also exist, see [156, 157, 158] and a general tension for obtaining controlled
quintessence in String Theory was pointed in [34]. A review on quintessence
attempts from string theory can be found in [32].

The claim that there is evidence for a landscape of de Sitter vacua in string
theory can only be justified when at least all of these criticisms are
carefully addressed. In the next section we explain how there are also reasons
to be more optimistic about dS scenarios because there exist new ideas and
some of the criticism above has been refuted, especially the criticism
regarding singular fluxes near anti-branes.

###  5.4 New arguments in favor of stringy de Sitter?

De Sitter model building in String Theory is an activity that started roughly
in the beginning of 2000 and never stopped since. It is fair to say that the
creation of models, or better said, suggestions for constructions, outpaces
the scrutiny. In other words, the amount of energy spend on suggesting how de
Sitter can be constructed outweighs the energy spend on cleaning up the
details who potentially render the suggestions inconsistent. Luckily with the
advent of the Swampland Program much more activity exists now in scrutinizing
existing de Sitter scenarios. Nonetheless it is important to come up with new
ideas as well. In what follows we present both the results of this scrutiny,
which partially refutes the previous criticism, and we present brief pointers
to the literature on new suggestions for dS model building since 2018, whereas
a classification of de Sitter attempts before 2018 has already been reviewed
in [24], see also [32].

#### No singular fluxes near anti-D3 branes in dS uplifts.

The first reference to argue that the singular flux-clumping around anti-D3
branes could be absent is [159] (see also [160]). The assumption in that
reference was that within supergravity the singularity seems unavoidable, but
that stringy corrections resolve it. This was argued on the basis of _brane
effective field theory_. The resulting corrected flux clumping would then not
be large enough to trigger direct brane-flux decay. This argument is very
sensible but some details have been criticised in [161, 162]. Nonetheless
references [163, 164] revised the supergravity arguments that lead to the
singular fluxes and found that exactly the ingredients used to derive brane-
flux decay provide the required mechanism to avoid the singularity _within
supergravity_.

The picture is not too complicated to understand intuitively; according to
Kachru, Pearson and Verlinde [127] the details of brane-flux decay proceed via
the anti-D3 brane “puffing” into a spherical 5 brane that behaves effectively
as the anti-D3 brane from some distance. The only effect is that the anti-D3
brane is smeared out a bit and stops being pointlike inside the extra
dimensions. Hence the manner in which that anti-brane attracts flux becomes
diluted. Using fully backreacted supergravity equations of motion references
[163, 164] found that this “polarisation” into a 5-brane seems a perfect way
around the singularity theorems derived in [130, 131].

This suggested resolution has then been established in much more details in
[165, 166, 167, 168, 169]. Even more, extra arguments for meta-stability were
found by showing that the anti-D3 or better said, the spherical 5-brane setup,
can be heated up with finite energy without destabilising the system. This is
what one expects from gapped states: a finite amount of energy could be added
without destabilising the system. The interplay between the singularity
(regularity) theorems of [163, 164] and the addition of temperature has lead
to a remarkably simple picture which is sketched informally in Figure 7.

Figure 7: _The left picture represents a spherical D5 brane at finite
temperature. The thickness of the doughnut is set by the amount of heat that
is added to the spherical 5-brane. If it gets heated up so much that the
doughnut becomes a Berliner, the system decays directly through brane-flux
decay._

The spherical 5-brane can be represented as a thin ring and when heated up it
becomes a fatter ring, that looks like a doughnut. The hole inside the
doughnut is a necessary requirement to evade the singularities since the
singularity theorems directly probe the topology of the brane setup. If the
doughnut is heated up too much such that the finite temperature horizon grows
to overtake the hole inside turning the doughnut into a Berliner, then the
singularities appear again and create an instability. This was confirmed in an
approximation beyond the probe approximation, known as the _blackfold
approach_ , see [165, 166, 168, 169].191919Another potential mechanism to
resolve the singular fluxes was suggested in [170], but shown to lead to an
unstable state in [165]. Hence the blackfold approach is a good test for the
meta-stability of the anti-brane state since it explicitly shows how it can be
heated up until a maximal temperature. In the particular case of anti-D6 brane
toy models these ideas can be verified completely explicit using fully
backreacted anti-brane solutions [167].

#### No singular uplifts in compact geometries?

A potential resolution to the singular bulk problem argued in [134] is
presented in [171]. The suggested resolution however relies on various
approximations, and if it takes place, it seems the corrections to the bulk
geometry are significant and it is unclear whether the KKLT effective field
theory is safe from these corrections.

#### No collapsing throat geometries?

It has been argued in a recent paper that the worry of a collapsing throat is
not necessary and that the instability is absent [172]. General remarks about
how warped down moduli in a throat alter the EFT can be found in [173, 174].

#### No flattening effects for uplifts?

The computation done in [146] that revealed the so-named flattening effects
was shrouded in some confusion due to a singular contribution of divergent
fluxes sourced by the gaugino condensate (and some sign issues [175]). How to
deal with such a divergent flux was explained in [176, 177] (see also [178,
179, 180]) and then the computation of the anti-brane uplift was revised in
[181, 182, 133]; whereas references [182, 133] came to the conclusion that the
flattening effect was gone, reference [181] did find a significant difference
between the 10d and the 4d approach, making it unclear the 4d EFT can be
trusted. The difference between the references arises due to a difference in
the way the 10d Einstein equation was computed. Reference [181] takes the 10d
action and varies with respect to the 10d metric to get the semi-classical
Einstein equation:

$G_{\mu\nu}=\langle T_{\mu\nu}\rangle\,,$ (5.25)

where afterwards the quantum vevs of the fields is filled in to obtain
$\langle T_{\mu\nu}\rangle$. The vevs of the fields is information obtained
from the 4d EFT since gaugino condensation is, by definition, an IR effect. On
the other hand, references [182, 133] notice that consistency with the 4d
SUGRA equations is only obtained if _in the 10d action_ the gaugino vevs
dependence on the volume is filled in, after which they take the variation of
the ‘10d action’ to arrive at the semi-classical Einstein equation (5.25).
Since the gaugino vev is metric dependent changing the order of the procedure
does change the results. Whereas [181] remains 10d covariant, references [182,
133] break this. Claiming that consistency with the 4d EFT is an argument for
the second procedure of [182, 133] to be correct is in some sense circular
reasoning since the whole idea was to see what can be learned from a 10d
approach. On the other hand, it is not strange that 10d covariance needs to be
sacrificed if one insists on writing an inherent large distance effect in
small distance variables. In other words: the whole 10d method is potentially
a moot point since gaugino condensation does not exist at scales as small as
the KK scale, so there is no reason one tries to understand how it backreacts
in 10d language.

#### Casimir energies and hyperbolic spaces:

Instead of using supersymmetric geometries, with their associated supergravity
effective field theories where dS is typically build from a combination of
quantum effects (fox example gaugino condensation) with classical
supersymmetry breaking sources (like anti-branes), one can simply break SUSY
at the KK scale. Then naively all bets are off since there is no supersymmetry
to protect the corrections to the effective field theory after
compactification. This means one has to be very careful. Using the fact that
compact hyperbolic geometries can have complicated circles inside them, non-
trivial Casimir energies arise from loop effects. Together with the classical
positive energy from the negative internal curvature, it seems de Sitter vacua
arise naturally [183]. In the next section we will discuss Casimir energies in
simple torus compactifications and how it can give rise to AdS vacua with
small cosmological constants.

#### Dark bubble scenario:

A very fresh out-of-the-box construction of de Sitter vacua was suggested in
[184] (see also [185], [186]). The idea is to take the Swampland tensions
seriously and assume this implies that standard compactifications tend to get
in trouble when you insist on de Sitter in the lower-dimension because it is
difficult to keep the extra dimensions stable while obtaining positive vacuum
energy. Instead Banerjee and collaborators contemplate _brane world scenarios_
which are vastly different from the standard Randall–Sundrum [187] or
Karch–Randall [188] kind. The idea is in fact based on a Swampland conjecture
known as the _non-SUSY AdS conjecture_ [189] and states that any non-
supersymmetric anti-de Sitter vacuum supported by fluxes must be unstable. In
absence of perturbative instabilities such decays tend to be of the Brown-
Teitelboim type [190]. This means that a spherical domain wall of sub-critical
tension forms and inside this bubble there is a vacuum with lower energy. The
insight of [184] is that matter can be localised on such a wall for instance
in the form of end-points of strings stretching from the boundary. Such matter
would obey 4d Einstein equations. A pictorial representation of this scenario
is presented in Figure 8.

Figure 8: _The left picture (a) represents the 5-dimensional description of
the 4D brane bubble word. The inside is the true stable vacuum and the outside
is the meta-stable AdS vacuum. The bubble wall separates the two. The right
picture (b) plots the 5D warp factor and compares it with the Randall-Sundrum
scenario, which makes clear that the dark bubble mechanism is vastly
different._

Even more striking is that the wall itself is expanding inside the bulk
spacetime and this expansion is exactly like a de Sitter expansion. Hence such
brane world scenarios are unavoidably de Sitter at late times (but also at
early times! [191]). Attempts to embed this scenario in actual string theory
examples are [192, 191]. This scenario has been criticised in [193], but then
it was explained in [186] how the criticism is flawed. Interestingly the model
is so concrete that even basic questions in quantum cosmology can be
addressed. One such a question concerns the quantum state of the universe.
There are many options but two canonical choices are the Vilenkin state [194]
or the Hartle–Hawking one [195]. The dark bubble scenario of [184] uniquely
selects the Vilenkin wave function [196].

#### Liberal singularities:

Allowing singularities of the orientifold kind that differ in their sub-
leading behavior seems a very simple manner to get de Sitter solutions at the
classical level which completely go around the no-go results constructed in
this review [197, 198]. The reason is that our no-go statements feature
integrated equations with certain assumptions of how the singularities
integrate. These assumptions were dropped in [197, 198], but criticised to be
unphysical in [199] or claimed to lead to instabilities as pointed out in
[200].

#### Other approaches:

Other interesting recent attempts to construct de Sitter which we do not
discuss due to lack of spacetime are for instance [201, 202, 203, 204, 205,
206, 207, 208, 209, 210, 211, 212, 213].

###  5.5 Ads with scale separation from supergravity/strings?

Holographic backgrounds known within String Theory are of the kind
AdS${}_{n}\times X_{m}$ which have a dual description in terms of a CFTn-1. An
important question that arises here is whether it is possible to decouple the
mass scale associated to the internal space (the mass of KK modes) from the
energy scale associated to the cosmological constant? In other words, can we
ignore the $X_{m}$? Or even other words [214, 215]; _how many space dimensions
emerge from a CFT in $d$ dimensions?_. Only one dimension (which is the usual
popular explanation) or $10-d$ dimensions (which is what happens in all known
cases)? While this question seems to be confined to the realms of the
holographic correspondence and not bear any consequence for phenomenological
model building this is incorrect. In several scenarios (like the KKLT
construction for instance) where a landscape of $10^{\text{some number}}$ dS
vacua is claimed to exist, always necessitates AdS vacua for which

$\displaystyle\frac{L^{2}_{KK}}{L^{2}_{\text{AdS}}}\ll 1\,.$ (5.26)

The reason for this should be evident from the previous section: most dS vacua
are obtained from uplifting AdS vacua and these AdS vacua are subsequently
obtained in some hybrid 10d string theory-4d EFT formalism. Such a hybrid
formalism can only make sense when the machinery of 4d EFT is applicable, and
that requires a separation of scales.

We now analyse whether or not AdS vacua can be scale separated using the
higher-dimensional equations of motion, in the simple setting of 11d
supergravity. After that we will analyse the same question in 10d supergravity
using our beloved $\rho,\tau$ scalings, which we argued in section 4.2 are
equivalent to solving certain integrated 10d equations of motion.

#### The supergravity perspective in 11d.

Here we follow arguments that first appeared in [18]. The starting point is to
use 11d supergravity whose action is

$\displaystyle S=\int
d^{11}x\sqrt{-g}\left[R_{11}-\frac{1}{2}\frac{1}{4!}F_{4}^{2}\right]+\dots\,,$
(5.27)

where the dots include the Chern–Simons terms that will be unimportant in the
following. We consider compactifications with fluxes down to 4d, and if we
assume there is no warping for simplicity we can write the traced Einstein
equations as follows:

$\displaystyle R_{4}$
$\displaystyle=-\frac{4}{3}\frac{F_{4}^{2}}{4!}-\frac{8}{3}\frac{F_{7}^{2}}{7!}\,,$
(5.28) $\displaystyle R_{7}$
$\displaystyle=\frac{5}{3}\frac{F_{4}^{2}}{4!}+\frac{7}{3}\frac{F_{7}^{2}}{7!}\,.$
(5.29)

We keep $F_{7}$ so that all fluxes can be written as “magnetic”. By that we
mean that all indices of $F_{4}$ and $F_{7}$ are inside the internal space.
This way of writing simplifies things since contractions like $F_{4}^{2}$ and
$F_{7}^{2}$ are only among spacelike indices and so the squares are strictly
positive. The above two equations then show that necessarily $R_{4}<0$ and
$R_{7}>0$. The fact that $R_{4}$ is negative is of course a consequence of the
Maldacena–Nuñez no-go theorem [91] and the fact that $R_{7}$ is positive is
part of what can be called the “Douglas-Kallosh” no-go theorem [89].

######  Exercise 5.6.

Verify equations (5.28) and (5.29).

For the sake of understanding scale separation it is instructive to consider
the ratio of integrated curvatures202020This kind of reasoning can be found in
older works like [216].

$\displaystyle\left|\frac{\int R_{7}}{\int
R_{4}}\right|=\frac{5\int\frac{F_{4}^{2}}{4!}+7\int\frac{F_{7}^{2}}{7!}}{4\int\frac{F_{4}^{2}}{4!}+8\int\frac{F_{7}^{2}}{7!}}\leq\frac{5}{4}\,.$
(5.30)

Notice that all integrals are done over 11d. We can define the two curvature
radii as follows

$\displaystyle L_{R}^{-2}$
$\displaystyle=\frac{1}{\text{vol}_{7}}\int_{\mathcal{M}_{7}}R_{7}\,,$ (5.31)
$\displaystyle L_{\text{AdS}}^{-2}$
$\displaystyle=\frac{1}{\text{vol}_{4}}\int_{\mathcal{M}_{4}}R_{4}\,.$ (5.32)

In defining $L_{\text{AdS}}$ one needs to take some care as the volume of
$\text{AdS}_{4}$ is clearly infinite so some regularization is needed. But
using these newly defined length scales we have that (5.30) can be rewritten
as

$\displaystyle\frac{L_{\text{AdS}}^{2}}{L_{R}^{2}}\leq\frac{5}{4}\,.$ (5.33)

Similar bounds can be derived in 10d whenever the ingredients are fluxes and
positive tension branes [18]. This means that, under these conditions, it is
not possible to make the AdS curvature length bigger than the curvature length
of the internal manifold. One can wonder to what extend this means there is no
scale separation between $L_{KK}$ and $L_{\text{AdS}}$? In some cases it is
possible to relate $L_{R}^{2}$ to $L_{KK}^{2}$, however it is not clear
whether this holds in general [21, 22, 217, 218, 64]. Note that for negatively
curved geometries it is rather simple to achieve such a separation, see [219]
or the appendix of [218], but the Einstein equations show that negatively
curved internal geometries are not possible within our assumptions (classical
supergravity with no negative tension sources).

######  Exercise 5.7.

Consider the 4d compact space given by the product of two 2-spheres. Call the
radii of these two spheres $R_{1}$ and $R_{2}$. Can one, at fixed total
volume, tune the curvature scale to be very large or very small?

So for all manifolds for which $L_{KK}$ can not be tuned small at fixed
$L_{R}$ we find a generalisation of the Maldacena–Nuñez theorem [91, 92]:

_There are no de Sitter nor Minkowski solutions with smooth internal static
compact manifolds including fluxes and positive tension sources. Furthermore,
for internal geometries with the property that $L_{KK}$ cannot be tuned small
at fixed $L_{R}$ there will not be AdS vacua that are scale separated. This
means we can also exclude AdS vacua with a small cosmological constant, where
small means small with respect to the KK scale._

There are possible ways out, like the introduction of orientifold planes [88]
(this introduces negative tension sources), or use of quantum corrections [74,
76]. Let us analyse how orientifolds can be a way out and for that we need to
be in 10 dimensions. Once we have done it we will be remarkably lead to
conclude that perhaps Einstein geometries with positive curvature exist for
which one can take $L_{KK}$ small at fixed $L_{R}$ exist.

#### The effective potential perspective from 10d

Let us try to formulate the problem- of scale separation within the framework
employed in these lectures: take the following metric 10d metric in string
frame

$\displaystyle
ds^{2}_{10}=\tau_{0}^{2}\tau^{-2}ds^{2}_{D}+\rho\,ds^{2}_{10-D}\,.$ (5.34)

Here $\tau_{0}$ is just the vev of $\tau$. Note that we need to require that
the $D$-dimensional metric is in Einstein frame, which is equivalent to the
condition (4.56)

After dimensional reduction we get the following action in $D$-dimensions

$\displaystyle S_{D}=\int
d^{D}x\sqrt{-g_{D}}\left[\tau_{0}^{D-2}R_{D}-\tau_{0}^{D}V+\dots\right]\,.$
(5.35)

With this action we can readily identify the $D$-dimensional Planck mass as
$M_{P}=\tau_{0}$. We wish to compare the length scale associated with the
potential at the minimum (that is the cosmological constant in $D$-dimensions)
to the mass scale of the lightest modes associated to the internal space.
While in principle there can be several modes whose mass scale needs to be
checked we will take as a proxy the mass scale of KK-modes. We can extract the
KK-scale to be $L^{2}_{KK}\sim\rho_{0}$ by simple estimation of the volume of
the internal manifold. Note that when fluxes and other ingredients are
introduced in compactifications this may distort the estimation of mass scale
of KK-modes but it is not possible to have a general estimate of this. Then,
we find from the $D$-dimensional Einstein equations that

$\displaystyle R_{D}=\frac{D}{D-2}M_{P}^{2}\,V\,,$ (5.36)

which implies that $L_{\text{AdS}}^{-2}=M_{P}^{2}\,V$. Therefore the ratio of
the scales entering in the problem is

$\displaystyle\frac{L^{2}_{KK}}{L^{2}_{\text{AdS}}}\sim\rho_{0}\,\tau_{0}^{2}\,V\,.$
(5.37)

The big looming question is whether this can be made arbitrarily small.

To try and get some insight we look at the setup studied in the previous
subsection; compactifications of type IIA on a Calabi–Yau threefold (that is
$D=4$ and $R_{6}=0$) with $F_{0}$, $H_{3}$, and $F_{4}$ fluxes as well as
O6-planes. There are two relevant Bianchi identities to be considered in this
case

$\displaystyle dF_{2}$ $\displaystyle=F_{0}H_{3}+\delta(\text{O6})\,,$ (5.38)
$\displaystyle dF_{4}$ $\displaystyle=F_{2}\wedge H_{3}+\delta(\text{O4})\,.$
(5.39)

Notice that existence of a solution for the first one bounds the amount of
$F_{0}$ and $H_{3}$ fluxes that can be added to the
compactification.212121Recall that the number of O6-planes is constrained by
topology and is not a free parameter. On the other hand the second Bianchi
identity is automatically satisfied since we have no O4/D4 sources and no
$F_{2}$ flux222222The generalisations with $F_{2}$ flux are such that
$F_{2}\wedge H_{3}$ vanishes.. Given that $F_{4}$ does not appear in any
Bianchi identity the amount of $F_{4}$-flux that can be added to the
compactification is unbounded. The scalar potential for this compactification
is

$\displaystyle
V=U_{H}\rho^{-3}\tau^{-2}+U_{0}\tau^{-4}\rho^{3}+U_{4}\tau^{-4}\rho^{-1}+U_{\text{O6}}\tau^{-3}\,.$
(5.40)

The minimization conditions are

$\displaystyle\rho\,\partial_{\rho}V=0\,,$
$\displaystyle\qquad\Longrightarrow\qquad-3V_{H}+3V_{0}-V_{4}=0\,,$ (5.41)
$\displaystyle\tau\,\partial_{\tau}V=0\,,$
$\displaystyle\qquad\Longrightarrow\qquad-2V_{H}-4(V_{0}+V_{4})-3V_{\text{O6}}=0\,.$
(5.42)

Using these conditions and replacing them in the scalar potential one gets
that at the minimum the scalar potential is

$\displaystyle V$ $\displaystyle=V_{H}+V_{0}+V_{4}+V_{\text{O6}}=$ (5.43)
$\displaystyle=-\frac{4}{9}V_{4}<0\,.$ (5.44)

######  Exercise 5.8.

Verify the above equations. Verify that without O6 planes there is no
solution.

This means that AdS solutions are perhaps possible if $F_{4}\neq 0$. We define
the flux quantum to be

$N=\int F_{4}\,,$

(ignoring factors of $\pi$) and consider the limit $N\rightarrow\infty$.
However, even if $N$ is not bounded by tadpole cancellation, it is still
necessary to find a solution of the equations of motion. We therefore make
sure that all terms in the full scalar potential balance each other, and we
will parametrize $\tau\sim N^{A}$ and $\rho\sim N^{B}$ with $A,\,B$ to be
determined. The scalar potential then scales as

$\displaystyle V\sim
U_{H}\,N^{-2A-3B}+U_{0}\,N^{-4A+3B}+N^{2-4A-B}+U_{\text{O6}}\,N^{-3A}\,.$
(5.45)

If we take $B=1/2$ and $A=3/2$ _all components scale in the same fashion with
$N$_. This is non-trivial since the system is over-constrained. Let us analyze
the consequences of this

  * •

The first observation is that $V\sim N^{-3A}\rightarrow 0$. With this we find
that $\rho_{0}\tau_{0}^{2}V\sim N^{-1}\rightarrow 0$. Therefore this
“solution” achieves scale separation;

  * •

The string coupling constant in the vacuum scales as
$e^{-2\phi_{0}}=\tau_{0}^{2}\rho_{0}^{-3}=N^{3/2}\rightarrow\infty$. Therefore
in the large $N$ limit we achieve a limit of weak coupling;

  * •

Given that $\rho_{0}\sim N^{\frac{1}{2}}\rightarrow\infty$ the volume becomes
infinite in the large $N$ limit.

######  Exercise 5.9.

Is a similar magic possible in a IIB set-up? Consider for instance IIB with
O5, O7 planes. What do you find? [Hint: It is not obvious, see for instance
[220]. To date no classical IIB solutions with scale separation are known. The
attempts in [220, 221] turn out problematic after all [218]]. See also [222].

######  Exercise 5.10.

Verify that reductions of massive IIA on Ricci-flat 7d manifolds with 06
planes can also be scale separated upon sending $F_{4}$ flux to infinity. (See
[223] for details (and [224, 225]).

It seems almost too good to be true: we seem to achieve a regime with scale
separation, weak string coupling and large volume. We have shown that the
equations of motion with respect to $\rho$ and $\tau$ are satisfied, however
the full set of equations of motion is larger. Luckily, in [88] it was shown
that a full solution does exist within a certain approximation; the derivation
in [88] (see also [226, 227, 228]) relied on the formalism of $\mathcal{N}=1$
supergravity in $D=4$ [229] instead of the 10d equations of motion. The 4d
equations show that there exists both supersymmetric and non-supersymmetric
AdS${}_{4}\times\text{CY}_{3}$ vacua with sometimes all moduli stabilized
[230, 231]. It was later shown that these solutions solve 10d equations of
motion with _smeared sources_ [232, 233, 87]. This means that the
$\delta$-sources in the Bianchi identities are replaced with finite forms,
just as we did for the Minkowski no-scale vacua [73]. In our case the
replacement leads to the new Bianchi identity

$\displaystyle dF_{2}=F_{0}\,H_{3}+Q\,\text{Re}(\Omega_{3})\,.$ (5.46)

where now the delta source is replaced by a finite regular form
$\text{Re}(\Omega_{3})$, which is usually interpreted as consisting of O6
planes intersecting in four different directions, see Appendix C of [234]:

$x_{0}$ | $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$ | $x_{5}$ | $x_{6}$ | $x_{7}$ | $x_{8}$ | $x_{9}$
---|---|---|---|---|---|---|---|---|---
$\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $-$
$\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $\times$ | $\times$ | $-$
$\times$ | $\times$ | $\times$ | $\times$ | $-$ | $\times$ | $-$ | $-$ | $\times$ | $\times$
$\times$ | $\times$ | $\times$ | $\times$ | $-$ | $-$ | $\times$ | $\times$ | $-$ | $\times$

The crosses describe directions wrapped by an O6 plane and the minus signs are
orthogonal. We see that the O6 planes intersect non-trivially inside the
internal manifold.

Smearing is actually _the_ source of criticism for these solutions [235, 236].
From the point of view of the equations of motion smearing is equivalent to
solving the integrated equations of motion as explained in section 4.2, and
corresponds therefore to the standard coarse-graining procedure of dimensional
reduction, which constructs an EFT valid at length scales below the KK scale.

A detailed (pedagogical) example of how to move between localized and smeared
sources appeared for instance in [237] (and the appendix of [238]) where the
background $\mathbb{R}^{1,6}\times\mathbf{T}^{3}/\mathbb{Z}_{2}$ was
considered in type IIA. There is $H_{3}$-flux on the
$\mathbf{T}^{3}/\mathbb{Z}_{2}$ factor as well as $F_{0}$, and some O6-planes
wrapping the Minkowski space factor. The upshot of that analysis is that the
smearing approximation becomes better and better as $g_{s}$ becomes smaller
and the volume becomes larger. For example the $\phi$ profile is depicted in
figure 9 and we notice that the regions where there is a deviation from the
smeared solution actually shrink to zero. This seems to suggest that smearing
might not be a problem in the context of the solutions of [88], however this
does not settle the question of its validity altogether.

Figure 9:  _A plot of $e^{\phi}/g_{s}$ inside the extra dimensions, taken from
[237]. Here $g_{s}$ is the zero mode of $e^{\phi}$, ie the vev that one
computes in the lower-dimensional theory. As $g_{s}/L$ becomes smaller the
function $e^{\phi}/g_{s}$ approximates unity quickly away from the O6-plane
singularities. ($L=L_{KK}$ is the size of the extra dimensions)._

One worrisome point is the fact that there is no localized picture for the
intersection of O6-planes. If we added the backreaction on the geometry of the
orientifold planes the Calabi–Yau threefold becomes a manifold with
$SU(3)\times SU(3)$ structure [239, 240, 238] for which backreaction is
understood at subleading order in a $1/N$ expansion. See also [241] for a
similar analysis applied to AdS3 vacua.

To summarize, we learned that O-planes are useful for obtaining scale
separated vacua, but there is only an understanding of their backreaction in
the smeared limit, and so there remains some doubt regarding the consistency
of scale separated vacua obtained that way.

#### Back to 11d and a mathematical conjecture.

In the beginning of the previous paragraph we deduced that simple Freund-Rubin
vacua could be scale separated on the condition we could shrink the KK scale
at fixed curvature scale and we mentioned no such geometries are known. To put
this in a bit more detail: we do not know of positively curved Einstein
geometries for which the first eigenvalue of the (generalised) Laplace
operator can be tuned arbitrarily high. Even more, recently Collins, Jafferis,
Vafa, Xu and Yau conjectured no such geometries can exist [64], at least for
the particular case of Einstein geometries obeying

$R_{ab}=\frac{1}{L_{R}^{2}}g_{ab}\,.$ (5.47)

One could be tempted to believe that orbifolding a sphere would do the job.
Consider for instance $S^{n}/\mathbb{Z}_{k}$ at large $k$. The volume goes
down at fixed curvature, but for this case the volume is not a good measure of
the KK scale since the only thing the orbifolding action does is killing
degeneracies in the eigenvalue spectrum, but it does not remove the
eigenvalues. It is remarkable how the simple computations we have done so far
brings us to the forefront of mathematics as well.

Interestingly, there is a string theory suggestion (no proof) that perhaps
this mathematical conjecture is false and even Freund–Rubin vacua can be scale
separated. Let us only briefly summarize this as this goes a bit beyond the
level of the lecture notes. The scale separated AdS vacuum we discussed before
in the limit of smeared sources relied on IIA string theory on a Calabi-Yau
with O6-planes, Romans mass $F_{0}$, $H_{3}$ flux and unbounded $F_{4}$ flux.
Interestingly from double T-duality [234] one can find new classes of
solutions in IIA string theory _without_ Romans mass at the expense of working
with generalised Calabi Yau spaces [218]. The corresponding AdS solution can
then be lifted to 11-dimensional supergravity (Romans mass would be an
obstacle to such a lift) and in this lift the O6-sources become pure geometry
and all the rest becomes only $F_{7}$ flux [218]. This suggests that there
exist 7-dimensional Einstein manifolds for which the KK scale can be shrunk to
zero at fixed curvature scale in violation of the mathematical conjecture of
[64]. Of course, the catch is in the word _suggests_ since no full 10d
description of the IIA solution of [218] is given due to the O6-planes being
smeared.

#### Scale separation from quantum corrections?

The discussion above about 11 supergravity only considered classical sources
in the compactification; curved extra dimensions and $p$-form fluxes. Yet, in
some cases it is claimed to be possible to find compactifications with scale
separation using quantum corrections, KKLT [74] and LVS [76] being prime
examples. We will follow the example of [217] where the authors are able to
build a non-supersymmetric AdS4 solution of M-theory with arbitrary scale
separation: the ingredient they use is Casimir energies in the internal space.
The example is a compactification of M-theory on a 7-dimensional torus
$\mathbf{T}^{7}$ which for simplicity is taken to be isotropic, that is all
circles are chosen to have the same size. The 11d metric has the form

$\displaystyle
ds^{2}_{11}=\tau^{-2}ds^{2}_{\text{AdS}}+\rho\,ds^{2}_{\mathbf{T}^{7}}\,.$
(5.48)

The internal metric is chosen such that vol$(\mathbf{T^{7}})=(2\pi)^{7}$. In
order to compute the Casimir energy coming from fields in the internal space
one should compute a one-loop diagram for the vacuum energy with the fields
running in it. While in principle all fields need to be taken into account the
massive states are exponentially suppressed and they can be safely neglected.
Typically, one expects that due to supersymmetry the one-loop contributions of
bosons and fermions cancel each other out, however in this example we will
take anti-periodic boundary conditions for the fermions in the circles of the
internal torus. This breaks supersymmetry by lifting the zero modes of the
fermions and therefore only leaves a contribution to the vacuum energy in 4d
coming only from the bosons. While there are techniques to perform such
computations (see for example the book [242]) in this setup it suffices to
parametrize the Casimir vacuum energy as the following correction to the
M-theory action

$\displaystyle
S^{\text{Casimir}}=2|\sigma_{c}|\int_{\mathcal{M}_{11}}d^{11}x\sqrt{-g_{11}}\,\rho^{-\frac{11}{2}}\,.$
(5.49)

Here $\sigma_{c}$ is a constant that can be in principle computed explicitly.
In addition to the Casimir energy we will add some 7-form flux threading the
internal torus. The flux has to be appropriately quantized, that is

$\displaystyle\frac{1}{(2\pi\ell_{11})^{6}}\int_{\mathbf{T}^{7}}F_{7}=N_{7}\,,$
(5.50)

where $N_{7}\in\mathbb{Z}$ is an integer and $\ell_{11}$ is the 11d Planck
length. We therefore have two contributions to the energy-momentum tensor,
that is the flux and the Casimir energy. It is possible to solve the 11d
Einstein equations and find the following solution

$\displaystyle\tau^{-2}\sim
N_{7}^{\frac{22}{3}}|\sigma_{c}|^{-\frac{14}{3}}\,,$ (5.51)
$\displaystyle\rho\sim N_{7}^{\frac{4}{3}}|\sigma_{c}|^{-\frac{2}{3}}\,,$
(5.52)

where we omitted some $\mathcal{O}(1)$ coefficients as we are interested in
the scaling in $N_{7}$. Remembering that $\sigma_{c}$ is $\mathcal{O}(1)$ we
find that in the large $N_{7}$ limit

$\displaystyle\frac{m_{\text{KK}}^{2}}{|\Lambda|}\sim N_{7}^{2}\,,$ (5.53)

thus achieving parametric separation in the limit of large $N_{7}$. This
solution potentially has instabilities, likely coming from metric deformations
of the torus or other effects. Another source of likely instabilities is
M2-brane bubble nucleation in AdS4, but no analysis of the stability of the
solution is available yet.

Another, less simple, manner to employ quantum corrections to construct scale
separated AdS vacua is the KKLT scenario [74]. Simply from equation (4.95) one
can see that scale separation is obtained in the limit of arbitrarily small
$W_{0}$ (albeit logarithmic in small $W_{0}$). Such small $W_{0}$ are
considered to be problematic in light of the tadpole conjecture since they
require many fluxes to cancel against each other and this might overshoot the
tadpole bound. Yet, recent breakthroughs suggest it could be achieved [120,
23, 121]. We remain agnostic about this highly-technical matter and hopefully
these lecture notes inspire the reader to study this issue in more depth.

###  5.6 Holography and scale separation

One may try to employ holographic techniques to settle the issue of whether or
not scale separation is possible by constructing or proving the absence of
CFTs dual to scale separated AdS vacua. The approach of translating Swampland
constraints to holography has turned out fruitful, see for example [243, 244,
245, 246, 247, 248, 249, 250, 251], but for the question of scale separation
we do not know for sure since the topic is in its infancy, yet some
interesting observations have appeared which we briefly summarise here.

Recall that holography dictates that CFT operators dual to scalars in 4d are
single trace scalar operators with conformal dimension

$\displaystyle\Delta=\frac{3}{2}+\frac{1}{2}\sqrt{9+4\,m^{2}L^{2}_{\text{AdS}}}\,.$
(5.54)

Here $m$ is the mass of the 4d scalar. Clearly an operator dimension is
therefore a useful lead to a dual sign of scale separation once the operators
dual to KK modes are identified since then we directly access the ratio
$L_{\text{AdS}}/L_{KK}=L_{\text{AdS}}m_{KK}$.

From this we conclude that putative CFT duals to AdS vacua used in
phenomenology have these peculiar features:

  * •

Standard properties of holographic CFTs such as a large gap to the higher spin
spectrum [252].

  * •

Lack of any relevant operators (that is, operators dual to scalars with
$m^{2}<0$) and lack of any marginal operators ($m^{2}=0$). This implies that
the putative dual conformal field theory would be the end point of an RG-flow,
known as dead-end CFTs. The lack of tachyons (above the Breitenlohner–Freedman
bound 232323In contrast to Minkowski space and dS space, tachyons are allowed
if they are not too tachyonic , which boils down to the square root appearing
in (5.54) to remain real. This is called the Breitenlohner–Freedman bound.) is
demanded for AdS vacua that allow an uplift to dS vacua, since dS vacua do not
allow for tachyons.

  * •

Very few low-lying relevant operators, a feature never observed in conformal
field theories.

The extreme version of this, i.e. a theory with all modes stabilised at a high
scale such that we end up with pure gravity, is believed to be in the
Swampland. This is one of the reasons that [175] conjectured there should
always be at least one low-lying operator dual to a bulk scalar with a mass of
the order of the AdS scale. This seems consistent with current moduli-
stabilisation scenarios. We will see below that some Swampland conjectures
take it much further and conjecture so many light operators dual to KK modes
that scale separation is excluded.

The idea of “bootstrapping” the landscape is not too old and started with
[215] but gained more attention recently in several works, such as [214, 253,
253, 250, 254, 64, 255, 256, 257]. The notion of bootstrapping here is used in
a general sense and does not necessarily invoke the crossing symmetry
constraints. In fact, crossing symmetry constraints seem always obeyed for any
low energy effective action in AdS [252]. Hence, to find out whether scale
separated AdS vacua exist using CFT constraints one has to go beyond the
standard crossing-symmetry constraints.

We briefly review some of the most recent results in this field:

  * •

Reference [214] remarks that loop diagrams in the bulk, are sensitive to scale
separation since they are effected by all KK modes. Such diagrams are dual to
certain non-planar CFT correlators and the main idea is then that there exists
a way to define the number of large bulk dimensions purely from CFT data. Such
large dimensions clearly count the number of AdS dimensions but would also
count internal dimensions when they are not scale separated. So this paper
does address the question of how many large bulk dimensions a holographic CFT
describes, it has not been used yet to settle the consistency of scale
separated AdS vacua.

  * •

In reference [126] a holographic QFT is identified that ought to flow to the
KKLT conformal field theory. It is possible to guess the central charge in
that ultraviolet QFT, and using that
$c_{\text{UV}}>c_{\text{IR}}=l^{2}_{\text{AdS}}$ it is possible to put a bound
on $c_{\text{UV}}$ by a small topological number contradicting
$c_{\text{UV}}>l^{2}_{\text{AdS}}$. If correct, this invalidates the KKLT
construction.

  * •

In [258] it was argued that extreme scale separation, meaning that one can
forget all together the extra dimensions, is inconsistent with holography in
the presence of enough supersymmetry. The reason being that the R-symmetry
would become a global symmetry in the bulk, which is impossible since quantum
gravity does not allow global symmetries that are not gauged.

  * •

In [64] after checking a large number of holographic conformal field theories,
a seemingly universal bound on the dimension of the first non-trivial spin 2
operator was found. This, in turn, puts bounds on the internal space which
turns out to have a minimal diameter in AdS units. Reference [64] then
conjectured to be true for all conformal field theories, which would make
scale separation impossible.

  * •

In a series of recent works [253, 255, 256, 257, 259], extending [260], it was
observed that the conformal dimensions obtained from supersymmetric DGKT
solutions are integers. This is absolutely non-obvious. What is less special
is that masses tend to be quantized in $1/L^{2}$ units, but that the square
root in (5.54) is highly special and requires explanation. An attempt to
explain this is presented in [257]. Notice however that in the case where
supersymmetry is broken the conformal dimensions sometimes cease to be
integers [261] unless one considers the scale separated non-SUSY solutions
[255].

###  5.7 Swampland arguments in favor of/against scale separation?

The main swampland conjecture relating to the existence of AdS solutions with
scale separation comes from [19] and is called the _AdS distance conjecture
(ADC)_ and reads242424A closely related conjecture that arrives at similar
conclusions invokes the mass of the gravitino, see [262, 263].:

_Consider quantum gravity on $d$-dimensional AdS space with cosmological
constant $\Lambda$. There exists an infinite tower of states with mass scale
$m$ which, as $\Lambda\rightarrow 0$, behaves (in Planck units) as_

$m\sim|\Lambda|^{\alpha}$ (5.55)

_where $\alpha$ is a positive order-one number. For SUSY AdS vacua
$\alpha=1/2$._252525A very interesting refinement of this conjecture has been
suggested in [20], and commented upon in [255].

This last statement about supersymmetry is known as the _strong AdS distance
conjecture_. Let us call the operator of vacuum energy in dimension $d$ to be
$M_{p}^{d-2}\Lambda$ so that $\Lambda$ has mass dimension two, in other words
$\Lambda=-1/L^{2}_{AdS}$. Then the strong ADC applied to a KK tower implies

$\displaystyle L_{KK}=c\,L_{\text{AdS}}\,,$ (5.56)

where $c$ is a $\mathcal{O}(1)$ constant and forbids scale separation for SUSY
vacua. If correct, then the IIA DGKT vacua are inconsistent, and KKLT with
exponentially parametric tuning of $W_{0}$ as well. The strong ADC is
therefore quite a strong conjecture, as the name suggests. The more general
ADC seems in line with all examples ever constructed within string theory, to
our knowledge.

The conjecture is based on an extension of the distance conjecture [58] for
scalar fields to distances on metric space. It is not clear why the distance
conjecture can be applied beyond scalar fields, but it is sensible. Yet, there
seems no particular argument for the strong ADC beside the observation that
well controlled examples in string theory obey it. Such examples typically
feature more than 4 conserved supercharges. Very intriguingly, [264, 265]
pointed out that the lack of scale separation for AdS vacua with more than 4
conserved supercharges is a direct consequence of the _magnetic Weak Gravity
Conjecture[266]_, which, unlike the strong ADC, is a very well established
conjecture. It states that in a quantum gravity theory which has a U(1) gauge
field at low energies with coupling $g$ then the cut-off $\Lambda_{C}$ of this
EFT is bounded by

$\Lambda_{C}<gM_{p}^{(d-2)/2}\,.$ (5.57)

What [264, 265] relied upon is to use that the vacuum energy for SUSY AdS
vacua with extended SUSY is fixed in terms of the gauge coupling and then
demanded that the AdS length scale is above the cut-off length scale, leading
to an effective absence of scale separation. Although in this case the tower
does not necessarily need to be a KK tower. It most likely is in the examples
we understand well, but that does not matter. The question at state is whether
the AdS vacuum can be seen as a vacuum within an EFT, and for that all towers
must decouple, KK or something else. The above relation with other Swampland
conjectures is typical to Swampland research; the conjectures seem to form a
very tied web, and conjectures that seem at first not related, often turn out
to be.

We already mentioned that the IIA vacua seem a counter-example to the strong
ADC, but then again, these vacua have been criticized on the basis of using
the approximation of smeared orientifolds.

Yet, there is in fact a Swampland argument _in favor_ of the consistency of
the IIA vacua, rather than against [267] (see also [225]). The basic argument
is rather simple. Instead of computing the distance in metric space as
$\Lambda\rightarrow 0$, we can compute the distance in scalar field space,
which is were most of the evidence for the distance conjecture comes from. The
reason we can do this, is that there exists an open-string scalar field that
continuously varies along the IIA vacua with different size of the energy. In
other words, the different IIA vacua can be seen as different minima of a
single potential energy function, once one includes this scalar field. It then
turns out that the distance conjecture is exactly obeyed! One could argue that
this is evidence for the ADC from the ordinary distance conjecture and that
the strong ADC somehow has no separate argument going for it.

###  5.8 The AdS moduli conjecture

The two main applications of flux compactifications discussed in these
lectures were 1) dS vacua and 2) AdS vacua with scale separation. We already
alluded to the fact that both topics are not disconnected and to end these
lecture notes we want to discuss that in a bit more detail.

The most obvious relation between the two has been explained before; scale
separation is a necessary condition for using the tools of 4d EFT to describe
the fluctuations around the vacuum $AdS_{4}\times\mathcal{M}_{6}$. In fact, it
is often used as a prior to construct the vacuum if the construction relies on
EFT methods. Then self-consistency requires the vacuum to be stabilised at a
volume for which scale separation is indeed realised and so the prior was
justified. This is crucial for dS vacua since almost all claimed de Sitter
compactifications start out with this assumption, where the majority of these
first goes about and constructs an AdS vacuum (often SUSY) and then uplifts.

Uplifts are often done using warped down anti-branes but can be generalised,
see for instance [268]. Yet, there are more conditions beyond scale separation
to guarantee a successful uplift. An obvious one is a SUSY-breaking energy
that is sufficiently small. Imagine this to be possible then there is still a
condition on the AdS vacuum: it needs to be sufficiently rigid and should not
be destabilised by the uplift. Interestingly this condition can be put into a
necessary condition on the mass spectrum, as first pointed out in [146]. The
condition is most easily visualised in a picture, see 10.

Figure 10: _A series of AdS potentials with ever increasing $m^{2}L^{2}_{AdS}$
combination._

One would like an AdS vacuum whose vacuum energy is a small negative number
and at the same time the minimum lies in a sharp enough potential. This means
that a small SUSY-breaking effect, which will backreact little on the assumed
approximations, will be able to generate a positive vacuum energy, without
causing a runaway. In practise this means that the dimensionless products

$m^{2}_{\phi}\,L^{2}_{AdS}\gg 1\,,$ (5.58)

should be large enough. Here $m^{2}_{\phi}$ is the mass squared of the
lightest scalar field. The factor $L^{2}_{AdS}$ in the product informs us how
close the vacuum is to Minkowski space and the other factor $m^{2}_{\phi}$ how
sharply peaked it is. In the picture 10 we have drawn various potentials with
ever increasing $m^{2}_{\phi}L_{AdS}^{2}$.

Interestingly this product is the one that appears in the formula (5.54) for
the conformal weight of the dual operator. And so vacua which can be easily
uplifted are such that the lowest weight is very high. But note that this is
not the case in KKLT for instance [173] and this is why uplifting is tedious.
Inspired by the absence of known CFTs obeying (5.58) and the difficulty in
achieving dS vacua it was conjectured that the condition (5.58) is a Swampland
condition in the sense that EFTs obeying (5.58) are in the Swampland [175]. In
other words the condition says there is always a scalar for which

$m^{2}_{\phi}L^{2}_{AdS}\sim\mathcal{O}(1)\,,$ (5.59)

This conjecture was later named the _AdS moduli conjecture_ in [269]. Note
that this conjecture could be considered weaker than the strong ADC since it
only requires that one scalar is light in AdS units (order 1 say) instead of a
whole tower.

Examples of stringy constructions that would violate this conjecture are the
so-called Kallosh–Linde racetrack fine tuning scenarios [270], which have no
clear controllable dS embedding so far, consistent with the AdS moduli
conjecture.

## Acknowledgments

We are grateful to Vincent Van Hemelryck for comments on an earlier draft and
we like to thank the organisers and participants of the _CERN Winter School on
Supergravity, Strings and Gauge Theory 2022_ and the _Tehran School on the
Swampland Program 2022_ for inspiration to write these notes. The work of GZ
is supported by the program Rita Levi Montalcini for young researchers (D.M.
n. 928, December 23rd 2020).

##  Appendix A Forms and Hodge star

For a general p-form we switch between component and form notation as follows:

$A_{p}=\frac{1}{p!}A_{M_{1}\ldots M_{p}}dx^{M_{1}}\wedge\ldots\wedge
dx^{M_{p}}\,.$ (A.1)

The exterior derivative works as usual:

$\displaystyle F_{p+1}=dA_{p}$
$\displaystyle=\frac{1}{p!}\partial_{N}A_{M_{1}\ldots M_{p}}dx^{N}\wedge
dx^{M_{1}}\wedge\ldots\wedge dx^{M_{p}}$ (A.2)
$\displaystyle\rightarrow\qquad$ $\displaystyle F_{M_{1}\ldots
M_{p+1}}=(p+1)\partial_{[M_{1}}A_{M_{2}\ldots M_{p+1}]}\,,$ (A.3)

where anti-symmetrization is taken with weight one, for example:
$T_{[ab]}=1/2\left(T_{ab}-T_{ba}\right)$.

In $D$ dimensions the epsilon-_symbol_ $\varepsilon_{M_{1}\ldots M_{D}}$ is
defined as

$\varepsilon_{1\ldots D}=1\,,$ (A.4)

and completely anti-symmetric in all indices: $\varepsilon_{[M_{1}\ldots
M_{D}]}=\varepsilon_{M_{1}\ldots M_{D}}$ . To get to a volume _tensor_ on a
space with metric $g_{\mu\nu}$ we define:

$\epsilon_{M_{1}M_{2}\ldots M_{D}}=\sqrt{|g|}\varepsilon_{M_{1}M_{2}\ldots
M_{D}}\,.$ (A.5)

Contractions of the epsilon tensor (and symbol) obey the following relations:

$\epsilon_{M_{1}\ldots M_{q}M_{q+1}\ldots M_{D}}\epsilon^{M_{1}\ldots
M_{q}N_{q+1}\ldots
N_{D}}=(-1)^{t}q!(D-q)!\delta^{[N_{q+1}}_{[M_{q+1}}\ldots\delta^{N_{D}]}_{M_{D}]}\,.$
(A.6)

where $t$ stands for the number of timelike dimensions of the $D$-dimensional
space. To define the Hodge star on all forms, it is enough to define on the
following set made from a coordinate base and then extend it by linearity:

$\star_{D}(dx^{M_{1}}\wedge\ldots\wedge
dx^{M_{p}})=\frac{1}{(D-p)!}\epsilon_{N_{1}\ldots N_{D-p}}^{\qquad M_{1}\ldots
M_{p}}dx^{N_{1}}\wedge\ldots\wedge dx^{N_{D-p}}\,.$ (A.7)

For example, we have $\star\Phi\equiv\Phi\,\epsilon\equiv\sqrt{|g|}\Phi
dx^{0}\wedge\ldots\wedge dx^{D}$ where $\Phi$ denotes some zero-form (scalar).
In other words, the volume tensor with all indices down is defined as:

$\epsilon=\star 1.$ (A.8)

The $\star$ operation obeys the following properties:

$\displaystyle(\star A_{p})\wedge B_{p}=(\star B_{p})\wedge
A_{p}=\frac{1}{p!}A_{\mu_{1}\ldots\mu_{p}}B^{\mu_{1}\ldots\mu_{p}}\star 1\,,$
(A.9) $\displaystyle\star\star A_{p}=(-1)^{t+p(D-p)}A_{p}\,.$ (A.10)

##  Appendix B Orientifold planes in the worldsheet

One ubiquitous ingredient in type II string compactifications is orientifold
planes. Their introduction is useful for two main reason: they play an
important role in tadpole cancellation providing a source of negative charge
without completely breaking supersymmetry, and they cut the supersymmetry
preserved by the background in half thus allowing to have $\mathcal{N}=1$
solutions in 4d of type II string theories. In this appendix we provide a
brief review of orientifolds, starting with their worldsheet description and
then moving to a spacetime perspective. For a more complete treatment which
also includes original references we point to [271].

The usual way orientifold planes are introduced is via worldsheet: they are
indeed pretty exotic objects and the natural path to their introduction is via
the study of un-oriented strings. When trying to introduce un-oriented strings
in the context of type II strings one encounters several consistency
constraints which lead to the formulation of type I string theory. Type I
string theory secretly already contains an orientifold plane, specifically an
O9-plane, and the remaining orientifold planes are found via toroidal
compactification and application of T-duality. We will start by discussing un-
oriented strings and then study T-duality of type I string theory to discover
all the other orientifold planes.

###  B.1 Un-oriented closed strings

In the description of un-oriented strings we will focus solely on the bosonic
worldsheet oscillations. In the case of superstrings there are also fermionic
worldsheet oscillations but it is sufficient to focus on the bosonic ones to
capture the salient features of un-oriented strings and orientifolds.

Recall that the fields describing the oscillation of strings are described as
scalar fields $X^{\mu}(t,\sigma)$ on the worldsheet, where the index $\mu$
runs over the number of spacetime dimensions. Here $t$ is the time coordinate
on the worldsheet and $\sigma$ is the space coordinate on the worldsheet which
is periodic in the case of closed strings, that is $\sigma\sim\sigma+2\pi$.
Equations of motion on the worldsheet decompose all oscillations of fields
into a linear superposition of left and right moving waves, that is

$\displaystyle
X^{\mu}(t,\sigma)=X^{\mu}_{L}(t+\sigma)+X_{R}^{\mu}(t-\sigma)\,,$ (B.1)

where the left movers $X^{\mu}_{L}(t+\sigma)$ are a function of only
$t+\sigma$ and the right movers are a function of $X_{R}^{\mu}(t-\sigma)$ are
a function of only $t-\sigma$. Having introduced this notation we can start
talking about the worldsheet parity operation $\Omega$: its action is to
simply map $X^{\mu}(t,\sigma)$ to $X^{\mu}(t,\pi-\sigma)$. This operation ends
up swapping left and right movers. This gives a classical definition of the
worldsheet parity, and it can be easily extended to an operator $\Omega$ on
the Hilbert space of string theory. Many states in the Hilbert space will not
be invariant under $\Omega$, however, given that clearly
$\Omega^{2}=1$,262626Some care has to be taken in the definition of $\Omega$
for the fermionic fields on the worldsheet. we can easily build a projection
operator that retains only states are invariant under $\Omega$. Calling this
projector $\Pi_{\Omega}$ its definition is obviously

$\displaystyle\Pi_{\Omega}=\frac{1}{2}\left(\mathbb{I}+\Omega\right)\,,$ (B.2)

where $\mathbb{I}$ is the identity operator. By applying $\Pi_{\Omega}$ on the
Hilbert state of string theory one retains only states that are worldsheet
parity invariant thus giving un-oriented strings. At the level of the spectrum
of closed strings the projection cuts out some string fluctuations from the
spectrum: for instance at the massless level of the bosonic string we have a
graviton, an anti-symmetric two-form and a dilaton. For the un-oriented string
the anti-symmetric two-form is projected out leaving therefore only the
graviton and the dilaton. Unfortunately the tachyon that plagues the bosonic
string is not projected out and therefore the projection $\Pi_{\Omega}$ does
not stabilise the bosonic string. Having discussed briefly what happens in the
bosonic string we can move to the case of superstrings: as usual here there
are some additional consistency constraints, first and foremost the projection
can be done only for type IIB string theory.272727The intuition is the
following: the operation $\Omega$ swaps left and right movers on the string
worldsheet, and therefore requires a worldsheet theory that is left-right
symmetric. That happens for type IIB given that the GSO projection is the same
for left and right movers, but fails for type IIA where the projection is
different in the two sectors. Moreover the projection requires the presence of
an open string sector for consistency, which leads to the formulation of type
I string theory eventually. Before moving onto how orientifold planes are
defined let us talk briefly about the case of un-oriented open strings.

###  B.2 Un-oriented open strings

When it comes to the definition of the operator $\Omega$ on open strings there
is not much of a difference: the action is that of a parity transformation
mapping the spatial coordinate $\sigma$ to $\pi-\sigma$. But to fully define
the action on any open string state it is necessary to define the action of
$\Omega$ on Chan–Paton factors. Recall that Chan–Paton factors are a set of
non-dynamical degrees of freedom attached to each endpoint of an open string.
To fully specify an open string state in addition to the oscillator content it
is necessary to choose the Chan–Paton factor, and their information can be
packaged in an $N\times N$ matrix $\lambda$ in the case where there are $N$
distinct Chan–Paton factors at the endpoints.282828The matrix $\lambda$ is
nothing but the tensor product of the two Chan–Paton labels at each endpoint
of a string. The system is invariant under $U(N)$ transformations that act on
$\lambda$ as $U\lambda U^{\dagger}$ which is a reflection of the $U(N)$ gauge
symmetry: indeed the dynamics of open strings is described at the massless
level by a $U(N)$ Yang–Mills theory. To fully define $\Omega$ it is necessary
to specify its action on $\lambda$: indeed $\Omega$ swaps the two endpoints of
an open string thus acting on $\lambda$ via transposition. However it is
possible to define a more general action as

$\displaystyle\Omega:\lambda\ \mapsto\ M\lambda^{T}M^{-1}\,,$ (B.3)

for some matrix $M$. Requiring that $\Omega^{2}=1$ forces $M$ to be either
symmetric or anti-symmetric. In the former case the system will end up having
a $SO(2N)$ gauge symmetry and in the latter case it will have a $USp(2N)$
gauge symmetry.292929Here we implicitly assumed that the number of Chan–Paton
factors is even. For the case of type I it is necessary to take $M$ to be
symmetric to have a consistent theory therefore giving a $SO(32)$ theory.

###  B.3 From un-oriented strings to orientifolds

The common way to discover orientifold planes is via T-duality of type I
string theory. We have discussed how to take type IIB string theory and do a
projection to un-oriented strings giving type I string theory. However it is
also known that type IIB string theory when defined on a manifold of the form
$\mathcal{M}_{9}\times S^{1}$ is equivalent to type IIA on
$\mathcal{M}_{9}\times S^{1}$ via T-duality. Therefore this suggest that
taking type I string theory on a circle and applying T-duality might give a
way to define un-oriented strings in type IIA string theory. Let us follow
this route then: we need to understand how to combine the action of $\Omega$
and a T-duality transformation on the worldsheet fields $X^{\mu}(t,\sigma)$.
Recall that performing a T-duality along the coordinate $i$ results in the
transformation

$\displaystyle X^{i}(t,\sigma)=X^{i}_{L}(t+\sigma)+X_{R}^{i}(t-\sigma)\
\mapsto\ \hat{X}^{i}(t,\sigma)=X_{L}^{i}(t+\sigma)-X_{R}^{i}(t-\sigma)\,.$
(B.4)

Moreover the effect of $\Omega$ is to swap left and right movers. Therefore in
terms of the dual coordinate $\hat{X}^{i}(t,\sigma)$ the worldsheet parity
acts as

$\displaystyle\hat{X}^{i}(t,\sigma)\rightarrow-\hat{X}^{i}(t,\pi-\sigma)\,.$
(B.5)

This is a combination of a pure worldsheet parity and a spacetime parity
transformation. Successive applications of T-duality along different
directions yield a similar result: in the dual frame a worldsheet parity will
also be combined with a spacetime parity along every dual coordinate. Notice
that we no longer have a pure un-oriented version of string theory: given that
we are performing a spacetime reflection any string state will have a mirror
version with opposite orientation and related by a spacetime transformation.
Only at the fixed point of the spacetime parity we do recover a pure un-
oriented string theory: this locus is called an _orientifold plane_. The
general action of an orientifold will be $\Omega\mathcal{R}$ where
$\mathcal{R}$ is a spacetime reflection, that is it is an operation that gives
the identity when applied twice. Orientifolds will act as a mirror for strings
relating states on one side to states on the other side. We can briefly add
some properties of orientifold planes that are specific to superstrings:

  * •

We only focused on the orientifold action on bosonic worldsheet fields. When
taking into account the action on fermions in order to make sure that the
operation squares to one it is necessary to change it to
$\Omega\mathcal{R}(-1)^{F_{L}}$ where $F_{L}$ is the left-moving space-time
fermion number;

  * •

The dimension of orientifold planes is fixed by which type II string theory we
are considering. The fixed locus we just encountered in type IIA has eight
space dimensions, making it a O8-plane. By successive application of T-duality
one finds that type IIA string theory has O$p$-planes with $p$ even and type
IIB has O$p$-planes with $p$ odd. By using the language of O-planes it is
possible to understand type I string theory as type IIB string theory with an
O$9$-plane;

  * •

Notice that when introducing an O$9$-plane in type IIB string theory we went
to type I string theory, which has half the supersymmetry. Applying T-duality
does not change the supersymmetry of the solution, implying that orientifold
planes break half of the supersymmetry.

###  B.4 Orientifold planes and the worldsheet topology expansion

Given a string diagram on a worldsheet $\Sigma$ its contribution to a given
correlation function will be weighted by a factor $g_{s}^{-\chi(\Sigma)}$
where $g_{s}=e^{\phi}$ is the string coupling constant and $\chi(\Sigma)$ is
the Euler characteristic of $\Sigma$. Therefore, similarly to how in quantum
field theory it is possible to expand correlation functions in terms of the
number of loops, it is possible to expand any string correlation function in
terms of the Euler number of the worldsheet provided one considers a regime
where $g_{s}\ll 1$. In the case of a string theory model with closed strings
only it is interesting to note that

$\displaystyle\chi(\Sigma)=2-2g(\Sigma)\,,$ (B.6)

where $g(\Sigma)$ is called the genus of $\Sigma$ and can be identified with
the number of holes in $\Sigma$. Therefore in this scenario it is possible to
identify the number of loops with the number of holes in the worldsheet. The
main advantage of string theory in this situation is that there is a unique
worldsheet for a given genus giving therefore a much reduced number of
diagrams to compute compared to the case of ordinary quantum field theory. The
worldsheet with no loops is nothing but the usual Riemann sphere and any
worldsheet with genus $g$ with $g>0$ is nothing but a doughnut with $g$ holes.
While this gives a concise description of how to organise any string amplitude
it does not cover the case we are interested in, that is the case of
orientifold planes (and it does not include any scenario with open strings
either). In order to cover these cases it is necessary to expand the set of
worldsheet topologies that are included in the computation of string
correlation functions. When only closed strings are included (without
orientifolds) the worldsheets that are included are simply closed manifolds of
complex dimension one (oftentimes called Riemann surfaces), and we just
discussed how to classify them for the sake of the genus expansion of string
amplitudes. To include orientifolds and open strings it is necessary to
introduce crosscaps and boundaries on the worldsheet. Let us cover these two
separately:

  * •

A crosscap is inserted on a two dimensional surface by excising a small disk
and closing back the surface by identifying the antipodal points of the
boundary of the disk. This is a local operation but it makes a surface un-
orientable. For instance a Möbius strip can be obtained by taking a disk and
inserting a crosscap in it. Similarly a Klein bottle is a Riemann sphere in
which two crosscaps have been added. Given that the worldsheet ceases to be
orientable in the presence of any number of crosscaps it means that strings
will no longer be oriented;

  * •

The introduction of boundaries on the worldsheet signals the presence of open
strings given that those boundaries are identified with open string endpoints.

Luckily enough it is still sufficiently easy to classify worldsheet topologies
in the presence of boundaries and crosscaps and compute their relative weight
in string amplitudes. The overall factor of each diagram still is
$g_{s}^{-\chi(\Sigma)}$, however the formula for the Euler characteristic is
modified to

$\displaystyle\chi(\Sigma)=2-2g-n_{b}-n_{c}\,,$ (B.7)

where here $n_{b}$ is the number of boundaries and $n_{c}$ is the number of
crosscaps. For example when computing the one-loop vacuum energy of type II
string theories one simply has to consider the torus as a diagram, the only
diagram with zero Euler characteristic. When going to type I string theory
which is un-orientable and has open strings one needs to consider all possible
diagrams with zero Euler characteristic, that is the Klein bottle (which has
$g=0$, $n_{b}=0$, and $n_{c}=2$), the cylinder (which has $g=0$, $n_{b}=2$,
and $n_{c}=0$), and the Möbius strip (which has $g=0$, $n_{b}=1$, and
$n_{c}=1$). All these four diagrams added up give the one-loop vacuum energy
of the system.

###  B.5 Orientifold properties: tension and charges

It is natural to wonder what kind of effect does the introduction of an
orientifold plane has on the geometry of space-time. Given that orientifolds
(and branes) are dynamical objects they will have their own energy-momentum
tensor thus implying that they source gravitational field. Moreover it is
important to understand how they couple to the remaining fields in the low-
energy action describing type II string theories. Quite remarkably it is
possible to write a quite compact action describing the coupling of O-planes
to the various bosonic fields of type II string theories. The resulting action
for a O$p$-plane wrapping a manifold $\Sigma_{p+1}$ is the sum of a DBI-action
and a Wess–Zumino action

$\displaystyle S=S_{\text{DBI}}+S_{\text{WZ}}\,,$ (B.8)

where we have that

$\displaystyle S_{\text{DBI}}$
$\displaystyle=-T_{p}\,\mu_{p}\int_{\Sigma_{p+1}}d^{p+1}xe^{-\phi}\sqrt{-\text{det}(g)}\,,$
(B.9) $\displaystyle S_{\text{WZ}}$
$\displaystyle=-Q_{p}\mu_{p}\int_{\Sigma_{p+1}}C\wedge\sqrt{\frac{\mathcal{L}(R_{T}/4)}{\mathcal{L}(R_{N}/4)}}\,.$
(B.10)

These formulas have a lot of information and we will spend some time unpacking
them. Let us start from the DBI-action: we can clearly see that O-planes
couple to the metric and the dilaton and therefore become sources of both
fields. The coupling to the metric is measured by the volume of the cycle
wrapped by the O-plane similarly to the case of other extended objects in
string theory. The quantity $\mu_{p}$ is the tension of a D$p$-brane and it is
given by

$\displaystyle\mu_{p}=\frac{2\pi}{(2\pi\sqrt{\alpha^{\prime}})^{p+1}}\,.$
(B.11)

The quantity $T_{p}$ is the ratio between the tension of an O-plane and a
D-brane and it is given by

$\displaystyle T_{p}=-2^{p-5}\,.$ (B.12)

Quite remarkably this is _negative_. This is not an issue because orientifold
planes are not dynamical, but it is this property that makes O-planes so
valuable in flux compactifications as they can cancel the positive energy
density sourced by branes and fluxes in a supersymmetric fashion. We will
shortly discuss how tensions are computed in string theory, but before that
let us discuss the Wess–Zumino term of the O-plane action. The quantity
$Q_{p}$ is again the ratio of the charge of an O$p$-plane and a D$p$-brane,
and its value matches exactly $T_{p}$. This is nothing but a manifestation of
the fact that orientifold planes are supersymmetric objects. The term $C$
inside the action is the formal sum of all Ramond–Ramond potentials, implying
that an orientifold plane will act as a source of these fields. In the formula
$R_{T}$ and $R_{N}$ denote the curvatures of the tangent and normal bundles
respectively, and the term $\mathcal{L}$ is the Hirzebruch
$\mathcal{L}$-polynomial, which can be expressed in terms of Pontryagin
classes as

$\displaystyle\sqrt{\mathcal{L}(R/4)}=1+\frac{(4\pi^{2}\alpha^{\prime})^{2}}{96}p_{1}(R)-\frac{(4\pi^{2}\alpha^{\prime})^{4}}{10240}p_{1}^{2}(R)+\frac{7(4\pi\alpha^{\prime})^{4}}{23040}p_{2}(R)+\dots\,.$
(B.13)

The main observation concerning the Wess–Zumino action is that again the
charge sourced by O-planes is negative, thus allowing to cancel all charges
sourced by D-branes and fluxes without having to introduce anti-branes that
would lead to a breakdown of supersymmetry.

As we highlighted the most remarkable property of O-planes is the fact that
their tensions and charges have the opposite sign of the one of D-branes. It
is quite odd to have a negative tension object in a theory with gravity, and
in the following we will briefly outline how the computation of tension and
charges are done in string theory. In keeping with the fact that string theory
has no free dimensionless parameters we will see that the fundamental theory
will completely fix the values of tensions and charges. We will discuss the
computation method for charges and tensions of both D-branes and O-planes
because of their similarity.

In order to compute tensions and charges one considers an exchange of closed
strings between two objects which can be two D-branes or one D-brane and an
O-plane: in the first case we will compute the charge and tension of D-branes
and in the second case the charge and tension of O-planes. The diagrams are
sketched in figure 11. In the case of an exchange between two D-branes the
diagram has the topology of an annulus, that is it has genus zero, two
boundaries, and no crosscap. For the case of the exchange between a D-brane
and an O-plane one simply replaces one boundary with a crosscap, thus giving a
Möbius strip.303030Looking at figure 11 it is clear that we are considering
the case in which one closed string is emitted from one object and lands on
the other one, which clearly computes the net force between them. However
there is a different perspective that can be useful to consider: if we
consider a stretched string between the two objects one and have its endpoints
move in a loop we recover the same exact diagram! This is an instance of open-
closed duality in which a closed string phenomenon can be interpreted dually
in terms of open strings physics. From the formula of the diagrams it is
possible to extract that the various objects exchange various fields between
each other (gravitons, dilaton, Ramond–Ramond fields), thus allowing to
extract the couplings constant directly. The result of the diagram as a matter
of fact is that the net force between the objects involved actually vanishes
which is nothing but a manifestation of supersymmetry. We will not go into the
details of the computation but simply point to [271] for those interested in
seeing how the diagrams are computed.

Figure 11: _The diagrams computing tension and charge of D-branes and
O-planes. On the left, the exchange of strings between two D-branes. On the
right, the exchange between a D-brane and an O-plane (with the O-plane on the
right side). The difference between the two diagrams is the presence of the
crosscap on top of the O-plane, drawn here with a cross inside the oval._

###  B.6 Geometrization of O6 planes

So far we have discussed some properties of orientifold planes as seen from
the string worldsheet, which implies that all the discussion is valid at weak
coupling. Accessing the fate of orientifold planes at strong coupling is a
difficult question given that a fully non-perturbative definition of String
Theory is still not completely available, however employing some dualities it
is possible to explore some strong coupling regimes and peek into how O-planes
behave in such regimes. While it is not important for the lectures we cite one
famous example of this: it is well known that non-perturbative effects break
the O7-plane of type IIB String Theory into a couple of 7-branes [272], see
[12] for a review. The other important example we would like to discuss is the
one of the O6-plane: in the main text we discuss several instances of
solutions that employ O6-planes and therefore it may be useful to understand
their behaviour at strong coupling, that is their uplift to M-theory. The
result is well known: the uplift of an O6-plane to M-theory is a smooth space
with a Atiyah–Hitchin metric [273]. We will not need the detailed metric, but
we would like to stress that the uplift to M-theory of O6-planes is purely
geometrical and smooth, and the same happens with D6-branes. Therefore any
question that one may have about type IIA compactifications that admit an
uplift to M-theory can be recast into a purely geometrical question without
the addition of localized sources. Some care has to be taken though: first,
the uplift is valid only at very large string coupling, so a solution of type
IIA at weak string coupling does not admit a well behaved uplift to M-theory
and therefore any question is better posed within the framework of type IIA
String Theory. Second, even in a situation of strong coupling, there may be
some hurdles in the uplift, most notably the presence of a non-zero Romans
mass.

## References

  * [1] C. Vafa, “The String landscape and the swampland,” arXiv:hep-th/0509212.
  * [2] T. D. Brennan, F. Carta, and C. Vafa, “The String Landscape, the Swampland, and the Missing Corner,” PoS TASI2017 (2017) 015, arXiv:1711.00864 [hep-th].
  * [3] E. Palti, “The Swampland: Introduction and Review,” Fortsch. Phys. 67 no. 6, (2019) 1900037, arXiv:1903.06239 [hep-th].
  * [4] M. van Beest, J. Calderón-Infante, D. Mirfendereski, and I. Valenzuela, “Lectures on the Swampland Program in String Compactifications,” Phys. Rept. 989 (2022) 1–50, arXiv:2102.01111 [hep-th].
  * [5] M. Graña and A. Herráez, “The Swampland Conjectures: A Bridge from Quantum Gravity to Particle Physics,” Universe 7 no. 8, (2021) 273, arXiv:2107.00087 [hep-th].
  * [6] N. B. Agmon, A. Bedroya, M. J. Kang, and C. Vafa, “Lectures on the string landscape and the Swampland,” arXiv:2212.06187 [hep-th].
  * [7] N. Cribiori and F. Farakos, “Supergravity EFTs and swampland constraints,”  4, 2023.  arXiv:2304.12806 [hep-th].
  * [8] E. Silverstein, “TASI / PiTP / ISS lectures on moduli and microphysics,” in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 2003): Recent Trends in String Theory, pp. 381–415.  5, 2004.  arXiv:hep-th/0405068.
  * [9] F. Denef, M. R. Douglas, and S. Kachru, “Physics of String Flux Compactifications,” Ann. Rev. Nucl. Part. Sci. 57 (2007) 119–144, arXiv:hep-th/0701050.
  * [10] M. Grana, “Flux compactifications in string theory: A Comprehensive review,” Phys. Rept. 423 (2006) 91–158, arXiv:hep-th/0509003.
  * [11] M. R. Douglas and S. Kachru, “Flux compactification,” Rev. Mod. Phys. 79 (2007) 733–796, arXiv:hep-th/0610102.
  * [12] F. Denef, “Les Houches Lectures on Constructing String Vacua,” Les Houches 87 (2008) 483–610, arXiv:0803.1194 [hep-th].
  * [13] L. E. Ibanez and A. M. Uranga, String theory and particle physics: An introduction to string phenomenology.  Cambridge University Press, 2, 2012.
  * [14] D. Baumann and L. McAllister, Inflation and String Theory.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 5, 2015.  arXiv:1404.2601 [hep-th].
  * [15] A. Hebecker, Naturalness, String Landscape and Multiverse: A Modern Introduction with Exercises, vol. 979 of Lecture Notes in Physics.  Springer, 3, 2021.
  * [16] A. Tomasiello, Geometry of String Theory Compactifications.  Cambridge University Press, 1, 2022.
  * [17] M. Cicoli, J. P. Conlon, A. Maharana, S. Parameswaran, F. Quevedo, and I. Zavala, “String Cosmology: from the Early Universe to Today,” arXiv:2303.04819 [hep-th].
  * [18] F. F. Gautason, M. Schillo, T. Van Riet, and M. Williams, “Remarks on scale separation in flux vacua,” JHEP 03 (2016) 061, arXiv:1512.00457 [hep-th].
  * [19] D. Lüst, E. Palti, and C. Vafa, “AdS and the Swampland,” Phys. Lett. B 797 (2019) 134867, arXiv:1906.05225 [hep-th].
  * [20] G. Buratti, J. Calderon, A. Mininno, and A. M. Uranga, “Discrete Symmetries, Weak Coupling Conjecture and Scale Separation in AdS Vacua,” JHEP 06 (2020) 083, arXiv:2003.09740 [hep-th].
  * [21] G. B. De Luca and A. Tomasiello, “Leaps and bounds towards scale separation,” JHEP 12 (2021) 086, arXiv:2104.12773 [hep-th].
  * [22] G. B. De Luca, N. De Ponti, A. Mondino, and A. Tomasiello, “Cheeger bounds on spin-two fields,” JHEP 12 (2021) 217, arXiv:2109.11560 [hep-th].
  * [23] M. Demirtas, M. Kim, L. McAllister, J. Moritz, and A. Rios-Tascon, “Small cosmological constants in string theory,” JHEP 12 (2021) 136, arXiv:2107.09064 [hep-th].
  * [24] U. H. Danielsson and T. Van Riet, “What if string theory has no de Sitter vacua?,” Int. J. Mod. Phys. D 27 no. 12, (2018) 1830007, arXiv:1804.01120 [hep-th].
  * [25] Supernova Cosmology Project Collaboration, S. Perlmutter et al., “Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae,” Astrophys. J. 517 (1999) 565–586, arXiv:astro-ph/9812133.
  * [26] Supernova Search Team Collaboration, A. G. Riess et al., “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” Astron. J. 116 (1998) 1009–1038, arXiv:astro-ph/9805201.
  * [27] Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209 [astro-ph.CO]. [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * [28] B. Ratra and P. J. E. Peebles, “Cosmological Consequences of a Rolling Homogeneous Scalar Field,” Phys. Rev. D 37 (1988) 3406.
  * [29] R. R. Caldwell, R. Dave, and P. J. Steinhardt, “Cosmological imprint of an energy component with general equation of state,” Phys. Rev. Lett. 80 (1998) 1582–1585, arXiv:astro-ph/9708069.
  * [30] E. J. Copeland, M. Sami, and S. Tsujikawa, “Dynamics of dark energy,” Int. J. Mod. Phys. D 15 (2006) 1753–1936, arXiv:hep-th/0603057.
  * [31] N. Kaloper and L. Sorbo, “Where in the String Landscape is Quintessence,” Phys. Rev. D 79 (2009) 043528, arXiv:0810.5346 [hep-th].
  * [32] M. Cicoli, S. De Alwis, A. Maharana, F. Muia, and F. Quevedo, “De Sitter vs Quintessence in String Theory,” Fortsch. Phys. 67 no. 1-2, (2019) 1800079, arXiv:1808.08967 [hep-th].
  * [33] M. C. David Marsh, “The Swampland, Quintessence and the Vacuum Energy,” Phys. Lett. B 789 (2019) 639–642, arXiv:1809.00726 [hep-th].
  * [34] A. Hebecker, T. Skrzypek, and M. Wittner, “The $F$-term Problem and other Challenges of Stringy Quintessence,” JHEP 11 (2019) 134, arXiv:1909.08625 [hep-th].
  * [35] D. Anninos, “De Sitter Musings,” Int. J. Mod. Phys. A 27 (2012) 1230013, arXiv:1205.3855 [hep-th].
  * [36] E. Witten, “Quantum gravity in de Sitter space,” in Strings 2001: International Conference.  6, 2001.  arXiv:hep-th/0106109.
  * [37] C. P. Burgess, Introduction to Effective Field Theory.  Cambridge University Press, 12, 2020.
  * [38] J. Martin, “Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask),” Comptes Rendus Physique 13 (2012) 566–665, arXiv:1205.3365 [astro-ph.CO].
  * [39] A. Padilla, “Lectures on the Cosmological Constant Problem,” arXiv:1502.05296 [hep-th].
  * [40] C. P. Burgess, “The Cosmological Constant Problem: Why it’s hard to get Dark Energy from Micro-physics,” in 100e Ecole d’Ete de Physique: Post-Planck Cosmology, pp. 149–197.  2015\.  arXiv:1309.4133 [hep-th].
  * [41] I. Bena, M. Graña, and T. Van Riet, “Trustworthy de Sitter compactifications of string theory: a comprehensive review,” arXiv:2303.17680 [hep-th].
  * [42] E. Mottola, “Particle Creation in de Sitter Space,” Phys. Rev. D 31 (1985) 754.
  * [43] A. M. Polyakov, “De Sitter space and eternity,” Nucl. Phys. B 797 (2008) 199–217, arXiv:0709.2899 [hep-th].
  * [44] A. M. Polyakov, “Decay of Vacuum Energy,” Nucl. Phys. B 834 (2010) 316–329, arXiv:0912.5503 [hep-th].
  * [45] A. M. Polyakov, “Infrared instability of the de Sitter space,” arXiv:1209.4135 [hep-th].
  * [46] S. P. de Alwis, “Radiative Generation of dS from AdS,” arXiv:2110.06967 [hep-th].
  * [47] S. P. De Alwis, “Wilsonian Effective Field Theory and String Theory,” arXiv:2103.13347 [hep-th].
  * [48] U. Danielsson, “The quantum swampland,” JHEP 04 (2019) 095, arXiv:1809.04512 [hep-th].
  * [49] G. Dvali, C. Gomez, and S. Zell, “Quantum Break-Time of de Sitter,” JCAP 06 (2017) 028, arXiv:1701.08776 [hep-th].
  * [50] R. Blumenhagen, C. Kneissl, and A. Makridou, “De Sitter quantum breaking, swampland conjectures and thermal strings,” JHEP 10 (2021) 157, arXiv:2011.13956 [hep-th].
  * [51] P. Breitenlohner and D. Z. Freedman, “Positive Energy in anti-De Sitter Backgrounds and Gauged Extended Supergravity,” Phys. Lett. B 115 (1982) 197–201.
  * [52] M. Dine and N. Seiberg, “Is the Superstring Weakly Coupled?,” Phys. Lett. B 162 (1985) 299–302.
  * [53] D. Junghans, “Weakly Coupled de Sitter Vacua with Fluxes and the Swampland,” JHEP 03 (2019) 150, arXiv:1811.06990 [hep-th].
  * [54] A. Banlaki, A. Chowdhury, C. Roupec, and T. Wrase, “Scaling limits of dS vacua and the swampland,” JHEP 03 (2019) 065, arXiv:1811.07880 [hep-th].
  * [55] A. Bedroya and C. Vafa, “Trans-Planckian Censorship and the Swampland,” JHEP 09 (2020) 123, arXiv:1909.11063 [hep-th].
  * [56] H. Ooguri, E. Palti, G. Shiu, and C. Vafa, “Distance and de Sitter Conjectures on the Swampland,” Phys. Lett. B 788 (2019) 180–184, arXiv:1810.05506 [hep-th].
  * [57] A. Hebecker and T. Wrase, “The Asymptotic dS Swampland Conjecture - a Simplified Derivation and a Potential Loophole,” Fortsch. Phys. 67 no. 1-2, (2019) 1800097, arXiv:1810.08182 [hep-th].
  * [58] H. Ooguri and C. Vafa, “On the Geometry of the String Landscape and the Swampland,” Nucl. Phys. B 766 (2007) 21–33, arXiv:hep-th/0605264.
  * [59] J. Stout, “Infinite Distance Limits and Information Theory,” arXiv:2106.11313 [hep-th].
  * [60] J. Stout, “Infinite Distances and Factorization,” arXiv:2208.08444 [hep-th].
  * [61] X. Gao, A. Hebecker, S. Schreyer, and G. Venken, “Loops, local corrections and warping in the LVS and other type IIB models,” JHEP 09 (2022) 091, arXiv:2204.06009 [hep-th].
  * [62] P. G. O. Freund and M. A. Rubin, “Dynamics of Dimensional Reduction,” Phys. Lett. B 97 (1980) 233–235.
  * [63] D. Roest, “M-theory and gauged supergravities,” Fortsch. Phys. 53 (2005) 119–230, arXiv:hep-th/0408175.
  * [64] T. C. Collins, D. Jafferis, C. Vafa, K. Xu, and S.-T. Yau, “On Upper Bounds in Dimension Gaps of CFT’s,” arXiv:2201.03660 [hep-th].
  * [65] E. Bergshoeff, R. Kallosh, T. Ortin, D. Roest, and A. Van Proeyen, “New formulations of D = 10 supersymmetry and D8 - O8 domain walls,” Class. Quant. Grav. 18 (2001) 3359–3382, arXiv:hep-th/0103233.
  * [66] U. H. Danielsson, S. S. Haque, G. Shiu, and T. Van Riet, “Towards Classical de Sitter Solutions in String Theory,” JHEP 09 (2009) 114, arXiv:0907.2041 [hep-th].
  * [67] M. P. Hertzberg, S. Kachru, W. Taylor, and M. Tegmark, “Inflationary Constraints on Type IIA String Theory,” JHEP 12 (2007) 095, arXiv:0711.2512 [hep-th].
  * [68] E. Silverstein, “Simple de Sitter Solutions,” Phys. Rev. D 77 (2008) 106006, arXiv:0712.1196 [hep-th].
  * [69] T. Wrase and M. Zagermann, “On Classical de Sitter Vacua in String Theory,” Fortsch. Phys. 58 (2010) 906–910, arXiv:1003.0029 [hep-th].
  * [70] T. Van Riet, “On classical de Sitter solutions in higher dimensions,” Class. Quant. Grav. 29 (2012) 055001, arXiv:1111.3154 [hep-th].
  * [71] S. B. Giddings, S. Kachru, and J. Polchinski, “Hierarchies from fluxes in string compactifications,” Phys. Rev. D 66 (2002) 106006, arXiv:hep-th/0105097.
  * [72] K. Dasgupta, G. Rajesh, and S. Sethi, “M theory, orientifolds and G - flux,” JHEP 08 (1999) 023, arXiv:hep-th/9908088.
  * [73] J. Blaback, U. H. Danielsson, D. Junghans, T. Van Riet, T. Wrase, and M. Zagermann, “Smeared versus localised sources in flux compactifications,” JHEP 12 (2010) 043, arXiv:1009.1877 [hep-th].
  * [74] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, “De Sitter vacua in string theory,” Phys. Rev. D 68 (2003) 046005, arXiv:hep-th/0301240.
  * [75] J. M. Maldacena and H. S. Nastase, “The Supergravity dual of a theory with dynamical supersymmetry breaking,” JHEP 09 (2001) 024, arXiv:hep-th/0105049.
  * [76] V. Balasubramanian, P. Berglund, J. P. Conlon, and F. Quevedo, “Systematics of moduli stabilisation in Calabi-Yau flux compactifications,” JHEP 03 (2005) 007, arXiv:hep-th/0502058.
  * [77] I. Bena, J. Blåbäck, M. Graña, and S. Lüst, “The tadpole problem,” JHEP 11 (2021) 223, arXiv:2010.10519 [hep-th].
  * [78] I. Bena, J. Blåbäck, M. Graña, and S. Lüst, “Algorithmically Solving the Tadpole Problem,” Adv. Appl. Clifford Algebras 32 no. 1, (2022) 7, arXiv:2103.03250 [hep-th].
  * [79] F. Marchesano, D. Prieto, and M. Wiesner, “F-theory flux vacua at large complex structure,” JHEP 08 (2021) 077, arXiv:2105.09326 [hep-th].
  * [80] E. Plauschinn, “The tadpole conjecture at large complex-structure,” JHEP 02 (2022) 206, arXiv:2109.00029 [hep-th].
  * [81] S. Lüst, “Large complex structure flux vacua of IIB and the Tadpole Conjecture,” arXiv:2109.05033 [hep-th].
  * [82] T. W. Grimm, E. Plauschinn, and D. van de Heisteeg, “Moduli stabilization in asymptotic flux compactifications,” JHEP 03 (2022) 117, arXiv:2110.05511 [hep-th].
  * [83] M. Graña, T. W. Grimm, D. van de Heisteeg, A. Herraez, and E. Plauschinn, “The tadpole conjecture in asymptotic limits,” JHEP 08 (2022) 237, arXiv:2204.05331 [hep-th].
  * [84] K. Tsagkaris and E. Plauschinn, “Moduli stabilization in type IIB orientifolds at h2,1 = 50,” JHEP 03 (2023) 049, arXiv:2207.13721 [hep-th].
  * [85] T. Coudarchet, F. Marchesano, D. Prieto, and M. A. Urkiola, “Symmetric fluxes and small tadpoles,” arXiv:2304.04789 [hep-th].
  * [86] S. Lüst and M. Wiesner, “The Tadpole Conjecture in the Interior of Moduli Space,” arXiv:2211.05128 [hep-th].
  * [87] M. Grana, R. Minasian, M. Petrini, and A. Tomasiello, “A Scan for new N=1 vacua on twisted tori,” JHEP 05 (2007) 031, arXiv:hep-th/0609124.
  * [88] O. DeWolfe, A. Giryavets, S. Kachru, and W. Taylor, “Type IIA moduli stabilization,” JHEP 07 (2005) 066, arXiv:hep-th/0505160.
  * [89] M. R. Douglas and R. Kallosh, “Compactification on negatively curved manifolds,” JHEP 06 (2010) 004, arXiv:1001.4008 [hep-th].
  * [90] S. S. Haque, G. Shiu, B. Underwood, and T. Van Riet, “Minimal simple de Sitter solutions,” Phys. Rev. D 79 (2009) 086005, arXiv:0810.5328 [hep-th].
  * [91] J. M. Maldacena and C. Nunez, “Supergravity description of field theories on curved manifolds and a no go theorem,” Int. J. Mod. Phys. A 16 (2001) 822–855, arXiv:hep-th/0007018.
  * [92] B. de Wit, D. J. Smit, and N. D. Hari Dass, “Residual Supersymmetry of Compactified D=10 Supergravity,” Nucl. Phys. B 283 (1987) 165.
  * [93] G. Shiu and Y. Sumitomo, “Stability Constraints on Classical de Sitter Vacua,” JHEP 09 (2011) 052, arXiv:1107.2925 [hep-th].
  * [94] R. Emparan and J. Garriga, “A Note on accelerating cosmologies from compactifications and S branes,” JHEP 05 (2003) 028, arXiv:hep-th/0304124.
  * [95] P. Marconnet and D. Tsimpis, “Universal accelerating cosmologies from 10d supergravity,” JHEP 01 (2023) 033, arXiv:2210.10813 [hep-th].
  * [96] J. Blåbäck, U. Danielsson, and G. Dibitetto, “Accelerated Universes from type IIA Compactifications,” JCAP 03 (2014) 003, arXiv:1310.8300 [hep-th].
  * [97] R. Flauger, S. Paban, D. Robbins, and T. Wrase, “Searching for slow-roll moduli inflation in massive type IIA supergravity with metric fluxes,” Phys. Rev. D 79 (2009) 086011, arXiv:0812.3886 [hep-th].
  * [98] C. Caviezel, P. Koerber, S. Kors, D. Lust, T. Wrase, and M. Zagermann, “On the Cosmology of Type IIA Compactifications on SU(3)-structure Manifolds,” JHEP 04 (2009) 010, arXiv:0812.3551 [hep-th].
  * [99] U. H. Danielsson, S. S. Haque, P. Koerber, G. Shiu, T. Van Riet, and T. Wrase, “De Sitter hunting in a classical landscape,” Fortsch. Phys. 59 (2011) 897–933, arXiv:1103.4858 [hep-th].
  * [100] D. Andriot, L. Horer, and P. Marconnet, “Exploring the landscape of (anti-) de Sitter and Minkowski solutions: group manifolds, stability and scale separation,” JHEP 08 (2022) 109, arXiv:2204.05327 [hep-th]. [Erratum: JHEP 09, 184 (2022)].
  * [101] D. Andriot, “Open problems on classical de Sitter solutions,” Fortsch. Phys. 67 no. 7, (2019) 1900026, arXiv:1902.10093 [hep-th].
  * [102] U. H. Danielsson, G. Shiu, T. Van Riet, and T. Wrase, “A note on obstinate tachyons in classical dS solutions,” JHEP 03 (2013) 138, arXiv:1212.5178 [hep-th].
  * [103] D. Andriot, “Tachyonic de Sitter Solutions of 10d Type II Supergravities,” Fortsch. Phys. 69 no. 7, (2021) 2100063, arXiv:2101.06251 [hep-th].
  * [104] D. Andriot, “On the de Sitter swampland criterion,” Phys. Lett. B 785 (2018) 570–573, arXiv:1806.10999 [hep-th].
  * [105] S. K. Garg and C. Krishnan, “Bounds on Slow Roll and the de Sitter Swampland,” JHEP 11 (2019) 075, arXiv:1807.05193 [hep-th].
  * [106] F. Denef, A. Hebecker, and T. Wrase, “de Sitter swampland conjecture and the Higgs potential,” Phys. Rev. D 98 no. 8, (2018) 086004, arXiv:1807.06581 [hep-th].
  * [107] M. Montero, T. Van Riet, and G. Venken, “Festina Lente: EFT Constraints from Charged Black Hole Evaporation in de Sitter,” JHEP 01 (2020) 039, arXiv:1910.01648 [hep-th].
  * [108] V. Guidetti, N. Righi, G. Venken, and A. Westphal, “Axionic Festina Lente,” JHEP 01 (2023) 114, arXiv:2206.03494 [hep-th].
  * [109] M. Montero, C. Vafa, T. Van Riet, and G. Venken, “The FL bound and its phenomenological implications,” JHEP 10 (2021) 009, arXiv:2106.07650 [hep-th].
  * [110] S. M. Lee, D. Y. Cheong, S. C. Hyun, S. C. Park, and M.-S. Seo, “Festina-Lente bound on Higgs vacuum structure and inflation,” JHEP 02 (2022) 100, arXiv:2111.04010 [hep-ph].
  * [111] G. Veneziano, “Large N bounds on, and compositeness limit of, gauge and gravitational interactions,” JHEP 06 (2002) 051, arXiv:hep-th/0110129.
  * [112] N. Arkani-Hamed, S. Dimopoulos, and S. Kachru, “Predictive landscapes and new physics at a TeV,” arXiv:hep-th/0501082.
  * [113] G. Dvali, “Black Holes and Large N Species Solution to the Hierarchy Problem,” Fortsch. Phys. 58 (2010) 528–536, arXiv:0706.2050 [hep-th].
  * [114] G. Dvali and M. Redi, “Black Hole Bound on the Number of Species and Quantum Gravity at LHC,” Phys. Rev. D 77 (2008) 045027, arXiv:0710.4344 [hep-th].
  * [115] A. Bedroya, R. Brandenberger, M. Loverde, and C. Vafa, “Trans-Planckian Censorship and Inflationary Cosmology,” Phys. Rev. D 101 no. 10, (2020) 103502, arXiv:1909.11106 [hep-th].
  * [116] T. Rudelius, “Conditions for (No) Eternal Inflation,” JCAP 08 (2019) 009, arXiv:1905.05198 [hep-th].
  * [117] T. Rudelius, “Dimensional reduction and (Anti) de Sitter bounds,” JHEP 08 (2021) 041, arXiv:2101.11617 [hep-th].
  * [118] D. Andriot and L. Horer, “(Quasi-) de Sitter solutions across dimensions and the TCC bound,” JHEP 01 (2023) 020, arXiv:2208.14462 [hep-th].
  * [119] C. P. Burgess, S. P. de Alwis, and F. Quevedo, “Cosmological Trans-Planckian Conjectures are not Effective,” JCAP 05 (2021) 037, arXiv:2011.03069 [hep-th].
  * [120] M. Demirtas, M. Kim, L. Mcallister, and J. Moritz, “Vacua with Small Flux Superpotential,” Phys. Rev. Lett. 124 no. 21, (2020) 211603, arXiv:1912.10047 [hep-th].
  * [121] M. Demirtas, M. Kim, L. McAllister, J. Moritz, and A. Rios-Tascon, “Exponentially Small Cosmological Constant in String Theory,” Phys. Rev. Lett. 128 no. 1, (2022) 011602, arXiv:2107.09065 [hep-th].
  * [122] R. Álvarez-García, R. Blumenhagen, M. Brinkmann, and L. Schlechter, “Small Flux Superpotentials for Type IIB Flux Vacua Close to a Conifold,” Fortsch. Phys. 68 (2020) 2000088, arXiv:2009.03325 [hep-th].
  * [123] M. Kim, “On string one-loop correction to the Einstein-Hilbert term and its implications on the Kahler potential,” arXiv:2302.12117 [hep-th].
  * [124] S. Sethi, “Supersymmetry Breaking by Fluxes,” JHEP 10 (2018) 022, arXiv:1709.03554 [hep-th].
  * [125] S. Kachru and S. P. Trivedi, “A comment on effective field theories of flux vacua,” Fortsch. Phys. 67 no. 1-2, (2019) 1800086, arXiv:1808.08971 [hep-th].
  * [126] S. Lüst, C. Vafa, M. Wiesner, and K. Xu, “Holography and the KKLT scenario,” JHEP 10 (2022) 188, arXiv:2204.07171 [hep-th].
  * [127] S. Kachru, J. Pearson, and H. L. Verlinde, “Brane / flux annihilation and the string dual of a nonsupersymmetric field theory,” JHEP 06 (2002) 021, arXiv:hep-th/0112197.
  * [128] I. R. Klebanov and M. J. Strassler, “Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities,” JHEP 08 (2000) 052, arXiv:hep-th/0007191.
  * [129] I. Bena, M. Grana, and N. Halmagyi, “On the Existence of Meta-stable Vacua in Klebanov-Strassler,” JHEP 09 (2010) 087, arXiv:0912.3519 [hep-th].
  * [130] F. F. Gautason, D. Junghans, and M. Zagermann, “Cosmological Constant, Near Brane Behavior and Singularities,” JHEP 09 (2013) 123, arXiv:1301.5647 [hep-th].
  * [131] J. Blåbäck, U. H. Danielsson, D. Junghans, T. Van Riet, and S. C. Vargas, “Localised anti-branes in non-compact throats at zero and finite $T$,” JHEP 02 (2015) 018, arXiv:1409.0534 [hep-th].
  * [132] J. Blaback, U. H. Danielsson, and T. Van Riet, “Resolving anti-brane singularities through time-dependence,” JHEP 02 (2013) 061, arXiv:1202.1132 [hep-th].
  * [133] F. Carta, J. Moritz, and A. Westphal, “Gaugino condensation and small uplifts in KKLT,” JHEP 08 (2019) 141, arXiv:1902.01412 [hep-th].
  * [134] X. Gao, A. Hebecker, and D. Junghans, “Control issues of KKLT,” Fortsch. Phys. 68 (2020) 2000089, arXiv:2009.03914 [hep-th].
  * [135] D. Junghans, “LVS de Sitter Vacua are probably in the Swampland,” arXiv:2201.03572 [hep-th].
  * [136] X. Gao, A. Hebecker, S. Schreyer, and G. Venken, “The LVS parametric tadpole constraint,” JHEP 07 (2022) 056, arXiv:2202.04087 [hep-th].
  * [137] I. Bena, E. Dudas, M. Graña, and S. Lüst, “Uplifting Runaways,” Fortsch. Phys. 67 no. 1-2, (2019) 1800100, arXiv:1809.06861 [hep-th].
  * [138] L. Randall, “The Boundaries of KKLT,” Fortsch. Phys. 68 no. 3-4, (2020) 1900105, arXiv:1912.06693 [hep-th].
  * [139] E. Dudas and S. Lüst, “An update on moduli stabilization with antibrane uplift,” JHEP 03 (2021) 107, arXiv:1912.09948 [hep-th].
  * [140] I. Bena, A. Buchel, and S. Lüst, “Throat destabilization (for profit and for fun),” arXiv:1910.08094 [hep-th].
  * [141] M. Scalisi, P. Soler, V. Van Hemelryck, and T. Van Riet, “Conifold dynamics and axion monodromies,” JHEP 10 (2020) 133, arXiv:2007.15391 [hep-th].
  * [142] R. Blumenhagen, D. Herschmann, and F. Wolf, “Challenges for Moduli Stabilization and String Cosmology near the Conifold,” PoS CORFU2016 (2017) 104, arXiv:1704.04140 [hep-th].
  * [143] P. Koerber and L. Martucci, “From ten to four and back again: How to generalize the geometry,” JHEP 08 (2007) 059, arXiv:0707.1038 [hep-th].
  * [144] M. Graña, N. Kovensky, and A. Retolaza, “Gaugino mass term for D-branes and Generalized Complex Geometry,” JHEP 06 (2020) 047, arXiv:2002.01481 [hep-th].
  * [145] M. Graña, N. Kovensky, and D. Toulikas, “Smearing and unsmearing KKLT AdS vacua,” JHEP 03 (2023) 015, arXiv:2212.05074 [hep-th].
  * [146] J. Moritz, A. Retolaza, and A. Westphal, “Toward de Sitter space from ten dimensions,” Phys. Rev. D 97 no. 4, (2018) 046010, arXiv:1707.08678 [hep-th].
  * [147] G. Dall’Agata, M. Emelin, F. Farakos, and M. Morittu, “Anti-brane uplift instability from goldstino condensation,” JHEP 08 (2022) 005, arXiv:2203.12636 [hep-th].
  * [148] F. Farakos and M. Morittu, “Goldstino condensation at large N,” Eur. Phys. J. C 83 no. 2, (2023) 166, arXiv:2211.12527 [hep-th].
  * [149] D. Kutasov, T. Maxfield, I. Melnikov, and S. Sethi, “Constraining de Sitter Space in String Theory,” Phys. Rev. Lett. 115 no. 7, (2015) 071305, arXiv:1504.00056 [hep-th].
  * [150] S. R. Green, E. J. Martinec, C. Quigley, and S. Sethi, “Constraints on String Cosmology,” Class. Quant. Grav. 29 (2012) 075006, arXiv:1110.0545 [hep-th].
  * [151] C. Quigley, “Gaugino Condensation and the Cosmological Constant,” JHEP 06 (2015) 104, arXiv:1504.00652 [hep-th].
  * [152] E. Plauschinn, “Moduli Stabilization with Non-Geometric Fluxes — Comments on Tadpole Contributions and de-Sitter Vacua,” Fortsch. Phys. 69 no. 3, (2021) 2100003, arXiv:2011.08227 [hep-th].
  * [153] T. W. Grimm, C. Li, and I. Valenzuela, “Asymptotic Flux Compactifications and the Swampland,” JHEP 06 (2020) 009, arXiv:1910.09549 [hep-th]. [Erratum: JHEP 01, 007 (2021)].
  * [154] E. Plauschinn, “Non-geometric backgrounds in string theory,” Phys. Rept. 798 (2019) 1–122, arXiv:1811.11203 [hep-th].
  * [155] P. Agrawal, G. Obied, P. J. Steinhardt, and C. Vafa, “On the Cosmological Implications of the String Swampland,” Phys. Lett. B 784 (2018) 271–276, arXiv:1806.09718 [hep-th].
  * [156] M. Cicoli, F. Cunillera, A. Padilla, and F. G. Pedro, “Quintessence and the Swampland: The Parametrically Controlled Regime of Moduli Space,” Fortsch. Phys. 70 no. 4, (2022) 2200009, arXiv:2112.10779 [hep-th].
  * [157] M. Cicoli, F. Cunillera, A. Padilla, and F. G. Pedro, “Quintessence and the Swampland: The Numerically Controlled Regime of Moduli Space,” Fortsch. Phys. 70 no. 4, (2022) 2200008, arXiv:2112.10783 [hep-th].
  * [158] G. Shiu, F. Tonioni, and H. V. Tran, “Accelerating universe at the end of time,” arXiv:2303.03418 [hep-th].
  * [159] B. Michel, E. Mintun, J. Polchinski, A. Puhm, and P. Saad, “Remarks on brane and antibrane dynamics,” JHEP 09 (2015) 021, arXiv:1412.5702 [hep-th].
  * [160] J. Polchinski, “Brane/antibrane dynamics and KKLT stability,” arXiv:1509.05710 [hep-th].
  * [161] I. Bena, M. Graña, S. Kuperstein, and S. Massai, “Giant Tachyons in the Landscape,” JHEP 02 (2015) 146, arXiv:1410.7776 [hep-th].
  * [162] I. Bena, J. Blåbäck, and D. Turton, “Loop corrections to the antibrane potential,” JHEP 07 (2016) 132, arXiv:1602.05959 [hep-th].
  * [163] D. Cohen-Maldonado, J. Diaz, T. van Riet, and B. Vercnocke, “Observations on fluxes near anti-branes,” JHEP 01 (2016) 126, arXiv:1507.01022 [hep-th].
  * [164] D. Cohen-Maldonado, J. Diaz, and F. F. Gautason, “Polarised antibranes from Smarr relations,” JHEP 05 (2016) 175, arXiv:1603.05678 [hep-th].
  * [165] J. Armas, N. Nguyen, V. Niarchos, N. A. Obers, and T. Van Riet, “Meta-stable non-extremal anti-branes,” Phys. Rev. Lett. 122 no. 18, (2019) 181601, arXiv:1812.01067 [hep-th].
  * [166] J. Armas, N. Nguyen, V. Niarchos, and N. A. Obers, “Thermal transitions of metastable M-branes,” JHEP 08 (2019) 128, arXiv:1904.13283 [hep-th].
  * [167] J. Blåbäck, F. F. Gautason, A. Ruipérez, and T. Van Riet, “Anti-brane singularities as red herrings,” JHEP 12 (2019) 125, arXiv:1907.05295 [hep-th].
  * [168] N. Nguyen, “Comments on the stability of the KPV state,” JHEP 11 (2020) 055, arXiv:1912.04646 [hep-th].
  * [169] N. Nguyen and V. Niarchos, “On matched asymptotic expansions of backreacting metastable anti-branes,” JHEP 06 (2022) 055, arXiv:2112.04514 [hep-th].
  * [170] G. S. Hartnett, “Localised Anti-Branes in Flux Backgrounds,” JHEP 06 (2015) 007, arXiv:1501.06568 [hep-th].
  * [171] F. Carta and J. Moritz, “Resolving spacetime singularities in flux compactifications & KKLT,” JHEP 08 (2021) 093, arXiv:2101.05281 [hep-th].
  * [172] S. Lüst and L. Randall, “Effective Theory of Warped Compactifications and the Implications for KKLT,” Fortsch. Phys. 70 no. 7-8, (2022) 2200103, arXiv:2206.04708 [hep-th].
  * [173] R. Blumenhagen, D. Kläwer, and L. Schlechter, “Swampland Variations on a Theme by KKLT,” JHEP 05 (2019) 152, arXiv:1902.07724 [hep-th].
  * [174] M.-S. Seo, “Uplift and towers of states in warped throat,” arXiv:2303.10237 [hep-th].
  * [175] F. F. Gautason, V. Van Hemelryck, and T. Van Riet, “The Tension between 10D Supergravity and dS Uplifts,” Fortsch. Phys. 67 no. 1-2, (2019) 1800091, arXiv:1810.08518 [hep-th].
  * [176] Y. Hamada, A. Hebecker, G. Shiu, and P. Soler, “On brane gaugino condensates in 10d,” JHEP 04 (2019) 008, arXiv:1812.06097 [hep-th].
  * [177] Y. Hamada, A. Hebecker, G. Shiu, and P. Soler, “Completing the D7-brane local gaugino action,” JHEP 11 (2021) 033, arXiv:2105.11467 [hep-th].
  * [178] R. Kallosh, “Gaugino Condensation and Geometry of the Perfect Square,” Phys. Rev. D 99 no. 6, (2019) 066003, arXiv:1901.02023 [hep-th].
  * [179] I. Bena, M. Graña, N. Kovensky, and A. Retolaza, “Kähler moduli stabilization from ten dimensions,” JHEP 10 (2019) 200, arXiv:1908.01785 [hep-th].
  * [180] S. Kachru, M. Kim, L. Mcallister, and M. Zimet, “de Sitter vacua from ten dimensions,” JHEP 12 (2021) 111, arXiv:1908.04788 [hep-th].
  * [181] F. F. Gautason, V. Van Hemelryck, T. Van Riet, and G. Venken, “A 10d view on the KKLT AdS vacuum and uplifting,” JHEP 06 (2020) 074, arXiv:1902.01415 [hep-th].
  * [182] Y. Hamada, A. Hebecker, G. Shiu, and P. Soler, “Understanding KKLT from a 10d perspective,” JHEP 06 (2019) 019, arXiv:1902.01410 [hep-th].
  * [183] G. B. De Luca, E. Silverstein, and G. Torroba, “Hyperbolic compactification of M-theory and de Sitter quantum gravity,” SciPost Phys. 12 no. 3, (2022) 083, arXiv:2104.13380 [hep-th].
  * [184] S. Banerjee, U. Danielsson, G. Dibitetto, S. Giri, and M. Schillo, “Emergent de Sitter Cosmology from Decaying Anti–de Sitter Space,” Phys. Rev. Lett. 121 no. 26, (2018) 261301, arXiv:1807.01570 [hep-th].
  * [185] S. Banerjee, U. Danielsson, G. Dibitetto, S. Giri, and M. Schillo, “de Sitter Cosmology on an expanding bubble,” JHEP 10 (2019) 164, arXiv:1907.04268 [hep-th].
  * [186] S. Banerjee, U. Danielsson, and S. Giri, “Dark Bubble: FAQs. Misconceptions, and why it is not Randall-Sundrum,” arXiv:2212.14004 [hep-th].
  * [187] L. Randall and R. Sundrum, “An Alternative to compactification,” Phys. Rev. Lett. 83 (1999) 4690–4693, arXiv:hep-th/9906064.
  * [188] A. Karch and L. Randall, “Locally localized gravity,” JHEP 05 (2001) 008, arXiv:hep-th/0011156.
  * [189] H. Ooguri and C. Vafa, “Non-supersymmetric AdS and the Swampland,” Adv. Theor. Math. Phys. 21 (2017) 1787–1801, arXiv:1610.01533 [hep-th].
  * [190] J. D. Brown and C. Teitelboim, “Neutralization of the Cosmological Constant by Membrane Creation,” Nucl. Phys. B 297 (1988) 787–836.
  * [191] U. Danielsson, O. Henriksson, and D. Panizo, “Stringy realization of a small and positive cosmological constant in dark bubble cosmology,” Phys. Rev. D 107 no. 2, (2023) 026020, arXiv:2211.10191 [hep-th].
  * [192] I. Basile and S. Lanza, “de Sitter in non-supersymmetric string theories: no-go theorems and brane-worlds,” JHEP 10 (2020) 108, arXiv:2007.13757 [hep-th].
  * [193] M. Mirbabayi, “A comment about the cosmology on a bubble wall,” arXiv:2210.14276 [hep-th].
  * [194] A. Vilenkin, “Creation of Universes from Nothing,” Phys. Lett. B 117 (1982) 25–28.
  * [195] J. B. Hartle and S. W. Hawking, “Wave Function of the Universe,” Phys. Rev. D 28 (1983) 2960–2975.
  * [196] U. H. Danielsson, D. Panizo, R. Tielemans, and T. Van Riet, “Higher-dimensional view on quantum cosmology,” Phys. Rev. D 104 no. 8, (2021) 086015, arXiv:2105.03253 [hep-th].
  * [197] C. Córdova, G. B. De Luca, and A. Tomasiello, “Classical de Sitter Solutions of 10-Dimensional Supergravity,” Phys. Rev. Lett. 122 no. 9, (2019) 091601, arXiv:1812.04147 [hep-th].
  * [198] C. Córdova, G. B. De Luca, and A. Tomasiello, “New de Sitter Solutions in Ten Dimensions and Orientifold Singularities,” JHEP 08 (2020) 093, arXiv:1911.04498 [hep-th].
  * [199] N. Cribiori and D. Junghans, “No classical (anti-)de Sitter solutions with O8-planes,” Phys. Lett. B 793 (2019) 54–58, arXiv:1902.08209 [hep-th].
  * [200] I. Bena, G. B. De Luca, M. Graña, and G. Lo Monaco, “Oh, wait, O8 de Sitter may be unstable!,” JHEP 03 (2021) 168, arXiv:2010.05936 [hep-th].
  * [201] F. Farakos, G. Tringas, and T. Van Riet, “Classical de Sitter solutions in three dimensions without tachyons?,” Eur. Phys. J. C 80 no. 10, (2020) 947, arXiv:2007.12084 [hep-th].
  * [202] P. Berglund, T. Hübsch, and D. Minic, “On de Sitter Spacetime and String Theory,” arXiv:2212.06086 [hep-th].
  * [203] K. Dasgupta, M. Emelin, M. Mehedi Faruk, and R. Tatar, “de Sitter Vacua in the String Landscape: La Petite Version,” in 11th International Symposium on Quantum Theory and Symmetries.  11, 2019.  arXiv:1911.12382 [hep-th].
  * [204] K. Dasgupta, M. Emelin, M. M. Faruk, and R. Tatar, “How a four-dimensional de Sitter solution remains outside the swampland,” JHEP 07 (2021) 109, arXiv:1911.02604 [hep-th].
  * [205] K. Dasgupta, M. Emelin, M. M. Faruk, and R. Tatar, “de Sitter vacua in the string landscape,” Nucl. Phys. B 969 (2021) 115463, arXiv:1908.05288 [hep-th].
  * [206] M. Cicoli, I. n. G. Etxebarria, F. Quevedo, A. Schachner, P. Shukla, and R. Valandro, “The Standard Model quiver in de Sitter string compactifications,” JHEP 08 (2021) 109, arXiv:2106.11964 [hep-th].
  * [207] P. Shukla, “On stable type IIA de-Sitter vacua with geometric flux,” Eur. Phys. J. C 83 no. 3, (2023) 196, arXiv:2202.12840 [hep-th].
  * [208] S. AbdusSalam, C. Crinò, and P. Shukla, “On K3-fibred LARGE Volume Scenario with de Sitter vacua from anti-D3-branes,” JHEP 03 (2023) 132, arXiv:2206.12889 [hep-th].
  * [209] B. V. Bento, D. Chakraborty, S. L. Parameswaran, and I. Zavala, “A new de Sitter solution with a weakly warped deformed conifold,” JHEP 12 (2021) 124, arXiv:2105.03370 [hep-th].
  * [210] I. Bena, E. Dudas, M. Graña, G. L. Monaco, and D. Toulikas, “Bare-Bones de Sitter,” arXiv:2202.02327 [hep-th].
  * [211] I. Bena and P. Heidmann, “The Uplifton,” arXiv:2212.04506 [hep-th].
  * [212] S. Alexander, K. Dasgupta, A. Maji, P. Ramadevi, and R. Tatar, “de Sitter State in Heterotic String Theory,” arXiv:2303.12843 [hep-th].
  * [213] G. K. Leontaris and P. Shukla, “Seeking de-Sitter Vacua in the String Landscape,”  3, 2023.  arXiv:2303.16689 [hep-th].
  * [214] L. F. Alday and E. Perlmutter, “Growing Extra Dimensions in AdS/CFT,” JHEP 08 (2019) 084, arXiv:1906.01477 [hep-th].
  * [215] J. Polchinski and E. Silverstein, “Dual purpose landscaping tools: small extra dimensions in ads/cft,” in Strings, gauge fields, and the geometry behind: The legacy of Maximilian Kreuzer, pp. 365–390.  World Scientific, 2013.
  * [216] D. Tsimpis, “Supersymmetric AdS vacua and separation of scales,” JHEP 08 (2012) 142, arXiv:1206.5900 [hep-th].
  * [217] G. B. De Luca, N. De Ponti, A. Mondino, and A. Tomasiello, “Gravity from thermodynamics: optimal transport and negative effective dimensions,” arXiv:2212.02511 [hep-th].
  * [218] N. Cribiori, D. Junghans, V. Van Hemelryck, T. Van Riet, and T. Wrase, “Scale-separated AdS4 vacua of IIA orientifolds and M-theory,” Phys. Rev. D 104 no. 12, (2021) 126014, arXiv:2107.00019 [hep-th].
  * [219] D. Andriot and D. Tsimpis, “Laplacian spectrum on a nilmanifold, truncations and effective theories,” JHEP 09 (2018) 096, arXiv:1806.05156 [hep-th].
  * [220] M. Petrini, G. Solard, and T. Van Riet, “AdS vacua with scale separation from IIB supergravity,” JHEP 11 (2013) 010, arXiv:1308.1265 [hep-th].
  * [221] C. Caviezel, T. Wrase, and M. Zagermann, “Moduli Stabilization and Cosmology of Type IIB on SU(2)-Structure Orientifolds,” JHEP 04 (2010) 011, arXiv:0912.3287 [hep-th].
  * [222] M. Emelin, F. Farakos, and G. Tringas, “Three-dimensional flux vacua from IIB on co-calibrated G2 orientifolds,” Eur. Phys. J. C 81 no. 5, (2021) 456, arXiv:2103.03282 [hep-th].
  * [223] F. Farakos, G. Tringas, and T. Van Riet, “No-scale and scale-separated flux vacua from IIA on G2 orientifolds,” Eur. Phys. J. C 80 no. 7, (2020) 659, arXiv:2005.05246 [hep-th].
  * [224] V. Van Hemelryck, “Scale-Separated AdS3 Vacua from G2-Orientifolds Using Bispinors,” Fortsch. Phys. 70 no. 12, (2022) 2200128, arXiv:2207.14311 [hep-th].
  * [225] F. Farakos, M. Morittu, and G. Tringas, “On/off scale separation,” arXiv:2304.14372 [hep-th].
  * [226] J.-P. Derendinger, C. Kounnas, P. M. Petropoulos, and F. Zwirner, “Superpotentials in IIA compactifications with general fluxes,” Nucl. Phys. B 715 (2005) 211–233, arXiv:hep-th/0411276.
  * [227] P. G. Camara, A. Font, and L. E. Ibanez, “Fluxes, moduli fixing and MSSM-like vacua in a simple IIA orientifold,” JHEP 09 (2005) 013, arXiv:hep-th/0506066.
  * [228] A. Font, A. Herráez, and L. E. Ibáñez, “On scale separation in type II AdS flux vacua,” JHEP 03 (2020) 013, arXiv:1912.03317 [hep-th].
  * [229] T. W. Grimm and J. Louis, “The Effective action of type IIA Calabi-Yau orientifolds,” Nucl. Phys. B 718 (2005) 153–202, arXiv:hep-th/0412277.
  * [230] D. Lust, F. Marchesano, L. Martucci, and D. Tsimpis, “Generalized non-supersymmetric flux vacua,” JHEP 11 (2008) 021, arXiv:0807.4540 [hep-th].
  * [231] F. Marchesano, D. Prieto, J. Quirant, and P. Shukla, “Systematics of Type IIA moduli stabilisation,” JHEP 11 (2020) 113, arXiv:2007.00672 [hep-th].
  * [232] P. Koerber and D. Tsimpis, “Supersymmetric sources, integrability and generalized-structure compactifications,” JHEP 08 (2007) 082, arXiv:0706.1244 [hep-th].
  * [233] B. S. Acharya, F. Benini, and R. Valandro, “Fixing moduli in exact type IIA flux vacua,” JHEP 02 (2007) 018, arXiv:hep-th/0607223.
  * [234] C. Caviezel, P. Koerber, S. Kors, D. Lust, D. Tsimpis, and M. Zagermann, “The Effective theory of type IIA AdS(4) compactifications on nilmanifolds and cosets,” Class. Quant. Grav. 26 (2009) 025014, arXiv:0806.3458 [hep-th].
  * [235] J. McOrist and S. Sethi, “M-theory and Type IIA Flux Compactifications,” JHEP 12 (2012) 122, arXiv:1208.0261 [hep-th].
  * [236] T. Banks and K. van den Broek, “Massive IIA flux compactifications and U-dualities,” JHEP 03 (2007) 068, arXiv:hep-th/0611185.
  * [237] S. Baines and T. Van Riet, “Smearing orientifolds in flux compactifications can be OK,” Class. Quant. Grav. 37 no. 19, (2020) 195015, arXiv:2005.09501 [hep-th].
  * [238] D. Junghans, “O-Plane Backreaction and Scale Separation in Type IIA Flux Vacua,” Fortsch. Phys. 68 no. 6, (2020) 2000040, arXiv:2003.06274 [hep-th].
  * [239] F. Saracco and A. Tomasiello, “Localized O6-plane solutions with Romans mass,” JHEP 07 (2012) 077, arXiv:1201.5378 [hep-th].
  * [240] F. Marchesano, E. Palti, J. Quirant, and A. Tomasiello, “On supersymmetric AdS4 orientifold vacua,” JHEP 08 (2020) 087, arXiv:2003.13578 [hep-th].
  * [241] M. Emelin, F. Farakos, and G. Tringas, “O6-plane backreaction on scale-separated Type IIA AdS3 vacua,” JHEP 07 (2022) 133, arXiv:2202.13431 [hep-th].
  * [242] M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir effect, vol. 145.  Oxford University Press, 2009.
  * [243] S. Giombi and E. Perlmutter, “Double-Trace Flows and the Swampland,” JHEP 03 (2018) 026, arXiv:1709.09159 [hep-th].
  * [244] F. Baume and J. Calderón Infante, “Tackling the SDC in AdS with CFTs,” JHEP 08 (2021) 057, arXiv:2011.03583 [hep-th].
  * [245] E. Perlmutter, L. Rastelli, C. Vafa, and I. Valenzuela, “A CFT distance conjecture,” JHEP 10 (2021) 070, arXiv:2011.10040 [hep-th].
  * [246] D. Harlow and H. Ooguri, “Constraints on Symmetries from Holography,” Phys. Rev. Lett. 122 no. 19, (2019) 191601, arXiv:1810.05337 [hep-th].
  * [247] D. Harlow, “Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture,” JHEP 01 (2016) 122, arXiv:1510.07911 [hep-th].
  * [248] D. Harlow and E. Shaghoulian, “Global symmetry, Euclidean gravity, and the black hole information problem,” JHEP 04 (2021) 175, arXiv:2010.10539 [hep-th].
  * [249] O. Aharony and E. Palti, “Convexity of charged operators in CFTs and the weak gravity conjecture,” Phys. Rev. D 104 no. 12, (2021) 126005, arXiv:2108.04594 [hep-th].
  * [250] J. P. Conlon and F. Revello, “Moduli Stabilisation and the Holographic Swampland,” LHEP 2020 (2020) 171, arXiv:2006.01021 [hep-th].
  * [251] F. Baume and J. Calderón-Infante, “On Higher-Spin Points and Infinite Distances in Conformal Manifolds,” arXiv:2305.05693 [hep-th].
  * [252] I. Heemskerk, J. Penedones, J. Polchinski, and J. Sully, “Holography from Conformal Field Theory,” JHEP 10 (2009) 079, arXiv:0907.0151 [hep-th].
  * [253] J. P. Conlon, S. Ning, and F. Revello, “Exploring the holographic Swampland,” JHEP 04 (2022) 117, arXiv:2110.06245 [hep-th].
  * [254] J. P. Conlon and F. Quevedo, “Putting the Boot into the Swampland,” JHEP 03 (2019) 005, arXiv:1811.06276 [hep-th].
  * [255] F. Apers, M. Montero, T. Van Riet, and T. Wrase, “Comments on classical AdS flux vacua with scale separation,” JHEP 05 (2022) 167, arXiv:2202.00682 [hep-th].
  * [256] F. Apers, J. P. Conlon, S. Ning, and F. Revello, “Integer conformal dimensions for type IIa flux vacua,” Phys. Rev. D 105 no. 10, (2022) 106029, arXiv:2202.09330 [hep-th].
  * [257] F. Apers, “Aspects of AdS flux vacua with integer conformal dimensions,” arXiv:2211.04187 [hep-th].
  * [258] M. Montero, M. Rocek, and C. Vafa, “Pure supersymmetric AdS and the Swampland,” JHEP 01 (2023) 094, arXiv:2212.01697 [hep-th].
  * [259] E. Plauschinn, “Mass spectrum of type IIB flux compactifications — comments on AdS vacua and conformal dimensions,” JHEP 02 (2023) 257, arXiv:2210.04528 [hep-th].
  * [260] O. Aharony, Y. E. Antebi, and M. Berkooz, “On the Conformal Field Theory Duals of type IIA AdS(4) Flux Compactifications,” JHEP 02 (2008) 093, arXiv:0801.3326 [hep-th].
  * [261] J. Quirant, “Noninteger conformal dimensions for type IIA flux vacua,” Phys. Rev. D 106 no. 6, (2022) 066017, arXiv:2204.00014 [hep-th].
  * [262] A. Castellano, A. Font, A. Herraez, and L. E. Ibáñez, “A gravitino distance conjecture,” JHEP 08 (2021) 092, arXiv:2104.10181 [hep-th].
  * [263] N. Cribiori, D. Lust, and M. Scalisi, “The gravitino and the swampland,” JHEP 06 (2021) 071, arXiv:2104.08288 [hep-th].
  * [264] N. Cribiori and C. Montella, “Quantum gravity constraints on scale separation and de Sitter in five dimensions,” arXiv:2303.04162 [hep-th].
  * [265] N. Cribiori and G. Dall’Agata, “Weak gravity versus scale separation,” JHEP 06 (2022) 006, arXiv:2203.05559 [hep-th].
  * [266] N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, “The String landscape, black holes and gravity as the weakest force,” JHEP 06 (2007) 060, arXiv:hep-th/0601001.
  * [267] G. Shiu, F. Tonioni, V. Van Hemelryck, and T. Van Riet, “AdS scale separation and the distance conjecture,” arXiv:2212.06169 [hep-th].
  * [268] A. Saltman and E. Silverstein, “The Scaling of the no scale potential and de Sitter model building,” JHEP 11 (2004) 066, arXiv:hep-th/0402135.
  * [269] R. Blumenhagen, M. Brinkmann, and A. Makridou, “Quantum Log-Corrections to Swampland Conjectures,” JHEP 02 (2020) 064, arXiv:1910.10185 [hep-th].
  * [270] R. Kallosh and A. D. Linde, “Landscape, the scale of SUSY breaking, and inflation,” JHEP 12 (2004) 004, arXiv:hep-th/0411011.
  * [271] J. Polchinski, “Tasi lectures on D-branes,” in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 96): Fields, Strings, and Duality, pp. 293–356.  11, 1996.  arXiv:hep-th/9611050.
  * [272] A. Sen, “F theory and orientifolds,” Nucl. Phys. B 475 (1996) 562–578, arXiv:hep-th/9605150.
  * [273] M. F. Atiyah and N. J. Hitchin, “Low-Energy Scattering of Nonabelian Monopoles,” Phys. Lett. A 107 (1985) 21–25.