11institutetext: Department of Physics, Lehigh University, Bethlehem, PA,
18018, USA22institutetext:  Kavli Institute of Theoretical Physics, University
of California Santa Barbara, Santa Barbara,CA, 93106, USA33institutetext:
Laboratoire d’Annecy-le-Vieux de Physique Théorique (LAPTh), CNRS, Université
Savoie Mont Blanc (USMB), 74940 Annecy, France

# On Asymptotic Dark Energy in String Theory

Sera Cremonini<EMAIL_ADDRESS>1 Eduardo Gonzalo
<EMAIL_ADDRESS>1 Muthusamy Rajaguru<EMAIL_ADDRESS>1 Yuezhang Tang<EMAIL_ADDRESS>1,3 and Timm Wrase<EMAIL_ADDRESS>
###### Abstract

We examine bounds on accelerated expansion in asymptotic regions of the moduli
space in string theory compactifications to four spacetime dimensions. While
there are conjectures that forbid or constrain accelerated expansion in such
asymptotic regions, potential counter examples have been discussed recently in
the literature. We check whether such counter examples can arise in explicit
string theory constructions, focusing in particular on non-geometric
compactifications of type IIB string theory that have no Kähler moduli. We
find no violation of the Strong Asymptotic dS Conjecture and thus provide
support for the absence of accelerated expansion in asymptotic regions of a
barely explored corner of the string landscape. Moreover, working in a
simplified setting, we point out a new mechanism for potentially connecting
the Sharpened Distance Conjecture and the Strong Asymptotic dS Conjecture. If
this argument could be generalized, it would mean that the Sharpened Distance
Conjecture is implied by the Strong Asymptotic dS Conjecture, and that their
exponential factors are naturally related by a factor of 2.

##  1 Introduction

One of the major challenges in string phenomenology is the construction of
universes with positive vacuum energy that explain the observed dark energy
SupernovaSearchTeam:1998fmf ; SupernovaCosmologyProject:1996grv . Current
cosmological data is compatible both with a constant and a time dependant
vacuum energy Planck:2018vyg . The first possibility means our universe is
described by either a stable or metastable de Sitter (dS) vacuum. From the
model building point of view this requires the stabilization of all moduli in
a non-supersymmetric vacuum. Without supersymmetry it is more difficult to
guarantee control over the low energy four-dimensional effective action
governing the theory. However, multiple efforts have been directed towards
finding dS vacua in string theory, starting with the seminal papers
Kachru:2003aw ; Balasubramanian:2005zx . These constructions stabilize moduli
at finite values and argue that the corrections that are neglected do not
change the results. Given that not all such corrections are explicitly known,
there is an on-going discussion about the validity of all the existing dS
constructions in string theory, see for example Danielsson:2018ztv ;
Gao:2020xqh ; Lust:2022lfc ; Junghans:2022exo ; Cicoli:2023opf . The second
scenario consists in building a so-called quintessence model, where scalar
fields are allowed to roll slowly Hellerman:2001yi ; Fischler:2001yj ;
Kaloper:2008qs ; Cicoli:2012tz ; Cicoli:2018kdo ; Hebecker:2019csg ;
ValeixoBento:2020ujr ; Brinkmann:2022oxy . From a model building point of view
this option needs to meet additional requirements; for example, there are
bounds on the size of the mass of the quintessence fields.

Before aspiring to build realistic quintessence models from string theory, it
is more prudent to establish whether the latter allows for quintessence at
all. Asymptotic limits in moduli space have the advantage of providing more
control due to suppressed corrections and are easier to study. As a first
step, it would therefore be useful to understand if scalar potentials arising
in string theory could asymptotically allow for quintessence. Such a question
fits perfectly into the theme of the swampland program Vafa:2005ui ;
vanBeest:2021lhn ; Agmon:2022thq ; Palti:2019pca ; Grana:2021zvf , which aims
to gain insights into the validity of low-energy effective field theories
(EFTs) using quantum gravity. A common procedure within the swampland program
is to collect sets of results and ideas and bundle them together in a single
statement in the form of a conjecture. The asymptotic behavior of scalar
potentials arising in string theory has been the subject of several swampland
investigations Ooguri:2006in ; Obied:2018sgi ; Banlaki:2018ayh ;
Junghans:2018gdb ; Ooguri:2018wrx ; Hebecker:2018vxz ; Lust:2019zwm  resulting
in the swampland dS Conjecture discussed in detail in the next paragraph. In
addition to this particular conjecture, there exist several lower bounds on
the slow-roll parameters which were obtained in the context of searches for dS
vacua and inflation in specific string theory setups Hertzberg:2007wc ;
Haque:2008jz ; Flauger:2008ad ; Caviezel:2009tu ; Wrase:2010ew ; Shiu:2011zt ;
Andriot:2016xvq ; Andriot:2018ept ; Andriot:2022xjh ; Andriot:2023isc . These
papers provide no-go theorems that apply at any point in moduli space and
usually forbid slow-roll inflation by showing that the slow-roll parameter
$\epsilon_{V}$ characterizing the slope of the potential obeys
$\epsilon_{V}\geq 1$. Hence they help narrow down the search for asymptotic
accelerated expansion from slowly rolling scalar fields by excluding large
classes of models.

The scalar potentials arising from string compactifications in the asymptotic
limits of moduli space are expected to take the following form, $V\sim
e^{-\gamma\phi}$, where $\phi$ is a canonical field and $\gamma$ is an $O(1)$
number. For accelerated expansion in four spacetime dimensions we would
require $\gamma=\frac{||\nabla V(\phi)||}{V(\phi)}<\sqrt{2}$. It was shown in
Cicoli:2021fsd  that type IIA/B and heterotic Calabi-Yau compactifications
cannot have $\gamma<\sqrt{2}$ in the large volume, weak coupling limit. The
swampland dS Conjecture in the asymptotic limit Ooguri:2018wrx ;
Hebecker:2018vxz , which is a generalization of the Dine-Seiberg problem
Dine:1985he , provides a rough bound of $\gamma\gtrsim\mathcal{O}(1)$ and
hence provides preliminary motivation for the potential absence of asymptotic
quintessence. Using dimensional reduction techniques Rudelius:2021azq ;
Rudelius:2021oaz ; Etheredge:2022opl , the bound was sharpened by Rudelius
Rudelius:2022gbz , see also Bedroya:2019snp . This led to the so-called Strong
Asymptotic dS Conjecture, which states that $\gamma\geq\sqrt{2}$ in four
dimensions and more generically $\gamma\geq 2/\sqrt{d-2}$ in $d$ dimensions.
This forbids in any asymptotic limit of string theory compactifications the
existence of solutions that are undergoing accelerated expansion (see also
Marconnet:2022fmx ; Apers:2022cyl ; Shiu:2023nph ; Shiu:2023rxt  for closely
related work).111Accelerated expansion is forbidden for $\gamma\geq\sqrt{2}$,
if the 4d spacetime has no curvature, which is the case we focus on in this
paper. For negative curvature it was shown in Marconnet:2022fmx  that
accelerated expansion is actually possible.

The authors of Calderon-Infante:2022nxb  have pointed out a potential loophole
that could then allow us to realize quintessence asymptotically in string
theory. In particular, they argue that having two terms compete asymptotically
in the scalar potential would provide a possible way out of the conjecture of
Rudelius:2022gbz , and they identify several potential examples that could
violate the bound of Rudelius:2022gbz . As already noted by the authors, the
examples of Calderon-Infante:2022nxb  do not account for the stabilization of
the volume modulus. If the latter is not stabilized then this runaway
direction is too steep to preserve accelerated expansion. The possible
loophole suggested in Calderon-Infante:2022nxb  is intriguing and we believe
requires additional investigation.

In this work, we attempt to take the first steps towards a realization of the
scenario described above, in which two terms in the potential compete
asymptotically. We will focus on non-geometric type IIB models whose moduli
space includes only the dilaton and the complex structure moduli. These models
were first introduced in Becker:2006ks ; Becker:2007dn  and were shown to be a
good testing ground for swampland conjectures in Ishiguro:2021csu ;
Bardzell:2022jfh ; Becker:2022hse . We show that in our explicit model there
is no violation of the Strong Asymptotic dS Conjecture. Thus, we are showing
the absence of accelerated expansion in asymptotic limits of moduli space in a
non-geometric and thus very stringy setup.

We also recall the previous observation that the minimal exponential
prefactors in the Strong Asymptotic dS Conjecture and the Sharpened Distance
Conjecture Ooguri:2006in ; Klaewer:2016kiy ; Etheredge:2022opl  are related by
a factor of 2. As mentioned above, for the Strong Asymptotic dS Conjecture we
have $\gamma\geq\gamma_{min}=2/\sqrt{d-2}$ in a generic $d$-dimensional
spacetime. For the Sharpened Distance Conjecture a tower of massive states
becomes light in the asymptotic limit $\phi\to\infty$, with masses of the form
$m(\phi)\sim e^{-\alpha\phi}$ with $\alpha\geq\alpha_{min}=1/\sqrt{d-2}$.
Somewhat surprisingly $\gamma_{min}=2\alpha_{min}$. An argument was given in
Etheredge:2022opl  for why the Sharpened Distance Conjecture, together with
the Emergent String Conjecture Lee:2019wij ; Lee:2019xtm , implies the Strong
Asymptotic dS Conjecture. In this paper, we provide a potential explanation
for this factor of 2, which is distinct from the argument given in
Etheredge:2022opl . If our argument can be generalized, then this would mean
that the Sharpened Distance Conjecture follows from the absence of asymptotic
accelerated expansion in string theory. Thus, this would be complementary to
the argument of Etheredge:2022opl , and provide an interesting new motivation
for the Sharpened Distance Conjecture.

The outline of this paper is as follows. In section 2 we review previous
results pertaining to slow-roll inflation and quintessence in string
compactifications. In section 3 we study the possibility of violating the
Strong Asymptotic dS Conjecture in non-geometric compactifications of type IIB
string theory. In section 4 we spell out evidence for a potential direct
connection between the Strong Asymptotic dS Conjecture and the Sharpened
Distance Conjecture. We summarize our results and conclude in section 5. Two
appendices contain technical details.

##  2 Review of previous results

In this section, we review the status of obtaining accelerated expansion in
asymptotic limits of the moduli space. We focus here on the case of slow-roll
type inflation/quintessence in asymptotic limits of string theory
compactifications to four dimensions. These setups are the most studied in the
literature and provide an interesting test bed for the conjecture that
accelerated expansion is forbidden in any asymptotic limit Rudelius:2022gbz .
In particular, it was argued in Calderon-Infante:2022nxb  that there could be
counter-examples if one studies infinite distance limits in which not a single
but multiple terms in the scalar potential are dominant in the asymptotic
limit. The caveat in Calderon-Infante:2022nxb  was however that the authors
studied only the complex structure sector of F-theory compactifications and
did not stabilize the Kähler moduli. In these setups, there was thus always a
steep direction associated with the overall volume of the CY4 manifold that
prevented any accelerated expansion. In the next section we study non-
geometric type IIB flux compactifications _without_ Kähler moduli and in
particular without a volume modulus Becker:2006ks ; Becker:2007dn  to see
whether the counter-examples of Calderon-Infante:2022nxb  can be realized in
such setups. Before doing that we review in this section the previous related
results in some detail.

Usually, the scalar potential falls off exponentially when we approach an
asymptotic point in moduli space, i.e. $V(\phi)\sim e^{-\gamma\phi}$ for very
large $\phi$ (see for example Ooguri:2018wrx ; Hebecker:2018vxz  for a recent
related discussion). While we find below cases where this is more subtle, it
makes sense to phrase this discussion in terms of the prefactor $\gamma$. The
requirement for slow-roll inflation, and a corresponding accelerated
expansion, is that the two slow-roll parameters are smaller than one

$\epsilon_{V}=\frac{1}{2}\left(\frac{|\nabla
V|}{V}\right)^{2}<1\,,\qquad\eta_{V}=\left|\text{min}\left(\frac{\nabla_{i}\nabla_{j}V}{V}\right)\right|<1\,.$
(1)

For a single scalar field with $V(\phi)\sim e^{-\gamma\phi}$ this amounts to
the bound $\gamma<\sqrt{2}$ in four dimensions. On the other hand, the Strong
Asymptotic dS Conjecture states that $\gamma\geq\sqrt{2}$ in asymptotic limits
of the scalar field space, see Rudelius:2021azq ; Rudelius:2021oaz ;
Etheredge:2022opl ; Rudelius:2022gbz .222The related Trans-Planckian
Censorship Conjecture (TCC) Bedroya:2019snp  imposes the weaker bound
$\gamma\geq\sqrt{2/3}$. However, notice that it was mentioned already in
Bedroya:2019snp  that their bound can become stronger for exponentially
decreasing potentials, implying the Strong Asymptotic dS Conjecture. For
consistency with the literature we will reserve the name TCC for the weaker
bound $\gamma\geq\sqrt{2/3}$. Thus, the question of whether or not the Strong
Asymptotic dS Conjecture is true is equivalent to the question of whether
accelerated expansion is possible in asymptotic limits of the moduli space.
Recently, the authors of Shiu:2023nph ; Shiu:2023rxt  pointed out that slow-
roll inflation/quintessence is non-generic since it requires
$\epsilon_{V}=\gamma^{2}/2\ll 1$, which is usually not the case. For a single
field with an exponential potential $\epsilon=\epsilon_{V}=\gamma^{2}/2$ even
if the slow-roll approximation is not valid. However, for more complicated
trajectories in generic multifield potentials, one has to solve the full set
of equations to check whether accelerated expansion is possible. This
difference will not be important to us below since we will not find
sufficiently shallow potentials.

Motivated by the observed accelerated expansion of our own universe
SupernovaSearchTeam:1998fmf ; SupernovaCosmologyProject:1996grv , string
theorists have been trying to construct corresponding models in string theory
for more than two decades. This has led to many interesting new insights and
also excluded certain classes of models that can never give rise to four
dimensional spacetimes that undergo accelerated expansion. Next we recall the
for us relevant no-go theorems and their loopholes.

In Hertzberg:2007wc  the authors proved the absence of dS vacua and slow-roll
inflation in flux compactifications of massive type IIA on CY3 manifolds. In
order to do that they focused only on two universal moduli, the 4d dilaton
$\tau$ and the overall volume modulus $\rho$. These scalar fields decouple
from the other scalar fields in the sense that the slow-roll parameter
$\epsilon_{V}$ takes the form 333$\tau$ and $\rho$ are the commonly used
scalar fields that are however not canonically normalized, leading to the
additional factors in the expression for $\epsilon_{V}$.

$\epsilon_{V}=\frac{1}{3}\left(\frac{\rho\partial_{\rho}V}{V}\right)^{2}+\frac{1}{4}\left(\frac{\tau\partial_{\tau}V}{V}\right)^{2}+|\ldots|^{2}\geq\frac{1}{3}\left(\frac{\rho\partial_{\rho}V}{V}\right)^{2}+\frac{1}{4}\left(\frac{\tau\partial_{\tau}V}{V}\right)^{2}\,,$
(2)

where $|\ldots|^{2}$ denotes the positive or zero contribution from all the
other scalar fields in the theory. Given the scaling of the scalar potential
in type IIA Calabi-Yau flux compactifications, one can show that the two terms
on the right-hand-side of equation (2) are larger or equal to 27/13
Hertzberg:2007wc . This means that accelerated expansion is forbidden
_anywhere_ in moduli space. Therefore the Strong Asymptotic dS Conjecture is
trivially satisfied in massive type IIA flux compactifications on Calabi-Yau
manifolds.

The above no-go theorem was extended to _massless_ type IIA compactifications
on internal manifolds with curvature and many more refined no-go theorems were
derived in Haque:2008jz ; Flauger:2008ad ; Caviezel:2009tu ; Wrase:2010ew ;
Andriot:2016xvq ; Andriot:2018ept . While these no-go theorems aimed to
excluded dS extrema, they usually prove a bound on $\epsilon_{V}$ that
requires it to be larger than one anywhere in moduli space. This means in all
such cases the Strong Asymptotic dS Conjecture holds. In cases where the no-
goes allow for $\epsilon_{V}$ to be smaller than one or in cases where there
is no no-go theorem at all, one can then ask the question of whether it is
possible to have accelerated expansion in an asymptotic region.

The most important and easiest to study asymptotic regions would be a large
volume and weak coupling regime in which both $\alpha^{\prime}$ and string
loop corrections are suppressed. This limit corresponds to sending $\rho$ and
$\tau$ above to infinity and it ensures that one can fully trust the low-
energy supergravity approximation to the particular string theory in question.
Whether dS extrema could exist in this limit was studied in Junghans:2018gdb ;
Banlaki:2018ayh , where it was shown that this is actually not possible in
large classes of type IIA string theory compactifications. The authors of
Cicoli:2021fsd  then asked whether quintessence could be realized with
parametric control using the above form of $\epsilon_{V}$. They found that in
the asymptotic large $\rho$ and $\tau$ limit of heterotic and type II string
theory the potential is always too steep to allow for accelerated expansion
from a slowly rolling scalar field Cicoli:2021fsd . This motivated the
question of whether any other asymptotic limit of string theory can give rise
to an accelerating universe.

Rudelius Rudelius:2022gbz  noted in many examples that any limit in which one
sends a single scalar field to zero leads to a scalar potential $V(\phi)\sim
e^{-\gamma\phi}$ that is so steep that slow-roll inflation and therefore
quintessence cannot arise. This then led to the Strong Asymptotic dS
Conjecture that forbids such accelerated expansion in any asymptotic region of
moduli space. However, as was noted in Calderon-Infante:2022nxb , it is
possible that two (or more) fields become large and approach an asymptotic
limit such that two terms in the scalar potential compete with each other. For
example, one could consider the limit $\phi_{1}=c\,\phi_{2}\to\infty$ for
$c\in\mathbb{R}_{>0}$ with the asymptotic scalar potential

$\displaystyle V(\phi_{1},\phi_{2})$ $\displaystyle=$ $\displaystyle
A_{1}e^{-\gamma_{1,1}\phi_{1}-\gamma_{1,2}\phi_{2}}+A_{2}e^{-\gamma_{2,1}\phi_{1}-\gamma_{2,2}\phi_{2}}+\ldots$
(3) $\displaystyle=$ $\displaystyle
A_{1}e^{-(c\,\gamma_{1,1}+\gamma_{1,2})\phi_{2}}+A_{2}e^{-(c\,\gamma_{2,1}+\gamma_{2,2})\phi_{2}}+\ldots\,,$
(4)

where $\ldots$ represents subleading terms. The first two terms above can
scale exactly in the same way, i.e.
$c\,\gamma_{1,1}+\gamma_{1,2}=c\,\gamma_{2,1}+\gamma_{2,2}$, but the
individual scalings with respect to the moduli $\phi_{1},\phi_{2}$ could be
very different in the two terms. Such competing terms open up the possibility
of getting accelerated expansion along the trajectory
$\phi_{1}=c\,\phi_{2}\to\infty$, even if two individual terms do not allow for
this Calderon-Infante:2022nxb . While this mechanism is extremely interesting,
the authors of Calderon-Infante:2022nxb  restricted themselves to the complex
structure sector and did not study the stabilization of the Kähler moduli. The
explicit potential had an overall $\rho^{-3}$ dependence, i.e.,

$V(\phi_{1},\phi_{2},\rho)=\frac{1}{\rho^{3}}\left(A_{1}e^{-(c\,\gamma_{1,1}+\gamma_{1,2})\phi_{2}}+A_{2}e^{-(c\,\gamma_{2,1}+\gamma_{2,2})\phi_{2}}+\ldots\right)\,.$
(5)

From equation (2) above, we see that this leads to $\epsilon_{V}\geq 3$ and
prohibits slow-roll inflation even if $\phi_{1}$ and $\phi_{2}$ contribute
less than 1 to $\epsilon_{V}$. One could now delve into stabilizing the Kähler
moduli while trying to stay in the asymptotic limits of the complex structure
moduli space that were identified as potential counter-examples in Calderon-
Infante:2022nxb . However, given the lively debate related to this topic, we
refrain from doing so and rather focus on string theory models without Kähler
moduli Becker:2006ks ; Becker:2007dn . Using Landau-Ginzburg techniques it was
shown to be possible to study flux compactifications on generalized Calabi-Yau
manifolds with $h^{1,1}=0$ that are the mirror dual of rigid Calabi-Yau
manifolds Becker:2006ks . This removes the requirement of having to stabilize
Kähler moduli in type IIB flux compactifications and leads to a variety of
interesting results Becker:2006ks ; Becker:2007dn ; Ishiguro:2021csu ;
Bardzell:2022jfh ; Becker:2022hse . In the next section we will review the
corresponding four-dimensional scalar field theories and try to exploit the
loopholes pointed out in Calderon-Infante:2022nxb . Before we do that we
quickly review the results and some notation from Calderon-Infante:2022nxb  in
the next subsection, so that we can build on these results in section 3.

###  2.1 A possible loophole

To find asymptotic trajectories that lead to accelerated expansion we must
solve the Friedmann equations under the slow roll approximation. Consider the
following 4-dimensional action for scalar fields minimally coupled to gravity,

$S=\int
d^{4}x\sqrt{-g}\left(\frac{R}{2}-\frac{1}{2}M_{ab}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}-V(\phi)\right),$
(6)

where $M_{ab}$ is the real-valued metric on the scalar field space. The slow-
roll Friedmann equations then amount to

$\frac{d\phi^{a}}{dt}=-\frac{1}{\sqrt{3V}}M^{ab}\partial_{b}V.$ (7)

Asymptotic trajectories can be parametrized as $\phi^{a}(\sigma)$ as
$\sigma\rightarrow\infty$. As pointed out in Calderon-Infante:2022nxb ,
solving (7) asymptotically can be simplified by taking ratios of pairs of
these equations.

In Calderon-Infante:2022nxb  the authors only study examples with two rolling
fields and note that in order to find a unique asymptotic trajectory it is
necessary to have two terms that compete asymptotically in the potential. For
a simple diagonal field space metric, we note here that a scalar potential
with $n$ moving fields requires $n$ terms competing asymptotically to get a
unique trajectory. Let us exemplify this by considering the following
potential

$V=A_{1}\frac{U_{2}}{U_{1}U_{3}}+A_{2}\frac{1}{U_{2}}\,.$ (8)

Such terms can arise in the scalar potential we discuss below in the next
section. The corresponding diagonal field space metric in that case is

$M^{-1}=\begin{pmatrix}2U_{1}^{2}&0&0\\\ 0&2U_{2}^{2}&0\\\ 0&0&2U_{3}^{2}\\\
\end{pmatrix}.\quad$ (9)

We parameterize the trajectory by
$U_{1}=\alpha_{1}\sigma^{\beta_{1}},U_{2}=\alpha_{2}\sigma^{\beta_{2}},U_{3}=\sigma$,
where $\sigma\rightarrow\infty$. Then from (7) we get,

$\displaystyle\alpha_{1}\beta_{1}\sigma^{\beta_{1}-1}\frac{d\sigma}{dt}$
$\displaystyle=\frac{2}{\sqrt{3V}}\left(A_{1}\alpha_{2}\sigma^{\beta_{2}-1}\right)$
$\displaystyle\alpha_{2}\beta_{2}\sigma^{\beta_{2}-1}\frac{d\sigma}{dt}$
$\displaystyle=\frac{2}{\sqrt{3V}}\left(A_{2}-A_{1}\frac{\alpha_{2}^{2}}{\alpha_{1}}\sigma^{2\beta_{2}-\beta_{1}-1}\right)$
(10) $\displaystyle\frac{d\sigma}{dt}$
$\displaystyle=\frac{2}{\sqrt{3V}}A_{1}\frac{\alpha_{2}}{\alpha_{1}}\sigma^{\beta_{2}-\beta_{1}}.$

Next, following Calderon-Infante:2022nxb  we take ratios of pairs of the above
equations and evaluate them in the limit $\sigma\rightarrow\infty$ to solve
for $\alpha_{1},\alpha_{2}$. The ratio of the first and the third equation
gives $\beta_{1}=1$. The ratio of the second and third equations then gives,

$\beta_{2}=\frac{A_{2}\sigma^{1-\beta_{2}}-A_{1}\frac{\alpha_{2}^{2}}{\alpha_{1}}\sigma^{\beta_{2}-1}}{A_{1}\frac{\alpha_{2}^{2}}{\alpha_{1}}\sigma^{\beta_{2}-1}}\,.$
(11)

For $\beta_{2}<1$ and $\sigma\rightarrow\infty$ the right-hand side evaluates
to $\infty$, whereas for $\beta_{2}>1$ and $\sigma\rightarrow\infty$ it
evaluates to $-1$. Thus, $\beta_{2}=1$ is the only solution to this equation
in the limit $\sigma\rightarrow\infty$. Plugging $\beta_{1}=\beta_{2}=1$ into
the equation (11) we see that we can only determine one of $\alpha_{1}$ and
$\alpha_{2}$. So, the path to infinity in the $(U_{1},U_{2},U_{3})$ moduli
space is not uniquely fixed.

Another way of seeing this is to define $\tilde{U}=U_{1}U_{3}$ and
$\tilde{U}_{T}=\frac{U_{1}}{U_{3}}$. The field space metric still remains
diagonal and is given by,

$M^{-1}=\begin{pmatrix}4\tilde{U}^{2}&0&0\\\ 0&2U_{2}^{2}&0\\\
0&0&4\tilde{U}_{T}^{2}\\\ \end{pmatrix}.\quad$ (12)

The scalar potential in terms of the redefined fields is independent of
$\tilde{U}_{T}$,

$V=A_{1}\frac{U_{2}}{\tilde{U}}+A_{2}\frac{1}{U_{2}}\,.$ (13)

As a result of this and the diagonal field space metric, $\tilde{U}_{T}$
remains a flat direction and the ratio of $U_{1},U_{3}$ is undetermined.
Hence, the asymptotic trajectory is not uniquely fixed. This means in turn
that all these trajectories have the same value of $\gamma$.

In Calderon-Infante:2022nxb  it was noted that situations where the trajectory
is not uniquely determined do not lead to $\gamma<\sqrt{2}$ when there are two
rolling scalar fields. Although we do not find any obstruction to obtaining
$\gamma<\sqrt{2}$ when only two terms compete with three rolling scalar
fields, we find that in practice such scenarios do not arise in the model we
study below. This is because either it is not possible to stabilize the
dilaton and the axions or because it is not possible to make terms that blow
up in the potential vanish without losing the competing terms that give rise
to $\gamma<\sqrt{2}$.

While for simple diagonal field space metrics one needs $n$ competing terms
for $n$ rolling scalar fields in order to have a unique trajectory, this is
not the case anymore for (generic) non-diagonal field space metrics. We do not
encounter those examples in our simple model below but it is not hard, for
example, to write down a toy model that uniquely fixes the trajectory for
three rolling fields with only two terms in the scalar potential.

##  3 Type IIB models with $\mathbf{h^{1,1}=0}$

In string theory it is possible to study compactifications on ‘non-geometric
spaces’ that have no Kähler moduli. In particular, in type IIB where it is not
possible to stabilize Kähler moduli with RR- or NSNS-fluxes this provides a
very interesting playground for studying truly stringy flux compactifications.
A detailed description of the effective action for these IIB flux vacua
arising from orientifolds of Landau-Ginzburg models was presented in
Becker:2006ks ; Becker:2007dn ; Bardzell:2022jfh . Even though these models do
not have Kähler moduli, they do have many complex structure moduli. These
could allow us to realize the loopholes pointed out in Calderon-
Infante:2022nxb  in full-fledged models, without having to worry about Kähler
moduli stabilization. In this section, we perform a systematic study of
possible asymptotic limits and check whether they allow for accelerated
expansion.

###  3.1 Effective action

We focus here on a specific orientifold studied in Becker:2006ks  and restrict
to those moduli which are mirror to the untwisted sector of the Type IIA
toroidal compactification. So, we restrict to a subset consisting of three
universal complex structure moduli.

The flux superpotential is given by the standard Gukov-Vafa-Witten formula
Gukov:1999ya ,

$W=\int_{M}G_{3}\wedge\Omega,$ (14)

where $G_{3}\equiv F_{3}-\tau H_{3}$, with the axio-dilaton $\tau=C_{0}+{\rm
i}\,e^{-\phi}$. When restricting to the untwisted sector this amounts to

$\displaystyle W$ $\displaystyle=$ $\displaystyle(f^{0}-\tau
h^{0})U_{1}U_{2}U_{3}-(f^{1}-\tau h^{1})U_{2}U_{3}-(f^{2}-\tau
h^{2})U_{1}U_{3}-(f^{3}-\tau h^{3})U_{1}U_{2}$ (15) $\displaystyle+(f_{1}-\tau
h_{1})U_{1}+(f_{2}-\tau h_{2})U_{2}+(f_{3}-\tau h_{3})U_{3}+(f_{0}-\tau
h_{0})\,,$

where $U_{1},U_{2}$ and $U_{3}$ are the complex structure moduli in the
untwisted sector. The Kähler potential is given by Becker:2007dn

$K=-4\log[-{\rm i}(\tau-\bar{\tau})]-\log[{\rm
i}\int_{M}\Omega\wedge\bar{\Omega}]\,.$ (16)

In terms of only the untwisted sector, we have

$K=-4\log[-{\rm i}(\tau-\bar{\tau})]-\log[{\rm
i}(U_{1}-\bar{U}_{1})(U_{2}-\bar{U}_{2})(U_{3}-\bar{U}_{3})]\,.$ (17)

The factor of 4 does not arise in the dimensional reduction of 10-dimensional
type IIB supergravity but can be understood by using mirror symmetry
Becker:2007dn .

The tadpole cancellation condition is given by

$\frac{1}{\tau-\bar{\tau}}\int_{M}G\wedge\bar{G}+N_{D3}=\frac{1}{2}N_{O3}\,.$
(18)

For us, this translates to

$N_{flux}+N_{D3}=-h^{0}f_{0}+\sum_{i=1}^{3}f^{i}h_{i}-\sum_{i=1}^{3}f_{i}h^{i}+h_{0}f^{0}+N_{D3}=12\,.$
(19)

Given all the above we can investigate if the scalar potential given by

$V=e^{K}\left(K^{I\bar{J}}D_{I}W\overline{D_{J}W}-3|W|^{2}\right),$ (20)

can lead to trajectories that support asymptotic accelerated expansion.

The above model is mirror dual to a type IIA compatification with geometric
and non-geometric fluxes. A particular such model was studied already in
subsection 3.3.2. of Calderon-Infante:2022nxb . That model corresponds in our
notation to $s=\tau$, $u=U_{1}=U_{2}=U_{3}$ and it was shown to not give rise
to asymptotic accelerated expansion. We will see below from a slightly
different point of view that the case where all three $U_{j}$ are equal cannot
give accelerated expansion. Thus, we will naturally be led to study the case
where the $U_{j}$ scale differently.444We thank Irene Valenzuela for pointing
out this important duality to us.

In the following discussions, we will rescale the fluxes in the scalar
potential such that we are at a point in field space where the axions are
vanishing and the saxions that are not rolling are set equal to one. This can
be achieved by temporarily giving up the quantization of the fluxes. This has
been employed before in the literature to make the analysis of scalar
potentials easier. One can calculate the scalar potential and its derivative
at the point where all non-rolling saxions are set equal to 1 and all axions
are set equal to zero. Then via rescaling of the fluxes, one can reach any
other points in moduli space. However, generically the tadpole cancellation
conditions in equation (19) prevent one from reaching a point of arbitrary
weak coupling or arbitrary large complex structure. We show the details of
this rescaling for our model in appendix A.

We will break up our analysis into two subsections. First, we study the
parametric weak coupling limit, i.e. a running dilaton
$\tau_{I}=e^{-\phi}\to\infty$, in the following subsection. After that we will
consider the case where the dilaton is stabilized. We will try to reproduce
settings from Calderon-Infante:2022nxb  that were argued to provide asymptotic
accelerated expansion.

###  3.2 Asymptotically weak coupling

We first show that it is impossible to get accelerated expansion when the
dilaton is rolling in the scalar potential arising from (15) and (17) (see
Shiu:2023nph  for a more generic argument). In terms of the real fields,
$\tau=\tau_{R}+{\rm i}\tau_{I}$ and $U_{j}=U_{jR}+{\rm i}U_{jI}$, where
$j=1,2,3$, the potential can be written as

$V=\frac{A_{\tau^{2}}}{\tau_{I}^{2}}-\frac{A_{\tau^{3}}}{\tau_{I}^{3}}+\frac{A_{\tau^{4}}}{\tau_{I}^{4}},$
(21)

where $A_{\tau^{2}}$, $A_{\tau^{3}}$ and $A_{\tau^{4}}$ are functions of the
fluxes, the axions ($U_{1R},U_{2R},U_{3R},\tau_{R}$) and the complex structure
moduli ($U_{1I},U_{2I},U_{3I}$). Here in the limit
$\tau_{I}\rightarrow\infty$, the $\frac{1}{\tau_{I}^{2}}$ term dominates and
leads to $\gamma=\sqrt{2}$ if all fields are stabilized and $\gamma>\sqrt{2}$
otherwise (see equation (2) above). If $A_{\tau^{2}}$ is zero or other fields
roll also in such a way that the $A_{\tau^{2}}$ term does not dominate, then
the potential would be even steeper and accelerated expansion certainly cannot
happen.

If we are able to actually stabilize all the axions and the complex structure
moduli such that the $A_{\tau^{2}}$ term dominates, then one can examine how
the subleading term $-A_{\tau^{3}}/\tau_{I}^{3}$ affects the total number of
e-folds we can get. This term is arising from localized sources and is
negative for a net number of O3-planes. That means it would make the
asymptotic potential for the dilaton from the $A_{\tau^{2}}/\tau_{I}^{2}$
flatter.

Unfortunately, one can show that one cannot stabilize the axions and all the
complex structure moduli while keeping the potential positive. To see this we
employ the rescaling discussed earlier and set $U_{1}=U_{2}=U_{3}={\rm i}$ at
the putative minimum. Since $\tau_{I}$ is a rolling field, we rescale the
fluxes so that $\tau_{R}=0$ only and do not rescale $\tau_{I}$. The potential
value at the minimum is then given by

$\displaystyle V$
$\displaystyle=\frac{\sum_{a=0}^{3}((h_{a})^{2}+(h^{a})^{2})+6(h^{0}(h_{1}+h_{2}+h_{3})-h_{0}(h^{1}+h^{2}+h^{3})-h_{1}h_{2}-h_{1}h_{3}-h_{2}h_{3}-h^{1}h^{2}-h^{1}h^{3}-h^{2}h^{3})}{128\tau_{I}^{2}}$
(22)
$\displaystyle\quad-\frac{\sum_{a=0}^{3}(f^{a}h_{a}-f_{a}h^{a})}{16\tau_{I}^{3}}+\frac{\sum_{a=0}^{3}((f_{a})^{2}+(f^{a})^{2})}{32\tau_{I}^{4}}\,.$

The first derivatives with respect to $U_{jI},U_{jR},\tau_{R}$ for $j=1,2,3$
must vanish. Since we are interested in the limit $\tau_{I}\to\infty$ we can
simplify the equations by expanding them around large $\tau_{I}$, keeping the
first non-trivial order in $\tau_{I}$. We find

$\displaystyle\partial_{\tau_{R}}V$
$\displaystyle=-\frac{\sum_{a=0}^{3}(f_{a}h_{a}+f^{a}h^{a})}{16\tau_{I}^{4}}\,,$
(23) $\displaystyle\partial_{U_{jR}}V$
$\displaystyle=\frac{3\sum_{a=0}^{3}h_{a}h^{a}-6h^{j}h_{j}+h_{0}h_{j}-h^{j}h^{0}-h_{k}h^{l}-h_{l}h^{k}}{64\tau_{I}^{2}}+\mathcal{O}(\tau_{I}^{-4})\,,$
(24) $\displaystyle\partial_{U_{jI}}V$
$\displaystyle=\frac{\sum_{a=0}^{3}((h^{a})^{2}\\!-(h_{a})^{2})\\!-2(h^{j})^{2}\\!+2(h_{j})^{2}\\!+6(h_{0}h^{j}+h_{j}h^{0}+h_{k}h_{l}-h^{k}h^{l})}{128\tau_{I}^{2}}+\mathcal{O}(\tau_{I}^{-4})\,,$

where $k,l\in\\{1,2,3\\},k\neq l\neq j\neq k$.

The derivative $\partial_{\tau_{R}}V=0$ can be solved via the $f_{a}$ or
$f^{a}$ fluxes that do not appear at leading order in the scalar potential for
large $\tau_{I}$. Thus, we can neglect this equation. We solve the three
equations $\partial_{U_{jR}}V=0$ for $j=1,2,3$ in terms of the three fluxes
$h_{j}$, finding a unique solution. We then plug this solution into
$\partial_{U_{1I}}V=0$ and solve it in terms of $h^{0}$. There are two
solutions for $h^{0}$. Both of these solutions actually ensure that
$\partial_{U_{2I}}V=\partial_{U_{3I}}V=0$ as well, so that we have no further
equations to solve.

For both of the solutions, we find that the fluxes fixed by the solutions are
real, iff

$(-7h_{0}+h^{1}+h^{2}+h^{3})(h_{0}-7h^{1}+h^{2}+h^{3})(h_{0}+h^{1}-7h^{2}+h^{3})(h_{0}+h^{1}+h^{2}-7h^{3})<0\,.$
(25)

We then calculate the value of $V$ at the minimum and find the same flux-
dependent expression for both solutions. If the above constraint in equation
(25) is imposed then it follows that $V_{min}<0$. So, the putative minimum
after stabilizing the complex structure moduli is AdS. There are a few special
cases to further be considered if the fluxes take on special values or vanish.
We discuss these in the appendix B and find that these are not viable either.
Thus, there is no solution in this model that has a rolling dilaton with
$\tau_{I}=e^{-\phi}\to\infty$ and $\gamma=\sqrt{2}$.

If both the dilaton and some or all of the complex structure moduli are to
roll, then we cannot find $\gamma\leq\sqrt{2}$ as any direction that the
fields would roll along would have to be as steep or steeper than the
$\tau_{I}$ direction. So in order to ever get $\gamma\leq\sqrt{2}$ we need to
stabilize the dilaton and investigate the scenario where the complex structure
moduli roll, which is what we are doing in the next subsection.

###  3.3 Finite coupling

The dependence of the scalar potential on the complex structure moduli is
shown below

$\displaystyle V$ $\displaystyle=$ $\displaystyle
A_{1}U_{1I}U_{2I}U_{3I}+A_{2}\frac{U_{1I}U_{2I}}{U_{3I}}+A_{3}U_{1I}+A_{4}\frac{U_{1I}U_{3I}}{U_{2I}}+A_{5}U_{2I}+A_{6}U_{3I}$
(26)
$\displaystyle+A_{7}+A_{8}\frac{U_{1I}}{U_{2I}U_{3I}}+A_{9}\frac{U_{2I}U_{3I}}{U_{1I}}+A_{10}\frac{U_{2I}}{U_{1I}U_{3I}}+A_{11}\frac{U_{3I}}{U_{1I}U_{2I}}+A_{12}\frac{1}{U_{1I}U_{2I}U_{3I}}$
$\displaystyle+A_{13}\frac{1}{U_{1I}}+A_{14}\frac{1}{U_{2I}}+A_{15}\frac{1}{U_{3I}}\,,$

where the $A_{i}$’s are functions of the dilaton, the axions, and the fluxes.
We will stabilize the axio-dilaton and due to the flux rescaling discussed
above we assume that the minimum is at $\tau_{R}=0,\tau_{I}=1$. Note that this
does not mean that the solution is at strong coupling. Neglecting flux
quantization any solution can be shifted to this point and one can always go
to weak coupling by a flux rescaling (modulo constraints coming from flux
quantization and the tadpole cancellation condition). Similarly, we can set
$U_{1R}=U_{2R}=U_{3R}=0$ by shifting the fluxes.

There is a permutation symmetry between the three complex structure moduli in
the above scalar potential. Given this, we can assume without loss of
generality the following rates of growth of the fields $U_{1I}\geq U_{2I}\geq
U_{3I}$ in any asymptotic limit. Such asymptotic limits could have either
$U_{1I}$ going to infinity or $U_{1I},U_{2I}$ going to infinity or all three
$U_{jI}$’s going to infinity. We see then that $A_{1},A_{2},A_{3}$ have to be
zero as these terms necessarily blow up asymptotically. This can be achieved
by setting $f^{0}=h^{0}=f^{3}=h^{3}=0$ in $W$ in equation (15). This in turn
actually implies that $A_{5}=0$ as well and the scalar potential reduces to

$\displaystyle V$ $\displaystyle=$ $\displaystyle
A_{4}\frac{U_{1I}U_{3I}}{U_{2I}}+A_{6}U_{3I}+A_{7}+A_{8}\frac{U_{1I}}{U_{2I}U_{3I}}+A_{9}\frac{U_{2I}U_{3I}}{U_{1I}}+A_{10}\frac{U_{2I}}{U_{1I}U_{3I}}$
(27)
$\displaystyle+A_{11}\frac{U_{3I}}{U_{1I}U_{2I}}+A_{12}\frac{1}{U_{1I}U_{2I}U_{3I}}+A_{13}\frac{1}{U_{1I}}+A_{14}\frac{1}{U_{2I}}+A_{15}\frac{1}{U_{3I}}\,.$

The term $A_{7}$ is the contribution from localized sources like O3-planes and
D3-branes and can be written in terms of the flux parameters using the tadpole
cancellation condition in equation (19) above. It is proportional to

$A_{7}\propto\frac{2N_{D3}-N_{O3}}{\tau_{I}^{3}}\,.$ (28)

One might be tempted to set it to zero to avoid having to deal with its
constant asymptotic value. However, it is the only term in the potential that
scales like $\tau_{I}^{-3}$ and it is crucial for stabilizing the dilaton,
which can be seen as follows: The scalar potential with contributions from the
NSNS-sector, localized sources, and the RR-sector take the schematic form as a
function of $\tau_{I}$ (cf. equation (22) above)

$V=\frac{A}{\tau_{I}^{2}}+\frac{B}{\tau_{I}^{3}}+\frac{C}{\tau_{I}^{4}}\,.$
(29)

Let us assume $B\propto A_{7}=0$ and assume we have rescaled $A$ and $C$ so
that the extremum is at $\tau_{I}=1$. Then $\partial_{\tau_{I}}V=0$ implies
that $A=-2C$. At this extremum, the potential and its second derivative have
opposite signs

$\displaystyle V|_{A=-2C}$ $\displaystyle=$ $\displaystyle-C\,,$
$\displaystyle\frac{d^{2}V}{d\tau_{I}^{2}}|_{A=-2C}$ $\displaystyle=$
$\displaystyle 8C\,.$ (30)

So, regardless of whether we can find an interesting asymptotic trajectory in
the potential in (27) or not, we cannot stabilize the dilaton and keep the
potential positive asymptotically if $A_{7}=0$. So, we have to keep $A_{7}$ in
the potential.

Having established the importance of the $A_{7}$ term, we now explore two
different options of canceling it asymptotically to ensure that we approach a
Minkowski solution. Let us first assume that all three $U_{jI}$ are rolling to
infinity. This means that in the scalar potential in equation (27) we have to
also set $A_{4}=A_{6}=0$, which can be achieved by setting $f^{2}=h^{2}=0$.
This leaves us with

$\displaystyle V$ $\displaystyle=$ $\displaystyle
A_{7}+A_{8}\frac{U_{1I}}{U_{2I}U_{3I}}+A_{9}\frac{U_{2I}U_{3I}}{U_{1I}}+A_{10}\frac{U_{2I}}{U_{1I}U_{3I}}+A_{11}\frac{U_{3I}}{U_{1I}U_{2I}}+A_{12}\frac{1}{U_{1I}U_{2I}U_{3I}}$
(31)
$\displaystyle+A_{13}\frac{1}{U_{1I}}+A_{14}\frac{1}{U_{2I}}+A_{15}\frac{1}{U_{3I}}\,.$

Given our hierarchy $U_{1I}\geq U_{2I}\geq U_{3I}$ we see from the scalar
potential above that only the terms $A_{8}$ and $A_{9}$ could be
asymptotically non-zero. If and only if $U_{1I}$ scales like $U_{2I}U_{3I}$,
do they both approach a constant that can cancel $A_{7}$. Let us assume the
following generic parameterization for $\sigma\to\infty$: $U_{1I}=\sigma$,
$U_{2I}=\alpha_{2}\sigma^{\beta}$ and $U_{3I}=\alpha_{3}\sigma^{1-\beta}$,
with $\tfrac{1}{2}\leq\beta<1$, $\alpha_{2},\alpha_{3}>0$. Let us now
calculate the derivatives of $V$ in equation (27) with respect to the complex
structure moduli in this limit to leading order

$\displaystyle\partial_{U_{1I}}V$
$\displaystyle=A_{8}\frac{1}{U_{2I}U_{3I}}-A_{9}\frac{U_{2I}U_{3I}}{U_{1I}^{2}}+\ldots=\left(\frac{A_{8}}{\alpha_{2}\alpha_{3}}-A_{9}\alpha_{2}\alpha_{3}\right)\sigma^{-1}+\ldots$
$\displaystyle\partial_{U_{2I}}V$
$\displaystyle=-A_{8}\frac{U_{1I}}{U_{2I}^{2}U_{3I}}+A_{9}\frac{U_{3I}}{U_{1I}}+\ldots=-\frac{1}{\alpha_{2}}\left(\frac{A_{8}}{\alpha_{2}\alpha_{3}}-A_{9}\alpha_{2}\alpha_{3}\right)\sigma^{-\beta}+\ldots$
(32) $\displaystyle\partial_{U_{3I}}V$
$\displaystyle=-A_{8}\frac{U_{1I}}{U_{2I}U_{3I}^{2}}+A_{9}\frac{U_{2I}}{U_{1I}}+\ldots=-\frac{1}{\alpha_{3}}\left(\frac{A_{8}}{\alpha_{2}\alpha_{3}^{2}}-A_{9}\alpha_{2}\right)\sigma^{\beta-1}+\ldots$

Given the simple diagonal field space metric arising from the Kähler potential
in equation (17) above, we have to demand that all derivatives are negative in
order to ensure that the three scalar fields actually do roll to infinity and
not away from it. Since $\alpha_{2}>0$ and $\alpha_{3}>0$ we clearly have
found a contradiction. It is not possible for all derivatives to be negative.
This means that all three complex structure moduli cannot roll asymptotically
to infinity and cancel the contribution from the localized sources, i.e. the
$A_{7}$ term.

As a result, we now focus on the scenario where two of the complex structure
moduli are rolling and the third one $U_{3I}$ along with the dilaton and the
axions needs to be stabilized. This allows us to use the terms $A_{6}U_{3I}$
and $A_{15}/U_{3I}$ in the scalar potential in equation (27) to cancel the
$A_{7}$ term to find asymptotic Minkowski solutions. I.e. at the minimum for
$\tau$ and $U_{3}$ we would have $A_{6}U_{3I}+A_{7}+A_{15}/U_{3I}=0$. Assuming
this, we can rewrite the scalar potential above in equation (27) as follows

$V=\hat{A}_{1}\frac{U_{1I}}{U_{2I}}+\hat{A}_{2}\frac{U_{2I}}{U_{1I}}+\hat{A}_{3}\frac{1}{U_{1I}}+\hat{A}_{4}\frac{1}{U_{2I}}+\hat{A}_{5}\frac{1}{U_{1I}U_{2I}}\,,$
(33)

where $\hat{A}_{i}$ are now functions of the axions, $\tau_{I}$ and $U_{3I}$.
As before, without loss of generality, we assume the rates of growth are
ordered, $U_{1I}\geq U_{2I}$. For the following parameterization
$U_{1I}=\alpha\sigma^{\beta},U_{2I}=\sigma$ with $\beta\geq 1$ we get from
equation (7) above that

$\displaystyle\alpha\beta\sigma^{\beta-1}\frac{d\sigma}{dt}$
$\displaystyle=\frac{2}{\sqrt{3V}}(\hat{A}_{1}\alpha^{2}\sigma^{2\beta-1}-\hat{A}_{2}\sigma-\hat{A}_{3}-\hat{A}_{5}\sigma^{-1})\,,$
(34) $\displaystyle\frac{d\sigma}{dt}$
$\displaystyle=\frac{2}{\sqrt{3V}}(-\hat{A}_{1}\alpha\sigma^{\beta}+\hat{A}_{2}\alpha^{-1}\sigma^{2-\beta}-\hat{A}_{4}-\hat{A}_{5}\alpha^{-1}\sigma^{-\beta})\,.$

Taking the ratio of these two equations we find

$\beta=\frac{\hat{A}_{1}\alpha\sigma^{\beta}-\hat{A}_{2}\alpha^{-1}\sigma^{2-\beta}-\hat{A}_{3}\alpha^{-1}\sigma^{1-\beta}-\hat{A}_{5}\alpha^{-1}\sigma^{-\beta}}{-\hat{A}_{1}\alpha\sigma^{\beta}+\hat{A}_{2}\alpha^{-1}\sigma^{2-\beta}-\hat{A}_{4}-\hat{A}_{5}\alpha^{-1}\sigma^{-\beta}}\,.$
(35)

For $\sigma\rightarrow\infty$, we examine two cases. For $\beta>1$, the right-
hand side evaluates to $-1$ which is a contradiction. And for $\beta=1$, again
the right-hand side evaluates to be $-1$. So, we return to (33) and set
$\hat{A}_{1}=0$. This requires us to set the following fluxes in equation (15)
for $W$ to zero $f^{2}=h^{2}=f_{1}=h_{1}=0$. Following the same procedure as
before we get

$\beta=\frac{-\hat{A}_{2}\alpha^{-1}\sigma^{2-\beta}-\hat{A}_{3}\alpha^{-1}\sigma^{1-\beta}-\hat{A}_{5}\alpha^{-1}\sigma^{-\beta}}{\hat{A}_{2}\alpha^{-1}\sigma^{2-\beta}-\hat{A}_{4}-\hat{A}_{5}\alpha^{-1}\sigma^{-\beta}}\,.$
(36)

For $1\leq\beta<2$, in the limit $\sigma\rightarrow\infty$, the right-hand
side evaluates to $-1$ which is a contradiction to our initial assumption. For
$\beta>2$, the limit of the right-hand side evaluates to $0$ which is again a
contradiction. For $\beta=2$ we get,

$2=\frac{-\hat{A}_{2}\alpha^{-1}}{\hat{A}_{2}\alpha^{-1}-\hat{A}_{4}}\,,$ (37)

which gives us $\alpha=\frac{3\hat{A}_{2}}{2\hat{A}_{4}}$. This in turn
uniquely fixes the trajectory to,
$U_{1I}=\frac{3\hat{A}_{2}}{2\hat{A}_{4}}\sigma^{2},U_{2I}=\sigma$. For this
trajectory we find $\gamma=\sqrt{\frac{2}{5}}<\sqrt{2}$. However, setting
$f^{2}=h^{2}=f_{1}=h_{1}=0$ sets $\hat{A}_{4}=0$ and hence this trajectory is
not realisable.

There are no further potential trajectories with two fields rolling to
infinity. Similarly, one cannot get accelerated expansion from a single
monomial with a single field rolling to infinity. Thus, we have proven that
the loophole pointed out in Calderon-Infante:2022nxb  cannot be realized in
these simple non-geometric models with four moduli. We will now proceed and
discuss some asymptotic behaviors for the scalar potentials that can arise in
asymptotic limits.

###  3.4 Asymptotic ridges and valleys

For exponential scalar potentials, it is natural to have a generalized weak
coupling region in which the scalar potential goes to zero if one sends a
scalar field, not necessarily the dilaton, to infinity. In a corresponding
generalized strong coupling region, the scalar potential grows exponentially.
Above we have been studying generalized weak coupling limits in which we send
combinations of scalar fields to infinity, while the scalar potential
approaches zero. However, in some cases we encountered terms that did not
vanish in our particular asymptotic limit, leaving finite values for the
scalar potential. One might ask the more general question of the possible
behaviors of scalar potentials in asymptotic limits. Clearly in principle
everything ranging from $-\infty$ to $+\infty$ could be a possible asymptotic
value. However, zero is being singled out as the value for which we can trust
the scalar potential even in the asymptotic limit. The reason is the Distance
Conjecture Ooguri:2006in , which states that for any infinite distance limit
in scalar moduli space, $\phi\to\infty$, there is a tower of states which
become light, with masses given by $m(\phi)\sim e^{-\alpha\phi}$. This leads
to a decreasing species scale, i.e. a decreasing UV cutoff of the low energy
effective theory Hebecker:2018vxz  (see also Hamada:2021yxy ;
vandeHeisteeg:2023ubh ; Andriot:2023isc ; vandeHeisteeg:2023uxj  for recent
related work). Thus, in the asymptotic limit any scalar potential that does
not go to zero will describe an EFT that necessarily breaks down after a
finite distance.555While this should be obvious for $V>0$, since the potential
energy cannot be larger than the cutoff of the theory, it should also apply to
negative $V<0$. Indeed, the energy scale associated with the scalar potential
should have a magnitude below the cutoff in order for the EFT to make sense in
AdS.

Assuming that the Distance Conjecture is correct, finite values for the scalar
potential in asymptotic limits are not trustworthy since the low energy
effective theory is breaking down. However, one might still be able to study
such a limit if the tower of states that are becoming light can be integrated
in. Even if that is not possible, it is also interesting to have a finite
range of scalar field space in which the scalar potential approaches a
constant value and therefore does not change much. For example, for models of
inflation we need a sufficiently flat scalar potential, and one that
exponentially approaches a constant could certainly be a viable candidate (see
Dias:2018pgj  for related work). Thus, in this section we study asymptotic
limits with finite scalar potential values for our scalar potential arising
from $W$ and $K$ in equations (15) and (17). Below we present two exemplary
cases, one where $V$ approaches a finite positive value and one where it
approaches a finite negative value. While these two examples arise in our
particular non-geometric model, we obtained similar features in other
geometric flux compactifications as well. Thus, we believe that these
asymptotic limits are generic and not specific to non-geometric models.

Consider the following choice of fluxes

$\displaystyle h^{0}=h^{1}=h^{2}=h^{3}=f^{0}=f^{2}=f^{3}=0\,,$ (38)
$\displaystyle h_{1}=\frac{12}{f^{1}}\,,\quad
f_{2}=\frac{f_{1}f^{1}h_{2}}{12}\,,\quad f_{3}=\frac{f_{1}f^{1}h_{3}}{12}\,.$
(39)

The tadpole cancellation condition is satisfied for the above choice without
needing to add any D3-branes, i.e. $N_{D3}=0$ in equation (19).

We can solve the axionic first derivative equations
$\partial_{\tau_{R}}V=\partial_{U_{jR}}=0$ and find that they are solved for

$\tau_{R}=\frac{f_{1}f^{1}}{12}\,,\qquad
U_{1R}=-\frac{f^{1}h_{0}}{12}\,,\qquad U_{2R}=U_{3R}=0\,.$ (40)

Now let us look at an asymptotic limit, $\sigma\to\infty$, with

$U_{1I}=\sigma^{2}\,,\qquad U_{2I}=U_{3I}=\sigma\,.$ (41)

In the large $\sigma$ limit there are two solutions when stabilizing the
dilaton, $\partial_{\tau_{I}}V=0$, namely

$\tau_{I}=\frac{(f^{1})^{2}\left(3\pm\sqrt{7}\right)}{6}+\mathcal{O}(\sigma^{-1})\,.$
(42)

The two solutions correspond to two different asymptotic constant values for
the scalar potential,

$\displaystyle V_{min}$
$\displaystyle=\frac{81\left(2\sqrt{7}+5\right)}{2\left(\sqrt{7}+3\right)^{4}(f^{1})^{6}}+\mathcal{O}(\sigma^{-1})\qquad\text{for}\qquad\tau_{I}=\frac{(f^{1})^{2}\left(3+\sqrt{7}\right)}{6}+\mathcal{O}(\sigma^{-1})\,,$
(43) $\displaystyle V_{min}$
$\displaystyle=-\frac{81\left(2\sqrt{7}-5\right)}{2\left(3-\sqrt{7}\right)^{4}(f^{1})^{6}}+\mathcal{O}(\sigma^{-1})\qquad\text{for}\qquad\tau_{I}=\frac{(f^{1})^{2}\left(3-\sqrt{7}\right)}{6}+\mathcal{O}(\sigma^{-1})\,.$

Thus, this single scalar potential has two asymptotic limits in which the
dilaton is extremized. In one of them, the scalar potential takes on a
positive constant value, and in the other a negative one.

In figure 1 we show the ridge-like shape that appears for positive values of
the scalar potential. For the plot, we have set all the free fluxes equal to
1, i.e. $f^{1}=f_{0}=f_{1}=h_{0}=h_{2}=h_{3}=1$.

Figure 1: Asymptotically for large $\sigma$ a ridge emerges in the scalar
potential.

The figure shows that the scalar potential approaches its asymptotic positive
value $V(\sigma\to\infty,\tau=(3+\sqrt{7})/6)\approx.41$ from below. This
means that the complex structure moduli $U_{1I}=\sigma^{2}$,
$U_{2I}=U_{3I}=\sigma$ would actually not roll to $\sigma=\infty$ along this
ridge but rather to small $\sigma$ values. Furthermore, we see that the
dilaton direction $\tau_{I}$ is a maximum. The $\eta_{V}$ value for $\tau_{I}$
on this ridge is larger than one and one cannot use this ridge to obtain slow-
roll inflation. Nevertheless, it is interesting – and probably known – that
such ridges can arise in asymptotic limits of the scalar field space. For
large values of $\sigma$ one expects that the species scale or the masses of
some tower of states drop below the value of the scalar potential and the
effective low energy theory breaks down.

Let us now follow the scalar potential above to smaller $\tau_{I}$ values,
where we encounter a valley-like shape that is shown in figure 2.

Figure 2: Asymptotically for large $\sigma$ a valley emerges in the scalar
potential at smaller values of the string coupling $\tau_{I}$.

We see that the scalar potential asymptotes in the valley to a constant
negative value for large $\sigma$. This time the dilaton $\tau_{I}$ is
stabilized at a minimum. Again, as before the complex structure moduli
$U_{1I}=\sigma^{2}$, $U_{2I}=U_{3I}=\sigma$ will not asymptotically roll to
large $\sigma$ but rather away from it since the potential approaches its
asymptotic value from below. However, contrary to the positive asymptotic
ridge, here all axions and the dilaton have positive masses and are stabilized
in a minimum.

This negative AdS valley seems at first sight in tension with the recently
proposed Anti-Trans-Planckian Censorship Conjecture (ATCC) Andriot:2022brg ,
which states that negative scalar potentials with positive slope should also
go to zero exponentially in any asymptotic limit. However, as in the case
above with a positive asymptotic value for the scalar potential, we expect
here as well that the low energy effective theory breaks down at some finite
value for $\sigma$, when the species scale or the masses of light towers
become of the same order as the absolute value of the scalar potential. Thus,
it seems plausible that this breakdown of the EFT prevents a violation of the
ATCC. This is an interesting point that deserves further study in the future.

##  4 Relating the Distance and Strong Asymptotic dS Conjecture

It was discussed previously in Etheredge:2022opl  that
$\gamma_{min}=2/\sqrt{d-2}$, the minimal coefficient allowed by the Strong
Asymptotic dS Conjecture, is twice the minimal value allowed by the Sharpened
Distance Conjecture Ooguri:2006in ; Klaewer:2016kiy ; Etheredge:2022opl . The
(refined) Distance Conjecture Ooguri:2006in ; Klaewer:2016kiy  states that for
any infinite distance limit in scalar moduli space, $\phi\to\infty$, there is
a tower of states becoming light, with $m(\phi)\sim e^{-\alpha\phi}$. The
Sharpened Distance Conjecture Etheredge:2022opl  furthermore states that
$\alpha\geq\alpha_{min}=1/\sqrt{d-2}$. The intriguing relationship
$\gamma_{min}=2\alpha_{min}$ was also discussed previously in Hebecker:2018vxz
; Andriot:2020lea ; Bedroya:2020rmd . It was shown in Etheredge:2022opl  that
the Sharpened Distance Conjecture together with the Emergent String Conjecture
Lee:2019wij ; Lee:2019xtm  implied the Strong Asymptotic dS Conjecture. Here
we give an intuitive argument for the opposite direction. If this argument is
correct and can be generalized, then the Strong Asymptotic dS Conjecture would
imply the Sharpened Distance Conjecture with the precise relationship
$\gamma_{min}=2\alpha_{min}$ above.

Let us quickly recall that for a free 5d scalar $\Phi(x^{\mu},y)$ on
$\mathbb{R}^{3,1}\times S^{1}$ we can write the action in 5d or in 4d via

$\displaystyle S$ $\displaystyle=-\frac{1}{2}\int
d^{4}x\,dy\sqrt{-g_{5}}\,\partial_{m}\Phi\partial^{m}\Phi$ (44)
$\displaystyle=-\frac{1}{2}(2\pi R)\int d^{4}x\sqrt{-g_{4}}\,\sum_{n\geq
0}\left(\partial_{\mu}\Phi_{n}\partial^{\mu}\Phi_{n}+\left(2\pi
n\right)^{2}\Phi_{n}^{2}\right)$ (45) $\displaystyle=-\frac{1}{2}\int
d^{4}x\sqrt{-g_{4}^{E}}\,\sum_{n\geq
0}\left(\partial_{\mu}\Phi_{n}\partial^{\mu}\Phi_{n}+\left(\frac{n}{R}\right)^{2}\Phi_{n}^{2}\right)\,.$
(46)

In compactifications to $d$ dimensions, one usually drops the heavy state
$\Phi_{n}$ for $n\neq 0$ but this is not necessary or required.666String
theory provides us with a UV complete theory. In simple toroidal orbifolds,
one can calculate the full mass spectrum and keep track of all fields. There
is no reason to discard for example KK-modes or higher string excitations. In
more complicated compactifications we cannot calculate the complete mass
spectrum but our argument below is independent of the explicit knowledge of
all the massive states. Having a lower dimensional $d$-dimensional theory one
could have likewise kept track of heavier states $\Phi_{n}$ like the KK-towers
that arise in the compactification and the massive string excitations. This
would then modify the $d$-dimensional scalar potential $V_{d}$ by additional
new terms as follows,777It is an interesting question at which energy scale
the low energy EFT breaks down, see for example Montero:2022prj ;
Rudelius:2022gbz . Here we include all the fields in all the towers, sending
the UV cutoff to infinity.

$V=V_{d}(\phi)+\sum_{\rm t}\left(\sum_{n\geq
0}\left(m^{(t)}_{n}(\phi)\right)^{2}(\Phi^{(t)}_{n})^{2}\right)\,,$ (47)

where $t$ labels the different towers. Recall that the logic of Ooguri:2006in
states that for any infinite distance limit $\phi\to\infty$ there is at least
one tower that satisfies $m(\phi)\sim e^{-\alpha\phi}$.888It does not matter
for our argument whether this direction corresponds to a massless modulus,
i.e. whether $V_{d}(\phi)=0$ or not. If $V_{d}(\phi)\neq 0$, we are dealing
with the refined Distance Conjecture Klaewer:2016kiy . Keeping only track of
towers $\tilde{t}$ that become light for $\phi\to\infty$ we have

$V=V_{d}(\phi)+\sum_{\rm\tilde{t}}\left(\sum_{n\geq
0}\left(m^{(\tilde{t})}_{n}\right)^{2}e^{-2\alpha^{(\tilde{t})}\phi}(\Phi^{(\tilde{t})}_{n})^{2}\right)\,.$
(48)

Now the Sharpened Distance Conjecture states that the _lightest_ tower in any
infinite distance limit $\phi\to\infty$ in $d$ dimensions satisfies
$\alpha_{min}\geq\frac{1}{\sqrt{d-2}}$ Etheredge:2022opl . In the above scalar
potential the masses appear quadratically, leading naturally to exponentially
decaying terms $\sim e^{-\gamma^{(\tilde{t})}\phi}$ for $\phi\to\infty$, with
$\gamma^{(\tilde{t})}=2\alpha^{(\tilde{t})}$. From the Strong Asymptotic dS
Conjecture we can then constrain
$2\alpha^{(\tilde{t})}\geq\gamma_{min}=2/\sqrt{d-2}$. While the Distance
Conjecture is very well established and tested, it is amusing to note that it
actually could be implied by the absence of asymptotic accelerated expansion
via the above argument.

The attentive reader will of course object to the above reasoning because of
the $(\Phi^{(\tilde{t})}_{n})^{2}$ factor in the above terms. Asymptotically
in the above example $\Phi^{(\tilde{t})}_{n}=0$ and the new terms in the
scalar potential all vanish. Thus, the above argument does not go through that
easily. However, when including massive fields from the tower we also need to
include their interactions and once those are included it is not clear at all
that $\Phi^{(\tilde{t})}_{n}=0$ is the only possible solution for the
$\Phi^{(\tilde{t})}_{n}$.

Thus a generic expectation for the $d$-dimensional scalar potential in the
asymptotic limit $\phi\to\infty$ would be

$V=V_{d}(\phi)+\sum_{\rm\tilde{t}}\left(\sum_{n}\left(m^{(\tilde{t})}_{n}\right)^{2}e^{-2\alpha^{(\tilde{t})}\phi}\,(\Phi^{(\tilde{t})}_{n})^{2}\right)+V_{int}(\phi,\Phi^{(\tilde{t})}_{n})\,.$
(49)

From the above scalar potential, it seems more reasonable to assume that the
Strong Asymptotic dS Conjecture should apply also when including all massive
towers and their interactions.

Given that we are very much used to integrating out heavy states and setting
the above $\Phi_{n}^{(t)}$ to zero, one might wonder how easy or hard it is to
generate a non-trivial scalar potential which has a minimum for
$\Phi_{n}^{(t)}$ away from zero. We will give here a simple toy example as a
proof of principle that it is indeed possible to do this. Let us return to the
above case of a real 5d scalar on $\mathbb{R}^{3,1}\times S^{1}$ and add to
the action in equation (44) two exponential terms999Similar actions arise in
10-dimensional supergravity for the dilaton.

$S=\int
d^{4}x\,dy\left(-\frac{1}{2}\partial_{m}\Phi\partial^{m}\Phi-\Lambda_{1}e^{-\gamma_{1}\Phi}-\Lambda_{2}e^{-\gamma_{2}\Phi}\right)\,.$
(50)

For a real scalar field we can make the following Ansatz

$\Phi(x^{\mu},y)=\Phi_{0}(x^{\mu})+\sqrt{2}\sum_{n=1}^{\infty}\Phi_{n}(x^{\mu})\cos(2\pi
ny)=\Phi_{0}(x^{\mu})+\sqrt{2}\Phi_{1}(x^{\mu})\cos(2\pi y)+\ldots$ (51)

For ease of presentation and because $\Phi_{1}$ is the lightest KK-mode, we
set $\Phi_{n\geq 2}=0$. We can then perform the integral over the
$y$-direction and find

$\displaystyle S$
$\displaystyle=\\!\\!\\!\int\\!\\!d^{4}x\,dy\sqrt{-g_{5}}\bigg{[}\\!\\!-\\!\frac{1}{2}\partial_{m}\\!\left(\Phi_{0}(x^{\mu})+\sqrt{2}\Phi_{1}(x^{\mu})\cos(2\pi
y)\right)\\!\partial^{m}\\!\\!\left(\Phi_{0}(x^{\mu})+\\!\sqrt{2}\Phi_{1}(x^{\mu})\cos(2\pi
y)\right)$ (52)
$\displaystyle\qquad\qquad\qquad\quad-\Lambda_{1}\,e^{-\gamma_{1}(\Phi_{0}(x^{\mu})+\sqrt{2}\Phi_{1}(x^{\mu})\cos(2\pi
y))}-\Lambda_{2}\,e^{-\gamma_{2}(\Phi_{0}(x^{\mu})+\sqrt{2}\Phi_{1}(x^{\mu})\cos(2\pi
y))}\bigg{]}$ (53) $\displaystyle=\int
d^{4}x\sqrt{-g_{4}^{E}}\;\bigg{[}-\frac{1}{2}\left(\partial_{\mu}\Phi_{0}\partial^{\mu}\Phi_{0}+\partial_{\mu}\Phi_{1}\partial^{\mu}\Phi_{1}+\frac{1}{R^{2}}(\Phi_{1})^{2}\right)$
(54)
$\displaystyle\qquad\qquad\qquad\quad-\frac{\Lambda_{1}}{4\pi^{2}R^{2}}e^{-\gamma_{1}\Phi_{0}}I_{0}(\sqrt{2}\gamma_{1}\Phi_{1})-\frac{\Lambda_{2}}{4\pi^{2}R^{2}}e^{-\gamma_{2}\Phi_{0}}I_{0}(\sqrt{2}\gamma_{2}\Phi_{1})\bigg{)}\bigg{]}\,.$

Thus, we see that interaction terms for the KK-modes appear naturally (here
via the $I$-Bessel function). The Bessel function $I_{0}$ has the following
expansion around $z=0$,

$I_{0}(z)=1+\frac{x^{2}}{4}+\frac{x^{4}}{64}+\frac{x^{6}}{2304}+\mathcal{O}(z^{8})\,.$
(55)

We can thus easily choose the parameters $\gamma_{i}\geq 0$ and the
$\Lambda_{i}$ with opposite signs to generate minima for $\Phi_{1}$ at which
$\Phi_{1}\neq 0$. In a full model we would have to also stabilize $\Phi_{0}$
but that is the modulus that appears in the lower dimensional scalar potential
anyways through a variety of terms. Note, that $1/R^{2}=m(\phi)^{2}\sim
e^{-2\alpha\phi}$ actually multiplies _all_ terms in the scalar potential for
$\Phi_{1}$. Thus, if we choose the parameters $\Lambda_{i}$ to generate a
minimum with $\Phi_{1}\neq 0$ then we can flow to $\phi\to\infty$ without
disturbing this minimum.

While the above is only a toy model it supports the general idea that in a UV
complete theory like string theory it is natural to include all massive fields
and their interactions in the lower dimensional scalar potential. The absence
of accelerated asymptotic expansion, as demanded by the Strong Asymptotic dS
Conjecture, is then related to the Sharpened Distance Conjecture. Their
prefactors naturally satisfy the same bound up to a factor of 2. The authors
of Etheredge:2022opl  argued that the Sharpened Distance Conjecture combined
with the Emergent String Conjecture Lee:2019wij ; Lee:2019xtm  implies the
Strong Asymptotic dS Conjecture. Together with our proposal this leads to an
intriguing interconnection between these three swampland conjectures.

We should note, however, that it is not clear to us at the moment how to
extend our argument to moduli spaces of more than one dimension.101010We thank
Tom Rudelius for pointing this out to us. Since the Strong Asymptotic dS
Conjecture only applies to gradient flows and the Sharpened Distance
Conjecture to any direction in field space, one would need to ensure that the
direction of interest is a gradient flow direction. Thus in the case where the
massive tower of interest depends on several scalar fields one would have to
be able to stabilize directions transverse to the flow in order for our
argument to apply. We leave it to the future to check explicit examples and
see whether this is possible or not.

##  5 Conclusions

Given the fact that our own universe is in a phase of accelerated expansion it
is of paramount importance to understand how one can obtain similar
cosmological histories from string theory. Given the difficulty of
constructing explicit dS vacua that are widely accepted, a lot of work in the
last several years has focused on the easier task of studying asymptotic
limits of moduli space in string theory. In this paper we have extended
previous work by studying asymptotic limits in a special non-geometric string
compactification of type IIB string theory without Kähler moduli.

Our study was motivated by potential counter-examples to the Strong Asymptotic
dS Conjecture Bedroya:2019snp ; Rudelius:2022gbz  put forth in the recent
paper Calderon-Infante:2022nxb . The authors of that paper found counter-
examples when neglecting the stabilization of the Kähler moduli and focusing
only on the axio-dilaton and the complex structure moduli. Thus, our setup
without Kähler moduli provides a natural way of realizing these counter-
examples in full-fledged string theory constructions. However, it turns out
that we cannot get sufficiently flat asymptotic regions in our scalar
potential. It would be interesting to study this further in other non-
geometric models and to extend the work of Calderon-Infante:2022nxb  to
include Kähler moduli stabilization.

While we find no asymptotic regions in the scalar potential that could give
rise to accelerated expansion, we do find limits in which the scalar potential
takes on finite positive or negative values. In the case of a positive scalar
potential the corresponding ridges in the scalar potential have large
tachyonic directions. In the case of negative asymptotic values this is not
the case. However, in both cases we find that these limits cannot be obtained
dynamically since the scalar fields would want to roll away from these limits,
towards the interior of moduli space.

Lastly, we discussed how the Strong Asymptotic dS Conjecture could potentially
imply the Sharpened Distance Conjecture. We show that massive towers of states
appear naturally in the scalar potential and thus provide extra terms that are
constrained by the Strong Asymptotic dS Conjecture. This leads naturally to a
previously observed factor of 2 between the lower bounds on the exponential
prefactors in these two conjectures. Our argument relies on the potential
existence of non-trivial extrema for the massive fields in the towers of
states that become light. While we provide a toy example that gives rise to
such non-trivial extrema, it would certainly be important to work out explicit
examples to further check the connection we observed between the two
conjectures. We hope to return to this in the near future.

## Acknowledgments

We like to thank David Andriot, Ben Heidenreich, Tom Rudelius, Flavio Tonioni
and Irene Valenzuela for useful discussions. The work of S. C. and Y. T. is
supported in part by the NSF grant PHY-2210271. The work of E. G., M. R. and
T. W. is supported in part by the NSF grant PHY-2013988. M.R. acknowledges the
support of the Dr. Hyo Sang Lee Graduate Fellowship from the College of Arts
and Sciences at Lehigh University. S.C. would like to thank the Harvard
University Department of Physics and the Kavli Institute for Theoretical
Physics for hospitality and support throughout part of this work. This
research was supported in part by the National Science Foundation under Grant
No. NSF PHY-1748958. T.W. would like to thank the LAPTh - CNRS for hospitality
during the completion of this work. His visit was made possible thanks to the
AAP USMB dSCordes (2023).

##  Appendix A Rescaling the fluxes

By giving up the quantization of fluxes temporarily and rescaling them, we can
shift the axions to $0$ and the saxions to $1$ such that the superpotential in
equation (15) is invariant. Such a rescaling would lead to the rescaling of
the scalar potential arising from equations (15) and (17), by an overall
positive factor. Consider a minimum of the scalar potential arising from (15)
and (17) which is at $\tau=\tau_{0R}+{\rm i}\tau_{0I}\,,U_{j}=U_{0jR}+{\rm
i}U_{0jI}$, where $j=1,2,3\,.$ We first shift the minimum to the point
$\tau^{\prime}={\rm i}\tau_{0I}\,,U_{j}^{\prime}={\rm i}U_{0jI}$ via the
following transformations of the fluxes,

$\displaystyle
h^{0^{\prime}}=h^{0}\,,\;\;f^{0^{\prime}}=f^{0}-h^{0}\tau_{0R}\,,$ (56)
$\displaystyle h^{j^{\prime}}=h^{j}-h^{0}\tau_{0R}\,,$ (57) $\displaystyle
f^{j^{\prime}}=f^{j}-U_{0jR}\tau_{0R}h^{0}-\tau_{0R}h^{j^{\prime}}-U_{0jR}f^{0^{\prime}}\,,$
(58) $\displaystyle
h_{j}^{{}^{\prime}}=h_{j}-\sum_{k\,,l}U_{0kR}h^{l}+U_{0kR}U_{0lR}h^{0}\,,$
(59) $\displaystyle
f_{j}^{{}^{\prime}}=f_{j}-\tau_{0R}h_{j}^{{}^{\prime}}-\sum_{k\,,l}U_{0kR}f^{l}+U_{0kR}U_{0lR}f^{0}\,,$
(60) $\displaystyle
h_{0}^{{}^{\prime}}=h_{0}+\sum_{j=1}^{3}U_{0jR}h_{j}-\frac{1}{2}\sum_{j,k,l}\rho_{jkl}U_{0jR}U_{0kR}h^{l}+U_{01R}U_{02R}U_{03R}h^{0}\,,$
(61) $\displaystyle
f_{0}^{{}^{\prime}}=f_{0}+\sum_{j}U_{0jR}f_{j}-\frac{1}{2}\sum_{j,k,l}\rho_{jkl}U_{0jR}U_{0kR}f^{l}+U_{01R}U_{02R}U_{03R}f^{0}-\tau_{0R}h_{0}^{{}^{\prime}}\,,$
(62)

where $j\,,k\,,l\in\\{1,2,3\\},k\neq l\neq j\neq k$ and $\rho_{jkl}=1$ iff
$k\neq j\neq l\neq k$ and is vanishing otherwise. It can be easily checked
that this leaves the superpotential in equation (15) invariant,

$\displaystyle W^{\prime}=$
$\displaystyle(f^{0^{\prime}}-\tau^{\prime}h^{0^{\prime}})U_{1}^{{}^{\prime}}U_{2}^{{}^{\prime}}U_{3}^{{}^{\prime}}-(f^{1^{\prime}}-\tau^{\prime}h^{1^{\prime}})U_{2}^{{}^{\prime}}U_{3}^{{}^{\prime}}-(f^{2^{\prime}}-\tau^{\prime}h^{2^{\prime}})U_{1}^{{}^{\prime}}U_{3}^{{}^{\prime}}-(f^{3^{\prime}}-\tau^{\prime}h^{3^{\prime}})U_{1}^{{}^{\prime}}U_{2}^{{}^{\prime}}$
$\displaystyle+(f_{1}^{{}^{\prime}}-\tau^{\prime}h_{1}^{{}^{\prime}})U_{1}^{{}^{\prime}}+(f_{2}^{{}^{\prime}}-\tau^{{}^{\prime}}h_{2}^{{}^{\prime}})U_{2}^{{}^{\prime}}+(f_{3}^{{}^{\prime}}-\tau^{\prime}h_{3}^{{}^{\prime}})U_{3}^{{}^{\prime}}+(f_{0}^{{}^{\prime}}-\tau^{\prime}h_{0}^{{}^{\prime}})=W\,.$
(63)

Now to shift the minimum from $\tau^{\prime}={\rm
i}\tau_{0I}\,,U_{j}^{\prime}={\rm i}U_{0jI}$ to $\hat{\tau}={\rm
i}\,,\hat{U_{j}}={\rm i}$ for $j=1\,,2\,,3$ we require the following
transformations,

$\displaystyle\hat{h}_{0}=h_{0}^{{}^{\prime}}\tau_{0I}\,,\;\;\hat{f}_{0}=f_{0}^{{}^{\prime}}\,,$
(64)
$\displaystyle\hat{h}_{j}=h_{j}^{{}^{\prime}}\tau_{0I}U_{0jI}\,,\;\;\hat{f}_{j}=f_{j}^{{}^{\prime}}U_{0jI}\,,$
(65)
$\displaystyle\hat{h}^{j}=h^{j^{\prime}}\tau_{0I}U_{0kI}U_{0lI}\,,\;\;\hat{f}^{j}=f^{j^{\prime}}U_{0kI}U_{0lI}\,,$
(66)
$\displaystyle\hat{h}^{0}=h^{0^{\prime}}\tau_{0I}U_{01I}U_{02I}U_{03I}\,,\;\;\hat{f}^{0}=f^{0^{\prime}}U_{01I}U_{02I}U_{03I}\,,$
(67)

where $j\,,k\,,l\in\\{1,2,3\\},k\neq l\neq j\neq k$. Once again, it is easy to
see that this leaves the superpotential in equation (15) invariant

$\displaystyle\hat{W}=$
$\displaystyle(\hat{f}^{0}-\hat{\tau}\hat{h}^{0})\hat{U}_{1}\hat{U}_{2}\hat{U}_{3}-(\hat{f}^{1}-\hat{\tau}\hat{h}^{1})\hat{U}_{2}\hat{U}_{3}-(\hat{f}^{2}-\hat{\tau}\hat{h}^{2})\hat{U}_{1}\hat{U}_{3}-(\hat{f}^{3}-\hat{\tau}\hat{h}^{3})\hat{U}_{1}\hat{U}_{2}$
$\displaystyle+(\hat{f}_{1}-\hat{\tau}\hat{h}_{1})\hat{U}_{1}+(\hat{f}_{2}-\hat{\tau}\hat{h}_{2})\hat{U}_{2}+(\hat{f}_{3}-\hat{\tau}\hat{h}_{3})\hat{U}_{3}+(\hat{f}_{0}-\hat{\tau}\hat{h}_{0})=W\,.$
(68)

##  Appendix B Special cases at asymptotic weak coupling

Above in section 3.2 we described how a generic solution that extremizes the
scalar potential with respect to the axions and complex structure moduli
necessarily leads to an AdS minimum with $V_{min}<0$. However, there are a few
special cases that need to be addressed to close all loopholes in our generic
argument.

Above we solved the three equations $\partial_{U_{jR}}V=0$ for $j=1,2,3$ in
terms of the three fluxes $h_{j}$. However, the $h_{j}$ appear in these
equations with a certain prefactor, and one needs to examine separately cases
where, for example, one or all of these prefactors are zero. This leads to
several special cases that we discuss below step by step.

The three axionic equations are of the form (see equations (3.3) above)

$\displaystyle 0=\partial_{U_{1R}}V$
$\displaystyle=\frac{1}{64\tau_{I}^{2}}\left(h_{1}(h_{0}-3h^{1})+\ldots\right)\,,$
(69) $\displaystyle 0=\partial_{U_{2R}}V$
$\displaystyle=\frac{1}{64\tau_{I}^{2}}\left(h_{2}(h_{0}-3h^{2})+\ldots\right)\,,$
(70) $\displaystyle 0=\partial_{U_{3R}}V$
$\displaystyle=\frac{1}{64\tau_{I}^{2}}\left(h_{3}(h_{0}-3h^{3})+\ldots\right)\,.$
(71)

If $h_{0}=3h^{1}=3h^{2}=3h^{3}$ we have a special case. Solving the equations
$\partial_{U_{jR}}V=\partial_{U_{jI}}V=0$ necessarily requires $h_{0}=0$. For
$h_{0}=0$ we have $\partial_{U_{jR}}V=0$ but we still need to solve
$\partial_{U_{jI}}V=0$. Up to permutations of the $j=1,2,3$ index there are
four different solutions. For three of them the scalar potential at the
minimum, $V_{min}$, is negative so we can discard them. The remaining case has

$h_{0}=h^{1}=h^{2}=h^{3}=0\,,\quad h_{1}=-h^{0}\,,\quad h_{2}=h_{3}=h^{0}\,.$
(72)

The scalar potential at the minimum is given by

$V_{min}=\frac{(h^{0})^{2}}{8\tau_{I}^{2}}-\mathcal{O}(\tau_{I}^{-3})\,.$ (73)

This seems to be in principle interesting. We have a single leading term in
the scalar potential that gives exactly $\gamma=\sqrt{2}$ from the dilaton
$\tau_{I}=e^{-\phi}$ running to infinity (see equation (2), which gives
$\epsilon_{V}=1$). This is a setting right at the boundary and leads to a
small number of e-folds. Now we however also have a subleading term that can
further flatten the potential if it is negative (see equation (28) above). One
could then hope to get a larger number of e-folds. However, if one calculates
the Hessian of the scalar potential above one finds that in this case $U_{1I}$
is a tachyonic direction with $\eta_{V}=1$ along this tachyonic direction.
Thus, there is no stable asymptotic trajectory in which only the dilaton
$\tau_{I}=e^{-\phi}$ rolls.

Now let us assume without loss of generality that the first prefactor
$(h_{0}-3h^{1})$ in equation (69) above is non-zero. We can then solve
$\partial_{U_{1R}}V=0$ for $h_{1}$ and plug the result into
$\partial_{U_{2R}}V=0$ and $\partial_{U_{3R}}V=0$ to get

$\displaystyle 0=\partial_{U_{2R}}V$
$\displaystyle=\frac{1}{64\tau_{I}^{2}}\left(h_{2}\frac{(h_{0}-h^{3})(h_{0}-3h^{1}-3h^{2}+h^{3})}{h_{0}-3h^{1}}+\ldots\right)\,,$
(74) $\displaystyle 0=\partial_{U_{3R}}V$
$\displaystyle=\frac{1}{64\tau_{I}^{2}}\left(h_{3}\frac{(h_{0}-h^{2})(h_{0}-3h^{1}-3h^{3}+h^{2})}{h_{0}-3h^{1}}+\ldots\right)\,.$
(75)

If the prefactors $(h_{0}-h^{3})(h_{0}-3h^{1}-3h^{2}+h^{3})$ and
$(h_{0}-h^{2})(h_{0}-3h^{1}-3h^{3}+h^{2})$ both vanish then we have more
special cases that we need to study in detail. We again find that solving all
equations either leads to $V_{min}<0$ or a tachyonic direction with large
$\eta_{V}$ value among the extremized complex structure moduli.

If one of the prefactors is non-zero, we can assume without loss of generality
that it is $(h_{0}-h^{3})(h_{0}-3h^{1}-3h^{2}+h^{3})$. Then we can solve
$\partial_{U_{2R}}V=0$ for $h_{2}$ and plug the answer into
$\partial_{U_{3R}}V=0$ to obtain

$\displaystyle 0=\partial_{U_{3R}}V=\frac{1}{64\tau_{I}^{2}}$
$\displaystyle\left(\frac{h_{3}}{(h_{0}-h^{3})(h_{0}-3h^{1}-3h^{2}+h^{3})}\cdot
C+\ldots\right)\,,$

with

$\displaystyle C=$ $\displaystyle
h_{0}^{3}+3(h^{1})^{3}-15(h^{1})^{2}(h^{2}+h^{3})-3h_{0}^{2}(h^{1}+h^{2}+h^{3})$
(76)
$\displaystyle+h^{1}\left(-15(h^{2})^{2}+106h^{2}h^{3}-15(h^{3})^{2}\right)+3(h^{2}+h^{3})\left((h^{2})^{2}-6h^{2}h^{3}+(h^{3})^{2}\right)$
(77) $\displaystyle-
h_{0}\left((h^{1})^{2}+(h^{2})^{2}-6h^{2}h^{3}+(h^{3})^{2}-6h^{1}(h^{2}+h^{3})\right)\,.$
(78)

Again, if the prefactor $C$ of $h_{3}$ vanishes then we have another special
case. This case is the hardest to analyze. In almost all subcases we find that
$V_{min}<0$. There are a few cases where a positive scalar potential is
possible. In some of those cases we could not prove in full generality that
there is a tachyonic direction, because the resulting expressions are
extremely complicated functions of the flux parameters. However, whenever we
plugged in explicit values for the fluxes we always found that $V_{min}>0$
only arises in the presence of a tachyonic direction among the complex
structure moduli. Thus, we have not found any interesting solutions with only
the dilaton running to infinity in this special case either.

If $C\neq 0$ we can solve $\partial_{U_{3R}}V=0$ for $h_{3}$. We then plug the
three solutions for the $h_{j}$ into $\partial_{U_{1I}}V$ and find

$0=\partial_{U_{1I}}V=\frac{1}{128\tau_{I}^{2}}\left(4(h^{0})^{2}\frac{A\cdot
B}{C}+\ldots\right)\,,$ (79)

with no linear term in $h^{0}$ and

$\displaystyle A$
$\displaystyle=\\!(-7h_{0}+h^{1}+h^{2}+h^{3})(h_{0}-7h^{1}+h^{2}+h^{3})(h_{0}+h^{1}-7h^{2}+h^{3})(h_{0}+h^{1}+h^{2}-7h^{3})\,,$
$\displaystyle B$
$\displaystyle=h_{0}^{2}-6h_{0}h^{1}+(h^{1})^{2}-(h^{2})^{2}+6h^{2}h^{3}-(h^{3})^{2}\,.$
(80)

Note that $A$ is the expression that appeared above in equation (25). If
$A\cdot B\neq 0$, then we have the generic case discussed above in subsection
3.2. So, let us address the special cases $A=0$ and/or $B=0$.

We start out with the case where $A=0$. Up to permutations of $j=1,2,3$ there
are two possibilities

$7h_{0}=h^{1}+h^{2}+h^{3}\,,\qquad\text{or}\qquad 7h_{1}=h_{0}+h^{2}+h^{3}\,.$
(81)

Solving all the remaining equations in terms of the fluxes leads to a variety
of solutions, all of which have $V_{min}<0$ and can therefore be discarded.

Lastly, we study the case $B=0$. Again there are many different subcases, most
of which are leading to a negative scalar potential at the minimum. The ones
that do lead to $V_{min}>0$ all have a tachyonic direction along one of the
complex structure moduli $U_{jI}$ and thus do not allow for stable
trajectories with only $\tau_{I}$ rolling. Thus, we have checked all special
cases and excluded them.

## References

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