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#  Dark energy: EFTs and supergravity

Francesc Cunillera

Summary

### Summary

The subject of this thesis is cosmological implications of string
compactifications understood in a broad sense. In the first half of the
thesis, we will begin by reviewing the four-dimensional description of the
tree-level perturbative type IIB action. We will then introduce a number of
open questions in cosmology and their relevance with regards to the remainder
of the thesis. We will first explore some of these questions from the
perspective of effective field theories motivated by supergravity. In
particular, we provide a description of a naturally light dark energy field in
terms of the clockwork mechanism and the Dvali-Kaloper-Sorbo four-form mixing.
We study its possible UV completion and show a no-go for its embedding within
perturbative type IIA supergravity. We also discuss the coincidence problem
for dynamical models of dark energy consistent with a quintessence field
slowly rolling down a potential slope, of the type one would expect from the
asymptotics of moduli space. As it rolls, a tower of heavy states will
generically descend, triggering a phase transition in the low energy
cosmological dynamics after at most a few hundred Hubble times. As a result,
dark energy domination cannot continue indefinitely and there is at least a
percentage chance that we find ourselves in the first Hubble epoch.

In the second half of the thesis, we introduce the effects of perturbative and
non-perturbative corrections to the tree-level type IIB action. We then focus
on obtaining a viable model of quintessence from the type IIB effective field
theory. However, we are able to show that such a model must have a non-
supersymmetric Minkowski vacuum at leading order. Furthermore, it must
necessarily take the form of axion hilltop quintessence. When we consider the
effects of quantum fluctuations during the early Universe, we see that such
models must have extremely fine-tuned initial conditions to describe a slow-
rolling scalar field at present times. We conclude that quintessence faces
more challenges than a true cosmological constant, to the point that
quintessence is very unattractive for model building modulo a ruling out of
the cosmological constant by observations. Following this line of reasoning,
we consider whether other perturbative corrections can generate de Sitter
solutions in an appropriate setting. In particular, we consider the effects of
higher curvature corrections in the Gauss-Bonnet term. Remarkably, we are able
to show that, for the particular setting of a fluxed runaway potential
motivated by heterotic supergravity, the curvature corrections reduce the
space of solutions. Declaration

### Declaration

This dissertation is the result of my own work and includes nothing which is
the outcome of work done in collaboration except where specifically indicated
in the text. No part of this thesis has previously been submitted for a degree
or other qualification at this or any other university.

The following chapters contain original research presented in the following
papers:

  * •

Chap. 4: A natural theory of dark energy, Lorenzo Bordin, FC, Antoine Lehébel
and Antonio Padilla, _Phys.Rev.D_ (2020), arXiv:1912.04905 [hep-th]

  * •

Chap. 5: A stringy perspective on the coincidence problem, FC and Antonio
Padilla, _JHEP_ (2021), arXiv:2105.03426 [hep-th]

  * •

Chap. 6: Quintessence and the Swampland. The parametrically controlled regime
of moduli space., Michele Cicoli, FC, Antonio Padilla and Francisco G. Pedro,
_Fortsch.Phys._ (2022), arXiv:2112.10779 [hep-th]

  * •

Chap. 8: Quintessence and the Swampland. The numerically controlled regime of
moduli space., Michele Cicoli, FC, Antonio Padilla and Francisco G. Pedro,
_Fortsch.Phys._ (2022), arXiv:2112.10783 [hep-th]

  * •

Chap. 9: Quadratic curvature corrections and the absence of de Sitter vacua,
FC, Will Emond, Antoine Lehébel and Antonio Padilla, _JHEP_ (2022),
arXiv:2112.05771 [hep-th]

These papers are references [1, 2, 3, 4, 5] in the bibliography, respectively.

Francesc Cunillera

Nottingham, August 28, 2024

Acknowledgements

### Acknowledgements

The first and biggest thanks goes to Tony. For some reason you decided that I
was worth your time and put up with 3 years of my painstaking mistakes and
dumb ideas. You gave me enough freedom to go on a random supergravity journey
for my PhD and taught me so much about Physics and what it means to be a good
physicist during that time. I will always be thankful for all the time you
have dedicated to nurture me as a budding physicist and, hopefully, some of
your innate intuition about Physics will have rubbed off.

I would also like to give a special thanks to Ed Copeland, Tasos Avgoustidis,
Michele Cicoli and Francisco Gil Pedro for many long discussions about strings
and physics in general. Ed, you also acted as a supervisor for me, making sure
the less obvious admin duties where fulfilled and entertained long discussions
about everything cosmological when I would randomly pop into your office.
Tasos, our chats and coffees at Choulia’s will be one of the things I miss the
most from my times at Nottingham. Michele and Francisco, you guys patiently
put up with many discussions over Zoom while I was getting up to speed with
model building aspects of supergravity. You also made both Tony and I jealous
of the beautiful Italian countryside, especially when comparing it to our
bleak English weather.

I would also like to thank Arthur Hebecker, Chad Briddon, Joe Conlon, Lorenzo
Bordin, Clare Burrage, Will Emond, Cristiano Germani, Oli Gould, Antoine
Lehébel, Pete Millington, Adam Moss, Paul Saffin and Marco Scalisi for many
useful and insightful discussions over the years.

###### Contents

  1. 1 Introduction and motivations
    1. 1.1 Outline
    2. 1.2 Units and conventions
  2. I String inspired EFTs
    1. 2 Perturbative supergravity in type IIB theories
      1. 2.1 Special geometry of Kähler manifolds
      2. 2.2 Perturbative $\mathcal{N}=2$ supergravity
      3. 2.3 Orientifolding the supergravity action
      4. 2.4 Flux compactifications
      5. 2.5 The Swampland programme
    2. 3 Some open questions in modern cosmology
      1. 3.1 The hierarchy problem and naturalness
      2. 3.2 The cosmological constant problem
      3. 3.3 Coincidence problem
    3. 4 A naturally light dark energy model
      1. 4.1 Clockwork mechanism
      2. 4.2 Dvali-Kaloper-Sorbo four-form model
      3. 4.3 Obtaining a naturally light dark energy scale
      4. 4.4 Warm-up: A string motivated toy model
      5. 4.5 A more in-depth look into achieving the embedding
      6. 4.6 Deconstructing clockwork dark energy
      7. 4.7 Conclusions
    4. 5 A stringy perspective on the coincidence problem
      1. 5.1 Generic idea
      2. 5.2 A toy model
      3. 5.3 Further discussion
    5. 6 Intermezzo: Supergravity and the Swampland
      1. 6.1 A simple example of the backreaction on moduli space
      2. 6.2 Calculations at the boundary of moduli space
      3. 6.3 Divergence of string diagrams and perturbative control or lack thereof
  3. II String models
    1. 7 Corrections to the supergravity action in type IIB
      1. 7.1 Review of type IIB effective field theory
      2. 7.2 Non-renormalisation theorems
      3. 7.3 $\alpha^{\prime 3}$ corrections
      4. 7.4 One-loop corrections
      5. 7.5 Higher derivative corrections
      6. 7.6 Non-perturbative corrections
      7. 7.7 Validity of the effective description
      8. 7.8 KKLT model building
      9. 7.9 Racetrack model building
      10. 7.10 LVS model building
    2. 8 Obstructions to quintessence model building
      1. 8.1 Old challenges for quintessence in string theory
      2. 8.2 The KL problem for quintessence
      3. 8.3 Axion hilltop quintessence and initial conditions
      4. 8.4 Discussion
    3. 9 Gauss-Bonnet curvature corrections and the absence of de Sitter vacua
      1. 9.1 Review of flux compactifications of the heterotic string action and stability conditions
      2. 9.2 Extending the Montero-van Riet-Venken bound in the small Gauss-Bonnet coupling limit
      3. 9.3 Non-perturbative conditions for the existence of de Sitter solutions with a Gauss-Bonnet term
      4. 9.4 Discussion
    4. 10 Conclusions and outlooks
    5. A Details and calculations for type IIB supergravity
      1. A.1 Calculations for the dimensional reduction of the $\mathcal{N}=2$ type IIB action
      2. A.2 Dualising the $C_{2}$ and $B_{2}$ axions
    6. B Type IIA perturbative supergravity
      1. B.1 $\mathcal{N}=2$ supergravity action
      2. B.2 Orientifolding and $\mathcal{N}=1$ type IIA supergravity
      3. B.3 The type IIA $\mathcal{N}=1$ effective action
      4. B.4 Type IIA flux compactifications
      5. B.5 Moduli stabilisation in type IIA flux supergravity

###  Chapter 1 Introduction and motivations

Let us begin by considering the age-old question: why string theory? To answer
it, we first consider the two underlying low-energy theories that string
theory aims to reproduce at the four-dimensional classical level: the Standard
Model (SM) and General Relativity (GR). These two theories have been extremely
successful in the many decades since their inception. Almost all of the
physical probes111As far as we are aware, the only measurement that requires
an explicit modification of the SM are the non-zero neutrino masses. The
anomalous dipole moment of the muon might also indicate some new physics, but
its interpretation is not as clear cut as the three-generations of massive
neutrinos, at the moment. currently available to physicists are consistent
with a description in terms of GR+SM, from gravitational waves to the low
energy particle spectrum of our Universe, at least to scales of order TeV[6].
It is astounding that these two effective low energy theories can explain such
a range of physical phenomena to such a degree of accuracy. In fact, one of
the current problems within particle physics is the inability to design
experiments able to probe regimes where we expect the SM description to break
down222Modulo the discovery of supersymmetry at a collider experiment, given
that the supersymmetry breaking scale for the matter sector could lie anywhere
between the mass of the top quark and $M_{\text{pl}}$ .333Between the writing
and defence of this thesis, the CDF collaboration [7] claimed a $\sim 7\sigma$
tension between their experimental results and the SM expectation for the mass
of the W boson. If confirmed by other experiments, this would be the strongest
signal for beyond the SM physics to date..

Nonetheless, despite the many successes, both the Standard Model and General
Relativity falter in some aspects. Regarding the SM, there are issues when one
considers it as a theory with some twenty free parameters to be fixed by
experiments. Although these parameters can be fixed by experiments, one can
ask the following two questions. First, what is the origin of the parameters
in the SM Lagrangian? Second, why do some of these parameters take unnatural
values, i.e. values that are much smaller than unity? Even if the physical
origin and smallness of the parameters in the SM Lagrangian were not a worry,
other issues arise when considering quantum effects [8]. Most notably the
Higgs, or any scalar not protected by some symmetry for that matter, is
unstable against radiative corrections. This is to say that loop corrections
couple the Higgs mass to the heaviest mass available in the theory, one of the
incarnations of the so-called hierarchy problem [9]. Therefore, unless new
physics comes in to protect the Higgs at some scale above the current collider
energies, there would be no reason to expect the Higgs to have its observed
mass.

On the other hand, the addition of gravity presents even more problems.
Setting aside the usual considerations about singularities of GR, like in the
interior of a black hole, the mismatch between theoretical and observational
predictions for the value of the vacuum energy density presents one of the
biggest open challenges in modern physics [10]. Indeed, in terms of
phenomenology we can think of the vacuum energy as a scalar field, which is
either a cosmological constant or a very slowly varying scalar dubbed
quintessence. Again, as a scalar field which is not protected by a symmetry at
the level of GR, the cosmological constant is radiatively unstable and will
tend to couple to the heavier masses available in the theory. In this case,
the presence of the SM particles alone would naively induce a correction to
the cosmological constant of $\mathcal{O}(\text{TeV}^{4})$, many orders of
magnitude above its observed value at $\mathcal{O}(\text{meV}^{4})$ scales.
This is the so-called cosmological constant problem [11].

Turning on gravity also introduces technical difficulties, which are not
entirely unrelated to the previous issues, in that the quantum field theory of
GR+SM becomes non-renormalisable. This is easily seen as soon as one realises
that the Ricci term in the Einstein-Hilbert action is parametrising the
strength of graviton interactions, with a coupling strength of
$M_{\text{pl}}^{-2}$. This is an irrelevant operator, growing weaker at low
energies and strong at large energies. It is certainly weak enough that at the
level of collider physics one seems to be able to turn gravity off and
accurately predict the scattering amplitudes of any interaction. The problem
with non-renormalisability becomes clear when considering internal graviton
loops. Indeed, in that case the integral over momentum space becomes divergent
for arbitrarily high energies. This divergence only becomes worse with higher
orders in the loop calculation [10]. This divergence can be mapped to position
space as the limit where graviton vertices become coincident[12].

String theory provides a beautiful description of graviton scatterings that
gets rid of these divergences. Heuristically, due to the extended nature of
strings, the scattering diagrams for gravitons are no longer coincident but
spread out over the worldsheet of the string. Incidentally, string
theory444Henceforth, by string theory we loosely refer to a complete theory of
extended objects as fundamental blocks of nature for which supergravity is a
low energy description. also has many desirables properties for a would-be
unification theory of the SM with gravity. Some of these are [12]

  1. 1.

String theory reproduces the low energy description of gravity, i.e. string
theory contains a massless spin-2 particle whose interactions reduce to
general relativity at the four-dimensional low energy level [13].

  2. 2.

Supersymmetry of the full theory follows from consistency requirements and it
provides a natural way of introducing fermions and bosons into the particle
spectrum of the theory, with a very clear physical interpretation in terms of
periodic and anti-periodic boundary conditions on the string worldsheet [14].
It could also aid in solving the SM hierarchy problem and ameliorate the
cosmological constant problem, to some extent [8].

  3. 3.

Further consistency considerations imply that the supersymmetric string theory
requires a definite number of spacetime dimensions. Supersymmetric string
compactifications allow a low-energy description of the spacetime in terms of
4 large external dimensions, that can host physics similar to those of the SM,
and 6 small internal dimensions, that can be integrated out at collider
energies [15].

  4. 4.

Supersymmetry can be partially broken so that the low energy dynamics of the
string theory are chiral, in agreement with the SM [15].

  5. 5.

String theory, in its final incarnation as described in the footnote, contains
no free parameters and is unique [16].

Of course, the lack of supersymmetry found at the LHC presents some subtleties
in engineering a phenomenologically viable low-energy description of string
theory. Nonetheless, these issues can, a priori, be fixed by softly breaking
supersymmetry in the matter sector at some scale between the energy limit at
collider searches and $M_{\text{pl}}$ . It is quite striking that the
consistency considerations of string theory can lead to such powerful
constraints and still yield a theory that is remarkably capable of describing
our Universe.

Clearly, string theory represents a powerful framework555And is possibly the
only one. to unify SM and GR. However, there must be some work left to be
done, otherwise theoretical physics would be solved. Most of the open
questions in this line of research can be recast as a lack of understanding of
the full non-perturbative description of string theory. Indeed, the work in
this thesis will concentrate on supergravity, i.e. the low-energy physics of
string theory. The supergravity action only contains the massless sector of
string theory, that is we will drop the dependence on oscillator modes and
keep only the string zero modes. This allows an exploration of the
perturbative aspects of string theory, where we see supergravity as an
expansion to some order in the genus of the worldsheet, parametrised by the
string coupling $g_{s}$, and to some order in worldsheet loops, parametrised
by the string length $l_{s}$, or equivalently by the
$\alpha^{\prime}$-parameter with $l_{s}:=\sqrt{2\pi\alpha^{\prime}}$. Thus,
supergravity inherits two expansion parameters666Note that, on top of the
expansion parameters, ten-dimensional supergravities admit a number of fluxes
being turned on, which at the low-energy effective level will effectively act
as continuous free parameters. in $g_{s}$ and $\alpha^{\prime}$. We also note
that our knowledge of non-perturbative corrections is quite limited.
Nonetheless, when trying to describe the low-energy dynamics of our four-
dimensional world, we would expect that in most instances supergravity
provides a close enough description. Keeping the perturbative expansion under
control, so that the supergravity predictions can be trusted, will be a
running theme in this thesis.

Within the supergravity approximation, we can begin to ask phenomenological
question about our Universe. The central question that we tackle in this work
is regarding the microscopic origin of the vacuum energy and its implications.
A number of the technical questions about naturalness and hierarchies that
concerned the SM, will also apply to the vacuum energy. Indeed, the ten-
dimensional supergravity action has a characteristic energy scale of
$\mathcal{O}(l_{s}^{-1})$, with $l_{s}^{-1}\lesssim M_{\text{pl}}$, whereas
the observational dark energy scale lies at
$\mathcal{O}(10^{-30}M_{\text{pl}})$. Thus, if supergravity is to explain the
scale of dark energy we will require a theory able to generate huge scale
hierarchies. Can these large scales hierarchies be generated in the first
place? If so, can it be done while avoiding extreme fine-tuning of the
supergravity parameters?

These are some important considerations that will play a key role in the
latter stages of this thesis. Before moving onto the bulk of our work, we
would like to touch upon two more points: EFTs and the Swampland (for a review
see [17, 18]). Computations in string theory and, by proxy, supergravity are
complex. We have a relatively small toolbox to compute quantities in
supergravity in comparison to, say, GR. This means that it is sometimes
convenient to explore an idea in a simplified framework within effective field
theories that are in a more or less rigorous sense motivated by supergravity.
In some sense, by working with supergravity we are doing just so. String
theory is too complicated to be treated in full generality, so we concentrate
on the massless sector and accept the limitations that this carries. Even
then, certain concepts might be far from the reach of our current
understanding of supergravity. It is then convenient to study such concepts
within effective field theory toy models acknowledging its limitations. When a
toy model proves promising, we can then try to find its supergravity
embedding.

An immediate question that can be raised is whether any self-consistent EFT
inspired by supergravity can find itself embedded in one corner or another of
the supergravity parameter space. The reasonable expectation is that this
would not be true. For example, one could imagine an EFT with more free
parameters than a particular supergravity theory, such that supergravity
imposes some non-trivial constraints on the free parameters of the EFT which
cannot be satisfied. These constraints are not, in general, accessible from
the EFT perspective as they require some knowledge of the UV embedding above
the EFT cut-off. The Swampland programme aims to provide conditions for any
EFT to be consistently embedded within a UV parent theory777This is usually
thought to be string theory/supergravity, although some aspects of the
Swampland are claimed to apply more generally.. This programme has become an
important part of the research done by the community in the past decade. Its
more fundamental parts concern M-theory/F-theory compactifications and, in
general, have been quite successful in improving our understanding of the ten-
dimensional description of supergravity (for example [19, 20, 21]). On the
other hand, there exists another section of the Swampland programme concerned
with deriving EFT constraints from the asymptotics of the ten-dimensional
theory. In this regime, the supergravity description becomes quite simplified
and strong phenomenological bounds on the resulting EFTs can be imposed [22,
23]. Critically, these results hold only in the asymptotic regime and claiming
that they hold anywhere else is a matter of speculation. However, due to the
powerful phenomenological constraints, this part of the Swampland has
attracted considerable research and statements about the asymptotic behaviour
of supergravity have turned into general conjectures anywhere in the parameter
space of the theory. In this thesis, we will briefly discuss the
phenomenological branch of the Swampland programme as some of its results
overlap with the research presented here. We will often refer to it as the
Swampland programme for convenience but we remark that our discussion concerns
only this second aspect of the Swampland.

Finally, one could worry whether our current description of our effective low-
energetic world, in the SM and GR, is at all consistent with a supergravity
embedding. We have already pointed out that string theory has many desirable
properties that would make us optimistic that such an embedding exists, we
also have constructions that closely resemble the low-energy dynamics of
SM+GR, and we have no evidence to the contrary, but one can never disregard
the possibility that this could not be the case. However, to conclude that
such embedding does not exists with any certain degree of confidence we will
require a more complete picture of supergravity. It is certainly not enough to
explore the asymptotics of the ten-dimensional description to claim anything
about the bulk of the theory.

####  1.1 Outline

The contents of this thesis are organised as follows. In Chap. 2, we will
obtain the effective field theory derived from the tree-level type IIB
supergravity action. In Chap. 3, we introduce a number of open question in
physics and cosmology that will be relevant to the contents of the following
sections. After this, in Chap. 4 and Chap. 5 concentrate on studying EFTs
motivated by supergravity that aim at answering some of the questions in Chap.
3. We conclude the first half of the thesis with a brief discussion about
perturbative control and the Swampland in Chap. 6.

The second half of the thesis is mostly devoted to the implications that the
microscopic nature of dark energy has on supergravity model building. In Chap.
7, we introduce a number of perturbative and non-perturbative corrections to
the tree-level type IIB action that will be necessary to build viable models
of dark energy. Then, in Chap. 8, we try to construct viable dynamical models
of dark energy and contrast them with a true cosmological constant in terms of
model building. We remark that the effects of higher order corrections might
prove crucial and begin to study whether this can be relevant in a fluxed
runaway scenario with Gauss-Bonnet corrections in Chap. 9. Finally, in Chap.
10, we conclude by reviewing our findings and provide some general thoughts as
to possible directions for future research.

####  1.2 Units and conventions

Unless explicitly stated, we take the usual unit conventions where
$\hbar=c=M_{\text{pl}}=1$. We will also set $\alpha^{\prime}=1$; this will in
turn fix the string scale to $M_{s}:=l_{s}^{-1}=1/\sqrt{2\pi}$. For the
metric, we take the mostly positive convention with Latin uppercase indices
($M,N...$) making reference to the ten-dimensional metric, Greek indices
($\mu,\nu,...$) referring to the external four-dimensional metric and Latin
lowercase indices ($i,j,...$) will represent the internal six-dimensional
metric. Note that this index convention only refers to metric indices and not
to the (co)homology indices which will follow particular conventions set out
in the pertinent chapters.

Where ambiguities may arise, we will also “hat” the ten-dimensional
quantities, for example $\hat{\phi}$ will be the ten-dimensional dilaton field
and $\phi$ will be its four-dimensional analogue.

##  Part I String inspired EFTs

###  Chapter 2 Perturbative supergravity in type IIB theories

In this chapter we explore the fundamentals of Calabi-Yau compactifications.
We begin in Sec. 2.1 by motivating the use and describing the geometry of
Calabi-Yau manifolds. In Sec. 2.2, we derive the moduli spectrum and scalar
potential for $\mathcal{N}=2$ type IIB Calabi-Yau compactifications. This
leaves us with 2 supersymmetry generators and a non-chiral theory.
Phenomenologically, we are interested in a chiral $\mathcal{N}=1$
supersymmetric theory. To achieve this, in Sec. 2.3, we orientifold our
$\mathcal{N}=2$ action and provide the $\mathcal{N}=1$ spectrum as well as its
effective scalar potential.

Finally, we would like to also discuss how to stabilise the moduli fields of
the theory. To do so, in Sec. 2.4, we introduce background fluxes through the
three-form flux piece in type IIB. This allows us to stabilise the axio-
dilaton and the complex structure sector. The stabilisation of the Kähler
sector is a challenging question that we will discuss in detail in Chap. 7, as
it requires non-perturbative corrections. We also briefly discuss localised
sources and tadpole cancellation conditions. In Sec. 2.5, we discuss the
Swampland programme and, in particular, the refined versions of the Swampland
distance conjecture and the de Sitter conjecture.

####  2.1 Special geometry of Kähler manifolds

Let $\mathcal{X}$ be a Calabi-Yau (CY) three-fold, i.e. a Kähler manifold in 3
complex dimensions with $SU(3)$ holonomy. By the famous theorem conjectured by
Calabi and proved by Yau [24], this description for a CY three-fold is
equivalent to that of a Kähler manifold in 3 complex dimensions that is Ricci-
flat, since a vanishing first Chern class implies an n-fold Kähler manifold
admits a Kähler metric with $SU(n)$ holonomy. A Kähler manifold is a hermitian
manifold with metric $g_{i\bar{\jmath}}$, i.e. the purely holomorphic and
anti-holomorphic components of the metric vanish, whose Kähler form

$J=-ig_{i\bar{\jmath}}dz^{i}\wedge d\bar{z}^{\bar{\jmath}}$ (2.1)

is closed, $dJ=0$, under the exterior derivative $d$, where $z^{i}$ denote
complex local coordinates and $\bar{z}^{\bar{\imath}}$ are their complex
conjugates.

There are a few properties that make studying CY three-folds attractive.
First, the critical superstring lives in a ten-dimensional spacetime. By
compactifying a six-dimensional subspace on a CY three-fold, we will be left
with an effective four-dimensional description of the low-energy physics of
the superstring, matching the low-energy dimensionality of our Universe.
Second, a compactification on a CY three-fold gets rid of $3/4$ of the
underlying supersymmetry of the theory, when compared to a toroidal
compactification. The number of preserved supersymmetries is related to the
decomposition of the spinor representation $\mathbf{16}\in SO(1,9)$. The
existence of a singlet in the decomposition of the internal spinor
representation will lead to the conservation of a supersymmetry in the
4-dimensional spectrum [25]. For a generic compact manifold $\mathcal{M}$ and
requiring 4-dimensional Poincaré invariance, the ten-dimensional Lorentz group
decomposes into

$SO(1,9)\longrightarrow SO(1,3)\times SO(6)\ ,$ (2.2)

and the spinor representation becomes

$\mathbf{16}\longrightarrow(\mathbf{2},\mathbf{4})\oplus(\mathbf{\bar{2}},\mathbf{\bar{4}})\
,$ (2.3)

where $\mathbf{2}$ is the Weyl spinor of $SO(1,3)$ and $\mathbf{4}$ is the
corresponding spinor of $SO(6)\cong SU(4)$, with
$(\mathbf{\bar{2}},\mathbf{\bar{4}})$ their respective conjugates. Since
$SU(4)$ does not contain a singlet in its representation, this would lead to a
non-supersymmetric four-dimensional theory. To preserve some supersymmetry we
require $\mathcal{M}$ to have $SU(n)$ holonomy with $n<4$, that is under
parallel transport around a closed loop the spinor singlet comes back to
itself with a rotation in $SU(n)\subset SO(6)\cong SU(4)$. For example, in the
case of a CY three-fold, which by definition has $SU(3)$ holonomy, the
internal spinor representation decomposes as

$\mathbf{4}\longrightarrow\mathbf{3}\oplus\mathbf{1}\ ,$ (2.4)

such that a singlet is left invariant and an $\mathcal{N}=1$ ten-dimensional
theory compactified on a CY three-fold will preserve $\mathcal{N}=1$
supersymmetry in four dimensions111Note that type IIB supergravity has
$\mathcal{N}=2$ in ten dimensions and thus preserves $\mathcal{N}=2$
supersymmetry after compactification on a CY three-fold. This is by virtue of
type IIB having both left and right moving states, which yield a spinor
decomposition in terms of $\mathbf{4}_{LR}\oplus\mathbf{4}_{RL}$.. Finally,
Ricci-flatness, or equivalently $SU(3)$ holonomy, allows to find a one-to-one
mapping between metric perturbations and the cohomology of the CY three-folds.
We explore this relation in the remainder of this section.

The p-th de Rham cohomology associated to the exterior derivative of the
three-fold $H^{p}(\mathcal{X})$ is defined to be the quotient space

$H^{p}(\mathcal{X}):=C^{p}(\mathcal{X})/Z^{p}(\mathcal{X})\ ,$ (2.5)

where $C^{p}(\mathcal{X})$ is the space of closed p-forms, i.e. $dA_{p}=0$,
and $Z^{p}(\mathcal{X})$ is the space of exact p-forms, i.e. $A_{p}=dA_{p-1}$.
Then, $H^{p}(\mathcal{X})$ is composed by the representatives of equivalency
classes of p-forms whose elements differ at most by an exact p-form. The
dimensionality of the $H^{p}(\mathcal{X})$ group is given by the Betti number
$b^{p}$, which is related to the Euler characteristic222This relation holds
even more generally for a CY n-fold $\mathcal{X}_{n}$ where the Euler
characteristic is then given by
$\chi(\mathcal{X}_{n})=\sum_{p=0}^{2n}(-1)^{p}b^{p}\ .$ of $\mathcal{X}$ by

$\chi=\sum_{p=0}^{6}(-1)^{p}b^{p}\ .$ (2.6)

Since $\mathcal{X}$ is a complex manifold, it will be useful to adopt a
holomorphic/anti-holomorphic index notation and to split the p-forms and
exterior derivative in holomorphic/anti-holomorphic pieces. Indeed, we will
define a (p,q)-form to be a form with p holomorphic and q anti-holomorphic
indices

$A_{p,q}=\frac{1}{p!q!}A_{m1,...,m_{p},\bar{n}_{1},...,\bar{n}_{q}}dz^{m_{1}}\wedge...\wedge
dz^{m_{p}}\wedge d\bar{z}^{\bar{n}_{1}}\wedge...\wedge d\bar{z}^{\bar{n}_{q}}\
.$ (2.7)

We also expand the exterior derivative in Dolbeault operators

$d=\partial+\bar{\partial}\ ,\qquad\partial=dz^{a}{\partial\over\partial
z^{a}}\
,\quad\bar{\partial}=d\bar{z}^{\bar{a}}{\partial\over\partial\bar{z}^{\bar{a}}}\
,$ (2.8)

where $(\partial,\bar{\partial})$ map (p,q)-forms to (p+1,q)-forms and
(p,q+1)-forms, respectively. The Dolbeault cohomology333In a slight abuse of
nomenclature, for the rest of the thesis we will use “cohomology” to refer to
either de Rham cohomology or Dolbeault cohomology equally. group
$H^{p,q}(\mathcal{X})$ follows from the definition of the operator444Note that
the Dolbeault cohomology of $\mathcal{X}$ can be defined with respect to
$\bar{\partial}$, since the two cohomologies will be mapped one-to-one by the
complex conjugation of their elements. This is by virtue of $\mathcal{X}$
being a Kähler manifold. ${\partial}$. Indeed, $H^{p,q}(\mathcal{X})$ is
composed by equivalency classes of closed (p,q)-forms under $\partial$, whose
elements differ at most by an exact (p,q)-form. The dimension of
$H^{p,q}(\mathcal{X})$ is the Hodge number $h^{p,q}$ and it follows that

$b^{q}=\sum_{p=0}^{q}h^{p,q-p}\ .$ (2.9)

It is convenient to represent cohomology structure of $\mathcal{X}$ through
the so-called the Hodge diamond, since this makes the CY symmetries manifest.
The Hodge diamond of any CY three-fold is given by

$\begin{array}[]{ccccccc}&&&1&&&\\\ &&0&&0&&\\\ &0&&h_{1,1}&&0&\\\
1&&h_{1,2}&&h_{1,2}&&1\\\ &0&&h_{1,1}&&0&\\\ &&0&&0&&\\\ &&&1&&&\end{array}\
.$ (2.10)

Note that the Hodge diamond, or the cohomology classes to be more precise, are
not a complete characterisation of a Calabi-Yau. Indeed, non-homeomorphic
Calabi-Yaus can have the same Hodge diamond. This can be better understood
with a simple example. Consider a strip with periodic boundary conditions (a
band) and one with anti-periodic boundary conditions (a Möbius strip). Both of
them are homotopically equivalent to a circle, and also cohomologically
equivalent, however there exists no one-to-one bijective map between the two
topologies due to the torsion. Therefore, the two topologies are
cohomologically equivalent but not homeomorphic. Although this is far from a
proof for generic Calabi-Yaus, we find the intuitive picture useful.

The following symmetries have been exploited to eliminate redundant groups in
the Hodge diamond:

  * •

Isomorphism of $H^{p}(\mathcal{X})$ and $H^{3-p}(\mathcal{X})$ requires
$h^{p,\,0}=h^{3-p,\,0}$.

  * •

Complex conjugation implies $h^{p,\,q}=h^{q,\,p}$.

  * •

Poincaré duality leads to $h^{p,\,q}=h^{3-p,\,3-q}$;

  * •

Simply connected CYs will have vanishing first homology group as every one-
cycle can be infinitesimally shrunk to a point such that $h^{0,1}=h^{1,0}=0$.

After applying these symmetries, we can see that the cohomological structure
of $\mathcal{X}$ will be determined by a pair of Hodge numbers
$h^{1,1},\,h^{2,1}$ which are related to its Euler characteristic through Eq.
(2.9)

$\chi=2\left(h^{1,1}-h^{2,1}\right)\ .$ (2.11)

One can then introduce a base of harmonic p-forms for each cohomology class.
We denote the basis of harmonic (1,1)-forms with $\omega_{A}$, its Poincaré
dual being the basis of (2,2)-forms $\tilde{\omega}^{A}$, with
$A=1,...,h^{1,1}$. Introducing the basis of $H^{3}(\mathcal{X})$ to be
$\left(\alpha_{\hat{K}},\beta^{\hat{L}}\right)$, which are related to each
other by complex conjugation and $\hat{K},\hat{L}=0,...,h^{1,2}$, the only
non-trivial intersection numbers are

$\int_{\mathcal{X}}\omega_{A}\wedge\tilde{\omega}^{B}=\delta_{A}^{B}\
,\qquad\int_{\mathcal{X}}\alpha_{\hat{K}}\wedge\beta^{\hat{L}}=\delta_{\hat{K}}^{\hat{L}}\
.$ (2.12)

It is also useful to breakdown the $H^{3}(\mathcal{X})$ basis in its
components as follows
$\left(\alpha_{\hat{K}},\beta^{\hat{L}}\right)=(\alpha_{0},\chi_{K},\beta^{0},\chi^{L})$
with $K,L=1,...,h^{2,1}$.

Cohomology group | Dimension | Basis
---|---|---
$H^{1,1}$ | $h^{1,1}$ | $\omega_{A}$
$H^{2,2}$ | $h^{1,1}$ | $\tilde{\omega}^{A}$
$H^{3}$ | $2+2h^{2,1}$ | $\left(\alpha_{\hat{K}},\beta^{\hat{L}}\right)$
$H^{2,1}$ | $h^{2,1}$ | $\chi_{K}$
$H^{3,3}$ | $1$ | $\text{dvol}(\mathcal{X})$
Table 2.1: Bases for a Calabi-Yau threefold.

At this point, one can expand any p-form of $\mathcal{X}$ as a product of
moduli fields and cohomology basis elements. For instance, a 2-form field
$B_{2}$ can generically be expanded as

$B_{2}=\bar{B}_{2}(x)+b^{A}(x)\omega_{A}\ ,$ (2.13)

where $x$ denotes external coordinates.

Aside from the field content present in the theory, one is free to consider
deformations $\delta g_{mn}$ of the CY metric $g_{mn}$ on $\mathcal{X}$ ,
where the indices $(m,n)$ cover both the holomorphic and anti-holomorphic
indices, that is $m=(i,\bar{\imath})$. In particular, we consider deformations
that preserve the Ricci flatness of the manifold, i.e. $R_{mn}(g+\delta g)=0$.
In this way, even after deforming the metric, the manifold is guaranteed to be
Calabi-Yau [26, 27, 28].

What are the set of deformations that preserve the Ricci flatness of
$\mathcal{X}$ ? First of all, notice that if the metric of the manifold
$g_{mn}$ is Ricci-flat so will be any other $g^{\prime}_{mn}$ related to the
original metric by diffeomorphisms. This diffeomorphism invariance allows us
to gauge fix $\nabla^{m}\delta g_{mn}={1\over 2}\nabla_{n}\delta g$, where
$\delta g=g^{mn}\delta g_{mn}$, such that the linearised Lichnerowicz equation
governing the deformations becomes

$\Delta\delta g_{mn}:=\nabla^{p}\nabla_{p}\delta g_{mn}+2R_{m\ n}^{\ \ p\
\,q}\delta g_{pq}=0\ ,$ (2.14)

and we can study the equations for the pure and mixed indices independently as
these decouple.

  * •

Mixed components: The Lichnerowicz equations reads

$\Delta\delta g_{i\bar{\jmath}}=0.$ (2.15)

Therefore, $\delta g_{i\bar{\jmath}}$ is a harmonic form of the Laplacian and
will admit an expansion in terms of $\omega_{A}$. These modes are related to
non-trivial deformations of the Kähler form $J$ through Eq. (2.1).

  * •

Pure components: In this case, the Lichnerowicz equations reads

$\Delta\delta g_{ij}=0,$ (2.16)

implying an expansion in (2,0)-forms. However, since we would like to recover
a CY manifold after the deformations, there should exist a one-to-one mapping
between the $\delta g_{ij}$ and some combination of mixed deformations.
Indeed, one can generate such a mapping between pure deformations and a new
set of mixed component deformations through the unique holomorphic (3,0)-form
$\Omega$ of $\mathcal{X}$ (cf. Subsec. 2.1.2).

All in all, one finds that the moduli space of deformations is spanned by [25,
29]

$\displaystyle\delta g_{i\bar{\jmath}}=-iv^{A}(\omega_{A})_{i\bar{\jmath}}\ ,$
(2.17) $\displaystyle\delta
g_{ij}=\frac{i}{||\Omega||^{2}}\overline{(z^{K}\chi_{K})}_{i\bar{\imath}\bar{\jmath}}\Omega_{jkl}g^{\bar{\imath}k}g^{\bar{\jmath}l}:=\frac{i}{||\Omega||^{2}}\bar{z}^{K}(\bar{\chi}_{K})_{i\bar{\imath}\bar{\jmath}}\Omega^{\bar{\imath}\bar{\jmath}}_{j}\
,$ (2.18)

where the last equality in Eq. (2.18) simply introduces shorthand notation and
we have defined555Again, in this shorthand notation, this definition really
means
$||\Omega||^{2}:=\frac{1}{3!}\Omega_{ijk}\bar{\Omega}_{\bar{\imath}\bar{\jmath}\bar{k}}g^{\bar{\imath}i}g^{\bar{\jmath}j}g^{\bar{k}k}$.
$||\Omega||^{2}:=\frac{1}{3!}\Omega_{ijk}\bar{\Omega}^{ijk}$. At this level,
the moduli space is locally a direct product of two Kähler manifolds

$\mathcal{M}^{ks}\times\mathcal{M}^{cs}\ ,$

this will continue being true at the 4D level so the potential generated by
either sector will decouple from the other and we can study them
independently666Note that the existence of this direct product of two
manifolds will be directly related to the no-scale structure of type IIB, as
we will discuss in Subsec. 2.3.2. Breaking the no-scale structure will mean
that the Kähler and complex structure sectors will not decouple anymore..
Furthermore, this implies that (at least locally) the metrics of both sectors
are the metrics of Kähler manifolds independently, i.e. there will exist a
pre-potential function for each sector such that the metric can be expressed
as a derivative of this pre-potential. In the following two section we explore
the structure of $\mathcal{M}^{ks}$ and $\mathcal{M}^{cs}$.

#####  2.1.1 The Kähler structure of $\mathcal{M}^{ks}$

The first of the two sectors is given by the deformations to the Kähler
structure. Intuitively these deformations can be understood as changes in the
size of the internal space. For example, in the simple case of a $T^{2}$ the
Kähler structure deformations are parametrised by the modulus
$\rho=iR_{1}R_{2}$ where $R_{i}$ are the two radii of the torus. A change in
$\rho$ corresponds to a change in the size of the torus.

Back to the more complicated case of a CY, we have $h^{1,1}$ Kähler moduli. It
is convenient to package the moduli coming from the Kalb-Ramond 2-form
together with the real scalar moduli of Eq. (2.17) in a new set of imaginary
scalars given by

$t^{A}:=v^{A}+ib^{A}\ .$ (2.19)

The metric of the moduli space is defined as

$G_{AB}:=\frac{1}{4\mathcal{V}}\int\omega_{A}\wedge\star\omega_{B}=-\frac{1}{4\mathcal{V}}\left(\mathcal{K}_{AB}-\frac{\mathcal{K}_{A}\mathcal{K}_{B}}{4\mathcal{V}}\right)=\partial_{t^{A}}\partial_{\bar{t}^{B}}K^{\text{ks}}\
,$ (2.20)

where we find the Kähler potential of the Kähler structure metric
$K^{\text{ks}}$ to be

$K^{\text{ks}}=-\ln\left[8\mathcal{V}\right]=-\ln\left[\frac{1}{6}\mathcal{K}_{ABC}(t+\bar{t})^{A}(t+\bar{t})^{B}(t+\bar{t})^{C}\right]$
(2.21)

with $\mathcal{V}$ the volume of $\mathcal{X}$ and we define the triple
intersection numbers $\mathcal{K}_{ABC}$ and their contractions as

$\displaystyle\mathcal{K}_{ABC}:=\int\omega_{A}\wedge\omega_{B}\wedge\omega_{C}\
,\qquad\mathcal{K}_{AB}:=\int\omega_{A}\wedge\omega_{B}\wedge
J=\mathcal{K}_{ABC}v^{C}\ ,$
$\displaystyle\mathcal{K}_{A}:=\int\omega_{A}\wedge J\wedge
J=\mathcal{K}_{ABC}v^{B}v^{C}\ ,\qquad\mathcal{K}:=\int J\wedge J\wedge
J=\mathcal{K}_{ABC}v^{A}v^{B}v^{C}=6\mathcal{V}\ .$

The metric $G_{AB}$ is fully encoded in a single function $K^{\text{ks}}$.
Even more so, the moduli space is a special Kähler manifold since the Kähler
potential itself is given in terms of a prepopential

$\displaystyle
K^{\text{ks}}=-\ln\left[2(f+\bar{f})-(t+\bar{t})^{A}(\partial_{t^{A}}f+\partial_{\bar{t}^{A}}\bar{f})\right]\
,$ (2.22) $\displaystyle f(t)=-\frac{1}{6}\mathcal{K}_{ABC}t^{A}t^{B}t^{C}\ ,$
(2.23)

To ensure that the metric is positive definite, even after the deformations,
we require $\mathcal{K}_{AB},\mathcal{K}_{A},\mathcal{K}>0\ \forall(A,B)$.
These conditions define the Kähler cone spanned by the moduli $t^{A}$.

Finally, every special Kähler manifold has a complex matrix defined as

$\mathcal{N}_{\hat{A}\hat{B}}:=\bar{\mathcal{F}}_{\hat{A}\hat{B}}+2i\frac{(\text{Im}\
\mathcal{F})_{\hat{A}\hat{C}}\;t^{\hat{C}}(\text{Im}\
\mathcal{F})_{\hat{B}\hat{D}}\;t^{\hat{D}}}{t^{\hat{C}}(\text{Im}\
\mathcal{F})_{\hat{C}\hat{D}}t^{\hat{D}}}\ ,$ (2.24)

where we use adapted coordinates $t^{\hat{A}}=\\{1,t^{A}\\}$,
$\mathcal{F}=f(t)$ and
$\mathcal{F}_{\hat{A}\hat{B}}=\\{0,\partial_{t^{A}}\partial_{t^{B}}\mathcal{F}\\}$.
Plugging in the form of $f(t)$ we obtain

$\displaystyle\text{Re}\mathcal{N}=\begin{pmatrix}-\frac{1}{3}\mathcal{K}_{ABC}b^{A}b^{B}b^{C}&\frac{1}{2}\mathcal{K}_{ABC}b^{A}b^{B}\\\
\frac{1}{2}\mathcal{K}_{ABC}b^{A}b^{B}&-\mathcal{K}_{ABC}b^{C}\end{pmatrix}\
,$ (2.25)
$\displaystyle\text{Im}\mathcal{N}=-\mathcal{V}\begin{pmatrix}1+4G_{AB}b^{A}b^{B}&-4G_{AB}b^{B}\\\
-4G_{AB}b^{B}&4G_{AB}\end{pmatrix}\ ,$ (2.26)
$\displaystyle(\text{Im}\mathcal{N})^{-1}=-\frac{1}{\mathcal{V}}\begin{pmatrix}1&b^{A}\\\
b^{A}&\frac{1}{4}G^{AB}+b^{A}b^{B}\end{pmatrix}\ .$ (2.27)

#####  2.1.2 The complex structure of $\mathcal{M}^{cs}$

The second sector is given by the deformations to the complex structure. Back
to the analogy with a $T^{2}$, these deformations are parametrised by the
modulus $\tau=i\frac{R_{2}}{R_{1}}$ and control the changes in shape of the
torus.

In the case of the CY, there will be $h^{2,1}$ complex structure fields. The
metric of the complex structure moduli space is defined as

$G_{K\bar{L}}:=-\frac{\int\chi_{K}\wedge\bar{\chi}_{L}}{\int\Omega\wedge\bar{\Omega}}\
.$ (2.28)

Kodaira’s formula gives the basis $\chi_{L}$ in terms of the holomorphic
(3,0)-form $\Omega$

$\chi_{K}(z,\bar{z})=\partial_{z^{K}}\Omega(z)+\Omega(z)\partial_{z^{K}}\left[-\ln\left(i\int\Omega\wedge\bar{\Omega}\right)\right]\
.$ (2.29)

The bracket is the Kähler potential for the complex structure metric since

$G_{K\bar{L}}=\partial_{z^{K}}\partial_{\bar{z}^{L}}K^{\text{cs}}\ ,\qquad
K^{\text{cs}}=-\ln\left(i\int\Omega\wedge\bar{\Omega}\right)=-\ln\
i\left(\mathcal{Z}^{\hat{K}}\bar{\mathcal{F}}_{\hat{K}}-\bar{\mathcal{Z}}^{\hat{K}}\mathcal{F}_{\hat{K}}\right)\
.$ (2.30)

Thus, the complex structure moduli space is Kähler as well. In the last
relation of Eq. (2.30) we have used that the harmonic (3,0)-form $\Omega$
benefits from an expansion in terms of the harmonic bases

$\Omega(z)=\mathcal{Z}^{\hat{K}}(z)\alpha_{\hat{K}}-\mathcal{F}_{\hat{K}}(z)\beta^{\hat{K}}\
.$ (2.31)

Since the Kähler potential is only defined up to a complex rescaling of
$\Omega$ by a holomorphic function

$\Omega\rightarrow\tilde{\Omega}=\Omega e^{-h(z)}\ ,\qquad
K^{\text{cs}}\rightarrow\tilde{K}^{\text{cs}}=K^{\text{cs}}+h+\bar{h}\ ,$
(2.32)

we are allowed to gauge fix $\mathcal{Z}^{0}=1$; this is the so-called Kähler
gauge. This leaves $h^{2,1}$ free components for the periods of $\Omega$ and
one can identify these periods with the $z^{K}$ moduli fields by defining
$z^{K}=\mathcal{Z}^{K}/\mathcal{Z}^{0}$. Furthermore, the manifold
$\mathcal{M}^{cs}$ is also a special Kähler manifold.

Indeed, one can choose
$\mathcal{F}=\frac{1}{2}\mathcal{Z}^{\hat{K}}\mathcal{F}_{\hat{K}}$ and the
relations set out in the previous section between special manifolds and its
special matrix still hold (now adapted to the special coordinates $z^{K}$
instead of $t^{A}$). Namely777We keep the $\mathcal{Z}^{0}$ dependence
explicit for the sake of generality

$\displaystyle K^{\text{cs}}=-\ln
i\left|\mathcal{Z}^{0}\right|^{2}\left[2(f-\bar{f})-(z-\bar{z})^{K}(\partial_{z^{K}}f+\partial_{\bar{z}^{K}}\bar{f})\right]\
,$ (2.33) $\displaystyle
f(z)={1\over\left(\mathcal{Z}^{0}\right)^{2}}\mathcal{F}\ ,$ (2.34)

and the special matrix (also known in this case as the period matrix) is given
by

$\mathcal{M}_{\hat{K}\hat{L}}:=\bar{\mathcal{F}}_{\hat{K}\hat{L}}+2i\frac{(\text{Im}\
\mathcal{F})_{\hat{K}\hat{M}}\;\mathcal{Z}^{\hat{M}}(\text{Im}\
\mathcal{F})_{\hat{L}\hat{N}}\;\mathcal{Z}^{\hat{N}}}{\mathcal{Z}^{\hat{M}}(\text{Im}\
\mathcal{F})_{\hat{M}\hat{N}}\mathcal{Z}^{\hat{N}}}\ .$ (2.35)

This matrix will appear in the dimensionally reduced type II supergravity
action as the kinetic metric for the complex structure vectors and is most
conveniently written down as

$\displaystyle\int\alpha_{\hat{K}}\wedge\star\alpha_{\hat{L}}=-\left[\text{Im}\
\mathcal{M}+(\text{Re}\ \mathcal{M})(\text{Im}\ \mathcal{M})^{-1}(\text{Re}\
\mathcal{M})\right]_{\hat{K}\hat{L}}\ ,$ (2.36)
$\displaystyle\int\beta^{\hat{K}}\wedge\star\beta^{\hat{L}}=-\left(\text{Im}\
\mathcal{M}\right)^{-1\ \hat{K}\hat{L}}\ ,$ (2.37)
$\displaystyle\int\alpha_{\hat{K}}\wedge\star\beta^{\hat{L}}=-\left[(\text{Re}\
\mathcal{M})(\text{Im}\ \mathcal{M})^{-1}\right]_{\hat{K}}^{\hat{L}}\ .$
(2.38)

With this, we finish our review on the geometry of moduli space. We will now
proceed to study the $\mathcal{N}=2$ bosonic sector for the type II
supergravity actions.

####  2.2 Perturbative $\mathcal{N}=2$ supergravity

The massless bosonic spectrum of type IIB supergravity is composed of two
sectors: the NSNS sector comprising of the metric $\hat{g}_{mn}$, the dilaton
$\hat{\phi}$ and the Kalb-Ramond two-form $\hat{B}_{2}$; and the RR sector
containing the even p-form gauge fields $\hat{C}_{0}$, $\hat{C}_{2}$,
$\hat{C}_{4}$.

Our starting point for this section is the ten dimensional type IIB action in
Einstein frame [25]

$\displaystyle\mathcal{S}_{IIB}=-\int\left({1\over
2}\hat{R}\star\mathbf{1}+{1\over 4}d\hat{\phi}\wedge\star d\hat{\phi}+{1\over
4}e^{-\hat{\phi}}\hat{H}_{3}\wedge\star\hat{H}_{3}\right)$ (2.39)
$\displaystyle-\int\frac{1}{4}\left(e^{-2\hat{\phi}}\hat{F}_{1}\wedge\star\hat{F}_{1}+e^{\hat{\phi}}\hat{F}_{3}\wedge\star\hat{F}_{3}+{1\over
2}\hat{F}_{5}\wedge\star\hat{F}_{5}+\hat{C}_{4}\wedge\hat{H}_{3}\wedge\hat{F}_{3}\right)\
,$

where the field strength are defined as follows

$\displaystyle\hat{H}_{3}=d\hat{B}_{2}\ ,\quad\hat{F}_{1}=d\hat{C}_{0}\
,\quad\hat{F}_{3}=d\hat{C}_{2}-\hat{C}_{0}\hat{H}_{3}\ ,$
$\displaystyle\hat{F}_{5}=d\hat{C}_{4}-{1\over
2}\hat{H}_{3}\wedge\hat{C}_{2}+{1\over 2}\hat{B}_{2}\wedge d\hat{C}_{2}\
,\quad$ (2.40)

and the self-duality of $\hat{F}_{5}$ must be imposed by hand at the level of
the equations of motion.

By making use of the harmonic bases, we can expand the field content of the
theory as

$\displaystyle\hat{B}_{2}=B_{2}(x)+b^{A}(x)\omega_{A}\
,\quad\hat{C}_{2}=C_{2}(x)+c_{2}^{A}\omega_{A}\ ,$
$\displaystyle\hat{C}_{4}=c_{4}^{A}(x)\omega_{A}+\xi^{\hat{K}}(x)\wedge\alpha_{\hat{K}}-\tilde{\xi}_{\hat{K}}(x)\wedge\beta^{\hat{K}}+\rho_{A}(x)\tilde{\omega}^{A}\
.$ (2.41)

We can further make use of the self-duality of $\hat{F}_{5}$ to get rid of
half of the degrees of freedom of $\hat{C}_{4}$ — we choose to keep the
$\xi^{\hat{K}}$ and $\rho_{A}$ fields. On top of these, we will also obtain a
number of moduli fields from the deformations of the metric as discussed in
the previous section.

Taking the metric ansatz to be

$ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}+g_{i\bar{\jmath}}(x,z)dz^{i}d\bar{z}^{\bar{\jmath}}\
,$ (2.42)

where $g_{\mu\nu}$ (with $\mu,\nu=0,...,3$) is the external metric and
$g_{i\bar{\jmath}}$ (with $i,\bar{\jmath}=1,...,3$) is the CY metric of
$\mathcal{X}$ and plugging the definitions of Eq. (2.40) in Eq. (2.2) one
finds the tree-level four-dimensional action888Where a further Weyl rescaling
has been made to reabsorb the internal volume $\mathcal{V}$, so that the
action is in Einstein frame. [30, 31] (see App. A.1 for further details)

$\displaystyle\mathcal{S}^{(4)}_{IIB}=\int-{1\over
2}R\star{\mathbf{1}}+{1\over 4}\text{Re}\
\mathcal{M}_{\hat{K}\hat{L}}F^{\hat{K}}\wedge F^{\hat{L}}+{1\over 4}\text{Im}\
\mathcal{M}_{\hat{K}\hat{L}}F^{\hat{K}}\wedge\star F^{\hat{L}}$
$\displaystyle-G_{K\bar{L}}dz^{K}\wedge\star
d\bar{z}^{\bar{L}}-h_{pq}dq^{p}\wedge\star dq^{q}\ ,$ (2.43)

where the real and imaginary parts of $\mathcal{M}_{\hat{K}\hat{L}}$ are found
from Eq. (2.35), $F^{\hat{K}}:=d\xi^{\hat{K}}$, the complex structure metric
$G_{K\bar{L}}$ is given in Eq. (2.30) and we define the four-dimensional
dilaton to be

$e^{\phi}:={1\over\mathcal{V}^{1/2}}e^{\hat{\phi}}\ ,$ (2.44)

such that the quaternionic metric $h_{pq}$ becomes999Where we have dualised
the $\left(C_{2},\,B_{2}\right)$ gauge fields into the scalars
$\left(h,\,\tilde{h}\right)$, as demonstrated in App. A.2. [32, 33]

$\displaystyle h_{pq}dq^{p}\wedge\star
dq^{q}:=(d\phi)^{2}+G_{AB}dT^{A}\wedge\star dT^{B}+{1\over
4}e^{2\phi}\mathcal{V}(dC_{0})^{2}$
$\displaystyle+e^{2\phi}\mathcal{V}G_{AB}\left(dc_{2}^{A}-C_{0}db^{A}\right)\wedge\star\left(dc_{2}^{B}-C_{0}db^{B}\right)$
$\displaystyle+{1\over
16\mathcal{V}}e^{2\phi}G^{AD}\left(d\rho_{A}-\mathcal{K}_{ABC}c_{2}^{B}db^{C}\right)\wedge\star\left(d\rho_{D}-\mathcal{K}_{DEF}c_{2}^{E}db^{F}\right)$
$\displaystyle+{1\over 4\mathcal{V}}e^{2\phi}\left[dh-{1\over
2}\left(\rho_{A}db^{A}-b^{A}d\rho_{A}\right)\right]^{2}$
$\displaystyle+{1\over
2}e^{4\phi}\left[d\tilde{h}+C_{0}dh+c_{2}^{A}d\rho_{A}+{1\over
2}C_{0}\left(\rho_{A}db^{A}-b^{A}d\rho_{A}\right)-{1\over
4}\mathcal{K}_{ABC}c_{2}^{A}c_{2}^{B}db^{C}\right]^{2}\ .$ (2.45)

In [32, 33], it was shown that the dimensionally reduced type IIB theory in
Eq. (2.43) admits a description in terms of a local direct product
$\mathcal{M}^{Q}\times\mathcal{M}^{\text{cs}}$, where $\mathcal{M}^{\text{Q}}$
is the quaternionic manifold described by the metric $h_{pq}$ and spanned by
the coordinates $q^{p}$, with $\mathcal{M}^{\text{ks}}$ a submanifold of
$\mathcal{M}^{\text{Q}}$, and $\mathcal{M}^{\text{cs}}$ the complex structure
manifold described in Subsec. 2.1.2. The ordered multiplets of Eq. (2.43) are
given in Tab. 2.2.

Multiplet | Dimension | Field content
---|---|---
Gravity multiplet | 1 | $\left(g_{\mu\nu},\ \xi^{0}\right)$
Vector multiplet | $h^{2,1}$ | $\left(\xi^{K},\ z^{K}\right)$
Hypermultiplet | $h^{1,1}+1$ | $\left(v^{A},b^{A},c_{2}^{A},\rho_{A}\right)+\left(h,\tilde{h},\phi,C_{0}\right)$
Table 2.2: Multiplets for the four-dimensional type IIB supergravity spectrum.

The four-dimensional effective theory described by Eq. (2.43) preserves
$\mathcal{N}=2$ supersymmetry. However, phenomenologically we are interested
in $\mathcal{N}=1$ theories. Half of the supersymmetry can be broken by
introducing an orientifold projection in an appropriate manner, as we will
discuss in the next section. Furthermore, the scalar potential for the four
dimensional scalars is flat. To be able to stabilise the moduli fields we will
require the use of background fluxes, which we describe in Sec. 2.4.

####  2.3 Orientifolding the supergravity action

In this section we will briefly discuss the orientifolded truncation of type
IIB supergravity. For a more complete description, we point the interested
reader to [29, 34, 31] and references therein.

We would like to consider the most generic field content invariant under the
action of the $\mathcal{N}=1$ orientifold operator101010A second choice of
orientifold projection is possible in $\tilde{\mathcal{O}}=\Omega_{p}\sigma$;
this would lead to the introduction of $O5$ and $O9$-planes in the theory.

$\mathcal{O}=\Omega_{p}(-1)^{\text{F}_{\text{L}}}\sigma\ ,$ (2.46)

which is the combined action of the world-sheet parity transformation
$\Omega_{p}$, i.e. the operator that transforms the world-sheet coordinates
$\left(\tau_{0},\tau_{1}\right)\rightarrow\left(\tau_{0},-\tau_{1}\right)$,
the left-moving fermion number operator $(-1)^{\text{F}_{\text{L}}}$ and an
isometric and holomorphic involution $\sigma$ acting solely on $\mathcal{X}$
$\left(\sigma^{2}=\mathbf{1}\right)$. Furthermore, $\sigma$ leaves the metric
and complex structure of $\mathcal{X}$ intact [31]. This implies that the
Kähler form will also be invariant under the involution $\sigma^{*}J=J$, where
$\sigma^{*}$ is the pullback of $\sigma$. Holomorphicity of $\sigma$ implies
that the Dolbeault decomposition for the cohomologies is respected, and in
particular $\sigma^{*}H^{3,\,0}=H^{3,\,0}$; together with its idempotency we
choose for the holomorphic (3,0)-form $\Omega$

$\sigma^{*}\Omega=-\Omega\ .$ (2.47)

This choice for the orientifold projection is consistent with the introduction
of $O3$ and $O7$-planes into the effective action [31].

Another effect of the involution is that the cohomology of $\mathcal{X}$ will
split in even and odd cohomologies under its action, by virtue of $\sigma$
being an involutive symmetry of $\mathcal{X}$ , and thus we write

$H^{p,\,q}=H^{p,\,q}_{+}\oplus H^{p,\,q}_{-}\ ,$ (2.48)

such that an element $\omega_{\pm}\in H^{p,\,q}_{\pm}$ transforms like
$\sigma^{*}\omega_{\pm}=\pm\omega_{\pm}$. Furthermore, we can see the
following [31]

  * •

The involution being isometric means that the metric and orientation of
$\mathcal{X}$ are preserved. The Hodge-star operator $\star$ then commutes
with the involution and $h^{1,\,1}_{\pm}=h^{2,\,2}_{\pm}$.

  * •

The holomorphicity of $\sigma$ implies that there will not be any mixing
between holomorphic and antiholomorphic groups
$h^{1,\,2}_{\pm}=h^{2,\,1}_{\pm}$.

  * •

The choice Eq. (2.47) leads to $h^{3,\,0}_{-}=h^{0,\,3}_{-}=1$ and
$h^{3,\,0}_{+}=h^{0,\,3}_{+}=0$.

  * •

The previous two points telll us that $\Omega\wedge\bar{\Omega}$ is invariant
under $\sigma^{*}$ such that $h^{3,\,3}_{-}=h^{0,\,0}_{-}=0$ and
$h^{3,\,3}_{+}=h^{0,\,0}_{+}=1$.

In Tab. 2.3 we summarise the notation for the truncated harmonic bases after
orientifolding.

Cohomology group | Dimension | Basis
---|---|---
$H^{1,\,1}_{-}$ | $h^{1,\,1}_{-}$ | $\omega_{a}$
$H^{1,\,1}_{+}$ | $h^{1,\,1}_{+}$ | $\omega_{\alpha}$
$H^{2,\,2}_{-}$ | $h^{1,\,1}_{-}$ | $\tilde{\omega}^{a}$
$H^{2,\,2}_{+}$ | $h^{1,\,1}_{+}$ | $\tilde{\omega}^{\alpha}$
$H^{2,\,1}_{-}$ | $h^{2,\,1}_{-}$ | $\chi_{k}$
$H^{2,\,1}_{+}$ | $h^{2,\,1}_{+}$ | $\chi_{\kappa}$
$H^{3}_{-}$ | $2h^{2,\,1}_{-}+2$ | $\left(\alpha_{\hat{k}},\beta^{\hat{l}}\right)$
$H^{3}_{+}$ | $2h^{2,\,1}_{+}$ | $\left(\alpha_{\kappa},\beta^{\lambda}\right)$
Table 2.3: Harmonic bases for the orientifolded $\mathcal{X}$ .

Finally, the non-trivial intersection numbers are given by truncation of Eq.
(2.12) as

$\int_{\mathcal{X}}\omega_{a}\wedge\tilde{\omega}^{b}=\delta_{a}^{b}\
,\qquad\int_{\mathcal{X}}\omega_{\alpha}\wedge\tilde{\omega}^{\beta}=\delta_{\alpha}^{\beta}\
,\qquad\int_{\mathcal{X}}\alpha_{\hat{k}}\wedge\beta^{\hat{l}}=\delta_{\hat{k}}^{\hat{l}}\
,\qquad\int_{\mathcal{X}}\alpha_{\kappa}\wedge\beta^{\lambda}=\delta_{\kappa}^{\lambda}\
.$ (2.49)

The action of the parity operator and the fermion number operator on the ten-
dimensional fields is summarised in Tab. 2.4 [35]. It is clear from it that
the invariant orientifolded states must obey

$\displaystyle\sigma^{*}\hat{\phi}=+\hat{\phi}\
,\qquad\sigma^{*}\hat{g}=+\hat{g}\ ,\qquad\sigma^{*}\hat{B_{2}}=-\hat{B_{2}}\
,$ $\displaystyle\sigma^{*}\hat{C_{0}}=+\hat{C_{0}}\
,\qquad\sigma^{*}\hat{C_{2}}=-\hat{C_{2}}\
,\qquad\sigma^{*}\hat{C_{4}}=+\hat{C_{4}}\ .$ (2.50)

Field | Under $\Omega_{p}$ | Under $(-1)^{\text{F}_{\text{L}}}$ | Under $\Omega_{p}\,(-1)^{\text{F}_{\text{L}}}$
---|---|---|---
$\hat{\phi}$ | $+$ | $+$ | $+$
$\hat{g}$ | $+$ | $+$ | $+$
$\hat{B_{2}}$ | $-$ | $+$ | $-$
$\hat{C_{0}}$ | $-$ | $-$ | $+$
$\hat{C_{2}}$ | $+$ | $-$ | $-$
$\hat{C_{4}}$ | $-$ | $-$ | $+$
Table 2.4: Actions of the parity and fermion number operators on the ten-
dimensional spectrum.

We are now able to expand the ten-dimensional spectrum in the new truncated
harmonic bases, much like we did in the $\mathcal{N}=2$ case. In the following
subsections we briefly study the truncation of the $\mathcal{N}=2$ Kähler
manifolds into their $\mathcal{N}=1$ analogues.

#####  2.3.1 Truncating the Kähler geometry

After orientifolding, we can expand the Kähler form and the Kalb-Ramond 2-form
as111111Note that the orientifold invariant states of $\hat{B}_{2}$ have to be
odd under $\sigma^{*}$. However, the external piece $B_{2}(x)$ is even since
$\sigma$ leaves the external space intact. Therefore, $B_{2}(x)$ gets
projected out by the orientifold projection.

$\sigma^{*}J=+J\ \longrightarrow\
J=v^{\alpha}\omega_{\alpha}\qquad\text{and}\qquad\sigma^{*}\hat{B}_{2}=-\hat{B}_{2}\
\longrightarrow\hat{B}_{2}=b^{a}\omega_{a}\ ,$ (2.51)

where $\alpha=1,...,h^{1,\,1}_{+}$ and $a=1,...,h^{1,\,1}_{-}$. The invariance
of the Kähler form under orientifolding, together with the condition that the
metric must remain semi-positive definite after orientifolding, implies that
some intersection numbers have to vanish. In particular, any intersection
number (or its contraction) with an odd number of odd 2-cycles will have to
vanish since orientifolding will flip the sign of the intersection number,
breaking the Kähler cone conditions. This means that

$\mathcal{K}_{abc}=\mathcal{K}_{a\alpha\beta}=\mathcal{K}_{a\alpha}=\mathcal{K}_{a}=0\
.$ (2.52)

Plugging these conditions in the definition of the metric in Eq. (2.20), we
find the $\mathcal{N}=1$ truncated metric for the Kähler sector

$G_{\alpha\beta}=-\frac{1}{4\mathcal{V}}\left(\mathcal{K}_{\alpha\beta}-\frac{\mathcal{K}_{\alpha}\mathcal{K}_{\beta}}{4\mathcal{V}}\right)\
,\quad G_{ab}=-{1\over 4\mathcal{V}}\mathcal{K}_{ab}\ ,\quad G_{\alpha
b}=G_{a\beta}=0\ ,$ (2.53)

where $\mathcal{V}:={1\over
6}\mathcal{K}_{\alpha\beta\gamma}v^{\alpha}v^{\beta}v^{\gamma}$ is the volume
of the truncated $\mathcal{X}$ and there is no ambiguity in the contractions
of the triple intersection numbers since $J\in H^{1,\,1}_{+}$. It will be
convenient for later calculations to also introduce the inverse metrics

$G^{\alpha\beta}=-4\mathcal{V}\mathcal{K}^{\alpha\beta}+2v^{\alpha}v^{\beta}\
,\quad G^{ab}=-4\mathcal{V}\mathcal{K}^{ab}\ .$ (2.54)

We now concentrate on the complex structure sector. After orientifolding, the
unique holomorphic form will be expanded as

$\sigma^{*}\Omega=-\Omega\ \longrightarrow\
\Omega=\mathcal{Z}^{\hat{k}}\alpha_{\hat{k}}-\mathcal{F}_{\hat{k}}\beta^{\hat{k}}\
,$ (2.55)

where $\hat{k}=0,...,h^{1,2}_{-}$. By proxy of Eq. (2.29), the even
deformations of the complex structure sector will be projected out, $z_{k}$
being the only ones left. The Kähler potential for the truncated complex
structure sector will become

$G_{k\bar{l}}=\partial_{z^{k}}\partial_{\bar{z}^{l}}K^{\text{cs}}\ ,\qquad
K^{\text{cs}}=-\ln\left(i\int\Omega\wedge\bar{\Omega}\right)=-\ln\
i\left(\mathcal{Z}^{\hat{k}}\bar{\mathcal{F}}_{\hat{k}}-\bar{\mathcal{Z}}^{\hat{k}}\mathcal{F}_{\hat{k}}\right)\
.$ (2.56)

#####  2.3.2 The effective $\mathcal{N}=1$ action

Let us begin by writing down the orientifold invariant spectrum

$\displaystyle\hat{B}_{2}=b^{a}(x)\omega_{a}\
,\quad\hat{C}_{2}=c_{2}^{a}\omega_{a}\ ,$
$\displaystyle\hat{C}_{4}=\xi^{k}(x)\wedge\alpha_{k}+\rho_{\alpha}(x)\tilde{\omega}^{\alpha}\
.$ (2.57)

The truncated effective action is directly inherited from the parent
$\mathcal{N}=2$ action, based on the considerations of the previous
subsection,

$\displaystyle\mathcal{S}^{(4)}_{IIB,\,\mathcal{N}=1}=\int-{1\over
2}R\star{\mathbf{1}}+{1\over 4}\text{Re}\
\mathcal{M}_{\kappa\lambda}F^{\kappa}\wedge F^{\lambda}+{1\over 4}\text{Im}\
\mathcal{M}_{\kappa\lambda}F^{\kappa}\wedge\star F^{\lambda}$ $\displaystyle-
G_{k\bar{l}}dz^{k}\wedge\star
d\bar{z}^{\bar{l}}-h_{\tilde{p}\tilde{q}}dq^{\tilde{p}}\wedge\star
dq^{\tilde{q}}\ ,$ (2.58)

where the quaternionic sector becomes

$\displaystyle h_{\tilde{p}\tilde{q}}dq^{\tilde{p}}\wedge\star
dq^{\tilde{q}}:=(d\phi)^{2}+G_{\alpha\beta}dv^{\alpha}\wedge\star
dv^{\beta}+G_{ab}db^{a}\wedge\star db^{b}+{1\over
4}e^{2\phi}\mathcal{V}(dC_{0})^{2}$
$\displaystyle+e^{2\phi}\mathcal{V}G_{ab}\left(dc_{2}^{a}-C_{0}db^{a}\right)\wedge\star\left(dc_{2}^{b}-C_{0}db^{b}\right)$
$\displaystyle+{1\over
16\mathcal{V}}e^{2\phi}G^{\alpha\beta}\left(d\rho_{\alpha}-\mathcal{K}_{\alpha
bc}c_{2}^{b}db^{c}\right)\wedge\star\left(d\rho_{\beta}-\mathcal{K}_{\beta
de}c_{2}^{d}db^{e}\right)\ .$ (2.59)

We can see that due to the effect of the orientifold projection, while the
$z^{k}$ coordinates are still the natural candidate that spans
$\mathcal{M}^{\text{cs}}$, the coordinates $t^{A}$ are no longer adequate to
describe $\mathcal{M}^{\text{ks}}$, and $\mathcal{M}^{\text{Q}}$ by proxy.
Finding the correct choice of coordinates is quite involved and we will only
present the result in the following. For $O3/O7$-planes we have [31, 36, 37,
38]

  * •

Axio-dilaton: $S=e^{-\phi}-iC_{0}$.

  * •

Complex structure: $U^{k}:=z^{k}$, with $k=1,...,h^{1,\,2}_{-}(\mathcal{X})$.

  * •

Odd Kähler moduli: $G^{a}=c^{a}-iSb^{a}$, with
$a=1,...,h^{1,\,1}_{-}(\mathcal{X})$

  * •

Even Kähler moduli:

$T_{\alpha}=\tau_{\alpha}-{1\over 2\left(S+\bar{S}\right)}\mathcal{K}_{\alpha
ab}G^{a}\left(G-\bar{G}\right)^{b}+i\rho_{\alpha}\ ,$ (2.60)

with $\alpha=1,...,h^{1,\,1}_{+}$ and where $\tau_{\alpha}$ are given as an
implicit function of $v^{\alpha}$ by

$\tau_{\alpha}={1\over 2}\int_{\mathcal{X}}\omega_{\alpha}\wedge J\wedge
J={1\over 2}\mathcal{K}_{\alpha\beta\gamma}v^{\beta}v^{\gamma}\ ,$ (2.61)

in this way, $\tau_{\alpha}$ is the volume of a four-cycle
$\tilde{\omega}^{\alpha}\in H^{2,\,2}_{+}(\mathcal{X})$, the Poincaré dual to
the two-cycle ${\omega}_{\alpha}\in H^{1,\,1}_{+}(\mathcal{X})$ with volume
$v^{\alpha}$. We note that in terms of the four-cycles, the volume has become
an implicit function of $\tau_{\alpha}$. In general, we would require to
invert the relation $\tau_{\alpha}={1\over
2}\mathcal{K}_{\alpha\beta\gamma}v^{\beta}v^{\gamma}$ but this cannot be done
in full generality for a Calabi-Yau with a large number $h^{1,\,1}$ of Kähler
moduli. In [39], some advances have been made in terms of studying Kähler
moduli stabilisation in the two-cycle frame allowing for a more general
approach to model building with, a priori, any number of Kähler moduli.

We also note that based on the definition of $\tau_{\alpha}$ in Eq. (2.61), we
have that the internal volume is a homogeneous function of degree $3/2$ on the
$\tau_{\alpha}$, or equivalently on $(T+\bar{T})_{\alpha}$. This is a powerful
result that holds for any CY based on the definition of the internal volume
derived from the Kähler geometry. This also implies, due to a nice theorem by
Euler, the useful relations

${1\over\mathcal{V}}{\partial\mathcal{V}\over\partial T_{\alpha}}={3\over
2}\left[\left(T+\bar{T}\right)^{-1}\right]^{\alpha}\
,\qquad{1\over\mathcal{V}}{\partial^{2}\mathcal{V}\over\partial
T_{\alpha}\partial T_{\beta}}={3\over
4}\left[\left(T+\bar{T}\right)^{-1}\right]^{\alpha}\left[\left(T+\bar{T}\right)^{-1}\right]^{\beta}\
,$ (2.62)

where $\left[\left(T+\bar{T}\right)^{-1}\right]^{\alpha}$ is taken to be the
inverse of $\left(T+\bar{T}\right)_{\alpha}$. The Kähler potential of the
truncated theory is then given by

$K=-\ln\left(S+\bar{S}\right)-\ln\left(-i\int_{\mathcal{X}}\Omega\wedge\bar{\Omega}\right)-2\ln\left(\mathcal{V}\right)\
.$ (2.63)

Here, and in the following, the volume is taken to be in the natural
coordinate frame in terms of four-cycles, i.e.
$\mathcal{V}=\mathcal{V}(T+\bar{T})$, and the $\mathcal{N}=1$ moduli space is
given locally by the direct product

$\mathcal{M}^{\text{cs}}_{h^{1,\,2}_{-}}\times\mathcal{M}^{\text{Q}}_{h^{1,\,1}+1}\
.$ (2.64)

In these coordinates, the $\mathcal{N}=1$ effective action, in terms of the
Kähler form $K$ and superpotential $W$, takes the well-known form [40, 41]

$\mathcal{S}^{(4)}_{IIB,\,\mathcal{N}=1}=-\int{1\over
2}R\star{\mathbf{1}}+K_{I\bar{J}}dM^{I}\wedge\star d\bar{M}^{\bar{J}}+{1\over
4}\text{Re}\ \mathcal{M}_{\kappa\lambda}F^{\kappa}\wedge F^{\lambda}+{1\over
4}\text{Im}\ \mathcal{M}_{\kappa\lambda}F^{\kappa}\wedge\star
F^{\lambda}+V\star\mathbf{1}\ ,$ (2.65)

where the scalar potential is given by

$V=e^{K}\left(K^{I\bar{J}}D_{I}WD_{\bar{J}}\bar{W}-3\left|W\right|^{2}\right)+{1\over
4}\text{Re}\ \mathcal{M}^{-1\ \kappa\lambda}D_{\kappa}D_{\lambda}\ ,$ (2.66)

where $D_{\kappa}$ are the would-be D-term contributions to the scalar
potential. We have used $M^{I}$ to denote all complex scalars in chiral
multiplets in the theory, $K_{I\bar{J}}$ is the Kähler metric obtained from
the Kähler potential of Eq. (2.63), i.e. $\partial_{I}\partial_{\bar{J}}K$;
and we have introduced the Kähler derivative
$D_{I}W=\partial_{I}W+W\partial_{I}K$.

Comparison of the actions in Eq. (2.58) and Eq. (2.65) immediately implies
that the scalar potential for the moduli fields is flat, $V=0$, or in other
words that the superpotential and D-terms vanish $W=D_{\kappa}=0$. We will
have to generate a potential to be able to stabilise the moduli fields. This
is no trivial task and, among other things, it will require the introduction
of background fluxes in our theory. We will dedicate the next section to the
study of flux compactifications. Before doing so, however, it is important to
remark a couple of things.

Orientifolding requires the introduction of $O$-planes. These objects, as well
as the presence of $D$-branes and background fluxes, will backreact onto the
Calabi-Yau geometry and the sourced solutions will, in general, not admit a
metric ansatz of the form Eq. (2.42). A warp factor must be included to
account for the backreaction and the new metric ansatz is [42, 43, 44]

$ds^{2}=e^{2A(z)}g_{\mu\nu}(x)dx^{\mu}dx^{\nu}+e^{-2A(z)}g_{i\bar{\jmath}}(x,z)dz^{i}d\bar{z}^{\bar{\jmath}}\
,$ (2.67)

In general, the effects of the warping can be dramatic, to the point where the
resultant geometry is not necessarily conformally Calabi-Yau anymore. In the
remainder of this thesis, however, we will concentrate on compactifications at
large volume, relative to the string length, where the warp factor tends to
unity and the warped and unwarped metric coincide [43]. In this limit, the
formulae derived so far hold. This is the limit of interest, in any case,
since we understand supergravity as a low-energy effective theory that is
well-defined only in the corner of parameter space with small string coupling
and large volume. Furthermore, the perturbative and non-perturbative
corrections that will become important in the second half of the thesis will
be only under control at large volume. It is important to note that although
we use the words “small” and “large” we really mean finitely small and
finitely large, not necessarily the strict limits $g_{s}\rightarrow 0$ and
$\mathcal{V}\rightarrow\infty$ that take us to the boundary of moduli space,
more on this later.

Finally, the addition of $Op$-planes and $Dp$-branes introduces a term into
the ten-dimensional action that is localized in $(p+1)$-dimensions, which we
have neglected in Eq. (2.65). This is consistent with considering a small
number of $O3/O7$-planes to be coincident with an equal number of
$D3/D7$-branes, such that their contributions to Eq. (2.65) cancel locally. At
the level of the perturbative action in Eq. (2.65), the main use of localised
objects is to satisfy tadpole cancellation conditions, allowing non-trivial
background fluxes, and to avoid a number of no-go theorems [45, 46]. We will
briefly mention these in some more detail in the next section.

Two other applications that will become relevant in the remainder of the
thesis are the introduction of non-perturbative corrections through a stack of
$N$ $D$-branes [45, 47] and the addition of uplifting contributions to the
potential through anti $D3$-branes [48]. In the former, a stack of $N$
$D$-branes will give rise to non-Abelian $U(N)$ gauge theories. The Kähler
moduli can then be charged under the action of such non-Abelian groups
obtaining a non-flat potential, this will be discussed in more detail in Chap.
7. In the latter, anti $D$-branes at the tip of a Klebanov-Strassler throat
are used to uplift four-dimensional anti-de Sitter (AdS4) vacua to de Sitter
(dS4) (for example, see [49] for an overview of the uplift mechanism).

Although the ten-dimensional description of such objects is very interesting
and of paramount importance, within the scope of this thesis we will only
treat these at an effective four-dimensional level. We will therefore not
dedicate more time to the ten-dimensional description of localised sources and
we point the interested reader to [50, 51, 52, 53] for fundamentals and to
[54, 55, 56, 57, 58, 59] for phenomenological applications.

####  2.4 Flux compactifications

As we have seen in the previous section, the scalar potential for the moduli
fields is flat. The introduction of background fluxes allows us to generate a
non-trivial superpotential at tree-level. The scalar potential obtained from
this non-zero superpotential will be able to stabilise the complex structure
sector as well as the axio-dilaton. However, we will also show that due to the
no-scale structure, the Kähler sector does not receive any contribution from
background fluxes and it remains flat. The stabilisation of Kähler moduli will
be the motivation to introduce corrections to the tree-level action in the
second half of the thesis. Finally, let us note that the introduction of
fluxes provides a way to spontaneously break the leftover $\mathcal{N}=1$
supersymmetry.

Take a p-form gauge field $A_{p}$ with field strength $F_{p+1}=dA_{p}$. We
will say, by analogy with the case of electromagnetism in 4-dimensions, that
the gauge field $A_{p}$ gives rise to a magnetic flux of the form

$\int_{\Sigma_{p+1}}F_{p+1}=m\in\mathbb{Z}\ ,$ (2.68)

where $\Sigma_{p+1}\subset\mathcal{X}$ is the (p+1)-cycle wrapping the
(p+1)-form field strength $F$. The same gauge field will also give rise to an
electric flux of the form

$\int_{\Sigma_{D-p-1}}\star F_{p+1}=n\in\mathbb{Z}\ ,$ (2.69)

where $D=\text{dim}(\mathcal{X})$ and $\Sigma_{D-p-1}\in\mathcal{X}$ is the
(D-p-1)-cycle wrapping the electric dual of $F$. We note that these fluxes are
quantised in units of ${1\over 2\pi\alpha^{\prime}}$ as a consequence of a
generalised Dirac quantisation condition [43], much like the electric and
magnetic fluxes are quantised in the case of electromagnetism. However, within
the context of the low energy supergravity description, these fluxes will be
treated as continuous parameters in the effective theory.

In type IIB supergravity, one can turn on background fluxes for the 3-form
field strengths in the NSNS and RR sector. For the background fluxes to be
consistent with our orientifold projection we see that the flux pieces must be
odd under the action of the involution $\sigma$. Thus, we expand the flux
pieces as

$\tilde{H}_{3}=m_{(H)}^{\hat{k}}\alpha_{\hat{k}}+n^{(H)}_{\hat{k}}\beta^{\hat{k}}\
,\quad\tilde{F}_{3}=m_{(F)}^{\hat{k}}\alpha_{\hat{k}}+n^{(F)}_{\hat{k}}\beta^{\hat{k}}\
,\qquad\hat{k}=0,...,h^{1,\,2}_{-}\ ,$ (2.70)

and the fluxed effective action can be obtained by substituting [31]

$\hat{H}_{3}\rightarrow\hat{H}_{3}^{(F)}=\hat{H}_{3}+\tilde{H}_{3}\
,\qquad\hat{F}_{3}\rightarrow\hat{F}_{3}^{(F)}=\hat{F}_{3}+\tilde{F}_{3}\ ,$
(2.71)

in the ten-dimensional action since the $\hat{F}_{5}$ terms that would involve
the 3-form fluxes are projected out due to the orientifold constraints. It is
convenient to repackage the two 3-form field strength in the imaginary self-
dual three-form field strength [43]

$\hat{G}_{3}=\hat{F}_{3}^{(F)}-iS\hat{H}_{3}^{(F)}\
,\qquad\star\hat{G}_{3}=i\hat{G}_{3}\ ,$ (2.72)

and we define the shorthand notation

$\tilde{G}_{3}=(m_{(F)}-iSm_{(H)})^{\hat{k}}\alpha_{\hat{k}}+(n_{(F)}-iSn^{(H)})_{\hat{k}}\beta^{\hat{k}}:=m^{\hat{k}}\alpha_{\hat{k}}+n_{\hat{k}}\beta^{\hat{k}}\
.$ (2.73)

The fluxed scalar potential is given by [60, 43, 61, 62, 29, 31]

$V=e^{K}\left(\int\Omega\wedge\bar{\tilde{G}}_{3}\int\bar{\Omega}\wedge\tilde{G}_{3}+G^{kl}\int\chi_{k}\wedge\tilde{G}_{3}\int\bar{\chi}_{l}\wedge\bar{\tilde{G}}_{3}\right)\
.$ (2.74)

From this expression, it is clear to see that setting the background fluxes to
zero will make the scalar potential vanish. To make a connection with Eq.
(2.65), it is also shown, for orientifolds without an odd-Kähler sector, the
GVW superpotential

$W_{GVW}=\int\Omega\wedge\tilde{G}_{3}\ ,$ (2.75)

generates the same scalar potential when making use of Eq. (2.66). For the
case of orientifold with an odd Kähler sector this same results holds as long
as the no-scale structure is unbroken. Generating a potential for the Kähler
sector directly relates to breaking this no-scale structure, since (at least)
the volume mode will couple to the other sectors as well as the Kähler moduli.
Given that our objective is to discuss four-dimensional phenomenology, we will
have to eventually break no-scale. In the remainder of this thesis, we will
concentrate in orientifolds that project out the odd Kähler sector, i.e.
$h^{1,1}=h^{1,1}_{+}$, and the results discussed so far will apply
straightforwardly. Now that we have generated a scalar potential, we would
like to see what sectors can be stabilised from it. As already anticipated,
only the axio-dilaton and complex structure moduli will receive a non-zero
potential. Let us see this now.

The idea is to show that the scalar potential obtained from GVW superpotential
is independent of the Kähler coordinates $T_{\alpha}$ Eq. (2.60), up to an
overall factor. We begin by noting that

$W_{GVW}=W_{GVW}(S,U)\ ,$ (2.76)

since $\Omega=\Omega(U)$ and $\tilde{G}_{3}=\tilde{G}_{3}(S)$. The F-terms for
$T_{\alpha}$ are

$D_{T}W_{GVW}=W_{GVW}K_{T}=-2W_{GVW}{\mathcal{V}_{T}\over\mathcal{V}}=-3W_{GVW}\left[(T+\bar{T})^{-1}\right]^{\alpha}\
,$ (2.77)

where $X_{T}:={\partial X\over\partial T_{\alpha}}$, we have also made use of
the form of the Kähler potential given in Eq. (2.63). The Kähler metric will
be block diagonal in the $\left(S,U^{k},T_{\alpha}\right)$ sectors such that
the scalar potential takes the form

$\displaystyle
V=e^{K}\left[K^{S\bar{S}}D_{S}WD_{\bar{S}}\bar{W}+K^{U\bar{U}}D_{U}WD_{\bar{U}}\bar{W}+K^{T\bar{T}}D_{T}WD_{\bar{T}}\bar{W}-3\left|W\right|^{2}\right]$
$\displaystyle=e^{K}\left[K^{S\bar{S}}D_{S}WD_{\bar{S}}\bar{W}+K^{U\bar{U}}D_{U}WD_{\bar{U}}\bar{W}+\left(K^{T\bar{T}}\partial_{T}K\partial_{\bar{T}}K-3\right)\left|W\right|^{2}\right]\
,$ (2.78)

where we remark that the first two terms in the brackets are independent of
$T_{\alpha}$. The Kähler metric for the $T_{\alpha}$ sector and its inverse
are given by

$\displaystyle
K_{T\bar{T}}={2\over\mathcal{V}}\left({\mathcal{V}_{T}\mathcal{V}_{\bar{T}}\over\mathcal{V}}-\mathcal{V}_{T\bar{T}}\right)=3\left[(T+\bar{T})^{-1}\right]^{\alpha}\left[(T+\bar{T})^{-1}\right]^{\beta}\
,$ (2.79)

where we have made use of the relations in Eq. (2.62). Contracting the inverse
metric we find

$K^{T\bar{T}}K_{T}K_{\bar{T}}=3\ ,$ (2.80)

so that the parentheses in Eq. (2.78) vanish identically and the scalar
potential has the no-scale structure

$V=e^{K}\left(K^{S\bar{S}}D_{S}WD_{\bar{S}}\bar{W}+K^{U\bar{U}}D_{U}WD_{\bar{U}}\bar{W}\right)\
,$ (2.81)

with the only $T_{\alpha}$ dependence in the overall exponential factor. From
the previous expression, it is clear that the Kähler sector remains flat at
tree-level.

Breaking the no-scale structure can be done in one of two ways, either the
superpotential gains a dependence on the Kähler moduli $W(S,U)\rightarrow
W(S,U,T)$ or by breaking the block diagonal structure of the Kähler metric.
The first option will be related to the introduction of non-perturbative
corrections to the superpotential. The second one can be associated to
perturbative corrections in the large volume/small coupling expansion. We will
dedicate Chap. 7 to the study of these effects.

By looking at Eq. (2.81) we see that, since any supersymmetric vacua must have
vanishing F-terms $D_{U}W=D_{S}W=0$, only supersymmetric Minkowski vacua are
allowed. Non-supersymmetric solutions with non vanishing F-terms are also
allowed by appropriate choices of fluxes. We can see this by turning once
again to the F-terms.

We already found the F-term for the Kähler sector in Eq. (2.77). Preserving
supersymmetry will require $W_{\text{GVW}}=0$, in other words the
(0,3)-component of $\tilde{G}_{3}$ must vanish. The remaining F-terms read

$D_{S}W={1\over S-\bar{S}}\int_{\mathcal{X}}\Omega\wedge\bar{\tilde{G}}_{3}=0\
,\qquad D_{U}W=\int_{\mathcal{X}}\chi_{k}\wedge\tilde{G}_{3}=0\ .$ (2.82)

These F-terms imply that the (3,0)-component and the (1,2)-component need to
vanish. Therefore, we have the following: if $\tilde{G}_{3}\in H^{2,\,1}_{-}$
all conditions are satisfied and supersymmetry is preserved, if
$\tilde{G}_{3}\in H^{0,\,3}_{-}$ we find some broken supersymmetry in
$D_{T}W\neq 0$ with $V=0$, and if $\tilde{G}_{3}\in H^{3,\,0}_{-}\oplus
H^{1,\,2}_{-}$ none of the above F-terms would vanish and we only find
unstable vacuum configurations.

Since the axio-dilaton and the complex structure sector are fixed at tree-
level, one is safe to consider them fixed by their vacuum expectations values.
We will take the tree-level superpotential to be

$W_{\text{tree}}:=W_{0}=\left\langle\int_{\mathcal{X}}\Omega\wedge\tilde{G}_{3}\right\rangle\
,$ (2.83)

for the remainder of the thesis, effectively integrating out the axio-dilaton
and the complex structure moduli. Any quantum corrections to their vevs in the
large volume limit will be subleading to the tree-level contributions and we
will neglect them safely. This also implies that the tree-level Kähler
potential will take the form

$K_{\text{tree}}:=-2\ln\left(\mathcal{V}\right)+K_{0}=-2\ln\left(\mathcal{V}\right)-\ln\left({2\over
g_{s}}\right)-\ln\left\langle-i\int_{\mathcal{X}}\Omega\wedge\bar{\Omega}\right\rangle\
.$ (2.84)

Finally, before moving on, we would like to discuss the tadpole cancellation
conditions in the presence of fluxes and localised sources. The choice of non-
trivial fluxes induces a new term in the Bianchi identity for the
$\hat{F}_{5}$ field strength. In particular,

$d\hat{F}_{5}=\tilde{H}_{3}\wedge\tilde{F}_{3}\ ,$ (2.85)

which leads to the no-go theorem of [63]. Since the fluxes are the only
contribution to the $\hat{C}_{4}$ tadpole, one would have to consider
compactifications with trivial fluxes only. However, the discovery of D-branes
[64] led to the realisation that charge densities from $D3/D7$ branes, and
also from orientifold planes, contribute to the tadpole [43]. To satisfy the
Bianchi identity and guarantee the absence of anomalies in the effective four-
dimensional theory, it is then required that the following tadpole
cancellation condition holds true

$N_{D3}-N_{\overline{D3}}+\int_{\mathcal{X}}\tilde{H}_{3}\wedge\tilde{F}_{3}={\chi(\mathcal{X}_{4})\over
24}\ ,$ (2.86)

where $\left(N_{D3},\,N_{\overline{D3}}\right)$ are the charge contributions
from $D3$-branes and $\overline{D3}$-branes, respectively, and
$\chi(\mathcal{X}_{4})$ is the contribution from $D7$-branes and orientifold
planes, which gains a geometrical interpretation in terms of the Euler
characteristic of the corresponding four-fold $\mathcal{X}_{4}$ [65].

####  2.5 The Swampland programme

In the last section we achieved the stabilisation of the non-Kähler moduli
through background fluxes. We highlighted that stabilising the Kähler sector
will generically require the introduction of perturbative and non-perturbative
correction to the tree-level Kähler potential and superpotential discussed
previously. These corrections are quintessential to supergravity model
building, as we will explore in Chap. 7. It turns out that obtaining a
systematic understanding of the stabilisation of the Kähler moduli is a highly
non-trivial task. Furthermore, since we lack a complete picture of non-
perturbative supergravity, it is complicated to define frameworks where
computations can be carried out under complete control. Defining complete
control is itself a hotly debated topic in the community. Another reason
behind the difficulty of the task at hand is that supergravity inherits two
perturbation parameter from string theory: $\alpha^{\prime}$ and the string
coupling $g_{s}$. In writing down quantum corrections to the tree-level
action, we will inevitably encounter terms of $\mathcal{O}(\alpha^{p}\cdot
g_{s}^{q})$, for some $p,q>0$. This can make it so that, at times, it can
become unclear which terms enter at a given order in perturbation theory. For
example, comparing terms $\mathcal{O}(\alpha^{\prime 3}g_{s}^{2})$ and
$\mathcal{O}(\alpha^{\prime 2}g_{s}^{3})$ can be hard. A very important part
of ongoing supergravity research is to systematically analyse these
corrections at all orders and ensure that Kähler moduli stabilisation is not
spoiled in the bulk of moduli space (see for example [66, 67, 49, 68, 69]).

The lack of a completely satisfactory answer to the challenges presented above
led to the creation of the so-called Swampland programme (see [70] for an in-
depth review). Its core goal is to differentiate between low energy effective
field theories that have a consistent completion within a prospective quantum
gravity UV theory and those that do not. In doing so, the Swampland programme
has formed a web of conjectures that attempt to delineate the space between
the consistent and inconsistent theories. In the next section we will briefly
describe two of these conjectures, the particular ones that concern the
contents of this thesis. One should remark that, although the ultimate goal is
for the programme to be independent of a prospective UV completion, most of
the conjectures will draw from string theoretical intuitions.

Figure 2.1: A pictorial representation of the web of conjectures in the
Swampland programme [71]

#####  2.5.1 The Swampland distance conjecture

In [72, 73], the authors proposed that a number of conjectures. Among them was
what would eventually become to be known as the Swampland distance conjecture
(SDC). A refined version of the SDC (rDC) claims the following [70, 74, 75],

######  Conjecture 1.

Consider a theory coupled to gravity with some moduli space parametrised by
the expectation values of some fields with no potential. Let the geodesic
distance between any two points $P\in\mathcal{M}$ and $Q\in\mathcal{M}$ in
moduli space be denoted by $d(P,Q)$. Then, there exists a tower of states in
theory with mass scale $M$ and scaling

$M(Q)<M(P)e^{-\alpha{d(P,Q)\over M_{\text{pl}}}}\ ,$ (2.87)

if $d(P<Q)\gtrsim M_{\text{pl}}$ and where $\alpha\sim\mathcal{O}(1)$ is a
positive model dependent coefficient.

A second claim of the rDC is that this statement holds true even for fields
with a potential by exchanging the moduli space with the field space of the
effective field theory.

The motivation for the existence of a light tower of states can be understood
from the mass spectrum in $D$ dimensions of a closed string when compactifying
one direction on a circle of radius $R$. Indeed, the $(D-1)$-dimensional mass
for the for the string is [17]

$M^{2}\sim n^{2}\left({1\over R^{2}}\right)^{{d-1\over
d-2}}+w^{2}\left({R^{2}\over\alpha^{\prime 2}}\right)^{{d-1\over d-2}}\ ,$
(2.88)

where $\left(n,w\right)\in\mathbb{N}$ are the Kaluza-Klein number and winding
number, respectively. Indeed, in the infinite distance limit
$R\rightarrow\infty$, a infinite tower of Kaluza-Klein modes labelled by $n$
will become massless. The existence of the winding modes ensures that in the
T-dual frame, where $R\rightarrow\tilde{R}=1/R$, there also exists another
tower becoming massless, in this case related to the winding modes with labels
$w$.

More detailed analysis of the conjecture has been carried out in a number of
string scenarios [74, 75, 76, 77, 78, 79, 80, 81]. Generically, the analyses
focus on finding solutions and then identifying the objects in the theory that
introduce an infinite tower of states which becomes light at the boundary of
moduli space. For the volume modulus, it is quite straightforward to show that
the tower of states can be associated with the KK modes like we did in the
simple example above. In more generality, the tower of states can be harder to
identify and related to, for example, D-branes becoming tensionless.

#####  2.5.2 The refined de Sitter conjecture

A second conjecture is connected to the existence of (meta)stable dS
backgrounds in an effective field theory obtained from string theory
compactifications. In [82], the authors introduced the following claim that a
potential of any theory coupled to gravity would have to satisfy

$\left|\nabla V\right|\geq{c\over M_{\text{pl}}}V\ .$ (2.89)

It was pointed out in [83], that this claim was in tension with the Higgs
potential. The previous statement was refined to its present form in [84, 85].
This refined de Sitter conjecture (rdSC) claims the following

######  Conjecture 2.

Given an effective field theory consistent with a theory of quantum gravity,
its potential must satisfy either

$\left|\nabla V\right|\geq{c\over M_{\text{pl}}}V\
,\qquad\text{or}\qquad\text{min}\left(\nabla_{i}\nabla_{j}V\right)\geq-{c^{\prime}\over
M_{\text{pl}}^{2}}V\ ,$ (2.90)

where $\left(c,\ c^{\prime}\right)$ are positive constants of $\mathcal{O}(1)$
and $\text{min}\left(\nabla_{i}\nabla_{j}V\right)$ is the minimum of the
eigenvalue of the corresponding Hessian.

This conjecture forbids the existence of stable de Sitter vacua in string
theory. Importantly, metastable vacua are allowed as long as a tachyonic
instability exists such that the lifetime of the vacua is bounded by its
Hubble time $H^{-1}\sim M_{\text{pl}}/\sqrt{V}$.

Whereas the refined distance conjecture has been well studied, the UV origin
of the refined de Sitter conjecture is more opaque. In part, this is related
to the fact that the de Sitter conjecture is entirely supported by cherry-
picked examples and difficulty of obtaining de Sitter vacua in string theory.
To some extent, it has been argued that the rdSC is consistent with the rSDC
[85], and thus the rdSC could be understood from an infinite distance limit
perspective in the boundary of moduli space.

If the statement of the de Sitter conjecture was the inability of obtaining
deSitter vacua in the boundary of moduli space, this would not be a surprise.
Indeed, it has been known for quite some time [86] that non-supersymmetric
degrees of freedom (like those of de Sitter vacua) are notably hard to access
from the point of view of weak coupling (perturbative) supergravity. In the
infinite distance limit all we have left are perturbative degrees of freedom.
Therefore, it would be naïve to expect that de Sitter vacua are hiding in that
corner of parameter space. However, the dS conjecture forbids dS vacua
existing also in the bulk of moduli space, where perturbative and non-
perturbative corrections become important.

Indeed, let us consider a heuristic scenario to exemplify this point —we will
make this point more formal in Chap. 6 and when discussing corrections to the
tree-level action in Chap. 7—. In [86], the authors describe the Dine-Seiberg
Minkowski vacua, the decompactification limit, for the dilaton modulus. A
similar argument for the canonical volume modulus, $\varphi$, can also be
made. Indeed, let us imagine that we have some de Sitter vacua generated
through corrections to the tree-level action as depicted in Fig. 2.2.

Figure 2.2: A heuristic example of a typical profile for the scalar potential
of a canonical saxion modulus $\phi$. The potential consists of three regions:
(i) the deep bulk of moduli space where the perturbative expansion breaks
down, (ii) the asymptotic bulk where numeric control can be retained and (iii)
the boundary of moduli space where the perturbative expansion can be made
exact.

At large field values, the potential decreases exponentially tending to the
Dine-Seiberg type vacuum, in the language of the canonical field $\varphi$.
This behaviour at large field values is quite generic since, in this limit, we
expect the terms in the action Eq. (2.58) to dominate over any corrections,
and the non-canonical saxions $(s)$ enter the theory as $d\phi\sim s^{-1}ds$,
based on the form of the block diagonal Kähler metric obtained from Eq.
(2.63).

We can observe two things. Moving the minimum in the bulk towards asymptotic
larger values of the field would imply tuning the corrections to the point
where the leading perturbative expansion is comparable with the tree-level
terms, at which point we lose computational control. In Chap. 6 we revisit the
discussion of the Swampland programme from the point of view of phenomenology
and point out that there exists a disconnect between the Swampland programme
and phenomenology. Secondly, an interesting phenomenological region exists in
the asymptotic bulk of moduli space, where the effective theory can remain
under numeric control. That region is where all the known de Sitter proposals
exists and where the corrections to the tree-level action remain under
control. This is where we will concentrate our efforts in the second half of
the thesis. Finally, a third region exists, the deep bulk of moduli space,
where the effective theory becomes strongly coupled and the perturbative
expansion breaks down. This regions maps to small volumes (in units of string
length) and/or a string coupling that becomes larger than unity.

###  Chapter 3 Some open questions in modern cosmology

Throughout this thesis we will concern ourselves with the consequences of a
number of important open questions in modern physics and more particularly,
its implications for cosmology. In Chap. 4 we will discuss how to generate a
naturally light dark energy field. In Chap. 5, we discuss the implications of
the coincidence problem and a possible stringy solution. Naturalness (in the
context of cosmology) and the coincidence problem are intimately related to
the cosmological constant problem. A problem which is, to a large extent, the
underlying motivation for a lot of the research we have undertaken.

In this chapter, we will begin by giving an overview of the naturalness and
hierarchy problem in Sec. 3.1. We will then shortly review some problems in
cosmology, diving into the well-known cosmological constant problem in Sec.
3.2. Finally, we will discuss the coincidence problem in Sec. 3.3.

####  3.1 The hierarchy problem and naturalness

In theoretical physics, the original setting for the hierarchy problem is
often thought of to be in the context of grand unified theories [9, 87]. To
see the hierarchy problem in action, we can take the much simpler example of a
scalar $\phi^{4}$ effective theory with some UV cut-off $\Lambda$ given by the
Lagrangian

$\mathcal{L}=-{1\over 2}(\partial_{\mu}\phi)^{2}-{1\over
2}m^{2}\phi^{2}-{\lambda\over 4!}\phi^{4}\ .$ (3.1)

We are interested in calculating the effects of one-loop quantum corrections
to the above Lagrangian. The tree-level Feynman diagrams for the theory are
given by

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a) at (-2,0) ;
\vertex(b) at ( 2,0) ; \diagram* { (a) -- [scalar] (b), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=D^{(2)}\qquad\qquad\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot] (m) at (0,
0); \vertex(a) at (-2,-1) ; \vertex(b) at ( 2,-1) ; \vertex(c) at (-2, 1) ;
\vertex(d) at ( 2, 1) ; \diagram* { (d) -- [scalar] (m) -- [scalar] (c), (b)
-- [scalar] (m) -- [scalar] (a), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=D^{(4)}\ .$ (3.2)

By standard calculation techniques [88], the two-point and four-point
propagators in Fourier space are given by

$D^{(2)}={-i\over p^{2}+m^{2}+i\epsilon}\ ,\qquad D^{(4)}=-i\lambda\ ,$ (3.3)

and the one loop-diagrams are

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(m) at (0, -1);
\vertex(a) at (-2,-1) ; \vertex(b) at ( 2,-1) ; \vertex(ll) at ( -1,0) ;
\vertex(lr) at ( 1,0) ; \vertex(lu) at (0,1) ; \diagram* { (a) -- [scalar] (m)
-- [scalar] (b), (m) -- [scalar, quarter left] (ll) -- [scalar, quarter left]
(lu) -- [scalar, quarter left] (lr) -- [scalar, quarter left] (m), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\qquad\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(ll) at (-1,0);
\vertex(lu) at (0,1); \vertex(lr) at (1,0); \vertex(ld) at (0,-1); \vertex(a)
at (2,1) ; \vertex(b) at ( 2,-1) ; \vertex(c) at (-2, 1) ; \vertex(d) at (-2,
-1) ; \diagram* { (d) -- [scalar] (ll) -- [scalar] (c), (ll) -- [scalar,
quarter left] (lu) -- [scalar, quarter left] (lr) -- [scalar, quarter left]
(ld) -- [scalar, quarter left] (ll), (b) -- [scalar] (lr) -- [scalar] (a), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$ (3.4)

The introduction of the one-loop diagrams corrects the tree-level expressions
such that the integral along a loop carrying internal momenta $q^{\mu}$ is

$\int_{0}^{\Lambda}{d^{4}q\over(2\pi)^{4}}{-i\over
q^{2}+m^{2}+i\epsilon}{-i\over(p-q)^{2}+m^{2}+i\epsilon}\propto
ln\left({\Lambda\over p}\right)\ .$ (3.5)

We can see that in the limit $\Lambda\rightarrow\infty$ the integral diverges.
More importantly for our discussion, the theory and the physical observables
now have knowledge of the cut-off scale $\Lambda$. For example, the scattering
amplitude for the 2-to-2 particle scattering diagrams is now
$i\mathcal{M}\propto-i\lambda+i\lambda^{2}\ln\left({\Lambda^{2}\over
m^{2}}\right)$. Another related problem is that the self-energy diagram
contributes to the two point correlation function with a term

$\Gamma_{(2)}^{1-\text{loop}}\propto\lambda\int_{0}^{\Lambda}{d^{4}q\over(2\pi)^{4}}{1\over
q^{2}+m^{2}+i\epsilon}\propto\lambda\left[\Lambda^{2}-m^{2}\ln\left({\Lambda^{2}\over
m^{2}}\right)\right]\ .$ (3.6)

The canonical solution to these issues is to introduce counterterms to
renormalise the theory. After doing so, the renormalised observables of the
theory are cut-off independent. For example, introducing the mass counterterm

$\mathcal{L}_{\delta m}=-{\delta m^{2}\over 2}\phi^{2}\ ,\qquad\delta
m^{2}=-{\lambda\over
32\pi^{2}}\left[\Lambda^{2}-m^{2}\ln\left({\Lambda^{2}\over
m^{2}}\right)\right],$ (3.7)

turns the two-point correlation function into its renormalised version

$\bar{\Gamma}_{(2)}=i\left(p^{2}+m^{2}\right)\ ,$ (3.8)

to one-loop, with higher orders in loop corrections being systematically
reabsorbed into the definition of the counterterms.

This has allowed us to decouple the effective description of the low-energy
physics from the cut-off of the theory, and got rid of the UV divergence that
appear by sending the cut-off to arbitrarily large values. This will require
an infinite number of counterterms but, a priori, we have cured the theory
from UV divergences. We would like to go one step further now and ask what
happens for a theory with a light scalar $\phi$ and a heavy scalar $\varphi$.
Let us now consider a theory with two scalar sectors coupled to each other

$\mathcal{L}=-{1\over 2}(\partial_{\mu}\phi)^{2}-{1\over
2}m^{2}\phi^{2}-{\lambda_{\phi}\over 4!}\phi^{4}-{1\over
2}(\partial_{\mu}\varphi)^{2}-{1\over
2}M^{2}\varphi^{2}-{\lambda_{\varphi}\over 4!}\varphi^{4}-{g\over
4}\varphi^{2}\phi^{2}\ ,$ (3.9)

here $m\ll M$. Much like in the previous case, one must regularise the theory.
After doing so, the tree-level one-to-one scatterings are decoupled from one
another since there is no quadratic term mixing $\phi$ and $\varphi$. However,
the one-loop diagrams for the light scalar now contain an extra contribution

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(m) at (0, -1);
\vertex(a) at (-2,-1) ; \vertex(b) at ( 2,-1) ; \vertex(ll) at ( -1,0) ;
\vertex(lr) at ( 1,0) ; \vertex(lu) at (0,1) ; \diagram* { (a) -- [scalar] (m)
-- [scalar] (b), (m) -- [quarter left] (ll) -- [quarter left] (lu) -- [quarter
left] (lr) -- [quarter left] (m), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\qquad,$ (3.10)

where the solid line represents the heavy scalar $\varphi$. By reproducing the
calculations for the renormalised two-point correlator, we find

$\bar{\Gamma}_{(2)}\sim i\left(p^{2}+m^{2}\right)-{i\lambda_{\phi}m^{2}\over
32\pi^{2}}-{igM^{2}\over 32\pi^{2}}\ .$ (3.11)

The physics associated to the light sector now depend on the heavy sector,
i.e. the light and heavy physics only decouple at tree-level. We could try to
retune the counterterms to reabsorb the heavy sector dependence. However, by
continuing our perturbative expansion to two-loops we would find diagrams of
the type

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(m1) at (-2, -1);
\vertex(a) at (-4,-1) ; \vertex(b) at ( 4,-1) ; \vertex(ll1) at ( -3,0) ;
\vertex(lr1) at ( -1,0) ; \vertex(lu1) at (-2,1) ; \vertex(m2) at (2,-1);
\vertex(ll2) at ( 1,0) ; \vertex(lr2) at ( 3,0) ; \vertex(lu2) at (2,1) ;
\diagram* { (a) -- [scalar] (m1) -- [scalar] (m2) -- [scalar] (b), (m1) --
[quarter left] (ll1) -- [quarter left] (lu1) -- [quarter left] (lr1) --
[quarter left] (m1), (m2) -- [quarter left] (ll2) -- [quarter left] (lu2) --
[quarter left] (lr2) -- [quarter left] (m2), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad\qquad,$ (3.12)

which would again contribute a term of $\mathcal{O}(gM^{2})$ to the two-point
correlation function. We would have to fine-tune to the same accuracy at every
order in perturbation theory, and we would say that the light scalar is
unstable against radiative corrections. This is effectively an infinite fine-
tuning and the theory loses all power of predictability. The hierarchy problem
can then be restated as: how can we keep the heavy physics decoupled from the
light physics?

A simple solution would be to argue that $g$ is strictly vanishing. Then the
theory with two scalars turns into two theories with one scalar each,
analogous to the one in Eq. (3.1).This implies the existence of an extra
symmetry in the theory. However, consider the case of the Higgs, for example,
where the interactions between different sectors are crucial to describe our
Universe, then $g\neq 0$ by necessity.

Another possibility is that the symmetry does not fix $g=0$ but rather
introduces new degrees of freedom such that the dependence on the heavy
physics drops off from the observables. Of course, this is the idea behind
supersymmetry [89] where each component in a supersymmetric multiplet
contributes with opposite sign to the n-point correlators, by virtue of
supersymmetry pairing fermions to bosons and vice versa. However, no
supersymmetric partner to the Standard Model particles has been found at
energies $\mathcal{O}(\text{TeV})\sim 10^{-15}M_{\text{pl}}$, and
supersymmetry must then be softly broken at some low-energy scale
$E_{SUSY}\gtrsim 10^{-15}M_{\text{pl}}$.

Finally, one could just set $g\ll 1$ by hand such that, even if the two
different sectors are coupled, the effects due to heavy physics are subleading
to the light sector physics. However, two considerations arise. One is that
the usual theoretical physics’ lore would tell us that the smallness of $g$
has to be protected by some approximate symmetry in the theory. Otherwise, the
RG flow running for $g$ towards arbitrarily high UV physics can spoil the
decoupling. A second one is that, in Physics, we do not find arbitrarily tuned
constants. Usually, couplings are $\mathcal{O}(1)$; if the couplings are very
small there is an underlying reason. This sentiment is the idea behind
naturalness. We say that a theory is natural when it contains only couplings
which are $\mathcal{O}(1)$. Setting $g\ll 1$ by hand would be unnatural.

More concretely, we introduce ‘t Hooft’s definition of naturalness [90],

######  Naturalness.

Consider an effective theory valid up to some cut-off $\Lambda$. Make any free
parameters dimensionless by measuring them in units of the cut-off. A theory
is natural if the only parameters that are much smaller than unity imply an
enhancement of the symmetry group of theory in the limit of those parameters
vanishing. Otherwise, the theory is unnatural.

To exemplify the meaning of an enhanced symmetry, we take our simple one-field
$\phi^{4}$ model and follow the discussion of [90]. Fixing $\lambda=0$ leads
to a non-interacting theory, a freely propagating massive particle with a
conserved particle number, $\lambda\neq 0$ breaks particle number
conservation. Therefore, a theory with $\lambda=0$ is natural despite having a
vanishingly small parameter since $\lambda=0$ induces a new symmetry in the
theory. On the other hand, $m=0$ does not enhance the symmetries of the theory
in any way, and thus fixing $m/\Lambda\ll 1$ by hand would render the theory
unnatural. Similarly, in the light-heavy Lagrangian Eq. (3.9) setting $g=0$
does not recover any symmetries, as the total number of particles is still not
conserved due to $\lambda_{\phi},\,\lambda_{\varphi}\neq 0$.

As discussed, within the framework of particle physics the hierarchy problem
and naturalness issues appear under the guise of the Higgs. The Higgs mass is
much smaller than the mass of the top gauge boson, the heaviest Standard Model
particle. As mentioned, supersymmetry aims to solve this. Another proposed
solution is technicolour [91, 92, 93], where the Higgs is a composite particle
of two new fermions and the underlying theory in terms of the two fermions is
natural. A more in-depth review of the historical developments and philosophy
around naturalness and the hierarchy problem can be found in [8].

We can also think of the hierarchy problem and naturalness in more general
terms as being a question regarding how to introduce parametrically large
scale differences in a theory. For example, the cosmological constant energy
scale is given by $E_{cc}\sim\sqrt{\Lambda}\sim 10^{-60}M_{\text{pl}}$ whereas
top collider physics have an associated energy scale $E_{\text{top}}\sim
10^{-16}M_{\text{pl}}$ while quantum gravity is often associated with
$E_{QG}\sim\mathcal{O}(M_{\text{pl}})$ physics. A full theory describing all
of these phenomena, and anything in between, would have to span $10^{60}$
orders of magnitude. If string theory is to describe our Universe, it will
certainly have to contain small numbers and whether it can do so in a natural
way is non-trivial.

####  3.2 The cosmological constant problem

In the previous section, we reviewed how light modes in the effective theory
can become unstable due to radiative corrections. This statement also affects
the cosmological constant, making the theoretical cosmological constant orders
of magnitude larger than the observed value for the vacuum energy of our de
Sitter Universe. This problem goes by the name of the cosmological constant
problem [11], for a modern formulation of this problem, and possible
resolutions, we direct the interested reader to [94, 10], as well as
references therein.

To see the problem in action, we consider the one-loop vacuum diagrams for the
$\phi^{4}$-theory of Eq. (3.1) in a flat spacetime

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(m) at (0, -1);
\vertex(ll) at ( -1,0) ; \vertex(lr) at ( 1,0) ; \vertex(lu) at (0,1) ;
\diagram* { (m) -- [scalar, quarter left] (ll) -- [scalar, quarter left] (lu)
-- [scalar, quarter left] (lr) -- [scalar, quarter left] (m), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\
=\int{d^{4}p\over(2\pi)^{4}}\log\left(p^{2}+m^{2}\right)\propto m^{4}\int
d^{4}x\subset-V_{\text{vac}}\int d^{4}x\ ,$ (3.13)

where we have renormalised the ${\epsilon}$ pole, obtained by calculating the
loop integral through the usual dimensional renormalisation methods, as this
will not play a role in our discussion. In a theory with $N$ particle species,
we might expect the vacuum energy to be given by

$V_{\text{vac}}\sim\sum_{0<i\leq N}c_{i}m^{4}_{i}\ ,$ (3.14)

where $c_{i}$ are arbitrary coefficients that by naturalness we would expect
to be $\mathcal{O}(1)$ and $m_{i}$ denotes the renormalised mass of the i-th
particle species. When gravity kicks in, the vacuum energy will gravitate in
the usual way

$V_{\text{vac}}\int d^{4}x\rightarrow V_{\text{vac}}\int d^{4}x\sqrt{-g}\ ,$
(3.15)

and in terms of Feynman diagrams this can be thought of as being given by the
sum of vacuum diagrams with an arbitrary number of graviton legs

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(m) at (0, -1);
\vertex(ll) at ( -1,0) ; \vertex(lr) at ( 1,0) ; \vertex(lu) at (0,1) ;
\diagram* { (m) -- [scalar, quarter left] (ll) -- [scalar, quarter left] (lu)
-- [scalar, quarter left] (lr) -- [scalar, quarter left] (m), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ +\ \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(m) at (0, -1);
\vertex(ll) at ( -1,0) ; \vertex(lr) at ( 1,0) ; \vertex(lu) at (0,1) ;
\vertex(ol) at (-2,0) ; \diagram* { (m) -- [scalar, quarter left] (ll) --
[scalar, quarter left] (lu) -- [scalar, quarter left] (lr) -- [scalar, quarter
left] (m), (ol) -- [photon] (ll), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ +\ \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(m) at (0, -1);
\vertex(ll) at ( -1,0) ; \vertex(lr) at ( 1,0) ; \vertex(lu) at (0,1) ;
\vertex(ol) at (-2,0) ; \vertex(or) at (2,0) ; \diagram* { (m) -- [scalar,
quarter left] (ll) -- [scalar, quarter left] (lu) -- [scalar, quarter left]
(lr) -- [scalar, quarter left] (m), (ol) -- [photon] (ll), (lr) -- [photon]
(or), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ +\ ...\sim-
V_{\text{vac}}\int d^{4}x\sqrt{-g}\ .$ (3.16)

After renormalisation, the one-loop contribution to the vacuum energy for this
scalar field will again be of $\mathcal{O}(m^{4})$. If we consider the matter
sector to be given by the Standard model, we would expect that
$V_{\text{vac}}\sim m^{4}_{\text{top}}\sim\mathcal{O}(\text{TeV})^{4}$ which
is much heavier than the observed value
$V_{\text{vac}}^{\text{observed}}\sim\mathcal{O}(\text{meV})^{4}$, this
constitutes a tuning to an accuracy of 60 orders of magnitude at one-loop.
However, we know that the problems do not stop here. Analogously to the
discussion of the last section, let us imagine that we do fine-tune the
counterterms to allow the theoretical value of the cosmological constant to
match the observational one at one-loop. Then, we consider the effects of two-
loop diagrams like

$\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(m) at (-1, -1);
\vertex(ll) at ( -2,0) ; \vertex(lr) at ( 0,0) ; \vertex(lu) at (-1,1) ;
\vertex(m2) at (1, -1); \vertex(lr2) at ( 2,0) ; \vertex(lu2) at (1,1) ;
\diagram* { (m) -- [scalar, quarter left] (ll) -- [scalar, quarter left] (lu)
-- [scalar, quarter left] (lr) -- [scalar, quarter left] (m), (m2) -- [scalar,
quarter left] (lr) -- [scalar, quarter left] (lu2) -- [scalar, quarter left]
(lr2) -- [scalar, quarter left] (m2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ =\
-\lambda\left(\int{d^{4}p\over(2\pi)^{4}}{1\over
p^{2}+m^{2}+i\epsilon}\right)^{2}\propto-\lambda m^{4}$ (3.17)

which contributes to the vacuum energy of the theory at the same order as the
one-loop diagrams. This will continue to hold true for higher orders loop
diagrams. Again, we find that the cosmological constant is unstable against
radiative corrections and that, to match the observations, we must fine-tune
the effective field theory at every level in the loop expansion. The
cosmological constant problem remains one of the most important and
challenging open questions in modern physics. Below, we briefly review some of
the proposed resolutions.

#####  3.2.1 The anthropic principle

The failure to obtain a microscopic explanation for the smallness of the
cosmological constant $\Lambda$ led researchers to look for a solution in the
form of the anthropic principle. The philosophy behind the anthropic principle
is best encapsulated in the statement: “$\Lambda$ should be small enough to
allow the Universe to evolve for long enough such that there are scientists
that would ask about its smallness”. If $\Lambda$ is much larger in magnitude
than its observed value, dark energy domination would start sooner and
structure formation could be affected. For concreteness, let us imagine a
Universe with $\Lambda\sim\pm M_{\text{pl}}^{4}$. In this Universe, we would
not fret about small numbers regarding $\Lambda$ and thus the cosmological
problem would be no more. However, no galaxy formation could have happened and
no civilisation could ask questions about the cosmological constant in the
first place. On the other hand, if galaxy formation bounds the cosmological
constant to small enough values, then the cosmological constant problem can be
solved by arguing that a small cosmological constant is the only acceptable
value that would sustain intelligent life, which would then go on to worry
about its smallness.

In [95], Weinberg builds upon previous anthropic considerations111See [96] for
a review of the work on anthropics prior to [95]. to provide a bound on the
available window for the cosmological constant. The anthropic bound on the
cosmological constant is given as

$-10^{-120}M_{\text{pl}}^{4}<\Lambda<10^{-118}M_{\text{pl}}^{4}\ ,$ (3.18)

where the upper bound is obtained from demanding that dark energy domination
must not begin until sufficiently late times such that structure formation can
go through and the lower bound follows from demanding that the Universe does
not collapse during matter domination, while stars and galaxies form.
Subsequent work has tried to strengthen the bound [97, 98, 99]. Work on
anthropics has also inspired other resolutions to the cosmological constant
problem and the coincidence problem, which we discuss in the following.

Although anthropic considerations might seem anti-climactic in nature, these
are by no means trivial statements. A number of underlying assumptions about
the true nature of our Universe are at work. For anthropics to solve the
cosmological constant problem we require that our Universe can select from
different values of the cosmological constant, that at least one of such
values is in the observational range and that there exists a dynamical
mechanism to allow some regions to attain such value. Since no concrete
example of such theories exists, anthropic considerations remain as a proposed
resolution.

#####  3.2.2 Sequestering the vacuum energy

A qualitatively different approach to solving the cosmological constant
problem relies on vacuum energy sequestering [100, 101, 102, 103, 104, 105].
In this case, the radiative corrections are reabsorbed in new rigid degrees of
freedom. The theory remains locally flat while global dynamics are modified.
The mechanism itself is reminiscent of decapitation [106, 107].

Let us consider a semiclassical action of the form

$\mathcal{S}=\int d^{4}x\sqrt{-g}\left[{1\over
2}R-\lambda^{4}\mathcal{L}_{m}(\lambda^{-2}g^{\mu\nu},\Psi)-\Lambda\right]+\sigma\left({\Lambda\over\lambda^{4}\mu^{4}}\right)\
,$ (3.19)

with $\mu$ some mass scale and where the classical gravitation sector has been
coupled to a quantum matter sector $\mathcal{L}_{m}$, which we take to contain
the Standard model. The global parameter $\lambda$ is introduced by hand to
couple matter $\Psi$ to the cosmological constant $\Lambda$, which has been
promoted to a global dynamical variable. Finally, $\sigma$ is a global
interaction term, which can be seen to be the dual of a top-form.

The equations of motion for the global variables $\lambda,\Lambda$ and the
metric are

$\displaystyle\sigma^{\prime}\left({\Lambda\over\lambda^{4}\mu^{4}}\right)=\lambda^{4}\mu^{4}\int
d^{4}x\sqrt{-g}\ ,$ (3.20)
$\displaystyle\sigma^{\prime}\left({\Lambda\over\lambda^{4}\mu^{4}}\right)={\lambda^{4}\mu^{4}\over
4\Lambda}\int d^{4}x\sqrt{-g}T^{\mu}_{\mu},$ (3.21) $\displaystyle
G_{\mu\nu}=-\Lambda g_{\mu\nu}+T_{\mu\nu}\ ,$ (3.22)

where $T_{\mu\nu}$ is the energy momentum tensor for the matter sector with
respect to the metric $g_{\mu\nu}$. The last equation is just Einstein’s
equations, the other two equations allow us to fix the cosmological constant
counterterm. By combining the first two equations of motion we find

$\Lambda={1\over 4}{\int d^{4}x\sqrt{-g}T^{\mu}_{\mu}\over\int
d^{4}x\sqrt{-g}}:={1\over 4}\langle T^{\mu}_{\mu}\rangle\ ,$ (3.23)

where $\langle T^{\mu}_{\mu}\rangle$ is understood as a spacetime average.
Furthermore, we write the energy-momentum tensor as
$T_{\mu\nu}=-V_{\text{vac}}\,g_{\mu\nu}+\tau_{\mu\nu}$ where $\tau_{\mu\nu}$
describes local excitations and $V_{\text{vac}}$ is the vacuum energy coming
from Standard model loops. Plugging this expansion for the energy-momentum
tensor and Eq. (3.23) into Eq. (3.22) we find

$G_{\mu\nu}=\tau_{\mu\nu}-{1\over 4}g_{\mu\nu}\langle\tau_{\mu\nu}\rangle\ .$
(3.24)

meaning that the radiative corrections due to Standard model loops always drop
out from the effective field theory. This holds true at any order in loop
expansion due to the diffeomorphism invariance of the action and the fact that
matter couples universally to $\lambda$ through the metric $g_{\mu\nu}$,
ensuring that at any order in loops the vacuum energy contribution scales like
$\lambda^{4}$.

Sequestering resolves the cosmological constant problem at an effective field
theory level, but the action has to take a very special form. Without a better
understanding of why the effective field theory would have to take that
particular form or what kind of UV embedding would yield a low-energy
effective action of this kind, we cannot justify sequestering as the one
panacea solving all our radiative problems.

#####  3.2.3 Supersymmetry

Another possible resolution could be supersymmetry. After all, supersymmetry
came to the rescue when we considered the hierarchy problem for the Standard
model. The problem here is that a de Sitter Universe is necessarily non-
supersymmetric222We note that supersymmetry may be realised non-linearly on de
Sitter space whenever the symmetry is broken spontaneously, as in [108, 109,
110, 111, 112]., meaning that supersymmetry must be (at least) softly broken
below some scale $E_{\text{SUSY}}$. In this case, the bosonic-fermionic
superpartners have a split in their mass spectrum that goes like

$m_{B}^{2}-m_{F}^{2}\simeq g_{s}^{2}E_{\text{SUSY}}^{2}\ ,$ (3.25)

and this ruins the cancellation for the vacuum loop diagrams giving raise to
vacuum energy contribution of $\mathcal{O}(g_{s}^{2}E_{\text{SUSY}}^{2})$.
Since we have not observed supersymmetry in the LHC, we can at most hope to
explain why the cosmological constant does not have an $M_{\text{pl}}^{4}$
energy scale, assuming $E_{\text{SUSY}}<M_{\text{pl}}$, but the coupling
problems to the standard model sector remain. It should be noted that one
could try to engineer a situation where supersymmetry is broken at two
different scales $E_{\text{brane}}\gg E_{\text{bulk}}$, so that the matter
sector living in a D3-brane has $E_{\text{SUSY}}=E_{\text{brane}}$ and gravity
propagating in the bulk has $E_{\text{SUSY}}=E_{\text{bulk}}$. This type of
scenario was already considered in braneworld constructions where the extra
dimensions had infinite volume [113, 114] and later in supersymmetric large
extra dimensions brane models [115, 116, 117]. It remains unclear how to
achieve this from a UV perspective. The problem stems from demanding a
(nearly) flat four-dimensional solution in the brane. This requires the brane
couplings to preserve bulk scale invariance and leads to a fine-tuning in the
parameters of the model to achieve the observed curvature scales [118, 119].

#####  3.2.4 The flux landscape

To some extend motivated by the anthropic prospects discussed previously,
Bousso and Polchinski [120] proposed how we may achieve a small enough
cosmological constant through four-form quantisation. Indeed, consider the
Einstein-Hilbert action minimally coupled to a four-form

$\mathcal{S}=\int d^{4}x\sqrt{-g}\left({1\over 2}R-\Lambda-{Z\over
48}F_{4}^{2}\right)\ ,$ (3.26)

where $Z$ is a normalisation constant. The four-form equations of motion fix
$F_{4}=c\epsilon_{4}$, where $\epsilon_{4}$ is the four-dimensional Levi-
Civita symbol and $c$ is a constant related to the Dirac quantisation
condition of the four-form, as we will see below. The effective cosmological
constant is the given by [120]

$\Lambda_{\text{eff}}=\Lambda-{Z\over 48}F_{4}^{2}=\Lambda+{Zc^{2}\over 2}\ .$
(3.27)

If $c$ could take any value, the bare cosmological constant $\Lambda$ could be
cancelled to any degree of accuracy. However, in string theory both the four-
form and its dual are quantised and for the dual of the four-form the
quantisation condition reads

$\star F_{4}=F_{0}={en\over Z}\ ,\quad n\in\mathbb{Z}\ ,$ (3.28)

where $e$ is the charge of the membrane coupled to the four-form. In turn,
this fixes our arbitrary constant $c$ to take values in the integer set, in
units of ${e\over Z}$. For the effective cosmological constant to be
efficiently cancelled by the four-form contribution, we require that

$\Lambda+{e^{2}n^{2}\over 2Z}\sim 0\qquad
n\sim{\sqrt{2Z\left|\Lambda\right|}\over e}\ ,$ (3.29)

and that the spacing between different effective cosmological constants,
associated to different values of $n$, is below the observed value
$\delta\Lambda\sim{e^{2}n\over Z}<10^{-120}$, so that a number of effective
cosmological constants can fit observations. All together, we find

$\sqrt{\left|\Lambda\right|\over Z}e\lesssim 10^{-120}\ .$ (3.30)

The problem is that the bare cosmological constant $\Lambda$ will see Standard
model loops, meaning that $\Lambda\sim\mathcal{O}(\text{TeV})^{4}$. This
implies that $e\lesssim\sqrt{Z}10^{-108}$, for the action in Eq. (3.26) we
would expect $Z\sim\mathcal{O}(1)$ which would require
$e\lesssim\mathcal{O}(10^{-108})$. One four-form is not enough since this is
just exchanging a fine-tuning on the cosmological constant by a slightly
smaller fine-tuning on the value of the membrane charge $e$.

However, in typical compactifications, we expect a large number of these four-
forms to be present. In [120], it was also shown that if we have some
$\mathcal{O}(100)$ of these four-forms arising from a string compactification,
then the individual charges only need to be of $\mathcal{O}(1)$ even if
$\Lambda$ couples to the Planck scale. Then, by anthropic considerations, our
Universe is one such that the cosmological constant picked the right value to
allow for structure formation. Overall, this argument is based on the idea
that flux compactifications generate a very large number of vacua such that
even exponentially small cosmological constant may exists in the landscape of
theories.

In supergravity models of our Universe, the smallness of the cosmological
constant is often related to a balance between an uplift contribution and a
tuned tree-level superpotential $W_{0}$. In [121, 122], it was argued that
statistics of classical flux vacua do indeed agree with the intuition that an
exponentially-tuned $W_{0}$ is achievable in the landscape such that the
cosmological constant can be tuned to a similar degree. For example, in the
case of KKLT models which we will discuss in Chap. 7, [123, 124] provide
realisations of AdS4 vacua with exponentially small $W_{0}$, fitting
observations in magnitude but not in sign as an uplift is still required to
obtain dS4. The idea of using the flux landscape to solve the cosmological
constant problem remains attractive but no concrete ten-dimensional
realisation of a dS4 vacua has been achieved yet.

####  3.3 Coincidence problem

A related but qualitatively distinct puzzle to the cosmological constant
problem is the cosmological coincidence problem [125]. This is usually
presented as a simple question: why now? Why do we happen to live at a time
when the energy density of matter and dark energy are roughly comparable?

Let us unpack the problem in a little more detail. The total age of the
universe is estimated to be around 13.7 Gyrs [126]. The early phase of
radiation domination represents only a small fraction of that history, giving
way to matter after around 51 Kyrs, at a redshift of $z_{eq}\approx 3400$. The
transition to dark energy domination occurred much later, after around 3.5
Gyrs, at a redshift of $z_{de}\approx 0.5$. This represents roughly a quarter
of the universe’s current lifetime [127]. This means that the universe has
been matter dominated for a significant fraction of its history, so much so
that its density today is still comparable to that of dark energy.

Why is this a problem? The issue is the rate at which dark energy dilutes in
comparison to matter, if indeed it dilutes at all. If dark energy is a
cosmological constant, as in the standard $\Lambda$CDM cosmology, its energy
density remains constant while the energy density of matter falls off
exponentially quickly once we enter the accelerating phase. If we take the