Brane Nucleation in Supersymmetric Models

Igor Bandos,111igor.bandos@ehu.eus,*,$\dagger$,$\ddagger$ Jose J. Blanco-
Pillado,222josejuan.blanco@ehu.eus,*,$\dagger$,$\ddagger$ Kepa
Sousa333kepa.sousa@uah.es,$\mathsection$
and Mikel A. Urkiola444mikel.alvarezu@ehu.eus,$\dagger$,$\mathparagraph$

*Department of Physics, University of the Basque Country, Bilbao, Spain
$\dagger$EHU Quantum Center, University of the Basque Country UPV/EHU, Bilbao,
Spain

$\ddagger$IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain

$\mathsection$University of Alcalá, Department of Physics and Mathematics,
28805 Alcalá de Henares (Madrid), Spain

$\mathparagraph$Department of Applied Mathematics, University of the Basque
Country UPV/EHU, Plaza Ingeniero Torres Quevedo 1, 48013 Bilbao, Spain

###### Abstract

This paper explores the process of vacuum decay in supersymmetric models
related to flux compactifications. In particular, we describe these
instabilities within supersymmetric Lagrangians for a single three-form
multiplet. This multiplet combines scalar fields, representing the moduli
fields in four dimensions, with 3-form fields that influence the potential for
these moduli via the integer flux of their associated 4-form field strength.
Furthermore, using supersymmetry as a guide we obtain the form of the
couplings of these fields to the membranes that act as sources to the 3-form
potentials. Adding small supersymmetry breaking terms to these Lagrangians one
can obtain instanton solutions describing the decay of the vacua in these
models by the formation of a membrane bubble. These instantons combine the
usual Coleman-de Luccia and the Brown-Teitelboim formalisms in a single
unified model. We study simple numerical examples of theories with and without
gravity in this new framework and generalize known Euclidean methods to
accomodate the simulataneous inclusion of scalar fields and charged membranes
to these instanton solutions. Moreover, we show explicitly in these examples
how one recovers the static supersymmetric solutions in the limiting case
where the supersymmetry breaking terms vanish. In this limit, the bubble
becomes infinite and flat and represents a hybrid between the usual
supersymmetric domain walls of field theory models and the brane solutions
interpolating between the supersymmetric vacua; a sort of dressed
supermembrane BPS solution. Finally, we briefly comment on the implications of
these solutions in cosmological models based on the String Theory Landscape
where these type of 4d effective theories could be relevant in inflationary
scenarios.

###### Contents

  1. 1 Introduction
  2. 2 Rigid supersymmetric scalar field theory models
    1. 2.1 Field theory supersymmetric domain walls
      1. 2.1.1 Example. Double well potential
    2. 2.2 Stability and Vacuum decay
  3. 3 Rigid supersymmetric models with 3-form potentials
    1. 3.1 Supersymmetric membranes coupled to 3-form potentials
    2. 3.2 Example with quadratic superpotential
    3. 3.3 Membrane nucleation
      1. 3.3.1 Numerical Example
  4. 4 Supergravity models
    1. 4.1 Flat membrane solutions in supergravity models
    2. 4.2 Example: quartic superpotential
  5. 5 Membrane nucleation in Supergravity
    1. 5.1 Example: quartic superpotential
      1. 5.1.1 AdS/Minkowski to AdS transitions
      2. 5.1.2 dS to AdS transitions
  6. 6 Conclusions
  7. A Three-form multiplets in supersymmetry
    1. A.1 Including 3-forms into the chiral superfields: special chiral superfields
  8. B Rigid supersymmetry, bosonic membranes, and BPS equations
  9. C Three-form multiplets in supergravity
    1. C.1 Supergravity interacting with 3-form multiplets
      1. C.1.1 Super-Weyl transformations

##  1 Introduction

Many high energy extensions of the Standard Model make use in some way or
another of supersymmetry. Furthermore, the presence of multiple vacua is quite
generic in these models. Some of these vacua preserve part of the original
supersymmetry while others break it completely. Understanding the stability of
these vacua and their possible decay processes is, therefore, an essential
aspect of the low energy description of these theories.

Some of the most interesting examples of this type of scenarios are the
effective four dimensional models obtained from string theory
compactifications. Compactifying $10d$ string theory to four dimensions leaves
us with a low energy effective theory with a collection of scalar fields (the
moduli fields) that parametrize the possible deformations of the internal
compact manifold. One can further impose that the compactification mechanism
preserves some supersymmetry so that we end up with a low energy theory that
can be classified as a supersymmetric scalar field theory.111More rigorously,
one should speak about supermultiplets involving scalar fields; in the case of
N=1 supersymmetry, these include not only the scalar supermultiplet but also
the so-called three form supermultiplets (see below).  This led the authors in
[1] to consider the non-perturbative stability of a N=1 model of supergravity.
They demonstrated that supersymmetric vacua are stable with respect to the
usual process of bubble nucleation. Furthermore, they also showed that this
stability is due to the restriction imposed by supersymmetry on the tension of
the wall that interpolates between the two vacua. In fact, this quenching
phenomenon is nothing more than the Coleman-deLuccia [2] suppression. In this
limit the bubble radius would be infinite and the solution would be a planar
domain wall that preserves part of the supersymmetry [3].

On the other hand, recent developments in models of string compactification
based on the use of fluxes along the internal dimensions has led us to the
idea of an extremely rich Landscape of possible $4d$ Effective Field Theories
(EFTs), also referred to as _flux vacua_ [4]. Each of these vacua is
characterized by the presence of a set of integer fluxes that thread some
cycles in the internal manifold. The stabilizing potential for the moduli in
each of these sectors of the theory is also quite complicated and could easily
have many local minima itself. One could therefore use the conclusions
discussed earlier for the scalar field potential in each of these sectors to
study their stability and find the bounce solutions using the techniques
derived by Coleman and collaborators [5, 6].

One can then ask whether there is an analogous process that would take us from
one sector with some set of fluxes to another one where one or several of
those fluxes have been changed. Naively this would seem impossible since the
flux is quantized and therefore cannot be continuously changed. However,
similarly to what happens in the Schwinger process [7], one could reduce the
flux by the creation of sources charged with respect to the same field that
produces it.222See [8] for a simple description of several field theory models
in different dimensions of spacetime that exhibit a similar behaviour to the
one described in this paper. In fact, this type of process had been already
discussed in a purely four dimensional context a number of years ago by Brown
and Teitelboim (BT) [9, 10]. In this model the presence of a 4-form field
strength in four dimensions induces an effective cosmological constant that
can only be changed by the nucleation of membranes charged with respect to its
3-form potential. The instanton solutions describing this type of instability
of the model were studied in detail in connection to the possible self-tuning
mechanism of the cosmological constant.

In models of string theory compactification the situation is quite similar,
and one can assume that each of the forms present in the 4d theory would have
an associated brane charged with respect to that specific form. Indeed one can
identify $4d$ membrane objects by wrapping higher dimensional branes along
some internal cycles. Similarly, one can also find the effective 4-form field
that couples to the 4-dimensional membrane and understand its higher
dimensional origin. Taking this point of view, these models of flux
compactification seem to lead to a model closely related to the Brown-
Teitelboim idea [4].

There is however an important difference between these two models. In the
original Brown-Teitelboim’s model the nucleation of the membranes would lead
to a jump in the value of the $4d$ cosmological constant. In models with
higher dimensions this change in the flux would lead us to a different sector
with a different moduli potential. This is interesting since it allows us to
avoid the so called "empty universe problem" in the purely $4d$ BT model by
postulating a period of inflation after the bubble nucleation driven by the
compactification potential (See for example [11]).

The previous arguments suggest the idea of combining both models into a
unifying picture where we can describe within the same theory the presence of
the scalar field moduli and the 4-form fields. Moreover, following what was
done before in the purely scalar field model, we will look for guidance in
supersymmetric models that include both types of degrees of freedom, the
4-form field as well as the moduli fields. The inclusion of form fields in
supersymmetric and supergravity multiplets has been done before in [12, 13,
14, 15, 16, 17, 18]. Furthermore, their interaction with supersymmetric
membranes (supermembranes) was studied in [14, 19, 20, 21] and their role in
the low energy description of flux compactifications has been recently
discussed in [18, 21, 22, 23]. In this paper we will study the non-
perturbative stability of these models and their modified versions once we
introduce small supersymmetry-breaking terms. More concretely, we will deform
the supersymmetric field-theoretic part of the action by including soft
supersymmetry breaking terms [24], while keeping the (super)membrane part of
the action untouched.

This paper is organized as follows. In section 2 we will review the results of
global supersymmetric scalar field theories (more concretely, self-interacting
scalar supermultiplet models which can be described in terms of a generic
chiral superfield), and the existence of supersymmetric domain wall solutions
in these models. We will show with explicit examples how these solutions
appear as limiting cases of a bubble decay process from a non-supersymmetric
vacua. In sec. 3 we will explain how to introduce 3-form gauge fields in our
supersymmetric theory, as well as membranes that naturally couple to these
forms. This will lead us to discuss the so-called 3-form supermultiplets
described by a special type of chiral superfields. Furthermore, we will obtain
the instanton solutions describing the decay processes in these theories, all
the while explicitly showing how they connect in a proper limit with the
supersymmetric solutions found in the previous section. Finally, in secs. 4
and 5 we will describe a similar situation in the case of supergravity and
find explicit solutions of the equations of motion describing the bubble
nucleation. We end with some conclusions in section 6.

##  2 Rigid supersymmetric scalar field theory models

As a warm up exercise, in this section, we will consider the stability of
vacua in a scalar field Lagrangian of an N=1 four dimensional globally
supersymmetric field theory of a single chiral superfield, which describes the
so-called scalar supermultiplet.333In the main part of this paper we will only
deal with the bosonic components of the supermultiplets. See, e.g., [25] for a
more complete treatment of the simplest cases.. Many of the results presented
in this section can be found in the literature, in particular in [1, 3]. In
the following sections we will discuss more complicated models following a
similar reasoning to the one presented here.

###  2.1 Field theory supersymmetric domain walls

The model we will be considering here is given by the most general Lagrangian
for a complex scalar field $(\phi)$ which is given by the leading component of
a generic chiral superfield (which describes the so-called scalar multiplet).
This Lagrangian reads

${\cal L}=-K_{\phi\bar{\phi}}\
\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-K^{\phi\bar{\phi}}|W_{\phi}|^{2}$
(2.1)

where we have introduced the two functions that define the model: the real
Kahler potential $K(\phi,\bar{\phi})$ and the complex holomorphic
superpotential $W(\phi)$. We will denote their derivatives as
$\partial_{\phi}\partial_{\bar{\phi}}K=K_{\phi\bar{\phi}}=1/K^{\phi\bar{\phi}}$
and $W_{\phi}=\partial_{\phi}W(\phi)$, thus following the conventions of
[25].444In this paper we will use the $(-+++)$ signature convention.

The equations of motion for this theory are

$K_{\phi\bar{\phi}}\partial_{\mu}\partial^{\mu}\phi-
K_{\phi\phi\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}+K_{\phi\bar{\phi}\bar{\phi}}(K^{\phi\bar{\phi}})^{2}|W_{\phi}|^{2}-K^{\phi\bar{\phi}}W_{\phi}\bar{W}_{\bar{\phi}\bar{\phi}}=0$
(2.2)

and its complex conjugate. These reduce to the usual Klein-Gordon equation for
a complex scalar field with a scalar potential given by
$V(\phi,\bar{\phi})=|W_{\phi}|^{2}$ if the kinetic term is canonical (which
particularly requires $K=\phi\bar{\phi}$).

We are looking for a domain wall solution in this model that interpolates
between two supersymmetric minima, in other words, between two points whose
superpotential satisfies $W_{\phi}(\phi_{\pm})=0$. Recall that all
supersymmetric minima have a vanishing potential, and therefore degenerate in
energy. For that matter, let us consider a flat domain wall whose transverse
direction is given by the coordinate $z$. One can then show [3] that the
static solution preserving half of the N=1 supersymmetry solves the first-
order equation

$\partial_{z}\phi(z)=e^{i\theta}K^{\phi\bar{\phi}}~{}\bar{W}_{\bar{\phi}}(\bar{\phi}(z)),$
(2.3)

known as the BPS equation, where the phase $\theta$ is given by

$e^{i\theta}=\frac{\Delta W}{|\Delta W|},\qquad\text{with}\qquad\Delta W\equiv
W(\phi(z=\infty))-W(\phi(z=-\infty)).$ (2.4)

Of course, given appropriate boundary conditions, both the first-order and
second-order equations should yield the same static solution for the
supersymmetric domain wall. The tension of the domain wall in this model can
be computed writing the energy per unit area as follows:

$\sigma=\left[\int_{-\infty}^{\infty}dz\
K_{\phi\bar{\phi}}~{}\left|\partial_{z}\phi(z)-e^{i\theta}K^{\phi\bar{\phi}}\bar{W}_{\bar{\phi}}(\phi(z))\right|^{2}\right]+2~{}\text{Re}[e^{-i\theta}\Delta
W]$ (2.5)

One can readily see that in the case of supersymmetric bosonic solutions,
where (2.3) holds, the tension becomes

$\sigma_{BPS}=2|\Delta W|.$ (2.6)

####  2.1.1 Example. Double well potential

Let us illustrate all of the above by considering the model defined by:

$K(\phi,\bar{\phi})=\phi\bar{\phi},\qquad
W(\phi)=\left(\frac{1}{3}\phi^{3}-a^{2}\phi\right)$ (2.7)

where we are assuming that $a>0$. For this particular model, the Lagrangian
(2.1) simplifies to

${\cal L}=-\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-|\phi^{2}-a^{2}|^{2}.$
(2.8)

The potential of the theory, restricted to the real part of $\phi$, has been
plotted in figure 1(a) (dashed line) where we see the double well potential
form with the two supersymmetric minima located at $\phi_{\pm}=\pm a$ . The
second-order equations of motion (2.2) read, in this case,

$\partial^{2}_{z}\phi(z)-2\bar{\phi}(\phi^{2}-a^{2})=0.$ (2.9)

On the other hand, the first-order BPS equation (2.3) reads

$\partial_{z}\phi(z)=-(\bar{\phi}^{2}-a^{2})~{},$ (2.10)

where we have chosen $W(\phi(z=\infty))<W(\phi(z=-\infty)$, which implies555On
the other hand, $e^{i\theta}=+1$ would yield the mirrored profile, often
denoted as the anti-domain wall. This simply corresponds to flipping the
boundary conditions imposed at $\pm\infty$. $e^{i\theta}=-1$. The solution of
this equation is given by the real field configuration,

$\phi(z)=a\tanh(az),$ (2.11)

while the tension of this domain wall is given by

$\sigma_{BPS}=2|\Delta W|=\frac{8}{3}a^{3}~{}.$ (2.12)

As we mentioned earlier, any solution to the BPS equation preserves some
supersymmetry by construction, so it is clear that it cannot represent the
decay of the vacuum. We can also see this noting that both vacua are
supersymmetric, degenerate in energy, and the wall is flat and infinite, so
there is no way these vacua can decay.

On the other hand, the solution we found here is purely real. This is
consistent with the potential we have constructed since its form is such that
perturbations around the solution in the imaginary field directions are
stabilized. We can check this by expanding the potential in the real and
imaginary parts of the field, namely

$\phi(z)=\psi(z)+i\ s(z)\ ,$ (2.13)

so the potential reads

$V(\psi,s)=(\psi^{2}-a^{2})^{2}+2(\psi^{2}+a^{2})s^{2}+s^{4}\ .$ (2.14)

This is why we can concentrate on the solution along the $s=0$ line. In all of
the examples we show here, we have checked that this is indeed the case;
therefore, in all of our illustrations we will simply draw the results
concerning the real part of the fields.

###  2.2 Stability and Vacuum decay

In the previous section we showed how one can find supersymmetric domain wall
configurations that interpolate between supersymmetric vacua in our model. The
fact that these vacua are stable is not surprising since supersymmetry imposes
them to be degenerate global minima of the potential. Let us now consider the
case where there is a small supersymmetry breaking term in our potential and
study the stability of the resultant vacua.

For simplicity let us assume that we introduce in the theory a couple of soft
supersymmetry breaking terms [24] of the form

$S_{\text{soft}}=-\int{d^{4}x\sqrt{-g}\left[\mu^{2}\phi\bar{\phi}+b\left(\phi^{3}+\bar{\phi}^{3}\right)\right]}~{},$
(2.15)

which break supersymmetry explicitly. In the following, we will consider the
coefficients to be small enough so that many of the properties of the solution
found earlier will still hold. This means we will consider the case where
$\mu^{2}\ll a^{2}$ as well as $b\ll a$, see the coloured curves of 1(a), where
for simplicity we have plotted some curves for the cases of $b=\mu$.

In this regime we can see that the theory still has two minima given by

$\phi_{\pm}=\pm a+\delta_{\pm}(\mu^{2},b)\ ,$ (2.16)

where the solutions have only shifted slightly, i.e. $|\delta_{\pm}/a|\ll 1$.
The interesting point now is that both of these minima break supersymmetry. If
one takes $b>0$, the potential at $\phi_{+}$ becomes slightly higher than the
other minimum; this means that this vacuum will be unstable with respect to
the nucleation of bubbles of the true vacuum at $\phi_{-}$. Furthermore, the
form of the supersymmetry-breaking terms allows for the tunneling to happen
along the real direction of the field, see fig. 1(a).

(a)  (b)

Figure 1: (a) Potential (2.19) with $a=1$, for several supersymmetry-breaking
parameters. (b) Solutions to (2.17), centered around the inflection point
corresponding to each profile, labeled as $\rho_{*}$. The BPS limit is shown
with a dashed line, representing the domain wall solution arising from the
dashed potential in (a).

In order to compute the probability of the decay and its bubble profile in
terms of the scalar field, we will resort to the usual methods developed by
Coleman and collaborators [5, 6] in the context of False Vacuum Decay. Thus,
we only have to extremize the Euclidean action associated to the Lagrangian
(2.1) completed by the supersymmetry-breaking terms. Assuming a standard
kinetic term ($K=\phi\bar{\phi}$) and an $O(4)$ symmetry in Euclidean space,
the equations of motion for the field are given by

$\phi^{\prime\prime}+\frac{3}{\rho}\phi^{\prime}=\frac{\partial
V}{\partial\bar{\phi}}$ (2.17)

where $\rho=\sqrt{\tau^{2}+\bm{x}^{2}}$, $\tau$ being the Euclidean time, and
primes denote derivatives with respect to $\rho$. This equation is to be
solved considering the boundary conditions

$\displaystyle\lim_{\rho\to\infty}\phi(\rho)=\phi_{+}\
,\qquad\phi^{\prime}(0)=0.$ (2.18)

Note that the Euclidean $O(4)$ symmetry will turn to $O(1,3)$ when
transporting the solution back to Lorentzian spacetime. Among other things,
this will mean that the profile $\phi(\rho)$ obtained via (2.17) will
correspond to the emergent scalar profile for the bubble at formation.
Furthermore, this symmetry implies that the radius of the bubble so formed
will expand outward following a constant acceleration trajectory.

Following the example presented in the previous subsection, we will work with
the following potential

$V(\phi,\bar{\phi})=(\phi^{2}-a^{2})(\bar{\phi}^{2}-a^{2})+\mu^{2}\phi\bar{\phi}+b(\phi^{3}+\bar{\phi}^{3})$
(2.19)

which already includes a contribution from supersymmetry-breaking terms within
its definition.

In this one-dimensional setup, the profile of the scalar field can be easily
found in Euclidean radial coordinates using an _undershoot/overshoot_
algorithm [5]. Essentially, since the boundary conditions (2.18) do not
specify the initial starting point $\phi(0)$, we can first obtain a couple of
points where the field either undershoots or overshoots the false vacuum at
$\phi_{+}$. Iteratively reducing this field range, we will eventually find a
starting point $\phi(0)$ which stays sufficiently close to the true vacuum at
$\phi_{-}$ for large values of $\rho$. The solutions found this way have been
contrasted with ones obtained using the software `AnyBubble` [26], which
applies an alternative multiple shooting method, and have been shown to be
essentially identical.

Using this numerical algorithm, we can check that indeed the form of the
domain wall forming the bubble does not change qualitatively in comparison to
the supersymmetric case given above. In order to visually make this comparison
we have plotted in figure 1(b) the resultant domain wall profiles for the
bubble nucleation centered on the same point. It is clear that dialling back
to zero the supersymmetry breaking coefficients $b,\mu$ one recovers the
profiles obtained with the first-order BPS equation (2.10), i.e., the
supersymmetric domain wall profile. Of course, this is consistent since in
this case the bubble radius is infinite and the tunnelling rate would be zero,
as this limit takes us back to the stable supersymmetric vacua presented in
the previous section.

##  3 Rigid supersymmetric models with 3-form potentials

As we described in the introduction, we are interested in studying
supersymmetric models with some new degrees of freedom beyond the complex
scalar field presented in the previous section. In particular, we would like
to introduce a 3-form potential into our model. Actually, using supersymmetry
as a guiding principle, we can describe both the scalar field and the 3-form
gauge field into a single multiplet described in terms of a special type of
chiral superfield. We show in Appendix A how one can construct such a special
chiral superfield describing the so-called single 3-form supermultiplet from
unconstrained real scalar superfields. In this case, the bosonic content of
the multiplet is simply given by complex scalar fields and auxiliary fields,
the real part of which will be related to a field strength of the 3-form
potential. Following this prescription we find that the simplest bulk action
for such fields is given by

$\displaystyle S_{\text{bulk}}$ $\displaystyle=\int
d^{4}x\left[-K_{\phi\bar{\phi}}~{}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-\frac{1}{4\cdot
4!}K_{\phi\bar{\phi}}~{}F^{\mu\nu\sigma\rho}F_{\mu\nu\sigma\rho}+\frac{1}{2\cdot
4!}\left(W_{\phi}+\bar{W}_{\bar{\phi}}\right)~{}\epsilon^{\mu\nu\sigma\rho}F_{\mu\nu\sigma\rho}\right.$
$\displaystyle\hskip
50.0pt\left.+\frac{1}{4}K^{\phi\bar{\phi}}\left(W_{\phi}-\bar{W}_{\bar{\phi}}\right)^{2}\right]\
,$ (3.1)

where $F_{\mu\nu\rho\sigma}=4\partial_{[\mu}A_{\nu\rho\sigma]}$ and as before
$K$ and $W$ denote the Kähler potential and superpotential which define the
model for the complex scalar field $\phi(x)$.666For the sake of simplicity,
will study models consisting of a single scalar field here. See [18] for a
study of similar systems with an arbitrary number of scalar fields .

As is usually the case in models involving gauge fields, this action must be
supplemented with some boundary terms that are required to make the variation
of the action well posed [10, 27]. In our present case, the boundary term is
given by

$S_{\text{bd}}=\frac{1}{2\cdot 3!}\int
d^{4}x\,\partial_{\mu}\left[A_{\nu\rho\sigma}\left(K_{\phi\bar{\phi}}F^{\mu\nu\rho\sigma}-\epsilon^{\mu\nu\rho\sigma}\left(W_{\phi}+\bar{W}_{\bar{\phi}}\right)\right)\right]\,,$
(3.2)

see appendix A for a derivation of this result.

The equation of motion for the 4-form field strength can be written as

$\partial_{\mu}\left[K_{\phi\bar{\phi}}F^{\mu\nu\rho\sigma}-\epsilon^{\mu\nu\rho\sigma}\left(W_{\phi}+\bar{W}_{\bar{\phi}}\right)\right]=0~{},$
(3.3)

which can be integrated to give,

$F^{\mu\nu\rho\sigma}=K^{\phi\bar{\phi}}\epsilon^{\mu\nu\rho\sigma}\left(W_{\phi}+\bar{W}_{\bar{\phi}}-2n\right)~{},$
(3.4)

where $n\in\mathbb{R}$ is a real integration constant. This expression allows
us to integrate out the 3-form potential from the original action to obtain a
new effective theory written in terms of the scalar field alone, namely,

$S=\int d^{4}x\left[-K_{\phi\bar{\phi}}\
\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-K^{\phi\bar{\phi}}\left(W_{\phi}-n\right)\left(\bar{W}_{\bar{\phi}}-n\right)\right]\;.$
(3.5)

It is important to stress that to arrive at this result the contribution
coming from the boundary term (3.2) on the surface of 3-form equations (3.4)
have to be taken into account.

Thus, quite interestingly, when the 3-form is set on-shell, the action reduces
to the usual theory for a complex scalar field, albeit with a potential
modified by a constant of motion that parametrizes the value of the 4-form
flux. As we mention in the introduction, we will assume that our $4d$ theory
is in fact obtained from a higher dimensional theory so we will consider these
fluxes to be quantized.

###  3.1 Supersymmetric membranes coupled to 3-form potentials

Incorporating a 3-form field in our model immediately suggests the presence of
sources of this field in the theory. For 3-form fields, these should be
fundamental membranes (2-branes), i.e., objects of codimension one in our 4d
setup. It is therefore clear that one should add new terms to the action in
order to describe the dynamics of these objects. In [10] the proposed terms
were

$S_{\text{memb
(BT)}}=-T_{m}\int_{\mathcal{M}}d^{3}\xi\sqrt{-h}~{}+~{}\frac{q}{3!}\int_{\mathcal{M}}d^{3}\xi
A_{\mu\nu\rho}\frac{\partial x^{\mu}}{\partial\xi^{a}}\frac{\partial
x^{\nu}}{\partial\xi^{b}}\frac{\partial
x^{\rho}}{\partial\xi^{c}}\epsilon^{abc}~{},$ (3.6)

where $\epsilon^{abc}$ denotes the fully antisymmetric unit tensor. The
physical origin of each these terms in the previous action is pretty clear.
The first one describes a Nambu-Goto-like contribution to the action, where
$T_{m}$ represents the tension of the membrane and $h$ is the determinant of
the induced metric on the brane whose worldvolume is parametrized by the
coordinates $\xi^{a}=(\xi^{0},\xi^{1},\xi^{2})$, that is, $h={\rm
det}\left(g_{\mu\nu}\frac{\partial x^{\mu}}{\partial\xi^{b}}\frac{\partial
x^{\nu}}{\partial\xi^{b}}\right)$. The second term shows that, as indicated
earlier, the branes are sources for the 3-form potential, in other words, they
are coupled (minimally or electrically) to $A_{\mu\nu\rho}$ and carry a charge
$q$ with respect to this field. In the case of supersymmetric membranes (i.e.,
supermembranes) this second contribution to the action (3.6) is also known as
the Wess-Zumino term.

However, our model is a little bit different from the one in [10] since we are
also interested in studying the dynamics of the complex scalar field and its
influence on the supermembrane. Therefore, one needs to consider the
possibility of having some new couplings of these branes to the scalar fields.
Here we will look for supersymmetry as a guiding principle and consider the
most generic supersymmetric action that describes our bulk model given by eq.
(3) coupled to a membrane. This problem was already addressed in [20] where
the authors concluded that the appropriate form of the brane terms in a global
supersymmetric action (after setting all fermions to zero) should be

$S_{\text{memb.}}=-\int_{\mathcal{M}}d^{3}\xi\sqrt{-h}~{}2|q\phi|~{}+~{}\frac{q}{3!}\int_{\mathcal{M}}d^{3}\xi
A_{\mu\nu\rho}\frac{\partial x^{\mu}}{\partial\xi^{a}}\frac{\partial
x^{\nu}}{\partial\xi^{b}}\frac{\partial
x^{\rho}}{\partial\xi^{c}}\epsilon^{abc}~{}.$ (3.7)

Note that the constant parameter in the previous action, $T_{m}$, has been
replaced by a scalar field dependent tension proportional to the modulus of
the 3-form charge $q$. This means that our branes will be coupled to the
scalar field as well. As we will see in the following, this will have
important consequences for the scalar field profile in the presence of these
branes. Moreover, since the tension of the brane is also linked to the value
of $q$, this is the only independent parameter of the membrane action.

In the following we will consider the bosonic part of the full action given by
the combination of all the terms given above, namely,

$S_{T}=S_{\text{bulk}}+S_{\text{bd}}+S_{\text{memb.}}~{}.$ (3.8)

The equation of motion for the 3-form field in the presence of these sources
gets now modified to be,

$\partial_{\mu}\left[K_{\phi\bar{\phi}}F^{\mu\nu\rho\sigma}-\epsilon^{\mu\nu\rho\sigma}\left(W_{\phi}+\bar{W}_{\bar{\phi}}\right)\right]=\,-2q\,\int
d^{3}\xi~{}\frac{\partial x^{\nu}}{\partial\xi^{a}}\frac{\partial
x^{\rho}}{\partial\xi^{b}}\frac{\partial
x^{\sigma}}{\partial\xi^{c}}\epsilon^{abc}\delta^{4}(x-x(\xi))~{}.$ (3.9)

Adding this source, one can show that the solution of the field strength will
change as we cross the membrane by a quantity that is proportional to the
charge, $q$. In other words, the solution is given by

$F^{\mu\nu\rho\sigma}=K^{\phi\bar{\phi}}\epsilon^{\mu\nu\rho\sigma}\left(W_{\phi}+\bar{W}_{\bar{\phi}}-2(n+qH(x))\right)~{},$
(3.10)

where the function $H(x)$ is defined implicitly as

$\partial_{\mu}H(x)=J_{\mu}=\frac{1}{3!}\epsilon_{\mu\nu\rho\sigma}\int
d^{3}\xi~{}\frac{\partial x^{\nu}}{\partial\xi^{a}}\frac{\partial
x^{\rho}}{\partial\xi^{b}}\frac{\partial
x^{\sigma}}{\partial\xi^{c}}\epsilon^{abc}\delta^{4}(x-x(\xi))\,.$ (3.11)

Thus, for a flat membrane perpendicular to the $z$ axes and placed at $z=0$
this function reduces to the usual Heaviside step function that changes its
value at the surface of the brane, namely,

$H(z)=\Theta(z)=\begin{cases}1,\quad z>0\;,\cr 0,\quad z<0\end{cases}.$ (3.12)

Using this solution we can rewrite the effective action in terms of the scalar
field alone to give

$S=\int
d^{4}x\sqrt{-g}\left[-K_{\phi\bar{\phi}}~{}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-K^{\phi\bar{\phi}}|\hat{W}_{\phi}|^{2}\right]-\int_{\mathcal{M}}d^{3}\xi\sqrt{-h}~{}2|q\phi|\,,$
(3.13)

where $\hat{W}(\phi)$ is the following effective superpotential:

$\hat{W}(\phi)\equiv W(\phi)-(n+qH(x))\phi.$ (3.14)

Just as in the case with no membrane source (see eq. (3.5)), the boundary term
contributes essentially to the effective action. Furthermore, in this case,
the Wess-Zumino term of the membrane action can be shown to be absorbed in the
above effective superpotential.

Let us now consider a flat brane located at $z=0$. The form of the effective
superpotential indicates that the membrane separates two regions of space
where the potential is different. However, one can still find a solution of
the second order equations that satisfies the first order BPS equations,
namely a solution of the form,

$\partial_{z}\phi(z)=e^{i\eta}K^{\phi\bar{\phi}}~{}\hat{\bar{W}}_{\bar{\phi}}(\bar{\phi}(z)),$
(3.15)

with the condition that

$e^{i\eta}=\frac{\Delta\hat{W}}{|\Delta\hat{W}|}=-\left.{\frac{q\phi}{|q\phi|}}\right|_{z=0}$
(3.16)

As it was shown in [21], these configurations preserve part of the
supersymmetry, see Appendix B for a discussion of this point. Furthermore,
equation (3.15) requires the derivative of the field to have a discontinuity
at the position of the brane given by,

$\left[\partial_{z}\phi(z)\right]_{|z=0}=-qe^{i\eta}K^{\phi\bar{\phi}}$ (3.17)

where we have introduced the notation for a jump of any quantity at some point
as $\left[A\right]_{|z=0}=A|_{z\rightarrow 0^{+}}-A|_{z\rightarrow 0^{-}}$.
One can easily check that this is indeed consistent with the second order
equations of motion obtained directly from the action given in eq. (3.13). In
the following we will present a simple example for such a membrane.

###  3.2 Example with quadratic superpotential

Let us consider the simplest model where we can study the type of solutions
discussed earlier. In particular, let us investigate the following Kähler
potential and superpotential777See [23] for a complete treatment of this
particular example.

$K(\phi,\bar{\phi})=\phi\bar{\phi},\qquad W(\phi)=\frac{1}{2}a\phi^{2},$
(3.18)

where $a$ is a constant with dimensions of energy which we will take for
simplicity to be $a=1$. Then, following the description of the previous
section, we integrate out the 3-form gauge field in the action (3.8). We thus
arrive at the effective action (3.13) where the effective superpotential is
given by

$\hat{W}(\phi)=\frac{1}{2}\phi^{2}-(n+q~{}\Theta(z))\phi\,$ (3.19)

where $q\in\mathbb{R}$ is the membrane charge and $n\in\mathbb{R}$ is the
constat 3-form flux. This means that the scalar field potential at each side
of the membrane is given by

$V_{-}(\phi)=|\phi-n|^{2},\qquad\qquad V_{+}(\phi)=|\phi-(n+q)|^{2}$ (3.20)

for $z<0$ and $z>0$, respectively. We show in figure 2(a) the potential around
their respective supersymmetric minima at $\phi_{-}=n$ and $\phi_{+}=n+q$. As
explained above, the perturbations of $\phi$ around its imaginary part are
completely stabilized, and thus we will only consider the physics of its real
part. This implies in our case that $e^{i\eta}=\pm 1$. We will take the
solution with $e^{i\eta}=-1$.

(a)  (b)

Figure 2: (a) Scalar potential, corresponding to eq. (3.20), with $n=1$ and
$q=2$. The darker line represents the potential for $z<0$, while the lighter
one is for $z>0$. (b) Scalar field profile interpolating between the minima of
the previous potential, with a membrane sitting at $z=0$, see eq. (3.22).

The BPS equation (3.16) can be now split into

$\left\\{\begin{array}[]{lr}\partial_{z}\phi(z)=-\bar{\phi}(z)+n&\quad
z<0,\\\\[5.0pt] \partial_{z}\phi(z)=-\bar{\phi}(z)+n+q&\quad
z>0.\end{array}\right.$ (3.21)

This system is easily solved by considering the field to be real, imposing
continuity of $\phi$ at $z=0$, and taking the field to asymptotically approach
the supersymmetric minimum of the potential at each side of the membrane. This
yields

$\phi(z)=n+\Theta(z)q\left(1-e^{-z}\right)$ (3.22)

which we have plotted in figure 2(b). Indeed, the profile interpolates between
the two different minima $\phi_{\pm}$ of their respective quadratic potentials
on both sides of the brane. Notice the jump in the first derivative of the
scalar field across the brane; as we will see below, this is entirely due to
the tension of the membrane involving the scalar field $\phi$.

One can also compute the tension of the membrane dressed with the scalar
field, by integrating the energy of the whole system across the membrane. In
the case at hand this becomes

$\sigma_{\text{DW+memb}}=2\left|\hat{W}_{z=\infty}-\hat{W}_{z=-\infty}\right|=2\left|\frac{1}{2}(n+q)^{2}-\frac{1}{2}n^{2}-2nq-q^{2}\right|=2nq+q^{2}$
(3.23)

where we have noted that the superpotential is actually different at each side
of the membrane. Note that this value is different from the one inferred from
the Nambu-Goto contribution of the membrane which is $T_{\text{NG}}=2nq$. If
we consider the fact that the original flux ($n$) should be quantized in units
of $q$ and assume the situation where $n\gg q$, we notice that the correction
to the tension is therefore small compared to the NG one.

###  3.3 Membrane nucleation

Now that we have studied the interplay of scalar fields and fluxes in the
presence of a static membrane, we are ready to take another step forward. In
the following, we will analyze how the Coleman-de Luccia [5, 2] and Brown-
Teitelboim [9, 10] schemes can be combined to yield a very interesting
perspective on membrane nucleation in the presence of scalar fields.

In order to have a potential with non-degenerate vacua, we will add soft
supersymmetry-breaking terms to the Lagrangian to get some shifted new minima
which will explicitly break supersymmetry. This will allow the false vacuum
solution (the highest of the two minima) to decay to the other vacuum by the
formation of a membrane bubble that interpolates between them. This membrane
bubble will have a structure locally similar to the flat membrane solution
found earlier and should approach the supersymmetric profile as the
supersymmetry-breaking terms are dialed down.

In order to study the tunnelling from a field configuration in the false
vacuum everywhere in space to a state with a spherical membrane coupled to the
field, we will apply Euclidean methods and assume $O(4)$ symmetry as in the
usual false vacuum decay process. In this case, the Euclidean action for our
problem becomes,

$\displaystyle S_{E}=2\pi^{2}\int d\rho\
\rho^{3}\left[K_{\phi\bar{\phi}}\left|\frac{d\phi}{d\rho}\right|^{2}+V(\phi,\bar{\phi})+2\left|q\phi\right|\delta(\rho-R)\right]~{},$
(3.24)

where $\rho$ denotes the radial coordinate in 4-dimensional Euclidean space,
and $V(\phi,\bar{\phi})$ is the potential for the scalar field which includes
the contribution from the supersymmetry-breaking terms in (2.15), namely

$\displaystyle
V=K^{\phi\bar{\phi}}\left|\hat{W}_{\phi}\right|^{2}+\mu^{2}\phi\bar{\phi}+b(\phi^{3}+\bar{\phi}^{3}),\qquad\hat{W}\equiv
W-(n+q\Theta(r-R))\phi~{}.$ (3.25)

Finally, the last term of the action describes the Euclidean contribution due
to the Nambu-Goto term of a spherical bubble of radius $R$.

The equation of motion for the complex scalar field is then

$\displaystyle\frac{1}{\rho^{3}}\partial_{\rho}\left(\rho^{3}K_{\phi\bar{\phi}}\frac{d\phi}{d\rho}\right)=K_{\phi\bar{\phi}\bar{\phi}}\left|\frac{d\phi}{d\rho}\right|^{2}+\frac{\partial
V}{\partial\bar{\phi}}-qe^{i\eta}\delta(\rho-R)$ (3.26)

where $e^{i\eta}=-\frac{q\phi}{|q\phi|}$. This equation is greatly simplified
if the scalar field has a canonical kinetic term, namely if
$K(\phi,\bar{\phi})=\phi\bar{\phi}$ so that $K_{\phi\bar{\phi}}=1$. In that
particular case, the equation of motion for the scalar field reads

$\displaystyle\frac{d^{2}\phi}{d\rho^{2}}+\frac{3}{\rho}\frac{d\phi}{d\rho}=\frac{\partial
V}{\partial\bar{\phi}}-qe^{i\eta}\delta(\rho-R)\ ,$ (3.27)

which should be supplemented with appropriate boundary conditions, namely the
ones described in eq. (2.18). This last equation is quite similar to the one
which is generally used for false vacuum decay, see eq. (2.17). The only
difference is that here we have derived it for a complex scalar field and that
it has a contribution proportional to a Dirac delta function, due to the
presence of a membrane of radius $R$ which appears to be charged with respect
to this scalar.

To find the instanton solution in this case we should proceed with a little
bit of care since the potential for the scalar field will be different on both
sides of the wall. Most notably, the true and false vacuum of the theory now
belong to two different potentials, as opposed to the usual case where they
are both local minima of the same function.

We should integrate this equation starting at the true vacuum state at the
center of the bubble at $\rho=0$ up to some distance $R$. At that point we
should change the potential and integrate the new equations of motion with the
new form of the potential $V_{+}(\phi)$ ending up in the false vacuum state.
There is, however, another important effect coming from the Nambu-Goto
contribution to the action and its dependence on the field $\phi$. As we can
see in the flat membrane case, the scalar field undergoes a jump on its first
derivative at the brane (see eq. (3.21)). In the case of the bubble profile we
should have the same situation. In fact, integrating (3.27) between
$R-\epsilon$ and $R+\epsilon$, where $R$ is the radius of the Euclidean
membrane, and making $\epsilon\to 0$, assuming $\phi$ is everywhere
continuous, we find

$\displaystyle\left[\partial_{\rho}\phi\right]_{|\rho=R}=-qe^{i\eta}$ (3.28)

across the membrane. Taking this jump in the first derivative we can now
integrate the profile for the field outside of the bubble all the way to
infinity.

This procedure for finding a solution of the Euclidean equation of motion with
spherical $O(4)$ symmetry has a free parameter: the radius of the membrane
bubble, $R$. There are actually several different methods one can use to fix
it in this non-gravitational case. Here we will use one that is easy to do
numerically and use some of the other methods to check for the validity of the
solution we find.

The method we use is based on the numerical evaluation of the Euclidean action
(3.24) at the solution as a function of this parameter, $R$. Indeed, one may
compute several profiles (and their corresponding Euclidean action) for
several values of $R$. Having done that, we only need to extremize this action
with respect to $R$. Recall that the Euclidean action describing the instanton
of bubble nucleation must be a maximum in terms of the radius of the bubble,
in the $R$ direction [5].

####  3.3.1 Numerical Example

Figure 3: Scalar potential (3.29) for $n=1$, $q=2$ and some values of the
supersymmetry-breaking parameters. The darker curve shows the potential inside
the membrane ($\rho<R$), while the lighter one represents the potential
outside it ($\rho>R$).

In order to visualize the procedure described above we have implemented the
numerical integration of the equation of motion in the model defined by
(3.18), albeit with the inclusion of soft supersymmetry-breaking terms. In
this case, the effective potential of the theory is

$V(\phi,\bar{\phi})=\left|\phi-\left(n+q\Theta(r-R)\right)\right|^{2}+\mu^{2}\phi\bar{\phi}+b(\phi^{3}+\bar{\phi}^{3})~{},$
(3.29)

which is shown in figure 3 along the real part of $\phi$. Once again, we will
not consider the imaginary part of the scalar field, since the system is
perturbatively stable along that direction. We are therefore interested in
finding the correct instanton that interpolates between the false vacuum at
higher energy density generated by the supersymmetry breaking terms and the
true vacuum.

As we explained in the previous section, the radius of the membrane
corresponding to the instanton solution is a priori unknown. Therefore, we
have computed the scalar field profiles for the bubbles with different values
of $R$. Using these profiles we can now calculate their action as a function
of $R$. We show the results of this procedure in fig. 4(a) for particular
values of the supersymmetry breaking parameters $b$ and $\mu$. We note in
these figures that the bounce action for each model reaches a maximum for a
particular value of the radius $R$. We take this to be the correct value of
the radius of the instanton that mediates the vacuum decay of interest to us.

(a)  (b)

Figure 4: (a) Euclidean action for solutions of the equation of motion (3.27),
with $n=1$, $q=2$ and several values for $R$, $b$ and $\mu$. (b) Energy and
Euclidean action difference with respect to the false vacuum state for several
radii, in the case where $\mu=b=0.03$. The maximum of $B$ coincides with the
root of $\Delta E$, as expected.

We also checked the variation in the total energy of the profile with respect
to the background at the time of nucleation. Indeed, there should be no energy
loss or gain for the instanton solution at the time of its emergence. Thus,
the correct bubble profile should correspond to the roots of the energy
density difference with respect to the false vacuum background. This quantity
is defined by

$\displaystyle\Delta E=4\pi\int_{0}^{\infty}dr\
r^{2}\left[K_{\phi\bar{\phi}}\left|\frac{d\phi}{dr}\right|^{2}+V(\phi,\bar{\phi})+2\left|q\phi\right|\delta(r-R)-V_{\text{fv}}\right]_{t=0}\,.$
(3.30)

This last expression can be easily evaluated from the Euclidean solution,
since the profile obtained in Euclidean space corresponds to the profile of
the emerging bubble in Lorentzian space at $t=0$. In all of the profiles that
we have computed, the value of R corresponding to no energy loss or gain with
respect to the background has been found to correspond with the maximum of the
Euclidean action, see figure 4(b) for an explicit example.

The profiles corresponding to the solutions which extremize $S_{E}$ have been
plotted in figure 5(a). As expected, as the supersymmetry-breaking parameters
are made smaller, the radius of the emerging membrane increases and the scalar
field profiles progressively tend towards the BPS solution derived above, as
shown in fig. 5(b).

A curious feature of these profiles is that, when considered as a particle in
the inverted potential $-V$, they first tend to get away from the false
vacuum, only to then be projected in the $r>R$ potential with enough velocity
to asymptotically reach the false vacuum.

(a)  (b)

Figure 5: (a) Scalar field profiles corresponding to each maximum Euclidean
action for some supersymmetry-breaking parameters. (b) The same profiles as
before, centered around the membrane radius corresponding to each profile,
denoted by $\rho_{\ast}$. The dashed line represents the BPS solution from eq.
(3.22), with $n=1$ and $q=2$, which is clearly the asymptotic behaviour of the
profiles as the supersymmetry-breaking parameters $b$ and $\mu$ tend to 0.

##  4 Supergravity models

We now turn to study the same setup as in the previous section, with gravity
taken into consideration. We will work in the context of N=1, $D=4$
Supergravity coupled to chiral matter. We will be interested in generalizing
the false vacuum decay discussed in the previous section including gravity.
However, before analysing more generic situations, we will first study the
supersymmetric limit of flat membrane solutions in supergravity. Thus, in this
section, we will start by analyzing the action of the system composed by
scalar fields, real three-forms and flat membranes in a spacetime of
Lorentzian signature.

The action we will be considering is once again given by the sum of the
following terms,

$S=S_{\text{bulk}}+S_{\text{membrane}}+S_{\text{boundary terms}}.$ (4.1)

As we review in appendix A, the bosonic part of the bulk action of the system,
which includes gravity, scalar fields and 3-forms, is given by the following
supergravity action

$\displaystyle S_{\text{bulk}}=\int d^{4}x\sqrt{-g}$
$\displaystyle\left[\frac{1}{2}\,\mathcal{R}-K_{\phi\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-\frac{1}{3}e^{-K}(M+K_{\bar{\phi}}\bar{\mathcal{F}})\,(\bar{M}+K_{\phi}\mathcal{F})-M\bar{W}-\bar{M}W\right.$
$\displaystyle+\left.e^{-K}K_{\phi\bar{\phi}}\mathcal{F}\bar{\mathcal{F}}+\mathcal{F}W_{\phi}+\bar{\mathcal{F}}\bar{W}_{\bar{\phi}}\right]$
(4.2)

where, as before, $K(\phi,\bar{\phi})$ and $W(\phi)$ describe the real Kähler
potential and the holomorphic superpotential functions of the scalar field and

$\displaystyle\mathcal{F}=\frac{1}{2}\left(D_{\mu}A^{\mu}+i{\rm
d}\right)+\frac{2}{3}\bar{\phi}{}M+\frac{1}{3}\phi\bar{M}.$ (4.3)

In the above equations we use $M_{\text{Pl}}=1$ and denote the Ricci scalar by
$\mathcal{R}$. Furthermore, $M$ is a complex scalar auxiliary field of the
minimal supergravity multiplet, d is a real scalar auxiliary field and
$A^{\mu}$ is the Hodge dual of the three-form. In our setting this latter
field belongs to the matter supermultiplet, which also includes the scalar
field888We note that a 3-form fields may also be included as an auxiliary
field of the supergravity supermultiplet [28, 29, 30, 31, 14, 32, 33, 34].
Even though this provides a more straightforward supersymmetric generalization
of the Brown-Teitelboim construction [9, 10], we will not consider it here
since the advantage of inclusion of the 3-form in the matter supermultiplet is
that this construction has a smooth and nontrivial flat space limit. .

Just as in the non-gravitational case, boundary terms must be included in the
full action to ensure the variation of the action with respect to the form
field is well posed. We leave their derivation for appendix C. Note that these
boundary terms will also include the Gibbons-Hawking term [35] in order for
the variational problem to be well defined also in the gravitational sector.

The action of the membrane can be fixed by requiring the interacting system
(4.1), with fermionic terms taken into account in the first two terms, to
remain invariant under a half of local supersymmetry after the bosonic
membrane is included [36, 37]. It reads

$\displaystyle S_{\text{memb.}}$
$\displaystyle=-\int_{\mathcal{M}}{d^{3}\xi\sqrt{-h}~{}2e^{K/2}|q\phi|}~{}+~{}\frac{q}{3!}\int_{\mathcal{M}}{d^{3}\xi
A_{\mu\nu\rho}\frac{\partial x^{\mu}}{\partial\xi^{a}}\frac{\partial
x^{\nu}}{\partial\xi^{b}}\frac{\partial
x^{\rho}}{\partial\xi^{c}}\epsilon^{abc}}$ (4.4)

which as before describes the coupling between the membrane and the 3-form
potential as well as the scalar field. Note however that in supergravity the
Nambu-Goto term receives a correction from the exponential of the Kähler
potential.999The exponential factor $e^{K/2}$ arises due to the super-Weyl
rescaling and field redefinitions required to bring the action to Einstein
frame, see appendix C for more detail.

It is easy to check that setting the form field and auxiliary fields on-shell
and taking into account the contribution of the boundary terms, the action
reads [21]

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{\mathcal{R}}{2}-K_{\phi\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-V(\phi,\bar{\phi})\right]+S_{\text{GH}}-\int_{\mathcal{M}}d^{3}\xi\sqrt{-h}\,2e^{K/2}|q\phi|$
(4.5)

where $S_{\text{GH}}$ represents the Gibbons-Hawking boundary term and
$V(\phi,\bar{\phi})$ is the N=1, $D=4$ matter-coupled supergravity scalar
potential of the form

$\displaystyle
V(\phi,\bar{\phi})=e^{K}\left[D_{\phi}\hat{W}K^{\phi\bar{\phi}}D_{\bar{\phi}}\hat{\bar{W}}-3|\hat{W}|^{2}\right]$
(4.6)

with the usual Kähler-covariant derivative denoted by
$D_{\phi}=\partial_{\phi}+K_{\phi}$. As in the global supersymmetric models
the information about the 3-form flux is encapsulated in the form of the
effective superpotential:

$\displaystyle\hat{W}=W-(n+qH(x))\phi~{},$ (4.7)

where, just as in the non-gravitational case, $H(x)$ is given by eq. (3.11).
This modified superpotential will allow us to find a profile for the dressed
membrane which interpolates between supersymmetric minima of _different_
potentials.

###  4.1 Flat membrane solutions in supergravity models

Similarly to what we did in the flat space case, one can find smooth domain
wall solutions in the supergravity context. This has been extensively studied
in the literature starting in [3] (see also a more recent discussion about
this topic in [38] and references therein). Here we show how one can
generalize these solutions to include the presence of a supermembrane as it
was described in [21].

Let us begin with the case of a flat membrane interpolating between two
supersymmetric vacua. Let us assume this static membrane sits at $z=0$. In
order to study the profile across such a membrane, we will assume the
following ansatz for the metric101010Note that with this choice of metric and
in the so-called static gauge $x^{a}(\xi)=\xi^{a}$, for flat membrane,
$z(\xi)=0$, we will have $\sqrt{-h}=\sqrt{-g}$, where the r.h.s. is calculated
at $z=0$. [38, 21]:

$\displaystyle ds^{2}=e^{2D(z)}(-dt^{2}+dx^{2}+dy^{2})+dz^{2}$ (4.8)

so that $\sqrt{-g}=e^{3D(z)}$. Let us turn to study the equation of motion for
the scalar field, which is

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}K_{\phi\bar{\phi}}g^{\mu\nu}\partial_{\nu}\phi\right)=K_{\phi\bar{\phi}\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}+\frac{\partial
V}{\partial\bar{\phi}}+e^{K/2}\left[K_{\bar{\phi}}|q\phi|-qe^{i\eta}\right]\delta(z),$
(4.9)

where $e^{i\eta}$ is defined as

$\displaystyle e^{i\eta}=-\left.{\frac{q\phi}{|q\phi|}}\right|_{z=0}.$ (4.10)

It is natural to assume that scalar field only depends on the transverse
coordinate to the membrane, i.e., $\phi=\phi(z)$. In that particular case, the
field obeys

$\displaystyle\partial_{z}(K_{\phi\bar{\phi}}\partial_{z}\phi)+3K_{\phi\bar{\phi}}\
\partial_{z}D\
\partial_{z}\phi=K_{\phi\bar{\phi}\bar{\phi}}\left|\partial_{z}\phi\right|^{2}+\frac{\partial
V}{\partial\bar{\phi}}+e^{K/2}\left[K_{\bar{\phi}}\left|q\phi\right|-qe^{i\eta}\right]\delta(z).$
(4.11)

On the other hand, the Einstein equations for the metric can be combined to
give

$\displaystyle\partial_{z}^{2}D+3(\partial_{z}D)^{2}$
$\displaystyle=-V-e^{K/2}\left|q\phi\right|\delta(z)$ (4.12)

Note that the deltas at $z=0$ will yield jumps in the first derivative of both
$\phi$, as in the non-gravitational case, and in the scale factor $D$.

A supersymmetric and static domain wall may interpolate between non-degenerate
minima, since essentially the gravitational contribution may compensate the
difference in scalar potential between both vacua [3]. If supersymmetry is
partly conserved across the profile of the domain wall, then the minima are
bound to be either Minkowski or AdS vacua (note, however, that no
supersymmetric domain wall may interpolate between two Minkowski vacua when
gravity is included [3]).

With these remarks at hand, the BPS equations acquire the form [3, 38, 21]

$\displaystyle\phi^{\prime}(z)$ $\displaystyle=\mp
e^{K/2}e^{i\arg(\hat{W})}K^{\bar{\phi}\phi}D_{\bar{\phi}}\hat{\overline{W}}$
(4.13) $\displaystyle D^{\prime}(z)$ $\displaystyle=\pm e^{K/2}|\hat{W}|$
(4.14)

where primes denote derivatives with respect to $z$, The second order
equations are obeyed by the solutions of the first-order BPS equations when
suitable boundary conditions are imposed. In particular, taking the derivative
of eq. (4.14) and using (4.13), we find that the second order equation (4.12)
is satisfied provided111111Actually this identification can be obtained
straightforwardly as it implies that on the worldvolume of the membrane the
supersymmetry preserved by the solution coincides with $\kappa$–symmetry of
the supermembrane action [39].

$\left.e^{i\arg(\hat{W})}\right|_{z=0}=\mp e^{i\eta}\;.$ (4.15)

It is convenient, following [38], to write these equations in terms of

$\displaystyle\mathcal{Z}\equiv e^{K/2}\hat{W}.$ (4.16)

Note that the value of the scalar potential at supersymmetric critical points
is then given by $V_{\text{susy}}=-3|\mathcal{Z}|^{2}$. In terms of
$\mathcal{Z}$, the BPS equations simplify to

$\displaystyle\phi^{\prime}(z)$ $\displaystyle=\mp
2K^{\bar{\phi}\phi}\partial_{\bar{\phi}}|\mathcal{Z}|$ (4.17) $\displaystyle
D^{\prime}(z)$ $\displaystyle=\pm|\mathcal{Z}|\;.$ (4.18)

The sign to use in the integration of the BPS equations is determined by the
value of $|\mathcal{Z}|$ at $z\to\pm\infty$. In order to see this, let us
firstly note that

$\displaystyle\frac{d|{\mathcal{Z}}|}{dz}=(\phi^{\prime}\partial_{\phi}|{\mathcal{Z}}|+\bar{\phi}{}^{\prime}\partial_{\bar{\phi}}|{\mathcal{Z}}|)\mp
e^{K/2}|q\phi|\delta(z)=\mp\left[4K^{\phi\bar{\phi}}\partial_{\phi}|{\mathcal{Z}}|\partial_{\bar{\phi}}|{\mathcal{Z}}|+e^{K/2}|q\phi|\delta(z)\right]$
(4.19)

where, in the second step, we have used (4.17). Then it is easy to observe
that, if for example we choose the lower sign in eqs. (4.17) and (4.18), which
then become

$\displaystyle\phi^{\prime}(z)$
$\displaystyle=2K^{\bar{\phi}\phi}\partial_{\bar{\phi}}|\mathcal{Z}|$ (4.20)
$\displaystyle D^{\prime}(z)$ $\displaystyle=-|\mathcal{Z}|\;,$ (4.21)

we find that the derivative of $|{\mathcal{Z}}|$ is positive

$\displaystyle\frac{d|{\mathcal{Z}}|}{dz}=4K^{\phi\bar{\phi}}\partial_{\phi}|{\mathcal{Z}}|\partial_{\bar{\phi}}|{\mathcal{Z}}|+e^{K/2}|q\phi|\delta(z)>0.$
(4.22)

Therefore, $|{\mathcal{Z}}|$ must increase monotonically, which implies
$|\mathcal{Z}|_{-\infty}<|\mathcal{Z}|_{+\infty}$. On the other hand, if
$|\mathcal{Z}|_{-\infty}>|\mathcal{Z}|_{+\infty}$, the lower sign of the BPS
equations eqs. (4.17)-(4.18) will apply.

Finally, it should be noted that in case $|\mathcal{Z}|$ has some root along
$z$, the situation becomes a bit more complicated. Indeed, the signs must be
swapped after crossing a root of the superpotential in order to have a
positive overall tension of the domain wall [38, 21]. However, as noted in
[3], such cases do not correspond to the limiting case of a bubble
instability. In fact the spacetime induced by these solutions is
asymptotically quite different and resembles the one in the Randall-Sundrum
scenario [40] with a single positive tension brane121212We will defer the
exploration of these types of solutions in our context for a future
publication..

###  4.2 Example: quartic superpotential

Figure 6: Scalar potential for the model described in this section, for two
different values of the flux value $n$.

(a)  (b)

Figure 7: Solution of the BPS equations to the model defined in (4.23), for
(a) the scalar field (b) the scale factor, for the metric defined in (4.8).
The potential at $z<0$ corresponds to the case $n=3$, while $z>0$ corresponds
to $n=2$; thus, the membrane is charged with $q=-1$.

The above equations can be used to find the profiles of a scalar field
interpolating between different minima of the model defined by

$\displaystyle K(\phi,\bar{\phi})=\phi\bar{\phi},\qquad
W(\phi)=(10M_{\text{Pl}})^{-1}\phi^{4}.$ (4.23)

where we have restored the Planck mass momentarily in order to show explicitly
the energy scales involved in this example. The scalar potential defined by
this Kähler potential and superpotential is shown in figure 6, once the 3-form
has been integrated out. The minimum featured by each branch can be shown to
be supersymmetric, i.e., it satisfies $D_{\phi}\hat{W}=0$.

Going back to units where $M_{\text{Pl}}=1$, the numerical BPS profile arising
from this potential for both the scalar field and the scale factor D is shown
in figure 7, where we have placed the lower minimum to the left (for easier
comparison later on). We have also checked that the second-order equations
(4.11) and (4.12), which explicitly incorporate the first-derivative jumps as
Dirac deltas, yield exactly the same profiles. Note that for this particular
solution the supersymmetric minima on both sides of the wall describe an anti-
deSitter vacua with different values of their cosmological constant.

##  5 Membrane nucleation in Supergravity

As we discussed in the introduction membrane nucleation in a model with a
4-form flux has been studied in the literature in the context of models
similar to Brown-Teitelboim [9, 10]. Our supergravity model includes a 3-form
field and a brane charged with respect to it, however, as we have shown in the
previous section it can also be casted exclusively in terms of a scalar field
by integrating out the 3-form potential. We can then ask whether there will be
bounce solutions similar to the flat space ones where the field interpolates
between two minima of the flux-dependent potential for the scalar field as one
crosses the wall.

In order to illustrate these vacuum decay processes in our model, we will
follow a similar procedure to the one we presented in the flat space limit in
section (3.3) where we introduced small corrections to the supersymmetric
Lagrangian. These terms will allow the possibility of having supersymmetry
breaking vacua that could be susceptible to decay. In particular, we will
study the Euclidean action given by131313We have omitted the supersymmetry-
breaking cubic term for simplicity, as its effect (in this model, at least)
was identical to turning on the quadratic term.

$\displaystyle S_{E}=\int
d^{4}x\sqrt{g}\left[-\frac{R}{2}+K_{\phi\bar{\phi}}g^{ab}\partial_{a}\phi\partial_{b}\bar{\phi}+\tilde{V}(\phi,\bar{\phi})\right]+2\int_{\mathcal{M}}d^{3}\xi\sqrt{h}e^{K/2}|q\phi|+S_{\text{GH}}$
(5.1)

where,

$\displaystyle\tilde{V}(\phi,\bar{\phi})=e^{K}\left(K^{\phi\bar{\phi}}\left|D_{\phi}\hat{W}\right|^{2}-3\left|\hat{W}\right|^{2}\right)+\mu^{2}\phi\bar{\phi},\qquad\hat{W}\equiv
W-(n+qH(x))\phi.$ (5.2)

As we will see afterwards, the Gibbons-Hawking boundary term will become
important once we evaluate the actual value of the Euclidean action.

Furthermore, assuming an $O(4)$ symmetric Euclidean solution for the
instanton, we introduce the following ansatz for the metric

$\displaystyle ds^{2}=d\chi^{2}+\rho(\chi)^{2}d\Omega_{3}^{2}~{}.$ (5.3)

Using these coordinates we search for a solution featuring a spherical
membrane sitting at a fixed value of the radial coordinate, which we call
$\chi=R$. With this setting, one arrives to the following equations of motion
for the metric function and the scalar field,

$\displaystyle\partial_{\chi}\left(K_{\phi\bar{\phi}}\partial_{\chi}\phi\right)+\frac{3\
\partial_{\chi}\rho}{\rho}K_{\phi\bar{\phi}}\partial_{\chi}\phi$
$\displaystyle=K_{\phi\bar{\phi}\bar{\phi}}\left|\partial_{\chi}\phi\right|^{2}+\frac{\partial\tilde{V}}{\partial\bar{\phi}}+e^{K/2}\left[K_{\bar{\phi}}|q\phi|-qe^{i\eta}\right]\delta(\chi-R)$
(5.4) $\displaystyle\rho^{\prime\prime}$
$\displaystyle=-\frac{1}{3}\rho\left(2K_{\phi\bar{\phi}}|\phi^{\prime}|^{2}+\tilde{V}\right)-\rho
e^{K/2}|q\phi|\delta(\chi-R),$ (5.5)

which in the case of a canonical kinetic term for the field $\phi$, that is,
$K(\phi,\bar{\phi})=\phi\bar{\phi}$, reduce to

$\displaystyle\phi^{\prime\prime}+\frac{3\rho^{\prime}}{\rho}\phi^{\prime}$
$\displaystyle=\frac{\partial\tilde{V}}{\partial\bar{\phi}}+e^{|\phi|^{2}/2}\left[\phi|q\phi|-qe^{i\eta}\right]\delta(\chi-R)$
(5.6) $\displaystyle\rho^{\prime\prime}$
$\displaystyle=-\frac{1}{3}\rho\left(2|\phi^{\prime}|^{2}+\tilde{V}\right)-\rho
e^{|\phi|^{2}/2}|q\phi|\delta(\chi-R),$ (5.7)

which are very similar to the set of equations to be solved in the case of a
purely scalar field bounce solution in curved space first found by Coleman and
de Luccia in [2] with the important difference that now both first derivatives
of the functions $\rho$ and $\phi$ present a jump at the position of the
membrane. This is to be expected since locally the behaviour of the functions
should be the same as the flat membrane case in these models.

Finally, note that the radius of the membrane, $R$, is still a free parameter
of our system. Therefore, in order to fix it, we can take a similar approach
as the one we took in the non-gravitational case. Namely, we can try solving
the equations for several choices of $R$ and find which one extremizes the
Euclidean action (5.1).

###  5.1 Example: quartic superpotential

In this subsection, we will apply the machinery described above to the simple
model of eq. (4.23). Taking into account the different values of the
supersymmetry breaking parameter $\mu$ one can encounter several situations
depending on the nature of the false vacuum. In particular, the geometry of
the Euclidean space will depend on the sign of the potential at the false
vacuum, i.e., while for $V_{\text{fv}}\leq 0$, the background will be a non-
compact space, the case with $V_{\text{fv}}>0$ will yield a compact de Sitter
space for the background. This different properties of the background will
become important for the way we handle our numerical solutions.

All the examples below were solved numerically using a simple overshoot-
undershoot algorithm to find the correct initial condition for the scalar
field. Note that, as far as the boundary conditions are concerned, requiring
the spacetime to be regular at the center of the bubble implies that
$\rho(\chi)=\chi+\mathcal{O}(\chi^{3})$ for $\chi\to 0$, see [2] for further
details.

####  5.1.1 AdS/Minkowski to AdS transitions

As we mentioned above, this tunnelling event occurs within a non-compact
space. Therefore, the integral of the Euclidean action corresponding to the
instanton with a membrane fixed at a particular value of the coordinate radius
$R$ is given by,

$\displaystyle S_{E}$
$\displaystyle=2\pi^{2}\int_{0}^{\infty}d\chi\left[\rho^{3}\left(|\phi^{\prime}|^{2}+\tilde{V}(\phi,\bar{\phi})\right)+3\left(\rho^{2}\rho^{\prime\prime}+\rho(\rho^{\prime})^{2}-\rho\right)\right]+4\pi^{2}\left[\rho^{3}e^{|\phi|^{2}/2}|q\phi|\right]_{\chi=R}+S_{\text{GH}}$
$\displaystyle=2\pi^{2}\int_{0}^{\infty}d\chi\left[\rho^{3}\left(|\phi^{\prime}|^{2}+\tilde{V}(\phi,\bar{\phi})\right)-3\left(\rho(\rho^{\prime})^{2}+\rho\right)\right]+4\pi^{2}\left[\rho^{3}e^{|\phi|^{2}/2}|q\phi|\right]_{\chi=R}$
(5.8)

where, in the last step, the term we have integrated out cancels the
contribution from the GH term (see [41, 42]).

On the other hand, the integral corresponding to the background is given by,

$\displaystyle
S_{E,\text{bg}}=2\pi^{2}\int_{0}^{\infty}d\chi\left[\rho_{\text{fv}}^{3}V_{\text{fv}}-3\left(\rho_{\text{fv}}(\rho_{\text{fv}}^{\prime})^{2}+\rho_{\text{fv}}\right)\right],$
(5.9)

where

$\displaystyle\rho_{\text{fv}}(\chi)=H^{-1}\sinh\left(H\chi\right)$ (5.10)

is the expression for the scale factor of Euclidean anti-deSitter space in the
particular slicing given by Eq. (5.3) and where we have introduced the Hubble
parameter $H=\sqrt{\frac{|V_{\text{fv}}|}{3}}$. We can take the Minkowski
limit of this expression to find that in the case of $V_{\text{fv}}=0$, the
scale factor becomes, $\rho_{\text{fv}}(\chi)=\chi$.

It is easy to see that these background Euclidean actions as well as the ones
obtained from the instanton solutions are, in fact, divergent. However, the
physically relevant quantity is the difference between the instanton action
and the background one. This action is finite.

The key to computing this difference correctly lies in performing the
integrations up to a certain $\rho_{\text{max}}$, such that its corresponding
radial coordinate $\chi_{\text{max}}$ satisfies $\chi_{\text{max}}\gg R$. See
[42] for more detail and an explicit proof of the convergence of this
difference.

(a)  (b)

Figure 8: (a) Scalar potential for $n=2$ (dashed) and $n=3$ (solid). (b)
Euclidean action for different fixed membrane radii $R$ for a list of
supersymmetry breaking parameters $\mu$. The color corresponding to each $\mu$
is the same for both figures. Note that the radius $R$ which extremizes the
action increases as we diminish the supersymmetry-breaking parameter $\mu$.

In figure 8 we show the result of computing this difference for several
membrane radii and supersymmetry-breaking parameter values. We can clearly see
that the difference between actions is finite and reaches a maximum at a
certain R, depending on $\mu$. Furthermore, as the potential tends towards its
original and supersymmetric form, the radius of the membrane interpolating
between both branches of the scalar potential gets bigger, which is consistent
with the fact that at the supersymmetric limit the tunneling transitions
becomes completely suppressed.

The profiles for both the scalar field and the scale factor corresponding to
the radii with maximum Euclidean action difference for each $\mu$ are shown in
figure 9. The scalar field profiles are all quite similar in shape, with the
only differences resting on the positions of the true and false vacuum, and in
the radius where the jump happens. On the other hand, the scale factor shows
very clearly where the jump happens as well, and the exponential behaviour
seems to pick up quite fast once the membrane has been crossed.

(a)  (b)

Figure 9: Evolution of (a) scalar field and (b) scale factor, for different
supersymmetry parameters $\mu$. Each case corresponds to the radius of maximum
Euclidean action obtained in figure 8. Figure 10: Evolution of scalar field
for different supersymmetry parameters $\mu$ and around the membrane (i.e.,
$R$ corresponds to the membrane’s radial coordinate). The BPS case has been
singled out with a dashed black line.

In order to compare all these profiles with the limiting BPS case, we show in
figure 10 a close-up plot around the membrane for all of them. Essentially, we
see that the BPS profile is actually a limiting case for the scalar field,
which concurs with our results in the non-gravitational case.

(a)  (b)

Figure 11: Scalar potential featuring (a) a Minkowskian false vacuum
($\mu=0.33$) (b) Scalar potential with dS false vacua, for several
supersymmetry breaking parameter values $\mu$.

(a)  (b)

Figure 12: (a) Scalar field profile and (b) scale factor, for the potential
shown in figure 11(a), corresponding to the membrane radius which maximized
the Euclidean action difference.

Tuning the supersymmetry-breaking parameter $\mu$, we can also analyze an
almost Minkowskian false vacuum ($\mu=0.33$), see figure 11(a). Note that
Minkowski false vacua are still non-compact spaces, and thus they should be
analyzed in exactly the same fashion as AdS vacua.

After computing the Euclidean action difference for several radii, we found
that the radius with maximum Euclidean action was $R=5.2$ in the Minkowskian
case. The profiles corresponding to a setting with such a membrane are shown
in figure 12.

Finally, as explicitly shown in [10, 42], the requirement of energy
conservation can be generalized to tunnelings where gravitational effects are
considered and the false vacuum is Minkowskian, by requiring the ADM mass
vanishes. After switching to Lorentzian signature, and integrating once the
$0-0$ component of Einstein’s equation (the Hamiltonian contraint), it is
possible to obtain the following expression for the ADM mass (see [42]):

$\displaystyle M_{\text{ADM}}=4\pi\int_{0}^{\infty}d\chi\
\rho^{\prime}\rho^{2}\left[\left|\phi^{\prime}\right|^{2}+\tilde{V}+2|q\phi|e^{K/2}\
\delta(\chi-R)\right]=0~{}.$ (5.11)

Here we have identified the Euclidean $\rho$ coordinate with the Lorentzian
radial one, $r$, and the integral is taken at the $\tau=0$ spacelike
surface141414Note that $\rho^{\prime}(R)$ should be evaluated as the mean
between the derivative of the scale factor right before and after the membrane
radius.. This means that the right hand side of this expression represents the
spatial integral of the energy density, that is, the $T_{00}$ component of the
energy-momentum tensor.

We have explicitly checked that this condition is satisfied for the solution
depicted in figure 12.

####  5.1.2 dS to AdS transitions

(a)  (b)

Figure 13: (a) Scalar field profile and (b) scale factor, for the dS $\to$ AdS
decays between the potentials depicted in fig. 11(b). All profiles plotted
here satisfy $\rho(\chi_{\text{max}})=0$ and
$\rho^{\prime}(\chi_{\text{max}})=-1$ simultaneously.

By making the supersymmetry-breaking parameter $\mu$ sufficiently big, we can
actually make the false vacuum lie in a positive potential value, while the
true vacuum rests at $V<0$. As opposed to the previous cases, the compact
geometry of this kind of instantons imposes boundary conditions at both the
origin and the maximum value of the radial coordinate (see, e.g., [41]), where
$\rho(0)=\rho(\chi_{\text{max}})=0$. Of course, the Euclidean action (5.8)
will be integrated up to $\chi_{\text{max}}$. Indeed, the background
consisting of a constant scalar potential $V_{\text{fv}}>0$ is described, in
terms of our ansatz (5.3), by the following scale factor:

$\displaystyle\rho_{\text{fv}}=H^{-1}\sin\left(H\chi\right)\
,\quad\chi\in[0,\chi_{\text{max}}]$ (5.12)

where $\chi_{\text{max}}$ represents the first positive and non-zero root of
$\rho_{\text{fv}}$. Furthermore, the integral of the background action can be
found analytically, and is given by

$\displaystyle S_{\text{bg}}=-\frac{24\pi^{2}}{V_{\text{fv}}}.$ (5.13)

Since the whole instanton is compact in dS $\to$ AdS decays, the only boundary
conditions we can impose on the scalar field are151515Note that, in our case
where a membrane is present, each of these boundaries is defined with respect
to a different scalar potential (since the flux is different at each side of
the membrane). Regardless, both $\phi(0)$ and $\phi(\chi_{\text{max}})$ should
lie close enough to the minimum of their respective potentials.
$\phi^{\prime}(0)=\phi^{\prime}(\chi_{\text{max}})$=0. On the other hand, in
order to find a non-singular instanton, for $\chi\to\chi_{\text{max}}$ one
must also require
$\rho^{\prime}(\chi)=-1+\mathcal{O}((\chi-\chi_{\text{max}})^{3})$. We have
found these requirements to be so restrictive, that for each potential in
terms of $\mu$ shown in figure 11(b), only a _single_ membrane radius solved
the equations of motion correctly with respect to all of these boundary
conditions.

The profiles of the scalar field and scale factor are shown in figure 13. The
geometries shown in fig. 13(b) have been numerically shown to satisfy
$\rho^{\prime}(\chi_{\text{max}})=-1$. In all cases, the scale factor clearly
shows an Euclidean AdS-like space starting at $\chi=0$, which evolves up to
the membrane radius, from where a compact Euclidean dS-like evolution
follows.161616Note that these profiles are full numerical solutions to the
instanton equations; i.e., no thin-wall approximation has been used in order
to compute either the scalar field or the scale factor.

##  6 Conclusions

In this paper, we have studied a combination of the Coleman-de Luccia [5, 2]
and Brown-Teitelboim [9, 10] formalisms for vacuum decay using both scalar
fields _and_ membranes, motivated from our study of flux compactifications in
supersymmetric theories. We have reviewed the existing theory on single three-
form multiplets [12, 13, 14, 15, 16, 17, 18] which combines scalar fields and
real three-form fields in a supersymmetric fashion, in the context of
$N=1,D=4$ supersymmetry and supergravity. Such a system has been extensively
studied in [21], with special focus on domain wall solutions with the
inclusion of the sources given by fundamental supersymmetric membranes. These
types of models provide a basis to quantitatively describe transitions between
vacua defined by different potentials related by changes of integer flux
contributions, such as the ones appearing in flux compactifications of String
Theory [4].

We have shown that adding soft supersymmetry-breaking terms to such a theory
allows for rich phenomenological applications. Indeed, these terms enable
false vacuum decays to occur through the nucleation of membranes. We have
studied such systems using Euclidean methods and found the instanton solutions
involving the form fields, scalar fields and membranes, both without and with
gravity included. Furthermore, we have checked that we retrieve the correct
supersymmetric limit as the supersymmetry-breaking terms are made smaller, so
the flat membrane case can be taken as the limiting instanton solution
corresponding to a membrane with infinite radius.

These processes have very interesting applications from a cosmological
viewpoint, as they can incorporate an emerging universe inside the created
bubble with an inflationary period. This type of models of open inflation have
been extensively studied in the context of the Landscape of String Theory [11,
43, 44, 45, 46, 47, 48]. In this work we have only investigated transitions
between very simple potentials, however, more phenomenologically interesting
results may be obtained by considering more realistic models. The transitions
studied in this work essentially generalize the false vacuum decays previously
discussed in the literature. These new ones naturally incorporate the
contributions of the three-forms to the flux potential and the membranes which
allow for transitions between vacua. Our formalism therefore includes any
backreaction of these objects to the instanton solution, selecting an initial
point for the open inflation period inside the bubble. This opens up the
possibility of studying the observational consequences of these models more
accurately.

On another note, recent proposals [49] have investigated the possibility of
embedding the Universe within a positively curved membrane created due to a
false vacuum decay in a higher-dimensional theory. While we have studied these
models in a four-dimensional environment, it could be interesting to
generalize our results to higher dimensions to check the compatibility of
these processes with the ones discussed in [49].

From a numerical perspective, we should note that the models we have solved
here were quite simple, in that they only involved a single (effectively real)
scalar field. Such models are quite easy to solve using an
undershoot/overshoot algorithm, even when membranes and gravity are present in
the setup. Solving non-perturbative decays with two or more scalar fields is
no simple matter. Thus, it might be useful to construct algorithms to compute
false vacuum decays which effectively incorporate the first-derivative jumps
on both the scalar fields and the scale factor. For this purpose it will also
be interesting to explore these kind of models within the new formalism to
study vacuum decay recently discussed in [50, 51].

Of course, the examples we have explored here can be generalized in several
ways, aside from using more involved superpotentials. For once, we have broken
supersymmetry by including explicit soft supersymmetry-breaking terms,
motivated from phenomenological considerations [24]. However, recall that the
tension of the membrane was determined by the requirement that the action of
whole system consisting of supergravity, matter supermultiplets (which contain
scalar fields and form fields) and the bosonic membrane preserves half of the
supersymmetry. Therefore, another interesting direction to explore would
consist on breaking supersymmetry by somehow deturning the coefficients
present in the Nambu-Goto and Wess-Zumino terms of the membrane.

Finally, in the models explored here we have considered single three-form
multiplets, which allow for one real three-form field for each complex scalar
field in the model (see appendix A and [18]). However, a general type IIB
compactification requires $2h^{1,2}+2$ real (and integer) fluxes, where
$h^{1,2}$ is the number of complex structure moduli. In order to build such a
model in our context, we can use the so-called double three-form
supermultiplet, which allows for the inclusion of _complex_ three-forms (i.e.,
two real fluxes for each complex scalar field). Such a system has been
considered in [21], and in [22] where some restrictions have been pointed out
on the possible EFTs. It may be interesting to re-check these restrictions
when considering false vacuum decay mediated by supersymmetry-breaking
spherical membranes.

Furthermore, extended objects such as effective strings and membranes have
recently made their way into the Swampland program [52, 53, 54]. In
particular, our results may be useful to test the AdS Instability Conjecture
[55] in completely backreacted systems.

Finally, the tunnelling processes we have discussed in the present paper
describe the transitions between different flux vacua. One may wonder whether
we can use this formalism to discuss the transitions where all the flux is
absorbed by the charged membrane. In this case the interior of the bubble will
not have any flux and therefore the geometry of the internal dimensions can
collapse. However, one can show that this collapse can be smooth in the higher
dimensional theory and the solution will resemble the bubble of nothing (BON)
geometry discussed by Witten in the simple 5-dimensional Kaluza-Klein model
[56]. These type of instabilities were first discussed in the context of a
higher dimensional field theory models of flux compactifications in a series
of papers in [57, 58, 59] (See also more recent discussions of these BONs in
related models in [60, 61, 62, 63, 64, 65, 66]). It would therefore be
interesting to study these solutions in our supersymmetric setup. These models
would be a perfect ground to study the conjecture put forward in [67] where it
was stated that the supersymmetric limit of these BON solutions would in fact
be some kind of static end of the world brane that preserves some
supersymmetry. We have initiated some work in this direction and we hope to be
able to report on these results in a future publication.

## Acknowledgments

We are grateful to Thibaut Coudarchet for interesting discussions. This study
is supported in part by the PID2021-123703NB-C21 and PID2021-125700NB-C21 (IB)
grants funded by MCIN/ AEI /10.13039/501100011033/ and by ERDF; ”A way of
making Europe”, the Basque Government grant (IT-1628-22) and the Basque
Foundation for Science (IKERBASQUE).

##  Appendix A Three-form multiplets in supersymmetry

In this appendix, we will briefly review the simplest way to construct the
supersymmetric Lagrangians used throughout the main text, which include
3-forms among their components. More specifically, we will work out the
expressions for _single_ 3-form multiplets, which are written in terms of
chiral superfields in which the real part of the F-component (the auxiliary
field) is substituted by a 3-form (as opposed to _double_ 3-form multiplets,
which are described by other special chiral superfields with both real and
imaginary parts of the F-component of which are replaced by 3-form fields).
Most of the following formalism may be found in [19, 18, 23], and we will try
to stick to the conventions of [25] unless otherwise stated.

In the superspace formalism, supersymmetric covariant derivatives are given by

$\displaystyle
D_{\alpha}=\partial_{\alpha}+i\sigma_{\alpha\dot{\alpha}}^{\mu}\bar{\theta}^{\dot{\alpha}}\partial_{\mu},\qquad\bar{D}_{\dot{\alpha}}=-\partial_{\dot{\alpha}}-i\theta^{\alpha}\sigma_{\alpha\dot{\alpha}}^{\mu}\partial_{\mu}.$
(A.1)

Their algebra, which is given by

${}\\{D_{\alpha},D_{\beta}\\}=0\;,\qquad\\{D_{\alpha},\bar{D}_{\dot{\alpha}}\\}=-2i\sigma_{\alpha\dot{\alpha}}^{\mu}\partial_{\mu}\;,\qquad\\{\bar{D}_{\dot{\alpha}},\bar{D}_{\dot{\beta}}\\}=0$
(A.2)

implies the following expressions:

$D_{\alpha}D_{\beta}=\frac{1}{2}\epsilon_{\alpha\beta}D^{2}\;,\qquad
D_{\alpha}D^{2}=0\;,\qquad\bar{D}_{\dot{\alpha}}\bar{D}_{\dot{\beta}}=-\frac{1}{2}\epsilon_{\dot{\alpha}\dot{\beta}}\bar{D}^{2}\;,\qquad\bar{D}_{\dot{\alpha}}\bar{D}^{2}=0\;.\qquad$
(A.3)

A _chiral_ supefield is defined by the constraint
$\bar{D}_{\dot{\alpha}}\Phi=0$, and has the following component expansion in
terms of the Grassmanian variables $\theta^{\alpha}$:

$\displaystyle\Phi$ $\displaystyle=\phi+\sqrt{2}\theta\psi+\theta\theta
F+i\theta\sigma^{\mu}\bar{\theta}\partial_{\mu}\phi-\frac{i}{\sqrt{2}}\theta\theta\partial_{\mu}\psi\sigma^{\mu}\bar{\theta}+\frac{1}{4}\theta\theta\bar{\theta}\bar{\theta}\square\phi$
(A.4)
$\displaystyle=e^{i\theta\sigma^{\mu}\bar{\theta}\partial_{\mu}}\left(\phi+\sqrt{2}\theta\psi+\theta\theta
F\right)\;,$ (A.5)

where $\phi(x)$ and $F(x)$ are complex scalar fields and $\psi(x)$ is a Weyl
spinor. It is convenient to use their equivalent definition in terms of the
leading components of the superfield and of its covariant derivatives:

$\displaystyle\left.\Phi\right|$ $\displaystyle=\phi$ (A.6)
$\displaystyle\frac{1}{\sqrt{2}}\left.D_{\alpha}\Phi\right|$
$\displaystyle=\psi_{\alpha}$ (A.7)
$\displaystyle-\frac{1}{4}\left.D^{2}\Phi\right|$ $\displaystyle=F,$ (A.8)

where the vertical line means that we are taking the superfield with
$\theta=\bar{\theta}=0$.

Given a set of chiral superfields
$\bm{\Phi}=\left\\{\Phi^{1},\ldots,\Phi^{n}\right\\}$, the most general
interacting supersymmetric theory describing their dynamics is given by the
Lagrangian

$\displaystyle\mathcal{L}$ $\displaystyle=\int d^{2}\theta d^{2}\bar{\theta}\
K(\bm{\Phi},\bar{\bm{\Phi}})+\left[\int d^{2}\theta\
W(\bm{\Phi})+\text{h.c.}\right]$ (A.9)
$\displaystyle=-K_{a\bar{b}}\partial_{\mu}\phi^{a}\partial^{\mu}\bar{\phi}^{\bar{b}}+K_{a\bar{b}}F^{a}\bar{F}^{\bar{b}}+F^{a}W_{a}+\bar{F}^{a}\bar{W}_{a}+\ldots$
(A.10)

where $a,b\in\\{1,\ldots,n\\}$, $K(\bm{\Phi},\bar{\bm{\Phi}})$ is an arbitrary
real function of the chiral superfields and thier complex conjugate anti-
chiral superfields called _Kähler potential_ , $W(\bm{\Phi})$ is an arbitrary
holomorphic function of the chiral superfields called _superpotential_ and
subscripts denote derivatives with respect to the scalar fields as in
$K_{a\bar{b}}=\frac{\partial}{\partial\phi^{a}}\frac{\partial}{\partial\phi^{b}}K$.
In (A.10) we have omitted the terms involving the fermionic fields for
simplicity; unless otherwise stated, we will not consider them below or also
in the main text.

It can be easily shown that, when the auxiliary fields are integrated out
applying their algebraic equations of motion, the above Lagrangian reads

$\displaystyle\left.\mathcal{L}\right|_{\text{bos., on-
sh.}}=-K_{a\bar{b}}\partial_{\mu}\phi^{a}\partial^{\mu}\bar{\phi}^{\bar{b}}-K^{a\bar{b}}W_{a}\bar{W}_{\bar{b}}.$
(A.11)

where $K^{a\bar{b}}$ is the inverse of $K_{a\bar{b}}$.

###  A.1 Including 3-forms into the chiral superfields: special chiral
superfields

The material of this subsection is based on the constructions of [12, 13, 15,
19, 21].

The chirality constraint $\bar{D}_{\dot{\alpha}}\Phi=0$ can be easily solved
in terms of an unconstrained complex scalar superfield $P=(\bar{P})^{*}$ which
is called _prepotential_ : $\Phi=\bar{D}\bar{D}P$ ($\bar{\Phi}=DD\bar{P}$).
The simplest special chiral superfield one may use which includes a 3–form as
the real part of its highest F-component is obtained by expressing this
complex prepotential by an uncostrained real superfield $V=\bar{V}$. It is
convenient to define its $\theta$-expansion as follows:

$\displaystyle V(x,\theta,\bar{\theta})=$ $\displaystyle
C+i\theta\chi-i\bar{\theta}\bar{\chi}+i\theta\theta\bar{\phi}-i\bar{\theta}\bar{\theta}\phi-\theta\sigma^{\mu}\bar{\theta}v_{\mu}$
$\displaystyle+i\theta\theta\bar{\theta}\left[\bar{\lambda}+\frac{i}{2}\bar{\sigma}^{\mu}\partial_{\mu}\chi\right]-i\bar{\theta}\bar{\theta}\theta\left[\lambda+\frac{i}{2}\sigma^{\mu}\partial_{\mu}\bar{\chi}\right]+\frac{1}{2}\theta\theta\bar{\theta}\bar{\theta}\left[D-\frac{1}{2}\square
C\right].$ (A.12)

Here $u(\bm{x})$ and $D(\bm{x})$ are real scalar fields, $\phi(\bm{x})$ is a
complex scalar field, $v_{\mu}(\bm{x})$ is a real vector field and
$\chi(\bm{x})$ and $\lambda(\bm{x})$ are Weyl spinors.

A special chiral superfield $Y$ can be obtained from the generic chiral
superfield $\Phi=\bar{D}\bar{D}P$ by expressing the complex prepotential by
$P=-\frac{i}{4}V$ the real superfield $V$, so that171717Note that this
definition allows for some freedom in choosing $V$, since (A.13) is invariant
under the gauge superspace symmetry $V\to V+L$, where $L$ is the so-called
_real linear superfield_ which satisfies $\bar{D}^{2}L=0=D^{2}L$ .
Furthermore, the definition of $Y$ may be generalized to the case where the
real superfield $V$ is not independent but composed from a so-called complex
linear superfield $\Sigma$, which obeys $D^{2}\Sigma=0$, and its complex
conjugate $\bar{\Sigma}$, which obys $\bar{D}^{2}\bar{\Sigma}=0$. This last
case may be used to construct special chiral superfields which describe a
double 3-form supermultiplet. See [18, 21, 23] for further detail.

$\displaystyle Y:=-\frac{i}{4}\bar{D}^{2}V$ (A.13)

(the prefactor is chosen for later convenience). It is easy to check that it
fulfills the condition $\bar{D}_{\dot{\alpha}}Y=0$ from its definition. Its
components can be projected using

$\displaystyle\left.Y\right|$
$\displaystyle=-\frac{i}{4}\left.\bar{D}^{2}V\right|=\phi$ (A.14)
$\displaystyle\left.D_{\alpha}Y\right|$
$\displaystyle=-\frac{i}{4}\left.D_{\alpha}\bar{D}^{2}V\right|=\lambda_{\alpha}$
(A.15) $\displaystyle-\frac{1}{4}\left.D^{2}Y\right|$
$\displaystyle=\frac{i}{16}\left.D^{2}\bar{D}^{2}V\right|=\frac{1}{2}\left(\partial^{\mu}v_{\mu}+iD\right)$
(A.16)

Note that, component-wise, $Y$ is almost identical to the original chiral
field $\Phi$ in that it contains a complex scalar field, a complex Weyl
fermion and a complex auxiliary field, albeit in this case we are mostly
interested in the real part of the latter, which is given by the devergence of
a 4-vector field $v^{\mu}$.

For our purposes, it will be convenient to consider $v_{\mu}$ as the one-form
associated through Hodge duality to a three-form. Indeed, the Hodge dual of
the three-form is given by the following vector field181818In our conventions,
$\epsilon^{0123}=-\epsilon_{0123}=1$ for Lorentzian signature (which we will
use throughout this appendix), while $\epsilon^{0123}=\epsilon_{0123}=1$ for
Euclidean one.

$\displaystyle(\ast A_{3})_{\mu}\equiv
A_{\mu}=\frac{1}{3!}\epsilon_{\mu\nu\rho\sigma}A^{\nu\rho\sigma}$ (A.17)

where the indices have been raised using the flat spacetime metric. The
divergence of this vector field is related to the Hodge dual of the 4-form
field strength $F_{4}$, associated to $A_{\nu\rho\sigma}$ through191919We have
omitted the contribution of the metric determinant in these expressions to
clarify the definitions. In the case of a curved spacetime, these expressions
are $(\ast A_{3})_{\mu}\equiv
A_{\mu}=\frac{1}{3!}\sqrt{-g}\epsilon_{\mu\nu\rho\sigma}A^{\nu\rho\sigma}$ and
$\ast F_{4}=\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}A^{\mu})$.

$\displaystyle\ast
F_{4}=\frac{1}{4!}\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu\rho\sigma}=\partial_{\mu}A^{\mu}$
(A.18)

With these expressions in hand, we find that if $v_{\mu}$ is identified with
$A_{\mu}$, i.e, with the Hodge dual of the 3-form field, the composite
auxiliary field of the special chiral superfield $Y$ reads

$\displaystyle F_{Y}=-\frac{1}{4}\left.D^{2}Y\right|=\frac{1}{2}\left(\ast
F_{4}+iD\right).$ (A.19)

The advantage of treating the vector $v_{\mu}$ as dual to 3-form is that a
membrane can be coupled ’electrically’ (or minially) to the 3-form field (see
below) and thus to the composite auxiliary field in the special chiral
superfield $Y$. Since $D$ does not enter the membrane part of the full action,
we will be able to remove it from the action by solving its algebraic
equations early on, leaving $\ast F_{4}$ untouched for its interplay with the
scalar field and the membrane.

We can easily apply these expressions to a single three-form multiplet $Y$,
with a certain Kähler potential $K(Y,\bar{Y})$ and superpotential $W(Y)$.
Plugging the component fields of $Y$ into the bosonic Lagrangian (A.10), we
find

$\displaystyle\left.\mathcal{L}\right|_{\text{bos.}}$
$\displaystyle=-K_{\phi\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}+\frac{1}{4}K_{\phi\bar{\phi}}\left(\ast
F_{4}+iD\right)\left(\ast F_{4}-iD\right)+\frac{1}{2}\left(\ast
F_{4}+iD\right)W_{\phi}+\frac{1}{2}\left(\ast
F_{4}-iD\right)\bar{W}_{\bar{\phi}}$
$\displaystyle=-K_{\phi\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}+\frac{1}{4}K_{\phi\bar{\phi}}\left(\ast
F_{4}\right)^{2}+\frac{1}{2}\left(\ast
F_{4}\right)\left(W_{\phi}+\bar{W}_{\bar{\phi}}\right)+\frac{1}{4}K_{\phi\bar{\phi}}D^{2}+\frac{i}{2}D\left(W_{\phi}-\bar{W}_{\bar{\phi}}\right)$
(A.20)

The equation of motion for the auxiliary field $D$ reads

$\displaystyle
D=-iK^{\phi\bar{\phi}}\left(W_{\phi}-\bar{W}_{\bar{\phi}}\right)$ (A.21)

which is completely algebraic, as expected. Plugging this into (A.20) yields

$\displaystyle\left.\mathcal{L}\right|_{\text{bos.}}$
$\displaystyle=-K_{\phi\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}+\frac{1}{4}K_{\phi\bar{\phi}}\left(\ast
F_{4}\right)^{2}+\frac{1}{2}\left(\ast
F_{4}\right)\left(W_{\phi}+\bar{W}_{\bar{\phi}}\right)+\frac{1}{4}K^{\phi\bar{\phi}}\left(W_{\phi}-\bar{W}_{\bar{\phi}}\right)^{2}$
(A.22)
$\displaystyle=-K_{\phi\bar{\phi}}~{}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-\frac{1}{4\cdot
4!}K_{\phi\bar{\phi}}~{}F^{\mu\nu\sigma\rho}F_{\mu\nu\sigma\rho}+\frac{1}{2\cdot
4!}\left(W_{\phi}+\bar{W}_{\bar{\phi}}\right)~{}\epsilon^{\mu\nu\sigma\rho}F_{\mu\nu\sigma\rho}+\frac{1}{4}K^{\phi\bar{\phi}}\left(W_{\phi}-\bar{W}_{\bar{\phi}}\right)^{2}$
(A.23)

where, in the second step, we have rewritten the Hodge duals in terms of their
original tensor fields for clarity.

In order to proceed, recall that the physical field we wish to extremise in
the action is not the field strength, but rather its antisymmetric tensor
potential $A_{\mu\nu\rho}$. In the following, it will be more convenient to
work with the Hodge-dual $A^{\mu}$, which is related to $\ast F_{4}$ by eq.
(A.18). The variation of (A.23) with respect to $A_{\mu}$ is

$\displaystyle\frac{1}{2}K_{\phi\bar{\phi}}\left(\ast
F_{4}\right)\left(\partial^{\mu}\delta
A_{\mu}\right)+\frac{1}{2}\left(\partial^{\mu}\delta
A_{\mu}\right)\left(W_{\phi}+\bar{W}_{\bar{\phi}}\right)$
$\displaystyle\quad=\delta
A_{\mu}\left[\frac{1}{2}\partial^{\mu}\left(-K_{\phi\bar{\phi}}\left(\ast
F_{4}\right)+W_{\phi}+\bar{W}_{\bar{\phi}}\right)\right]+\partial^{\mu}\left[\delta
A_{\mu}\frac{1}{2}\left(K_{\phi\bar{\phi}}\left(\ast
F_{4}\right)-W_{\phi}-\bar{W}_{\bar{\phi}}\right)\right]\;.$ (A.24)

We can clearly see that the vanishing of the first term in the r.h.s. will
give us an equation of motion for the 3-form gauge field, while the second one
will produce a boundary term. As originally discussed in [10], in order to
deal with the second term in (A.24) we will need to add a boundary term to our
original Lagrangian to convert this contribution into an expression containing
the variations of gauge invariant quantities ($F_{4}$ and scalar fields).
Indeed, otherwise the variational problem with respect to gauge potential
would not be well posed (see [18] for recent discussion). This boundary term
not only ensures the consistency of the variational problem but, as we will
see shortly, it will have a noticeable effect on the final, on-shell result.

The required boundary term is given by

$\displaystyle\mathcal{L}_{bd}$
$\displaystyle=-\frac{1}{2}\partial^{\mu}\left[A_{\mu}\left(K_{\phi\bar{\phi}}\left(\ast
F_{4}\right)-W_{\phi}-\bar{W}_{\bar{\phi}}\right)\right]$
$\displaystyle=\frac{1}{2\cdot
3!}\partial^{\mu}\left[A^{\nu\rho\sigma}\left(K_{\phi\bar{\phi}}F_{\mu\nu\rho\sigma}+\epsilon_{\mu\nu\rho\sigma}\left(W_{\phi}+\bar{W}_{\bar{\phi}}\right)\right)\right]$
(A.25)

while the equation of motion for the form field is

$\displaystyle\partial_{\mu}\left(\frac{1}{2}K_{\phi\bar{\phi}}(\ast
F_{4})+{\rm Re}\,W_{\phi}\right)=0\quad\Rightarrow\quad\ast
F_{4}=-2K^{\phi\bar{\phi}}\left(\text{Re}W_{\phi}-n\right)$ (A.26)

where $n\in\mathbb{R}$ is an arbitrary real (integration) constant. As this
constant appears in the expression for the 4-form field strength, it can be
identified with the flux of the 3-form gauge field.

Plugging this result into (A.23) and taking into account the non-vanishing
contribution of the boundary term yields

$\displaystyle\left.\mathcal{L}\right|_{\text{bos.,on-
sh.}}=-K_{\phi\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-K^{\phi\bar{\phi}}\left(W_{\phi}-n\right)\left(\bar{W}_{\bar{\phi}}-n\right).$
(A.27)

From this final form of the Lagrangian we can conclude that the contribution
of 3-forms in a supersymmetric setup results in a linear contribution to our
original superpotential in which the field is multiplied by a constant
associated with the flux of the 3-form gauge field. Thus, setting the 3-forms
on shell reduces the original theory to a model of a scalar field described by
the original Kähler potential and an effective superpotential given by

$\displaystyle\hat{W}(\phi)\equiv W(\phi)-n\phi.$ (A.28)

##  Appendix B Rigid supersymmetry, bosonic membranes, and BPS equations

As it was shown in [36, 37], the action of the interacting system composed of
the supergravity multiplet and a bosonic membrane is invariant under a half of
local supersymmetry, provided the membrane term of the interacting action is
given by the bosonic ‘limit’ of a supermembrane. Furthermore, the preserved
part of the local supersymmetry reflects the local fermionic $\kappa$–symmetry
of the original supermembrane action.

In the case of an interacting system composed of supersymmetric matter and a
supermembrane, there is no rigorous way to find rigid supersymmetry invariance
of the interacting system including the matter supermultiplet and a bosonic
membrane. However, as we will show below, there exists a trick allowing us to
see the $\kappa$-symmetry of the original supermembrane part of the action for
the interacting system. It implies the manipulation of the supersymmetry
transformation of the bosonic membrane action with the use of some ansatze
both for the fields and for membrane configuration. Actually, such a
possibility, when it exists, reflects the existence of purely bosonic
supersymmetric solutions of the complete supersymmetric system.202020The
relation of the $\kappa$-symmetry of a super-p-brane with supersymmetry
preserved by solutions of equations was described for the first time in [39].
In the following we will show how using this arguments we obtained the simple
form of the BPS equations for our brane in our model.

Let us start by looking at the supersymmetry transformation in our model.
Although we did not write those transformations of the 3-form supermultiplet
in the main text, they can be easily restored from the association of the
spacetime fields with the superfield components in eqs. (A.14, A.15, A.16) and
the identification of supersymmetry as fermionic supertranslations in
superspace. This leads to the following supersymmetric transformations for the
fields,

$\displaystyle\delta_{\epsilon}\phi$ $\displaystyle=$
$\displaystyle\epsilon^{\alpha}D_{\alpha}Y|=\epsilon^{\alpha}\lambda_{\alpha}\;,\qquad$
(B.1) $\displaystyle\delta_{\epsilon}\lambda_{\alpha}$ $\displaystyle=$
$\displaystyle\epsilon^{\beta}D_{\beta}D_{\alpha}Y|+\bar{\epsilon}_{\dot{\beta}}\bar{D}^{\dot{\beta}}D_{\alpha}Y|=2\epsilon_{\alpha}F_{Y}+2i(\sigma^{\mu}\bar{\epsilon})_{\alpha}\partial_{\mu}\phi\qquad$
(B.2) $\displaystyle=\epsilon_{\alpha}(\partial_{\mu}A^{\mu}+i{\rm
D})+2i(\sigma^{\mu}\bar{\epsilon})_{\alpha}\partial_{\mu}\phi\;,\qquad$

and

$\delta_{\epsilon}F_{Y}=\frac{1}{2}(\partial_{\mu}\delta_{\epsilon}A^{\mu}+i\delta_{\epsilon}{\rm
D})=-\frac{1}{4}\bar{\epsilon}_{\dot{\beta}}\bar{D}^{\dot{\beta}}D^{2}Y|=i(\partial_{\mu}\lambda\sigma^{\mu}\bar{\epsilon})\,.$
(B.3)

These transformations also imply

$\delta_{\epsilon}{\rm
D}=\partial_{\mu}\lambda\sigma^{\mu}\bar{\epsilon}+{\epsilon}\sigma^{\mu}\partial_{\mu}\bar{\lambda}\;,\qquad\delta_{\epsilon}A^{\mu}=i(\partial_{\mu}\lambda\sigma^{\mu}\bar{\epsilon}-{\epsilon}\sigma^{\mu}\partial_{\mu}\bar{\lambda})\;,\qquad$
(B.4)

as well as

$\delta_{\epsilon}A_{\mu\nu\rho}=i\epsilon_{\mu\nu\rho\sigma}(\partial_{\mu}\lambda\sigma^{\sigma}\bar{\epsilon}-{\epsilon}\sigma^{\sigma}\partial_{\mu}\bar{\lambda})~{},$
(B.5)

for the dual 3-form $A_{\mu\nu\rho}=\epsilon_{\mu\nu\rho\sigma}A^{\sigma}$.

Let us now perform the above supersymmetry transformation to the scalar and
3-form field in the bosonic membrane action (3.7):

$\delta_{\epsilon}S_{{\rm membr.}}=-2|q|\int d^{3}\xi\sqrt{-h}\,{\rm
Re}\,\left(\epsilon^{\alpha}\lambda_{\alpha}\,\frac{\bar{\phi}}{|\bar{\phi}|}\right)+2q\int
d^{3}\xi\,{\rm
Im}\,(\lambda\sigma^{\mu}\bar{\epsilon})\epsilon_{\mu\nu\rho\sigma}\partial_{0}x^{\nu}\partial_{1}x^{\rho}\partial_{0}x^{\sigma}\;.$
(B.6)

Generically, this last expression does not vanish. However, if we consider a
flat membrane lying at $x^{3}=z=0$ in the static gauge, i.e.,

$x^{a}(\xi)=\xi^{a}\;,\qquad x^{3}(\xi):=z(\xi)=0\;,$ (B.7)

then $\sqrt{-h}=1$ and eq. (B.6) reduces to

$\delta_{\epsilon}S_{{\rm membr.}}=-2|q|\int d^{3}\xi{\rm
Re}\,\left(\epsilon^{\alpha}\lambda_{\alpha}\,\frac{\bar{\phi}}{|\bar{\phi}|}-i\frac{q}{|q|}(\bar{\epsilon}\tilde{\sigma}{}^{3}\lambda)\right)\;.$
(B.8)

This formally vanishes if we set

$\epsilon^{\alpha}=i(\bar{\epsilon}\tilde{\sigma}{}^{3})^{\alpha}\,\left.\frac{q\phi}{|q\phi|}\right|_{z=0}\;.$
(B.9)

This equation has nontrivial solutions for a constant fermionic spinor
$\epsilon^{\alpha}$ if $\left.\frac{{q\phi}}{|{q\phi}|}\right|_{z=0}$ is
independent of $\xi^{a}$ (i.e., as long as it represents a constant phase).
This condition is clearly satisfied if we assume the scalar field to depend on
the $z$ coordinate only, that is,

$\phi(x^{a},z)=\phi(z)\;.$ (B.10)

This and the restriction to a flat membrane given by the last equation in
(B.7) define the ansatz for the supersymmetric bosonic solution of the
interacting system of the 3-form multiplet and supermembrane.

Let us repeat that these formal calculations reflect the $\kappa$–symmetry of
the complete supermembrane action, which guarantees the existence of the
purely bosonic supersymmetric solution of the equations for the interacting
system of a supermembrane and the single 3-form matter supermultiplet.

Let us look at the transformation of the fields in the bulk. For a purely
bosonic solution obtained with the ansatz (B.10), the preservation of
supersymmetry implies $\delta_{\epsilon}\lambda=0$ which, after using (B.2)
and the auxiliary field’s equation

$\displaystyle 2F_{Y}=(*F_{4}+i{\rm
D})=-K^{\phi\phi}\hat{\bar{W}}_{\bar{\phi}}\,$ (B.11)

reduces to

$i(\sigma^{3}\bar{\epsilon})_{\alpha}\partial_{z}\phi=\epsilon_{\alpha}K^{\phi\phi}\hat{\bar{W}}_{\bar{\phi}}\;.$
(B.12)

This equation has a nontrivial solution with the constant fermionic spinor
obeying projecting condition

$\epsilon_{\alpha}=e^{i\eta}i(\sigma^{3}\bar{\epsilon})_{\alpha}\qquad\Leftrightarrow\qquad\epsilon^{\alpha}=-e^{i\eta}i(\bar{\epsilon}\tilde{\sigma}^{3})^{\alpha}$
(B.13)

if the scalar field obeys the BPS equation

$\partial_{z}\phi(z)=e^{i\eta}K^{\phi\bar{\phi}}\,\hat{\bar{W}}_{\bar{\phi}}\;.$
(B.14)

Furthermore, the supersymmetry preserved by the bosonic configuration will
also preserve the supersymmetry of the static flat bosonic membrane
confirguration given by (B.7) if (B.13) coincides with (B.9), i.e. when

$\displaystyle e^{i\eta}=-\left.\frac{q\phi}{|q\phi|}\right|_{z=0}\,.$ (B.15)

Given the BPS equation (B.14), it is easy to check that the phase of
$\partial_{z}\hat{W}$ remains constant _even across the membrane_. Indeed, if
we multiply both sides of the equation by $\hat{W}_{\phi}$, we find

$\displaystyle(\partial_{z}\phi)\hat{W}_{\phi}=e^{i\eta}\hat{W}_{\phi}K^{\phi\bar{\phi}}\,\hat{\bar{W}}_{\bar{\phi}}\qquad\Rightarrow\qquad\partial_{z}\hat{W}=e^{i\eta}\left[V(\phi,\bar{\phi})+|q\phi|\delta(z)\right]$
(B.16)

Therefore, one finds that $e^{i\eta}$ may be written as

$\displaystyle
e^{i\eta}=-\left.\frac{q\phi}{|q\phi|}\right|_{z=0}=\frac{\Delta\hat{W}}{|\Delta\hat{W}|}$
(B.17)

where $\Delta\hat{W}\equiv\hat{W}_{z=+\infty}-\hat{W}_{z=-\infty}$. Note that
this last result exactly reproduces the phase obtained in [3], albeit in our
case it includes the effect of the membrane localized at $z=0$, which
effectively changes the superpotential when crossing it.

One may also use this result in order to find a closed expression for the
tension of the domain wall solution in the presence of a supermembrane. As
shown in [23], one may rewrite the action (3.13) as

$\displaystyle S=$ $\displaystyle-\int d^{3}x\int
dzK_{\phi\bar{\phi}}\left[\partial_{z}\phi-e^{i\beta}K^{\phi\bar{\phi}}\hat{\bar{W}}_{\bar{\phi}}\right]\left[\partial_{z}\bar{\phi}-e^{-i\beta}K^{\phi\bar{\phi}}\hat{W}_{\phi}\right]$
$\displaystyle-\int d^{3}x\left(2|q\phi|_{z=0}+2{\rm
Re}\,\left[e^{-i\beta}(\Delta\hat{W}+q\phi|_{z=0})\right]\right).$ (B.18)

where the phase $e^{i\beta}$ is, at this stage, arbitrary. However, upon
choosing $e^{i\beta}=e^{i\eta}$ one can see that the expression for the
tension of any field configuration is maximized by the solution of the BPS
equation (B.14). Taking this into account the tension becomes

$\displaystyle\mathcal{T}_{\text{DW+memb.}}=|2\Delta\hat{W}|\ .$ (B.19)

Note that even though this expression is formally similar to the one in the
scalar field model in [3] the result is now written in terms of the effective
superpotential $\hat{W}$ which includes the jump due to the different fluxes
on both sides of the membrane.

As described in [21] and references therein, similar arguments can be applied
to obtain the counterpart of (B.14) for dynamical systems including
supergravity, which lead to eqs. (4.13)-(4.14).

##  Appendix C Three-form multiplets in supergravity

In this section we will follow a similar reasoning as the one above, with
gravity included. We will first present the action for scalar multiplets
described by generic chiral superfields, which we will later on generalize to
include special chiral superfields which have vector or 3-form components
among their bosonic ingredients. All of the results we present here have also
been derived in [18, 23] using a super-Weyl invariant approach to matter-
coupled supergravity, reaching the same conclusions.

We recall that supergravity can be described in terms of the superspace
supervielbein $E_{M}^{A}(z)$ and the spin connection
$\omega_{M}^{ab}(z)=-\omega_{M}^{ba}(z)$ subject to a set of torsion
constraints. After fixing the so-called Wess-Zumino gauge, the field content
reduces to

  * •

$e_{a}^{\mu}$, the vielbein,

  * •

$\psi_{\alpha}^{\mu}$, the gravitino,

  * •

$b^{\mu}$, a real vector auxiliary field,

  * •

$M$, a complex scalar auxiliary field.

The most general superspace action of interacting supergravity and scalar
supermultiplets in terms of $E_{M}^{A}(z)$ and generic chiral superfields
(defined in curved supergravity superspace) can be found e.g. in [25]. The
spacetime action for the component fields is then obtained upon fixing the
Wess-Zumino gauge and integrating over the fermionic coordinates of
superspace. In the conventions of [25], the spacetime Lagrangian of the
bosonic sector of such an action reads

$\displaystyle\frac{1}{\sqrt{-g}}{\cal
L}=\frac{1}{2}\,Re^{-\frac{1}{3}K}+\Omega_{a\bar{b}}\partial_{\mu}\phi^{a}\partial^{\mu}\bar{\phi}^{\bar{b}}-\frac{1}{3}e^{-\frac{1}{3}K}\tilde{M}\tilde{\bar{M}}-\tilde{M}\bar{W}-\tilde{\bar{M}}W$
$\displaystyle+e^{-\frac{1}{3}K}K_{a\bar{b}}F^{a}\bar{F}^{\bar{b}}+F^{a}(W_{a}+K_{a}W)+\bar{F}^{\bar{b}}(\bar{W}_{\bar{b}}+K_{\bar{b}}\bar{W})$
$\displaystyle-\frac{1}{9}\Omega
b_{\mu}b^{\mu}-\frac{i}{3}b^{\mu}(\partial_{\mu}\phi^{a}\,\Omega_{a}-\partial_{\mu}\bar{\phi}^{\bar{b}}\,\Omega_{\bar{b}})\;,\qquad$
(C.1)

where

$\displaystyle\Omega(\Phi,\bar{\Phi})=-3e^{-\frac{1}{3}K(\Phi,\bar{\Phi})},\qquad\tilde{M}=M+K_{\bar{a}}\bar{F}^{\bar{a}}\;,\qquad\tilde{\bar{M}}={\bar{M}}+K_{a}F^{a}\;.$
(C.2)

This action is not written in Einstein frame. Therefore, it is customary to
rescale the vielbein as follows:

$\displaystyle e_{\mu}^{a}\;\mapsto\;e_{\mu}^{a}\,e^{\frac{1}{6}K}\;,\qquad$
(C.3)

and to supplement this with a suitable transformation of the spin connection.
Furthermore, rescaling the auxiliary fields as

$\displaystyle F^{i}\,\mapsto\,F^{i}\,e^{-\frac{1}{6}K}\;,\qquad
M\,\mapsto\,M\,e^{-\frac{1}{6}K}\;,\qquad$ (C.4)

we arrive at the following action in Einstein frame

$\displaystyle\frac{1}{\sqrt{-g}}{\cal L}=$
$\displaystyle\frac{1}{2}\,R-K_{a\bar{b}}\partial_{\mu}\phi^{a}\partial^{\mu}\bar{\phi}{}^{\bar{b}}-\frac{1}{3}\tilde{M}\tilde{\bar{M}}-e^{\frac{1}{2}K}\tilde{M}\bar{W}-e^{\frac{1}{2}K}\tilde{\bar{M}}W\qquad$
$\displaystyle+K_{a\bar{b}}F^{a}\bar{F}{}^{\bar{b}}+e^{\frac{1}{2}K}F^{a}(W_{a}+K_{a}W)+e^{\frac{1}{2}K}\bar{F}{}^{\bar{b}}(\bar{W}_{\bar{b}}+K_{\bar{b}}\bar{W})\;.\qquad$
(C.5)

Notice that in this last expression the auxiliary fields $b^{\mu}$ have been
integrated out using their equations of motion. If we are dealing with minimal
supergravity and scalar multiplets, $\tilde{M}$ and all $F^{i}$ are
independent, and the auxiliary field equations read

$\displaystyle\tilde{M}=-3e^{\frac{1}{2}K}W\;,\qquad
F^{a}K_{a\bar{b}}=-e^{\frac{1}{2}K}(\bar{W}_{\bar{b}}+K_{\bar{b}}\bar{W})\;.$
(C.6)

Substituting this into (C) we find the well known matter-coupled N=1, $D=4$
supergravity Lagrangian

$\displaystyle\frac{1}{\sqrt{-g}}{\cal L}=\frac{1}{2}\,{\cal
R}-K_{a\bar{b}}\partial_{\mu}\phi^{a}\partial^{\mu}\bar{\phi}{}^{\bar{b}}-V(\phi,\bar{\phi})$
(C.7)

where the potential is given by

$\displaystyle
V(\phi,\bar{\phi})=e^{K}\left(D_{a}WK^{a\bar{b}}D_{\bar{b}}\bar{W}-3|W|^{2}\right),$
(C.8)

and $D_{a}=\partial_{a}+K_{a}$ are the so-called Kähler-covariant derivatives.

###  C.1 Supergravity interacting with 3-form multiplets

Just as in the non-gravitational case, we will implicitly introduce three-
forms by passing from generic to special chiral superfields. In this case,
special chiral superfields describing single three-form multiplets are given
by

$\displaystyle S=-\frac{i}{4}(\bar{D}^{2}-8\mathcal{R})\,{\cal
P}\;,\qquad{\cal P}=({\cal P})^{*}$ (C.9)

where $\mathcal{R}$ is the so-called main chiral superfield of minimal
supergravity, whose leading component is proportional to the complex scalar
auxiliary field, $\mathcal{R}|=-\frac{1}{6}M$, and ${\cal P}$ is an
unconstrained real superfield. The latter is defined up to shift by a real
linear superfield (C.9)

$\displaystyle{\cal P}\mapsto{\cal
P}+L,\qquad(\bar{D}^{2}-8\mathcal{R})L=0=(D^{2}-8\bar{\mathcal{R}})L.$ (C.10)

This freedom is the manifestation of a gauge symmetry which can be used to fix
the Wess-Zumino gauge, where

$\displaystyle{\cal P}|=0\;,\qquad{D}_{\alpha}{\cal
P}|=0\;,\qquad\bar{D}_{\dot{\alpha}}{\cal P}|=0\;.\qquad$ (C.11)

The remaining part of the $L$ symmetry, preserving this gauge, coincides with
the 2-form gauge symmetry for the 3-form dual to vector component of the
prepotential superfield,

$\displaystyle\sigma^{a}_{\alpha\dot{\alpha}}[D^{\alpha},\bar{D}{}^{\dot{\alpha}}]{\cal
P}|=4A^{a}\;,\qquad A^{a}=*(A_{3})^{a}\;.$ (C.12)

In the gauge (C.11), the highest component of the special chiral superfield of
$S$ simplifies to (in the notation of [25])

$\displaystyle\mathcal{F}\equiv
F_{S}=-\frac{1}{4}\,D^{2}S|=\frac{i}{16}\,D^{2}\bar{D}{}^{2}{\cal
P}|-\frac{i}{2}\,\mathcal{R}{D}{}^{2}{\cal
P}|=\frac{1}{2}\left(D_{\mu}A^{\mu}+i{\rm
d}\right)+\frac{1}{3}(s\bar{M}+2\bar{s}{}{M})\;.$ (C.13)

where $s=\left.S\right|$, $M$ and $\bar{M}$ are the scalar auxiliary fields of
minimal supergravity, $\rm d$ is a real auxiliary scalar field, and

$\displaystyle\ast
F_{4}=D_{\mu}A^{\mu}=\frac{1}{e}\partial_{\mu}(eA^{\mu})\;,\qquad e=\det
e_{\mu}^{a}=\sqrt{-g}\;.$ (C.14)

Fixing the WZ gauge and integrating over the fermionic coordinates of
superspace in the superfield action we arrive at the following Lagrangian in
the bosonic limit:

$\displaystyle\frac{1}{\sqrt{-g}}{\cal
L}=\frac{1}{2}\,Re^{-\frac{1}{3}K}+\Omega_{s\bar{s}}\partial_{\mu}s\partial^{\mu}\bar{s}-\frac{1}{3}e^{-\frac{1}{3}K}(M+K_{\bar{s}}\bar{\mathcal{F}})\,({\bar{M}}+K_{s}\mathcal{F})-{M}\bar{W}-{\bar{M}}W$
$\displaystyle+e^{-\frac{1}{3}K}K_{s\bar{s}}\mathcal{F}\bar{\mathcal{F}}+\mathcal{F}W_{s}+\bar{\mathcal{F}}\bar{W}_{\bar{s}}-\frac{1}{9}\Omega
b_{\mu}b^{\mu}-\frac{i}{3}b^{\mu}(\partial_{\mu}s\,\Omega_{s}-\partial_{\mu}\bar{s}\Omega_{\bar{s}})+\frac{1}{\sqrt{-g}}\mathcal{L}_{bd}\,$
(C.15)

This Lagrangian formally coincides with that of scalar multiplets interacting
with supergravity (C) up to boundary term, and up to the composite nature of
the F-component ${\cal F}$ of the special chiral superfields (C.13).

Before substituting the expression for the F-components of the special chiral
superfields, (C.13), it is convenient to perform a Weyl rescaling of the
fields with (C.3), which will bring our action to the Einstein frame. In the
follwoing, will carefully consider the case of supergravity interacting with
special chiral superfields describing 3-form multiplets, as this case has some
peculiarities with respect to the case of generic chiral superfields
describing scalar supermultiplets.

####  C.1.1 Super-Weyl transformations

At the beginning of this appendix, when considering the interaction of
supergravity an chiral superfields $\Phi$, we assumed that each superfield
$\Phi$ and its components $\phi$ and $F$ are inert under Weyl transformations.
This is a consistent assumption in the case of a generic chiral
superfield.212121This can be checked by writing the chiral field in terms of
an unconstrained complex superfield (with a similar definition as (C.9)).
Choosing the Weyl weights of the transformation accordingly, it can be shown
that a general chiral superfield $\Phi$ is invariant under these rescalings.
However, since the chiral superfield $S$ has been written in terms of a real
superfield $\mathcal{P}$, it is important to check how _super-Weyl
transformations_ , act on them. These are defined via the following
transformations of supervielbein [25, 68]:

$\displaystyle E^{a}$ $\displaystyle\mapsto$
$\displaystyle\tilde{E}{}^{a}=e^{\Upsilon+\bar{\Upsilon}}E^{a}\;,\qquad$
(C.16) $\displaystyle E^{\alpha}$ $\displaystyle\mapsto$
$\displaystyle\tilde{E}{}^{\alpha}=e^{2\bar{\Upsilon}-{\Upsilon}}\left(E^{\alpha}-\frac{i}{4}E^{a}\bar{\cal
D}_{\dot{\alpha}}\bar{\Upsilon}\tilde{\sigma}{}_{a}^{\dot{\alpha}{\alpha}}\right)\;,\qquad$
(C.17) $\displaystyle\bar{E}{}^{\dot{\alpha}}$ $\displaystyle\mapsto$
$\displaystyle\tilde{\bar{E}}{}^{\dot{\alpha}}=e^{2{\Upsilon}-\bar{\Upsilon}}\left(\bar{E}{}^{\dot{\alpha}}+\frac{i}{4}E^{a}\tilde{\sigma}{}_{a}^{\dot{\alpha}{\alpha}}{\cal
D}_{\alpha}{\Upsilon}\right)\;,\qquad$ (C.18)

where ${\Upsilon}$ is a chiral superfield

$\displaystyle\bar{D}_{\dot{\alpha}}{\Upsilon}=0\;,\qquad
D_{\alpha}\bar{\Upsilon}=0\;.$ (C.19)

These transformations must be supplemented by a suitable transformations of
the spin connection; however, their explicit form is not needed for our
purposes (see [25, 68] for more detail).

It is important to note that the supergravity chiral projector transforms in
an inhomogeneous way under super-Weyl transformations:

$\displaystyle(\bar{D}\bar{D}-8{\mathcal{R}})\mapsto
e^{-4{\Upsilon}}(\bar{D}\bar{D}-{\mathcal{R}})e^{2\bar{\Upsilon}},\qquad(DD-8\bar{\mathcal{R}})\mapsto
e^{-4\bar{\Upsilon}}(DD-8\bar{\mathcal{R}})e^{2\Upsilon}.$ (C.20)

In the case of a generic chiral superfield
$\Phi=(\bar{D}\bar{D}-8{\mathcal{R}})P$, this projector acts on the generic
complex superfield potential $P$. Choosing the transformation of this
superfield to be $P\mapsto e^{+4\bar{\Upsilon}}e^{-2\Upsilon}P$ we can
actually make $\Phi$ inert under the super-Weyl tranformations.

On the other hand, this is not possible in the case of a special chiral
superfield $S$ (C.9) constructed from the real superfield prepotential ${\cal
P}=({\cal P})^{*}$. In this case, in the light of (C.20), the only way to
obtain a covariant super-Weyl transformation of the special chiral superfields
(C.9) is to attribute to its real prepotential the transformation rule

$\displaystyle{\cal P}\,\mapsto{\cal P}\,e^{-2\Upsilon-2\bar{\Upsilon}}$
(C.21)

which results in

$\displaystyle S\,\mapsto S\,e^{-6\Upsilon}$ (C.22)

and in the following transformations of its leading component222222We do not
write transformation of $P|$ explicitly because it vanishes in the Wess-Zumino
gauge (C.11).

$\displaystyle s\,\mapsto s\,e^{-6\Upsilon|}\;.$ (C.23)

We will be interested in the purely bosonic part of the super-Weyl
transformations with

$\displaystyle\Upsilon|=\frac{1}{12}K=\bar{\Upsilon}|\;,\qquad
D_{\alpha}\Upsilon|=0\;,\qquad D^{2}\Upsilon|=0\;,$ (C.24)

since, in that case,

$\displaystyle e_{\mu}^{a}\;\mapsto\;e_{\mu}^{a}\,e^{\frac{1}{6}K}\;$ (C.25)

as needed to write the Lagrangian in Einstein frame, just as in the case of
supergravity interacting with chiral multiplets. However, in this scenario,
both $M$ and $\mathcal{F}$ will be affected by this transformation. This can
be seen from

$\displaystyle(D^{2}-8\bar{\mathcal{R}})S|=-4\mathcal{F}-8\mathcal{R}S|=-4\mathcal{F}+\frac{4}{3}s\bar{M}$
(C.26)

whose transformation with the use of (C.23) and (C.24) results in

$\displaystyle s\ $ $\displaystyle\mapsto\ e^{-\frac{1}{2}K}s$ (C.27)
$\displaystyle\mathcal{F}\ $ $\displaystyle\mapsto\
e^{-\frac{2}{3}K}\mathcal{F}$ (C.28) $\displaystyle M\ $
$\displaystyle\mapsto\ e^{-\frac{1}{6}K}M\;.$ (C.29)

As far as the scalar field is concerned, it is convenient to combine the Weyl
rescaling with the field redefinition

$\displaystyle\phi:=e^{-\frac{1}{2}K}s$ (C.30)

so that the kinetic term of $\phi$ field remains in a canonical form.

Taking all of the above into account and integrating out the auxiliary field
$b^{\mu}$ using its algebraic equations of motion, we find that

$\displaystyle\frac{1}{\sqrt{-g}}{\cal L}=$
$\displaystyle\frac{1}{2}\,R-K_{\phi\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}{}^{J}-\frac{1}{3}(M+e^{-\frac{1}{2}K}K_{\bar{\phi}}\bar{\mathcal{F}})\,({\bar{M}}+e^{-\frac{1}{2}K}K_{\phi}\mathcal{F})$
$\displaystyle-e^{\frac{1}{2}K}{M}\bar{W}-e^{\frac{1}{2}K}{\bar{M}}W+e^{-K}K_{\phi\bar{\phi}}\mathcal{F}\bar{\mathcal{F}}+\mathcal{F}W_{\phi}+\bar{\mathcal{F}}\bar{W}_{\bar{\phi}}+\frac{1}{\sqrt{-g}}\mathcal{L}_{bd}.$
(C.31)

It is convenient to further redefine the supergravity auxiliary field $M$ as

$\displaystyle\check{M}\equiv Me^{\frac{1}{2}K}$ (C.32)

so that the Einstein frame Lagrangian reads

$\displaystyle\frac{1}{\sqrt{-g}}{\cal L}=$
$\displaystyle\frac{1}{2}\,R-K_{\phi\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-\frac{1}{3}e^{-K}(\check{M}+K_{\bar{\phi}}\bar{\mathcal{F}})\,(\check{\bar{M}}+K_{\phi}\mathcal{F})-\check{M}\bar{W}-\check{\bar{M}}W$
$\displaystyle+e^{-K}K_{\phi\bar{\phi}}\mathcal{F}\bar{\mathcal{F}}+\mathcal{F}W_{\phi}+\bar{\mathcal{F}}\bar{W}_{\bar{\phi}}+\frac{1}{\sqrt{-g}}\mathcal{L}_{bd}$
(C.33)

with the composite $\mathcal{F}$ field of the form

$\displaystyle\mathcal{F}=\frac{1}{2}\left(D_{\mu}A^{\mu}+i{\rm
d}\right)+\frac{2}{3}\bar{\phi}{}\check{M}+\frac{1}{3}\phi\check{\bar{M}}.$
(C.34)

The boundary term can be obtained in the same fashion as in the previous
section. Since the 3-form field enters our action only through
$D_{\mu}A^{\mu}=\frac{1}{e}\partial_{\mu}(eA^{\mu})$, varying the action with
respect to this field gives

$\displaystyle\delta S$ $\displaystyle=\int d^{4}x\ e\
\frac{\partial\mathcal{L}}{\partial(D_{\mu}A^{\mu})}\
\left(\frac{1}{e}\partial_{\mu}(e\ \delta A^{\mu})\right)$
$\displaystyle=-\int d^{4}x\ e\ \delta A^{\mu}\
\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial(D_{\mu}A^{\mu})}\right)+\int
d^{4}x\ \partial_{\mu}\left(e\ \delta
A^{\mu}\frac{\partial\mathcal{L}}{\partial(D_{\mu}A^{\mu})}\right).$ (C.35)

Therefore, we conclude that the boundary term, which makes the variational
problem well posed, includes, besides the Gibbons-Hawking term [35] and its
superpartner (see [69, 70]),

$\displaystyle\mathcal{L}_{bd}=-\int d^{4}x\ \partial_{\mu}\left(e\
A^{\mu}\frac{\partial\mathcal{L}}{\partial(D_{\mu}A^{\mu})}\right).$ (C.36)

The auxiliary field and 3-form equations of motion are then

$\displaystyle\frac{\partial\mathcal{L}}{\partial{\rm
d}}=0,\qquad\frac{\partial\mathcal{L}}{\partial\check{M}}=0,\quad\text{and}\quad\partial_{\nu}\left(\frac{\partial\mathcal{L}}{\partial(D_{\mu}A^{\mu})}\right)=0\
\Rightarrow\ \frac{\partial\mathcal{L}}{\partial(D_{\mu}A^{\mu})}=n,$ (C.37)

where $n\in\mathbb{R}$. These algebraic equations are solved by

$\displaystyle{\rm d}$
$\displaystyle=-iK^{\phi\bar{\phi}}e^{K}\left[D_{\phi}(W-n\phi)-\text{c.c.}\right]$
(C.38) $\displaystyle\check{M}$
$\displaystyle=e^{K}\left[D_{\phi}(W-n\phi)K^{\phi\bar{\phi}}K_{\bar{\phi}}-3(W-n\phi)\right]$
(C.39) $\displaystyle D_{\mu}A^{\mu}$
$\displaystyle=e^{K}\left[3(W-n\phi)\bar{\phi}-(1+\bar{\phi}K_{\bar{\phi}})K^{\phi\bar{\phi}}D_{\phi}(W-n\phi)+\text{c.c.}\right]$
(C.40)

where $D_{\phi}=\partial_{\phi}+K_{\phi}$ is the usual Kähler-covariant
derivative. Plugging (C.38)– (C.40) into (C.33) and taking into account the
contribution of the boundary term yields

$\displaystyle\frac{1}{\sqrt{-g}}\mathcal{L}=\frac{1}{2}R-K_{\phi\bar{\phi}}\partial_{\mu}\phi\partial^{\mu}\bar{\phi}-e^{K}\left(D_{\phi}\hat{W}K^{\phi\bar{\phi}}D_{\bar{\phi}}\hat{\bar{W}}-3\left|\hat{W}\right|\right)$
(C.41)

where, exactly as in the non-gravitational case, the superpotential is shifted
my a term linear in the scalar field multiplied by the constant flux of the
3-form field:

$\displaystyle\hat{W}\equiv W-n\phi.$ (C.42)

Therefore, in summary, after all the auxiliary fields’ equations of motion
have been solved and used in the action, the contribution of the 3-form fields
amount to adding a simple linear term to the superpotential, which is
proportional to the constant flux $n$ of the 3-form field. The effective
action for the scalar fields and gravity coincides with the usual N=1, $D=4$
matter-coupled supergravity, albeit with a potential constructed from the new
effective superpotential (C.42).

## References

  * [1] M. Cvetic, S. Griffies and S.-J. Rey, _Nonperturbative stability of supergravity and superstring vacua_ , _Nucl. Phys. B_ 389 (1993) 3 [hep-th/9206004].
  * [2] S.R. Coleman and F. De Luccia, _Gravitational Effects on and of Vacuum Decay_ , _Phys. Rev. D_ 21 (1980) 3305.
  * [3] M. Cvetic, S. Griffies and S.-J. Rey, _Static domain walls in N=1 supergravity_ , _Nucl. Phys. B_ 381 (1992) 301 [hep-th/9201007].
  * [4] R. Bousso and J. Polchinski, _Quantization of four form fluxes and dynamical neutralization of the cosmological constant_ , _JHEP_ 06 (2000) 006 [hep-th/0004134].
  * [5] S.R. Coleman, _The Fate of the False Vacuum. 1. Semiclassical Theory_ , _Phys. Rev. D_ 15 (1977) 2929.
  * [6] C.G. Callan, Jr. and S.R. Coleman, _The Fate of the False Vacuum. 2. First Quantum Corrections_ , _Phys. Rev. D_ 16 (1977) 1762.
  * [7] J.S. Schwinger, _On gauge invariance and vacuum polarization_ , _Phys. Rev._ 82 (1951) 664.
  * [8] J.J. Blanco-Pillado, D. Schwartz-Perlov and A. Vilenkin, _Quantum Tunneling in Flux Compactifications_ , _JCAP_ 12 (2009) 006 [0904.3106].
  * [9] J.D. Brown and C. Teitelboim, _Dynamical Neutralization of the Cosmological Constant_ , _Phys. Lett. B_ 195 (1987) 177.
  * [10] J.D. Brown and C. Teitelboim, _Neutralization of the Cosmological Constant by Membrane Creation_ , _Nucl. Phys. B_ 297 (1988) 787.
  * [11] B. Freivogel, M. Kleban, M. Rodriguez Martinez and L. Susskind, _Observational consequences of a landscape_ , _JHEP_ 03 (2006) 039 [hep-th/0505232].
  * [12] S.J. Gates, Jr., _Super p-form gauge superfields_ , _Nucl. Phys. B_ 184 (1981) 381.
  * [13] P. Binetruy, F. Pillon, G. Girardi and R. Grimm, _The Three form multiplet in supergravity_ , _Nucl. Phys. B_ 477 (1996) 175 [hep-th/9603181].
  * [14] B.A. Ovrut and D. Waldram, _Membranes and three form supergravity_ , _Nucl. Phys. B_ 506 (1997) 236 [hep-th/9704045].
  * [15] S.M. Kuzenko, _Variant supercurrent multiplets_ , _JHEP_ 04 (2010) 022 [1002.4932].
  * [16] S.M. Kuzenko, _Variant supercurrents and Noether procedure_ , _Eur. Phys. J. C_ 71 (2011) 1513 [1008.1877].
  * [17] I.A. Bandos and C. Meliveo, _Three form potential in (special) minimal supergravity superspace and supermembrane supercurrent_ , _J. Phys. Conf. Ser._ 343 (2012) 012012 [1107.3232].
  * [18] F. Farakos, S. Lanza, L. Martucci and D. Sorokin, _Three-forms in Supergravity and Flux Compactifications_ , _Eur. Phys. J. C_ 77 (2017) 602 [1706.09422].
  * [19] I.A. Bandos and C. Meliveo, _Superfield equations for the interacting system of D=4 N=1 supermembrane and scalar multiplet_ , _Nucl. Phys. B_ 849 (2011) 1 [1011.1818].
  * [20] I.A. Bandos and C. Meliveo, _Supermembrane interaction with dynamical D=4 N=1 supergravity. Superfield Lagrangian description and spacetime equations of motion_ , _JHEP_ 08 (2012) 140 [1205.5885].
  * [21] I. Bandos, F. Farakos, S. Lanza, L. Martucci and D. Sorokin, _Three-forms, dualities and membranes in four-dimensional supergravity_ , _JHEP_ 07 (2018) 028 [1803.01405].
  * [22] S. Lanza, F. Marchesano, L. Martucci and D. Sorokin, _How many fluxes fit in an EFT?_ , _JHEP_ 10 (2019) 110 [1907.11256].
  * [23] S. Lanza, _Exploring the Landscape of effective field theories_ , Ph.D. thesis, Padua U., 2019.  1912.08935.
  * [24] D.J.H. Chung, L.L. Everett, G.L. Kane, S.F. King, J.D. Lykken and L.-T. Wang, _The Soft supersymmetry breaking Lagrangian: Theory and applications_ , _Phys. Rept._ 407 (2005) 1 [hep-ph/0312378].
  * [25] J. Wess and J. Bagger, _Supersymmetry and supergravity_ , Princeton University Press, Princeton, NJ, USA (1992).
  * [26] A. Masoumi, K.D. Olum and B. Shlaer, _Efficient numerical solution to vacuum decay with many fields_ , _JCAP_ 1701 (2017) 051 [1610.06594].
  * [27] E. Dyer and K. Hinterbichler, _Boundary Terms, Variational Principles and Higher Derivative Modified Gravity_ , _Phys. Rev. D_ 79 (2009) 024028 [0809.4033].
  * [28] K.S. Stelle and P.C. West, _Minimal Auxiliary Fields for Supergravity_ , _Phys. Lett. B_ 74 (1978) 330.
  * [29] V. Ogievetsky and E. Sokatchev, _Equation of Motion for the Axial Gravitational Superfield_ , _Sov. J. Nucl. Phys._ 32 (1980) 589\.
  * [30] S.J. Gates, Jr. and W. Siegel, _VARIANT SUPERFIELD REPRESENTATIONS_ , _Nucl. Phys. B_ 187 (1981) 389.
  * [31] M.T. Grisaru and W. Siegel, _SUPERGRAPHITY. PART 1. BACKGROUND FIELD FORMALISM_ , _Nucl. Phys. B_ 187 (1981) 149.
  * [32] S.M. Kuzenko and S.A. McCarthy, _On the component structure of N=1 supersymmetric nonlinear electrodynamics_ , _JHEP_ 05 (2005) 012 [hep-th/0501172].
  * [33] I.A. Bandos and C. Meliveo, _On supermembrane supercurrent and special minimal supergravity_ , _Fortsch. Phys._ 60 (2012) 868.
  * [34] S.M. Kuzenko and G. Tartaglino-Mazzucchelli, _Complex three-form supergravity and membranes_ , _JHEP_ 12 (2017) 005 [1710.00535].
  * [35] G. Gibbons and S. Hawking, _Action Integrals and Partition Functions in Quantum Gravity_ , _Phys.Rev._ D15 (1977) 2752.
  * [36] I.A. Bandos, J.A. De Azcarraga and J.M. Izquierdo, _Supergravity interacting with bosonic p-branes and local supersymmetry_ , _Phys. Rev. D_ 65 (2002) 105010 [hep-th/0112207].
  * [37] I. Bandos and J.A. de Azcarraga, _Dirac equation for the supermembrane in a background with fluxes from a component description of the D=11 supergravity-supermembrane interacting system_ , _JHEP_ 09 (2005) 064 [hep-th/0507197].
  * [38] 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 [hep-th/0605266].
  * [39] E. Bergshoeff, R. Kallosh, T. Ortin and G. Papadopoulos, _Kappa symmetry, supersymmetry and intersecting branes_ , _Nucl. Phys. B_ 502 (1997) 149 [hep-th/9705040].
  * [40] L. Randall and R. Sundrum, _An Alternative to compactification_ , _Phys. Rev. Lett._ 83 (1999) 4690 [hep-th/9906064].
  * [41] E.J. Weinberg, _Classical solutions in quantum field theory: Solitons and Instantons in High Energy Physics_ , Cambridge Monographs on Mathematical Physics, Cambridge University Press (9, 2012), 10.1017/CBO9781139017787.
  * [42] A. Masoumi, S. Paban and E.J. Weinberg, _Tunneling from a Minkowski vacuum to an AdS vacuum: A new thin-wall regime_ , _Phys. Rev. D_ 94 (2016) 025023 [1603.07679].
  * [43] J.J. Blanco-Pillado, M. Gomez-Reino and K. Metallinos, _Accidental Inflation in the Landscape_ , _JCAP_ 02 (2013) 034 [1209.0796].
  * [44] A. Masoumi, A. Vilenkin and M. Yamada, _Inflation in random Gaussian landscapes_ , _JCAP_ 05 (2017) 053 [1612.03960].
  * [45] A. Masoumi, A. Vilenkin and M. Yamada, _Initial conditions for slow-roll inflation in a random Gaussian landscape_ , _JCAP_ 07 (2017) 003 [1704.06994].
  * [46] A. Masoumi, A. Vilenkin and M. Yamada, _Inflation in multi-field random Gaussian landscapes_ , _JCAP_ 12 (2017) 035 [1707.03520].
  * [47] J.J. Blanco-Pillado, A. Vilenkin and M. Yamada, _Inflation in Random Landscapes with two energy scales_ , _JHEP_ 02 (2018) 130 [1711.00491].