aainstitutetext: Rudolf Peierls Centre for Theoretical Physics
Beecroft Building, Clarendon Laboratory, Parks Road, University of Oxford, OX1
3PU, UK

# Exploring The Holographic Swampland

Joseph P. Conlon a Sirui Ning a Filippo Revello
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We extend studies of holographic aspects of moduli stabilisation scenarios to
both fibred versions of LVS and the type IIA DGKT flux vacua. We study the
holographic properties of the low-energy moduli Lagrangian that describes both
the AdS vacuum and also small perturbations about it. For type IIA vacua in
the large-volume regime, the CFT data (operator dimensions and higher-point
interactions) take a universal form independent of the many arbitrary flux
choices, as was previously found for LVS stabilisation. For these IIA vacua
the conformal dimensions of the dual operators are also, surprisingly, all
integers, although we do not understand a deeper reason why this is so. In
contrast to behaviour previously found for LVS and KKLT, the fibred models
also admit cases of mixed double-trace operators (for two different axion
fields) where the anomalous dimension is positive.

##  1 Introduction

If string theory is the true underlying theory of this world, then the
observed universe exists as a vacuum state of string theory. This makes it
important for physicists to understand the space of allowed vacua of string
theory; in particular, vacua with three large spatial dimensions with all
remaining dimensions compactified.

The traditional approach to understanding string compactifications was based
on top-down analyses and constructions of string vacua (for reviews see Refs.
Douglas:2006es ; Gra_a_2006 ; Conlon:2006gv ; Denef:2008wq ). This aimed at
obtaining such 4-dimensional vacua by starting with the full 10-dimensional
theory and then proceeding through a series of controlled approximations to
emerge eventually with controlled 4-dimensional solutions. Typically, the
progression was from 10-dimensional string theory to 10-dimensional
supergravity to 4-dimensional supergravity effective field theories – and
then, finally, to a vacuum derived within this framework of 4-dimensional
effective field theory.

The claim to have a controlled solution was provided in three ways. The first
means of control was through working with approximately supersymmetric
compactifications (typically with Calabi-Yau internal manifolds) and having
supersymmetry broken only at scales far below the compactification scale. This
allowed higher-dimensional questions to remain within the framework of
supersymmetry and benefit from the additional control provided. The second
means of control was to insist that the internal manifold was in the geometric
regime, so that the $\alpha^{\prime}$ expansion was parametrically controlled.
The final means of control was to work at small string coupling, thereby
allowing control of the string loop expansion.

In terms of obtaining string vacua, these restrictions are (mostly) not ones
of principle. They are imposed not because of any belief that vacua _cannot_
exist at strong coupling with badly-broken supersymmetry, but rather through
the (correct) sense that it is extremely hard to _establish in a convincing
fashion_ that such vacua exist.

While there are clear limitations in restricting ourselves to vacua which are
at weak coupling, there is one important physical reason to focus on such
vacua – the fact that the observed universe, both in particle physics and
cosmology, is characterised by small numbers and weak couplings. This
statement becomes more, rather than less, true as we move to the highest
energy scales. Examples are the inflationary density perturbations
(characterised by $\frac{\delta\rho}{\rho}\sim 10^{-5}$), the gauge couplings
of the Standard Model $\alpha_{SU(3)}$, $\alpha_{SU(2)}$ and
$\alpha_{U(1)_{Y}}$ (which all become weakly coupled at high but sub-Planckian
energies) or the Yukawa couplings of the Standard Model.

Recent years have seen an alternative approach to understanding string vacua:
the Swampland program Vafa:2005ui ; Ooguri:2006in ; Palti:2019pca . This aims
at determining general methods to exclude many low-energy Lagrangians as
possible vacua of string theory, by finding principles that must _always_ be
satisfied within quantum gravity. In this way it can (potentially) bypass the
many steps involved in more traditional approaches using dimensional reduction
and effective field theory. While such bold conjectures offer hopes of
circumventing the need for laborious construction and control of individual
vacua, it replaces the hard task of ensuring parametric control of vacua with
the equally hard task of proving, or at least establishing significant
evidence for, such general conjectures.

Nonetheless – new perspectives on old problems are still beneficial. This is
one motivation for our work here, which is focused on analysing ‘traditional’
vacua from a holographic perspective. This has two purposes. The first is to
determine whether the holographic perspective reveals any interesting
structure within established models of moduli stabilisation which may be
opaque in more traditional treatments. The second is to determine whether
manifestly ‘swampland’ modifications within a supergravity effective field
theory can be related to defined modifications within a dual CFT. In an ideal
world, this may provide a hope of understanding swampland constraints in terms
of allowed behaviour within CFTs Conlon:2018vov ; Conlon:2020wmc . While CFT
approaches to the swampland have already attracted a significant degree of
attention in the literature Nakayama:2015hga ; Benjamin:2016fhe ;
Montero:2016tif ; Giombi:2017mxl ; Urbano:2018kax ; Harlow:2018tng ;
Harlow:2018jwu ; Montero:2018fns ; Baume:2020dqd ; Perlmutter:2020buo ;
Kundu:2021qpi ; Aharony:2021mpc ; Antipin:2021rsh , not much work has been
carried out in the context of specific realisations. There was some discussion
in older literature about properties of duals to explicit moduli stabilisation
constructions Silverstein:2003jp ; Aharony:2008wz ; deAlwis:2014wia , but the
description is very complex and many details still remain to be worked out. In
this work, we shall take a different approach and exploit parametric limits
(such as large volume) which allow one to isolate treatable, effective CFTs
with a large gap to the heavier modes.

This paper is structured as follows. In section 2 we review some aspects of
CFT and the computation of anomalous dimensions. In section 3 we discuss the
case of fibred versions of LVS, where there are multiple light moduli compared
to ‘standard’ LVS, whereas in section 4 we discuss type IIA flux
compactifications and the corresponding operator dimensions. In section 5 we
conclude.

##  2 CFT Preliminaries

The subset of Conformal Field Theories (CFTs) with a weakly coupled gravity
dual go by the name of Holographic CFTs, and display some peculiar properties
which we attempt to summarize here. While much of the discussion carries over
from Conlon:2020wmc , this concise introduction aims to make the presentation
as self-contained as possible.

In weakly coupled scenarios, correlation functions in AdS can be calculated
using standard perturbation theory, and the results mapped to the CFT side
with the aid of the holographic dictionary. For the former to be possible,
both curvature and string coupling corrections must be under control:
$R_{AdS}\gg\ell_{S}$ and $g_{s}\ll 1$. In the most celebrated incarnation of
the correspondence - type IIB String Theory on $AdS_{5}\times S^{5}$ dual to
$\mathcal{N}=4$ SYM - this amounts to the requirement $N\gg 1$, since the
t’Hooft coupling $\lambda=g^{2}N$ is proportional to $(R/\ell_{s})^{d}$. More
generally, Holographic CFTs can be characterized by the following two
properties:

  * •

A finite number of primary operators with conformal dimensions of
$\mathcal{O}(1)$, which are separated from the rest of the spectrum by a large
gap $\Delta_{\text{gap}}\gg 1$.

  * •

The existence of an expansion parameter playing the role of large-$N$, around
which a perturbative series can be organised. By analogy, this is always
denoted by $N$ in the literature.

It has also been conjectured that these property are both necessary and
sufficient conditions for a CFT to be holographic Heemskerk:2009pn . At the
first non-trivial order in $1/N$, the operator content of such theories
consists of the primaries $\\{\mathcal{O}_{i}\\}$ and the double trace
operators,111In a realistic setup, one of course also has to include the
stress energy tensor $T_{\mu\nu}$ alongside any double trace operators that
can be constructed using it. which are (schematically) built out of the former
as

$[\mathcal{O}_{i}\mathcal{O}_{j}]_{n,\ell}\equiv\mathcal{O}_{i}\Box^{n}\partial_{\mu_{1}}...\partial_{\mu_{\ell}}\mathcal{O}_{j}.$
(1)

For each value of $n$ and $\ell$, only one such operator exists. Their
conformal dimension is given by

$\Delta_{n,\ell}=\Delta_{i}+\Delta_{j}+2n+\ell+\gamma(n,\ell),$ (2)

which is the sum of the classical dimension, $\Delta_{i}+\Delta_{j}+2n+\ell$,
and the anomalous dimension $\gamma(n,\ell)$ of magnitude
$\mathcal{O}(1/N^{2})$. Alongside the OPE coefficients, the anomalous
dimensions (which can be viewed as the binding energy in AdS space) encode all
the dynamical information about the theory and will be central quantities in
our analysis. In principle they can be extracted from the expression of AdS
4-point correlators or using a Bootstrap approach, but the procedure is not
always straightforward. In section 2.2, we will however show how they can be
systematically computed in the particular case that we will be concerned with,
i.e. anomalous dimensions $\gamma(n,\ell)$ with $n=0$. The relevance of such a
scenario is explained in the next section, in the context of CFT positivity
bounds.

###  2.1 Positivity Bounds

The requirements of conformal and crossing symmetry alone, cornerstones of the
Bootstrap program Ferrara:1973yt ; Polyakov:1974gs ; Mack:1975jr , are not
sufficient to impose any non-trivial bounds on the OPE data for holographic
CFTs. Indeed, it was shown in Heemskerk:2009pn  how all the higher order
operator coefficients in the Lagrangian of a single scalar on AdS map to a
single solution of the bootstrap equations. On the other hand, the physical
principles of causality and unitarity often impose additional constraints in
the space of possible theories, and CFTs make no exception. By demanding that
every state in a conformal multiplet has a non-negative norm, for example, one
is led to the following bounds for the conformal dimensions of primary
operators Mack:1975je ; Minwalla:1997ka :

$\Delta\geq\frac{d-2}{2}\quad{\rm{for}}\,\,\ell=0,\quad\quad\quad\Delta\geq\ell+d-2\quad{\rm{for}}\,\,\ell>0$
(3)

More elaborate consequences of these fundamental principles have been
pioneered in Komargodski:2012ek ; Fitzpatrick:2012yx , resulting in the
formulation of positivity bounds for towers of operators with minimal
twist.222The twist of an operator is denoted by is defined as
$\tau=\Delta-\ell$. Heuristically, the latter give the dominant contribution
to the Lorentzian OPE in certain kinematic regimes, and thus can be
constrained more easily. More precisely, if one denotes by $\tau^{*}(\ell)$
the twist of the minimal twist operators with spin $\ell$ appearing in the OPE
$\mathcal{O}\mathcal{O}^{{\dagger}}$ of a scalar with itself, the following
bound applies for $\ell$ even and larger than a critical value $\ell_{2}\geq
2$ (with $l_{3}>l_{2}$):

$\frac{\tau^{*}(\ell_{3})-\tau^{*}(\ell_{1})}{\ell_{3}-\ell_{1}}\leq\frac{\tau^{*}(\ell_{2})-\tau^{*}(\ell_{1})}{\ell_{2}-\ell_{1}}.$
(4)

Although the authors of Refs Komargodski:2012ek ; Fitzpatrick:2012yx  had to
invoke additional assumptions on the behaviour of off-shell amplitudes, these
were later dropped in subsequent proofs Costa:2017twz , where the result was
also generalized to any $\ell>1$ through the OPE inversion formula of Caron-
Huot:2017vep . In cases where gravity can be decoupled – i.e. an effective
theory on AdS characterized by a cutoff scale $\Lambda_{c}\ll M_{P}$ – the
leading twist operators in the $\mathcal{O}\mathcal{O}^{{\dagger}}$ OPE are
the double traces $[\mathcal{O}\mathcal{O}]_{0,\ell}$, and (5) can be turned
into a convexity statement on their anomalous dimension:

$\frac{\gamma(0,\ell_{3})-\gamma(0,\ell_{1})}{\ell_{3}-\ell_{1}}\leq\frac{\gamma(0,\ell_{2})-\gamma(0,\ell_{1})}{\ell_{2}-\ell_{1}}.$
(5)

Given that the anomalous dimensions asymptote to zero as
$\ell\rightarrow\infty$, they will also have to be negative for $\ell>1$. In a
different context, similar results concerning the negativity of the
$\gamma(0,2)$ anomalous dimension were also proven in Hartman:2015lfa . More
recently, such convexity relations have also been generalized to treat the OPE
of two non-identical scalar primaries Kundu:2020gkz  and other closely related
bounds for EFTs on AdS (which include the effect of dynamical gravity) have
been obtained in Kundu:2021qpi ; Caron-Huot:2021enk .

These results served as a partial motivation in our previous paper
Conlon:2020wmc  to conjecture that a similar negativity property for anomalous
dimensions would correlate with Swampland conditions on AdS, supported from a
number of examples in 4d string compactifications.333The original condition
(5) leads to trivial constraints for the type of Lagrangian we considered
Conlon:2020wmc  . In particular, we proposed that the mixed anomalous
dimensions $\gamma^{\rm{mix}}_{12}(0,\ell)$, corresponding to the double trace
operators of non-identical scalars $[\mathcal{O}_{1}\mathcal{O}_{2}]_{0,\ell}$
also be negative in CFTs dual to scenarios of moduli stabilization (based on
results for racetrack, LVS and KKLT stabilisation scenarios). For simplicity,
we considered anomalous dimensions for $\ell\gg 1$, which are not affected by
contact terms in the lagrangian and can be easily computed using a formula
shown in the next section. The rest of this paper is devoted to the
exploration of this condition in a different class of scenarios.

###  2.2 Anomalous Dimensions at Large Spin

The purpose of this section is to show explicitly how large spin anomalous
dimensions of the form $\gamma(0,\ell)$ can be calculated from the effective
action on AdS. Although there are no general, analytic formulas to extract
anomalous dimensions for any $(n,\ell)$ and arbitrary values of the external
dimensions, the case we are interested is rather special. In particular, Ref.
Costa:2014kfa  derived the formal expression

$\begin{split}\gamma^{\rm{mix}}_{12}(0,\ell)=-\int_{-i\infty}^{+i\infty}&\frac{dt}{2\pi
i}M_{12\rightarrow
12}(0,t)\,\,_{3}F_{2}(-\ell,\Delta_{1}+\Delta_{2}+\ell-1,\frac{t}{2};\Delta_{1},\Delta_{2};1)\\\
&\times\Gamma\Big{(}\Delta_{1}-\frac{t}{2}\Big{)}\,\Gamma\Big{(}\Delta_{2}-\frac{t}{2}\Big{)}\,\Gamma\Big{(}\frac{t}{2}\Big{)}^{2},\end{split}$
(6)

where $M(s,t)_{12\rightarrow 12}$ is the Mellin amplitude corresponding to the
$\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{1}\mathcal{O}_{2}\rangle$
correlator.444 We follow the kinematic conventions of Costa:2014kfa , which
differ from those adopted in Conlon:2020wmc  by an exchange of the s- and
t-channels. Mellin amplitudes Mack:2009mi ; Mack:2009gy ; Penedones:2010ue ;
Penedones:2016voo  are particular representations of conformal correlators
which most closely resemble scattering amplitudes in flat spacetime, and in
this case $M(s,t)_{12\rightarrow 12}$ is the analogue of a $12\rightarrow 12$
scattering amplitude. For a fuller development of Mellin amplitudes in the
context of AdS moduli stabilisation scenarios, we refer readers to our earlier
work Conlon:2020wmc .

Using (6), we will later see that only exchange diagrams contribute to the sum
at leading order in the $1/\ell$ expansion, and in particular only in the
t-channel. Therefore, the final result only depends on certain cubic
couplings. Contact terms, for example, contribute up to a finite spin
$\ell_{f}=N_{\partial}/2$, where $N_{\partial}$ is equal to number of
derivatives it contains. In particular, the exchange of a given primary
operator $\mathcal{O}$ induces an anomalous dimension for the mixed double
trace operators $[\mathcal{O}_{1}\mathcal{O}_{2}]_{0,\ell}$ of

$\gamma^{\rm{mix}}_{12}(0,\ell)\underset{\ell\gg
1}{\simeq}-\frac{2\Gamma(\Delta_{\mathcal{O}})}{\Gamma(\frac{\Delta_{\mathcal{O}}}{2})^{2}}\frac{\Gamma(\Delta_{1})\Gamma(\Delta_{2})}{\Gamma(\Delta_{1}-\frac{\Delta_{\mathcal{O}}}{2})\Gamma(\Delta_{2}-\frac{\Delta_{\mathcal{O}}}{2})}\frac{C_{11\mathcal{O}}C_{22\mathcal{O}}}{\ell^{\Delta_{\mathcal{O}}}},$
(7)

where $C_{11\mathcal{O}},C_{22\mathcal{O}}$ are the coefficients of the three-
point functions
$\langle\mathcal{O}_{1(2)}\mathcal{O}_{1(2)}\mathcal{O}\rangle$. Although the
same result could have been more easily obtained by solving the bootstrap
crossing equation at leading order Li:2015rfa , we emphasize that formula (6)
also provides a systematic way to expand the answer in subsequent powers of
$1/\ell$, and eventually resum the series to obtain the finite spin result. It
also allows one to keep the analogy with positivity bounds from flat space
more manifest, given the similarity of Mellin amplitudes to scattering
amplitudes in flat spacetime.

The three-point function coefficients in (7) can be computed by evaluating the
corresponding Witten diagram in AdS, which is most easily performed in the
embedding formalism Penedones:2010ue ; Penedones:2016voo . Given the
particular structure of the Lagrangians we are going to consider, the cubic
coupling will always be of the general form

$\mathcal{L}\supset
c_{1}\varphi_{A}^{2}\varphi_{B}+c_{2}\partial_{\mu}\varphi_{C}\partial_{\mu}\varphi_{C}\varphi_{D}.$
(8)

Then, the corresponding coefficients take the form Conlon:2005ki

$C_{AAB}=c_{1}(2+4\delta_{AB})\frac{\pi^{d/2}\Gamma\big{(}\frac{2\Delta_{A}+\Delta_{B}-d}{2}\big{)}}{\Gamma(\Delta_{A})^{2}\Gamma(\Delta_{B})}.$
(9)
$C_{CCD}=c_{2}\frac{\pi^{d/2}\Gamma\big{(}\frac{2\Delta_{C}+\Delta_{D}-d}{2}\big{)}}{\Gamma(\Delta_{C})^{2}\Gamma(\Delta_{D})}(\Delta_{C}^{2}-2\Delta_{C}-\Delta_{D}+3).$
(10)

###  2.3 A Special Case: Degenerate Conformal Dimensions

When a CFT contains two primaries with integer spaced conformal dimensions,
the spectra of the corresponding double trace operators overlap, and one needs
to be more careful in the computation of the anomalous dimensions. To be
explicit, let us consider two scalars $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$
with scaling dimensions $\Delta_{1}$, $\Delta_{2}$. At zeroth order in the
large-$N$ expansion, the double trace operators
$[\mathcal{O}_{1}\mathcal{O}_{1}]_{n_{1},\ell}$ and
$[\mathcal{O}_{2}\mathcal{O}_{2}]_{n_{2},\ell}$ will be degenerate if the
dimensions satisfy

$2(n_{1}-n_{2})=\Delta_{2}-\Delta_{1},\quad\quad n_{1},n_{2}\in\mathbb{N}$
(11)

and analogous conclusions hold with the mixed operators
$[\mathcal{O}_{1}\mathcal{O}_{2}]_{n,\ell}$ too. While from an AdS perspective
Eq. (11) would seemingly correspond to a fine-tuning conspiracy on the value
of the masses,555In the type IIA models of section 4 we will surprisingly find
examples where this condition is satisfied. However, we expect it to be broken
by subleading volume effects. there is one notable exception where this
condition naturally occurs, namely when two fields are massless and
$\Delta_{1}=\Delta_{2}=d$. One such example is provided by the case where the
potential has flat directions corresponding to certain moduli or,
alternatively, for axions protected by shift invariance. In large volume
models, the latter scenario is quite robust, since any additional non-
perturbative effects lifting the axion mass are suppressed as $m\propto
e^{-b\mathcal{V}^{2/3}}$, which does not appear at any order in a $1/N$
expansion.666Since the volume plays the role of large-$N$ in such models.

Of course, in any realistic stringy example – such as those we will consider
in sections 3 and 4 – particle masses (and hence the dimensions of primaries)
may receive higher order contributions that break the degeneracy. Since we are
working in a perturbative framework, this is only relevant if the corrections
to the conformal dimensions dominate parametrically over the anomalous
dimensions, i.e. they have to be strictly larger than $\mathcal{O}(1/N^{2})$.
If this happens for $\Delta_{1}\neq\Delta_{2}$, the operator mixing is
effectively eliminated, and anomalous dimensions can be calculated as usual.
On the other hand, if $\Delta_{1}=\Delta_{2}$ the degeneracy is still resolved
in principle (unless the symmetries of the problem conspire to leave the
degeneracy still unbroken at this subleading order), but mixing effects can
appear between $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. Indeed, the true
primaries are the eigenstates of the degeneracy-breaking perturbation, which
can be rotated from the original fields by an unknown angle.777Crucially, this
angle remains finite if the magnitude of the perturbation approaches zero.
Therefore, the form of the perturbation becomes necessary to calculate the
anomalous dimensions even at leading order, and in such cases we will not be
able to provide an expression for $\gamma(0,\ell)$. Otherwise, the operators
should effectively be considered degenerate, and it will be necessary to use
the techniques outlined in this section.

####  2.3.1 Mixing Matrix

For concreteness, let us focus on the case where
$\Delta_{1}=\Delta_{2}\equiv\Delta$ and no degeneracy breaking is present. For
each given pairing $(n,\ell)$, there is a three dimensional degenerate space,
corresponding to the operators

$[\mathcal{O}_{1}\mathcal{O}_{1}]_{n,\ell}\,;\,\,[\mathcal{O}_{2}\mathcal{O}_{2}]_{n,\ell}\,;\,\,[\mathcal{O}_{1}\mathcal{O}_{2}]_{n,\ell}.$
(12)

They will be denoted by the double indices $[ij]$, taking values in
${[11],[22],[12]}$. In this subspace, the dilatation operator is diagonal only
at order zero in the $1/N$ expansion, and the scaling dimensions are given by
its eigenvalues: this is totally analogous to what happens in standard QM
perturbation theory. In the approach of Fitzpatrick:2010zm , anomalous
dimensions are calculated by applying QM perturbation theory to the dilatation
operator, and in a non-degenerate system this amounts to evaluating the
expectation value of the interaction term $\Delta\hat{D}$ on a given state. In
a degenerate system, one instead has to diagonalize

$\bar{\gamma}_{[ij]\,[kl]}(n,\ell)=\,_{ij}\bra{n,\ell}\Delta\hat{D}\ket{n,\ell}_{kl}$
(13)

where $\ket{n,\ell}_{ij}$ is the state created by acting with
$[\mathcal{O}_{i}\mathcal{O}_{j}]_{n,\ell}(0)$ on the vacuum, and the bar is
used to distinguish between the matrix elements and the actual anomalous
dimensions, given by the eigenvalues. The eigenvectors of the matrix determine
the basis of double trace operators in which the dilatation operator is
diagonal at the first non-trivial order in $1/N$, as a linear combination of
the three operators in (12).

Although our approach to the evaluation of large spin anomalous dimensions is
technically (and conceptually) different, anomalous dimensions can still be
obtained as the eigenvalues of an analogous mixing matrix
$\tilde{\gamma}_{[ij]\,[kl]}(0,\ell)$. In particular, we show in Appendix A
that the matrix elements represent the “effective” anomalous dimension
extracted from the mixed correlators

$\langle\mathcal{O}_{i}\mathcal{O}_{j}\mathcal{O}_{k}\mathcal{O}_{l}\rangle\supset\frac{f_{0}(n,\ell)^{2}\tilde{\gamma}_{[ij][kl]}(n,\ell)}{N^{2}}\partial_{\Delta}G_{2\Delta+2n+\ell,\ell},$
(14)

where the $f_{0}(0,\ell)$ are the generalized free theory OPE coefficients.
They can be calculated with the analogue of Eq.(7) for the Mellin amplitude
corresponding to the
$\langle\mathcal{O}_{i}\mathcal{O}_{k}\mathcal{O}_{j}\mathcal{O}_{l}\rangle$
correlator.

If we assume that at least one of the two primaries is odd with respect to a
$\mathbb{Z}_{2}$ symmetry, as is the case for operators dual to an axion on
AdS, the system becomes two-dimensional. This is the only case we will
encounter in practice, as the degeneracy will always be broken in other
systems. In this case, only the
$[\mathcal{O}_{1}\mathcal{O}_{1}]^{i}_{n,\ell}$ and
$[\mathcal{O}_{2}\mathcal{O}_{2}]^{i}_{n,\ell}$ operators have the same
transformation property under the discrete symmetry and can mix amongst
themselves. In this particular case, the eigenvalues take the simplified form

$\gamma_{\pm}=\frac{\tilde{\gamma}_{[11][11]}+\tilde{\gamma}_{[22][22]}\pm\sqrt{(\tilde{\gamma}_{[11][11]}-\tilde{\gamma}_{[22][22]})^{2}+4\tilde{\gamma}_{[11][22]}^{2}}}{2},$
(15)

where the indices $n,\ell$ have been suppressed for simplicity. An interesting
consequence of the modified formula (15) is that one of the anomalous
dimensions can apparently be positive,888Since $\tilde{\gamma}_{[11][11]}$ and
$\tilde{\gamma}_{[22][22]}$ will still be positive by construction. when
$\tilde{\gamma}^{2}_{[11][22]}>\tilde{\gamma}_{[11][11]}\tilde{\gamma}_{[22][22]}$.
Even though we are dealing with identical operators, there does not seem any
reason why this behaviour should be inconsistent. The reason is that in the
OPE of two identical primaries, only the lowest twist operators have to obey a
convexity condition (implying negativity) - therefore it is enough to have
just one double trace operator with negative anomalous dimension appearing for
every value of $\ell$. For instance, the OPE of $\mathcal{O}_{1}$ with itself
will contain two classes of double trace operators, with positive and negative
anomalous dimensions respectively. The lowest twist operators will be those
belonging to the latter class, and these satisfy the required condition
irrespective of the existence of those with a positive anomalous dimension.

##  3 Fibred Calabi-Yau Scenarios of Moduli Stabilization

In our earlier work Conlon:2020wmc  we studied holographic aspects of certain
moduli stabilisation scenarios, in particular LVS, KKLT and racetrack
scenarios. These displayed interesting properties. LVS displayed a
universality of conformal dimensions in the large-volume limit, and all
scenarios had an interesting negative behaviour for the mixed anomalous
dimensions. We are therefore interested more scenarios, and in this section we
extend these studies to fibred versions of the original LVS scenario.

###  3.1 A Brief Review of Fibred Calabi-Yau Scenarios

One interesting variation on the original LVS is to consider models with
fibred Calabi-Yau (CY) structures. In such models, there are multiple large
moduli associated both to base and fibre moduli Cicoli:2008gp ; Cicoli:2011it
. We shall take the form of the volume of a fibred Calabi-Yau to be given as
the following (based on Cicoli:2008gp ):

$\mathcal{V}=f_{\frac{3}{2}}(\tau_{j})-\sum_{i=1}^{N_{small}}\lambda_{i}\tau_{i}^{3/2},$
(16)

where $f$ is a homogeneous function of degree 3/2, $j$ runs from 1 to
$N_{large}$ and $h_{1,1}=N_{small}+N_{large}$, where $N_{large}$ is the number
of large moduli and $N_{small}$ is the number of blow up moduli.

The fibred CY shares a similar feature with the “Swiss Cheese” form used in
the original LVS, in terms of the blow-up moduli included in the volume. The
difference lies in the function $f$, as this now involves multiple moduli:
both the overall volume modulus and the fibration moduli. Different moduli
play different roles. The volume modulus controls the LVS potential. The fibre
modulus is a flat direction of the LVS potential, which means it will be
absent in the LVS potential at leading order. However, it will appear in the
kinetic terms, and also at subleading order in the string loop expansion. The
blow up modulus controls the heavy mode which will subsequently be integrated
out of the LVS potential within the low-energy effective field theory.

In the next two subsections, we will use two specific examples to illustrate
these ideas which will involve respectively one or two fibre moduli.

###  3.2  Fibred Calabi-Yau: Example with One Fibre Modulus

We will first look at moduli stabilisation in order to establish the AdS
effective action (essentially a review of Gra_a_2006 ), before considering the
holographic implications of this effective action.

####  3.2.1 Moduli Stabilisation

We assume as in Cicoli:2008gp  that the volume takes the form:

$\mathcal{V}=\alpha(\sqrt{\tau_{1}}\tau_{2})-\gamma\tau_{3}^{3/2}.$ (17)

Here $\tau_{1}$ controls the volume of the fibre, $\tau_{2}$ controls the
volume of the base and $\tau_{3}$ describes the blow up modulus.

The Large Volume Scenario is a specific type IIB flux compactification moduli
stabilisation scenario, which stabilises all moduli in a non-SUSY AdS vacuum
at an exponentially large volume $\mathcal{V}$ (as per Balasubramanian:2005zx
; Conlon:2005ki ; Conlon:2006gv ). Following the usual approach in LVS, after
stabilising and integrating out the complex structure moduli and dilaton, the
Kähler potential and superpotential read:

$K=-2\textrm{ln}(\mathcal{V}+\hat{\xi}),$ (18) $W=W_{0}+A_{3}e^{-a_{3}T_{3}},$
(19) $\hat{\xi}=\frac{\xi}{g_{s}^{\frac{3}{2}}},$ (20)
$\xi=-\frac{\chi(X)\zeta(3)}{2(2\pi)^{3}},$ (21)

Here $\chi(X)$ is the Euler number of the Calabi-Yau manifold X, $\zeta$ is
the Riemann zeta function with $\zeta(3)\approx 1.2$. $T_{3}$ is the chiral
multiplet for the blow-up Kähler modulus, $T_{3}=\tau_{3}+iC_{3}$.

The $\mathcal{N}=1$ F-term supergravity scalar potential takes the following
form:

$V=e^{K}(K^{i\overline{j}}D_{i}WD_{\overline{j}}\overline{W}-3|W|^{2}),$ (22)

where we have set $M_{P}=1$. After substituting the Kähler potential and
superpotential into the scalar potential, the effective potential becomes:

$V_{scalar}=\frac{Aa_{3}^{2}\sqrt{\tau_{3}}}{\mathcal{V}}e^{-2a_{3}\tau_{3}}-\frac{BW_{0}a_{3}A_{3}\tau_{3}}{\mathcal{V}^{2}}e^{-a_{3}\tau_{3}}+\frac{C\xi
W_{0}^{2}}{g_{s}^{\frac{3}{2}}\mathcal{V}^{3}},$ (23)

where $A,B,C$ are numerical constants. The important point is that the fibre
modulus does not appear in the effective potential at tree level, and so at
this order is a flat direction of the scalar potential. Therefore there is an
additional light scalar degree of freedom, which is an interesting feature of
fibred Calabi-Yau models compared to the original LVS.

As with standard LVS, this scalar potential has a minimum at exponentially
large volumes,

$\langle\tau_{3}\rangle\sim\frac{\xi^{\frac{2}{3}}}{g_{s}},$ (24)
$\langle\mathcal{V}\rangle\sim e^{a_{3}\langle\tau_{3}\rangle}.$ (25)

After integrating out the blow up modulus $\tau_{3}$, the effective potential
for the LVS volume modulus reads Conlon:2018vov :

$V_{scalar}=[-A^{{}^{\prime}}\left(\textrm{ln}(f\mathcal{V})\right)^{\frac{3}{2}}+B^{{}^{\prime}}]\frac{W_{0}^{2}}{\mathcal{V}^{3}}.$
(26)

where $A^{{}^{\prime}}$, $B^{{}^{\prime}}$ are numerical constants.

####  3.2.2 Kinetic Terms and Canonical Fields

The effective field theory of these fibred models contains four light moduli:
one overall volume modulus which is massive, plus one fibre modulus and two
axions, which are approximately massless. In order to see how they interact
with each other, we write down the kinetic terms for the theory in the large
volume limit using standard EFT techniques. Neglecting higher order terms, the
kinetic terms read:

$L_{kin}=\frac{1}{4\tau_{1}^{2}}\partial_{u}\tau_{1}\partial^{u}\tau_{1}+\frac{1}{2\tau_{2}^{2}}\partial_{u}\tau_{2}\partial^{u}\tau_{2}+\frac{1}{4\tau_{1}^{2}}\partial_{u}a_{1}\partial^{u}a_{1}+\frac{1}{2\tau_{2}^{2}}\partial_{u}a_{2}\partial^{u}a_{2}.$
(27)

As a first step in diagonalising the action, we re-express the mode $\tau_{2}$
in terms of the overall LVS volume:

$\tau_{2}=\frac{\mathcal{V}}{\alpha\sqrt{\tau_{1}}}.$ (28)

The kinetic terms then become

$\mathcal{L}\supset\frac{3}{8\tau_{1}^{2}}\partial_{u}\tau_{1}\partial^{u}\tau_{1}-\frac{1}{2\tau_{1}\mathcal{V}}\partial_{u}\tau_{1}\partial^{u}\mathcal{V}+\frac{1}{2\mathcal{V}^{2}}\partial_{u}\mathcal{V}\partial^{u}\mathcal{V}+\frac{1}{4\tau_{1}^{2}}\partial_{u}a_{1}\partial^{u}a_{1}+\frac{\alpha^{2}\tau_{1}}{2\mathcal{V}^{2}}\partial_{u}a_{2}\partial^{u}a_{2}.$
(29)

We next diagonalise both kinetic terms and mass terms in order to get a
canonically normalised theory.

$\textrm{ln}\,\tau_{1}=a\Phi_{1}+b\Phi_{2},$ (30)
$\textrm{ln}\,\mathcal{V}=c\Phi_{1}+d\Phi_{2}.$ (31)

Here $\Phi_{1}$ and $\Phi_{2}$ will be the canonically normalised fields.
Before we go into the details of the calculation, we make an observation. As
the mass term only depends on the overall LVS volume, we must choose either
$c=0$ or $d=0$ when diagonalising. Here we choose $d=0$, so our ansatz
becomes:

$\displaystyle\textrm{ln}\,\tau_{1}=a\Phi_{1}+b\Phi_{2},$ (32)
$\displaystyle\textrm{ln}\,\mathcal{V}=c\Phi_{1}.$

We refer interested readers to appendix B.1 for the details. Here we just
state the final results:

$\displaystyle\textrm{ln}\,\tau_{1}=\sqrt{\frac{2}{3}}(\Phi_{1}+\sqrt{2}\Phi_{2}),$
(33) $\displaystyle\textrm{ln}\,\mathcal{V}=\sqrt{\frac{3}{2}}\Phi_{1}.$

The kinetic terms then read:

$\mathcal{L}\supset\frac{1}{2}(\partial_{u}\Phi_{1})^{2}+\frac{1}{2}(\partial_{u}\Phi_{2})^{2}+\frac{1}{2}e^{-2\sqrt{\frac{2}{3}}(\Phi_{1}+2\Phi_{2})}\partial_{u}a_{1}\partial^{u}a_{1}+\frac{1}{2}e^{\left(\sqrt{\frac{2}{3}}-2\sqrt{\frac{3}{2}}\right)\Phi_{1}+2\sqrt{\frac{2}{3}}\Phi_{2}}\partial_{u}a_{2}\partial^{u}a_{2},$
(34)

where the axions have also been rescaled by a constant factor to achieve
canonical normalization. The potential only depends on $\Phi_{1}$ and so is
diagonal in this basis. Having expressed the Lagrangian in canonical form, we
can now move to a discussion of the interactions and their holographic
interpretation.

####  3.2.3 Holographic Interpretation

From the couplings in (34) and in the scalar potential, we can easily read off
the OPE coefficients following Eqs (9)-(10). The results are the following:

$\displaystyle
f_{a_{1}a_{1}}^{\Phi_{1}}=\sqrt{\frac{2}{3}}\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{1}}+\Delta_{\Phi_{1}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{1}}\right)\Gamma\left(\Delta_{a_{1}}\right)^{2}}\left(\Delta_{\Phi_{1}}+2\Delta_{a_{1}}-\Delta_{a_{1}}^{2}-3\right),$
(35) $\displaystyle
f_{a_{1}a_{1}}^{\Phi_{2}}=2\sqrt{\frac{2}{3}}\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{1}}+\Delta_{\Phi_{2}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{2}}\right)\Gamma\left(\Delta_{a_{1}}\right)^{2}}\left(\Delta_{\Phi_{2}}+2\Delta_{a_{1}}-\Delta_{a_{1}}^{2}-3\right),$
$\displaystyle
f_{a_{2}a_{2}}^{\Phi_{1}}=\left(\sqrt{\frac{3}{2}}-\frac{1}{2}\sqrt{\frac{2}{3}}\right)\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{2}}+\Delta_{\Phi_{1}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{1}}\right)\Gamma\left(\Delta_{a_{2}}\right)^{2}}\left(\Delta_{\Phi_{1}}+2\Delta_{a_{2}}-\Delta_{a_{2}}^{2}-3\right),$
$\displaystyle
f_{a_{2}a_{2}}^{\Phi_{2}}=-\sqrt{\frac{2}{3}}\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{2}}+\Delta_{\Phi_{2}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{2}}\right)\Gamma\left(\Delta_{a_{2}}\right)^{2}}\left(\Delta_{\Phi_{2}}+2\Delta_{a_{2}}-\Delta_{a_{2}}^{2}-3\right),$
$\displaystyle
f_{\Phi_{1}\Phi_{1}}^{\Phi_{1}}=\frac{81\sqrt{6}}{2}\frac{3\pi^{\frac{d}{2}}\Gamma\left(\frac{3\Delta_{\Phi_{1}}-3}{2}\right)}{\Gamma\left(\Delta_{\Phi_{1}}\right)^{3}}.$

The key difference between the fibred Calabi-Yau LVS model and the original
LVS model is that there are a total of 3 massless moduli in the effective
field theory rather than just one. This leads to operator mixing in the
operator product expansion. On the other hand, the volume modulus behaves
exactly as in the pure LVS scenario, obtaining a conformal dimension of
Conlon:2018vov ; Conlon:2020wmc

$\Delta_{\Phi_{1}}=\frac{3(1+\sqrt{19})}{2},$ (36)

and with the same self-couplings.

In principle, the following double trace operators are degenerate

$[a_{1}a_{1}]_{n,\ell},\quad[a_{2}a_{2}]_{n,\ell},\quad[a_{1}a_{2}]_{n,\ell},\quad[\Phi_{2}\Phi_{2}]_{n,\ell},\quad[a_{1}\Phi_{2}]_{n,\ell},\quad[a_{2}\Phi_{2}]_{n,\ell},$
(37)

while

$\quad[a_{1}\Phi_{1}]_{n,\ell},\quad[a_{2}\Phi_{1}]_{n,\ell}$ (38)

also mix within themselves. Each axion is invariant under an independent
discrete $\mathbb{Z}_{2}$ symmetry acting as $a_{i}\rightarrow-a_{i}$, which
results in selection rules for the double trace operators that can appear in a
given OPE. This implies that states in different symmetry classes cannot mix
with one another even though their classical dimension is the same. This is
shown in Table 1.

| $\mathbb{Z}_{2}(a_{1})$ | $\mathbb{Z}_{2}(a_{2})$
---|---|---
$[a_{1}\Phi_{2}]_{n,\ell}$ | -1 | 1
$[a_{2}\Phi_{2}]_{n,\ell}$ | 1 | -1
$[a_{1}a_{2}]_{n,\ell}$ | -1 | -1
$[a_{1}a_{1}]_{n,\ell},[a_{2}a_{2}]_{n,\ell},[\Phi_{2}\Phi_{2}]_{n,\ell}$ | 1 | 1
$[a_{1}\Phi_{1}]_{n,\ell}$ | -1 | 1
$[a_{2}\Phi_{1}]_{n,\ell}$ | 1 | -1
Table 1: Irreducible representation of the double trace operators under the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry. The two different blocks
correspond to distinct subspaces, where the degeneracies are partly broken by
the discrete symmetries.

For the non-degenerate operators involving different fields
$[a_{1}\Phi_{1}]_{0,\ell},[a_{2}\Phi_{1}]_{0,\ell}[a_{1}\Phi_{2}]_{0,\ell},$
$[a_{2}\Phi_{2}]_{0,\ell},[a_{1}a_{2}]_{0,\ell}$, we can directly refer to (7)
to work out the anomalous dimension. In particular, since the kinematic
factors are equal for the two axions

$\displaystyle\gamma(0,\ell)_{a_{1}\Phi_{1}}\propto-
f_{a_{1}a_{1}\Phi_{1}}f_{\Phi_{1}\Phi_{1}\Phi_{1}}<0,$ (39)
$\displaystyle\gamma(0,\ell)_{a_{2}\Phi_{1}}=0,$
$\displaystyle\gamma(0,\ell)_{a_{1}\Phi_{2}}=0,$
$\displaystyle\gamma(0,\ell)_{a_{2}\Phi_{2}}=0.$
$\displaystyle\gamma(0,\ell)_{a_{1}a_{2}}\propto-
f_{a_{1}a_{1}\Phi_{2}}f_{a_{2}a_{2}\Phi_{2}}>0,$

The first two anomalous dimensions are exactly those of LVS without fibred
directions, and their agreement with our original conjecture should not come
as a surprise. However, $\gamma(0,\ell)_{a_{1}a_{2}}$ is positive, violating
the expected behaviour. The interpretation for this is that maintaining a
fixed volume requires one cycle to increase in size while the other decreases,
and so the coefficients reflecting the coupling of the associated axions to
the modulus have opposite signs.

Let us now analyse the properly degenerate subspace. Taking perturbative
corrections into account Berg:2005ja ; vonGersdorff:2005bf ; Berg:2005yu ;
Cicoli:2007xp , D-brane loops lift the flat potential for $\Phi_{2}$ with a
scaling

$V_{\rm{loop}}\sim\frac{1}{\mathcal{V}^{10/3}},$ (40)

which is parametrically faster than $\mathcal{O}(1/N)$ and thus dominates over
anomalous dimensions. Therefore the degeneracy between $a_{1},a_{2}$ and
$\Phi_{2}$ breaks to $a_{1}$ and $a_{2}$ only. Since there are no interactions
mixing $a_{1}$ and $a_{2}$ in the Lagrangian, the resulting matrix is
diagonal, with

$\displaystyle\gamma(0,\ell)_{a_{1}a_{1}}\propto-
f_{a_{1}a_{1}\Phi_{2}}^{2}<0,$ (41)
$\displaystyle\gamma(0,\ell)_{a_{2}a_{2}}\propto-
f_{a_{2}a_{2}\Phi_{2}}^{2}<0.$

Similarly, the anomalous dimension is negative (by construction) for
$[\Phi_{2}\Phi_{1}]_{0,\ell}$.

###  3.3 Fibred Calabi Yau: Example with Two Fibred Moduli

A similar analysis can be carried out for other fibred Calabi-Yau models (here
Appendix B.2) gives the full details). One direct extension is to consider a
model with two fibred moduli, in which the volume of the fibred Calabi-Yau
manifold reads:

$\mathcal{V}=\sqrt{\tau_{1}\tau_{2}\tau_{3}}-\tau_{s}^{3/2}.$ (42)

Here $\tau_{1},\tau_{2},\tau_{3}$ control the volumes of individual 4-cycles,
with the overall volume direction corresponding in the rescaling
$\tau_{i}\rightarrow\lambda\tau_{i}$. The $\sqrt{\tau_{1}\tau_{2}\tau_{3}}$
form is reminiscent of toroidal orbifolds based on $T_{2}\times T_{2}\times
T_{2}$, while $\tau_{s}$ is the blow up mode necessary to realize LVS.

In the large volume limit, the Kähler metric reads:

$K_{i\overline{j}}=\left(\begin{matrix}\frac{1}{4\tau_{1}^{2}}&\frac{\tau_{s}^{\frac{3}{2}}}{8\tau_{1}^{\frac{3}{2}}\tau_{2}^{\frac{3}{2}}\tau_{3}^{\frac{1}{2}}}&\frac{\tau_{s}^{\frac{3}{2}}}{8\tau_{1}^{\frac{3}{2}}\tau_{2}^{\frac{1}{2}}\tau_{3}^{\frac{3}{2}}}&-\frac{3\tau_{s}^{\frac{1}{2}}}{8\tau_{1}\mathcal{V}}\\\
\\\
\frac{\tau_{s}^{\frac{3}{2}}}{8\tau_{1}^{\frac{3}{2}}\tau_{2}^{\frac{3}{2}}\tau_{3}^{\frac{1}{2}}}&\frac{1}{4\tau_{2}^{2}}&\frac{\tau_{s}^{\frac{3}{2}}}{8\tau_{1}^{\frac{1}{2}}\tau_{2}^{\frac{3}{2}}\tau_{3}^{\frac{3}{2}}}&-\frac{3\tau_{s}^{\frac{1}{2}}}{8\tau_{2}\mathcal{V}}\\\
\\\
\frac{\tau_{s}^{\frac{3}{2}}}{8\tau_{1}^{\frac{3}{2}}\tau_{2}^{\frac{1}{2}}\tau_{3}^{\frac{3}{2}}}&\frac{\tau_{s}^{\frac{3}{2}}}{8\tau_{1}^{\frac{1}{2}}\tau_{2}^{\frac{3}{2}}\tau_{3}^{\frac{3}{2}}}&\frac{1}{4\tau_{3}^{2}}&-\frac{3\tau_{s}^{\frac{1}{2}}}{8\tau_{3}\mathcal{V}}\\\
\\\
-\frac{3\tau_{s}^{\frac{1}{2}}}{8\tau_{1}\mathcal{V}}&-\frac{3\tau_{s}^{\frac{1}{2}}}{8\tau_{2}\mathcal{V}}&-\frac{3\tau_{s}^{\frac{1}{2}}}{8\tau_{3}\mathcal{V}}&\frac{3}{8}\frac{1}{\tau_{s}^{\frac{1}{2}}\mathcal{V}}\\\
\\\ \end{matrix}\right),$

while the inverse matrix is:

$K^{-1}=\left(\begin{matrix}4\tau_{1}^{2}&-2\frac{\tau_{s}^{\frac{3}{2}}\tau_{1}^{\frac{1}{2}}\tau_{2}^{\frac{1}{2}}}{\tau_{3}^{\frac{1}{2}}}&-2\frac{\tau_{s}^{\frac{3}{2}}\tau_{1}^{\frac{1}{2}}\tau_{3}^{\frac{1}{2}}}{\tau_{2}^{\frac{1}{2}}}&4\tau_{1}\tau_{s}\\\
\\\
-2\frac{\tau_{s}^{\frac{3}{2}}\tau_{1}^{\frac{1}{2}}\tau_{2}^{\frac{1}{2}}}{\tau_{3}^{\frac{1}{2}}}&4\tau_{2}^{2}&-2\frac{\tau_{s}^{\frac{3}{2}}\tau_{2}^{\frac{1}{2}}\tau_{3}^{\frac{1}{2}}}{\tau_{1}^{\frac{1}{2}}}&4\tau_{2}\tau_{s}\\\
\\\
-2\frac{\tau_{s}^{\frac{3}{2}}\tau_{1}^{\frac{1}{2}}\tau_{3}^{\frac{1}{2}}}{\tau_{2}^{\frac{3}{2}}}&-2\frac{\tau_{s}^{\frac{3}{2}}\tau_{3}^{\frac{1}{2}}\tau_{2}^{\frac{1}{2}}}{\tau_{1}^{\frac{1}{2}}}&4\tau_{3}^{2}&4\tau_{3}\tau_{s}\\\
\\\
4\tau_{1}\tau_{s}&4\tau_{2}\tau_{s}&4\tau_{3}\tau_{s}&\frac{8}{3}V\tau_{s}^{\frac{1}{2}}\\\
\\\ \end{matrix}\right).$

The kinetic term for the moduli $\tau_{1},\tau_{2},\tau_{3}$ and their axion
partners can be expressed as the following:

$\displaystyle\mathcal{L}\supset$
$\displaystyle\frac{1}{4\tau_{1}^{2}}\partial^{u}\tau_{1}\partial_{u}\tau_{1}+\frac{1}{4\tau_{1}^{2}}\partial^{u}a_{1}\partial_{u}a_{1}+\frac{1}{4\tau_{2}^{2}}\partial^{u}\tau_{2}\partial_{u}\tau_{2}+$
(43)
$\displaystyle\frac{1}{4\tau_{2}^{2}}\partial^{u}a_{2}\partial_{u}a_{2}+\frac{1}{4\tau_{3}^{2}}\partial^{u}\tau_{3}\partial_{u}\tau_{3}+\frac{1}{4\tau_{3}^{2}}\partial^{u}a_{3}\partial_{u}a_{3}.$

Following similar techniques to the last section, we can substitute one of the
flat moduli with the volume modulus, diagonalise the kinetic term and write
down the scalar potential. After putting all the constituents together, the
resulting effective field theory reads:

$\displaystyle\mathcal{L}=$
$\displaystyle\frac{1}{2}(\partial_{u}\Phi_{1})^{2}+\frac{1}{2}(\partial_{u}\Phi_{2})^{2}+\frac{1}{2}(\partial_{u}\Phi_{3})^{2}+\frac{1}{2}e^{-2\sqrt{\frac{2}{3}}\Phi_{1}-2\sqrt{\frac{1}{3}}\Phi_{2}-2\Phi_{3}}(\partial_{u}a_{1})^{2}$
(44)
$\displaystyle+\frac{1}{2}e^{-2\sqrt{\frac{2}{3}}\Phi_{1}-2\sqrt{\frac{1}{3}}\Phi_{2}+2\Phi_{3}}(\partial_{u}a_{2})^{2}+\frac{1}{2}e^{\left(2\sqrt{\frac{2}{3}}-2\sqrt{\frac{3}{2}}\right)\Phi_{1}+4\sqrt{\frac{1}{3}}\Phi_{2}}(\partial_{u}a_{3})^{2}-V_{LVS},$

where the axions have also been rescaled to give canonical kinetic terms. As
before, the OPE coefficients can be determined with Eqs (9)-(10). In
particular, they are

$\displaystyle
f_{a_{1}a_{1}}^{\Phi_{1}}=\sqrt{\frac{2}{3}}\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{1}}+\Delta_{\Phi_{1}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{1}}\right)\Gamma\left(\Delta_{a_{1}}\right)^{2}}\left(\Delta_{\Phi_{1}}+2\Delta_{a_{1}}-\Delta_{a_{1}}^{2}-3\right),$
(45) $\displaystyle
f_{a_{1}a_{1}}^{\Phi_{2}}=\sqrt{\frac{1}{3}}\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{1}}+\Delta_{\Phi_{2}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{2}}\right)\Gamma\left(\Delta_{a_{1}}\right)^{2}}\left(\Delta_{\Phi_{2}}+2\Delta_{a_{1}}-\Delta_{a_{1}}^{2}-3\right),$
$\displaystyle
f_{a_{1}a_{1}}^{\Phi_{3}}=\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{1}}+\Delta_{\Phi_{3}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{3}}\right)\Gamma\left(\Delta_{a_{1}}\right)^{2}}\left(\Delta_{\Phi_{3}}+2\Delta_{a_{1}}-\Delta_{a_{1}}^{2}-3\right),$
$\displaystyle
f_{a_{2}a_{2}}^{\Phi_{1}}=\sqrt{\frac{2}{3}}\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{2}}+\Delta_{\Phi_{1}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{1}}\right)\Gamma\left(\Delta_{a_{2}}\right)^{2}}\left(\Delta_{\Phi_{1}}+2\Delta_{a_{2}}-\Delta_{a_{2}}^{2}-3\right),$
$\displaystyle
f_{a_{2}a_{2}}^{\Phi_{2}}=\sqrt{\frac{1}{3}}\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{2}}+\Delta_{\Phi_{2}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{2}}\right)\Gamma\left(\Delta_{a_{2}}\right)^{2}}\left(\Delta_{\Phi_{2}}+2\Delta_{a_{2}}-\Delta_{a_{2}}^{2}-3\right),$
$\displaystyle
f_{a_{2}a_{2}}^{\Phi_{3}}=-\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{2}}+\Delta_{\Phi_{3}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{3}}\right)\Gamma\left(\Delta_{a_{2}}\right)^{2}}\left(\Delta_{\Phi_{3}}+2\Delta_{a_{2}}-\Delta_{a_{2}}^{2}-3\right),$
$\displaystyle
f_{a_{3}a_{3}}^{\Phi_{1}}=\left(\sqrt{\frac{3}{2}}-\sqrt{\frac{2}{3}}\right)\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{3}}+\Delta_{\Phi_{1}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{1}}\right)\Gamma\left(\Delta_{a_{3}}\right)^{2}}\left(\Delta_{\Phi_{1}}+2\Delta_{a_{3}}-\Delta_{a_{3}}^{2}-3\right),$
$\displaystyle
f_{a_{3}a_{3}}^{\Phi_{2}}=-2\sqrt{\frac{1}{3}}\frac{\pi^{\frac{d}{2}}\Gamma\left(\frac{2\Delta_{a_{3}}+\Delta_{\Phi_{2}}-d}{2}\right)}{\Gamma\left(\Delta_{\Phi_{2}}\right)\Gamma\left(\Delta_{a_{3}}\right)^{2}}\left(\Delta_{\Phi_{2}}+2\Delta_{a_{3}}-\Delta_{a_{3}}^{2}-3\right),$
$\displaystyle f_{a_{3}a_{3}}^{\Phi_{3}}=0,$ $\displaystyle
f_{\Phi_{1}\Phi_{1}}^{\Phi_{1}}=\frac{81\sqrt{6}}{2}\frac{3\pi^{\frac{d}{2}}\Gamma\left(\frac{3\Delta_{\Phi_{1}}-3}{2}\right)}{\Gamma\left(\Delta_{\Phi_{1}}\right)^{3}}.$

Analogously to what happens in the previous examples, all the axions and the
two fibre moduli are massless at zeroth order, leading to a large degeneracy
between the double trace operators. A first simplification is again provided
by the existence of a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, symmetry, which
partially breaks it down to subset of operators in the same representations. A
classification is provided in Table 2.

| $\mathbb{Z}_{2}(a_{1})$ | $\mathbb{Z}_{2}(a_{2})$ | $\mathbb{Z}_{2}(a_{3})$
---|---|---|---
$[a_{1}a_{2}]_{n,\ell}$ | -1 | -1 | 1
$[a_{1}a_{3}]_{n,\ell}$ | -1 | 1 | -1
$[a_{2}a_{3}]_{n,\ell}$ | 1 | -1 | -1
$[a_{1}\Phi_{2}]_{n,\ell},[a_{1}\Phi_{3}]_{n,\ell}$ | -1 | 1 | 1
$[a_{2}\Phi_{2}]_{n,\ell},[a_{2}\Phi_{3}]_{n,\ell}$ | 1 | -1 | 1
$[a_{3}\Phi_{2}]_{n,\ell},[a_{3}\Phi_{3}]_{n,\ell}$ | 1 | 1 | -1
$[a_{1}a_{1}]_{n,\ell},[a_{2}a_{2}]_{n,\ell},[a_{3}a_{3}]_{n,\ell},[\Phi_{2}\Phi_{2}]_{n,\ell},[\Phi_{3}\Phi_{3}]_{n,\ell},,[\Phi_{2}\Phi_{3}]_{n,\ell}$ | 1 | 1 | 1
$[a_{1}\Phi_{2}]_{n,\ell},[a_{1}\Phi_{1}]_{n,\ell}$ | -1 | 1 | 1
$[a_{2}\Phi_{2}]_{n,\ell},[a_{2}\Phi_{1}]_{n,\ell}$ | 1 | -1 | 1
$[a_{3}\Phi_{2}]_{n,\ell},[a_{3}\Phi_{1}]_{n,\ell}$ | 1 | 1 | -1
Table 2: Irreducible representation of the double trace operators under the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry. The two
different blocks correspond to distinct subspaces, where the degeneracies are
partly broken by the discrete symmetries.

We are now ready to discuss the anomalous dimensions. For operators involving
axions only,

$\displaystyle\gamma_{a_{1}a_{2}}\propto-(f_{a_{1}a_{1}}^{\Phi_{2}}f_{a_{2}a_{2}}^{\Phi_{2}}+f_{a_{1}a_{1}}^{\Phi_{3}}f_{a_{2}a_{2}}^{\Phi_{3}})>0,$
(46)
$\displaystyle\gamma_{a_{1}a_{3}}\propto-(f_{a_{1}a_{1}}^{\Phi_{2}}f_{a_{3}a_{3}}^{\Phi_{2}})>0,$
$\displaystyle\gamma_{a_{2}a_{3}}\propto-(f_{a_{2}a_{2}}^{\Phi_{2}}f_{a_{3}a_{3}}^{\Phi_{2}})>0.$

We have obtained the same, puzzling result encountered in the previous section
- positive anomalous dimensions for operators involving two axions. Moreover,
all three anomalous dimensions are identical in this case. On the other hand,
for the ones involving the volume modulus

$\displaystyle\gamma_{a_{1}\Phi_{1}}\propto-
f_{a_{1}a_{1}}^{\Phi_{1}}f_{\Phi_{1}\Phi_{1}}^{\Phi_{1}}<0,$ (47)
$\displaystyle\gamma_{a_{2}\Phi_{1}}\propto-
f_{a_{2}a_{2}}^{\Phi_{1}}f_{\Phi_{1}\Phi_{1}}^{\Phi_{1}}<0,$
$\displaystyle\gamma_{a_{3}\Phi_{1}}\propto-
f_{a_{3}a_{3}}^{\Phi_{1}}f_{\Phi_{1}\Phi_{1}}^{\Phi_{1}}<0$

are all negative as in previous studies. Again, all of the operators of the
form $[a_{i}\Phi_{2}]_{0,\ell},[a_{j}\Phi_{3}]_{0,\ell}$ decouple through a
combination of the discrete symmetries and sub-leading volume correction,
which lift the degeneracy between $\Phi_{2}$ and $\Phi_{3}$ (and with the
axions too). In particular they are all zero at this order in perturbation
theory. The same happens for
$[\Phi_{2}\Phi_{2}]_{0,\ell},[\Phi_{2}\Phi_{3}]_{0,\ell},[\Phi_{3}\Phi_{3}]_{0,\ell}$,
which do not mix with the axion double trace operators for the same reason.
Finally,

$[a_{1}a_{1}]_{n,\ell},\quad[a_{2}a_{2}]_{n,\ell},\quad[a_{3}a_{3}]_{n,\ell}$

are still degenerate, but the corresponding anomalous dimension matrix is
diagonal and all entries are necessarily negative, since they are proportional
to a sum of squared OPE coeffcients (with a minus sign in front).

##  4 Type IIA Models

Up to this point, we have exclusively focused on scenarios arising in type IIB
compactifications: LVS, KKLT and variations thereof. However, it is clearly
right that we also explore qualitatively different moduli stabilisation
scenarios, in particular those arising from IIA flux compactifications. One
significant difference with these is that, because IIA string theory has both
2- and 3-form flux field strengths in its spectrum, it allows for tree-level
flux stabilisation of the moduli in a way that is not possible in type IIB
scenarios.

In particular, we therefore consider the vacua of DeWolfe:2005uu  (for earlier
work on the effective action of IIA orientifolds see Grimm:2004ua ;
Kachru:2004jr ). This found that type IIA Calabi-Yau orientifolds with fluxes
admit stable $AdS_{4}$ vacua in the controlled regime of large volume and weak
coupling, and presented an explicit example where all moduli are stabilized,
which will form the basis of our analysis. This construction is based on the
toroidal orientifold $T^{6}/\mathbb{Z}^{2}_{3}$, where the discrete symmetries
fix the shape of the tori and thus eliminate all complex structure moduli.
Aside from the blow-up modes associated to the singularities, the only moduli
are the Kähler moduli determining the size of the three tori, with their
associated $B$-field axions, and the dilaton, with an axionic partner coming
from the $C_{3}$ three-form.

In 4D, the low energy theory is then characterized by a small number of fields
which are light in the holographic sense; i.e. whose conformal dimensions
remain finite in the limit of an infinite volume. While previous
investigations into the nature of the dual to this scenario Aharony:2008wz
have concentrated on more generic properties of the whole spectrum, we again
focus on the detailed properties of this low-lying CFT subsector. From our
perspective, the low number of moduli is thus crucial to ensure a manageable
effective CFT.

###  4.1 Effective Action

In 4D, the dynamics of the type IIA vacua can be conveniently analysed with
the formalism of $\mathcal{N}=1$ supergravity. The superfield coordinates
corresponding to the Kähler moduli and the dilaton can be written as

$t_{i}=b_{i}+iv_{i}\quad\quad S=e^{-D}+i\frac{\xi}{\sqrt{2}},$ (48)

and they are related to the overall volume of the manifold $\mathcal{V}$ and
the 10-dimensional dilaton $\phi$ by

$\mathcal{V}=\kappa\,v_{1}v_{2}v_{3},\quad\quad\quad
e^{D}=\frac{e^{\phi}}{\sqrt{\mathcal{V}}}.$ (49)

In terms of these variables, the Kähler potential is Grimm:2004ua ;
DeWolfe:2005uu

$\mathcal{K}=-\log(\mathcal{V})-4\log(S+\bar{S}).$ (50)

This gives rise to a kinetic term

$\mathcal{L}\supset\partial_{\mu}D\partial^{\mu}D+\frac{1}{2}e^{2D}\partial_{\mu}\xi\partial^{\mu}\xi+\sum_{i}\frac{1}{2}\partial_{\mu}\phi_{i}\partial^{\mu}\phi_{i}+\frac{1}{4}e^{-2\sqrt{2}\phi_{i}}\partial_{\mu}b_{i}\partial^{\mu}b_{i},$
(51)

where the canonically normalized fields
$\phi_{i}=\frac{1}{\sqrt{2}}\log(v_{i})$ have been used to make contact with
the conventions of DeWolfe:2005uu .999For the same reason the 4-dimensional
dilaton has not yet been normalized. The moduli are then stabilized by turning
on background fluxes for the NS-NS 3-form $H_{3}$, a constant $F_{0}$ and the
four-form $F_{4}$, quantized according to

$\int F_{p}=(2\pi)^{p-1}\alpha^{\prime(p-1)/2}f_{p}.$ (52)

These three fluxes can also be rewritten in terms of integers $f_{p}$ as

$m_{0}=\frac{f_{0}}{2\sqrt{2}\pi\sqrt{\alpha^{\prime}}},\quad
p=(2\pi)^{2}\alpha^{\prime}h_{3},\quad
e_{i}=\frac{\kappa^{1/3}}{\sqrt{2}}\left(2\pi\sqrt{\alpha^{\prime}}\right)^{3}f_{4}^{i},$
(53)

where the index $i$ in the last equation refers to the three non-trivial
4-cycles of the manifold. To ensure tadpole cancellation, the first two fluxes
cannot be independent and must satisfy

$m_{0}p=-2\left(\sqrt{2}\pi\sqrt{\alpha^{\prime}}\right).$ (54)

In terms of these quantities, one can finally write a scalar potential for the
moduli DeWolfe:2005uu , which reads

$V=\frac{p^{2}}{4}\frac{e^{2D}}{k}e^{-\sqrt{2}\sum_{i}\phi_{i}}+\Big{(}\sum_{i}e_{i}^{2}e^{2\sqrt{2}\phi_{i}}\Big{)}\frac{e^{4D-\sqrt{2}\sum_{i}\phi_{i}}}{2k}+\frac{m_{0}^{2}}{2}e^{4D}ke^{\sqrt{2}\sum_{i}\phi_{i}}-\sqrt{2}|m_{0}p|e^{D}.$
(55)

In particular, the minimum occurs for

$\bar{\phi}_{i}=\frac{1}{\sqrt{2}}\log\Bigg{(}\frac{1}{|e_{i}|}\sqrt{\frac{5}{3}\left\lvert\frac{e_{1}e_{2}e_{3}}{km_{0}}\right\rvert}\,\Bigg{)}$
(56)
$e^{D}=|p|\sqrt{\frac{27}{160}\left\lvert\frac{km_{0}}{e_{1}e_{2}e_{3}}\right\rvert},$
(57)

where the potential takes the value

$V|_{\text{min}}\equiv-\frac{3}{R^{2}_{AdS}}=-\frac{243\sqrt{\frac{3}{5}}k^{3/2}p^{4}\left|m_{0}\right|{}^{5/2}}{12800\lvert
e_{1}e_{2}e_{3}\rvert^{3/2}}.$ (58)

In the presence of background fluxes, dimensional reduction of the $B_{2}$ and
$C_{3}$ forms gives rise to a potential for the axions. The effective
Lagrangian of the form DeWolfe:2005uu

$\begin{split}\mathcal{L}\
\supset\,&\,\frac{1}{4}\sum_{i=1}^{3}e^{-2\sqrt{2}\phi_{i}}\partial_{\mu}b_{i}\partial^{\mu}b_{i}+\frac{1}{2}e^{2D}\partial_{\mu}\xi\partial^{\mu}\xi-\frac{e^{4D}}{\mathcal{V}}\left(b_{1}e_{1}+b_{2}e_{2}+b_{3}-p\xi\right)^{2}\\\
&-\frac{e^{4D}}{2}\sum_{i=1}^{3}\Big{(}m_{0}^{2}e^{-2\sqrt{2}\phi_{i}}\mathcal{V}\,b_{i}^{2}-2m_{0}e^{2\sqrt{2}\phi_{i}-\sqrt{2}(\phi_{1}+\phi_{2}+\phi_{3})}b_{1}b_{2}b_{3}\frac{e_{i}}{b_{i}}\Big{)},\end{split}$
(59)

###  4.2 Integer Conformal Dimensions

Expanding around the vacuum solution discussed in the previous section, we can
obtain the masses and couplings for the moduli. In particular, the mass matrix
is non diagonal; full details of the mixing are provided in Appendix C. Here,
we stress a surprising fact: after diagonalization, the dilatation operator
eigenstates obtain conformal dimensions which do not depend on any of the
compactification parameters (e.g. the values of the fluxes) and furthermore
are all integers:

$\Delta_{\varphi}=(10,6,6,6).$ (60)

Similarly, the couplings appearing in the interaction terms are fixed only by
the AdS radius, and all come with the same factor of $R_{AdS}^{-2}$ in front.
As in LVS, this again hints at the fact that the dual CFT at large volumes can
be thought as small perturbation above some universal generalized free theory.
There is also a high degree of degeneracy between the various modes. Naively
this would appear to translate into mixing between the corresponding double
trace operators at order $\mathcal{O}(1/N^{2})$.101010Notice that is also true
for the first scalar, since the spectra of different double trace operators
overlap when their conformal dimensions are integer spaced (and not only when
they are equal).  However, in practice we expect this degeneracy to be lifted
by the effects of higher order operators. The degeneracy partially comes as a
result of the exchange symmetries between pairs of tori and their fluxes
$b_{i}\rightarrow b_{j}$ and $e_{i}\rightarrow e_{j}$111111And the
independence from the fluxes, which effectively turns the symmetry into
$b_{i}\rightarrow b_{j}$ only., but the fact that there is also a third
identical eigenvalue seems to be coincidental.

Surprisingly, the exact same phenomenon occurs for the Kähler axions, with the
only difference that there is a residual imprint of the flux signs, and one
can distinguish between a few discrete cases. Expanding (59) around the
solution (56) and (57), the mass matrix is almost independent of the values
taken by the fluxes, which only enter through the signs

$s_{i}\equiv\text{sgn}(m_{0}e_{i}).$ (61)

While different choices of signs do not affect the vacuum, they do affect its
supersymmetry - the only choice that leads to a supersymmetric vacuum is
complete negativity, $s=(-1,-1,-1)$ (e.g. see Narayan:2010em ). However, for
our purposes the presence of supersymmetry is not crucial either way.

The diagonalisation procedure is reported in Appendix C, where we also give
the full set of cubic interactions relevant for the computation of anomalous
dimensions. Except for a discrete sign choice associated to the $s_{i}$’s,
couplings are entirely determined up to a global $R_{AdS}^{-2}$ factor. For
the conformal dimensions, the signs $s=(1,1,1)$ and $s=(1,-1,-1)$ yield

$\Delta_{a}=(8,8,8,2)\quad\quad\text{or}\quad\quad\Delta_{a}=(8,8,8,1).$ (62)

Two choices are available for the last eigenvalue, since both solutions for
the conformal dimension in terms of the mass satisfy the unitarity bound
$\Delta\geq d/2-1$. For the other two cases $s=(-1,1,1)$ and $s=(-1,-1,-1)$,
there is only one possibility:

$\Delta_{a}=(11,5,5,5).$ (63)

In both instances, the considerations about degeneracy apply as before, and
again we only see integer conformal dimensions.

While we cannot offer an explanation of this fact within our framework, it is
certainly surprising to see the emergence of such a peculiar pattern. Given
the quadratic nature of the $\Delta(\Delta-d)=m^{2}R_{AdS}^{2}$ relation, the
origin of these integers cannot simply be traced back to integers on the AdS,
giving more credibility to the idea that the CFT duals might offer a
complementary perspective on the description of such vacua. Moreover, the
(almost) complete independence on the details of the compactification is
already a striking feature, also shared with LVS (but not KKLT). It is very
much in line with the philosophy of the Swampland program, according to which
consistency conditions should become more and more stringent in certain
limits, and eventually culminate with a very restricted set of “special”
theories.

Interestingly, something similar happens if we consider what happens for more
general Calabi-Yaus with $h^{2,1}$ complex structure moduli $U_{i}$. The above
example concerned a $\mathbb{T}^{6}/\mathbb{Z}_{3}\times\mathbb{Z}_{3}$
geometry with no complex structure moduli at all. For more general IIA flux
vacua, an orientifold projection reduces the original hypermultiplet down to
two scalars. The real part corresponds to the calibrated volume of a 3-cycle
$\Sigma_{i}$ weighted by $g_{s}^{-1}$, and the imaginary part is a ‘true’
axion arising from the reduction of the RR 3-form on the cycle,
$\int_{\Sigma_{i}}C_{3}$. For models of particle physics arising from
intersecting D6-branes, these $U_{i}$ moduli represent the gauge kinetic
functions for the gauge groups living on the D6 branes.

$f_{a}=\sum\lambda_{i}U_{i}.$

The use of the expression ‘true’ axion refers to the origin from an RR 3-form.
Within weakly coupled string theory, the only physics sensitive to the value
of such axions are non-perturbative effects - either D-brane instantons or
gaugino condensation - associated to the cycles wrapped by the D6 branes. All
such effects are suppressed by non-perturbative exponentials,
$e^{-2\pi\textrm{Vol}(\Sigma_{i})/g_{s}}$, which are non-perturbative in both
$\alpha^{\prime}$ and $g_{s}$ expansions. This implies that, within
perturbation theory, the shift symmetry
$\textrm{Im}(U_{i})\to\textrm{Im}(U_{i})+\epsilon$ is exact: if it is there at
tree-level, it is also there at all orders in perturbation theory.

In IIA flux compactifications, the superpotential for the complex structure
moduli is linear in the complex structure moduli. Equivalently, the presence
of the fluxes _explicitly_ breaks the shift symmetry for one – and only one –
linear combination of the axions. By taking linear combinations of the moduli,
one can regard the superpotential as depending on only one of these moduli,
while the remaining moduli do not appear in the superpotential and also retain
the shift symmetry, $K=K(U_{i}+\bar{U}_{i})$.

For such moduli, their axionic components are of course massless (as nothing
in the potential depends on their vev). However, at the minimum the properties
of the saxions are also fixed. In fact, one can show that for a supersymmetric
minimum, the above properties imply (Conlon:2006tq  for a general argument,
Marchesano:2019hfb ; Marchesano:2020uqz  for explicit spectra and some non-
supersymmetric examples)

$M_{saxion}^{2}=-\frac{2}{3}V_{min}\equiv\frac{-2}{R_{AdS}^{2}}.$

On a holographic interpretation, this immediately implies

$\Delta_{saxion}\left(\Delta_{saxion}-3\right)=-2$

and so

$\Delta_{saxion}=1\qquad\textrm{ or }\qquad\Delta_{saxion}=2.$

###  4.3 Degeneracy Lifting and Anomalous Dimensions

As explained in 2.3, the presence of degeneracies can significantly complicate
the calculation of anomalous dimensions, because of the mixing between the
double trace operators occurring at $\mathcal{O}(1/N^{2})$. However, the
effect is not present if the conformal dimensions of the single trace
operators receive corrections of order $\mathcal{O}(1/N)$ or higher, as this
effectively lifts the degeneracy. This is what we would expect to happen in
this case, as can be inferred from the scaling of higher derivative
supergravity corrections. From Eq. (56), the controlled limit of large volume
is obtained for large four-form fluxes, when $\lvert e_{i}\rvert>>\lvert
m_{0}\rvert$. Assuming all fluxes to be of the same order $e_{i}\sim\bar{e}$,
we then have the following scalings:

$v_{i}\sim\bar{e}^{\frac{1}{2}}\quad\quad
e^{D}\sim\bar{e}^{-\frac{3}{2}}\quad\quad
R_{AdS}\sim\bar{e}^{\frac{9}{4}}\quad\quad\frac{1}{N}\sim\bar{e}^{-\frac{9}{2}},$
(64)

which can be used to estimate the impact of higher dimensional operators. As
noted in Grimm:2004ua , the scalar potential will receive relative corrections
of order $\bar{e}^{-\frac{3}{2}}$ from $\lvert F_{4}\rvert^{4}$ terms in the
Lagrangian, taking all the metric contractions and $g_{s}$ factors into
account. This will result in a mass shift for the moduli

$\delta\Delta_{\varphi}=\frac{2\Delta_{\varphi}(\Delta_{\varphi}-3)}{2\Delta_{\varphi}-3}\bigg{(}\frac{\delta
m^{2}}{m^{2}}\bigg{)}\sim\bar{e}^{-\frac{3}{2}}\gg\frac{1}{N}.$ (65)

Similarly, $\lvert F_{4}\rvert^{2}\lvert B_{2}\rvert^{2}$ terms will be
multiplied by additional factors of $g_{s}$ and $R^{-4}$, and give

$\delta\Delta_{a}=\frac{2\Delta_{a}(\Delta_{a}-3)}{2\Delta_{a}-3}\bigg{(}\frac{\delta
m^{2}}{m^{2}}\bigg{)}\sim\bar{e}^{-\frac{5}{2}}\gg\frac{1}{N}.$ (66)

for the axions. This would suggest that the degeneracy is lifted in both
cases, and anomalous dimensions can be calculated using the standard
procedure.121212We should give one note of caution here. As we do not
understand the origin of integer conformal dimensions, we cannot exclude the
possibility that whatever structure gave rise to them also holds for the
higher order terms - in which case the higher order terms would fail to lift
the degeneracy.

Unfortunately, it also means that the single trace operators, out of which the
double trace ones are constructed, will be rotated with respect to the basis
we are using. Within an exactly degenerate subspace (with two or more
identical conformal dimensions), the eigenstates will depend on the
degeneracy-splitting hamiltonian at leading order, and it is impossible to
deduce what they are without knowing the explicit form of the perturbation.
Thus, the only meaningful anomalous dimensions correspond to the double trace
operators constructed with singlets of the original Hamiltionian:
$\varphi_{1}$ and $a_{1}$ or $a_{4}$ (corresponding to cases 1 and 2, which
depend on the flux signs). As shown in appendix C, cases 3 and 4 can be
related to and 2 respectively by some discrete transformations which leave the
anomalous dimensions invariant, so we can focus only on the first two. In the
first case, formula (7) dictates that the anomalous dimension will be
proportional to a factor

$\gamma_{\varphi_{1}a_{1}}\propto\frac{1}{\Gamma(\Delta_{a_{1}}-\frac{\Delta_{\mathcal{O}}}{2})},$
(67)

where $\mathcal{O}$ is the lowest dimensional operator exchanged (one of
$\varphi_{2},\varphi_{3},\varphi_{4}$ in this case, with
$\Delta_{\varphi_{i}}=6$). For the values of $\Delta$ found above this term
diverges, and it is only regulated by sub-leading corrections to the conformal
dimensions. However, given the behaviour of the gamma function near its poles,
the sign of the anomalous dimension will depend on the signs of such
corrections, and cannot be determined at this point. The only thing that can
be said is that the two cases $\Delta_{a}=1$ and $\Delta_{a}=2$ the overall
sign will be the same, since any correction will shift their magnitude in
opposite directions. On the other hand, the relevant anomalous dimension in
case 2 can be calculated meaningfully. Using the couplings derived in C,

$C_{\varphi_{1}\varphi_{1}\varphi_{2}}=-\frac{315\sqrt{2}}{169}\quad\quad
C_{\varphi_{1}\varphi_{1}\varphi_{3}}=\frac{105\sqrt{2}}{169}\quad\quad
C_{\varphi_{1}\varphi_{1}\varphi_{4}}=\frac{105\sqrt{2}}{169}\quad\quad$ (68)

and

$C_{a_{1}a_{1}\varphi_{2}}=-\frac{60(3+100\sqrt{2})}{2197}\quad\,\,C_{a_{1}a_{1}\varphi_{3}}=\frac{20(3+100\sqrt{2})}{2197}\quad\,\,C_{a_{1}a_{1}\varphi_{4}}=\frac{20(3+100\sqrt{2})}{2197}\quad\,\,.$
(69)

Since the kinematic factors appearing in (7) are always positive in this
example,

$\gamma_{\varphi_{1}a_{1}}(0,\ell)\propto-\sum_{i=2}^{4}C_{\varphi_{1}\varphi_{1}\varphi_{i}}C_{a_{1}a_{1}\varphi_{i}}<0$
(70)

Thus the only anomalous dimensions we can calculate, which involve an axion
and a modulus (as in LVS), are negative.

##  5 Conclusions

We summarise here what we regard as the most interesting results and open
questions from these investigations.

First, in the context of the type IIA models of flux stabilisation the
conformal dimensions of low-lying operators dual to moduli fields take on a
form that is both universal and surprising, in that in a large-volume limits
the dimensions are all integer and (modulo some discrete flux choices) fixed.
As well as the conformal dimensions, it is also the case that the higher-point
interactions take on a universal form in the large-volume limit and ‘forget’
about the precise details of flux quantum numbers.

There are two particular aspects of this that are striking. First, the
universality of the large-volume limit is reminiscent of similar behaviour for
LVS. There, although the dual moduli dimensions were non-integer, the CFT also
took a universal form in the large-volume limit. It is worth stressing that
this runs counter to the landscape picture of string vacua being able to scan
over enormously large parameter spaces in the low-energy effective field
theory. Instead, these two examples of moduli stabilisation scenarios
producing vacua that are in the extreme limit of Kähler moduli space instead
lead to almost unique forms for the corresponding dual CFT.

We note this is _not_ true of KKLT or racetrack stabilisation, which do not
stabilise in the asymptotic regions of moduli space. Even with extraordinarily
small values of $W_{0}$ (such as the $10^{-95}$ reported in Demirtas:2021nlu )
the volume in KKLT is only logarithmic in $W_{0}$, and so is never in a limit
of asymptotically large volume.

It is intriguing to speculate that any stabilisation method producing vacua in
the asymptotic large-volume limit (or, more generally, in the extreme limits
of moduli space) corresponds to a very limited set of choices for the CFT
properties. This sounds vaguely similar to ideas for classifying the extremal
limits of moduli space (e.g. see Grimm:2019ixq ) and a more precise
correspondence would be interesting.

The other striking aspect is the presence of integer dual conformal dimensions
for the moduli in IIA flux vacua. This is surprising, as we can discern no
obvious reason why this should be the case. In information, the conformal
dimensions are equivalent to the moduli masses, but it is only when expressed
in this particular fashion that an integer structure appears - the masses are
not, for example, integer multiples of $R_{AdS}^{-1}$ or $m_{BF}$. This hints
at some deeper structure, but we can offer no explanation as to what it is.

A second interesting result concerns the sign of the anomalous dimensions. In
our previous work, we found that all anomalous dimensions for the double-trace
operators in the moduli sector were negative, and this negativity correlated
to signs in the low-energy effective field theory that were required from
swampland considerations (for example, a sign change would correspond to a
divergent axion decay constant in the infinite volume limit). However, in this
case (in the IIB fibred construction), this is only true for operators
involving an axion and a modulus. Indeed, all the double trace operators made
out of two separate axions now have positive anomalous dimensions.

## Acknowledgements

We thank Sandipan Kundu and especially Pietro Ferrero for illuminating
conversations and STFC for financial support of the Oxford particle theory
group. SN acknowledges funding support from the China Scholarship Council-
FaZheng Group- University of Oxford. FR is supported by the Dalitz Graduate
Scholarship, jointly established by the Oxford University Department of
Physics and Wadham College.

##  Appendix A Anomalous Dimension Matrix for the Degenerate Case

In this appendix we show how the anomalous dimensions of a degenerate system
can be obtained by diagonalizing the matrix of the “effective” anomalous
dimensions $\tilde{\gamma}_{[ij]\,[kl]}(n,\ell)$, as stated in section 2.3.1.
The latter are obtained from the mixed correlators
$\langle\mathcal{O}_{i}\mathcal{O}_{j}\mathcal{O}_{k}\mathcal{O}_{l}\rangle$,
and will be defined precisely in the following.

From a bootstrap perspective, the crucial point is that at zeroth order in the
large-$N$ expansion the contribution of the various double traces in any of
the OPEs cannot be distiguished from one another. Schematically, the
$\mathcal{O}_{1}\mathcal{O}_{1},\,\mathcal{O}_{2}\mathcal{O}_{2}$ and
$\,\mathcal{O}_{1}\mathcal{O}_{2}$ OPEs’ can be decomposed as

$\mathcal{O}_{i}(x)\mathcal{O}_{j}(0)\supset
c_{ijA}(n,\ell)\,[\mathcal{O}\mathcal{O}]^{A}_{n,\ell}(x);\,\,c_{ijB}(n,\ell)\,[\mathcal{O}\mathcal{O}]^{B}_{n,\ell}(x);\,\,c_{ijC}(n,\ell)\,[\mathcal{O}\mathcal{O}]^{C}_{n,\ell}(x)$
(71)

where the $[\mathcal{O}\mathcal{O}]^{A,B,C}_{n,\ell}$ operators are orthogonal
linear combinations of the double trace operators in absence of mixing, which
diagonalize the Hamiltonian at order $1/N^{2}$.

In terms of these variables, the conformal block expansion of a correlator
involving pairs of identical operators is

$\langle\mathcal{O}_{i}\mathcal{O}_{j}\mathcal{O}_{i}\mathcal{O}_{j}\rangle=\sum_{\Delta,\ell}c_{ijA}^{2}(n,\ell)G^{A}_{\Delta,\ell}+c_{ijB}^{2}(n,\ell)G^{B}_{\Delta,\ell}+c_{ijC}^{2}(n,\ell)G^{C}_{\Delta,\ell},$
(72)

where the $G^{X}_{\Delta,\ell}$ are the conformal blocks corresponding to the
$[\mathcal{O}\mathcal{O}]^{X}_{n,\ell}(x)$ double trace operators. The
functional form of the conformal blocks is identical in all cases, and the
subscript is just to remind they are a function of different conformal
dimensions. Up to order $\mathcal{O}\big{(}\frac{1}{N^{4}}\big{)}$, each term
can be expanded as:

$\begin{split}&\big{[}c_{ijA}^{(0)\,2}(n,\ell)+c_{ijB}^{(0)\,2}(n,\ell)+c_{ijC}^{(0)\,2}(n,\ell)\big{]}G_{2\Delta+2n+\ell,\ell}\\\
+&\big{[}2c^{(0)}_{ijA}c^{(1)}_{ijA}+2c^{(0)}_{ijB}c^{(1)}_{ijB}+2c^{(0)}_{ijC}c^{(1)}_{ijC}\big{]}\frac{G_{2\Delta+2n+\ell,\ell}}{N^{2}}\\\
+&\big{[}c_{ijA}^{(0)\,2}\gamma_{A}(n,\ell)+c_{ijB}^{(0)\,2}(n,\ell)\gamma_{B}(n,\ell)+c_{ijC}^{(0)\,2}\gamma_{C}(n,\ell)\big{]}\frac{\partial_{\Delta}G_{2\Delta+2n+\ell,\ell}}{N^{2}}\\\
\end{split}$ (73)

Comparing this with the standard expansion for the correlator

$f^{(0)\,2}(n,\ell)G^{11}_{2\Delta+2n+\ell,\ell}+2f^{(0)}(n,\ell)f^{(1)}(n,\ell)\frac{G_{2\Delta+2n+\ell,\ell}}{N^{2}}+f^{(0)\,2}(n,\ell)\tilde{\gamma}_{[ij][ij]}(n,\ell)\frac{\partial_{\Delta}G_{2\Delta+2n+\ell,\ell}}{N^{2}},$
(74)

we obtain the equations

$\displaystyle
c_{ijA}^{(0)\,2}(n,\ell)+c_{ijB}^{(0)\,2}(n,\ell)+c_{ijC}^{(0)\,2}(n,\ell)=f^{(0)\,2}(n,\ell)$
(75) $\displaystyle
c_{ijA}^{(0)\,2}\gamma_{A}(n,\ell)+c_{ijB}^{(0)\,2}(n,\ell)\gamma_{B}(n,\ell)+c_{ijC}^{(0)\,2}\gamma_{C}(n,\ell)=f^{(0)\,2}(n,\ell)\tilde{\gamma}_{[ij][ij]}(n,\ell)$
(76)

The same can be done for a mixed correlator
$\langle\mathcal{O}_{i}\mathcal{O}_{j}\mathcal{O}_{k}\mathcal{O}_{l}\rangle$,
with different pairs $(i,j)\neq(k,l)$. Then, the conformal block decomposition
reads

$\langle\mathcal{O}_{i}\mathcal{O}_{j}\mathcal{O}_{i}\mathcal{O}_{j}\rangle=\sum_{\Delta,\ell}c_{ijA}(n,\ell)c_{klA}(n,\ell)G^{A}_{\Delta,\ell}+c_{ijB}(n,\ell)c_{klB}(n,\ell)G^{B}_{\Delta,\ell}+c_{ijC}c_{klC}(n,\ell)G^{C}_{\Delta,\ell}.$
(77)

and can be expanded as

$\begin{split}&\big{[}c_{ijA}^{(0)}(n,\ell)c_{klA}^{(0)}(n,\ell)+c_{ijB}^{(0)}(n,\ell)c_{klB}^{(0)}(n,\ell)+c_{ijC}^{(0)}(n,\ell)c_{klC}^{(0)}(n,\ell)\big{]}G_{2\Delta+2n+\ell,\ell}\\\
+&\big{[}c^{(0)}_{ijA}c^{(1)}_{klA}+c^{(1)}_{ijA}c^{(0)}_{klA}+c^{(0)}_{ijB}c^{(1)}_{klB}+c^{(1)}_{ijB}c^{(0)}_{klB}+c^{(0)}_{ijC}c^{(1)}_{klC}+c^{(1)}_{ijC}c^{(0)}_{klC}\big{]}\frac{G_{2\Delta+2n+\ell,\ell}}{N^{2}}\\\
+&\big{[}c_{ijA}^{(0)}c_{klA}^{(0)}\gamma_{A}(n,\ell)+c_{ijB}^{(0)}c_{klB}^{(0)}\gamma_{B}(n,\ell)+c_{ijC}^{(0)}c_{klC}^{(0)}\gamma_{C}(n,\ell)\big{]}\frac{\partial_{\Delta}G_{2\Delta+2n+\ell,\ell}}{N^{2}}.\\\
\end{split}$ (78)

Since the disconnected four-point function only receives a contribution from
the identity, the same correlator can be written as

$2f^{(0)}(n,\ell)f^{(1)}(n,\ell)\frac{G_{2\Delta+2n+\ell,\ell}}{N^{2}}+f^{(0)\,2}(n,\ell)\tilde{\gamma}_{[ij][kl]}(n,\ell)\frac{\partial_{\Delta}G_{2\Delta+2n+\ell,\ell}}{N^{2}},$
(79)

resulting in the equations

$\displaystyle
c_{ijA}^{(0)}(n,\ell)c_{klA}^{(0)}(n,\ell)+c_{ijB}^{(0)}(n,\ell)c_{klB}^{(0)}(n,\ell)+c_{ijC}^{(0)}(n,\ell)c_{klC}^{(0)}(n,\ell)=0$
(80) $\displaystyle
c_{ijA}^{(0)}(n,\ell)c_{klA}^{(0)}(n,\ell)\gamma_{A}(n,\ell)+c_{ijB}^{(0)\,2}(n,\ell)\gamma_{B}(n,\ell)+c_{ijC}^{(0)\,2}\gamma_{C}(n,\ell)=f^{(0)\,2}(n,\ell)\tilde{\gamma}_{[ij][ij]}(n,\ell).$
(81)

This system admits a simple geometric interpretation. Equations (75) and (80)
imply that the vectors

$\vec{c}_{ij}^{(0)}(n,\ell)=\frac{1}{f^{(0)}(n,\ell)}\big{(}c_{ijA}^{(0)}(n,\ell),c_{ijB}^{(0)}(n,\ell),c_{ijC}^{(0)}(n,\ell)\big{)}$
(82)

are an orthonormal basis in $\mathbb{R}^{3}$. In this basis, the original
anomalous dimension matrix
$\mathbf{M}(n,\ell)=\rm{Diag}(\gamma_{A}(n,\ell),\gamma_{B}(n,\ell),\gamma_{C}(n,\ell))$
has matrix elements

$\mathbf{\tilde{M}}(n,\ell)_{[ij][kl]}=\bra{c_{ij}^{(0)}(n,\ell)}\mathbf{M}(n,\ell)\ket{c_{kl}^{(0)}(n,\ell)}=\tilde{\gamma}_{[ij][kl]},$
(83)

where the second equality derives from (76) and (81). Therefore, the
$\tilde{\gamma}_{[ij][kl]}$ are the coefficients of the anomalous dimension
matrix $\mathbf{M}(n,\ell)$ expressed in a different orthonormal basis,
implying the two share the same eigenvalues.

##  Appendix B Fibred Calabi-Yau Effective Lagrangians

###  B.1 One Fibre Modulus

From equation(34), the Lagrangian for the fibred one-modulus Calabi Yau
scenario is :

$L_{kin}=\frac{3}{8\tau_{1}^{2}}\partial_{u}\tau_{1}\partial^{u}\tau_{1}-\frac{1}{2\tau_{1}\mathcal{V}}\partial_{u}\tau_{1}\partial^{u}\mathcal{V}+\frac{1}{2\mathcal{V}^{2}}\partial_{u}\mathcal{V}\partial^{u}\mathcal{V}+\frac{1}{4\tau_{1}^{2}}\partial_{u}a_{1}\partial^{u}a_{1}+\frac{\alpha^{2}\tau_{1}}{2\mathcal{V}^{2}}\partial_{u}a_{2}\partial^{u}a_{2}.$
(84)

To put this in a canonically normalised form, we use the ansatz:

$\displaystyle\textrm{ln}\tau_{1}=a\Phi_{1}+b\Phi_{2},$ (85)
$\displaystyle\textrm{ln}\mathcal{V}=c\Phi_{1}.$

We subsitute this ansatz into the Lagrangian:

$\displaystyle\frac{3}{8\tau_{1}^{2}}\partial_{u}\tau_{1}\partial^{u}\tau_{1}$
$\displaystyle=\frac{3}{8}(\partial_{u}\textrm{ln}\tau_{1})^{2}=\frac{3}{8}(a\partial_{u}\Phi_{1}+b\partial_{u}\Phi_{2})^{2}$
(86)
$\displaystyle=\frac{3}{8}a^{2}(\partial_{u}\Phi_{1})^{2}+\frac{3}{8}b^{2}(\partial_{u}\Phi_{2})^{2}+\frac{3}{4}ab\partial_{u}\Phi_{1}\partial^{u}\Phi_{2},$
$\displaystyle-\frac{1}{2\tau_{1}\mathcal{V}}\partial_{u}\tau_{1}\partial^{u}\mathcal{V}$
$\displaystyle=-\frac{1}{2}\partial_{u}\textrm{ln}\tau_{1}\partial^{u}\textrm{ln}\mathcal{V}=-\frac{1}{2}(a\partial_{u}\Phi_{1}+b\partial_{u}\Phi_{2})c\partial^{u}\Phi_{1}$
(87)
$\displaystyle=-\frac{1}{2}ac(\partial_{u}\Phi_{1})^{2}-\frac{1}{2}bc\partial_{u}\Phi_{2}\partial^{u}\Phi_{1},$
$\displaystyle\frac{1}{2\mathcal{V}^{2}}\partial_{u}\mathcal{V}\partial^{u}\mathcal{V}=\frac{1}{2}(\partial_{u}\textrm{ln}\mathcal{V})^{2}=\frac{1}{2}c^{2}(\partial_{u}\Phi_{1})^{2}.$
(88)

Putting them together, the kinetic term reads:

$L_{kin}=\left(\frac{3}{8}a^{2}-\frac{1}{2}ac+\frac{1}{2}c^{2}\right)(\partial_{u}\Phi_{1})^{2}+\left(\frac{3}{4}ab-\frac{1}{2}bc\right)\partial_{u}\Phi_{1}\partial^{u}\Phi_{2}+\frac{3}{8}b^{2}(\partial_{u}\Phi_{2})^{2}$
(89)

Canonical normalisation then requires satisfying the following conditions:

$\frac{3}{8}a^{2}-\frac{1}{2}ac+\frac{1}{2}c^{2}=\frac{1}{2},$ (90)
$\frac{3}{4}ab-\frac{1}{2}bc=0,$ (91) $\frac{3}{8}b^{2}=\frac{1}{2}.$ (92)

A solution is:

$\displaystyle a=\sqrt{\frac{2}{3}},$ (93) $\displaystyle
b=\frac{2}{\sqrt{3}},$ $\displaystyle c=\sqrt{\frac{3}{2}}.$

After substituting this into the kinetic term, we have:

$\frac{3}{8\tau_{1}^{2}}\partial_{u}\tau_{1}\partial^{u}\tau_{1}=\frac{1}{4}(\partial_{u}\Phi_{1})^{2}+\frac{1}{2}(\partial_{u}\Phi_{2})^{2}+\frac{\sqrt{2}}{2}\partial_{u}\Phi_{1}\partial^{u}\Phi_{2},$
(94)
$-\frac{1}{2\tau_{1}\mathcal{V}}\partial_{u}\tau_{1}\partial^{u}\mathcal{V}=-\frac{1}{2}(\partial_{u}\Phi_{1})^{2}-\frac{\sqrt{2}}{2}\partial_{u}\Phi_{1}\partial^{u}\Phi_{2},$
(95)
$\frac{1}{2\mathcal{V}^{2}}\partial_{u}\mathcal{V}\partial^{u}\mathcal{V}=\frac{1}{2}(\partial_{u}\ln\mathcal{V})^{2}=\frac{3}{4}(\partial_{u}\Phi_{1})^{2}.$
(96)

Adding these together, we have both diagonal kinetic terms and diagonal mass
terms

$\frac{1}{2}(\partial_{u}\Phi_{1})^{2}+\frac{1}{2}(\partial_{u}\Phi_{2})^{2}+V(\Phi).$
(97)

Where the canonical fields $\Phi_{1},\Phi_{2}$ relate to the original fields
as:

$\mathcal{V}=e^{\sqrt{\frac{3}{2}}\Phi_{1}},$ (98)
$\tau_{1}=e^{\sqrt{\frac{2}{3}}(\Phi_{1}+2\Phi_{2})}.$ (99)

###  B.2 Two Fibre Moduli

We again start from the kinetic terms in the Lagrangian:

$\displaystyle L_{kin}=$
$\displaystyle\frac{1}{4\tau_{1}^{2}}\partial^{u}\tau_{1}\partial_{u}\tau_{1}+\frac{1}{4\tau_{1}^{2}}\partial^{u}a_{1}\partial_{u}a_{1}+\frac{1}{4\tau_{2}^{2}}\partial^{u}\tau_{2}\partial_{u}\tau_{2}+$
(100)
$\displaystyle\frac{1}{4\tau_{2}^{2}}\partial^{u}a_{2}\partial_{u}a_{2}+\frac{1}{4\tau_{3}^{2}}\partial^{u}\tau_{3}\partial_{u}\tau_{3}+\frac{1}{4\tau_{3}^{2}}\partial^{u}a_{3}\partial_{u}a_{3}$

To keep the mass term diagonal, we must have:

$\ln\mathcal{V}=a\Phi_{1},$ (101)

and we also anticipate:

$\displaystyle\textrm{ln}\tau_{1}=b\Phi_{1}+c\Phi_{2}+d\Phi_{3},$ (102)
$\displaystyle\textrm{ln}\tau_{2}=b\Phi_{1}+e\Phi_{2}+f\Phi_{3}.$

Similarly to the previous subsection, canonical normalisation leads to the
following conditions:

$\displaystyle a^{2}+\frac{3}{2}b^{2}-2ab=\frac{1}{2},$ (103) $\displaystyle
c^{2}+e^{2}+ce=1,$ $\displaystyle d^{2}+f^{2}+df=1,$ $\displaystyle
a(c+e)=\frac{3}{2}b(c+e),$ $\displaystyle a(d+f)=\frac{3}{2}b(d+f),$
$\displaystyle c(d+\frac{1}{2}f)+e(\frac{1}{2}d+f)=0.$

A solution is:

$\displaystyle a=\sqrt{\frac{3}{2}},$ (104) $\displaystyle
b=\sqrt{\frac{2}{3}},$ $\displaystyle c=e=\sqrt{\frac{1}{3}},$ $\displaystyle
d=-f=1.$

The Lagrangian then reads:

$\displaystyle L_{eff}=$
$\displaystyle\frac{1}{2}(\partial_{u}\Phi_{1})^{2}+\frac{1}{2}(\partial_{u}\Phi_{2})^{2}+\frac{1}{2}(\partial_{u}\Phi_{3})^{2}+\frac{1}{4}e^{-2\sqrt{\frac{2}{3}}\Phi_{1}-2\Phi_{3}-2\sqrt{\frac{1}{3}}\Phi_{2}}(\partial_{u}a_{1})^{2}$
(105)
$\displaystyle+\frac{1}{4}e^{-2\sqrt{\frac{2}{3}}\Phi_{1}+2\Phi_{3}-2\sqrt{\frac{1}{3}}\Phi_{2}}(\partial_{u}a_{2})^{2}+\frac{1}{4}e^{\left(2\sqrt{\frac{2}{3}}-2\sqrt{\frac{3}{2}}\right)\Phi_{1}+4\sqrt{\frac{1}{3}}\Phi_{2}}(\partial_{u}a_{3})^{2}-V_{lvs}.$

Other solutions are also possible but are physically equivalent under field
redefinitions.

##  Appendix C Type IIA Effective Lagrangian

###  C.1 Scalar Potential

At the minimum, the potential takes the value in (49) and through the relation
$V_{0}=-\frac{3M_{P}^{2}}{R_{AdS}}$ one can express the various couplings as a
function of the AdS radius. Surprisingly, all of the dependence on the values
of the fluxes drops out, and the the mass matrix is simply given by131313Since
the kinetic term (51) is not canonically normalized, we have been careful to
use $\tilde{D}=\frac{D}{\sqrt{2}}$.

$M^{2}_{ab}=\frac{1}{R^{2}_{AdS}}\left(\begin{array}[]{cccc}34&16&16&\,8\\\
16&34&16&\,8\\\ 16&16&34&\,8\\\ \,8&\,8&\,8&22\\\ \end{array}\right).$ (106)

It can be diagonalised in terms of new fields $\varphi=U\phi$, where

$U=\left(\begin{array}[]{cccc}\,\,2&\,\,2&\,\,2&\,\,1\\\
-1&\,\,0&\,\,0&\,\,2\\\ -1&\,\,0&\,\,1&\,\,0\\\ -1&\,\,1&\,\,0&\,\,0\\\
\end{array}\right),$ (107)

yielding the mass matrix

$M^{2}_{D}=\frac{1}{R^{2}_{AdS}}\text{Diag}(70,18,18,18).$ (108)

Using the standard relation $\Delta(\Delta-d)=m^{2}R^{2}_{AdS}$ one
obtains141414This equation can have two real solutions, but we only consider
the one satisfying the unitarity bound.

$\Delta_{1}=10\quad\quad\Delta_{2}=\Delta_{3}=\Delta_{4}=6.$ (109)

It is interesting to notice how the exchange symmetry
$\phi_{i}\leftrightarrow\phi_{j}$ for the first three fields is only
responsible for part of the degeneracy between the conformal dimensions.
Indeed, for a generic matrix

$M=\begin{pmatrix}a&b&b&c\\\ b&a&b&c\\\ b&b&a&c\\\ c&c&c&d\\\ \end{pmatrix}$
(110)

with the same symmetry properties only two of the eigenvalues are identical:

$\lambda_{1}=\lambda_{2}=a-b\quad\lambda_{3,4}=\frac{1}{2}\left(a+2b+d\pm\sqrt{(a+2b-d)^{2}+12c^{2}}\,\right).$
(111)

In order for three (or more) to be equal, the matrix elements must satisfy the
additional constraint

$(a-b-d)b+c^{2}=0,$ (112)

which in the case at hand appears to be coincidental.

Expanding at higher orders, one can obtain the n-point self interactions for
the moduli. Given the exponential structure of the potential, derivatives with
respect to any of the moduli evaluated at the minimum will be proportional to
$V|_{\text{min}}$; hence all vertices will all come with the same factor of
$R_{AdS}^{-2}$ in front. This means that also the couplings are independent of
the flux choice, and universally fixed apart from an overall constant
(corresponding to the $1/N$ expansion parameter) which vanishes in the
infinite volume limit. We report here the cubic terms in the potential, the
only ones necessary to compute large-spin anomalous dimensions.

$\begin{split}&V^{(3)}R^{2}_{AdS}=12\sqrt{2}\tilde{D}^{2}\phi_{1}+12\sqrt{2}\tilde{D}^{2}\phi_{2}+12\sqrt{2}\tilde{D}^{2}\phi_{3}+\frac{27\tilde{D}^{3}}{\sqrt{2}}+30\sqrt{2}\tilde{D}\phi_{1}^{2}+30\sqrt{2}\tilde{D}\phi_{2}^{2}\\\
&+30\sqrt{2}\tilde{D}\phi_{3}^{2}+24\sqrt{2}\tilde{D}\phi_{1}\phi_{2}+24\sqrt{2}\tilde{D}\phi_{1}\phi_{3}+24\sqrt{2}\tilde{D}\phi_{2}\phi_{3}+18\sqrt{2}\phi_{1}\phi_{2}\phi_{3}\end{split}.$
(113)

In terms of the diagonal fields $\varphi_{i}$, it becomes

$\begin{split}&V^{(3)}R^{2}_{AdS}=\frac{\sqrt{2}}{169}\Big{(}1080\,\varphi_{2}\varphi_{4}^{2}-360\,\varphi_{3}\varphi_{4}^{2}-120\,\varphi_{4}^{3}-105\,\varphi_{1}^{2}\varphi_{4}-198\,\varphi_{2}^{2}\varphi_{4}-360\,\varphi_{3}^{2}\varphi_{4}\\\
&-882\,\varphi_{2}\varphi_{3}\varphi_{4}+135\,\varphi_{1}^{3}+198\,\varphi_{2}^{3}-120\,\varphi_{3}^{3}+1080\,\varphi_{2}\varphi_{3}^{2}+315\,\varphi_{1}^{2}\varphi_{2}-105\,\varphi_{1}^{2}\varphi_{3}-198\,\varphi_{2}^{2}\varphi_{3}\Big{)}\end{split}.$
(114)

We notice that, at least up to cubic order, the potential is invariant under
the exchange of $\varphi_{3}$ and $\varphi_{4}$.

###  C.2 Axion Kinetic Term

The same procedure can be carried out for axions, expanding the lagrangian
(59) about the usual vacuum solution in terms of the canonically normalized
fields

$b^{\prime}_{i}=\frac{b_{i}}{\sqrt{2}\bar{v}_{i}}\quad\quad\quad\xi^{\prime}=\xi
e^{\bar{D}}$ (115)

and the moduli fluctuations

$\phi_{i}^{\prime}=\phi_{i}-\bar{\phi}_{i}\quad\quad D^{\prime}=D-\bar{D},$
(116)

where it is understood that all primes will be omitted in the following. The
resulting expression can be divided into a kinetic term

$\mathcal{L}_{kin}=\frac{1}{2}e^{-2\sqrt{2}\phi_{i}}\partial^{\mu}b_{i}\partial_{\mu}b^{i}+\frac{1}{2}e^{\sqrt{2}\tilde{D}}\partial_{\mu}\xi\partial^{\mu}\xi$
(117)

and a potential

$\begin{split}\mathcal{L}_{\phi
b}&=\frac{15}{R^{2}_{AdS}}e^{2\sqrt{2}\tilde{D}}\Big{[}s_{1}b_{2}b_{3}e^{\sqrt{2}(\phi_{1}-\phi_{2}-\phi_{3})}+s_{2}b_{3}b_{1}e^{\sqrt{2}(\phi_{2}-\phi_{3}-\phi_{1})}+s_{3}b_{1}b_{2}e^{\sqrt{2}(\phi_{3}-\phi_{1}-\phi_{2})}\Big{]}\\\
&-\frac{9}{2R^{2}_{AdS}}e^{\sqrt{2}(2\tilde{D}-\phi_{1}-\phi_{2}-\phi_{3})}\Big{[}b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+2s_{1}s_{2}b_{1}b_{2}+2s_{1}s_{3}b_{1}b_{3}+2s_{2}s_{3}b_{2}b_{3}\Big{]}\\\
&-\frac{12}{R^{2}_{AdS}}e^{\sqrt{2}(2\tilde{D}-\phi_{1}-\phi_{2}-\phi_{3})}\Big{[}s_{1}b_{1}+s_{2}b_{2}+s_{3}b_{3}\Big{]}\xi-\frac{8}{R^{2}_{AdS}}e^{\sqrt{2}(2\tilde{D}-\phi_{1}-\phi_{2}-\phi_{3})}\xi^{2}\\\
&-\frac{25}{2R^{2}_{AdS}}e^{2\sqrt{2}\tilde{D}}\Big{[}b_{1}^{2}e^{\sqrt{2}(\phi_{2}+\phi_{3}-\phi_{1})}+b_{2}^{2}e^{\sqrt{2}(\phi_{3}+\phi_{1}-\phi_{2})}+b_{3}^{2}e^{\sqrt{2}(\phi_{1}+\phi_{2}-\phi_{3})}\Big{]}.\end{split}$
(118)

####  C.2.1 Spectrum

Setting all of the moduli fluctuations to zero, this gives the mass matrix

$M^{2}_{ab}=\frac{1}{R^{2}_{AdS}}\begin{pmatrix}34&9\,s_{1}s_{2}-15\,s_{3}&9\,s_{1}s_{3}-15\,s_{2}&12\,s_{1}\\\
9\,s_{1}s_{2}-15\,s_{3}&34&9\,s_{2}s_{3}-15\,s_{1}&12s_{2}\\\
9\,s_{1}s_{3}-15\,s_{2}&9\,s_{2}s_{3}-15\,s_{1}&34&12\,s_{3}\\\
12\,s_{1}&12\,s_{2}&12\,s_{3}&16\end{pmatrix},$ (119)

as reported in [DeWolfe:2005uu, ]. Since the matrix depends on the signs
$s_{i}\equiv\text{sgn}(m_{0}e_{i})$, we can distinguish four different
combinations (permutations are indistinguishable):

##### Case 1:

s = (1,1,1)

The matrix is diagonalised in term of the new fields $a=U_{1}b$, where

$U_{1}=\begin{pmatrix}\,\,2&\,\,0&\,\,0&\,\,1\\\ -1&\,\,0&\,\,1&\,\,0\\\
-1&\,\,1&\,\,0&\,\,0\\\ -1&-1&-1&\,\,2\\\ \end{pmatrix}.$ (120)

The mass matrix is given by

$M^{2}_{D}=\frac{1}{R^{2}_{AdS}}\text{Diag}(40,40,40,-2),$ (121)

resulting in

$\Delta_{a}=(8,8,8,2)\quad\quad\text{or}\quad\quad\Delta_{a}=(8,8,8,1).$ (122)

##### Case 2:

s = (1,1,-1)

The matrix is diagonalised in term of the new fields $a=U_{2}b$, where

$U_{2}=\begin{pmatrix}\,\,2&\,\,2&-2&\,\,1\\\ -1&\,\,0&\,\,0&\,\,2\\\
\,\,1&\,\,0&\,\,1&\,\,0\\\ -1&\,\,1&\,\,0&\,\,0\\\ \end{pmatrix}.$ (123)

The mass matrix is given by

$M^{2}_{D}=\frac{1}{R^{2}_{AdS}}\text{Diag}(88,10,10,10),$ (124)

resulting in

$\Delta_{a}=(11,5,5,5).$ (125)

##### Case 3:

s = (1,-1,-1)

The matrix is diagonalised in term of the new fields $a=U_{3}b$, where

$U_{3}=\begin{pmatrix}\,\,2&\,\,0&\,\,0&\,\,1\\\ \,\,1&\,\,0&1\,\,&\,\,0\\\
\,\,1&\,\,1&\,\,0&\,\,0\\\ -1&\,\,1&\,\,1&\,\,2\\\ \end{pmatrix}.$ (126)

The mass matrix is given by

$M^{2}_{D}=\frac{1}{R^{2}_{AdS}}\text{Diag}(40,40,40,-2),$ (127)

resulting in

$\Delta_{a}=(8,8,8,2)\quad\quad\text{or}\quad\quad\Delta_{a}=(8,8,8,1).$ (128)

##### Case 4:

s = (-1,-1,-1)

The matrix is diagonalised in term of the new fields $a=U_{4}b$, where

$U_{4}=\begin{pmatrix}-2&-2&-2&\,\,1\\\ \,\,1&\,\,0&\,\,0&\,\,2\\\
-1&\,\,0&\,\,1&\,\,0\\\ -1&\,\,1&\,\,0&\,\,0\\\ \end{pmatrix}.$ (129)

The mass matrix is given by

$M^{2}_{D}=\frac{1}{R^{2}_{AdS}}\text{Diag}(88,10,10,10),$ (130)

resulting in

$\Delta_{a}=(11,5,5,5).$ (131)

####  C.2.2 Interactions

For the interactions, let’s start from the ones originating from the kinetic
term (117). Again, we can distinguish different depending on the flux signs:

##### Case 1:

s = (1,1,1)

$\begin{split}&\mathcal{L}^{1}_{kin}=-\frac{1}{637\sqrt{2}}\Big{[}\\\
&(\partial
b_{1})^{2}\big{(}39\varphi_{1}-78\varphi_{2}+26\varphi_{3}+26\varphi_{4}\big{)}+\partial
b_{1}\partial
b_{2}\big{(}52\varphi_{1}-104\varphi_{2}+520\varphi_{3}-208\varphi_{4}\big{)}\\\
&+\partial b_{1}\partial
b_{3}\big{(}52\varphi_{1}-104\varphi_{2}-208\varphi_{3}+520\varphi_{4}\big{)}+\partial
b_{1}\partial
b_{4}\big{(}-60\varphi_{1}-48\varphi_{2}+16\varphi_{3}+16\varphi_{4}\big{)}\\\
&+(\partial
b_{2})^{2}\big{(}148\varphi_{1}-100\varphi_{2}+640\varphi_{3}-270\varphi_{4}\big{)}+\partial
b_{2}\partial
b_{3}\big{(}-96\varphi_{1}-4\varphi_{2}-120\varphi_{3}-120\varphi_{4}\big{)}\\\
&+\partial b_{2}\partial
b_{4}\big{(}-40\varphi_{1}-32\varphi_{2}-232\varphi_{3}+132\varphi_{4}\big{)}+(\partial
b_{3})^{2}\big{(}148\varphi_{1}-100\varphi_{2}-270\varphi_{3}+640\varphi_{4}\big{)}\\\
&+\partial b_{3}\partial
b_{4}\big{(}-40\varphi_{1}-32\varphi_{2}+132\varphi_{3}-232\varphi_{4}\big{)}+(\partial
b_{4})^{2}\big{(}8\varphi_{1}-30\varphi_{2}+10\varphi_{3}+10\varphi_{4}\big{)}\Big{]}\end{split}$
(132) $\begin{split}&\mathcal{L}^{1}_{\varphi b}=\frac{1}{637}\Big{[}\\\
&b_{1}^{2}\Big{(}(1560\sqrt{2}-360)\varphi_{1}+(180-6240\sqrt{2})\varphi_{2}+(2080\sqrt{2}-60)\varphi_{3}+(2080\sqrt{2}-60)\varphi_{4}\Big{)}\\\
&+b_{1}b_{2}\Big{(}(2080\sqrt{2}-480)\varphi_{1}+(240-8320\sqrt{2})\varphi_{2}+(8840\sqrt{2}-3720)\varphi_{3}+(1740-260\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{1}b_{3}\Big{(}(2080\sqrt{2}-480)\varphi_{1}+(240-8320\sqrt{2})\varphi_{2}+(1740-260\sqrt{2})\varphi_{3}+(8840\sqrt{2}-3720)\varphi_{4}\Big{)}\\\
&+b_{1}b_{4}\Big{(}(360+120\sqrt{2})\varphi_{1}-(180+60\sqrt{2})\varphi_{2}+(60+20\sqrt{2})\varphi_{3}+(60+20\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{2}^{2}\Big{(}(330-1430\sqrt{2})\varphi_{1}-(165+10205\sqrt{2})\varphi_{2}+(10985\sqrt{2}+1875)\varphi_{3}-(855+390\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{2}b_{3}\Big{(}(3510\sqrt{2}-810)\varphi_{1}+(4050+1885\sqrt{2})\varphi_{2}-2145\sqrt{2}\varphi_{3}-2145\sqrt{2}\varphi_{4}\Big{)}\\\
&+b_{2}b_{4}\Big{(}(240+80\sqrt{2})\varphi_{1}-(120+40\sqrt{2})\varphi_{2}\Big{)}+(1860-3020\sqrt{2})\varphi_{3}+(1530\sqrt{2}-870)\varphi_{4}\Big{)}\\\
&+b_{3}^{2}\Big{(}(330-1430\sqrt{2})\varphi_{1}-(165+10205\sqrt{2})\varphi_{2}-(855+390\sqrt{2})\varphi_{3}+(10985\sqrt{2}+1875)\varphi_{4}\Big{)}\\\
&+b_{3}b_{4}\Big{(}(240+80\sqrt{2})\varphi_{1}-(120+40\sqrt{2})\varphi_{2}\Big{)}+(1530\sqrt{2}-870)\varphi_{3}+(1860-3020\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{4}^{2}\Big{(}-(90+58\sqrt{2})\varphi_{1}+(45+120\sqrt{2})\varphi_{2}-(15+40\sqrt{2})\varphi_{3}-(15+40\sqrt{2})\varphi_{4}\Big{)}\Big{]}.\end{split}$
(133)

Both are symmetric under the simultaneous exchanges

$\varphi_{3}\longleftrightarrow\varphi_{4}\quad\quad\quad
b_{2}\longleftrightarrow b_{3}.$ (134)

##### Case 2:

s = (1,-1,1)

$\begin{split}&\mathcal{L}^{2}_{kin}=-\frac{1}{2197\sqrt{2}}\Big{[}\\\
&(\partial
b_{1})^{2}\big{(}47\varphi_{1}-30\varphi_{2}+10\varphi_{3}+10\varphi_{4}\big{)}+\partial
b_{1}\partial
b_{2}\big{(}-60\varphi_{1}-48\varphi_{2}+16\varphi_{3}+16\varphi_{4}\big{)}\\\
&+\partial b_{1}\partial
b_{3}\big{(}-20\varphi_{1}-16\varphi_{2}-896\varphi_{3}+456\varphi_{4}\big{)}+\partial
b_{1}\partial
b_{4}\big{(}20\varphi_{1}+16\varphi_{2}+456\varphi_{3}-896\varphi_{4}\big{)}\\\
&+(\partial
b_{2})^{2}\big{(}-24\varphi_{1}-222\varphi_{2}+74\varphi_{3}+74\varphi_{4}\big{)}+\partial
b_{2}\partial
b_{3}\big{(}-16\varphi_{1}-148\varphi_{2}+500\varphi_{3}-176\varphi_{4}\big{)}\\\
&+\partial b_{2}\partial
b_{4}\big{(}16\varphi_{1}+148\varphi_{2}+176\varphi_{3}-500\varphi_{4}\big{)}+(\partial
b_{3})^{2}\big{(}448\varphi_{1}-250\varphi_{2}+1210\varphi_{3}-480\varphi_{4}\big{)}\\\
&+\partial b_{3}\partial
b_{4}\big{(}456\varphi_{1}-176\varphi_{2}+960\varphi_{3}+960\varphi_{4}\big{)}+(\partial
b_{4})^{2}\big{(}448\varphi_{1}-250\varphi_{2}-480\varphi_{3}+1210\varphi_{4}\big{)}\Big{]}\end{split}$
(135) $\begin{split}&\mathcal{L}^{2}_{\varphi b}=\frac{1}{2197}\Big{[}\\\
&b_{1}^{2}\Big{(}(8\sqrt{2}+360)\varphi_{1}-(180+7440\sqrt{2})\varphi_{2}+(60+2480\sqrt{2})\varphi_{3}+(60+2480\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{1}b_{2}\Big{(}(2080\sqrt{2}-480)\varphi_{1}+(240-8320\sqrt{2})\varphi_{2}+(380\sqrt{2}-60)\varphi_{3}+(380\sqrt{2}-60)\varphi_{4}\Big{)}\\\
&+b_{1}b_{3}\Big{(}(760\sqrt{2}-120)\varphi_{1}+(60-380\sqrt{2})\varphi_{2}-(6780+11140\sqrt{2})\varphi_{3}+(3360+5760\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{1}b_{4}\Big{(}(-760\sqrt{2}+120)\varphi_{1}+(-60+380\sqrt{2})\varphi_{2}-(3360+5760\sqrt{2})\varphi_{3}+(6780+11140\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{2}^{2}\Big{(}(90+210\sqrt{2})\varphi_{1}-(45+2640\sqrt{2})\varphi_{2}+(15+880\sqrt{2})\varphi_{3}+(15+880\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{2}b_{3}\Big{(}(60+140\sqrt{2})\varphi_{1}-(30+1760\sqrt{2})\varphi_{2}+(3390+6220\sqrt{2})\varphi_{3}-(1680+2230\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{2}b_{4}\Big{(}-(60+140\sqrt{2})\varphi_{1}+(30+1760\sqrt{2})\varphi_{2}+(1680+2230\sqrt{2})\varphi_{3}-(3390+6220\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{3}^{2}\Big{(}-(1680+3920\sqrt{2})\varphi_{1}+(840-5645\sqrt{2})\varphi_{2}+(15965\sqrt{2}-13800)\varphi_{3}+(6480-5160\sqrt{2})\varphi_{4}\Big{)}\\\
&+b_{3}b_{4}\Big{(}-(1710+3990\sqrt{2})\varphi_{1}+(855-4765\sqrt{2})\varphi_{2}\Big{)}+(12855\sqrt{2}-15945)\varphi_{3}+(12855\sqrt{2}-15945)\varphi_{4}\Big{)}\\\
&+b_{4}^{2}\Big{(}-(1680+3920\sqrt{2})\varphi_{1}+(840-5645\sqrt{2})\varphi_{2}+(15965\sqrt{2}-13800)\varphi_{3}+(6480-5160\sqrt{2})\varphi_{4}\Big{)}\Big{]}.\end{split}$
(136)

Both are symmetric under the simultaneous exchanges

$\varphi_{3}\longleftrightarrow\varphi_{4}\quad\quad\quad
b_{3}\longleftrightarrow-b_{4}.$ (137)

##### Case 3:

s = (1,-1,-1)

$\mathcal{L}_{kin}^{3}(b_{1},b_{2},b_{3},b_{4})=\mathcal{L}_{kin}^{1}(b_{1},-b_{2},-b_{3},b_{4})$
(138)
$\mathcal{L}_{\varphi_{b}}^{3}(b_{1},b_{2},b_{3},b_{4})=\mathcal{L}_{\varphi
b}^{1}(b_{1},-b_{2},-b_{3},b_{4})$ (139)

##### Case 4:

s = (-1,-1,-1)

$\mathcal{L}_{kin}^{4}(b_{1},b_{2},b_{3},b_{4})=\mathcal{L}_{kin}^{2}(b_{1},b_{2},b_{3},-b_{4})$
(140) $\mathcal{L}_{\varphi
b}^{4}(b_{1},b_{2},b_{3},b_{4})=\mathcal{L}_{\varphi
b}^{2}(b_{1},b_{2},b_{3},-b_{4})$ (141)

The relations between cases 1 & 3 and 2 & 4 descend from the relations between
the matrices $U_{1}$ & $U_{2}$ and $U_{3}$ & $U_{4}$ introduced above.

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