aainstitutetext: Jefferson Physical Laboratory, Harvard University, Cambridge,
MA 02138, USAbbinstitutetext: Instituto de Física Teórica UAM-CSIC and
Departamento de Física Teórica, Universidad Autónoma de Madrid, Cantoblanco,
28049 Madrid, Spainccinstitutetext: CERN, Theoretical Physics Department, 1211
Meyrin, Switzerland

# On Measuring Distances in the Quantum Gravity Landscape

Amineh Mohsenia,c Miguel Monterob Cumrun Vafaa Irene Valenzuelab,c
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this note, we propose a generalized notion of distance between vacua in the
theory of a scalar field $\phi$ with scalar potential $V(\phi)$ coupled to
gravity. We propose the normalized tension of the domain wall connecting
different field values, with a varying normalization relative to a local
energy scale, as the distance. We show this definition reproduces the usual
moduli space distance for zero potential, as well as the
$d\propto|\log\Lambda|$ behavior with the vacuum energy $-\Lambda$ in the AdS
case, previously proposed in the literature. In the case of large AdS we also
obtain the expected exponent of mass versus distance in one particular case,
when the mass of the light tower is $m\sim\sqrt{\Lambda}$ and there is a
single extra dimension decompactifying. We also discuss the features and
shortcomings of alternative but related proposals.

††preprint: IFT-24-097 , CERN-TH-2024-101

##  1 Introduction

Quantum gravity theories possess particular features, which are the object of
study of the Swampland program palti2019swampland ; van2022lectures ;
agmon2022lectures . An important subset of these pertains to the physics of
asymptotic regimes in field space. For example at large distances in the
moduli space of Minkowski theories one expects to get an exponentially light
tower of particles ooguri2007geometry , where distances are measured using the
kinetic term of the scalars in the Einstein frame action as a metric. The same
situation is expected to continue holding in the case with non-vanishing
vacuum energy, although defining a proper notion of distance in that case is
more challenging. If we consider two AdS vacua connected by a moduli space, we
can still use the Zamolodchikov distance of the conformal manifold of the dual
Conformal Field Theory to recover the exponential drop-off of the tower of
states as distance becomes parametrically large Baume:2020dqd ;
Perlmutter:2020buo ; Baume:2023msm ; Ooguri:2024ofs . The notion of distance
has also been proposed in the AdS/dS case lust2019ads  with a number of
followup works Kehagias:2019akr ; DeBiasio:2022nsd ; DeBiasio:2022omq ;
Velazquez:2022eco ; DeBiasio:2020xkv ; DeBiasio:2022zuh ; Shiu:2022oti ;
Shiu:2023bay ; Li:2023gtt ; Basile:2023rvm ; Palti:2024voy  including its
interplay with the asymptotic towers of states. Most of these notions of
distance in field space are motivated by the kinetic term in the action and do
not take into account the existence of potential as a functions of fields
(see, however, Palti:2024voy  for an example that also takes into account the
potential). One of the main aims of this paper is to make progress in filling
this gap.

Defining a notion of distance between different quantum field theories is
certainly a long-standing open challenge, and interesting proposals have been
investigated in the literature (see e.g.douglas2013spaces ; Bachas:2013nxa ;
Balasubramanian:2014bfa ; Stout:2022phm ). In this work, we will take
inspiration from the Cobordism conjecture mcnamara2019cobordism  to define a
distance between theories. This Swampland constraint implies the existence of
a domain wall of finite tension connecting every two quantum gravity theories
of the same dimension. This is proposed to avoid the existence of global
symmetries, which cannot be exact in quantum gravity. Hence, can we use the
domain wall tension (or some related quantity) to define a distance between
the theories? If so, this would connect nicely with another quantum gravity
expectation: the finiteness of the quantum gravity landscape. If all theories
are connected by finite tension domain walls, they would all be at finite
distance from each other, implying they are all part of a single, universal
quantum theory of gravity.

Given two effective field theories (EFTs), we can consider the path of minimal
euclidean action in the space of field configurations (for short: the
configuration space) that interpolates between the two EFTs. Whenever this
euclidean solution admits a “thin wall’ description, the action can be written
in terms of the domain wall tension. We will use this euclidean action
(normalized by the euclidean energy density of the configuration) as our
proposal for a distance between theories. As we will see, it reproduces the
moduli space distance in the absence of a scalar potential. However, our
notion ends up having some intriguing properties when the potential is
included. For instance, the distance becomes energy dependent, as it depends
on the energy of the path in configuration space connecting the two theories.
Hence, it approximates the moduli space distance in the UV (when the potential
becomes negligible) and as we decrease the energy, the potential becomes
important. So in a sense it is a notion of distance provided the available
energy at a starting point. Therefore it is not a distance in the mathematical
sense, since it will not satisfy standard properties like the triangular
inequality as it also depends on the initial energy. In some sense, it is more
appropriate to regard it as a “cost function”, since it quantifies the
difficulty in reaching faraway regions in moduli space given an initial finite
resource (energy), but we refer to it as distance throughout this manuscript.

In the case of supersymmetric AdS vacua separated by a field-theory domain
wall, our notion of distance also reproduces the expected AdS distance in
lust2019ads . We view this as a nice consistency check of our proposal.
Assuming a relationship $m\sim\sqrt{\Lambda}$ for the mass of the tower of
particles that becomes light in the $\Lambda\rightarrow 0$ limit, we can
reconstruct a relationship between the distance and the mass of the tower. We
are able to recover the expected exponent of the Distance Conjecture, only for
the case when there is exactly one extra dimension decompactifying. This is
consistent with top-down holographic examples because in the formalism that we
develop here we restrict ourselves to scalar potentials and scalar moduli
spaces, and ignore e.g. effects of higher-dimensional p-form fields and other
degrees of freedom. With these restrictions, indeed, the only possible flux
compactification we can access is when there is a 1-form flux for an axion,
and indeed, making this large leads one to decompactification of a single
extra dimensions. Understanding how cases like e.g. $AdS_{5}\times S^{5}$,
which involve 5-form flux, enter in this picture, is left for future work.

The organization of this paper is as follows: In section 2 we discuss the
domain wall tension as a measure of moduli space distance. We then study the
distance between vacua with zero vacuum energy, leading to Minkowski
backgrounds. In section 3 we study the extensions of this to the case where
the vacuum energy changes between vacua, and in this case we need to take into
account the gravitational backreaction. In section 4, we offer a number of
examples. In particular, we show why the supersymmetric case offers a
particularly nice example where the distance can be readily computed in the
case of minimal initial energy. In section 5, we end the paper with some
concluding thoughts.

##  2 Proposal for a distance in a fixed background

###  2.1 Domain wall tension as a moduli space distance

To begin with, let us ignore the effects of gravity, and consider two
different vacua with zero vacuum energy in a d-dimensional effective theory
with a fixed background. For simplicity, let us restrict ourselves to the case
of homogeneous configurations, such that the two vacua only differ by the
vacuum expectation values of some scalar fields. The goal is to find a notion
of distance between these two vacua. In the case in which no potential barrier
exists, namely the scalar fields are massless, there is a well-defined notion
of distance measured by the field metric of the kinetic term of the scalars.
This is known as the moduli space distance and it is given by

$\Delta\phi=\int\sqrt{g_{ij}d\phi^{i}d\phi^{j}}\ ,\quad\text{with}\
\mathcal{L}\supset\frac{1}{2}g_{ij}\partial\phi^{i}\partial\phi^{j}+\dots$ (1)

We now explain how this moduli space distance can be recovered from the
tension of the domain wall connecting the two vacua, so that we can use the
domain wall tension to define a generalised notion of distance in the presence
of a potential in the next subsection.

For the sake of simplicity, let us first focus on the case of a single
massless scalar field which is canonically normalised (i.e. $g_{ii}=1$).
Consider a domain wall connecting the two vacua characterized by the values
$\phi_{i}$ and $\phi_{f}$. The tension of the domain wall is given by the
minimal energy (per unit volume) of the path field configuration $\phi(\tau)$
interpolating between the two vacua, where $\tau$ is a spatial coordinate.

Since there is no potential, the solution to the equation of motion is simply
a trajectory of constant speed

$\frac{1}{2}\left(\frac{d\phi}{d\tau}\right)^{2}=\rho_{E}\quad\rightarrow\quad\phi(\tau)=\phi_{0}+\sqrt{2\rho_{E}}\tau,$
(2)

with $\sqrt{2\rho_{E}}=\Delta\phi/\Delta\tau$ and
$\Delta\phi=\phi_{f}-\phi_{i}$. Hence, the tension of the solution
interpolating between the two EFTs reads

$T=\int\left(\frac{d\phi}{d\tau}\right)^{2}d\tau=\int
2\rho_{E}d\tau=\frac{(\Delta\phi)^{2}}{\Delta\tau}.$ (3)

As a final step to get the domain wall tension, we need to minimize over all
possible trajectories with different values of $\Delta\tau$, which implies
that the tension vanishes when connecting two vacua in the same moduli space.
This is because the minimal value occurs when the domain wall becomes
infinitely thick ($\Delta\tau\rightarrow\infty$), i.e. when it spreads over
the entire space, forcing the tension to vanish. However, we can recover the
moduli space distance from the following _normalized_ tension

$\Delta\equiv\frac{T}{\sqrt{2\rho_{E}}}=\sqrt{2\rho_{E}}\Delta\tau=\Delta\phi,$
(4)

This normalized tension will be our starting point to define a generalized
notion of distance in more complicated scenarios in the following. At the
moment, dividing by $\sqrt{2\rho_{E}}$ can simply be seen as a rescaling of
the distance which is required both to get the correct dimensions (i.e. a
distance with units of energy) and to reproduce the moduli space distance in
the absence of the potential.

We also notice that, strictly speaking, the solution (2) does not interpolate
between the two vacua (in the sense of Lorentz invariant states where the
scalar field is homogeneous in space and time), but rather between the field
_values_ $\phi_{i}$ and $\phi_{f}$. This feature will be present in the rest
of the paper, when scalar potential and the effects of gravity are included,
and sets apart our approach from e.g. the usual solitonic Coleman-De Luccia
domain wall coleman1977fate ; coleman1980gravitational , where the vacuum is
approached asymptotically for large values of $\tau$. By demanding only that
the solution interpolates between the two vacua only at a particular value of
the $\tau$ coordinate, we have a much more flexible framework, and can define
the generalized distance in situations where the soliton and Coleman-De Luccia
solutions do not exist (such as when interpolating between two vacua with
different asymptotic values of the scalar potential, as we will see
momentarily).

###  2.2 Distance in the presence of a scalar potential

Let us now move beyond moduli spaces by adding a scalar potential to the
theory. To start with, let us still ignore gravity and consider two field
values $\phi_{i}$ and $\phi_{f}$ where the potential still vanishes (so
$V(\phi_{i})=V(\phi_{f})=0$), separated now by a potential barrier as
illustrated in Figure 1. This is the usual setup where one can construct a
one-dimensional soliton domain wall solution rajaraman1982solitons  that
interpolates between the vacua, since they have the same energy.

Figure 1: Two Minkowski vacua, separated by a barrier much smaller than the
Planck mass.

The tension of the domain wall separating the two vacua is given by

$T=\int\left(\frac{d\phi}{d\tau}\right)^{2}d\tau,$ (5)

and the scalar field configuration satisfies the equations of motion

$\frac{d^{2}\phi}{d\tau^{2}}=V^{\prime},$ (6)

subject to the following boundary conditions

$\phi(-\infty)=\phi_{i}\ ,\ \phi(\infty)=\phi_{f}\ ,\
\left.\frac{d\phi}{d\tau}\right|_{\pm\infty}=0\ .$ (7)

In general, when $V(\phi_{i})\neq V(\phi_{f})$, the minimization problem (6)
does not have a solution with boundary conditions (7); however, as explained
at the end of the previous subsection, in this case we relax (7) to demand
only that $\phi_{i}$ and $\phi_{f}$ are attained for some values of $\tau$. In
general, this yields the following domain wall tension

$T=\int
2(\rho_{E}+V(\phi))d\tau=\int_{\phi_{i}}^{\phi_{f}}\sqrt{2(\rho_{E}+V(\phi))}d\phi,$
(8)

where we have used that the solution to (6) is given by

$\frac{1}{2}\left(\frac{d\phi}{d\tau}\right)^{2}=V+\rho_{E}.$ (9)

The integration constant $\rho_{E}$ must vanish in this case where
$V(\phi_{i})=V(\phi_{f})=0$ (or more generally, be equal to the minimal vacuum
energy $V_{min}$ at the endpoints, $\rho_{E}=-V(\phi_{1,2})$) in order to
satisfy the boundary conditions (7) and asymptote to the two vacua. This
recovers the usual domain wall tension of Coleman de Luccia coleman1977fate ;
coleman1980gravitational .

Notice that the above tension is equivalent to the on-shell euclidean action
of the path of minimal action in configuration space that interpolates between
$\phi_{1}$ and $\phi_{2}$ at a fixed energy $\rho_{E}$. Interestingly, this
variational principle is known in classical mechanics as the Maupertuis
principle. It differs from the Hamilton’s principle in that it finds the
trajectory of minimal action by varying over time for a fixed energy, instead
of varying over energies for a fixed time duration (as in Hamilton’s). Thus,
the Maupertuis functional $S_{M}$ can be written as the Legendre transform of
Hamilton’s action: $S_{H}=S_{M}-\rho_{E}T$, where $T$ is the time duration of
the trajectory. For instance, consider the euclidean action for a single
scalar in one dimension (i.e., on classical mechanics),

$S_{M}=\int d\tau(\frac{1}{2}\dot{\phi}^{2}+V(\phi))+\int d\tau\rho_{E}\ .$
(10)

This yields the following equations of motion at fixed energy $\rho_{E}$

$\frac{1}{2}\dot{\phi}^{2}-V(\phi)=\rho_{E}\
,\quad\ddot{\phi}=V^{\prime}(\phi),$ (11)

where $\dot{}=\partial_{\tau}$, and ${}^{\prime}=\partial_{\phi}$. The above
equations are solved by $\dot{\phi}=\sqrt{2(\rho_{E}+V)}$. Plugging this into
the action we indeed get

$S_{M}=\int d\tau\dot{\phi}^{2}=\int d\phi\,\sqrt{2(\rho_{E}+V)}\ ,$ (12)

reproducing the domain wall tension when the euclidean energy
$\rho_{E}=-V_{min}=0$.

Following the reasoning of the previous section, we propose the distance
between two theories that only differ by the vev of one scalar field as

$\Delta(\rho_{E})=\frac{1}{\sqrt{\rho_{E}}}\int_{\phi_{i}}^{\phi_{f}}d\phi\,\sqrt{\rho_{E}+V},$
(13)

where we have added the normalization factor $\sqrt{2\rho_{E}}$ to recover the
moduli space distance in the absence of the potential. This can be easily
generalized to the case of multiple scalar fields with field metric $g_{ij}$,
yielding

$\Delta(\rho_{E})=\frac{1}{\sqrt{\rho_{E}}}\int_{\phi_{i}}^{\phi_{f}}\sqrt{g_{ij}(\rho_{E}+V)d\phi^{i}d\phi^{j}}\
.$ (14)

where we integrate over the path in configuration space that minimizes the
euclidean action. This formulation in terms of the Maupertuis action makes
manifest that our proposed notion of distance comes from a well-defined
variational principle, so it will satisfy the required properties to behave as
a geodesic distance in metric spaces. In fact, notice that (14) looks very
similar to the geodesic distance in field space, except for the fact that the
metric gets multiplied by a conformal factor equal to $\sqrt{\rho_{E}+V}$.
Interestingly, this new metric $J_{ij}=2(\rho_{E}+V)g_{ij}$ is known as the
Jacobi metric in classical mechanics, and it is well known that geodesics of
this metric are equivalent to the trajectories of constant energy $\rho_{E}$
solving the equations of motion goldstein1980classical .

Since the Jacobi metric is a metric in the traditional sense, the above
definition satisfies all required properties to behave as a proper distance:

  * •

It is positive definite. As long as the kinetic energy is quadratic in the
velocity components, the Maupertuis action is always positive for all
trajectories:

$\Delta=\int\sqrt{1+\frac{V}{\rho_{E}}}d\phi=\frac{1}{\sqrt{2\rho_{E}}}\int\dot{\phi}\,d\phi=\frac{1}{\sqrt{2\rho_{E}}}\int\dot{\phi}^{2}d\tau\geq
0.$ (15)

  * •

For a fixed $\rho_{E}$, it is symmetric so that
$\Delta(\phi_{1},\phi_{2})=\Delta(\phi_{2},\phi_{1})$ and
$\Delta(\phi_{1},\phi_{1})=0$.

  * •

It satisfies the triangular inequality
$\Delta(\phi_{1},\phi_{2})+\Delta(\phi_{2},\phi_{3})\geq\Delta(\phi_{1},\phi_{3})$.
We are evaluating the on-shell action along the trajectory that minimizes the
action, so it satisfies the triangular inequality by default as long as we are
comparing trajectories with the same euclidean energy $\rho_{E}$.

Recall that the above distance is only related to the CdL domain wall tension
when $\rho_{E}=-V_{\text{m}in}$ (namely, $\rho_{E}=0$ for Minkowski vacua).
However, it behaves as a geodesic distance for any value of $\rho_{E}$ due to
the above properties. This opens up an interesting possibility: we can use the
on-shell Euclidean Maupertuis action to define a generalized notion of
distance $\Delta(\rho_{E})$ between theories which is $\rho_{E}$-dependent.
This is interesting for the following reason. First of all, in field theory,
it allows us to define a distance between theories even in the case in which
they cannot be connected by a planar domain wall (e.g. because they have
different vacuum energies). If $\rho_{E}\gg V$, the distance (14) reproduces
the moduli space distance, i.e. $\Delta(\rho_{E}\gg V)\approx\Delta\phi$,
while if $\rho_{E}\sim V$, the effect of the potential barrier becomes more
important and the distance between the theories increases, until it reaches
its maximum value given by the normalized tension of the CdL domain wall. This
makes sense intuitively, as the potential barrier makes it more difficult to
connect the two theories. In the case of connecting two Minkowski vacua, the
distance diverges in the limit $\rho_{E}\rightarrow 0$. Hence, we get a notion
of distance that interpolates between the moduli space distance and the
normalized CdL domain wall tension. Furthermore, it reproduces the field
distance in the moduli space in the absence of a scalar potential, independely
of the value of $\rho_{E}$. The cobordism conjecture (i.e. the existence of a
finite energy domain wall interpolating between any two vacua of the same
space-time dimension) translates then to the statement that any two theories
should be at finite distance for non-vanishing $\rho_{E}$. Notice that
$\rho_{E}$ only measures the energy density of the Euclidean configuration. In
Lorentzian signature, it corresponds to a non-vanishing spatial derivative of
the scalar field.

To summarize, in the context of field theory (i.e. ignoring gravitational
backreaction) we have a perfectly reasonable one-parameter notion of distance
$\Delta(\rho_{E})$, where the reference energy scale $\rho_{E}$ provides a
notion of “scale dependence” of the distance. In the next section we will see,
however, that this is significantly complicated by gravitational effects:
since there is no notion of energy in general relativity for general
spacetimes, the scale $\rho_{E}$ loses its meaning as a conserved quantity,
and indeed, it can change with $\tau$.

##  3 Generalization of distance including gravitational effects

In this section we investigate the case in which the background is also
dynamical, to account for gravitational effects. Consider a d-dimensional
scalar field theory weakly coupled to gravity with a scalar potential,
$V(\phi)$. Assume the scalar potential has two minima at $\phi_{i}$, and
$\phi_{f}$ with $V_{f}\leq V_{i}$ (see figure 2). For simplicity, let us
assume that we want to compute the distance between the minima of this scalar
potential with negative energy, i.e, between two AdS vacua. We also allow the
minima to have different vacuum energies, such that $V_{f}\leq V_{i}$. We will
try to construct the configuration that we discussed in the previous
subsection, this time taking into account an Einstein-Hilbert piece of the
action. As we will see, general relativity implies that the two vacua are no
longer connected by a configuration with constant euclidean energy. Hence, the
conservation of energy can no longer be assumed, implying that we cannot use
the Maupertuis variational principles of the previous section.

Figure 2: The scalar potential, $V(\phi)$, has minima at $\phi_{i}$, and
$\phi_{f}$ which are the false and true vacua respectively.

The metric ansatz for a domain wall is given by

$ds^{2}=d\tau^{2}+a^{2}(\tau)d\sigma_{\Sigma}^{2},$ (16)

where $d\sigma_{\Sigma}^{2}$ is the metric on the space $\Sigma_{d-1}$ spanned
by the domain wall, which we assume for the moment to be maximally symmetric
and flat, while $\tau$ is the spatial coordinate transverse to the domain
wall.

We will also define

$\Lambda(\phi)\equiv V(\phi)-\frac{1}{2}\dot{\phi}^{2}.$ (17)

Notice that $\Lambda(\phi)$ has the same definiton as $\rho_{E}$ in the
previous subsection; however, in the presence of gravity, the Euclidean
equations of motion that we need to satisfy read

$\ddot{\phi}+(d-1)\frac{\dot{a}}{a}\dot{\phi}=V^{\prime}(\phi),$ (18)
$\frac{\dot{a}^{2}}{a^{2}}=\frac{2\kappa^{\frac{d-2}{2}}}{(d-1)(d-2)}\left(\frac{1}{2}\dot{\phi}^{2}-V(\phi)\right),$
(19)

where $\kappa=8\pi G$. In terms of $\Lambda(\phi)$, the above two equations of
motion can be simply recast as

$\Lambda^{\prime
2}=\alpha\kappa^{\frac{(d-2)}{2}}\Lambda(\Lambda-V),\quad\text{ with
}\quad\alpha=4\frac{d-1}{d-2}\ .$ (20)

We see immediately that $\Lambda(\phi)$ is not a conserved quantity, as
advertised.

The equation of motion is first order, and so it needs a boundary condition.
We will take

$\Lambda(\phi_{i})=V(\phi_{i})-\left.\frac{1}{2}\dot{\phi}^{2}\right|_{\phi=\phi_{i}}\equiv\Lambda_{i},$
(21)

for some constant $\Lambda_{i}$. This parameter is the _initial_ value of
$\Lambda$, and it plays the same role as the $\rho_{E}$ in the theory without
gravity. This leads to a unique solution for $\Lambda(\phi)$ up to a sign
ambiguity in choosing the square root in equation (20) which will be explained
below.

Once the equations of motion are resolved, the question remains of determining
which functional of the on-shell solution we should use as a notion of
distance. As illustrated in Section 2, in the case without gravity we divided
the domain wall tension by $\sqrt{\rho_{E}}$ to ensure agreement with the
moduli space metric. This is a property we would like to preserve in the
presence of gravity, but now $\rho_{E}=-\Lambda$ depends on $\phi$. We would
also like to recover the formula of the previous subsection in the limit where
$\kappa\rightarrow 0$. The simplest generalization of the flat space formula
with these properties is to define our scale-dependent distance between the
theories with scalar vevs $\phi_{i}$ and $\phi_{f}$ (including gravitational
effects) as

$\displaystyle\Delta(\Lambda_{i})\equiv\int_{\phi_{i}}^{\phi_{f}}\sqrt{1-\frac{V(\phi)}{\Lambda(\phi)}}d\phi.$
(22)

for $\Lambda$ satisfying (20) subject to the boundary condition (21).

The outcome is that we obtain a scale-dependent distance, which depends on the
initial value of $\Lambda_{i}$, such that it again reproduces the moduli space
distance in the limit $\Lambda_{i}\gg V$. Also, in the gravitational
decoupling limit, $|\Lambda|,|V|\ll\kappa^{-\nicefrac{{d}}{{2}}}$, we recover
the Maupertuis distance. In this limit, we have $\kappa\rightarrow 0$, and
$\nicefrac{{\dot{a}}}{{a}}\rightarrow 0$ such that

$\frac{(d-1)(d-2)}{2}\kappa^{-\frac{d-2}{2}}\left(\frac{\dot{a}}{a}\right)^{2}\equiv\rho_{E},$
(23)

remains constant. Upon retrieving the gravitational coupling and taking the
decoupling limit, the equations of motion reduce to (11), and we recover the
Jacobi metric. We can interpret (22) as a recipe to compute the
gravitationally backreacted distance by first chopping up the problem into
many infinitesimal field displacements where gravity should not matter, using
(14) in each of these, and add up the results such that
$\Delta=\int\Delta_{\text{flat space}}\,d\phi$. As we explain in more detail
later, this varying $\Lambda(\phi)$ has both advantages and disadvantages. On
the one hand, we recover the AdS distance proposed in lust2019ads . On the
other, we will lose the connection with a metric distance (satisfying the
triangular inequality) in certain cases.

The initial value $\Lambda_{i}$ is bounded above, since the initial Euclidean
kinetic energy must be large enough to guarantee

$\displaystyle\Lambda-V\leq 0,\quad\text{for }\Lambda\leq 0\ ,$ (24)

so that the distance is well-defined. Otherwise, there is no physical solution
connecting the two theories. To guarantee monotonicity, the condition is that
$\Lambda-V<0,\Lambda<0$ between the initial and final vacua. This ensures that
the right-hand side of (20) does not cross zero in between $\phi_{i}$ and
$\phi_{f}$, so $\Lambda(\phi)$ becomes a monotonic function of the scalar
field. For a generic potential, a sufficient (but not necessary) condition
that guarantees (24) is

$\Lambda_{i}<\min\\{V_{\text{min}},0\\},$ (25)

where $V_{\text{min}}$ represents the minimum value of the scalar potential in
the interval defined by $\phi_{i}$ and $\phi_{f}$. This condition can be
relaxed depending on the shape of the scalar potential.

Interestingly, this notion of distance is also related to a suitable
generalized notion of domain wall tension beyond the thin-wall approximation,
valid also in the gravitiational context Espinosa:2023tuj . The tension of the
(possibly thick) domain wall solution is given by

$T=\int_{\phi_{i}}^{\tilde{\phi}_{f}}\sqrt{2(V(\phi)-\Lambda(\phi))}d\phi,$
(26)

where $\Lambda(\phi)$ satisfies111This function $\Lambda(\phi)$ is known in
Espinosa:2018hue ; Espinosa:2018voj ; Espinosa:2023tuj  as the tunneling
potential $V_{t}(\phi)$. precisely (20) and the field value
$\tilde{\phi}_{2}$, known in the literature as the “tunneling exit point”, is
defined by the condition $\Lambda(\tilde{\phi}_{2})=V(\tilde{\phi}_{2})$ –
namely, vanishing euclidean kinetic energy. Strictly speaking, this implies
that $\tilde{\phi}_{2}$ in (26) can be smaller than the value of the minimum
$\phi_{2}$ (if it is larger, then the domain wall solution does not exist). In
the thin wall approximation, one has $V_{f}\sim V_{i}$ so that $\Lambda(\phi)$
is approximately constant and $\tilde{\phi}_{2}\approx\phi_{2}$, so that (26)
reproduces the thin wall tension result (8).

Finally, note that the differential equation (20) has two possible solutions
that differ on the sign of $\Lambda^{\prime}$,

$\Lambda^{\prime}=\pm\sqrt{\alpha\kappa^{\frac{(d-2)}{2}}\Lambda(\Lambda-V)}.$
(27)

Since
$\Lambda^{\prime}=\kappa^{\nicefrac{{1}}{{2}}}(d-1)\frac{\dot{a}}{a}\dot{\phi}$,
the choice of sign is correlated to providing a gravitational boost or
friction to the accelaration of the scalar field. Since $\Lambda^{\prime}$ is
a monotonic function, the sign will be well defined along the entire path,
yielding a priori two possible values for the distance. To have a unique well-
define notion of distance, we choose the sign in correlation to the sign of
the variation of the scalar potential, namely

$sign(\Lambda^{\prime})=sign(V_{f}-V_{i}),$ (28)

which corresponds to the boosting solution. For a fixed initial $\Lambda_{i}$,
this actually corresponds to the larger of the two possible distances between
the two vacua. This again guarantees that we can always find a solution, since
the solution associated with the shorter distance does not always exist. For
$V_{f}=V_{i}$, we again choose the larger notion of distance if the two
choices differ. When $V(\phi)=0$ the choice of sign does not matter, and we
recover the expected result with either sign.

In Appendix A we generalize this distance to multiple scalar fields, obtaining
the following result as expected,

$\displaystyle\Delta(\Lambda_{i})\equiv\int_{\phi_{i}}^{\phi_{f}}\sqrt{g_{ij}(1-\frac{V}{\Lambda})d\phi^{i}d\phi^{j}}.$
(29)

with
$\Lambda_{i}=V_{i}-\frac{1}{2}g_{ij}\dot{\phi}^{i}\dot{\phi}^{j}|_{\phi=\phi_{i}}$.

###  3.1 Recovering the AdS Distance

In this section we show that our proposal reproduces the AdS Distance proposed
in lust2019ads  in the limit in which $\dot{\phi}$ approximately vanishes at
the endpoints. Hence, our proposed distance interpolates between the moduli
space distance when $\Lambda_{i}\gg V_{i}$ and the AdS Distance when
$\Lambda_{i}\sim V_{i}$, at least in the cases where the AdS potential is
driven by scalar fields.

By plugging (27) into (29), we can write the distance simply in terms of
variations of the euclidean energy density

$\Delta(\Lambda_{i})=\frac{1}{\kappa^{\frac{d-2}{4}}\sqrt{\alpha}}\int_{\Lambda_{i}}^{\Lambda_{f}}\frac{d\Lambda}{\Lambda},$
(30)

which becomes

$\Delta(\Lambda_{i})=\frac{1}{\kappa^{\frac{d-2}{4}}\sqrt{\alpha}}\text{Log}\left(\frac{\Lambda_{f}}{\Lambda_{i}}\right),$
(31)

so that it only depends on the value of $\Lambda$ at the endpoints. Recall
that $\Lambda_{i}$ is an initial arbitrary condition, while $\Lambda_{f}$ gets
fixed by solving the equation of motion (20). Whenever there exists a solution
such that $\Lambda_{f}=V_{f}$ starting with $\Lambda_{i}=V_{i}$, one exactly
recovers the AdS Distance $\Delta\sim\log(V_{f}/V_{i})$ from (31). This occurs
whenever there exists a static planar domain wall between the two AdS vacua,
as occurs in supersymmetric vacua connected by a BPS domain wall (see Section
4.1).

Known supersymmetric AdS vacua arising from string theory do not exhibit scale
separation between the AdS scale and the size of the extra dimensions,
implying that there is a Kaluza-Klein tower of states scaling as $m\sim
V_{0}^{1/2}$. Assuming the scaling $m\sim V^{\frac{1}{2}}$, we can derive the
exponential rate of this tower in terms of the distance $\Delta$, thus
obtaining

$m\sim\Lambda^{1/2}\sim{\rm
exp}(-{\kappa^{\frac{(d-2)}{4}}\sqrt{\alpha}\Delta/2})\sim{\rm
exp}\left[-\kappa^{\frac{(d-2)}{4}}\sqrt{\frac{d-1}{d-2}}\Delta\right].$ (32)

The exponential mass decay rate for a KK tower decompactifying from a lower
$d$-dimensional vacuum to a higher $D$-dimensional vacuum is
$\sqrt{\frac{D-2}{(d-2)(D-d)}}$. Therefore, the exponential rate in (32)
coincides with the rate of a KK tower associated to the decompactification of
one extra dimension. It is remarkable that using our definition of distance
inspired by the domain wall tension we are getting a recognizable exponent for
moduli space distances. Of course, it is not clear whether the exponential
rates in the AdS case should match those associated to the moduli space
distance. But given the coincidence, one may wonder why we are getting an
exponent which only corresponds to one extra dimension opening up. One
explanation may be that since we are not studying the situation with fluxes in
AdS and only with scalar, and since an axionic scalar can only give rise to
flux with one higher dimension, this could be the reason we only are getting
this situation.

###  3.2 Properties and drawbacks of the scale-dependent distance

This proposal for distance has a couple of interesting properties:

  * •

It boils down to moduli distance, $\Delta_{\Lambda_{i}}=\int d\phi$, for
$V\rightarrow 0$.

  * •

It is positive definite

$\displaystyle\Delta_{\Lambda_{i}}=\int\frac{\dot{\phi}^{2}}{\sqrt{-2\Lambda}}d\tau\geq
0.$ (33)

  * •

It depends on the initial scale $\Lambda_{i}$, such that it interpolates
between the moduli space distance for $\Lambda_{i}\gg V_{i}$ and the AdS
Distance for $\Lambda_{i}\sim V_{i}$

However, this latter scale-dependent property also comes with some drawbacks.
The distance is not symmetric anymore, since it depends on the initial value
of $\Lambda_{i}$. Therefore, we need to adjust the initial conditions suitably
to obtain a symmetric solution. In other words, we have
$\Delta_{\Lambda^{B}_{i}}(B,A)=\Delta_{\Lambda^{A}_{i}}(A,B)$, which can be
checked to be consistent with our sign choice for $\Lambda^{\prime}$ in (28).
Related to this, the triangular inequality is not guaranteed in general,
although it holds whenever we have a solution with $V_{i}=\Lambda_{i}$ at both
endpoints (like in the supersymmetric case in section 4.1). In that case, the
distance only depends on the initial and final values of the potential, and it
therefore satisfies triangular equality (as in one-dimensional spaces).
However, in more generic cases, our proposal does not behave as a distance in
a metric space. In this sense, it should be better understood as a kind of
cost function, which is scale-dependent. In the next section, we comment on
other possible generalizations to try to overcome this issue, although it is
not clear to us to what extent this failure of behaving as a metric distance
is really a problem or a necessary feature. Intuitively, it is not
unreasonable that the notion of distance loses its symmetric property, since
roughly speaking, it costs more energy to go up a potential than going down,
as any mountain hiker will confirm.

###  3.3 Other possible generalizations

The generalized distance we just proposed was relatively ad hoc, motivated
mostly by the connection to the AdS distance and domain wall tension, and it
did not come from an action principle in the gravitational case. One could try
instead to directly use the gravitational generalization of the Maupertuis
action or Jacobi metric that we used in the field theory case. The Jacobi
metric for the time-dependent system at hand is given by
cariglia2018cosmological

$J_{ij}{d\phi}^{i}{d\phi}^{j}=2\kappa^{\nicefrac{{d}}{{2}}}a^{d-1}V(\phi)(-(d-1)(d-2)\kappa^{-\frac{d-2}{2}}a^{d-3}{da}^{2}+a^{d-1}{d\phi}^{2}),$
(34)

and the line element in field space is

$dl=\sqrt{J_{ij}{d\phi}^{i}{d\phi}^{j}}.$ (35)

Notice that, unlike in the field theory case, this metric is not positive
definite. This happens whenever one has a system with constraints; in the case
of gravity, not every initial condition is valid, since they must satisfy the
Hamiltonian constraint of GR wald2010general . In other words, for the above
metric to be equivalent to the solution of Einstein’s equations, is that the
physical time is defined via

$\frac{(d-1)(d-2)}{2}\kappa^{-\frac{(d-2)}{2}}\frac{\dot{a}^{2}}{a^{2}}=\frac{1}{2}\dot{\phi}^{2}-V(\phi).$
(36)

It turns out that, when this constraint is taken into account and the on-shell
action is evaluated, the dynamical component of gravity cancels the
contribution of matter fields to the distance. This line element vanishes when
$V=0$. Therefore, although it is a good metric, it does not straightforwardly
lead to a good measure of distance. Extracting a notion of distance from this
metric is left for future work. Our proposed distance of the previous
subsections corresponds to the piece of the above Maupertuis action associated
only to the scalar field, which recovers the moduli space distance when $V=0$
at the cost of losing the triangular inequality (although we have triangular
equality for supersymmetric configurations).

A related, perhaps more physical proposal starts by trying to construct not an
Euclidean solution in $d$-dimensions, but instead, a valid _initial_ condition
for general relativity, and try to minimize its mass. More specifically, we
could look at the ADM mass of a generic bubble of radius $R$ (meaning it
bounds a sphere of area $4\pi R^{2}$ interpolating between the two
geometries), and find the field configuration that minimizes $M_{\text{ADM}}$.
Details are provided in Appendix B. The energy functional that must be
minimized resembles the Maupertuis euclidean action of Section 2.2, although
the equations of motion for the scalars are evaluated in one dimension less,
since we are only looking for a solution at a initial time slice. However, we
find that in general (e.g. when the potential is too steep) an event horizon
may form, in which case the formalism stops being valid. On top of this
problem, we face the difficulty that the ADM mass is not universally positive
in general relativity, and only in restricted circumstances (e.g. whenever a
positive energy theorem applies) this will happen. Since the effective tension
could happen to be negative (as e.g. for orientifolds, or in some sense for
bubbles of nothing Friedrich:2023tid ; Friedrich:2024aad ), it is not clear
that this definition leads to the expected properties of a distance for
metastable vacua. On the other hand, these difficulties might be ameliorated
for supersymmetric, exactly stable vacua, where positivity of the ADM mass is
expected to hold.

Because of these difficulties, we have not explored this notion of distance
any further, but the simple underlying result and direct physical
interpretation in terms of a minimal mass are quite appealing. Furthermore,
the connection to BPS domain walls and minima works in the same way as for the
proposal described in the previous Section, since the notion of thin wall is
similar in this case. We hope to return to this question in the future.

As a final comment, let us remark that we have only investigated the case in
which the different field configurations can be connected by a planar domain
wall. This is why we have focused on Section 3 in the case of AdS vacua, since
they admit maximally symmetric flat slices in euclidean signature. However, if
we want to have solutions interpolating between more general configurations
(like e.g. de Sitter vacua, which are positively curved), we will have to add
non-vanishing intrinsic curvature to the metric (16). Upon doing so, the
equation of motion for the scalar field (18) does not change, while (19) will
be modified by a curvature term. If we keep our definition of $\Lambda$ as
minus the energy density of the scalar field, which accounts for both
intrinsic and induced sources of curvature, the equation (20) governing the
dynamics of $\Lambda$ is modified to $\Lambda^{\prime
2}=\alpha\kappa^{\frac{(d-2)}{2}}(\Lambda+\frac{(d-1)(d-2)}{2}\frac{k}{a^{2}})(\Lambda-V)$,
and consequently the logarithm (31) is also modified by a term containing the
effects of the intrinsic curvature222The distance would apparently get
modified to
$\Delta=\frac{1}{\kappa^{\frac{d-2}{4}}\sqrt{\alpha}}\int\frac{d\Lambda}{\sqrt{\Lambda\left(\Lambda+\frac{(d-1)(d-2)}{2}\frac{k}{a^{2}}\right)}}$
where $\frac{k}{a^{2}}$ is the intrinsic curvature.. However, when written as
in (22), the formula for the distance remains unchanged. Since additional
subtleties could appear in these cases, we leave the investigation of these
curved solutions for future work.

##  4 Examples

In this Section, we offer a couple of examples in the decoupling limit of
gravity, $V\ll\kappa^{-\nicefrac{{d}}{{2}}}$, using our metric of Section 2
(sections 4.3, and 4.5). We also illustrate the generalized notion of distance
explained in Section 3 with some examples, including the $\mathcal{N}=1$
supersymmetric example (sections 4.1, 4.2 and 4.4).

###  4.1 Distance between Supersymmetric Vacua

Consider a four-dimensional $\mathcal{N}=1$ supersymmetric theory

$S=\int dv_{\sigma}d\tau
a^{3}\left[-3\frac{\dot{a}^{2}}{a^{2}}+K_{I\bar{J}}\dot{\phi}^{I}\dot{\bar{\phi}}^{\bar{J}}+V(\phi^{I},\bar{\phi}^{\bar{I}})\right].$
(37)

with the scalar potential of the theory given by

$V=e^{K}\left(K^{I\bar{J}}D_{I}WD_{\bar{J}}\overline{W}-3\,\left|W\right|^{2}\right).$
(38)

We set $\kappa=1$ throughout this section. Each superfield can contribute to
supersymmetry breaking through a non-vanishing F-term, $F_{I}\equiv
D_{I}W=(\partial_{I}+\partial_{I}K)W$. We are interested in the solution
interpolating between two supersymmetric AdS minima with
$F_{I}^{i}=F_{I}^{f}=0$. In between these two minima the supersymmetry is
broken, $D_{I}W\neq 0$ (we can have a positive barrier for example).

There exists a BPS domain wall solution connecting the vacua. The solution
preserves $\mathcal{N}=\frac{1}{2}$ supersymmetry. The supersymmetry
equations, $\delta\psi^{\mu}=\delta\chi^{I}=0$, have the following solutions

$\displaystyle\frac{\dot{a}}{a}$ $\displaystyle=$
$\displaystyle\pm\left|e^{\nicefrac{{K}}{{2}}}W\right|,$ (39)
$\displaystyle\dot{\phi}^{I}$ $\displaystyle=$ $\displaystyle\mp
e^{\nicefrac{{K}}{{2}}}K^{I\bar{J}}\frac{W\overline{D_{J}W}}{|W|},$ (40)

which also satisfy equations of motion (74), and Einstein’s equation (75)
ceresole2006domain ; cvetivc1992static .

Similar to the previous section, we can reparametrize the path in the
following way

$d\varphi^{2}=2K_{I\bar{J}}d\phi^{I}d\bar{\phi}^{\bar{J}}.$ (41)

For brevity, we define
$\left|DW\right|^{2}=K^{I\bar{J}}D_{I}WD_{\bar{J}}\overline{W}$. From (40), we
get

$\frac{1}{2}\dot{\varphi}^{2}=K_{I\bar{J}}\dot{\phi}^{I}\dot{\bar{\phi}}^{\bar{J}}=e^{K}|DW|^{2},$
(42)

so the kinetic term exactly cancels the F-term contribution to the scalar
potential. Therefore, the total kinetic energy of the scalar fields is a
measure of supersymmetry breaking in the original potential, and we have

$\displaystyle\Lambda=-3e^{K}|W|^{2}.$ (43)

At the initial and final supersymmetric vacua we have
$\nicefrac{{1}}{{2}}\dot{\varphi}^{2}=e^{K}|DW|^{2}=0$. Therefore we get

$\displaystyle\Lambda(\phi^{I}_{f})$ $\displaystyle=$ $\displaystyle
V(\phi^{I}_{f}),$ (44) $\displaystyle\Lambda(\phi^{I}_{i})$ $\displaystyle=$
$\displaystyle V(\phi^{I}_{i}).$ (45)

In this case, the value of $\Lambda$ is uniquely fixed at initial and final
points.

We can rewrite the solution for the trajectory in the following way

$\displaystyle\frac{d\phi^{I}}{d\varphi}=\pm\frac{K^{I\bar{J}}W\overline{D_{J}W}}{\sqrt{2}|W||DW|},$
(46)
$\displaystyle\frac{d\bar{\phi}^{\bar{I}}}{d\varphi}=\pm\frac{K^{\bar{I}J}\overline{W}D_{J}W}{\sqrt{2}|W||DW|}.$
(47)

We note that the trajectory aligns with the gradient of $\Lambda$

$\displaystyle\frac{\partial\Lambda}{\partial\phi^{I}}=-3e^{K}\overline{W}D_{I}W,$
(48)
$\displaystyle\frac{\partial\Lambda}{\partial\bar{\phi}^{\bar{I}}}=-3e^{K}W\overline{D_{I}W}.$
(49)

As expected, we recover

$\displaystyle\frac{d\Lambda}{d\varphi}$ $\displaystyle=$
$\displaystyle\frac{\partial\Lambda}{\partial\phi^{I}}\frac{d\phi^{I}}{d\varphi}+\frac{\partial\Lambda}{\partial\bar{\phi}^{\bar{I}}}\frac{d\bar{\phi}^{\bar{I}}}{d\varphi},$
(50) $\displaystyle=$ $\displaystyle\mp 3\sqrt{2}e^{K}|W||DW|,$ (51)
$\displaystyle=$ $\displaystyle\mp\sqrt{6\Lambda(\Lambda-V)},$ (52)

As long as $W,\,DW\neq 0$ (which is the case in between the vacua), the sign
does not flip and is determined according to equation (28). This allows us to
switch between the square root and logarithm forms of the distance. The
distance between two supersymmetric vacua is then given by

$\Delta=\sqrt{1-\frac{V}{\Lambda}}d\varphi=\frac{1}{\sqrt{6}}\text{Log}\left(\frac{\Lambda_{f}}{\Lambda_{i}}\right)=\frac{1}{\sqrt{6}}\text{Log}\left(\frac{V_{f}}{V_{i}}\right),$
(53)

where we have used the equation of motion (52) in the first equality, and the
boundary conditions (44), and (45) in the second equality. We can rewrite the
distance in terms of the tension of the BPS wall,

$\sigma=\frac{2}{\sqrt{3}}\left(|\Lambda_{f}|^{1/2}-|\Lambda_{i}|^{1/2}\right),$
(54)

in the following way

$\Delta=\sqrt{\frac{2}{3}}\text{Log}\left(1+\frac{\sqrt{3}}{2}\frac{\sigma}{|V_{i}|^{1/2}}\right),$
(55)

which can be approximated by

$\Delta\simeq\frac{1}{\sqrt{2}}\frac{\sigma}{|V_{i}|^{1/2}}+\dots.$ (56)

Therefore, the proposed distance reproduces the normalized tension of the BPS
domain wall in the supersymmetric case, as long as one chooses the minimal
value for the energy scale at the initial point (as given in (45)). In this
case, the distance (53) satisfies the triangular inequality in a kind of
trivial way, since it only depends on the value of the vacuum energy at the
endpoints. This is specific to supersymmetric setups in which there exists a
domain wall solution that starting with $\dot{\varphi}=0$ at the initial
vacuum reaches again $\dot{\varphi}=0$ at the final vacuum.

Finally, we note that another typical property of a distance function is that
it only vanishes when evaluated between a point and itself. This is not
automatically true of our solutions. Indeed, there could be in principle $W=0$
solitonic domain walls between a theory an itself. Such domain walls were
considered, and constructed, purely in the supergravity context in
Cvetic:1992bf ; Cvetic:1993xe  (see also the review Cvetic:1996vr ). For such
a domain wall, the asymptotic value of $\Lambda_{i}$ would be equal to
$\Lambda_{f}$, and the expression for the distance above would vanish.
Interestingly, however, there are no known top-down examples where the
phenomenon described in Cvetic:1992bf  takes place333We thank Mirjam Cvetič
for pointing this out to us., which we take to be an encouraging sign for our
proposal.

###  4.2 Constant Scalar Potential; $V=V_{0}$

Take a d-dimensional theory with a constant scalar potential, $V=V_{0}\leq 0$.
In this case the equation of motion is

$\Lambda^{\prime}=-\sqrt{\alpha\kappa^{\frac{(d-2)}{2}}\Lambda(\Lambda-
V_{0})},$ (57)

which can be easily solved. To start with, we set the initial condition
$\Lambda(0)=V_{0}$, where we have assumed $\phi_{i}=0$ for simplicity. This
initial condition implies that we start with the minimal possible energy at
the initial point. We find

$\Lambda=V_{0}\left[\text{cosh}\left(\frac{\sqrt{\alpha}}{2}\kappa^{\frac{(d-2)}{4}}\,\phi\right)\right]^{2},$
(58)

so that the distance gets independent of the value of the potential:

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\int_{0}^{\phi}\text{tanh}\left(\frac{\sqrt{\alpha}}{2}\kappa^{\frac{(d-2)}{4}}\,\phi\right)\,d\phi,$
(59) $\displaystyle=$
$\displaystyle\frac{2}{\kappa^{\frac{(d-2)}{4}}\sqrt{\alpha}}\text{Log}\left(\text{cosh}\left(\frac{\sqrt{\alpha}}{2}\kappa^{\frac{(d-2)}{4}}\,\phi\right)\right).$
(60)

On the other hand, if we choose the initial condition such that it includes a
non-zero contribution $\rho_{K}$ from the kinetic energy at the initial point

$\Lambda(0)=-\rho_{K}+V_{0},$ (61)

with $\rho_{K}\gg V_{0}$, we get $\Lambda\simeq-\rho_{K}$ and we recover

$\Delta\simeq\int d\phi,$ (62)

which is, as expected, the distance in moduli space. We note that, although
the scalar potential is constant, the euclidean energy $\Lambda(\phi)$ changes
due to the gravitational backreaction.

###  4.3 Distance between two $T^{3}$ compactifications connected by a massive
deformation

Let us consider next the distance between two EFTs arising from a toroidal
compactification of the same theory, such that they are no longer connected by
a moduli space. For instance, we can consider a $T^{3}$ compactification with
two different metrics related by a conformal transformation of the form

$\tilde{g}_{\tilde{T}^{3}}=e^{\varphi}g_{T^{3}},$ (63)

where $\varphi$ is a deformation which does not solve the Einstein’s equations
on $T^{3}$. Hence, the field $\varphi$ is not a modulus of this
compactification. As an example, we can take
$e^{2\varphi}=\phi(\lambda)(1+\cos(kx))$. The relevant part of the action is
given by

$S=\int_{T^{3}\times\mathbb{R}_{\lambda}}\mathcal{R}=\int_{T^{3}\times\mathbb{R}_{\lambda}}e^{2\varphi}\nabla_{i}\varphi\nabla^{i}\varphi+\dots=\int
d\lambda\left((\partial_{\lambda}\phi)^{2}+\frac{k^{2}}{4}\phi^{2}\right)+\dots$
(64)

where we have already integrated over $T^{3}$ in the last step. The resulting
EFT is that of a massive scalar field in 0+1 euclidean dimensions. The on-
shell action for the path solving the equation of motion at fixed energy
$\rho_{E}$ is given by (12) with potential $V=m^{2}\phi^{2}$ and
$m^{2}=k^{2}/4$. We work in the decoupling limit,
$V\ll\kappa^{-\nicefrac{{d}}{{2}}}$, so that we can use (13) to obtain the
distance between the two EFTs

$\Delta(\rho_{E})=\frac{1}{\sqrt{\rho_{E}}}\int_{\phi_{i}}^{\phi_{f}}\sqrt{g_{ij}(\rho_{E}+V)d\phi^{i}d\phi^{j}}=\int_{\phi_{i}}^{\phi_{f}}\sqrt{1+\frac{m^{2}\phi^{2}}{\rho_{E}}}d\phi,$
(65)

Due to the potential, the result for the distance is energy dependent. If
$\rho_{E}\gg V$, then we recover the moduli space distance again. However, as
$\rho_{E}$ decreases, the distance between the two EFTs increases and diverges
in the limit $\rho_{E}\rightarrow 0$. The interpretation of this IR divergence
is that the potential obstructs the path, so that there is no domain wall
solution that interpolates between the two theories unless we allow for some
non-vanishing energy $\rho_{E}>0$.

###  4.4 Exponential runaway potential

Consider a scalar potential of exponential form. This behavior is expected in
the asymptotic region of the moduli space.

$V(\phi)=-V_{0}e^{c\,\kappa^{\frac{(d-2)}{4}}\phi},$ (66)

where $c>0$, and $V_{0}>0$. For $0<c<\sqrt{\alpha}$ with $\alpha$ given in
(20), there is a solution for $\Lambda(\phi)$ for any arbitrary range in
$\phi$.

$\Lambda(\phi)=-V_{0}\left(1-\frac{c^{2}}{\alpha}\right)^{-1}e^{c\,\kappa^{\frac{(d-2)}{4}}\phi}.$
(67)

This is known as the attractor solution in cosmology. The distance is then
given by

$\Delta=\frac{c}{\sqrt{\alpha}}\phi,$ (68)

which has a nice physical interpretation: as $c$ (the slope of the potential)
increases, the distance increases. Finally, it matches the moduli distance for
$c=\sqrt{\alpha}$, where kination happens, and the distance is expected to
match that of a free field. We note that the potentials obtained from string
theory typically satisfy the above condition on $c$.

###  4.5 Distance between Calabi-Yau’s connected by a conifold topological
transition

We are now ready to try a slightly more involved but interesting case. One of
the most interesting applications of our metric is that it can be used to
provide a distance between different Calabi-Yau compactifications to Minkowski
space. It is well known that different Calabi-Yau’s can be often connected by
a topological transition in which some cycles blow up and others blow down.
From a physics perspective, if we have string theory compactified on these
CYs’, what happens is that a finite number of states coming from wrapping
branes become massless at the finite distance singularity where the
topological transition occurs.

As an illustrative example, let us consider a toy model for two Calabi-Yau’s
(CY1 and CY2) connected by a conifold-like topological transition. In such
cases, on one branch one field is massless and on the other side, a new field
becomes massless. In particular, we can view the relevant potential for the
two fields $r,\rho$ by

$V=\frac{1}{2}r^{2}\rho^{2}.$ (69)

When one of the fields $r$ or $\rho$ is non-zero, it serves as mass for the
other field.

Let us now compute the distance between some regular point in the moduli space
of CY1 and another regular point in the moduli space of CY2, such that the two
points are in the vicinity of the conifold singularity. For this, let us work
in the decoupling limit, $V\ll\kappa^{-\nicefrac{{d}}{{2}}}$, so that we can
use the notion of distance given in (14). In Figure 3(a) we represent
(schematically) the two-field space parametrized by $r$ and $\rho$. Using
(13), the distance between the point $\phi_{i}=\\{\rho=0,r=r_{0}\\}$ and
$\phi_{f}=\\{\rho=\rho_{0},r=0\\}$ is given by

$\Delta(\rho_{E})=\int_{\phi_{i}}^{\phi_{f}}\sqrt{\left(1+\frac{r^{2}\rho^{2}}{2\rho_{E}}\right)\left(dr^{2}+d\rho^{2}\right)}.$
(70)

The equations of motion are

$\displaystyle\frac{1}{2}\left(\dot{r}^{2}+\dot{\rho}^{2}\right)-\frac{1}{2}r^{2}\rho^{2}=\rho_{E},$
(71) $\displaystyle\ddot{\rho}=r^{2}\rho,\;\ddot{r}=\rho^{2}r,$ (72)

depending on the value of $\rho_{E}$, the path $r(\tau),\rho(\tau)$ minimizing
the action will be different. Solutions for different values of $\rho_{E}$ are
shown in Figure 3(b). As the figure shows, for small values of the energy
$\rho_{E}$, the trajectory is mainly controlled by the effect of the scalar
potential, and the trajectories match the equipotential lines (the blue line).
As $\rho_{E}$ increases, the effect of the potential becomes more and more
negligible, and the trajectory asymptotes towards the free trajectory, which
is the straight line connecting the two points (the red line). The two extreme
cases are given by:

(a) Contour plot of $V(\rho,r)$.

(b) Constant energy trajectories interpolating between the points $\\{1,0\\}$
and $\\{0,1\\}$. The euclidean energy $\rho_{E}$ increases from blue to red.

Figure 3: Details of the scalar potential and distance for the conifold toy
model discussed in the main text.

  * •

As $\rho_{E}\rightarrow 0$, the path approaches the valley of the potential
(i.e. moving along the horizontal and vertical axes) and the distance behaves
as
$\Delta\simeq\frac{1}{\sqrt{\rho_{E}}}\int_{\phi_{i}}^{\phi_{f}}r\rho\sqrt{dr^{2}+d\rho^{2}}$
for large $r$ and $\rho$, and approximates the flat one near $r,\rho\sim 0$
evaluated on the moduli space geodesic.

  * •

As $\rho_{E}\rightarrow\infty$, it approaches the expected field space
distance, $\Delta\simeq\int_{\phi_{i}}^{\phi_{f}}\sqrt{dr^{2}+d\rho^{2}}$ and
moves along the straight line as if the potential would not exist.

The distance is a function of energy, that varies between the above two
values.

##  5 Discussion

In this paper we have shown that the notion of moduli space distance in
quantum field theory (when gravity is decoupled) can be naturally extended to
the cases where there is a scalar potential on the field space using the
Maupertuis action principle. This provides a new metric (the Jacobi metric)
that takes into account the scalar potential and whose geodesics are the paths
of fixed energy solving the euclidean equations of motion and therefore
extremizing the euclidean Maupertuis action. Our notion of distance thus
depends on an auxiliary energy scale, which plays the role of an “RG scale”
for measuring distance. When this scale is much higher than any potential
barrier, the distance reproduces the moduli space distance. On the other hand,
when it gets comparable to the potential barrier, the distance gets larger and
can even diverge. When this energy scale takes its minimal possible value, the
distance approaches the value of the tension of the domain wall between vacua
normalized by this energy scale.

The above notion of distance can also be generalized in the presence of
gravity. Although in this case the resulting quantity does not obey in general
triangular inequality or other expected properties of distance (and is,
therefore, a “generalized” distance between field configurations at best), the
quantity we found still quantifies a notion of “distance to a given point” in
the space of vacua, providing a monotonic, positive function that increases
away from our given reference point. The appealing feature of this generalized
distance is that, as we vary the initial energy of the euclidean configuration
connecting the two theories, it interpolates between the moduli space distance
and the distance proposed in the AdS case for discrete families of vacua
lust2019ads , recovering this way the expected $\log V_{0}$ behavior with the
vacuum energy $V_{0}$. Moreover, it behaves as a distance in the mathematical
sense when connecting supersymmetric AdS vacua, as it again gets directly
related to the tension of the BPS domain wall between the vacua in that case.

We believe this notion has physical significance. One important application of
defining a distance on moduli spaces is to extend the Distance Conjecture to
situations with a scalar potential. To do this, one does not obviously need
the exact mathematical notion of distance (symmetric, positive, and satisfying
triangle inequality); perhaps all that is needed is a notion of “point at
infinite distance”, as we provided. Since our notion of distance reproduces
the moduli space distance and the AdS distance in certain limits, it seems
well suited to provide such generalization of the Distance Conjecture. In this
line, it is perhaps interesting that we get the expected value of the
exponential decay rate of the tower for one particular case corresponding to
decompactifying one extra dimension in non-scale separated AdS vacua. It
remains an open question how to completely generalize the Distance Conjecture
using this notion of distance and how the mass of the tower should behave
depending on the chosen value of the initial auxiliary energy scale.

Finally, we have also discussed other possible generalizations of the distance
that come from variational principles including gravity but fail to reproduce
the moduli space distance and/or the AdS Distance of lust2019ads . Moreover,
in this work we have focused on defining the distance between theories that
only differ by the vev of some scalar fields, so that the domain wall can be
smoothly described within the EFT using the effective potential for these
scalars. It would be interesting to generalize this notion of distance to
other field configurations, such as e.g. when we have gauge fields, fluxes,
etc. In those cases, the domain walls might include singular defects that
cannot be resolved within the EFT and one should probably consider the full
euclidean action to define the distance.

Acknowledgements: We are grateful to Costas Bachas, Alek Bedroya, José
Calderón Infante, Mirjam Cvetič, Jose Ramón Espinosa and Ignacio Ruiz for
illuminating discussions and comments. The work of AM and CV is supported in
part by a grant from the Simons Foundation (602883,CV), the DellaPietra
Foundation, and by the NSF grant PHY-2013858. AM is thankful to the TH
department of CERN, where most of the research on this project was carried
out. M.M. is supported by an Atraccion del Talento Fellowship 2022-
T1/TIC-23956 from Comunidad de Madrid. M.M. and I.V acknowledge the
hospitality of the Department of Physics of Harvard University during the
different stages of this work, as well as the Simons Summer workshop and the
Erwin Schrödinger International Institute for Mathematics and Physics of the
University of Vienna. MM and IV thanks the Spanish Research Agency (Agencia
Estatal de Investigación) through the grants IFT Centro de Excelencia Severo
Ochoa CEX2020-001007-S and PID2021-123017NB-I00, funded by
MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe. The work of
I.V. is also partly supported by the grant RYC2019-028512-I from the MCI
(Spain) and the ERC Starting Grant QGuide-101042568 - StG 2021.

##  Appendix A Generalization to Multiple Scalars

Assume we have several scalars $\phi_{k}$ with $k=1,\dots,n$, the action is

$S=\int dv_{\sigma}d\tau
a^{d-1}\left[-\frac{1}{2}R+\frac{1}{2}g_{ij}(\phi^{k})\dot{\phi}^{i}\dot{\phi}^{j}+V(\phi^{k})\right],$
(73)

equations of motion for the scalar fields are as follows

$\ddot{\phi}^{k}+(d-1)\frac{\dot{a}}{a}\dot{\phi}^{k}+\Gamma^{k}_{ij}\dot{\phi}^{i}\dot{\phi}^{j}=g^{ki}V_{,\phi^{i}},$
(74)

and Einstein’s equation implies

$\Lambda(\tau)\equiv-\frac{(d-1)(d-2)}{2}\frac{\dot{a}^{2}}{a^{2}}=-\frac{1}{2}g_{ij}(\phi^{k})\dot{\phi}^{i}\dot{\phi}^{j}+V(\phi^{k}).$
(75)

We can solve these $n+2$ equations ((74), and (75)) for $\phi^{k}(\tau)$,
$a(\tau)$, and $\Lambda(\tau)$. We define the field measuring length of the
field space trajectory in the following way

$d\varphi^{2}\equiv g_{ij}(\phi^{k})d\phi^{i}d\phi^{j}.$ (76)

We note that

$g_{ij}\frac{d\phi^{i}}{d\varphi}\frac{d\phi^{j}}{d\varphi}=1,$ (77)

we can take a derivative with respect to $\varphi$ to get

$g_{ij}\frac{d^{2}\phi^{i}}{d\varphi^{2}}\frac{d\phi^{j}}{d\varphi}+g_{al}\Gamma^{a}_{ij}\frac{d\phi^{i}}{d\varphi}\frac{d\phi^{j}}{d\varphi}\frac{d\phi^{l}}{d\varphi}=0.$
(78)

Upon projecting (74) on $g_{kl}\nicefrac{{d\phi^{l}}}{{d\varphi}}$ (tangent to
the path), and
$\left(g_{kl}-g_{ki}g_{lj}\nicefrac{{d\phi^{i}}}{{d\varphi}}\nicefrac{{d\phi^{j}}}{{d\varphi}}\right)$
(orthogonal to the path) after using the above identities, we get

$\displaystyle\ddot{\varphi}+(d-1)\frac{\dot{a}}{a}\dot{\varphi}=\sum_{i}\frac{\partial
V}{\partial\phi^{i}}\frac{d\phi^{i}}{d\varphi}\equiv\frac{dV}{d\varphi},$ (79)
$\displaystyle\left(\frac{d^{2}\phi^{i}}{d\varphi^{2}}+\Gamma^{i}_{jk}\frac{d\phi^{j}}{d\varphi}\frac{d\phi^{k}}{d\varphi}\right)\dot{\varphi}^{2}=\left(g^{ik}-\frac{d\phi^{i}}{d\varphi}\frac{d\phi^{k}}{d\varphi}\right)\frac{dV}{d\phi^{k}},$
(80)

also, (75) can be rewritten as

$\Lambda=-\frac{1}{2}\dot{\varphi}^{2}+V,$ (81)

combining equations (79), and (81), we get

$\left(\frac{d\Lambda}{d\varphi}\right)^{2}=\alpha\Lambda\left(\Lambda-V\right),$
(82)

which is similar the equation we had for one direction. We again choose the
negative sign for $\partial_{\varphi}\Lambda$

$\frac{d\Lambda}{d\varphi}=-\sqrt{\alpha\Lambda\left(\Lambda-V\right)}.$ (83)

The proposed distance generalizes to

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\int\sqrt{g^{ij}(\phi^{k})\left(1-\frac{V}{\Lambda}\right)d\phi_{i}d\phi_{j}},$
(84) $\displaystyle=$
$\displaystyle\int\sqrt{\left(1-\frac{V}{\Lambda}\right)}d\varphi,$ (85)

incorporating equation (83), we get

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\alpha}}\text{Log}\left(\frac{\Lambda_{f}}{\Lambda_{i}}\right).$
(86)

we therefore recover the AdS distance conjecture in multi-field case.

To define $\Lambda(\phi)$ at each point in the field space, we begin at a base
point, $\phi^{i}_{0}$, with some initial $\Lambda(\phi^{i}_{0})=\Lambda_{0}$.
Subsequently, we can move to any arbitrary point in the field space and assign
the value of $\Lambda$ at that point as $\Lambda(\phi^{i})$. Note that this
definition of $\Lambda(\phi^{i})$ thus defined will depend on the choice of
both a base point and an initial value for $\Lambda$.

##  Appendix B ADM mass as a distance

In this Appendix, we develop the idea discussed in the main text of using the
ADM mass of an asymptotically flat or AdS configuration as a notion of
distance. Specifically, consider the case $d=4$, and again the problem of
defining a distance between two vacua with energies $V_{1}$ and $V_{2}$. To do
this, we will look at the ADM mass of a generic bubble of radius $R$ (meaning
it bounds a sphere of area $4\pi R^{2}$ interpolating between the two
geometries). On general grounds, we expect that, for general $R$, the ADM mass
will go as

$M\sim 4\pi\,TR^{2}-(\Delta V)\text{Vol}(R),$ (87)

where $\text{Vol}(R)$ is the volume of a sphere of radius $R$. From this we
can determine $T$, in the limit $R\rightarrow\infty$ this will define yet
another notion of distance that will reduce to the domain wall tension.

We will not solve the equations of motion, and look only at initial conditions
at a moment of time-reversal symmetry. In that case, we only need to solve the
Hamiltonian constraint from general relativity, which takes the form

$G^{00}=8\pi G\,T^{00}.$ (88)

Specifically, consider a transition from vacuum $V_{\text{in}}$ to
$V_{\text{out}}$ for a scalar field $\phi$. We will take a metric ansatz

$ds_{3}^{2}=\left(1-\frac{8\pi
GV(\phi)}{3}r^{2}-\frac{2Gm(r)}{r}\right)^{-1}dr^{2}+r^{2}d\Omega_{3}^{2}.$
(89)

The ADM mass of the configuration is simply $M(\infty)$ in the AdS/flat space
cases (and presumably, $M$ evaluated at the horizon at the de Sitter case).
With this metric ansatz, the Hamiltonian constraint takes the form (for
spherically symmetric configurations)

$\frac{2G\dot{m}}{r^{2}}+\frac{8}{3}\pi GrV^{\prime}\dot{\phi}+8\pi GV=8\pi
G\left(\frac{1}{2}\dot{\phi}^{2}+V\right),$ (90)

which simplifies to

$\frac{\dot{m}}{4\pi\,r^{2}}+\frac{r}{3}V^{\prime}\dot{\phi}=\frac{1}{2}\dot{\phi}^{2},$
(91)

Note that our ansatz allows us to get rid of the pure potential terms,
similarly to what happens in the classical Maupertuis action.

This is a first-order ODE for $\dot{m}$, which allows one to find an integral
expression for the $T$,

$TR^{2}=\int_{R}^{\infty}dr\,\dot{m}(r)=4\pi\int_{R}^{\infty}dr\,r^{2}\left[\frac{1}{2}\dot{\phi}^{2}-\frac{r}{3}V^{\prime}\dot{\phi}\right].$
(92)

Now that $m(r)$ has been eliminated from the problem, our task is simply to
find the on-shell value of the action $TR^{2}$ at an extremum, with the
boundary conditions that $\phi=\phi_{1}$ at $r=R$, and $\phi=\phi_{2}$ at
$R\rightarrow\infty$. We can integrate by parts

$\int_{R}^{\infty}dr\,r^{3}V^{\prime}\dot{\phi}=Vr^{3}|^{\infty}_{R}-3\int
r^{2}V,$ (93)

to obtain

$TR^{2}+\text{Vol}(R)\Delta
V=\int_{R}^{\infty}dr\,\dot{m}(r)=4\pi\int_{R}^{\infty}dr\,r^{2}\left[\frac{1}{2}\dot{\phi}^{2}+V\right]-(Vr^{3}|^{\infty}_{R}).$
(94)

In other words, the final expression we obtain is again the classical
expression for the energy (while substracting a diverging boundary term if the
asymptotic potential is nonzero). So we recover the non-gravitational problem.
But not exactly; we need to check whether horizons form, i.e. whether we ever
get that the $dr^{2}$ term in (89) changes sign. In other words, we need to
impose

$m(r)\leq\frac{r}{2G}-\frac{4\pi}{3}Vr^{3}.$ (95)

This is satisfied at $r=R$. Comparing derivatives, we get

$m^{\prime}(r)=4\pi
r^{2}\left[\frac{1}{2}\dot{\phi}^{2}-\frac{r}{3}V^{\prime}\dot{\phi}\right]\leq\frac{1}{2G}-\frac{4\pi}{3}V^{\prime}\dot{\phi}r^{3}-4\pi
r^{2}\,V,$ (96)

which rearranges to

$4\pi r^{2}\left[\frac{1}{2}\dot{\phi}^{2}+V\right]\leq\frac{1}{2G}.$ (97)

If this satisfied, horizon never forms. If it is not, one may fix it by going
to very large $R$, so that the rhs of (95) is initially large, and so there is
a lot of leeway to be satisfied. For fixed tension $T$, $TR^{2}$ is however
larger than $R/G$ for large enough $R$, so unfortunately a horizon always
forms in the limit we wish to take, unless perhaps in the AdS case. As
described in the main text, this presents a hurdle for this notion being
completely universal.

## References

  * (1) E. Palti, _The swampland: introduction and review_ , _Fortschritte der Physik_ 67 (2019) 1900037.
  * (2) M. van Beest, J. Calderón-Infante, D. Mirfendereski and I. Valenzuela, _Lectures on the swampland program in string compactifications_ , _Physics Reports_ 989 (2022) 1.
  * (3) N.B. Agmon, A. Bedroya, M.J. Kang and C. Vafa, _Lectures on the string landscape and the swampland_ , _arXiv preprint arXiv:2212.06187_ (2022) .
  * (4) H. Ooguri and C. Vafa, _On the geometry of the string landscape and the swampland_ , _Nuclear physics B_ 766 (2007) 21.
  * (5) F. Baume and J. Calderón Infante, _Tackling the SDC in AdS with CFTs_ , _JHEP_ 08 (2021) 057 [2011.03583].
  * (6) E. Perlmutter, L. Rastelli, C. Vafa and I. Valenzuela, _A CFT distance conjecture_ , _JHEP_ 10 (2021) 070 [2011.10040].
  * (7) F. Baume and J. Calderón-Infante, _On higher-spin points and infinite distances in conformal manifolds_ , _JHEP_ 12 (2023) 163 [2305.05693].
  * (8) H. Ooguri and Y. Wang, _Universal Bounds on CFT Distance Conjecture_ , 2405.00674.
  * (9) D. Lüst, E. Palti and C. Vafa, _Ads and the swampland_ , _Physics Letters B_ 797 (2019) 134867.
  * (10) A. Kehagias, D. Lüst and S. Lüst, _Swampland, Gradient Flow and Infinite Distance_ , _JHEP_ 04 (2020) 170 [1910.00453].
  * (11) D. De Biasio, J. Freigang, D. Lust and T. Wiseman, _Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes_ , _JHEP_ 03 (2023) 074 [2210.14705].
  * (12) D. De Biasio and D. Lust, _Geometric flow of bubbles_ , _Nucl. Phys. B_ 980 (2022) 115812 [2201.01679].
  * (13) D.M. Velázquez, D. De Biasio and D. Lust, _Cobordism, singularities and the Ricci flow conjecture_ , _JHEP_ 01 (2023) 126 [2209.10297].
  * (14) D. De Biasio and D. Lüst, _Geometric Flow Equations for Schwarzschild-AdS Space-Time and Hawking-Page Phase Transition_ , _Fortsch. Phys._ 68 (2020) 2000053 [2006.03076].
  * (15) D. De Biasio, _On-Shell Flow_ , 2211.04231.
  * (16) G. Shiu, F. Tonioni, V. Van Hemelryck and T. Van Riet, _Ads scale separation and the distance conjecture_ , _Journal of High Energy Physics_ 2023 (2023) 1.
  * (17) G. Shiu, F. Tonioni, V. Van Hemelryck and T. Van Riet, _Connecting flux vacua through scalar field excursions_ , _Physical Review D_ 109 (2024) 066017.
  * (18) Y. Li, E. Palti and N. Petri, _Towards ads distances in string theory_ , _Journal of High Energy Physics_ 2023 (2023) 1.
  * (19) I. Basile and C. Montella, _Domain walls and distances in discrete landscapes_ , _JHEP_ 02 (2024) 227 [2309.04519].
  * (20) E. Palti and N. Petri, _A positive metric over dgkt vacua_ , _Journal of High Energy Physics_ 2024 (2024) 1.
  * (21) M.R. Douglas, _Spaces of quantum field theories_ , in _Journal of Physics: Conference Series_ , vol. 462, p. 012011, IOP Publishing, 2013.
  * (22) C.P. Bachas, I. Brunner, M.R. Douglas and L. Rastelli, _Calabi’s diastasis as interface entropy_ , _Phys. Rev. D_ 90 (2014) 045004 [1311.2202].
  * (23) V. Balasubramanian, J.J. Heckman and A. Maloney, _Relative Entropy and Proximity of Quantum Field Theories_ , _JHEP_ 05 (2015) 104 [1410.6809].
  * (24) J. Stout, _Infinite Distances and Factorization_ , 2208.08444.
  * (25) J. McNamara and C. Vafa, _Cobordism classes and the swampland_ , _arXiv preprint arXiv:1909.10355_ (2019) .
  * (26) S. Coleman, _Fate of the false vacuum: Semiclassical theory_ , _Physical Review D_ 15 (1977) 2929.
  * (27) S. Coleman and F. De Luccia, _Gravitational effects on and of vacuum decay_ , _Physical Review D_ 21 (1980) 3305.
  * (28) R. Rajaraman, _Solitons and instantons. an introduction to solitons and instantons in quantum field theory_ , .
  * (29) H. Goldstein, C. Poole and J. Safko, _Classical mechanics_ , vol. 426, Addison-Wesley, Reading, MA (1980).
  * (30) J. Espinosa, _Thick wall from thin walls_ , _Physical Review D_ 108 (2023) 103503.
  * (31) J.R. Espinosa, _A Fresh Look at the Calculation of Tunneling Actions_ , _JCAP_ 07 (2018) 036 [1805.03680].
  * (32) J.R. Espinosa, _Fresh look at the calculation of tunneling actions including gravitational effects_ , _Phys. Rev. D_ 100 (2019) 104007 [1808.00420].
  * (33) M. Cariglia, A. Galajinsky, G. Gibbons and P. Horvathy, _Cosmological aspects of the eisenhart–duval lift_ , _The European Physical Journal C_ 78 (2018) 1.
  * (34) R.M. Wald, _General relativity_ , University of Chicago press (2010).
  * (35) B. Friedrich, A. Hebecker and J. Walcher, _Cobordism and bubbles of anything in the string landscape_ , _JHEP_ 02 (2024) 127 [2310.06021].
  * (36) B. Friedrich and A. Hebecker, _The Boundary Proposal_ , 2403.18892.
  * (37) A. Ceresole, G. Dall’Agata, A. Giryavets, R. Kallosh and A. Linde, _Domain walls, near-bps bubbles, and probabilities in the landscape_ , _Physical Review D_ 74 (2006) 086010.
  * (38) M. Cvetič, S. Griffies and S.-J. Rey, _Static domain walls in n= 1 supergravity_ , _Nuclear Physics B_ 381 (1992) 301.
  * (39) M. Cvetic, S. Griffies and S.-J. Rey, _Static domain walls in N=1 supergravity_ , _Nucl. Phys. B_ 381 (1992) 301 [hep-th/9201007].
  * (40) M. Cvetic, S. Griffies and H.H. Soleng, _Local and global gravitational aspects of domain wall space-times_ , _Phys. Rev. D_ 48 (1993) 2613 [gr-qc/9306005].
  * (41) M. Cvetic and H.H. Soleng, _Supergravity domain walls_ , _Phys. Rept._ 282 (1997) 159 [hep-th/9604090].