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## Contribution

Sungjoon Park led the project as project manager, initiated the project, made
decisions on overall progress of this project, secured financial resources,
signed up with ACROFAN for the articles, and organized IRB submission and
research paper.

Jihyung Moon led the project as project manager, managed overall datasets,
models, and ethical concerns, signed up with ACROFAN for the articles,
prepared IRB, as well as contributed to NER, STS, NLI, MRC dataset
constructions, AIRBNB, POLICY corpora collection, and leaderboard design.

Sungdong Kim managed overall fine-tuning of language models, served as a
person in charge (PIC) of DST, and contributed to the dataset construction of
TC, STS, and RE.

Won Ik Cho managed the overall dataset construction of TC, STS, NLI, RE, MRC,
and DST, provided the original corpus of PARAKQC, and served as a PIC of STS.

Jiyoon Han managed the overall dataset construction of DP and NER, served as a
PIC of NLI, contributed to the dataset construction of STS, and took part in
preparing IRB.

Jangwon Park served as a PIC of model pretraining, contributed to the text
collection of YNA, collected and pre-processed MODU, CC-100-Kor, NAMUWIKI,
NEWSCRAWL, and PETITION, and conducted the fine-tuning of TC.

Chisung Song served as a PIC of NER and contributed to the dataset
construction of DP and DST.

Junseong Kim served as a PIC of MRC, conducted the fine-tuning of MRC, and
contributed to the text collection of WIKITREE and WIKIPEDIA.

Youngsook Song served as a PIC of TC and contributed to the dataset
construction of NER and DST.

Taehwan Oh served as a PIC of DP, contributed to the dataset construction of
NER and NLI, and took part in preparing IRB.

Joohong Lee served as a PIC of RE, and conducted the fine-tuning of RE.

Juhyun Oh contributed to the dataset construction of NLI, NER, DP and STS,
took part in ethical considerations, IRB preparation and setup for model
pretraining.

Sungwon Lyu contributed to the dataset construction of STS, NLI, RE, and MRC,
taking part in the overall model pretraining and task-wise fine-tuning.

Younghoon Jeong contributed to the text collection of WIKINEWS, modeling of
DP, and pre-processing of the pretraining corpus.

Inkwon Lee contributed to the text collection, modeling of DP and pre-
processing of the pretraining corpus.

Sangwoo Seo contributed to the dataset construction of RE, and took part in
preparing IRB.

Dongjun Lee contributed to the construction of the fine-tuning pipeline and
the modeling of STS.

Hyunwoo Kim contributed to the dataset construction of MRC, and took part in
preparing IRB.

Myeonghwa Lee contributed to the dataset construction of STS and TC.

Seongbo Jang contributed to the dataset construction of RE, and took part in
preparing IRB.

Seungwon Do contributed to the dataset construction of DST and text
collection.

Sunkyoung Kim contributed to the dataset construction of RE and MRC, and
modeling of MRC.

Kyungtae Lim contributed to the dataset construction of DP.

Jongwon Lee contributed to the dataset construction and modeling of DST.

Kyumin Park contributed to the dataset construction of DST, and took part in
preparing IRB.

Jay Shin contributed to the dataset construction of DST.

Seonghyun Kim contributed to the dataset construction and modeling of NER.

Lucy Park contributed to the dataset construction of MRC and provided the
original corpus of NSMC.

Alice Oh advised the project, sponsored the project via KAIST, provided
feedback and suggested better way to improve the quality of our dataset and
models, and helped with the final manuscript.

Jung-Woo Ha advised the project, sponsored annotation cost and computing cloud
credits the project via NAVER, provided license-free news articles from The
Korea Economy Daily, and helped with the final manuscript.

Kyunghyun Cho advised the project, provided critical feedback, suggested
better way to improve the quality of our dataset and models, and helped a lot
with polishing and rewriting the final manuscript.

All participants contributed to this manuscript.

##  Appendix A Dev Set Results

Table 36: Performances of our pretrained LMs and other baselines on KLUE
benchmark dev set. The notations are same with Table 32.

YNAT KLUE-STS KLUE-NLI KLUE-NER KLUE-RE KLUE-DP KLUE-MRC WoS Model F1 RP F1
ACC F1E F1C F1mic AUC UAS LAS EM ROUGE JGA F1S $\text{mBERT}_{\text{BASE}}$
82.64 82.97 75.93 72.90 75.14 87.51 58.39 56.41 88.53 86.04 49.96 55.57 35.27
88.60 $\text{XLM-R}_{\text{BASE}}$ 84.52 88.88 81.20 78.23 80.73 91.37 57.62
57.05 93.12 87.23 26.76 53.36 41.54 89.81 $\text{XLM-R}_{\text{LARGE}}$ 87.30
93.08 87.17 86.40 81.43 92.30 58.75 63.53 92.87 87.82 35.23 66.55 42.44 89.88
$\text{KR-BERT}_{\text{BASE}}$ 85.36 87.50 77.92 77.10 74.76 90.06 62.83 65.42
92.87 87.13 48.95 58.38 45.60 90.82 $\text{KoELECTRA}_{\text{BASE}}$ 85.99
93.14 85.89 86.87 86.38 92.70 62.67 57.46 90.93 87.07 59.54 65.64 39.83 88.91
$\text{KLUE-BERT}_{\text{BASE}}$ 86.59 91.01 83.44 79.87 84.42 91.19 65.58
68.11 93.07 87.25 62.42 68.15 49.34 92.07 $\text{KLUE-RoBERTa}_{\text{SMALL}}$
85.53 91.23 85.54 79.90 84.47 91.00 61.54 60.79 92.45 86.61 57.83 63.23 45.92
91.25 $\text{KLUE-RoBERTa}_{\text{BASE}}$ 86.03 92.70 85.54 85.50 85.18 91.81
66.01 69.56 93.22 87.56 69.65 74.72 47.40 91.67 $\text{KLUE-
RoBERTa}_{\text{LARGE}}$ 87.21 93.26 85.60 89.70 85.15 91.55 69.99 75.05 93.74
87.91 76.46 81.19 50.25 92.03

In order to prevent early saturation of KLUE benchmark, we limit users to
submit their models once per day. We thus present the dev set results to
provide a reference for future work and local tests, in Table 36. Models we
used are same as in test set.


IFT-UAM/CSIC-22-35††joan.quirant _at_ estudiante.uam.es

# Non-integer conformal dimensions for type IIA flux vacua

Joan Quirant Instituto de Física Teórica UAM-CSIC, Cantoblanco, 28049 Madrid,
Spain

###### Abstract

We study the conformal dimensions of the would-be dual operators to the
stabilized moduli for two non-supersymmetric DGKT vacua. For the first of
them, related to the standard SUSY DGKT vacuum by
$G_{4}=-G_{4}^{\text{SUSY}}$, we obtain integer conformal dimensions. For the
second of them, which has a non-zero harmonic component in $G_{2}$, we obtain
both integers and real numbers.

##  I Introduction

Scale separation and moduli stabilisation are two indispensable requirements
in any EFT constructed from string theory that aims to describe our universe.
As of today, some of the most studied models featuring both characteristics
are the so-called DGKT vacua [1, 2], AdS4 flux compactifications of massive
IIA in Calabi-Yau orientifolds.

Under the approach of the swampland program [3] -for reviews see [4, 5, 6,
7]\- these scenarios have received a renewal interest and are now being
scrutinised, specially since it was conjectured in [8] that (SUSY) scale
separated vacua may lie in the swampland -see [9] for previous discussion and
[10] for a refinement of the conjecture111DGKT vacua are compatible with this
refinement and also with the proposal of [11]-. In this regard, its 10d uplift
was addressed in [12, 13, 14, 15, 16] whereas more discussion on scale
separation in IIA can be found in [17, 18, 19, 20, 21, 22, 23, 24]. The study
of the holographic description of these vacua was initiated recently in [25,
26, 22, 27]. Scale separation has a clear interpretation in terms of
parametric gaps in the spectrum of masses of the dual theory, and it may be
clarified if DGKT vacua are consistent by looking at the kind of CFT that they
would correspond to. Surprisingly, it was first noticed in [26] for the
original toroidal example, and then generalized in [27] for any CY
orientifold, that the conformal dimension for the low-lying scalar primaries
in the dual CFT of SUSY DGKT vacua is always an integer, independently of the
details of the compactification. In the same spirit, [22] found that this is
also true for the toroidal non-SUSY DGKT models examined there222But not for
the duals of AdS3 DGKT-type vacua and for the example studied in [17] where
there is no scale separation..

In this letter, we will continue in this direction by examining two of the
non-SUSY branches of general (no specific CY orientifold is assumed) DGKT
vacua derived in [28]. As we will show below, for the non-SUSY branch related
to the SUSY one by changing $G_{4}^{\text{non-SUSY}}=-G_{4}^{\text{SUSY}}$,
the conformal dimension of the scalars dual to the stabilised moduli are
integers as well. Nevertheless, for the so-called branch A2-S1 in that paper,
characterised for having a harmonic component in $G_{2}$ different from zero,
$G_{2}^{\text{harmonic}}\neq 0$, these same conformal dimensions are no longer
integers.

Before presenting the results, it is worth pointing out that non-SUSY AdS
vacua of this kind are conjectured to be unstable [29]. Regarding the status
of the two branches at hand, its perturbative stability was verified in [30,
28]. In addition to this, the non-perturbative stability of the first one
($-G_{4}^{\text{SUSY}}$) was first studied in [30], where decays were found to
be at best marginal, and then in [31], where a D8 domain wall instability was
found if space-time filling D6s are used to cancel the tadpole.

##  II DGKT vacua in a nutshell

Consider massive type IIA string theory compactified on an orientifold of
$\mathds{R}^{1,3}\times\mathcal{M}_{6}$ with $\mathcal{M}_{6}$ a compact
Calabi-Yau threefold. Dimensional reduction leads to a $\mathcal{N}=1$ 4d
supergravity theory whose massless scalar field content is organised as
follows [32]. On the one hand, there are the complexified Kähler moduli,
coming from integrating the Kähler 2-form $J=t^{a}\omega_{a}$ and the
$B=b^{a}\omega_{a}$ field

$\displaystyle T^{a}=b^{a}+it^{a}\,,\quad
a\in\left\\{1,\dots,h^{1,1}_{-}\right\\}\,,$ (1)

where $l_{s}^{-2}\omega_{a}$ are harmonic representatives of
$H_{-}^{2}\left(\mathcal{M}_{6},\left(\mathds{Z}\right)\right)$ and $l_{s}$
the string length. The metric appearing in the kinetic terms of these moduli
is obtained from the Kälher potential

$\displaystyle
K_{K}=-\log\left(\frac{4}{3}{\mathcal{K}}_{abc}t^{a}t^{b}t^{c}\right)=-\log\left(\frac{4}{3}{\mathcal{K}}\right)\,,$
(2)

with
${\mathcal{K}}_{abc}=l_{s}^{-6}\int\omega_{a}\wedge\omega_{b}\wedge\omega_{b}$
and ${\mathcal{K}}=6{\text{Vol}}_{{\mathcal{M}}_{6}}$. On the other hand,
there are the complex structure moduli, coming from the complex 3-form
$\Omega$, the axio-dilaton and the RR 3-form potential $C_{3}$. Introducing
$\Omega_{c}\equiv C_{3}+i{\text{Re}}\left(\mathcal{C}\Omega\right)$ where
$\mathcal{C}=e^{-\phi}e^{\frac{1}{2}\left(K_{cs}-K_{K}\right)}$ is a
compensator, with
$K_{cs}=-\log\left(-il_{s}^{-6}\int\Omega\wedge\bar{\Omega}\right)$ and $\phi$
the 10d dilaton, the complex structure moduli are defined as

$\displaystyle
U^{\mu}=\xi^{\mu}+iu^{\mu}=l_{s}^{-3}\int\Omega_{c}\wedge\beta^{\mu}\,,\quad\mu\in\left\\{0,\dots,h^{2,1}\right\\}\,,$
(3)

where we are taking a symplectic basis $\beta^{\mu}\in
H_{-}^{3}\left({\mathcal{M}}_{6},\mathds{Z}\right)$. Finally, the metric
appearing in the kinetic terms of this sector is constructed from the
following Kähler potential

$\displaystyle
K_{Q}=4\log{\left(\frac{e^{\phi}}{\sqrt{{\text{Vol}}_{{\mathcal{M}}_{6}}}}\right)}\equiv-\log\left(e^{-4D}\right)\,.$
(4)

On top of this background one can add RR and NSNS background fluxes. Following
the conventions of [28] their flux quanta are

$\displaystyle l_{s}\bar{G}_{0}$ $\displaystyle=-m\,,$
$\displaystyle\frac{1}{l_{s}}\int_{\tilde{\pi}^{a}}\bar{G}_{2}$
$\displaystyle=m^{a}\,,$
$\displaystyle\frac{1}{l_{s}^{3}}\int_{\pi_{a}}\bar{G}_{4}$
$\displaystyle=-e_{a}\,,$
$\displaystyle\frac{1}{l_{s}^{5}}\int_{{\mathcal{M}}_{6}}\tilde{G}_{6}$
$\displaystyle=e_{0}\,,$
$\displaystyle\frac{1}{l_{s}^{2}}\int_{B_{\mu}}\bar{H}$
$\displaystyle=h_{\mu}\,,$ (5)

with $\left[\pi_{a}\right]\in
H_{4}^{+}\left({\mathcal{M}}_{6},{\mathds{Z}}\right)$ Poincaré dual to
$\left[l_{s}^{-2}\omega_{a}\right]$ and $\left[\tilde{\pi}^{a}\right]\in
H^{2}_{-}\left({\mathcal{M}}_{6},{\mathds{Z}}\right)$ Poincaré dual to
$\left[l_{s}^{-4}\tilde{\omega}_{a}\right]$, where
$l_{s}^{-6}\int_{X_{6}}\omega^{a}\wedge\tilde{\omega}^{b}=\delta^{b}_{a}$.
$B_{\mu}$ is the 3-cycle de Rham dual to $\beta^{\mu}$. In the 4d action the
presence of fluxes is encoded through the superpotential

$\displaystyle l_{s}W=$ $\displaystyle
e_{0}+e_{a}T^{a}+\frac{1}{2}{\mathcal{K}}_{abc}m^{a}T^{b}T^{c}+\frac{m}{6}{\mathcal{K}}_{abc}T^{a}T^{b}T^{c}$
$\displaystyle+h_{\mu}U^{\mu}\,,$ (6)

which involves both the Kähler and the complex structure moduli. As shown in
[33], the F-term scalar potential generated by this superpotential exhibits a
remarkable factorisation between the saxionic and the axionic components.
Namely, the potential can be written in full generality as

$\displaystyle V=\frac{1}{\kappa_{4}^{2}}\vec{\rho}^{\
t}\,\textbf{Z}\,\vec{\rho}\,,$ (7)

where the vector $\vec{\rho}$ depends only on the flux quanta and the axions
$\\{b,\xi\\}$:

$\displaystyle l_{s}\rho_{0}$ $\displaystyle=$ $\displaystyle
e_{0}+e_{a}b^{a}+\frac{1}{2}{\mathcal{K}}_{abc}m^{a}b^{b}b^{c}+\frac{m}{6}{\mathcal{K}}_{abc}b^{a}b^{b}b^{c}$
$\displaystyle+h_{\mu}\xi^{\mu}\,,$ $\displaystyle l_{s}\rho_{a}$
$\displaystyle=$ $\displaystyle
e_{a}+{\mathcal{K}}_{abc}m^{b}b^{c}+\frac{m}{2}{\mathcal{K}}_{abc}b^{b}b^{c}\,,$
$\displaystyle l_{s}\tilde{\rho}^{a}$ $\displaystyle=$ $\displaystyle
m^{a}+mb^{a}\,,$ $\displaystyle l_{s}\tilde{\rho}$ $\displaystyle=$
$\displaystyle m\,,$ $\displaystyle l_{s}\hat{\rho}_{\mu}$ $\displaystyle=$
$\displaystyle h_{\mu}\,;$ (8)

whereas the matrix Z depends only on the saxions $\\{t,u\\}$:

$\displaystyle\textbf{Z}=e^{K}\left(\begin{matrix}4&&&&\\\ &K^{ab}&&&\\\
&&\frac{4}{9}{\mathcal{K}}^{2}K_{ab}&&\\\
&&&\frac{1}{9}{\mathcal{K}}^{2}&\frac{2}{3}{\mathcal{K}}u^{\mu}\\\
&&&\frac{2}{3}{\mathcal{K}}u^{\mu}&K^{\mu\nu}\\\ \end{matrix}\right)\,,$ (9)

with $K=K_{K}+K_{Q}$,
$K_{ab}=\frac{1}{4}\partial_{t^{a}}\partial_{t^{b}}K_{K}$ and
$K_{\mu\nu}=\frac{1}{4}\partial_{u^{\mu}}\partial_{u^{\nu}}K_{Q}$. This
factorisation is maintained even when D6-brane moduli [34] and
$\alpha^{\prime}$ corrections [35] are included.

Though we will use the $\vec{\rho}$ language in this note, it may be useful to
recall how their components are related with the more familiar gauge invariant
field strengths [33]. Using the democratic formulation and calling
$\textbf{C}\equiv C_{1}+C_{3}+C_{5}+C_{7}+C_{9}$ and $\textbf{G}\equiv
d\textbf{C}-H\wedge\textbf{C}+\bar{\textbf{G}}\wedge e^{B}$, where
$\bar{\textbf{G}}$ is just the sum of the previously introduced flux quanta,
then

$\displaystyle\rho_{0}$ $\displaystyle=\int_{{\mathcal{M}}_{6}}G_{6}\,,$
$\displaystyle\rho_{a}$ $\displaystyle=\int_{\pi_{a}}G_{4}\,,$
$\displaystyle\tilde{\rho}^{a}=$ $\displaystyle\int_{\tilde{\pi}^{a}}G_{2}\,,$
$\displaystyle\tilde{\rho}$ $\displaystyle=G_{0}\,,$
$\displaystyle\hat{\rho}_{\mu}$ $\displaystyle=\int_{B_{\mu}}H_{3}\,.$ (10)

##  III Non-susy vacua and conformal dimensions

Writing the potential as a bilinear expression is a very useful tool in the
search for vacua, as it was exploited in [34, 35] and especially in [28],
where a systematic search of extrema for this setup was performed. Among the
different branches of vacua found in that paper we will focus on two cases:

  1. A)

Branch Non-SUSY${}_{G_{4}}$, characterised by
$\rho_{a}=-\rho_{a}^{\text{SUSY}}$. In this vacuum

$\displaystyle\hat{\rho}_{\mu}$
$\displaystyle=\frac{1}{15}\tilde{\rho}{\mathcal{K}}\partial_{u^{\mu}}K\,,$
$\displaystyle\rho_{a}$
$\displaystyle=-\frac{3}{10}\tilde{\rho}{\mathcal{K}}_{a}\,,$
$\displaystyle\tilde{\rho}^{a}$ $\displaystyle=0\,,$ $\displaystyle\rho_{0}$
$\displaystyle=0\,,$ (11) $\displaystyle V$
$\displaystyle=-\frac{4e^{K}}{75}{\mathcal{K}}^{2}{\tilde{\rho}}^{2}\,;$

  2. B)

Branch Non-SUSY${}_{G_{2}}$, characterised by ${\tilde{\rho}}^{a}\neq 0$. In
this vacuum

$\displaystyle\hat{\rho}_{\mu}$
$\displaystyle=\frac{1}{12}\tilde{\rho}{\mathcal{K}}\partial_{u^{\mu}}K\,,$
$\displaystyle\rho_{a}$
$\displaystyle=-\frac{1}{4}\tilde{\rho}{\mathcal{K}}_{a}\,,$
$\displaystyle\tilde{\rho}^{a}$ $\displaystyle=\pm\frac{1}{2}{\tilde{\rho}}\
t^{a}\,,$ $\displaystyle\rho_{0}$ $\displaystyle=0\,,$ (12) $\displaystyle V$
$\displaystyle=-\frac{e^{K}}{18}{\mathcal{K}}^{2}{\tilde{\rho}}^{2}\,;$

with ${\mathcal{K}}_{a}={\mathcal{K}}_{abc}t^{b}t^{c}$. In addition to this,
the SUSY branch was studied recently in [27], agreeing with what we obtain. We
will not discuss it here.

###  III.1 Branch Non-SUSY${}_{G_{4}}$

All the necessary features of this solution, including the physical mass of
the stabilized moduli, were calculated in [28], namely in appendix B. We can
limit ourselves to compute the conformal dimension
$\Delta\left(\Delta-d\right)=m^{2}R_{AdS}^{2}$ of the correspondent fields in
the would-be CFT3 dual. Regarding the saxions of the compactification, its
conformal dimension would be

$\displaystyle\Delta$ $\displaystyle=10\,,$ $\displaystyle\Delta_{i}$
$\displaystyle=1\text{ or }2\,,$ $\displaystyle\Delta_{a}$
$\displaystyle=6\,,$ (13)

with $i=1,\dots,h^{2,1}$ and $a=1,\dots,h_{-}^{1,1}$ taking into account that
some fields acquire the same mass. Notice that, as expected, these same
results were obtained in [27] for the saxionic spectrum of the supersymmetric
case, since the mass matrix in both cases is block diagonal and shares the
entries of this part. The difference comes from the axions, whose conformal
dimensions in the dual theory would be

$\displaystyle\Delta$ $\displaystyle=1\text{ or }2\,,$
$\displaystyle\Delta_{i}$ $\displaystyle=3\,,$ $\displaystyle\Delta_{a}$
$\displaystyle=8\,,$ (14)

with again $i=1,\dots,h^{2,1}$ and $a=1,\dots,h_{-}^{1,1}$. The $\Delta_{i}=3$
comes from the fact that only a linear combination of axions appear in the
superpotential, so $h^{2,1}$ of them remain massless. These results agree and
generalize the work of [22], who looked at this kind of vacua but only in
toroidal examples.

###  III.2  Branch Non-SUSY${}_{G_{2}}$

Unlike the previous case, the mass acquired by the stabilized moduli was not
explicitly computed in the original reference, so some intermedia steps have
to be done. We relegate the diagonalization of the Hessian and all the details
needed to appendix A. Using again the relation
$\Delta\left(\Delta-d\right)=m^{2}R_{AdS}^{2}$, the conformal dimension of the
dual operators for this branch should be

$\displaystyle\Delta$ $\displaystyle=\frac{1}{2}\left(3+\sqrt{393}\right)\,,$
$\displaystyle\Delta_{a}$
$\displaystyle=\frac{1}{2}\left(3+\sqrt{201}\right)\,,$
$\displaystyle\Delta_{i}$ $\displaystyle=3\,,$ $\displaystyle\Delta$
$\displaystyle=\frac{1}{2}\left(3+\sqrt{33}\right)\,,$
$\displaystyle\Delta_{a}$ $\displaystyle=6\,,$ $\displaystyle\Delta_{i}$
$\displaystyle=3\,,$ (15)

with again $i=1,\dots,h^{2,1}$ and $a=1,\dots,h_{-}^{1,1}$. In this case there
are $2h^{2,1}$ fields with conformal dimension $\Delta=3$, since the saxionic
partners of the usual massless axions do not acquire a mass333They develop a
quartic potential.

##  IV Conclusions

In this work we have computed the conformal dimension of the low-lying
operators of the putative CFT3 dual of two different non-supersymmetric DGKT
vacua.

In the first place, we focused on what we called the Non-SUSY${}_{G_{4}}$
vacuum, which has the property of being related to the SUSY one by
$G_{4}=-G_{4}^{\text{SUSY}}$. We obtained that these dimensions are always
integers -see expressions (13) for the saxionic sector and (14) for the
axionic sector-, and totally independent of the details of the
compactification. With respect to the saxions this is not new, since this part
of the mass matrix is shared with the SUSY branch and the same numbers were
obtained in [27]. For the axions, this result extends the work of [22] -who
only looked at toroidal models- to any CY orientifold.

In the second place, we studied what we named the Non-SUSY${}_{G_{2}}$ vacuum,
called in this way by having $G_{2}^{\text{harmonic}}\neq 0$. Unlike the SUSY
and the Non-SUSY${}_{G_{4}}$ vacua, the conformal dimensions of the operators
of the dual theory would not be only integers, see (III.2). The result is
again quite simple and does not depend on the details of the compactification.
The 10d uplift of this branch has not been studied in detail and it could
happen that problems arise when looked at from the 10d point of view.
Following the analysis of [15], there should not be any obstructions in
constructing first, a smearing uplift, and then, expanding the solution and
localising the source at first order. We have already checked explicitly that
indeed the smearing uplift exists. These and more details are being studied in
an upcoming work [36].

Non-supersymmetric vacua of the kind studied here are conjectured to be
unstable [29]. This could imply that it would not make much sense to study
their CFT duals, since these theories could be sick or ill-defined. Indeed in
[31] we showed that for the Non-SUSY${}_{G_{4}}$ branch there seems to be an
instability if space-time filling D6s are used to cancel the tadpole. Opposite
to this reasoning, studying these non-supersymmetric vacua from their putative
CFT dual could be useful to show that they are unstable, which makes this
analysis interesting per se.

In fact, constructing the would-be CFT duals of scale separated AdS vacua
could be a way of understanding if scenarios of this type are consistent or if
they lie in the swampland. In this note, we have shown that not only integers
appear when we study the conformal dimensions of the low-lying operators of
DGKT vacua. But we still do not understand why they do appear for the SUSY
case or even if the conformal theories we are trying to construct really
exist. We hope that both questions could be answered in the not too distant
future.

Acknowledgements

We would like to thank M. Sasieta for very helpful discussions and
explanations and F. Marchesano for very useful discussions and comments on the
manuscript. This work is partially supported by the Spanish Research Agency
(Agencia Estatal de Investigación) through the Grant IFT Centro de Excelencia
Severo Ochoa No CEX2020-001007-S and PGC2018-095976-B-C21, funded by
MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe. The author
is supported through the FPU grant No. FPU17/04293.

##  Appendix A Mass spectrum for the non-SUSY${}_{G_{2}}$ vacua

The Hessian for this branch was obtained explicitly in appendix B of [28], see
equation (B.6). To compute the physical mass spectrum one has to express the
fields on a canonical basis. As explained there, for this we decompose the
Kähler metrics for the Kähler and complex structure fields as

$\displaystyle K_{ab}$
$\displaystyle=\frac{3}{2\mathcal{K}}\left(\frac{3\mathcal{K}_{a}\mathcal{K}_{b}}{2\mathcal{K}}-\mathcal{K}_{ab}\right)$
$\displaystyle=\frac{3}{4}\frac{{\mathcal{K}}_{a}{\mathcal{K}}_{b}}{{\mathcal{K}}^{2}}+\frac{3}{2{\mathcal{K}}}\left(\frac{{\mathcal{K}}_{a}{\mathcal{K}}_{b}}{{\mathcal{K}}}-{\mathcal{K}}_{ab}\right)=K_{ab}^{\rm
NP}+K_{ab}^{\rm P}\,,$ (16) $\displaystyle K_{\mu\nu}$
$\displaystyle=\frac{1}{16}\frac{{\partial}_{\mu}G{\partial}_{\nu}G}{G^{2}}+\frac{1}{4}\left(\frac{3}{4}\frac{{\partial}_{\mu}G{\partial}_{\nu}G}{G^{2}}-\frac{{\partial}_{\mu}{\partial}_{\nu}G}{G}\right)$
$\displaystyle=K_{\mu\nu}^{\rm NP}+K_{\mu\nu}^{\rm P}\,,$ (17)

with ${\mathcal{K}}_{ab}={\mathcal{K}}_{abc}t^{c}$,
$K=K_{K}+K_{Q}=-\log{\left(\mathcal{G}\right)}$ and
$\partial_{\mu}\equiv\partial_{u^{\mu}}$. Here $K^{\rm P}$ and $K^{\rm NP}$
refers to the primitive and non-primitive parts of the metric, which act on
orthogonal subspaces. One can then express the fields in the canonically
normalised basis

$\displaystyle(\xi^{\mu}\quad b^{a}\quad u^{\mu}\quad
t^{a})\rightarrow(\hat{\xi}\quad\hat{b}\quad\xi^{\hat{\mu}}\quad
b^{\hat{a}}\quad\hat{u}\quad\hat{t}\quad u^{\hat{\mu}}\quad t^{\hat{a}})\,,$
(18)

where $\hat{\xi}\left(\hat{b}\right)$ is the vector along the subspace
corresponding to $K_{\mu\nu}^{\rm NP}\rvert_{\rm vac}\left(K_{ab}^{\rm
NP}\rvert_{\rm vac}\right)$, with unit norm, and
$\xi^{\hat{\mu}}\left(b^{\hat{a}}\right)$ correspond to vectors of unit norm
with respect to $K_{\mu\nu}^{\rm P}\rvert_{\rm vac}\left(K_{ab}^{\rm
P}\rvert_{\rm vac}\right)$. Analogously, $\left\\{\hat{u}\quad\hat{t}\quad
u^{\hat{\mu}}\quad t^{\hat{a}}\right\\}$ are defined in the same way. In this
canonically normalised basis the Hessian reads

$\displaystyle\textbf{H}={\mathcal{K}}^{2}e^{K}{\tilde{\rho}}^{2}\left(\begin{matrix}\frac{8}{9}&\frac{4}{3\sqrt{3}}&0&0&0&0&0&0\\\
\\\ \frac{4}{3\sqrt{3}}&\frac{14}{9}&0&0&0&\pm\frac{8}{9}&0&0\\\ \\\
0&0&0&0&0&0&0&0\\\ \\\ 0&0&0&\frac{14}{9}&0&0&0&\mp\frac{4}{9}\\\ \\\
0&0&0&0&\frac{8}{9}&-\frac{4}{3\sqrt{3}}&0&0\\\ \\\
0&\pm\frac{8}{9}&0&0&-\frac{4}{3\sqrt{3}}&\frac{26}{9}&0&0\\\ \\\
0&0&0&0&0&0&0&0\\\ \\\ 0&0&0&\mp\frac{4}{9}&0&0&0&\frac{8}{9}\\\
\end{matrix}\right)\,,$ (19)

and the physical masses can be obtained straightforwardly by diagonalising
this matrix and dividing by two. Expressed in terms of
$R_{AdS}^{2}=\frac{3}{|\Lambda|}=\frac{54}{e^{K}{\mathcal{K}}^{2}{\tilde{\rho}}^{2}}$
they are

$\displaystyle m^{2}=R_{AdS}^{-2}\left\\{96\,,\quad 6\,,\quad 48\,,\quad
18\,,\quad 0\right\\}\,,$ (20)

with multiplicity $\left\\{1\,,\quad 1\,,\quad h^{1,1}_{-}\,,\quad
h^{1,1}_{-}\,,\quad 2h^{2,1}\right\\}$ respectively.

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