\tilde{\zeta}_{+}$ are independent four-dimensional spinors
of positive chiraliy and $\eta_{+}$ is the unique six-dimensional covariantly
constant positive-chirality spinor of the Calabi-Yau. Further,
$\zeta_{-}=(\zeta_{+})^{c},\tilde{\zeta}_{-}=(\tilde{\zeta}_{+})^{c}$,
$\eta_{-}=(\eta_{+})^{c}$. The eight real degrees of freedom of
$\zeta_{+},\tilde{\zeta}_{+}$ lead to the familiar $\mathcal{N}=2$ theory in
4d. Using similar arguments as in [89] chapter 9.2 and adopting the same
conventions, one finds that the condition

$\displaystyle\tilde{\epsilon}=\Gamma_{\parallel}^{D3}\epsilon\,,$ (106)

where the D3 brane is spacetime filling, results in the relation

$\displaystyle\tilde{\zeta}_{+}=i\zeta_{+}\,.$ (107)

As a result, the preserved supercharges are reduced from eight to four. The
second condition

$\displaystyle\tilde{\epsilon}=\Gamma_{\parallel}^{D5}\epsilon\,,$ (108)

with the D5 brane being extended in the $0,1,2$ directions of the non-compact
space and wrapping a three-cycle $S_{3}$ of the CY, implies

$\displaystyle\zeta_{-}=\beta\gamma_{3}\zeta_{+}\,,\quad\eta_{-}=\beta^{-1}\gamma_{\parallel}\eta_{+}\,,$
(109)

with $\beta$ a constant to be determined. Here, following [89] chapter 9.2 and
5.6, $\gamma_{3}$ is to be seen as a 4d gamma matrix and $\gamma_{\parallel}$
is the product of gamma matrices along the directions of the three-cycle
$S_{3}$. Since $\gamma_{\parallel}$ is unitary, the second condition in (109)
determines $|\beta|=1$. Hence, $\beta=e^{i\phi}$. It is now straightforward to
see that the first condition of (109) always has a solution. For example, in
the chiral basis, we have the following representation of $\gamma_{3}$ and a
general spinor $\zeta_{+},\zeta_{-}$

$\displaystyle\gamma_{3}=i\begin{pmatrix}0&-\sigma^{3}\\\
\sigma^{3}&0\end{pmatrix}\,,\qquad\zeta_{+}=\begin{pmatrix}a\\\ b\\\ 0\\\
0\end{pmatrix}\,,\qquad\zeta_{-}\begin{pmatrix}0\\\ 0\\\ -\overline{b}\\\
\overline{a}\end{pmatrix}\,,\qquad a,b\in\mathbb{C}\,.$ (110)

The first condition of (109) then becomes

$\displaystyle\begin{pmatrix}0\\\ 0\\\ -\overline{b}\\\
\overline{a}\end{pmatrix}=e^{i\phi}\begin{pmatrix}0\\\ 0\\\ ia\\\
-ib\end{pmatrix}\,.$ (111)

We see that this system can be solved for any $\phi$ by choosing
$a=ie^{-i\phi}\overline{b}$. This condition further reduces the number of
preserved supercharges from four to two, resulting in $\mathcal{N}=1$
supersymmetry in 3d. Finally, we need to check whether the second condition of
(109) is fulfilled. Following the logic of [89] chapter 5.6, 9.2 and adopting
it to our case of interest, one finds that
$\eta_{-}=e^{-i\phi}\gamma_{\parallel}\eta_{+}$ is obeyed when the cycle
$S_{3}$ is calibrated with respect to the form $Re(e^{i\theta}\Omega)$

$\displaystyle Re(e^{i\phi}\Omega)|_{S_{3}}=\text{vol}_{S_{3}}\,.$ (112)

The condition (112) precisely states that the three cycle $S_{3}$ needs to be
special Lagragian. We conclude that our D3/D5 system preserves $\mathcal{N}=1$
SUSY in 3d if the D5 brane is wrapped on a special Lagrangian cycle. Since D7
branes wrapped on holomorphic four-cycles of the CY preserve the same
supercharges as D3 branes, our results straightforwardly generalize to such
D7/D5 models. This can further be seen by again using techniques from [89]
chapter 9 and by noting that the condition

$\displaystyle\tilde{\epsilon}=\Gamma_{\parallel}^{D7}\epsilon\,,$ (113)

with the D7 wrapped on a holomorphic cycle, also leads to (107). Then, from
the arguments below (107), it is clear that two supercharges are preserved if
the $D5$ brane is wrapped on a special Lagrangian cycle of the CY.

##  Appendix D AdS bubble transitions in the limit of small AdS length

In this Appendix we discuss whether and how tunneling to/from a deep AdS may
be related to tunneling to/from ‘nothing’. This analogy has been suggested and
analysed in[7], but we disagree with a key detail which is decisive in our
context.

We start by considering thin-wall tunneling instantons for the decay of
Minkowski space to AdS as a possible model for bubbles of nothing. We denote
the bubble radius by $R$, the AdS length by $\ell_{AdS}$, the matter tension
by $T$ and we work in 4d Planck units. The tunneling exponent prior to radius
extremization is given by (see e.g. [90])

$\displaystyle B_{\text{Mink. $\to$
AdS}}=6\pi^{2}R^{2}+2\pi^{2}TR^{3}-4\pi^{2}\ell_{AdS}^{2}\left(\left(1+\frac{R^{2}}{\ell_{AdS}^{2}}\right)^{3/2}-1\right)\,.$
(114)

The instanton corresponds to a patch of AdS of radius $R$ glued into Minkowski
space. The first term on the r.h.s. of (114) represents the extrinsic
curvature contribution at the bubble wall on the Minkowski side, the second
term is the effect of the wall tension, and the third term combines both the
AdS bulk and the AdS-side extrinsic curvature contributions. The extrinsic
curvature term at infinity is cancelled by the Minkowski vacuum contribution.
In the limit $R\gg\ell_{AdS}$ the last term in (114) reduces to
$2\pi^{2}(-2/\ell_{AdS})R^{3}$. As a result, the total effect of the AdS side
is to correct the brane tension by $-2/\ell_{AdS}$. Thus, at $R\gg\ell_{AdS}$
one has

$\displaystyle B_{\text{Mink. $\to$
AdS}}=6\pi^{2}R^{2}+2\pi^{2}T_{eff}R^{3}\,,$ (115)

with $T_{eff}=T-2/\ell_{AdS}$. One immediately sees that this is the exponent
for a bubble-of-nothing tunneling process in flat space.

The action (114) has an extremum only for $T_{eff}<0$, in which case the
critical bubble radius reads

$\displaystyle R_{\text{Mink. $\to$
AdS}}=\frac{4T}{\frac{4}{\ell_{AdS}^{2}}-T^{2}}=-\frac{4\left(\frac{T_{eff}\ell_{AdS}}{2}+1\right)}{\frac{\ell_{AdS}T_{eff}^{2}}{2}+2T_{eff}}\,.$
(116)

In the limit $\ell_{AdS}\to 0$, one then obtains

$\displaystyle R_{\text{Mink. $\to$ AdS}}=-\frac{2}{T_{eff}}\,.$ (117)

This could have equivalently been found by extremizing (115). We see that
tunneling becomes possible if the effective tension is negative. This
effective tension may be thought of as a physical quantity by taking the
renormalization-type limit $\ell_{AdS}\to 0$, $T\to\infty$ while keeping
$T_{eff}$ finite.

For the bubble of something, the corresponding AdS $\to$ Minkowski tunneling
exponent takes the form

$\displaystyle B_{\text{AdS $\to$
Mink.}}=-6\pi^{2}R^{2}+2\pi^{2}TR^{3}+4\pi^{2}\ell_{AdS}^{2}\left(\left(1+\frac{R^{2}}{\ell_{AdS}^{2}}\right)^{3/2}-1\right)\,.$
(118)

The instanton geometry is a patch of Minkowski space of radius $R$ glued into
AdS, which in contrast to [7] we take to be infinite. Extremizing the
expression in (118) gives

$\displaystyle R_{\text{AdS $\to$
Mink.}}=\frac{4T}{T^{2}-\frac{4}{\ell_{AdS}^{2}}}$ (119)

for $T<0$ and no extremum otherwise. Moreover, demanding a positive bubble
radius one finds the additional requirement $T_{eff}\equiv T+2/\ell_{AdS}>0$.
In the limit $\ell_{AdS}\to 0$, the tunneling exponent (118) becomes

$\displaystyle B_{\text{AdS $\to$
Mink.}}=-6\pi^{2}R^{2}+2\pi^{2}T_{eff}R^{3}\,.$ (120)

It is extremized by

$\displaystyle R_{\text{AdS $\to$ Mink.}}=\frac{2}{T_{eff}}\,,$ (121)

which is also the $\ell_{AdS}\to 0$ limit of (119) if $T_{eff}$ is kept
finite. We see that the limit $\ell_{AdS}\to 0$ can only be sensibly taken
when the ‘bare tension’ $T$ is negative and diverges. While negative-tension
ETW-branes are known to exist, it is not clear to us whether the negative
tension brane between AdS and Minkowski required above can be realized in
string theory or even makes sense more generally, in an EFT.

Our key conclusion for the present paper is that bubbles of something are not
ruled out since, if negative tension branes are forbidden, then up-tunneling
from AdS is not a good model for bubbles of something. If, on the contrary,
negative-tension branes are allowed, then up-tunneling from AdS reproduces the
simpler bubble-of-something results.

## References

  * [1] S. R. Coleman and F. De Luccia, “Gravitational Effects on and of Vacuum Decay,” Phys. Rev. D 21 (1980) 3305.
  * [2] S. W. Hawking and I. G. Moss, “Supercooled Phase Transitions in the Very Early Universe,” Phys. Lett. B 110 (1982) 35–38.
  * [3] E. Witten, “Instability of the Kaluza-Klein Vacuum,” Nucl. Phys. B 195 (1982) 481–492.
  * [4] J. J. Blanco-Pillado and B. Shlaer, “Bubbles of Nothing in Flux Compactifications,” Phys. Rev. D 82 (2010) 086015, arXiv:1002.4408 [hep-th].
  * [5] J. J. Blanco-Pillado, B. Shlaer, K. Sousa, and J. Urrestilla, “Bubbles of Nothing and Supersymmetric Compactifications,” JCAP 10 (2016) 002, arXiv:1606.03095 [hep-th].
  * [6] A. R. Brown and A. Dahlen, “Bubbles of Nothing and the Fastest Decay in the Landscape,” Phys. Rev. D 84 (2011) 043518, arXiv:1010.5240 [hep-th].
  * [7] A. R. Brown and A. Dahlen, “On ’nothing’ as an infinitely negatively curved spacetime,” Phys. Rev. D 85 (2012) 104026, arXiv:1111.0301 [hep-th].
  * [8] J. J. Blanco-Pillado, H. S. Ramadhan, and B. Shlaer, “Decay of flux vacua to nothing,” JCAP 10 (2010) 029, arXiv:1009.0753 [hep-th].
  * [9] G. Dibitetto, N. Petri, and M. Schillo, “Nothing really matters,” JHEP 08 (2020) 040, arXiv:2002.01764 [hep-th].
  * [10] P. Draper, I. Garcia Garcia, and B. Lillard, “De Sitter decays to infinity,” JHEP 12 (2021) 154, arXiv:2105.10507 [hep-th].
  * [11] P. Draper, I. G. Garcia, and B. Lillard, “Bubble of nothing decays of unstable theories,” Phys. Rev. D 104 no. 12, (2021) L121701, arXiv:2105.08068 [hep-th].
  * [12] P. Draper, B. Lillard, and C. Skye, “Neutralizing Topological Obstructions to Bubbles of Nothing,” arXiv:2305.17838 [hep-th].
  * [13] D. Brill and G. T. Horowitz, “Negative energy in string theory,” Phys. Lett. B 262 (1991) 437–443.
  * [14] M. Fabinger and P. Horava, “Casimir effect between world branes in heterotic M theory,” Nucl. Phys. B 580 (2000) 243–263, arXiv:hep-th/0002073.
  * [15] G. T. Horowitz, J. Orgera, and J. Polchinski, “Nonperturbative Instability of AdS(5) x S**5/Z(k),” Phys. Rev. D 77 (2008) 024004, arXiv:0709.4262 [hep-th].
  * [16] H. Ooguri and L. Spodyneiko, “New Kaluza-Klein instantons and the decay of AdS vacua,” Phys. Rev. D 96 no. 2, (2017) 026016, arXiv:1703.03105 [hep-th].
  * [17] B. S. Acharya, “Supersymmetry, Ricci Flat Manifolds and the String Landscape,” JHEP 08 (2020) 128, arXiv:1906.06886 [hep-th].
  * [18] I. n. García Etxebarria, M. Montero, K. Sousa, and I. Valenzuela, “Nothing is certain in string compactifications,” JHEP 12 (2020) 032, arXiv:2005.06494 [hep-th].
  * [19] P. Bomans, D. Cassani, G. Dibitetto, and N. Petri, “Bubble instability of mIIA on $\mathrm{AdS}_{4}\times S^{6}$,” SciPost Phys. 12 no. 3, (2022) 099, arXiv:2110.08276 [hep-th].
  * [20] J. J. Blanco-Pillado, H. S. Ramadhan, and B. Shlaer, “Bubbles from Nothing,” JCAP 01 (2012) 045, arXiv:1104.5229 [gr-qc].
  * [21] J. B. Hartle and S. W. Hawking, “Wave Function of the Universe,” Phys. Rev. D 28 (1983) 2960–2975.
  * [22] A. D. Linde, “Quantum Creation of the Inflationary Universe,” Lett. Nuovo Cim. 39 (1984) 401–405.
  * [23] A. Vilenkin, “Cosmic Strings and Domain Walls,” Phys. Rept. 121 (1985) 263–315.
  * [24] S. W. Hawking and N. Turok, “Open inflation without false vacua,” Phys. Lett. B 425 (1998) 25–32, arXiv:hep-th/9802030.
  * [25] N. Turok and S. W. Hawking, “Open inflation, the four form and the cosmological constant,” Phys. Lett. B 432 (1998) 271–278, arXiv:hep-th/9803156.
  * [26] J. Garriga, “Smooth ’creation’ of an open universe in five-dimensions,” arXiv:hep-th/9804106.
  * [27] J. Garriga, “Open inflation and the singular boundary,” Phys. Rev. D 61 (2000) 047301, arXiv:hep-th/9803210.
  * [28] R. Bousso and A. Chamblin, “Open inflation from nonsingular instantons: Wrapping the universe with a membrane,” Phys. Rev. D 59 (1999) 063504, arXiv:hep-th/9805167.
  * [29] S. Cespedes, S. de Alwis, F. Muia, and F. Quevedo, “Quantum Transitions, Detailed Balance, Black Holes and Nothingness,” arXiv:2307.13614 [hep-th].
  * [30] J. McNamara and C. Vafa, “Cobordism Classes and the Swampland,” arXiv:1909.10355 [hep-th].
  * [31] M. Montero and C. Vafa, “Cobordism Conjecture, Anomalies, and the String Lamppost Principle,” JHEP 01 (2021) 063, arXiv:2008.11729 [hep-th].
  * [32] M. Dierigl, J. J. Heckman, M. Montero, and E. Torres, “IIB string theory explored: Reflection 7-branes,” Phys. Rev. D 107 no. 8, (2023) 086015, arXiv:2212.05077 [hep-th].
  * [33] A. Debray, M. Dierigl, J. J. Heckman, and M. Montero, “The Chronicles of IIBordia: Dualities, Bordisms, and the Swampland,” arXiv:2302.00007 [hep-th].
  * [34] G. Buratti, J. Calderón-Infante, M. Delgado, and A. M. Uranga, “Dynamical Cobordism and Swampland Distance Conjectures,” JHEP 10 (2021) 037, arXiv:2107.09098 [hep-th].
  * [35] 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 [hep-th].
  * [36] 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 [hep-th].
  * [37] 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 [hep-th].
  * [38] D. Andriot, N. Carqueville, and N. Cribiori, “Looking for structure in the cobordism conjecture,” SciPost Phys. 13 no. 3, (2022) 071, arXiv:2204.00021 [hep-th].
  * [39] R. Blumenhagen, N. Cribiori, C. Kneissl, and A. Makridou, “Dimensional Reduction of Cobordism and K-theory,” JHEP 03 (2023) 181, arXiv:2208.01656 [hep-th].
  * [40] R. Blumenhagen, N. Cribiori, C. Kneissl, and A. Makridou, “Dynamical cobordism of a domain wall and its companion defect 7-brane,” JHEP 08 (2022) 204, arXiv:2205.09782 [hep-th].
  * [41] R. Blumenhagen, C. Kneissl, and C. Wang, “Dynamical Cobordism Conjecture: Solutions for End-of-the-World Branes,” arXiv:2303.03423 [hep-th].
  * [42] M. Dine, P. J. Fox, and E. Gorbatov, “Catastrophic decays of compactified space-times,” JHEP 09 (2004) 037, arXiv:hep-th/0405190.
  * [43] F. Denef and M. R. Douglas, “Distributions of flux vacua,” JHEP 05 (2004) 072, arXiv:hep-th/0404116.
  * [44] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, “De Sitter vacua in string theory,” Phys. Rev. D 68 (2003) 046005, arXiv:hep-th/0301240.
  * [45] V. Balasubramanian, P. Berglund, J. P. Conlon, and F. Quevedo, “Systematics of moduli stabilisation in Calabi-Yau flux compactifications,” JHEP 03 (2005) 007, arXiv:hep-th/0502058.
  * [46] J. P. Conlon, F. Quevedo, and K. Suruliz, “Large-volume flux compactifications: Moduli spectrum and D3/D7 soft supersymmetry breaking,” JHEP 08 (2005) 007, arXiv:hep-th/0505076.
  * [47] S. Kachru, J. Pearson, and H. L. Verlinde, “Brane / flux annihilation and the string dual of a nonsupersymmetric field theory,” JHEP 06 (2002) 021, arXiv:hep-th/0112197.
  * [48] B. Freivogel and M. Lippert, “Evidence for a bound on the lifetime of de Sitter space,” JHEP 12 (2008) 096, arXiv:0807.1104 [hep-th].
  * [49] S. de Alwis, R. Gupta, E. Hatefi, and F. Quevedo, “Stability, Tunneling and Flux Changing de Sitter Transitions in the Large Volume String Scenario,” JHEP 11 (2013) 179, arXiv:1308.1222 [hep-th].
  * [50] S. R. Coleman, V. Glaser, and A. Martin, “Action Minima Among Solutions to a Class of Euclidean Scalar Field Equations,” Commun. Math. Phys. 58 (1978) 211–221.
  * [51] D. Garfinkle, “General Relativistic Strings,” Phys. Rev. D 32 (1985) 1323–1329.
  * [52] T. Takayanagi, “Holographic Dual of BCFT,” Phys. Rev. Lett. 107 (2011) 101602, arXiv:1105.5165 [hep-th].
  * [53] M. Fujita, T. Takayanagi, and E. Tonni, “Aspects of AdS/BCFT,” JHEP 11 (2011) 043, arXiv:1108.5152 [hep-th].
  * [54] E. Witten, “A Simple Proof of the Positive Energy Theorem,” Commun. Math. Phys. 80 (1981) 381.
  * [55] X. Dai, “A positive mass theorem for spaces with asymptotic SUSY compactification,” Communications in Mathematical Physics 244 no. 2, (Jan, 2004) 335–345. https://doi.org/10.1007%2Fs00220-003-0986-2.
  * [56] X. Dai, “A note on positive energy theorem for spaces with asymptotic SUSY compactification,” Journal of Mathematical Physics 46 no. 4, (Mar, 2005) 042505. https://doi.org/10.1063%2F1.1862095.
  * [57] S. Giri, L. Martucci, and A. Tomasiello, “On the stability of string theory vacua,” JHEP 04 (2022) 054, arXiv:2112.10795 [hep-th].
  * [58] J. Garriga, D. Schwartz-Perlov, A. Vilenkin, and S. Winitzki, “Probabilities in the inflationary multiverse,” JCAP 01 (2006) 017, arXiv:hep-th/0509184.
  * [59] A. De Simone, A. H. Guth, M. P. Salem, and A. Vilenkin, “Predicting the cosmological constant with the scale-factor cutoff measure,” Phys. Rev. D 78 (2008) 063520, arXiv:0805.2173 [hep-th].
  * [60] J. Garriga and A. Vilenkin, “Watchers of the multiverse,” JCAP 05 (2013) 037, arXiv:1210.7540 [hep-th].
  * [61] B. Friedrich, A. Hebecker, M. Salmhofer, J. C. Strauss, and J. Walcher, “A local Wheeler-DeWitt measure for the string landscape,” Nucl. Phys. B 992 (2023) 116230, arXiv:2205.09772 [hep-th].
  * [62] A. Vilenkin, “Quantum Creation of Universes,” Phys. Rev. D 30 (1984) 509–511.
  * [63] J. Feldbrugge, J.-L. Lehners, and N. Turok, “No smooth beginning for spacetime,” Phys. Rev. Lett. 119 no. 17, (2017) 171301, arXiv:1705.00192 [hep-th].
  * [64] J. Feldbrugge, J.-L. Lehners, and N. Turok, “No rescue for the no boundary proposal: Pointers to the future of quantum cosmology,” Phys. Rev. D 97 no. 2, (2018) 023509, arXiv:1708.05104 [hep-th].
  * [65] J. Feldbrugge, J.-L. Lehners, and N. Turok, “Inconsistencies of the New No-Boundary Proposal,” Universe 4 no. 10, (2018) 100, arXiv:1805.01609 [hep-th].
  * [66] A. Vilenkin and M. Yamada, “Tunneling wave function of the universe,” Phys. Rev. D 98 no. 6, (2018) 066003, arXiv:1808.02032 [gr-qc].
  * [67] A. Vilenkin and M. Yamada, “Tunneling wave function of the universe II: the backreaction problem,” Phys. Rev. D 99 no. 6, (2019) 066010, arXiv:1812.08084 [gr-qc].
  * [68] A. Strominger, S.-T. Yau, and E. Zaslow, “Mirror symmetry is T duality,” Nucl. Phys. B 479 (1996) 243–259, arXiv:hep-th/9606040.
  * [69] M. Cvetic, S. Griffies, and S.-J. Rey, “Static domain walls in N=1 supergravity,” Nucl. Phys. B 381 (1992) 301–328, arXiv:hep-th/9201007.
  * [70] M. Cvetic and H. H. Soleng, “Supergravity domain walls,” Phys. Rept. 282 (1997) 159–223, arXiv:hep-th/9604090.
  * [71] A. Ceresole, G. Dall’Agata, A. Giryavets, R. Kallosh, and A. D. Linde, “Domain walls, near-BPS bubbles, and probabilities in the landscape,” Phys. Rev. D 74 (2006) 086010, arXiv:hep-th/0605266.
  * [72] J. Bagger and D. V. Belyaev, “Supersymmetric branes with (almost) arbitrary tensions,” Phys. Rev. D 67 (2003) 025004, arXiv:hep-th/0206024.
  * [73] F. F. Gautason, M. Schillo, T. Van Riet, and M. Williams, “Remarks on scale separation in flux vacua,” JHEP 03 (2016) 061, arXiv:1512.00457 [hep-th].
  * [74] F. F. Gautason, V. Van Hemelryck, and T. Van Riet, “The Tension between 10D Supergravity and dS Uplifts,” Fortsch. Phys. 67 no. 1-2, (2019) 1800091, arXiv:1810.08518 [hep-th].
  * [75] D. Lüst, E. Palti, and C. Vafa, “AdS and the Swampland,” Phys. Lett. B 797 (2019) 134867, arXiv:1906.05225 [hep-th].
  * [76] A. Westphal, “Lifetime of Stringy de Sitter Vacua,” JHEP 01 (2008) 012, arXiv:0705.1557 [hep-th].
  * [77] 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, arXiv:hep-th/0007191.
  * [78] I. Bena, E. Dudas, M. Graña, and S. Lüst, “Uplifting Runaways,” Fortsch. Phys. 67 no. 1-2, (2019) 1800100, arXiv:1809.06861 [hep-th].
  * [79] R. Blumenhagen, D. Kläwer, and L. Schlechter, “Swampland Variations on a Theme by KKLT,” JHEP 05 (2019) 152, arXiv:1902.07724 [hep-th].
  * [80] S. Lüst and L. Randall, “Effective Theory of Warped Compactifications and the Implications for KKLT,” Fortsch. Phys. 70 no. 7-8, (2022) 2200103, arXiv:2206.04708 [hep-th].
  * [81] F. Carta, J. Moritz, and A. Westphal, “Gaugino condensation and small uplifts in KKLT,” JHEP 08 (2019) 141, arXiv:1902.01412 [hep-th].
  * [82] X. Gao, A. Hebecker, and D. Junghans, “Control issues of KKLT,” Fortsch. Phys. 68 (2020) 2000089, arXiv:2009.03914 [hep-th].
  * [83] D. Junghans, “LVS de Sitter vacua are probably in the swampland,” Nucl. Phys. B 990 (2023) 116179, arXiv:2201.03572 [hep-th].
  * [84] X. Gao, A. Hebecker, S. Schreyer, and G. Venken, “The LVS parametric tadpole constraint,” JHEP 07 (2022) 056, arXiv:2202.04087 [hep-th].
  * [85] A. Hebecker, S. Schreyer, and G. Venken, “Curvature corrections to KPV: do we need deep throats?,” JHEP 10 (2022) 166, arXiv:2208.02826 [hep-th].
  * [86] S. Schreyer and G. Venken, “$\alpha$’ corrections to KPV: an uplifting story,” JHEP 07 (2023) 235, arXiv:2212.07437 [hep-th].
  * [87] J. Feldbrugge, J.-L. Lehners, and N. Turok, “Lorentzian Quantum Cosmology,” Phys. Rev. D 95 no. 10, (2017) 103508, arXiv:1703.02076 [hep-th].
  * [88] L. Martucci and P. Smyth, “Supersymmetric D-branes and calibrations on general N=1 backgrounds,” JHEP 11 (2005) 048, arXiv:hep-th/0507099.
  * [89] A. Tomasiello, Geometry of String Theory Compactifications.  Cambridge University Press, 1, 2022.
  * [90] K. Eckerle, “A unified system for Coleman–De Luccia transitions,” Annals Phys. 424 (2021) 168362, arXiv:2003.04365 [hep-th].