Moduli stabilization with non-geometric fluxes
—
comments on tadpole contributions
and de-Sitter vacua

Erik Plauschinn

Institute for Theoretical Physics, Utrecht University
Princetonplein 5, 3584CE Utrecht
The Netherlands

###### Abstract

We study moduli stabilization for type IIB orientifold compactifications on
Calabi-Yau three-folds with (non-)geometric fluxes. For this setting it is
possible to stabilize all closed-string moduli classically without the need
for non-perturbative contributions, and examples of stable de-Sitter
constructions can be found in the literature which violate a prominent
swampland conjecture.

In this paper we derive general properties of non-geometric flux-
compactifications, we argue that the contribution of fluxes to the tadpole
cancellation conditions should be similar to D-branes (and not anti-D-branes),
and we exclude supersymmetric Minkowski vacua for certain cases. We also
reassess known stable de-Sitter constructions with (non-)geometric $H$-, $F$-,
$Q$\- and $R$-fluxes and argue that these are not consistent in string theory.

###### Contents

  1. 1 Introduction
  2. 2 Type IIB compactifications with fluxes
    1. 2.1 Type IIB orientifolds with O3-/O7-planes
    2. 2.2 Fluxes
    3. 2.3 Scalar potential
  3. 3 Tadpole charges
  4. 4 Moduli stabilization
    1. 4.1 Non-geometric flux vacua
    2. 4.2 Constraints on the Hodge numbers
  5. 5 The minimal case: $h^{2,1}_{-}=h^{1,1}$
    1. 5.1 Manifest tadpole charges
    2. 5.2 No supersymmetric Minkowski vacua
    3. 5.3 Stabilizing the radial Kähler-sector modulus
  6. 6 Comments on de-Sitter vacua
  7. 7 Summary, conclusions, outlook

##  1 Introduction

Constructing stable de-Sitter vacua in string theory is notoriously difficult.
An overview of existing de-Sitter constructions and discussions of their
validity can be found in the reviews [1] and [2], but to the best of our
knowledge no generally-accepted, fully-consistent and explicitly worked-out
example of a stable de-Sitter vacuum in string theory is known. This
observation may have led the authors of [3, 4, 5] to conjecture that de-Sitter
vacua cannot be realized in string theory, or any consistent theory of quantum
gravity. As a potential counter-example to this conjecture non-geometric flux-
compactifications are often mentioned, which are the topic of this work.

Non-geometric backgrounds are configurations which cannot be described in
terms of Riemannian geometry. Such spaces are inconsistent for point particles
but are well-defined for strings, and for these backgrounds the transition
functions between local charts are required to include T-duality
transformations [6]. For a review of this topic we refer to [7]. The standard
example for a non-geometric background is obtained by applying successive
T-duality transformations to a three-torus with $H$-flux, leading to a twisted
torus with geometric $F$-flux [8, 9], a T-fold with non-geometric $Q$-flux [6]
and a non-associative space with a non-geometric $R$-flux [10]. When
compactifying string theory in the presence of fluxes, the $H$-flux generates
the familiar Gukov-Vafa-Witten superpotential [11] in the lower-dimensional
theory. Applying then T-duality transformations leads to contributions of the
geometric $F$\- and of the non-geometric $Q$\- and $R$-fluxes [10, 12, 13],
but an uplift of these four-dimensional theories to ten-dimension is often not
known. However, we note that certain compactifications with geometric and non-
geometric fluxes fit into the framework of $SU(3)\times SU(3)$-structure
compactifications which have been studied in [14, 15, 16, 17].

Coming back to de-Sitter vacua, for compactifications including positive-
tension objects such as D-branes the Maldacena-Nuñez no-go theorem excludes
the existence of de-Sitter vacua [18]. However, this no-go theorem does not
hold in the presence of orientifold planes or for non-geometric backgrounds,
and the purpose of the present work is to investigate the latter case. We
focus on type IIB string-theory compactifications on Calabi-Yau orientifolds
with O3-/O7-planes and (non-)geometric $H$-, $F$-, $Q$\- and $R$-fluxes which,
in the context of de-Sitter vacua, have been considered before for instance in
[19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Unfortunately, we were
neither able to show that stable de-Sitter vacua from non-geometric fluxes
cannot exists — nor able to construct a consistent de-Sitter model for this
setting. Nevertheless, we made progress in understanding non-geometric flux
vacua and, in particular, we argue that all stable de-Sitter constructions
with $H$-, $F$-, $Q$\- and $R$-fluxes known to us are inconsistent in string
theory. Our findings are reported in the present paper, which is organized as
follows:

  * •

In section 2 we review type IIB Calabi-Yau orientifold compactifications with
O3-/O7-planes and non-geometric $H$-, $F$-, $Q$\- and $R$-fluxes. This section
contains no new results, but it clarifies for instance how non-geometric
fluxes should be quantized and it may serve as a modern introduction to the
topic.

  * •

In section 3 we discuss how (non-)geometric fluxes contribute to the tadpole
cancellation conditions. By requiring the rank of the four-dimensional gauge
group to be bounded (in agreement with [31]), we present arguments that fluxes
should contribute in the same way as supersymmetric D-branes and not as anti-
D-branes. This condition excludes a known de-Sitter construction with non-
geometric fluxes.

  * •

In section 4 we study the scalar potential induced by non-geometric fluxes. We
derive a simple expression for the scalar potential at the minimum, and we
derive the necessary conditions $h^{1,1}_{\vphantom{-}}\leq h^{2,1}_{-}$ and
$h^{2,1}_{+}=0$ on the Hodge numbers for stabilization of all closed-string
moduli.

  * •

In section 5 we specialize to the situation
$h^{1,1}_{\vphantom{-}}=h^{2,1}_{-}$. We rewrite the scalar potential and
eliminate the contribution of the Ramond-Ramond three-form flux in favor of
tadpole charges, which is a form suitable for computer-aided scans for vacua.
We furthermore show that supersymmetric Minkowski vacua do not exists and we
discuss necessary conditions for stable de-Sitter minima.

  * •

In section 6 we review known de-Sitter constructions with non-geometric
fluxes. For the stable models which in addition to the Ramond-Ramond three-
form flux only involve $H$-, $F$-, $Q$\- and $R$-fluxes we conclude – using
the constraint from section 3 – that none of them are consistent in string
theory.

  * •

In section 7 we summarize our findings.

##  2 Type IIB compactifications with fluxes

We start by introducing our notation and conventions for compactifications of
type IIB string theory on Calabi-Yau orientifolds with fluxes. This section
contains no new results, and for the reader familiar with the topic it is safe
to only skim over the formulas and move to the next section.

###  2.1 Type IIB orientifolds with O3-/O7-planes

In this subsection we summarize some properties of type IIB orientifold
compactifications with O3- and O7-planes which are relevant for our subsequent
discussion.

#### Cohomology

We consider type IIB string theory on $\mathbb{R}^{3,1}\times\mathcal{X}$
subject to an orientifold projection of the form $(-1)^{F_{\rm L}}\Omega_{\rm
P}\hskip 1.0pt\sigma$. $F_{\rm L}$ denotes the left-moving fermion number and
$\Omega_{\rm P}$ is the world-sheet parity operator, and their action on the
type IIB field content can be found for instance in [32]. The compact space
$\mathcal{X}$ is chosen to be a Calabi-Yau three-fold, and we impose a
holomorphic involution $\sigma$ on $\mathcal{X}$ such that its Kähler and
holomorphic three-form satisfy $\sigma^{*}J=+J$ and $\sigma^{*}\hskip
1.0pt\Omega=-\Omega$. The fixed loci of $\sigma$ in
$\mathbb{R}^{3,1}\times\mathcal{X}$ correspond to O3- and O7-planes.

Since $\sigma$ is an involution, the cohomology groups of $\mathcal{X}$ split
into even and odd eigenspaces as
$H^{p,q}(\mathcal{X})=H^{p,q}_{+}(\mathcal{X})\oplus H^{p,q}_{-}(\mathcal{X})$
with dimensions $h^{p,q}=h^{p,q}_{+}+h^{p,q}_{-}$. For the even cohomology
groups of $\mathcal{X}$ we introduce bases in the following way

$\begin{split}\begin{array}[]{lcl@{\hspace{30pt}}lcl@{\hspace{60pt}}lcl}\omega_{0}&\in&H^{3,3}_{+}(\mathcal{X})\,,\hfil\hskip
30.\\\\[2.0pt] \omega_{a}&\in&H^{1,1}_{+}(\mathcal{X})\,,\hfil\hskip
30.&\omega_{\alpha}&\in&H^{1,1}_{-}(\mathcal{X})\,,\hfil\hskip
60.&a&=&1,\ldots,h^{1,1}_{+}\,,\\\\[2.0pt]
\sigma^{a}&\in&H^{2,2}_{+}(\mathcal{X})\,,\hfil\hskip
30.&\sigma^{\alpha}&\in&H^{2,2}_{-}(\mathcal{X})\,,\hfil\hskip
60.&\alpha&=&1,\ldots,h^{1,1}_{-}\,,\\\\[2.0pt]
\sigma^{0}&\in&H^{0,0}_{+}(\mathcal{X})\,,\hfil\hskip
30.\end{array}\end{split}$ (2.1)

which can be chosen such that the only non-vanishing pairings satisfy the
relations $\int_{\mathcal{X}}\omega_{0}\wedge\sigma^{0}=1$,
$\int_{\mathcal{X}}\omega_{a}\wedge\sigma^{b}=1$ and
$\int_{\mathcal{X}}\omega_{\alpha}\wedge\sigma^{\beta}=\delta_{\alpha}{}^{\beta}$.
For the third cohomology group we introduce symplectic bases as

$\begin{split}\begin{array}[]{lcl@{\hspace{60pt}}lcl}\\{\alpha_{I},\beta^{I}\\}&\in&H^{3}_{-}(\mathcal{X})\,,\hfil\hskip
60.&I&=&0,\ldots,h^{2,1}_{-}\,,\\\\[2.0pt]
\\{\alpha_{\lambda},\beta^{\lambda}\\}&\in&H^{3}_{+}(\mathcal{X})\,,\hfil\hskip
60.&\lambda&=&1,\ldots,h^{2,1}_{+}\,,\end{array}\end{split}$ (2.2)

with the only non-vanishing pairings
$\int_{\mathcal{X}}\alpha_{I}\wedge\beta^{J}=\delta_{I}{}^{J}$ and
$\int_{\mathcal{X}}\alpha_{\kappa}\wedge\beta^{\lambda}=\delta_{\kappa}{}^{\lambda}$.

#### Moduli

The effective theory obtained after compactification can be described in terms
of four-dimensional $\mathcal{N}=1$ supergravity. The bosonic field content
originating from the closed-string sector is summarized in table 1, and for
details we refer to [33].

fields | multiplicity | symbol
---|---|---
four-dimensional metric | $1$ | $g$
$U(1)$ vector fields | $h^{2,1}_{+}$ | $U^{\lambda}$
complex-structure moduli | $h^{2,1}_{-}$ | $z^{i}$
axio-dilaton | $1$ | $\tau$
Kähler moduli | $h^{1,1}_{+}$ | $T_{a}$
axionic moduli | $h^{1,1}_{-}$ | $G^{\alpha}$
Table 1: Bosonic massless field content corresponding to the closed-string
sector after compactifying type IIB string theory on Calabi-Yau orientifolds
with O3-/O7-planes.

The complex-structure moduli $z^{i}$ with $i=1,\ldots,h^{2,1}_{-}$ are encoded
in the holomorphic three-form $\Omega$, while the axio-dilaton, Kähler moduli
and axionic moduli are contained in the multiform [34]

$\begin{split}\Phi=e^{B}\mathcal{C}+i\,\mbox{Re}\left(e^{-\phi}\hskip
1.0pt\lambda\left[e^{B-i\hskip 1.0ptJ}\right]\right).\end{split}$ (2.3)

Here, $\mathcal{C}$ denotes the sum over all Ramond-Ramond potentials
$\mathcal{C}=\sum_{p}C_{p}$, $B$ denotes the Kalb-Ramond field and we
introduced an operator $\lambda$ acting on a $p$-form $A_{(p)}$ as
$\lambda[A_{(p)}]=(-1)^{[\frac{p}{4}]}A_{(p)}$ with $[\ldots]$ denoting the
integral part. Expanding $\Phi$ in the bases (2.1) defines the moduli in the
following way

$\begin{split}\Phi=\tau+G^{\alpha}\omega_{\alpha}+T_{a}\hskip
1.0pt\sigma^{a}\,,\end{split}$ (2.4)

and the precise form of $\tau$, $G^{\alpha}$ and $T_{a}$ can be found for
instance in [33] or by working out the definition of $\Phi$. For later
convenience we group the Kähler-sector moduli together as

$\begin{split}\mathsf{T}^{A}=(\hskip 1.0pt\tau,G^{\alpha},T_{a})\,,\hskip
50.0pt\mathsf{T}^{A}=c^{A}+i\hskip 1.0pt\tau^{A}\,,\end{split}$ (2.5)

where $c^{A}$ and $\tau^{A}$ denote the real and imaginary parts of
$\mathsf{T}^{A}$ and where we introduced a collective index
$A=0,\ldots,h^{1,1}$.

#### Kähler potential

The moduli-space geometry is captured by a Kähler potential $K$, which splits
into the Kähler sector containing the axio-dilaton, Kähler moduli and axionic
moduli and the complex-structure sector as follows

$\begin{split}K=K_{\rm K}+K_{\rm cs}\,,\hskip 50.0pt\begin{array}[]{lcl}K_{\rm
K}&=&\displaystyle-\log\left[\vphantom{\bigl{(}}-i\hskip
1.0pt(\tau-\overline{\tau})\right]-2\log\left[\mathcal{V}+\frac{\xi}{2}\right],\\\\[10.0pt]
K_{\rm cs}&=&\displaystyle-\log\left[+\hskip
1.0pti\int_{\mathcal{X}}\Omega\wedge\overline{\Omega}\hskip
1.0pt\right].\end{array}\end{split}$ (2.6)

For the Kähler sector we note that the Einstein-frame volume of the Calabi-Yau
three-fold is denoted by $\mathcal{V}$ which depends implicitly on $\tau$,
$G^{\alpha}$ and $T_{a}$, and we included $\alpha^{\prime}$-corrections
encoded in $\xi=-\frac{\zeta(3)\hskip 1.0pt\chi(\mathcal{X})\hskip
1.0pt(\tau-\overline{\tau})^{3/2}}{2\hskip 1.0pt(2\pi)^{3}\hskip
1.0pt(2i)^{3/2}}$ [35]. In the following we will employ the notation

$\begin{split}\begin{array}[]{lcl@{\hspace{50pt}}lcl}K_{A}&=&\partial_{\mathsf{T}^{A}}K\,,\hfil\hskip
50.&G_{A\overline{B}}&=&\partial_{\mathsf{T}^{A}\vphantom{\overline{\mathsf{T}}^{B}}}\partial_{\overline{\mathsf{T}}^{B}}K\,,\\\\[6.0pt]
K^{A}&=&G^{A\overline{B}}K_{\overline{B}}\,,\hfil\hskip
50.&G^{A\overline{B}}&\equiv&(G^{-1})^{A\overline{B}}\,,\end{array}\end{split}$
(2.7)

and from (2.6) we can determine the relations

$\begin{split}K^{A}=-(\mathsf{T}-\overline{\mathsf{T}})^{A}\,,\hskip
50.0ptK_{A\vphantom{\overline{B}}}\hskip
1.0ptG^{A\overline{B}}K_{\overline{B}}=4\,.\end{split}$ (2.8)

We emphasize that our analysis below depends on (2.8), which we have checked
explicitly for the $\alpha^{\prime}$-corrected Kähler potential $K_{\rm K}$
shown in (2.6). However, we do not know wether (2.8) also holds for the full
quantum expression.

In the complex-structure sector $\Omega$ denotes again the holomorphic three-
form of $\mathcal{X}$, and the Kähler covariant derivatives
$D_{i}=\partial_{z^{i}}+\partial_{z^{i}}K$ acting on $\Omega$ provide a basis
$\chi_{i}=D_{i}\hskip 1.0pt\Omega$ for $H^{2,1}_{-}(\mathcal{X})$. In addition
to (2.2), an alternative basis for $H^{3}(\mathcal{X})$ is therefore given by
$\\{\Omega,\chi_{i\vphantom{\overline{i}}},\overline{\chi}_{\overline{i}},\overline{\Omega}\\}$
where the bar denotes complex conjugation. A general three-form can then be
expanded as

$\begin{split}A=a^{0}\hskip
1.0pt\Omega+a^{i}\chi_{i\vphantom{\overline{i}}}+a^{\overline{i}}\hskip
1.0pt\overline{\chi}_{\overline{i}}+a^{\overline{0}}\hskip
1.0pt\overline{\Omega}\,,\end{split}$ (2.9)

and, with the Hodge-star operator acting as $\star\Omega=-i\hskip 1.0pt\Omega$
and $\star\chi_{i}=+i\hskip 1.0pt\chi_{i}$, for two three-forms $A$ and $B$ we
obtain the relation

$\begin{split}a^{\overline{0}}b^{0}+a^{i\vphantom{\overline{j}}}G_{i\overline{j}}\hskip
1.0ptb^{\overline{j}}&=\frac{e^{K_{\rm
cs}}}{2}\int_{\mathcal{X}}A\wedge\bigl{(}\star B-iB\bigr{)}\,.\end{split}$
(2.10)

###  2.2 Fluxes

To generate a potential for the moduli in the effective theory we introduce
fluxes, and to preserve four-dimensional Poincaré invariance they are chosen
to extend only along the compact space. The type of fluxes we consider in the
Ramond-Ramond (R-R) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) sectors are

$\begin{split}\begin{array}[]{l@{\hspace{4pt}}l@{\hspace{50pt}}l}\mbox{R-R}\hfil\hskip
4.0&\mbox{sector:}\hfil\hskip 50.&F_{3}\,,\\\\[4.0pt] \mbox{NS-NS}\hfil\hskip
4.0&\mbox{sector:}\hfil\hskip 50.&H\,,\hskip 4.0ptF\,,\hskip 4.0ptQ\,,\hskip
4.0ptR\,.\end{array}\end{split}$ (2.11)

#### (Non-)geometric fluxes

In the NS-NS sector $H$ denotes the familiar $H$-flux, and its T-dual
completions are the geometric $F$-flux and the non-geometric $Q$\- and
$R$-fluxes [8, 9, 6, 10]. A detailed discussion of non-geometric fluxes can be
found for instance in [7], but here we only note that they can be interpreted
as operators acting on the cohomology as

$\begin{split}\begin{array}[]{l@{\hspace{6pt}}c@{\hspace{12pt}}lcl}H\,\wedge\hfil\hskip
6.0&:\hfil\hskip 12.0&\mbox{$p$-form}&\to&\mbox{$(p+3)$-form}\,,\\\
F\,\circ\hfil\hskip 6.0&:\hfil\hskip
12.0&\mbox{$p$-form}&\to&\mbox{$(p+1)$-form}\,,\\\ Q\,\bullet\hfil\hskip
6.0&:\hfil\hskip 12.0&\mbox{$p$-form}&\to&\mbox{$(p-1)$-form}\,,\\\
R\,\llcorner\hfil\hskip 6.0&:\hfil\hskip
12.0&\mbox{$p$-form}&\to&\mbox{$(p-3)$-form}\,.\end{array}\end{split}$ (2.12)

These operators can be conveniently summarized using a generalized derivative
of the form [10]

$\begin{split}\mathcal{D}=d+H\wedge+F\circ+Q\bullet+R\hskip
1.0pt\llcorner\,,\end{split}$ (2.13)

where $d$ is the ordinary exterior derivative. We furthermore note that $H$,
$Q$ and $F_{3}$ are odd under the action of the combined world-sheet parity
and left-moving fermion number $\Omega_{\rm P}(-1)^{F_{\rm L}}$ whereas $F$
and $R$ are even [10, 36]. When acting with $\mathcal{D}$ on the cohomology
bases shown in (2.1) we obtain

$\begin{split}&\mathcal{D}\hskip
1.0pt\bigl{(}\sigma^{0},\omega_{\alpha},\sigma^{a}\bigr{)}\equiv\bigl{(}\mathsf{H},\mathsf{F}_{\alpha},\mathsf{Q}^{a}\bigr{)}\equiv\Xi_{A}\in
H^{3}_{-}(\mathcal{X})\,,\\\\[4.0pt] &\mathcal{D}\hskip
1.0pt\bigl{(}\omega_{0},\sigma^{\alpha},\omega_{a}\bigr{)}\equiv\bigl{(}\mathsf{R},\mathsf{Q}^{\alpha},\mathsf{F}_{a}\bigr{)}\equiv\Theta^{A}\in
H^{3}_{+}(\mathcal{X})\,,\end{split}$ (2.14)

where we employ the same collective index $A$ as in (2.5). Note also that the
three-forms $F_{3}$, $\Xi_{A}$ and $\Theta^{A}$ can be expanded in the
integral bases (2.2), for which the expansion coefficients are quantized due
to the familiar flux-quantization condition. This includes in particular the
quantization of non-geometric fluxes.

#### Bianchi identities and tadpole cancellation condition

In our setting we assume the absence of localized NS-NS sources such as
NS5-branes, Kaluza-Klein monopoles or non-geometric branes, and hence the
Bianchi identities for the NS-NS fluxes can be expressed as
$\mathcal{D}^{2}=0$ [10, 15]. Using the three-forms defined in (LABEL:hi_001),
these conditions can be written as

$\begin{split}\int_{\mathcal{X}}\Xi_{A}\wedge\Xi_{B}=0\,,\hskip
40.0pt\int_{\mathcal{X}}\Theta^{A}\wedge\Theta^{B}=0\,,\hskip
40.0pt\Xi_{A}\otimes\Theta^{A}=0\,.\end{split}$ (2.15)

Turning to the R-R sector, performing an orientifold projection typically
gives rise to orientifold planes and requires the introduction of D-branes.
Both are charged under the R-R potentials and therefore the Bianchi identities
for the R-R field strengths are in general non-trivial. Their integrated
versions – or equivalently their formulation in cohomology – are known as the
tadpole cancellation conditions and read (see for instance [7] for more
details of this formulation)

$\begin{split}\mathcal{D}\mathcal{F}=\sum_{{\rm D}p+{\rm
D}p^{\prime}}\mathcal{Q}_{{\rm D}p}+\sum_{{\rm O}p}\mathcal{Q}_{{\rm
O}p}\,,\end{split}$ (2.16)

where the combined R-R field strength is given by
$\mathcal{F}=\mathcal{D}\hskip 1.0pt\mathcal{C}$, $\mathcal{Q}_{{\rm D}p}$ and
$\mathcal{Q}_{{\rm O}p}$ are multi-forms encoding the D-brane and O-plane
charges, and the sums are over all D-branes, their orientifold images and
O-planes present in the background. We discuss these conditions in more detail
in section 3 below. In our situation the only non-trivial R-R flux is $F_{3}$
and we obtain for the left-hand side in (2.16)

$\begin{split}\mathcal{D}\mathcal{F}=N^{0}\hskip
1.0pt\omega_{0}+N_{\alpha}\hskip 1.0pt\sigma^{\alpha}+N^{a}\hskip
1.0pt\omega_{a}\,,\end{split}$ (2.17)

where

$\begin{split}N_{A}=\int_{\mathcal{X}}\Xi_{A}\wedge F_{3}=\Bigl{(}(H\wedge
F_{3})^{0}\hskip 1.0pt,\,(F\circ F_{3})_{\alpha}\hskip 1.0pt,\,(Q\bullet
F_{3})^{a}\Bigr{)}\end{split}$ (2.18)

are the contributions of the $H$-, $F$\- and $Q$-fluxes to the D3-, D5- and
D7-brane tadpoles, respectively. We will refer to the $N_{A}$ also as tadpole
charges, and we note that since the $\Xi_{A}$ and $F_{3}$ are quantized the
$N_{A}$ take integer values.

###  2.3 Scalar potential

The fluxes introduced above generate a scalar potential $V=V_{\rm F}+V_{\rm
D}$ in the effective four-dimensional $\mathcal{N}=1$ supergravity theory. The
F-term potential $V_{\rm F}$ is expressed in terms of a superpotential $W$ and
the Kähler potential (2.6) as

$\begin{split}V_{\rm F}=e^{K}\left[D_{i\vphantom{\overline{j}}}W\hskip
1.0ptG^{i\overline{j}}D_{\overline{j}}\overline{W}+D_{A\vphantom{\overline{B}}}W\hskip
1.0ptG^{A\overline{B}}D_{\overline{B}}\overline{W}-3|W|^{2}\right],\end{split}$
(2.19)

where $D_{i}W$ and $D_{A}W$ are the Kähler-covariant derivatives and
$G^{i\overline{j}}$ and $G^{A\overline{B}}$ denote the inverse Kähler metrics
in the complex-structure and Kähler sector, respectively. The fluxes generate
a generalization of the Gukov-Vafa-Witten superpotential of the form
$W=\int_{\mathcal{X}}\Omega\wedge(F_{3}-\mathcal{D}\hskip 1.0pt\Phi)$ [10, 12,
13] which, using (LABEL:hi_001) and (2.4), can be expressed as

$\begin{split}W=\int_{\mathcal{X}}\Omega\wedge G\,,\hskip
50.0ptG=F_{3}-\Xi_{A}\hskip 1.0pt\mathsf{T}^{A}\,.\end{split}$ (2.20)

The D-term potential in the effective theory takes the form $V_{\rm
D}=[\mbox{Re}\hskip 1.0ptf\hskip
1.0pt]^{-1|\kappa\lambda}D_{\kappa}D_{\lambda}$, where $f_{\kappa\lambda}$ is
the gauge-kinetic function for the $U(1)$ vector fields shown in table 1. We
emphasize that $f_{\kappa\lambda}$ only depends on the complex-structure
moduli, and we note that the D-term potential is generated by the fluxes
$\Theta^{A}$ [37, 38, 36]. It can be written as

$\begin{split}V_{\rm
D}=2\int_{\mathcal{X}}\bigl{(}\widetilde{\Theta}^{A}K_{A\vphantom{\overline{B}}}\bigr{)}\wedge\star\bigl{(}\widetilde{\Theta}^{\overline{B}}K_{\overline{B}}\bigr{)}\,,\end{split}$
(2.21)

where, due to the self-duality of the R-R four-form $C_{4}$ (see [33] for
details), only the restricted three-forms
$\widetilde{\Theta}^{A}=-\Theta^{A}{}_{\lambda}\hskip 1.0pt\beta^{\lambda}$
appear.

##  3 Tadpole charges

In equation (2.18) we have defined tadpole charges $N_{A}$, which play an
important role for moduli stabilization. They connect the open- and closed-
string sectors to each other and give rise to strong restrictions (see for
instance [39, 40, 41, 42, 43, 44, 45, 46, 47] for recent discussions of this
question). In this section we present arguments that the $N_{A}$ should
contribute to the tadpole cancellation conditions (2.16) in the same way as
D-branes and not as anti-D-branes. This implies

$\begin{split}N_{A}\leq 0\,.\end{split}$ (3.1)

#### D-branes and orientifold planes

Let us start by extending our review of type IIB orientifold compactifications
from section 2.1. When performing an orientifold projection of the form
$(-1)^{F_{\rm L}}\Omega_{\rm P}\hskip 1.0pt\sigma$ the fixed loci of $\sigma$
correspond to orientifold planes which fill-out four-dimensional space-time
$\mathbb{R}^{3,1}$ and wrap cycles $\Gamma_{{\rm O}p}$ in the compact space
$\mathcal{X}$. In our case these are O3- and O7-planes which are point-like
and are wrapping four-cycles in $\mathcal{X}$, respectively. When orientifold
planes are present then D-branes can be introduced, which in our situation are
D3- and D7-branes filling-out four-dimensional space-time $\mathbb{R}^{3,1}$
and wrapping cycles $\Gamma_{{\rm D}p}$ in $\mathcal{X}$. Additionally,
D-branes can have a non-trivial gauge flux $\mathcal{F}_{{\rm
D}p}=\mathsf{F}_{{\rm D}p}+B\in H^{2}(\Gamma_{{\rm D}p})$ on their world-
volume, where $\mathsf{F}_{{\rm D}p}$ denotes the open-string gauge flux and
$B$ is understood as pulled-back to the D-brane. We do not consider discrete
torsion for the $B$-field.

The data characterizing D-branes and O-planes can be encoded in the charges
$\mathcal{Q}_{{\rm D}p}$ and $\mathcal{Q}_{{\rm O}p}$ which already appeared
in (2.16). These are multi-forms defined as

$\begin{split}\mathcal{Q}_{{\rm D}p}&=\lambda\left[[\Gamma_{{\rm
D}p}]\wedge\mbox{tr}\left(e^{\mathsf{F}_{{\rm
D}p}}\right)\wedge\sqrt{\frac{\hat{\mathcal{A}}(\mathcal{R}_{T})}{\hat{\mathcal{A}}(\mathcal{R}_{N})}}\right],\\\
\mathcal{Q}_{{\rm O}p}&=Q_{p}\,\lambda\left[[\Gamma_{{\rm
O}p}]\wedge\sqrt{\frac{\mathcal{L}(\mathcal{R}_{T}/4)}{\mathcal{L}(\mathcal{R}_{N}/4)}}\right],\end{split}$
(3.2)

where $[\Gamma_{{\rm D}p}]$ is the Poincaré dual of the cycle wrapped by the
D-brane, $\mathsf{F}_{{\rm D}p}$ denotes the quantized open-string gauge-flux
and the trace is over the fundamental representation. The expressions
$\mathcal{R}_{T}$ and $\mathcal{R}_{N}$ stand for the restrictions of the
curvature two-form $\mathcal{R}$ to the tangent and normal bundle of
$\Gamma_{{\rm D}p}$, and we used the $\hat{\mathcal{A}}$-genus and the
Hirzebruch polynomial $\mathcal{L}$. The charge $Q_{p}$ of the orientifold
planes is given by $Q_{p}=-2^{p-4}$, and the operator $\lambda$ was introduced
below equation (2.3).

#### Calibrations

In order for D-branes to preserve $\mathcal{N}=1$ supersymmetry, calibration
conditions have to be satisfied. In our situation they are of the form [48]
(see for instance [49] for a review)111 It would be desirable to include
curvature terms in the calibration condition, but we are not aware of any work
addressing this question.

$\begin{split}{\rm Vol}(\Gamma_{{\rm D}p})=e^{i\hskip 1.0pt\theta_{{\rm
D}p}}\int_{\Gamma_{{\rm D}p}}e^{-\phi}\,\lambda\left[\hskip
1.0pt\mbox{tr}\bigl{(}e^{\mathcal{F}_{{\rm D}p}}\bigr{)}\wedge e^{-i\hskip
1.0ptJ}\right],\end{split}$ (3.3)

where the volume is computed using the DBI action and $J$ denotes the Kähler
form of $\mathcal{X}$. The operator $\lambda$ was introduced below equation
(2.3), and $\theta_{{\rm D}p}$ is an up to now undetermined phase which
selects a particular $\mathcal{N}=1$ combination of $\mathcal{N}=2$
supercharges. Turning to orientifold planes, since the involution $\sigma$ is
holomorphic the cycles $\Gamma_{{\rm O}p}$ are holomorphic and satisfy the
calibration conditions

$\begin{split}{\rm Vol}(\Gamma_{{\rm O}p})=e^{i\hskip 1.0pt\theta_{{\rm
O}p}}\int_{\Gamma_{{\rm O}p}}e^{-\phi}\,\lambda\left[e^{-i\hskip
1.0ptJ}\right],\hskip 60.0pt\theta_{{\rm O}p}=0\,.\end{split}$ (3.4)

Note that since a gauge flux $\mathcal{F}_{{\rm D}p}$ is odd under the
orientifold projection $\sigma$ it vanishes when restricted to the fixed-point
set of $\sigma$. We can therefore trivially extend (3.4) by $\mbox{tr}\hskip
1.0pt(e^{\mathcal{F}_{{\rm D}p}})$ and compare with (3.3). In order for all
D-branes and O-planes to preserve the same $\mathcal{N}=1$ supersymmetry, we
then have to require

$\begin{split}\theta_{{\rm D}p}=\theta_{{\rm O}p}=0\,.\end{split}$ (3.5)

#### Tadpole cancellation conditions

The tadpole cancellation conditions have been stated schematically already in
(2.16), but here we want to make them more precise. For our setting with
O3-/O7-planes and D3-/D7-branes the tadpole cancellation conditions can be
written in the following way (see for instance [50] for details of the
derivation)

$\displaystyle N^{a}\omega_{a}=\sum_{{\rm D}7_{\mathsf{i}}}N_{{\rm
D}7_{\mathsf{i}}}\Bigl{(}\hskip 1.0pt[\Gamma_{{\rm
D}7_{\mathsf{i}}}]+[\Gamma^{\prime}_{{\rm D}7_{\mathsf{i}}}]\Bigr{)}-8\hskip
1.0pt\sum_{{\rm O}7_{\mathsf{j}}}\>[\Gamma_{{\rm O}7_{\mathsf{j}}}]\,,$ (3.6a)
$\displaystyle N_{\alpha}\sigma^{\alpha}=-\sum_{{\rm
D}7_{\mathsf{i}}}\Bigl{(}\mbox{tr}\left[\mathsf{F}_{{\rm
D}7_{\mathsf{i}}}\right]\wedge\left[\Gamma_{{\rm
D}7_{\mathsf{i}}}\right]+\mbox{tr}\left[\mathsf{F}^{\prime}_{{\rm
D}7_{\mathsf{i}}}\right]\wedge\left[\Gamma^{\prime}_{{\rm
D}7_{\mathsf{i}}}\right]\Bigr{)}\,,$ (3.6b)
$\displaystyle\frac{N^{0}}{2}=N_{{\rm D}3}-\frac{N_{{\rm O}3}}{4}-\sum_{{\rm
D}7_{\mathsf{i}}}\left(\frac{1}{2}\int_{\Gamma_{{\rm
D}7_{\mathsf{i}}}}\mbox{tr}\left[\mathsf{F}^{2}_{{\rm
D}7_{\mathsf{i}}}\right]+N_{{\rm D}7_{\mathsf{i}}}\hskip
1.0pt\frac{\chi(\Gamma_{{\rm D}7_{\mathsf{i}}}\bigr{)}}{24}\right)-\sum_{{\rm
O}7_{\mathsf{j}}}\frac{\chi\bigl{(}\Gamma_{{\rm
O}7_{\mathsf{j}}}\bigr{)}}{12}\,,$ (3.6c)

where $N_{{\rm D}7_{\mathsf{i}}}$ denotes the number of D7-branes in a stack
labelled by $\mathsf{i}$ and $N_{{\rm D}3}$ is the total number of D3-branes.
Both of these numbers are counted without the orientifold images which are
denoted by a prime. Furthermore, $N_{{\rm O}3}$ is the total number of
O3-planes, and $\chi(\Gamma)$ denotes the Euler number of the cycle
$\Gamma$.222 In case the D7-branes have double intersection points the
corresponding Euler number has to be corrected by the number of pinch points
[51, 52]. This subtlety will however not be relevant here.

#### Gauge groups

Let us also recall that D-branes give rise to gauge theories on their world-
volume. In type IIB the corresponding gauge groups are either $U(N_{{\rm
D}p})$, $SO(2N_{{\rm D}p})$ or $Sp(2N_{{\rm D}p})$, where $N_{{\rm D}p}$
denotes the number of D$p$-branes on top of each other. Since we can have
several such stacks of D-branes wrapping cycles in the compact space, the
four-dimensional gauge group is of the schematic form

$\begin{split}\mathsf{G}=\prod_{\mathsf{i}}U(N_{{\rm
D}p_{\mathsf{i}}})\times\prod_{\mathsf{j}}SO(2N_{{\rm
D}p_{\mathsf{j}}})\times\prod_{\mathsf{k}}Sp(2N_{{\rm D}p_{\mathsf{k}}})\times
U(1)^{h^{2,1}_{+}}\,,\end{split}$ (3.7)

where the last factor corresponds to the closed-string $U(1)$ gauge fields
$U^{\lambda}$ mentioned in table 1.

Now, in [31] conjectures have been presented which imply that the rank of the
total gauge group in a consistent quantum-gravity theory should be bounded
from above. This requirement is supported for instance by extensive experience
in string-theory model building, and progress in establishing such a bound for
different settings has been made for instance in [53, 54, 55]. And indeed, in
the absence of fluxes we see from (3.6) that the rank of the gauge group
$\mathsf{G}$ is bounded by the orientifold data. More concretely, for a fixed
compactification space and orientifold projection, for supersymmetric D-brane
configurations and in the absence of closed-string fluxes, the rank of
$\mathsf{G}$ is bounded by the number of O-planes, by the Euler numbers of
four-cycles in $\mathcal{X}$ and by $h^{2,1}_{+}$. Our main requirement is
then that also when introducing fluxes, the rank of the gauge group
$\mathsf{G}$ remains bounded from above.

#### Contribution of closed-string fluxes

We now want to argue that the closed-string fluxes encoded in the tadpole
charges $N_{A}$ should contribute in the same way as supersymmetric D-branes
to the tadpole cancellation conditions (3.6) and not as anti-D-branes or
orientifold planes. Loosely speaking, in string theory one should not be able
to generate arbitrarily-large gauge groups.

  * •

Let us start by considering the D3-brane tadpole condition and assume that we
found a configuration of branes and fluxes which solves (3.6c), say for
$N^{0}=0$. A new solution can be obtained by changing the fluxes and number of
D3-branes as333 As a physical process this is known as brane-flux
transmutation [56], however, in general (3.8) will lead to a physically-
different configuration solving the tadpole condition (3.6c).

$\begin{split}N^{0}\to N^{0}+2\hskip 1.0pt\alpha\,,\hskip 30.0ptN_{{\rm
D}3}\to N_{{\rm D}3}+\alpha\,,\hskip 60.0pt\alpha\in\mathbb{Z}\,.\end{split}$
(3.8)

Since for supersymmetric D-branes $N_{{\rm D3}}$ is non-negative (cf. the
calibration condition (3.3)), the integer $\alpha$ is bounded from below.
Requiring furthermore the rank of the gauge group to be bounded implies that
$\alpha$ also has to be bounded from above. These restrictions then translate
into bounds on $N^{0}$. It is not clear to us how to derive an upper bound on
$N^{0}$ from first principles, but since the tadpole cancellation conditions
contain only topological data in the form of integers, the only natural bound
is

$\begin{split}N^{0}\leq 0\,.\end{split}$ (3.9)

We explain this point in more detail below. This means that the combination of
fluxes contained in $N^{0}$ contributes like a D-brane to the tadpole
conditions and not like an anti-D-brane or orientifold plane. Note also that
(3.9) is of course realized in standard flux compactifications.

  * •

A similar reasoning can be applied to the D7-brane tadpole (3.6a). For
simplicity we consider a setting with $h^{1,1}_{-}=0$ and $\chi(\Gamma_{{\rm
D}7})=0$, which is satisfied for instance in toroidal-orbifold
compactifications. Let us then again assume a configuration of D$7$-branes has
been found which solves (3.6a) for $N^{a}=0$. A new solution can be obtained
by replacing $N^{a}\to N^{a}+2\hskip 1.0pt\alpha^{a}$, where
$\alpha^{a}\in\mathbb{Z}$, and introducing/removing $\alpha^{a}$ D$7$-branes
without gauge flux wrapping a cycle dual to $\omega_{\hat{a}}$. The
$\alpha^{a}$ are bounded from below by the requirement that $N_{{\rm D}7}\geq
0$, and from above by requiring the rank of the gauge group to be bounded. It
is again not clear to us how to derive an upper bound on the $N^{a}$, but a
natural choice is

$\begin{split}N^{a}\leq 0\,,\end{split}$ (3.10)

and we discuss this point below in more detail. Let us emphasize that this
restriction can be derived from (3.9) via T-duality. From (3.10) we then see
that the flux combinations $N^{a}$ contribute like D-branes to the tadpole
conditions and not like anti-D-branes or orientifold planes. Finally, for more
general situations with $h^{1,1}_{-}\neq 0$ or $\chi(\Gamma_{{\rm
D}7}\bigr{)}\neq 0$ also the D3- and D5-brane tadpole might have to be
adjusted, but the main conclusion doesn’t change.

  * •

For the D5-brane tadpole (3.6b) we have to slightly extend our setting by
including D5-branes. In this case the corresponding tadpole condition has been
determined for instance in [50], and following similar arguments as above we
can infer that

$\begin{split}N_{\alpha}\leq 0\,.\end{split}$ (3.11)

To summarize, we have argued that in order for the total rank of the gauge
group (shown in (3.7)) to be bounded – in agreement with the swampland
conjectures made in [31] – the tadpole charges $N_{A}$ should be bounded from
above. We identified

$\begin{split}\framebox{$\displaystyle{N_{A}\leq 0}$}\end{split}$ (3.12)

as the most natural choice for such a bound, which is consistent with known
constructions as well as with T-duality. This implies that fluxes should
contribute in the same way as supersymmetric D-branes to the tadpole
cancellation conditions.

#### Identifying a natural bound for integers

We mentioned above that for integers a natural threshold is given by zero,
which we now would like to explain in more detail. To do so, let us first
consider a setting with a dimension-full quantity such as an energy $E$ which
is bounded from above. If for this setting there exists a single
characteristic energy scale $E_{0}$, then one would naturally choose $E_{0}$
as the upper bound for $E$. That is $E\leq E_{0}$, or equivalently
$E/E_{0}\leq 1$. Similarly, let us consider a setting with a positive
dimension-less quantity $\epsilon\in\mathbb{R}^{+}$ which is known to be
bounded from above. In the absence of any related characteristic quantity
(such as $2\pi$), a natural upper bound for $\epsilon$ would be given by
$\epsilon\leq 1$. This would, for instance, typically allow for a power-series
expansion. Finally, we consider a setting with a dimension-less integer
$p\in\mathbb{Z}$ without a related characteristic quantity. This situation is
different from the previous two cases, since here we would not be able to
argue via a natural scale in the problem or via a power-series expansion.
However, the only special point is $p_{0}=0$ where the sign of $p$ changes,
and we can identify this point as a natural threshold. Hence, when forced to
choose an upper bound for integers (taking positive as well as negative
values) in a setting without any characteristic quantity, we identify $p\leq
0$ is the most natural choice.

#### Discussion

As we have emphasized above, with our current knowledge we are not able to
derive the requirement $N_{A}\leq 0$ on the tadpole charges from first
principles. This condition has therefore the character of a (swampland)
conjecture, which is however based on plausability and duality arguments and
on requiring consistency with other swampland conjectures.

Naturally, if $N_{A}\leq 0$ is a non-empty statement then some constructions
which have appeared in the literature will not satisfy this condition. Let us
discuss a selection of them:

  * •

In some of the examples in [57] and [58] closed-string fluxes contribute as
anti-D-branes to the tadpole cancellation condition, and hence violate our
requirement $N_{A}\leq 0$. In these papers the possibility to have fluxes
contributing as anti-D-branes is considered to be an advantage which allows
for more flexibility in model building, but we take the opposite point of
view. However, with our current knowledge we are not able to exclude these
models from first principles.

  * •

Some of the AdS vacua found in [59] and [22] do not satisfy the requirement
$N_{A}\leq 0$, but some examples in these papers do indeed satisfy this
condition. This shows that one can find non-trivial solutions to the condition
$N_{A}\leq 0$.

  * •

Similarly, in [39] we studied supersymmetric AdS vacua for the
$\mathbb{T}^{6}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ type IIB orientifold and
found solutions with $N_{A}\leq 0$ and all moduli stabilized (see equation
(5.14) in [39] and note that our conventions are related as $N_{\rm
here}=-\mathsf{Q}_{\rm there}$). For this paper we randomly generated $3\cdot
10^{6}$ stable AdS vacua with all moduli stabilized, but of those only a
fraction of $1.57\cdot 10^{-4}$ satisfy the requirement $N_{A}\leq 0$. This is
a considerable reduction of this part of the string-theory landscape.

#### Remarks

We close this section with the following comments and remarks concerning the
tadpole charges $N_{A}$:

  * •

Let us recall the general form of the tadpole cancellation conditions (2.16),
multiply both sides with $e^{-\phi}\hskip 1.0pt\lambda\hskip
1.0pt[e^{B-i\hskip 1.0ptJ}]$ and integrate over $\mathcal{X}$. Using the
definition of the moduli shown in (2.4) and the calibration conditions (3.3)
and (3.4), we obtain

$\begin{split}\frac{N_{A}\hskip
1.0pt\tau^{A}}{2}=&\begin{array}[t]{cl@{\op}lclr@{\op}lcll}&\displaystyle
e^{-\phi}\hfil\hskip 1.0pt&\displaystyle N_{{\rm
D}3}&+&\displaystyle\sum_{{\rm D}7_{\mathsf{i}}}\,\biggl{[}&\hskip
1.0pt&\displaystyle\mbox{Vol}(\Gamma_{{\rm D}7_{\mathsf{i}}})&-&\displaystyle
e^{-\phi}\,N_{{\rm D}7_{\mathsf{i}}}\,\frac{\chi(\Gamma_{{\rm
D}7_{\mathsf{i}}})}{24}&\displaystyle\biggr{]}\\\ -&\displaystyle
e^{-\phi}\hfil\hskip 1.0pt&\displaystyle\frac{N_{{\rm
O}3}}{4}&-&\displaystyle\sum_{{\rm O}7_{\mathsf{j}}}\,\biggl{[}&4\hskip
1.0pt&\displaystyle\hskip 1.0pt\mbox{Vol}(\Gamma_{{\rm
O}7_{\mathsf{j}}})&+&\displaystyle e^{-\phi}\,\frac{\chi(\Gamma_{{\rm
O}7_{\mathsf{j}}})}{12}&\displaystyle\biggr{]}\,.\end{array}\end{split}$
(3.13)

Our requirement of having the closed-string fluxes contributing in the same
way as supersymmetric D-branes to the tadpole conditions (the first line on
the right-hand side) then implies

$\begin{split}N_{A}\tau^{A}\leq 0\,.\end{split}$ (3.14)

  * •

In the absence of non-geometric fluxes, the presence of O7-planes requires the
introduction of D7-branes via the tadpole cancellation condition. However,
with non-vanishing $Q$-flux we can obtain configurations without D-branes. In
particular, in the absence of D-branes the tadpole conditions (3.6) read

$\begin{split}N^{a}\omega_{a}=-8\hskip 1.0pt\sum_{{\rm
O}7_{\mathsf{j}}}\,[\Gamma_{{\rm O}7_{\mathsf{j}}}]\hskip 1.0pt,\hskip
22.25ptN_{\alpha}=0\hskip 1.0pt,\hskip 22.25ptN^{0}=-\frac{N_{{\rm
O}3}}{2}-\sum_{{\rm O}7_{\mathsf{j}}}\frac{\chi\bigl{(}\Gamma_{{\rm
O}7_{\mathsf{j}}}\bigr{)}}{6}\hskip 1.0pt,\end{split}$ (3.15)

which satisfy the requirement $N_{A}\leq 0$. Note that especially with respect
to stabilizing all moduli, this is a desirable situation since no open-string
moduli (typically coming with the introduction of D-branes) need to be
stabilized.

  * •

In this section we have considered effective quantum-gravity theories which
arise from compactifying string theory in the presence of fluxes and space-
time filling D-branes. We argued that in this situation the rank of the gauge
group should be bounded, however, in other situations this does not need to be
true. For instance, the standard example for the AdS/CFT correspondence is
given by taking $N_{{\rm D}3}$ coincident D3-branes in flat ten-dimensional
space-time and making $N_{{\rm D}3}$ large. Here there is no restriction on
$N_{{\rm D}3}$, though the resulting theory is dual to a non-gravitational
theory.444We thank D. Junghans for bringing this question to our attention.

##  4 Moduli stabilization

In this section we discuss some general features of moduli stabilization with
non-geometric fluxes. This question has been studied before, for instance in
[10, 13, 60, 19, 20, 23, 24, 27] for toroidal orbifold compactifications and
in [16, 61, 62, 63, 64, 65] for more general settings. Our results in this
section are the following: in subsection 4.1 we determine for Calabi-Yau
orientifold compactifications the form of the scalar potential at an extremum
as

$\begin{split}V\hskip 1.0pt\Bigr{\rvert}_{\rm ext}=-\frac{e^{K_{\rm
K}}}{2}\left[\int_{\mathcal{X}}\bigl{(}\mbox{Re}\,G\wedge\star\mbox{Re}\,G\bigr{)}+N_{A}\hskip
1.0pt\tau^{A}\right]_{\rm ext},\end{split}$ (4.1)

where $G$ denotes the three-form flux defined in (2.20),
$\tau^{A}=\mbox{Im}\,\mathsf{T}^{A}$ and $N_{A}$ denotes the tadpole charges.
This surprisingly simple form will be employed later on. In subsection 4.2 we
show that in order to stabilize all moduli by non-geometric fluxes, the Hodge
numbers have to satisfy

$\begin{split}h^{1,1}\leq h^{2,1}_{-}\,,\hskip
60.0pth^{2,1}_{+}=0\,.\end{split}$ (4.2)

###  4.1 Non-geometric flux vacua

In this section we discuss necessary conditions for (non-)geometric flux vacua
for the setting introduced in section 2. We first determine an expression for
extrema of the scalar potential and derive a necessary relation for stable
minima.

#### Rewriting the F-term potential

The scalar potential in $\mathcal{N}=1$ supergravity is given by $V=V_{\rm
F}+V_{\rm D}$, where the D-term potential has been given in (2.21). The F-term
potential (2.19) for the superpotential (2.20) can be worked out using the
relation (2.10) as

$\begin{split}V_{\rm F}=e^{K}\left[\,\frac{e^{-K_{\rm
cs}}}{2}\int_{\mathcal{X}}G\wedge\bigl{(}\star\overline{G}+i\hskip
1.0pt\overline{G}\bigr{)}+e^{-2K_{\rm
cs}}\,\Xi^{\overline{0}}_{A\vphantom{\overline{B}}}\,\omega^{A\overline{B}}\,\Xi^{0}_{\overline{B}}\right],\end{split}$
(4.3)

where the three-form flux reads $G=F_{3}-\mathsf{T}^{A}\hskip 1.0pt\Xi_{A}$
and where $\Xi^{\overline{0}}_{A\vphantom{\overline{B}}}$ and
$\Xi^{0}_{\overline{B}}$ denote the $\overline{\Omega}$ and $\Omega$
components of $\Xi_{A\vphantom{\overline{B}}}$ and $\Xi_{\overline{B}}$ as in
(2.9). These components together with the matrix $\omega^{A\overline{B}}$ are
given by

$\begin{split}\Xi^{\overline{0}}_{A\vphantom{\overline{a}}}=i\hskip
1.0pte^{K_{\rm cs}}\int_{\mathcal{X}}\Omega\wedge\Xi_{A}\,,\hskip
60.0pt\omega^{A\overline{B}}=G^{A\overline{B}}-K^{A}K^{\overline{B}}\,,\end{split}$
(4.4)

where the former only depends on the complex-structure moduli and the latter
only depends on the imaginary parts $\tau^{A}$ of the Kähler-sector moduli.
(In equation (4.21) we show the form of $\omega^{A\overline{B}}$ for a
particular example.) In order to make (4.3) more symmetric we can use
$\star^{2}=-1$ when acting on three-forms on an Euclidean six-dimensional
space and rewrite the scalar potential as

$\begin{split}V_{\rm F}=e^{K}\left[\,\frac{e^{-K_{\rm
cs}}}{4}\int_{\mathcal{X}}(G+i\star
G)\wedge\star\bigl{(}\overline{G}-i\star\overline{G}\bigr{)}+e^{-2K_{\rm
cs}}\,\Xi^{\overline{0}}_{A}\,\omega^{A\overline{B}}\,\Xi^{0}_{\overline{B}}\right].\end{split}$
(4.5)

When ignoring $\alpha^{\prime}$-corrections to the Kähler potential and only
considering $H$-flux, the second term in (4.5) vanishes. This corresponds to
the situation studied by Giddings, Kachru and Polchinski in [66].

#### Some identities

In order to proceed, we determine some identities for derivatives of the
Kähler potential and the Kähler metric in the Kähler-moduli sector. We first
note that the expression $K_{A}$ and $K^{A}$ are purely imaginary and hence
$K_{A\vphantom{\overline{A}}}=-K_{\overline{A}}$ and
$K^{A\vphantom{\overline{A}}}=-K^{\overline{A}}$. Using then the relations
shown in equation (2.8), we compute

$\begin{split}&K^{C}\partial_{C}K^{A}=K^{C}(-\delta_{C}{}^{A})=-K^{A}\,,\\\
&K^{C}\partial_{C}K^{\overline{A}}=K^{C}(+\delta_{C}{}^{\overline{A}})=-K^{\overline{A}}\,,\end{split}$
(4.6)

and in a similar way we determine for the first derivatives of the Kähler
potential

$\begin{split}K^{C}\partial_{C}K_{A}=-K^{C}G_{C\overline{A}}=+K_{A}\,,\\\
K^{C}\partial_{C}K_{\overline{A}}=+K^{C}G_{C\overline{A}}=+K_{\overline{A}}\,.\end{split}$
(4.7)

Furthermore, for the Kähler metric $G_{A\overline{B}}$ and for the matrix
$\omega^{A\overline{B}}$ defined in (4.4) we compute

$\begin{split}\begin{array}[]{lclcl}\displaystyle K^{C}\partial_{C}\hskip
1.0ptG_{A\overline{B}}&=&\displaystyle
K^{C}\partial_{A}G_{C\overline{B}}=\partial_{A}(K^{C}G_{C\overline{B}})-(\partial_{A}K^{C})G_{C\overline{B}}&=&+2\hskip
1.0ptG_{A\overline{B}}\,,\\\\[4.0pt] \displaystyle K^{C}\partial_{C}\hskip
1.0pt\omega^{A\overline{B}}&=&\displaystyle-G^{A\overline{M}}(K^{C}\partial_{C}G_{\overline{M}N})G^{N\overline{B}}+2K^{A\vphantom{\overline{B}}}K^{\overline{B}}&=&-2\hskip
1.0pt\omega^{A\overline{B}}\,.\end{array}\end{split}$ (4.8)

#### A necessary condition for extrema

In order to obtain extrema of the combined potential $V=V_{\rm F}+V_{\rm D}$
we have to solve $\partial_{i}V=0$ and $\partial_{A}V=0$, where $i$ and $A$
label the complex-structure and combined Kähler moduli. These are in general
complicated equations which are typically solved numerically, however, let us
consider the necessary condition

$\begin{split}0=K^{A}\partial_{A}V\,.\end{split}$ (4.9)

With the help of the relations (LABEL:lu_015), (4.7) and (4.8) we find that
the D-term potential satisfies $K^{A}\partial_{A}V_{\rm D}=2\hskip 1.0ptV_{\rm
D}$, and for the full potential we obtain

$\begin{split}0=K^{A}\partial_{A}V=2\hskip 1.0ptV+\frac{e^{K_{\rm
K}}}{2}\int_{\mathcal{X}}(G+\overline{G})\wedge\bigl{(}\star\overline{G}+i\hskip
1.0pt\overline{G}\bigr{)}\,.\end{split}$ (4.10)

Note that the last term in (4.10) contains an imaginary contribution which,
since the scalar potential $V$ is real, implies

$\begin{split}0=\int_{\mathcal{X}}(G+\overline{G})\wedge\star(G-\overline{G})\,.\end{split}$
(4.11)

We now use this relation together with the Bianchi identity (2.15) and the
definition of $N_{A}$ and $\tau^{A}$ given in (2.18) and (2.5) to evaluate the
scalar potential at the extremum as

$\begin{split}\framebox{$\displaystyle{V\hskip 1.0pt\Bigr{\rvert}_{\rm
extr}=-\frac{e^{K_{\rm
K}}}{2}\left[\int_{\mathcal{X}}\bigl{(}\mbox{Re}\,G\wedge\star\mbox{Re}\,G\bigr{)}+N_{A}\hskip
1.0pt\tau^{A}\right]_{\rm extr}\,.}$}\end{split}$ (4.12)

#### A necessary condition for stability

Next, we derive a necessary condition for stability of the extremum. The
combined mass matrix for the Kähler-sector moduli $\mathsf{T}^{A}$ and
complex-structure moduli $z^{i}$ takes the following general form

$\begin{split}M^{2}=\left[\begin{array}[]{cc|cc}m^{2}_{A\overline{B}}&m^{2}_{AB\vphantom{\overline{B}}}&m^{2}_{A\overline{j}\vphantom{\overline{B}}}&m^{2}_{Aj\vphantom{\overline{B}}}\\\
m^{2}_{\overline{A}\overline{B}}&m^{2}_{\overline{A}B\vphantom{\overline{B}}}&m^{2}_{\overline{A}\overline{j}\vphantom{\overline{B}}}&m^{2}_{\overline{A}j\vphantom{\overline{B}}}\\\
\hline\cr
m^{2}_{i\overline{B}}&m^{2}_{iB\vphantom{\overline{B}}}&m^{2}_{i\overline{j}\vphantom{\overline{B}}}&m^{2}_{ij\vphantom{\overline{B}}}\\\
m^{2}_{\overline{i}\overline{B}}&m^{2}_{\overline{i}B\vphantom{\overline{B}}}&m^{2}_{\overline{i}\overline{j}\vphantom{\overline{B}}}&m^{2}_{\overline{i}j\vphantom{\overline{B}}}\end{array}\right],\end{split}$
(4.13)

where
$m^{2}_{A\overline{B}}=\partial_{A\vphantom{\overline{B}}}\partial_{\overline{B}}V\rvert_{\rm
extr}$ and similarly for the other entries. For our purpose we do not need to
determine the complete mass matrix, but we are only interested in the
expression

$\begin{split}K^{A\vphantom{\overline{B}}}\hskip
1.0ptm^{2}_{A\overline{B}}\hskip
1.0ptK^{\overline{B}}=K^{A\vphantom{\overline{B}}}K^{\overline{B}}\partial_{A\vphantom{\overline{B}}}\partial_{\overline{B}}V\bigr{\rvert}_{\rm
extr}\,.\end{split}$ (4.14)

Using again the relations (LABEL:lu_015), (4.7) and (4.8) we compute

$\begin{split}K^{A\vphantom{\overline{B}}}K^{\overline{B}}\partial_{A\vphantom{\overline{B}}}\partial_{\overline{B}}V&=K^{A\vphantom{\overline{B}}}\partial_{A\vphantom{\overline{B}}}V+K^{\overline{B}}\partial_{\overline{B}}\bigl{(}K^{A\vphantom{\overline{B}}}\partial_{A\vphantom{\overline{B}}}V\bigr{)}\\\
&=K^{A\vphantom{\overline{B}}}\partial_{A\vphantom{\overline{B}}}V+2K^{\overline{B}}\partial_{{\overline{B}}}V\\\
&\hskip 20.0pt+\frac{e^{K_{\rm K}}}{2}\int_{\mathcal{X}}\Bigl{[}\hskip
1.0ptG\wedge\star
G+5G\wedge\star\overline{G}+2\overline{G}\wedge\star\overline{G}+3i\hskip
1.0ptG\wedge\overline{G}\hskip 1.0pt\Bigr{]}\,,\end{split}$ (4.15)

and, employing (4.10) and (4.11), we evaluate (4.14) at the extremum as

$\begin{split}K^{A\vphantom{\overline{B}}}\hskip
1.0ptm^{2}_{A\overline{B}}\hskip 1.0ptK^{\overline{B}}=\left[-6\hskip
1.0ptV+e^{K_{\rm K}}\int_{\mathcal{X}}G\wedge\star\overline{G}\,\right]_{\rm
extr}\,.\end{split}$ (4.16)

For a stable minimum of the potential the mass matrix (4.13) has to be
positive definite, which implies that the left-hand side of (4.16) has to be
positive. This allows us to determine the following necessary condition

$\begin{split}V\hskip 1.0pt\bigr{\rvert}_{\rm min}<\frac{e^{K_{\rm
K}}}{6}\int_{\mathcal{X}}G\wedge\star\overline{G}\;\biggr{\rvert}_{\rm
min}\,.\end{split}$ (4.17)

#### Comments on de-Sitter vacua

Let us briefly return to our discussion of the tadpole charges in section 3,
where we argued that $N_{A}\hskip 1.0pt\tau^{A}\leq 0$. From the form of the
potential in the minimum (4.12), we see that a necessary condition for a de-
Sitter minimum is

$\begin{split}0>\int_{\mathcal{X}}\bigl{(}\mbox{Re}\,G\wedge\star\mbox{Re}\,G\bigr{)}+N_{A}\hskip
1.0pt\tau^{A}\,.\end{split}$ (4.18)

Since the Hodge star is positive definite the first term on the right-hand
side is always non-negative, and hence $N_{A}\tau^{A}$ has to be negative for
a de-Sitter solution. This shows that our requirement (3.14) of having fluxes
contributing like D-branes to the tadpole cancellation conditions does not
exclude de-Sitter vacua. Furthermore, together with the stability condition
(4.17) we find the following constraint at the minimum

$\begin{split}0<-\int_{\mathcal{X}}\bigl{(}\mbox{Re}\hskip
1.0ptG\wedge\star\mbox{Re}\hskip 1.0ptG\bigr{)}-N_{A}\hskip
1.0pt\tau^{A}<\frac{1}{3}\int_{\mathcal{X}}\bigl{(}\mbox{Re}\hskip
1.0ptG\wedge\star\mbox{Re}\hskip 1.0ptG+\mbox{Im}\hskip
1.0ptG\wedge\star\mbox{Im}\hskip 1.0ptG\bigr{)}.\end{split}$ (4.19)

These inequalities restrict stable de-Sitter minima, but they do not exclude
them. We come back to this relation below.

#### Remark

To provide some intuition for the matrix $\omega^{A\overline{B}}$ defined in
(4.4), let us consider the type IIB
$\mathbb{T}^{6}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orientifold with
O3-/O7-planes (where we ignore the twisted sectors). This compactification
space is characterized by $h^{1,1}_{+}=h^{2,1}_{-}=3$ and
$h^{1,1}_{-}=h^{2,1}_{+}=0$, and the Kähler potential for the combined Kähler-
sector moduli is given by

$\begin{split}K=-\sum_{A=0}^{3}\log\left[-i\hskip
1.0pt(\mathsf{T}^{A}-\overline{\mathsf{T}}{}^{A}\bigr{)}\right].\end{split}$
(4.20)

With $\tau^{A}=\mbox{Im}\hskip 1.0pt\mathsf{T}^{A}$ the matrix
$\omega^{A\overline{B}}$ in (4.4) then takes the form

$\begin{split}\omega^{A\overline{B}}=-4\left[\begin{array}[]{cc}0&\tau^{A}\tau^{B}\\\\[2.0pt]
\tau^{A}\tau^{B}&0\end{array}\right].\end{split}$ (4.21)

###  4.2 Constraints on the Hodge numbers

Let us now require that all complex-structure moduli $z^{i}$ and combined
Kähler moduli $\mathsf{T}^{A}$ are stabilized classically by the fluxes
introduced in section 2.2. In order to achieve that, Bianchi identities imply
constraints on the Hodge numbers which we derive in this section.

#### Notation

To derive constraints on the Hodge numbers for stabilizing all closed-string
moduli, it is useful to introduce some more notation. First, for ease of
presentation, we define the two numbers

$\begin{split}d_{1}=2(h^{2,1}_{-}+1)\,,\hskip
50.0ptd_{2}=h^{1,1}+1\,.\end{split}$ (4.22)

Next, for the symplectic basis $\\{\alpha_{I},\beta^{I}\\}\in
H^{3}_{-}(\mathcal{X})$ shown in (2.2) we define two $d_{1}\times d_{1}$
matrices as follows (we suppress indices most of the time)

$\begin{split}\begin{array}[]{rcl@{\hspace{40pt}}rcl}\mathcal{M}&=&\displaystyle\int_{\mathcal{X}}\left(\begin{array}[]{r}\alpha\\\
-\beta\end{array}\right)\wedge\star\left(\begin{array}[]{r@{\hspace{2pt}}c@{\hspace{2pt}}l}\alpha\hskip
2.0&,\hfil\hskip 2.0&-\beta\end{array}\right),\hfil\hskip
40.&\mathcal{M}^{T}&=&+\mathcal{M}\,,\\\\[12.0pt]
\eta&=&\displaystyle\int_{\mathcal{X}}\left(\begin{array}[]{r}\alpha\\\
-\beta\end{array}\right)\wedge\hphantom{\star}\left(\begin{array}[]{r@{\hspace{2pt}}c@{\hspace{2pt}}l}\alpha\hskip
2.0&,\hfil\hskip
2.0&-\beta\end{array}\right)=\left(\begin{array}[]{cc}0&-1\\\\[2.0pt]
+1&0\end{array}\right),\hfil\hskip
40.&\eta^{T}&=&-\eta\,,\end{array}\end{split}$ (4.23)

where $\mathcal{M}$ is positive definite. We also expand the three-forms
$F_{3}$ and $\Xi_{A}$ introduced in (2.11) and (LABEL:hi_001) into
$\\{\alpha_{I},\beta^{I}\\}$ as

$\begin{split}\begin{array}[]{lcl@{\hspace{50pt}}lcl}F_{3}&=&\displaystyle\left(\begin{array}[]{r@{\hspace{2pt}}c@{\hspace{2pt}}l}\alpha\hskip
2.0&,\hfil\hskip 2.0&-\beta\end{array}\right)\hat{F}_{3}\,,\hfil\hskip
50.&\hat{F}_{3}&=&\displaystyle\binom{F_{3\hskip 1.0pt}{}^{I}}{F_{3\hskip
1.0ptI}}\,,\\\\[14.0pt]
\Xi_{A}&=&\displaystyle\left(\begin{array}[]{r@{\hspace{2pt}}c@{\hspace{2pt}}l}\alpha\hskip
2.0&,\hfil\hskip 2.0&-\beta\end{array}\right)\hat{\Xi}_{A}\,,\hfil\hskip
50.&\hat{\Xi}_{A}&=&\displaystyle\binom{\Xi^{I}{}_{A}}{\Xi_{IA}}\,,\end{array}\end{split}$
(4.24)

and to avoid confusion let us state that vector $\hat{F}_{3}$ has $d_{1}$
integer components and that $\hat{\Xi}$ is a $d_{1}\times d_{2}$ dimensional
matrix with integer components.

#### Stabilizing the $c^{A}$

We now turn to the stabilization of the real parts $c^{A}$ of the Kähler-
sector moduli $\mathsf{T}^{A}$. The Kähler potential $K_{\rm K}$ shown in
(2.6), its derivatives $K_{A}=\partial_{A}K$ as well as the Kähler metric
$G_{A\overline{B}}$ do not depend on the $c^{A}$, and in the F-term potential
(4.3) we can isolate the $c^{A}$ contributions as follows

$\begin{split}V_{\rm F}&=\frac{e^{K_{\rm
K}}}{2}\int_{\mathcal{X}}\bigl{(}F_{3}-\Xi_{A}c^{A}\bigr{)}\wedge\star\bigl{(}F_{3}-\Xi_{B}c^{B}\bigr{)}+\ldots\\\
&=\frac{e^{K_{\rm
K}}}{2}\,\bigl{(}\hat{F}^{T}_{3}-c^{T}\Xi^{T}\bigr{)}\,\mathcal{M}\,\bigl{(}\hat{F}_{3}-\Xi\hskip
1.0ptc\bigr{)}+\ldots\,,\end{split}$ (4.25)

where we used the first Bianchi identity in (2.15) and employed the matrix
notation introduced in (4.23) and (4.24). By similar arguments we see that the
D-term potential (2.21) does not depend on $c^{A}$. The extremum of the
combined scalar potential for the $c^{A}$ is then determined as

$\begin{split}0=\partial_{c^{A}}V\hskip 40.0pt\Longrightarrow\hskip
40.0pt0=\Xi^{T}\mathcal{M}\hskip 1.0pt\hat{F}_{3}-(\hskip
1.0pt\Xi^{T}\mathcal{M}\,\Xi\hskip 1.0pt)\hskip 1.0ptc\,,\end{split}$ (4.26)

and in order to stabilize all of the $c^{A}$ the $d_{2}\times d_{2}$ matrix
$\Xi^{T}\mathcal{M}\,\Xi$ has to have maximal rank equal to $d_{2}$. Since
$\mathcal{M}$ is a $d_{1}\times d_{1}$-dimensional matrix, we therefore obtain
the restriction

$\begin{split}d_{2}\leq d_{1}\,.\end{split}$ (4.27)

#### Bianchi identities I

We can however make (4.27) more precise by taking into account the Bianchi
identities for the fluxes. To do so, we first perform a singular value
decomposition of the flux matrix $\hat{\Xi}$ as

$\begin{split}\hat{\Xi}=U\hskip 1.0pt\Sigma\hskip 1.0ptV^{T}\,,\hskip
40.0pt\Sigma=\left(\begin{array}[]{c}\sigma\\\
0\end{array}\right),\end{split}$ (4.28)

where $U$ is a $d_{1}\times d_{1}$ dimensional orthogonal matrix, $V$ is a
$d_{2}\times d_{2}$ dimensional orthogonal matrix and $\sigma$ is a
$d_{2}\times d_{2}$-dimensional diagonal matrix with the singular values of
$\hat{\Xi}$ on the diagonal. Note that since we require $(\hskip
1.0pt\Xi^{T}\mathcal{M}\,\Xi\hskip 1.0pt)$ to have maximal rank, $\sigma$ has
to have maximal rank and therefore is invertible. Turning then to the first
Bianchi identity in (2.15), we see that using matrix notation we can write

$\displaystyle 0=\hat{\Xi}^{T}\eta\>\hat{\Xi}\hskip 40.0pt$
$\displaystyle\Longrightarrow\hskip
40.0pt0=V\left(\begin{array}[]{r@{\hspace{2pt}}c@{\hspace{2pt}}l}\sigma\hskip
2.0&,\hfil\hskip
2.0&0\end{array}\right)U^{T}\eta\>U\left(\begin{array}[]{c}\sigma\\\
0\end{array}\right)V^{T}\,,$ (4.32) $\displaystyle\Longrightarrow\hskip
40.0ptU^{T}\eta\>U=\left(\begin{array}[]{cc}0&B\\\\[2.0pt]
-B^{T}&C\end{array}\right),$ (4.35)

where the upper-left block of $U^{T}\eta\>U$ has dimensions $d_{2}\times
d_{2}$ and $B$ has dimensions $d_{2}\times(d_{1}-d_{2})$. Now, the determinant
of the left-hand side in (4.35) is equal to one, and hence also the right-hand
side has to have a non-vanishing determinant (equal to one). This is however
only possible if $(d_{1}-d_{2})\geq d_{2}$, that is $d_{2}\leq d_{1}/2$, which
means

$\begin{split}\framebox{$\displaystyle{h^{1,1}_{\vphantom{-}}\leq
h^{2,1}_{-}\,.}$}\end{split}$ (4.36)

Hence, in order to stabilize all closed-string moduli by fluxes and satisfy
the Bianchi identities, a necessary requirement is given by (4.36).

#### Bianchi identities II

Let us also consider the third Bianchi identity in (2.15). Expanding the
three-form $\Theta^{A}$ in the basis
$\\{\alpha_{\lambda},\beta^{\lambda}\\}\in H^{3}_{+}(\mathcal{X})$ similarly
as in (4.24) gives rise to a matrix $\hat{\Theta}$ of dimensions $2\hskip
1.0pth^{2,1}_{+}\times(h^{1,1}+1)$. Using matrix notation we then have

$\begin{split}0=\hat{\Xi}\,\hat{\Theta}^{T}\hskip 30.0pt\Longrightarrow\hskip
30.0pt0=U\hskip 1.0pt\Sigma\hskip 1.0ptV^{T}\hskip 1.0pt\hat{\Theta}^{T}\hskip
30.0pt\Longrightarrow\hskip 30.0pt0=\hat{\Theta}^{T}\,,\end{split}$ (4.37)

where we used that $\hat{\Xi}$ has maximal rank and hence the diagonal matrix
$\sigma$ in $\Sigma$ is invertible. We can therefore conclude that when
stabilizing all closed-string moduli by fluxes, Bianchi identities imply that
$\Theta^{A}=0$ and therefore the D-term potential vanishes. Without loss of
generality, for the purpose of moduli stabilization we can therefore restrict
ourselves to

$\begin{split}\framebox{$\displaystyle{h^{2,1}_{+}=0\,.}$}\end{split}$ (4.38)

##  5 The minimal case: $h^{2,1}_{-}=h^{1,1}$

We now consider the situation $h^{2,1}_{-}=h^{1,1}_{\vphantom{+}}$ and
$h^{2,1}_{+}=0$, which is the minimal case for stabilizing all closed-string
moduli by fluxes. In this section we discuss the following questions:

  * •

In section 5.1 we rewrite the scalar potential and replace the R-R three-form
flux $F_{3}$ by the tadpole charges $N_{A}$. This formulation is suitable for
performing computer-based scans for flux vacua.

  * •

In section 5.2 we show that supersymmetric Minkowski vacua with all moduli
stabilized by fluxes are always singular and should be excluded, and

  * •

in section 5.3 we discuss moduli stabilization of the radial Kähler-sector
modulus and determine restrictions for de-Sitter vacua.

###  5.1 Manifest tadpole charges

We start by showing how, after the moduli $c^{A}=\mbox{Re}\,\mathsf{T}^{A}$
have been stabilized, the R-R three-form flux $F_{3}$ can be replaced by the
tadpole charges $N_{A}$. Such a rewriting is useful for eliminating the
dependence on $F_{3}$ in computer-based searches for flux vacua.

#### Singular value decomposition

Let us recall and expand our discussion from section 4.2. In equation (4.28)
we have performed a singular value decomposition of the flux matrix
$\hat{\Xi}$ as

$\begin{split}\hat{\Xi}=U\hskip 1.0pt\Sigma\hskip 1.0ptV^{T}\,,\hskip
40.0pt\Sigma=\left(\begin{array}[]{c}\sigma\\\
0\end{array}\right),\end{split}$ (5.1)

where $\sigma$ is an invertible diagonal matrix. Here we specialize to the
situation $h^{1,1}=h^{2,1}_{-}$, and for ease of presentation we use
$d=h^{1,1}+1=h^{2,1}_{-}+1$. The orthogonal $2d\times 2d$ matrix $U$ can be
expressed in terms of two $2d\times d$ blocks $u_{1}$ and $u_{2}$, and we
introduce the notation

$\begin{split}U=(u_{1},u_{2})\,,\hskip
30.0ptU^{T}\hat{F}_{3}=\left(\begin{array}[]{c}f^{1}\\\
f^{2}\end{array}\right),\hskip 30.0ptU^{T}\mathcal{M}\hskip
1.0ptU=\left(\begin{array}[]{cc}\mathsf{M}_{11}&\mathsf{M}_{12}\\\
\mathsf{M}_{21}&\mathsf{M}_{22}\end{array}\right),\end{split}$ (5.2)

where the $2d\times 2d$ matrix $\mathcal{M}$ was defined in (4.23) and the
$2d$ vector $\hat{F}_{3}$ was defined in (4.24). Note that since $\mathcal{M}$
is positive definite, also $\mathsf{M}_{11}$ is positive definite.
Furthermore, we have argued before that $\hat{\Xi}$ has to satisfy the Bianchi
identity $0=\hat{\Xi}^{T}\eta\>\hat{\Xi}$, which puts restrictions on $u_{1}$
and $u_{2}$. In particular, in the present situation we find that

$\begin{split}U^{T}\eta\,U=\left(\begin{array}[]{cc}0&b\\\\[2.0pt]
-b^{T}&0\end{array}\right),\hskip 60.0pt\det b=\pm 1\,,\end{split}$ (5.3)

where $b=u_{1}^{T}\hskip 1.0pt\eta\hskip 1.0ptu_{2}$ is an invertible $d\times
d$ matrix, and the lower-right block vanishes by requiring
$(U^{T}\eta\,U)^{2}=\mathds{1}$. With a bit of algebra we can then compute

$\begin{split}&\int_{\mathcal{X}}\mbox{Re}\,G\wedge\star\mbox{Re}\,G=\mathsf{F}^{T}\mathsf{M}_{11}\hskip
1.0pt\mathsf{F}+N^{T}m^{-1}N\,,\\\
&\int_{\mathcal{X}}\mbox{Im}\,G\wedge\star\mbox{Im}\,G=\tau^{T}m\,\tau\,,\end{split}$
(5.4)

where $N_{A}$ denotes again the vector of tadpole charges, where
$\tau^{A}=\mbox{Im}\,\mathsf{T}^{A}$ are the imaginary parts of the combined
Kähler-sector moduli, and where we defined the $d\times d$ matrix and
$d$-vector

$\begin{split}m=\Xi^{T}\mathcal{M}\,\Xi\,,\hskip
60.0pt\mathsf{F}=f^{1}+\mathsf{M}_{11}^{-1}\mathsf{M}_{12}^{\vphantom{-1}}f^{2}-\sigma\,V^{T}c\,.\end{split}$
(5.5)

#### Stabilizing the $c^{A}$

Next, we turn to the scalar F-term potential shown in (4.3). As mentioned
already in section 4.2, the moduli $c^{A}=\mbox{Re}\,\mathsf{T}^{A}$ only
appear through $\mbox{Re}\,G$, and using (LABEL:rw_838) we obtain

$\begin{split}V_{F}=\frac{e^{K_{\rm
K}}}{2}\,\int_{\mathcal{X}}\mbox{Re}\,G\wedge\star\mbox{Re}\,G+\ldots=\frac{e^{K_{\rm
K}}}{2}\,\mathsf{F}^{T}\mathsf{M}_{11}\hskip
1.0pt\mathsf{F}+\ldots\,.\end{split}$ (5.6)

Since by assumption the $d\times d$ matrix $\mathsf{M}_{11}$ has maximal rank
$d$, minimizing (5.6) with respect to $c^{A}$ fixes all $c^{A}$ in terms of
the flux matrix $\hat{\Xi}$ and complex-structure moduli as

$\begin{split}\mathsf{F}=0\hskip 40.0pt\Longrightarrow\hskip
40.0ptc=V\sigma^{-1}\bigl{(}f^{1}+\mathsf{M}_{11}^{-1}\mathsf{M}^{\vphantom{-1}}_{12}f^{2}\bigr{)}\,.\end{split}$
(5.7)

However, for our purpose the precise values of the stabilized $c^{A}$ are not
important. We furthermore note that with $\phi$ denoting all real moduli
except the $c^{A}$, the mass matrix takes the form

$\begin{split}M^{2}=\left[\begin{array}[]{cc}e^{K_{\rm K}}m&0\\\
0&M^{2}_{\phi\phi}\end{array}\right],\hskip
60.0ptm=\Xi^{T}\mathcal{M}\,\Xi\,,\end{split}$ (5.8)

where $m$ is positive definite and $M^{2}_{\phi\phi}$ denotes the mass matrix
for all moduli $\phi$. Hence, the stabilization of the $c^{A}$ can be
separated from the problem of stabilizing the remaining moduli.

#### Scalar potential

We finally turn to the full form of the F-term potential (4.3) after the
$c^{A}$ moduli have been stabilized. We denote this potential by
$\tilde{V}_{F}=V_{F}\hskip 1.0pt\rvert_{{\rm min}\,c^{A}}$. Separating the
three-form flux $G$ into real and imaginary components, using the first
Bianchi identity in (2.15) and employing equation (LABEL:rw_838), this
potential can be written as

$\begin{split}\framebox{$\displaystyle{\tilde{V}_{F}=e^{K_{\rm
K}}\biggl{[}\,\frac{1}{2}\hskip 1.0ptN^{T}m^{-1}N+N^{T}\tau+\frac{1}{2}\hskip
1.0pt\tau^{T}m\,\tau+e^{-K_{\rm
cs}}\,\Xi^{\overline{0}}_{A\vphantom{\overline{B}}}\,\omega^{A\overline{B}}\,\Xi^{0}_{\overline{B}}\,\biggr{]}\,.}$}\end{split}$
(5.9)

From here we see that indeed, after the $c^{A}$ have been stabilized, we can
express the F-term potential using the tadpole charges $N_{A}$ instead of the
R-R three-form flux $F_{3}$. However, of course one has to check whether for a
given $N$ and flux matrix $\hat{\Xi}$ there exists a corresponding
$\hat{F}_{3}$ with integer components such that $N=\hat{\Xi}^{T}\eta\hskip
1.0pt\hat{F}_{3}$. Coming then back to the expression for the scalar potential
at the minimum derived in (4.12), using (LABEL:rw_838) together with (5.7) we
find

$\begin{split}V\hskip 1.0pt\bigr{\rvert}_{\rm extr}=-\frac{e^{K_{\rm
K}}}{2}\Bigl{[}\hskip 1.0ptN^{T}m^{-1}N+N^{T}\tau\Bigr{]}_{\rm
extr}\,,\end{split}$ (5.10)

where we recall that the matrix $m=\Xi^{T}\mathcal{M}\,\Xi$ was defined in
(5.5). We discuss this expression in more detail below.

###  5.2 No supersymmetric Minkowski vacua

We now want to show that in the case $h^{2,1}_{-}=h^{1,1}_{\vphantom{+}}$,
supersymmetric Minkowski vacua require the complex-structure moduli to be
stabilized at the boundary of moduli space. Such solutions should therefore be
excluded. Note that for type IIA compactifications with non-geometric fluxes a
similar result has been derived in [16].

#### Analysis

To show the above statement, we recall the general form of the F-term
potential (2.19) and note that for supersymmetric Minkowski vacua we have to
require

$\begin{split}0=W\,,\hskip 40.0pt0=F_{A}=\partial_{A}W\,,\hskip
40.0pt0=F_{i}=\partial_{i}W\,,\end{split}$ (5.11)

where $A=0,\ldots,h^{1,1}$ labels the combined Kähler-sector moduli and
$i=1,\ldots,h^{2,1}_{-}$ labels the complex-structure moduli. The
superpotential $W$ was defined in (2.20), for which the second condition in
(5.11) reads

$\begin{split}0=\int_{\mathcal{X}}\Xi_{A}\wedge\Omega\,.\end{split}$ (5.12)

Let us now expand the holomorphic three-form $\Omega$ into the symplectic
basis (2.2) and define

$\begin{split}\Omega=(\alpha,-\beta)\,\binom{\mathsf{X}}{\mathsf{F}}\,,\hskip
40.0ptU^{T}\binom{\mathsf{X}}{\mathsf{F}}=\binom{x^{1}}{x^{2}}\,,\end{split}$
(5.13)

where $\mathsf{X}$ and $\mathsf{F}$ denote the periods of the Calabi-Yau
three-fold, the $2d\times 2d$ matrix $U$ appeared in the singular value
decomposition (5.1) and where we suppressed indices. Using this notation
together with the expansion of $\Xi_{A}$ given in (4.24), the singular value
decomposition of $\hat{\Xi}$ shown in (5.1) and the form of $U^{T}\eta\,U$
given in (5.3) allows us to write (5.12) as

$\begin{split}0=\int_{\mathcal{X}}\Xi\wedge\Omega=V\hskip
1.0pt\bigl{(}\sigma,0\bigr{)}\,U^{T}\eta\,U\,U^{T}\binom{\mathsf{X}}{\mathsf{F}}=V\hskip
1.0pt\sigma\hskip 1.0ptb\hskip 1.0ptx^{2}\hskip 20.0pt\Longrightarrow\hskip
20.0ptx^{2}=0\,,\end{split}$ (5.14)

where we used that the $d\times d$ matrices $V$, $\sigma$ and $b$ are
invertible. We can now use this solution to evaluate

$\begin{split}\int_{\mathcal{X}}\Omega\wedge\overline{\Omega}=\bigl{(}\mathsf{X},\mathsf{F}\bigr{)}\,\eta\,\binom{\overline{\mathsf{X}}}{\overline{\mathsf{F}}}=\bigl{(}x^{1T},x^{2T}\bigr{)}\left(\begin{array}[]{cc}0&b\\\\[2.0pt]
-b^{T}&0\end{array}\right)\binom{\overline{x}{}^{1}}{\overline{x}{}^{2}}\overset{x^{2}=0}{=}0\,.\end{split}$
(5.15)

This implies that the Kähler potential $K_{\rm cs}$ shown in (2.6) is
singular, and hence the complex-structure moduli are stabilized at the
boundary of their moduli space. Note that since we did not specify the form of
the periods this result holds for the quantum-corrected periods, and we have
to exclude such situations.

#### Remark

We close this subsection with the following remark. For
$h^{1,1}_{\vphantom{-}}<h^{2,1}_{-}$ the argument presented above is modified
and supersymmetric Minkowski vacua stabilizing all moduli cannot be excluded.
The condition (5.12) is equal to $\Xi_{A}^{\overline{0}}=0$, which when
inserted in (4.5) gives the condition

$\begin{split}\mbox{supersymmetric Minkowski vacuum:}\hskip 40.0pt\star
G=i\hskip 1.0ptG\,.\end{split}$ (5.16)

This is the well-known requirement of $G$ being imaginary self-dual for the
case of only $H$-flux present [66].

###  5.3 Stabilizing the radial Kähler-sector modulus

In this section we study moduli stabilization and stability conditions for the
radial Kähler-sector modulus, and discuss conditions for obtaining de-Sitter
minima.

#### Moduli stabilization

To do so, we first split the combined Kähler-sector modulus $\tau^{A}$ and the
tadpole-charge vector $N_{A}$ into radial and angular parts, that is, using
the Euclidean norm $\delta\equiv\delta_{AB}$ and suppressing indices we define

$\begin{split}\begin{array}[c]{lcl}\tau&=&\mathsf{r}\,e_{\tau}\,,\\\\[4.0pt]
N&=&\mathsf{n}\,e_{N}\,,\end{array}\hskip 40.0pt\mbox{where}\hskip
40.0pt\begin{array}[c]{r@{\op}l@{\op}lcl@{\hspace{25pt}}r@{\op}l@{\op}lcl}\tau^{T}\hskip
1.0pt&\delta\hfil\hskip 1.0pt&\tau&=&\mathsf{r}^{2}\,,\hfil\hskip 25.0&\
e_{\tau}^{T}\hskip 1.0pt&\delta\hfil\hskip
1.0pt&e^{\vphantom{T}}_{\tau}&=&1\,,\\\\[4.0pt] N^{T}\hskip
1.0pt&\delta^{-1}\hfil\hskip 1.0pt&N&=&\mathsf{n}^{2}\,,\hfil\hskip 25.0&\
e_{N}^{T}\hskip 1.0pt&\delta^{-1}\hfil\hskip
1.0pt&e^{\vphantom{T}}_{N}&=&1\,.\end{array}\end{split}$ (5.17)

Our convention is $\mathsf{r}\geq 0$ and $\mathsf{n}\geq 0$. Next, we note
that the Kähler potential $K_{\rm K}$ and the matrix $\omega^{A\overline{B}}$
appearing in the scalar potential (5.9) depend on the radial modulus
$\mathsf{r}$ as $e^{K_{\rm K}}\sim\mathsf{r}^{-4}$ and
$\omega\sim\mathsf{r}^{2}$. We can then express the scalar potential (5.9) as

$\begin{split}\tilde{V}_{F}=\frac{1}{\mathsf{r}^{4}}\left[\,\frac{a}{2}+b\,\mathsf{r}+\frac{c}{2}\hskip
1.0pt\mathsf{r}^{2}\,\right]\,,\end{split}$ (5.18)

where the coefficients $a,b,c$ are independent of $\mathsf{r}$. Using
$e^{\tilde{K}_{\rm K}}=\mathsf{r}^{4}\hskip 1.0pte^{K_{\rm K}}$ and
$\tilde{\omega}^{A\overline{B}}=\mathsf{r}^{-2}\omega^{A\overline{B}}$, they
are given by

$\begin{split}\begin{array}[c]{lcl}a&=&\displaystyle e^{\tilde{K}_{\rm
K}}\hskip 1.0pt\mathsf{n}^{2}\,e_{N}^{T}m^{-1}e_{N}\,,\\\\[4.0pt]
b&=&\displaystyle e^{\tilde{K}_{\rm K}}\hskip
1.0pt\mathsf{n}^{\hphantom{2}}\,e_{N}^{T}\,e_{\tau}\,,\end{array}\hskip
60.0ptc=e^{\tilde{K}_{\rm
K}}\Bigl{(}e_{\tau}^{T}m\,e_{\tau}^{\vphantom{T}}+2\hskip 1.0pte^{-K_{\rm
cs}}\,\Xi^{\overline{0}}\,\tilde{\omega}\,\Xi^{0}\Bigr{)}\,,\end{split}$
(5.19)

where the matrix $m$ was defined in (5.5), and since $m$ is positive definite
we have $a\geq 0$. We can then minimize the potential (5.18) with respect to
$\mathsf{r}$ by solving $\partial_{\mathsf{r}}\tilde{V}_{F}=0$. Imposing in
addition the necessary condition $\partial_{\mathsf{r}}^{2}\tilde{V}_{F}>0$
for the minimum to be stable, we obtain the following necessary conditions for
stable minima:

$\begin{split}\begin{array}[]{|l@{\hspace{50pt}}lll|}\hline\cr\hbox{\multirowsetup\mbox{AdS
minimum}}\hfil\hskip 50.&a>0&b\leq 0&c<\frac{b^{2}}{a}\\\ \hfil\hskip
50.&a>0&b>0&c<0\\\ \hline\cr\mbox{Minkowski minimum}\hfil\hskip
50.&a>0&b<0&c=\frac{b^{2}}{a}\\\ \hline\cr\mbox{dS minimum}\hfil\hskip
50.&a>0&b<0&\frac{b^{2}}{a}<c<\frac{9}{8}\hskip 1.0pt\frac{b^{2}}{a}\\\
\hline\cr\end{array}\end{split}$ (5.20)

Note however that our requirement (3.14) excludes AdS minima with $b>0$.
Furthermore, when ignoring $\alpha^{\prime}$-corrections to the Kähler
potential and only considering $H$-flux, the term
$\Xi^{\overline{0}}\,\tilde{\omega}\,\Xi^{0}$ vanishes and $c>0$.

#### Stable de-Sitter minima

Let us now focus on stable de-Sitter minima. As we can see from (5.20), after
$\mathsf{r}$ has been stabilized the remaining moduli have to be fixed in a
rather restricted region. This is in agreement with the observation made in
[23, 24] that stable de-Sitter vacua are located in a narrow band in moduli
space. For the stabilized radial modulus $\mathsf{r}$ the constraints on $c$
shown in (5.20) translate into

$\begin{split}\mbox{dS minimum:}\hskip 40.0pt\frac{a}{|b|}<\mathsf{r}_{\rm
min}<\frac{4}{3}\,\frac{a}{|b|}\hskip 25.0pt\mbox{with}\hskip
25.0pt\frac{a}{|b|}=\mathsf{n}\,\frac{e_{N}^{T}\hskip
1.0ptm^{-1}e_{N}}{|e_{N}^{T}\,e_{\tau}|}\,,\end{split}$ (5.21)

and from here we see that a large value for $\mathsf{r}_{\rm min}$ – necessary
for a large-volume and small-coupling limit – can be achieved in three
different ways:

  1. 1.

The first possibility to make $a/|b|$ large is to require $\mathsf{n}\gg 1$.
This means that the tadpole charges have to be large, requiring a large number
of O3- and O7-planes to be present. Such configurations exists – but these
usually have a complicated topology and hence a large number of moduli have to
be stabilized. Problems related to the latter issue have been discussed
recently for instance in [44, 45, 46].

  2. 2.

A second possibility for obtaining a large $\mathsf{r}_{\rm min}$ is to
stabilize the complex-structure moduli such that $e_{N}^{T}\hskip
1.0ptm^{-1}e_{N}\gg 1$, where $m$ was defined in (5.5). Since $e_{N}$ is
normalized to one, such a condition is possible only near a boundary of
complex-structure moduli space.

  3. 3.

A third way to make $a/|b|$ large is to have $|e_{N}^{T}\,e_{\tau}|\ll 1$.
Demanding the moduli to be stabilized in a large-volume and weak-coupling
regime implies that all components of $\tau^{A}$ have to be large (excluding
the $h^{1,1}_{-}$ moduli). This means that approximately we have
$\displaystyle e_{\tau}\sim(1,1,\ldots,1)/\sqrt{h^{1,1}}$. Since we require
all components of the tadpole charges to be non-positive, to achieve
$|e_{N}^{T}\,e_{\tau}|\ll 1$ we therefore may look for compactifications with
$e_{N}=(0,\ldots,0,-1,0,\ldots)$ and $h^{1,1}\gg 1$.

To summarize, for stable de-Sitter vacua the moduli have to be stabilized in a
narrow region of moduli space and one typically has to consider non-generic
situations. This explains why it is difficult to find de-Sitter solutions,
however, from our analysis here we do not see any reason why they cannot
exist.

##  6 Comments on de-Sitter vacua

A number of approaches for constructing de-Sitter vacua using non-geometric
fluxes have appeared in the literature [19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30]. We briefly review them in this section. For the stable
constructions which in addition to the R-R three-form flux $F_{3}$ only use
(non-)geometric $H$-, $F$-, $Q$\- and $R$-fluxes to stabilize all closed-
string moduli, we argue that none of them satisfy all the consistency
conditions discussed above.

#### The de Carlos-Guarino-Moreno model

The authors of [19, 20] have constructed a family of stable de-Sitter minima
for type IIB string theory compactified on the isotropic
$\mathbb{T}^{6}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orientifold with
O3-/O7-planes in the presence of geometric and non-geometric fluxes. Let us
recall a particular representative of this family using our conventions:

  * •

The isotropic torus can be seen as a compactification manifold with topology
characterized by $h^{1,1}_{+}=h^{2,1}_{-}=1$ and $h^{1,1}_{-}=h^{2,1}_{+}=0$.
Up to an irrelevant additive constant, the corresponding Kähler potential in
the large-volume, small-coupling and large-complex-structure limit is given by

$\begin{split}K=-\log\bigl{[}-i\hskip
1.0pt(\mathsf{T}^{0}-\overline{\mathsf{T}}{}^{0}\bigr{)}\bigr{]}-3\log\bigl{[}-i\hskip
1.0pt(\mathsf{T}^{1}-\overline{\mathsf{T}}{}^{1}\bigr{)}\bigr{]}-3\log\bigl{[}-i\hskip
1.0pt(z^{1}-\overline{z}{}^{1}\bigr{)}\bigr{]}.\end{split}$ (6.1)

  * •

The R-R three-form flux, the NS-NS $H$\- and $Q$-flux and the tadpole charges
can be expressed using the matrix notation introduced in (4.24). In
particular, we have

$\begin{split}\hat{F}_{3}=\left(\begin{array}[]{r}-47\\\ -44\\\ 44\\\
51\end{array}\right),\hskip
40.0pt\hat{\Xi}=\left(\begin{array}[]{r@{\hspace{8pt}}r}44\hskip 8.0&0\\\
1\hskip 8.0&-1\\\ -1\hskip 8.0&-3\\\ 132\hskip 8.0&0\end{array}\right),\hskip
40.0ptN=\left(\begin{array}[]{r}-7748\\\ 192\end{array}\right),\end{split}$
(6.2)

where the first column in $\hat{\Xi}$ corresponds to the $H$-flux and the
second column to the non-geometric $Q$-flux. Note also that the fluxes
contained in $\hat{\Xi}$ satisfy the Bianchi identity (2.15) and that
$\hat{\Xi}$ is of maximal rank equal to $2$.

  * •

Using this data for instance in (5.9) and minimizing the resulting potential,
the authors of [19, 20] find a de-Sitter minimum with $V\rvert_{\rm
min}=1.43\cdot 10^{-3}$ at

$\begin{split}\tau^{0}_{\rm min}=2.008\,,\hskip 30.0pt\tau^{1}_{\rm
min}=48.684\,,\hskip 30.0ptz^{1}_{\rm min}=-1.034+1.144i\,.\end{split}$ (6.3)

(As explained above, the $c^{A}$ dependence can be decoupled.) This minimum is
stable, and the eigenvalues of the corresponding canonically-normalized mass
matrix are

$\begin{split}\mbox{eigenvalues of $M^{2}_{\rm can}$\,:}\hskip
20.0pt\bigl{\\{}\,4.25\hskip 1.0pt,\;1.75\hskip 1.0pt,\;1.46\hskip
1.0pt,\;2.29\cdot 10^{-3}\,\bigr{\\}}\,.\end{split}$ (6.4)

This example illustrates that the supergravity equations following from non-
geometric flux compactifications can have stable de-Sitter solutions. However,
we believe that this model is not consistent in string theory: in section 3 we
argued that in order to avoid arbitrary-large gauge groups the tadpole charges
$N_{A}$ should be non-positive, and from (6.2) we see that the present model
violates this requirement. (This observation extends to the whole family of
models.) We therefore do not consider it to be a consistent string-theory
construction.

#### Other de-Sitter constructions with non-geometric fluxes I

Other papers known to us which construct de-Sitter vacua – using in addition
to the R-R three-form flux $F_{3}$ only (non-)geometric $H$-, $F$-, $Q$\- and
$R$-fluxes – are the following:

  * •

From a supergravity point of view the gauge algebras originating from non-
geometric fluxes haven been studied in [21, 22]. However, no stable de-Sitter
models have been found.

  * •

In [23] a search for de-Sitter vacua has been performed for the
$\mathbb{T}^{6}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orientifold, but no fully
stable de-Sitter extrema without any tachyonic directions have been found.

  * •

In [24] the authors found three stable de-Sitter minima with the help on an
evolutionary algorithm. However, one can check that these models (summarized
in appendix A of [24]) do not satisfy the Bianchi identities (2.15). We
therefore do not consider them to be consistent string-theory models.

  * •

In [25] as well as in [26] the authors considered compactifications of type
IIB string theory on the $\mathbb{T}^{6}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
orientifold with O3-/O7-planes and (non-)geometric fluxes. With the help of a
genetic algorithm a number of stable de-Sitter minima are obtained for this
setting, however, we were not able to reproduce these results.555We thank C.
Damian and O. Loaiza-Brito for correspondence on this question.

  * •

In [28] Bianchi identities for toroidal type IIB compactifications have been
analyzed, but no stable de-Sitter solutions were found.

  * •

In [30] the authors consider non-geometric flux compactifications of type IIB
string theory on the isotropic
$\mathbb{T}^{6}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orientifold. Using
machine-learning techniques the authors obtain de-Sitter extrema which,
however, have at least one tachyonic direction.

We conclude that none of the non-geometric de-Sitter models reviewed here are
fully-stable and consistent in string theory. This observation is in agreement
with the swampland de-Sitter conjecture made in [3].

#### Other de-Sitter constructions with non-geometric fluxes II

In addition to the $H$-, $F$-, $Q$\- and $R$-fluxes discussed in this paper,
one can consider the non-geometric $P$-flux. This flux is the the S-dual
completion of the $Q$-flux [58], which generates quadratic couplings among the
Kähler-sector moduli in the superpotential.

  * •

In [26, 27, 29] the authors include non-geometric $P$-flux for the
construction of de-Sitter vacua. We have not considered such fluxes in our
work and therefore do not comment on these examples.

##  7 Summary, conclusions, outlook

In this paper we have studied compactifications of type IIB string theory on
Calabi-Yau orientifolds with O3-/O7-planes and (non-)geometric $H$-, $F$-,
$Q$\- and $R$-fluxes. For this setting it is possible to stabilize all closed-
string moduli classically without the need for non-perturbative contributions,
and examples of stable de-Sitter constructions can be found in the literature.

#### Summary

Our motivation for this work was to investigate whether non-geometric flux
compactifications allow for stable de-Sitter vacua. Although we were neither
able to show that de-Sitter vacua cannot be obtained from non-geometric fluxes
nor able to construct a consistent de-Sitter solution, we made progress in
understanding non-geometric flux compactifications. More concretely,

  * •

based on the requirement that the total rank of the gauge group should be
bounded from above (cf. [31]), in section 3 we argued that the contributions
of closed-string fluxes to the tadpole cancellation conditions should be
similar to D-branes and not anti-D-branes. In our conventions this means
$N_{A}\leq 0$.

  * •

In section 4 we derived a simple expression for the scalar potential at the
minimum (cf. equation (4.12)). We furthermore showed that in order to
stabilize all closed-string moduli classically by $H$-, $F$-, $Q$-, $R$\- and
$F_{3}$-fluxes, the topology of the compactification manifold has to be
restricted as $h^{1,1}_{\vphantom{-}}\leq h^{2,1}_{-}$ and we can choose
$h^{2,1}_{+}=0$ without loss of generality.

  * •

When specializing to the case $h^{1,1}_{\vphantom{-}}=h^{2,1}_{-}$, we were
able to eliminate the dependence on the R-R three-form flux $F_{3}$ in favor
of the tadpole charges $N_{A}$ in the scalar potential (cf. equation (5.9)).
This form promises to be very useful for performing computer-based searches
for flux vacua. We also showed that for $h^{1,1}=h^{2,1}_{-}$ it is not
possible to obtain supersymmetric Minkowski vacua, and we derived constraints
for stable de-Sitter vacua. In particular, for the latter we concluded that
moduli have to be stabilized in a rather restricted region of moduli space.

  * •

In section 6 we reviewed known de-Sitter constructions based on non-geometric
flux compactifications. For the fully-stable constructions with only $H$-,
$F$-, $Q$\- and $R$-fluxes we concluded that these are not consistent in
string theory.

#### Conclusions

The swampland conjecture made in [3] proposes that stable de-Sitter vacua
cannot exist in any consistent theory of quantum gravity. Non-geometric flux
compactifications provide potential counter-examples to this conjecture,
however, in this paper we have argued that all examples known to us (which
only involve $H$-, $F$-, $Q$-, $R$\- and $F_{3}$-fluxes) are not consistent in
string theory. On the other hand, we were not able to show in general that
stable de-Sitter vacua cannot be obtained from non-geometric flux-
compactifications. In fact, the scalar potential we derived shows a close
resemblance to ordinary Calabi-Yau compactifications with the familiar $H$\-
and $F_{3}$-flux. Whether non-geometric stable de-Sitter vacua exist in string
theory remains an open question.

#### Outlook

The results obtained in this paper provide a starting point for further
research on non-geometric flux-compactifications:

  * •

Our requirement from section 3 on the tadpole charges $N_{A}$ has been
motivated by plausibility and duality arguments and by extending existing
swampland conjectures. It would be desirable to derive the condition
$N_{A}\leq 0$ from first principles.

  * •

For the case $h^{1,1}_{\vphantom{-}}=h^{2,1}_{-}$ we derived an expression for
the scalar potential where the R-R three-form flux $F_{3}$ has been replaced
by the tadpole charges $N_{A}$. Using our requirement from section 3, for a
given compactification manifold there are finitely-many choices for $N_{A}$
which in turn simplifies computer-based searches for flux vacua.

  * •

The complex-structure moduli sector is subject to a
$Sp(2h^{2,1}_{-}+2,\mathbb{Z})$ duality acting on the third cohomology [67].
This implies that the NS-NS $H$-, $F$-, $Q$\- and $R$-fluxes contained in the
flux matrix (4.24) are unique up to $Sp(2h^{2,1}_{-}+2,\mathbb{Z})$
transformations. As a preliminary study we considered
$h^{1,1}_{\vphantom{-}}=h^{2,1}_{-}=1$ and flux quanta with values
$-3,\ldots,+3$ appearing in $\hat{\Xi}$. In this case there are $318\,784$
combinations for which $\hat{\Xi}$ has maximal rank, which by
$Sp(2h^{2,1}_{-}+2,\mathbb{Z})$ transformations are related to $70$
inequivalent choices. We therefore have a reduction of the size of the flux
landscape as

$\begin{split}\mbox{$318\,784$ flux choices}\quad\xrightarrow{\hskip
10.0ptSp(4,\mathbb{Z})\hskip 10.0pt}\quad\mbox{$70$ inequivalent flux
choices}\,.\end{split}$ (7.1)

Note however that $Sp(2h^{2,1}_{-}+2,\mathbb{Z})$ also acts on the periods of
the holomorphic three-form in a non-trivial way and typically does not
preserve for instance a large-volume limit. It would be interesting to extend
this observation to more general NS-NS flux configurations.

We are planning to come back to these questions in the future.

#### Acknowledgments

We thank Johan Blåbäck, Cesar Damian, Adolfo Guarino, Daniel Junghans, Oscar
Loaiza-Brito and Thomas van Riet for helpful discussions and communications.
The work of EP is supported by a Heisenberg grant of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) with project-number
430285316.

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