New families of scale separated vacua

Massive type IIA flux compactifications of the form AdS4 × X6, where X6 admits a Calabi-Yau metric and O6-planes wrapping three-cycles, display families of vacua with parametric scale separation between the compactification scale and the AdS4 radius, generated by an overall rescaling of internal four-form fluxes. For toroidal orbifolds one can perform two T-dualities and map this background to an orientifold of massless type IIA compactified on an SU(3)-structure manifold with fluxes. Via a 4d EFT analysis, we generalise this last construction and embed it into new branches of supersymmetric and non-supersymmetric vacua with similar features. We apply our results to propose new infinite families of vacua based on elliptic fibrations with metric fluxes. Parametric scale separation is achieved by an asymmetric flux rescaling which, however, in general is not a simple symmetry of the 4d equations of motion. At this level of approximation the vacua are stable but, unlike in the Calabi-Yau case, they display a non-universal mass spectrum of light fields.


Introduction and summary
The structure of AdS solutions in string theory plays nowadays a central role in the quest to determine which EFTs are compatible with Quantum Gravity [1][2][3][4][5].This is mostly due to two current proposals for Swampland criteria known as the AdS instability [6,7] and AdS distance [8] conjectures, which severely constrain those AdS vacua that can have a holographic CFT dual.
The first one proposes that all non-supersymmetric AdS backgrounds are unstable, while the strong version of the second forbids infinite families of supersymmetric AdS vacua with an increasing separation between the inverse AdS radius and the compactification scale.
A class of AdS 4 constructions that have become under scrutiny partially due to these conjectures is the so-called DGKT-CFI family of vacua [9,10] (see also [11,12]), which are based on massive type IIA string theory compactified on a smooth Calabi-Yau or a toroidal orbifold X 6 , with additional p-form fluxes threading X 6 , and an orientifold projection that introduces O6-planes wrapping its three-cycles.It turns out that the vacua equations obtained in this setting lead to infinite families of supersymmetric and non-supersymmetric solutions, generated by an overall rescaling of the internal four-form flux, and which display increasing separation between the AdS 4 and compactification scales for larger flux quanta and weaker string coupling.
These constructions, which in the toroidal setting can be supplemented by metric fluxes [10][11][12], can be understood as a set of vacua obtained at a 4d EFT level, in the sense that they only solve the 10d equations of motion and Bianchi identities if localised sources like D6branes and O6-planes are smeared at wavelengths below the compactification scale [13].This fact opened the possibility that solving the actual 10d string theory equations would lead to modifications of the 4d EFT, as it happens in some instances for the Kähler potential due to flux backreaction [14], and this would modify the final set of vacua.A closer look into the 10d description of these backgrounds showed that this does not seem to be the case, at least for the Calabi-Yau setting.Indeed, it was found in [15,16] that the smeared-source approximation can be understood as the leading term of a perturbative expansion of the 10d supergravity equations in either the AdS radius or the string coupling.Such 10d equations were solved up to next-to-leading order, displaying source localisation and an AdS radius and compactification scale in agreement with 4d EFT expectations.Moreover, up to this level of approximation a subset of non-supersymmetric vacua was found to be perturbatively [17] and (marginally) nonperturbatively [18][19][20] stable.Extending these results to higher orders in the said perturbative expansion seems a challenging task, since beyond the current level of accuracy one should also take into account α ′ and string loop effects that take us away from the 10d supergravity description.As a result, it seems involved to provide a more accurate microscopic description of these backgrounds, as it is oftentimes the case with massive type IIA compactifications [21].
Since, as of today, the holographic dual of DGKT-CFI vacua has not been found, the debate of the interplay of these constructions with the above Swampland conjectures remains open.One possibility that could make all of them compatible is that an instability is generated at a level of accuracy beyond the one tested so far, which for the supersymmetric branch of vacua would in particular imply that supersymmetry is secretly broken.A more dramatic possibility that would favour the Swampland criteria is that these constructions are inconsistent to begin with, since they mix ingredients like the Romans mass and O6-planes that are separately microscopically well understood, but not combined.To circumvent this second caveat, [22] proposed to analyse these massive IIA constructions from a dual viewpoint, namely after applying two T-dualities on a T 2 factor of a toroidal orbifold, which leads to massless IIA string theory with O6-planes and internal two-form and six-form fluxes, albeit compactified on a twisted torus, non-Calabi-Yau geometry.Subsequently, [23] embedded this T-dual frame in the context of smeared SU(3)structure compactifications, which allows one to connect with the 4d EFT description of these orbifolded twisted torus vacua.More recently, [24] implemented the perturbative techniques of [15,16,25] in this T-dual setup, finding the expected source-localisation as a correction to the initial SU(3)-structure metric.Moreover, they pointed out the existence of infinite families of vacua related by flux rescalings, that also display parametric scale separation for larger flux quanta.Unlike in the massive IIA frame, for some families of vacua rescaling the fluxes also takes us to a strong 10d string coupling regime, suggesting a family of M-theory flux vacua with scale separation which has nevertheless been challenged in [26].
Clearly, testing the properties of AdS string vacua, in particular their consistency, stability and scale separation properties, plays a key role in shaping our vision of the string theory Landscape.However, the general questions under debate will not be answered until we have a global perspective over the whole set of AdS vacua.The aim of this paper is to proceed in the classification of type IIA geometric flux compactifications that lead to AdS 4 vacua, addressed from a 4d EFT viewpoint, as already initiated in [17,27].This time we focus on understanding the supersymmetric and non-supersymmetric branches of vacua that include type IIA on (orientifolds of) twisted tori with RR background fluxes but no Romans mass, as in [22][23][24], and in particular on those constructions that lead to parametric scale separation upon flux rescaling.
Our approach leads us to unveil four new branches of 4d vacua, of which one is supersymmetric and three non-supersymmetric, that can be interpreted as a 10d smeared-source compactification on a SU(3)-structure manifold X 6 with p-form fluxes and non-trivial intrinsic torsion classes, more precisely with the structure of a half-flat manifold.According to the criterion of [23], any of these branches may contain vacua displaying scale separation.However, we focus on those cases that correspond to backgrounds without Romans mass, as in principle these are easier to analyse microscopically, even beyond the 10d supergravity description.This restriction leads us to consider two subbranches of vacua, one supersymmetric and one non-supersymmetric, that contain the nilmanifold-based compactifications considered in [23,24], and as such are good candidates to implement the flux scalings of [24] in a more general setting.Indeed, one of the main results of this work is to propose new families of AdS 4 backgrounds, where scale separation is achieved via a family of vacua generated by flux rescaling, in an analogous manner to the nilmanifold case in [24].A key observation to arrive to this result is the understanding of such a flux rescaling via the original DGKT-CFI setup.Instead of considering an overall rescaling of all four-form flux quanta as in [9,10], one must implement an asymmetric rescaling in which all Kähler moduli grow except for that corresponding to a shrinking T 2 , so that upon two T-dualities on this T 2 one is taken to a large volume compactification.
With this picture in mind one may implement in new settings some of the flux scalings considered in [24], that lead to an asymptotic behaviour for the compactification and AdS scale featuring parametric scale separation.An obvious setup in which to do so are Calabi-Yau threefolds that are smooth elliptic fibrations and realise DGKT-like vacua.Upon a Fourier-Mukai transformation on the fibre, these configurations should be taken to SU(3)-structure manifolds with a similar structure of triple intersection numbers and non-vanishing torsion classes, encoded in a metric flux matrix of rank-one.These topological data are essentially all that one needs to implement our 4d EFT moduli stabilisation techniques, finding similarities but also novelties with respect to the results in [24].More precisely, we find that in this more general setup the scaling of flux quanta is not a simple symmetry of the equations of motion, so in order to find the family of vacua generated by such a rescaling one must resort to solve them via a perturbative expansion or to a numerical analysis.Nevertheless, the feature of parametric scale separation upon rescaling is still present.Similarly, one finds that the massive spectrum for the stabilised moduli is in general not universal, unlike in DGKT vacua [17], and that one only approaches such a spectrum asymptotically.One possible interpretation could be that with these constructions we are testing the dual DGKT-like vacua in small volume regimes where curvature corrections are important, causing such deviations.In any event, since the massive spectrum is not far from the one found in [17] these vacua are perturbatively stable.In addition, at the smeared-source level of approximation at which we are working, these families of vacua are stable against nonperturbative membrane nucleation process, a statement that holds both for the supersymmetric and non-supersymmetric branches that we consider.Some of these results are perhaps not surprising given the T-dual relation of such constructions to DGKT-like vacua with similar properties.It is however important to stress that, in general, T-duality is only expected to hold at the world-sheet perturbative level [28], and that any relevant effect that is found for DGKT-CFI vacua beyond that level should be reconsidered for the present set of vacua.This, together with the possibility of finding new infinite families of scale-separated vacua with no T-dual relation to the ones in the literature, lead us to believe that the present construction merit attention on their own, and may account for an important sector of the Landscape of AdS vacua in string theory.
The rest of the paper is organised as follows.In section 2 we review the main features of DGKT-CFI vacua from the 4d EFT perspective and discuss four-form flux rescalings different from the ones typically considered in the literature.These shrink some two-tori at substringy areas and are thus better described from a T-dual perspective.In section 3 we discuss, from a 4d EFT viewpoint, new branches of type IIA AdS 4 vacua that were missed by the analysis of [27], and which include the T-duals of DGKT-CFI vacua mentioned above, as follows from their smeared-source 10d description.In section 4 we apply the techniques of [27] to these new branches, and describe the vevs of their stabilised moduli in terms of flux quanta, which for factorised geometries translate into scaling symmetries generating infinite families of vacua with parametric scale separation, reproducing the results of [24].Section 5 applies the same approach to more involved geometries, namely elliptic fibrations with rank-one metric fluxes.In this case the flux scaling symmetry is only approximate, which prompts a perturbative expansion in order to solve the vacua equations.Section 6 studies the perturbative and non-perturbative stability for the set of vacua considered in the previous two sections, finding stability up to our level of approximation.Finally, Appendix A verifies that the leading-order results on moduli stabilisation and perturbative stability for vacua based on elliptical fibrations are a good approximation to the actual solution, by focusing on two specific models and solving them numerically.

DGKT-CFI vacua and T-duality
Let us consider type IIA string theory compactified on an orientifold of X 4 × X 6 with X 6 a compact six-manifold that admits an SU(3)-structure metric defined by the invariant two-and three-form J and Ω, and with volume form − 1 6 J 3 = 1 4 Im Ω ∧ Re Ω.We take the standard orientifold quotient by Ω p (−1) F L R [29][30][31][32][33], where Ω p is the worldsheet parity-reversal operator, F L is the spacetime fermion number for left movers, and R a is an involution of the metric acting as R(J, Ω) = −(J, Ω).This quotient induces a set of O6-planes wrapping three-cycles of X 6 which, when backreacted, are presumed to deform the SU(3) structure to an SU(3)× SU (3) structure.One however expects that in the limit of smeared sources the initial SU(3)-structure metric is recovered, as it happens in [15,16,24].Therefore, in the following we will take such an SU(3)-structure metric as a proxy for the orientifold compactification, following the standard practice in the type IIA moduli stabilisation literature.
In this framework, one can expand the SU(3)-structure invariant forms (J, Ω) to define the metric deformations of X 6 .Following [34][35][36][37][38] one performs the expansion to define the complexified Kähler fields T a ≡ b a + it a .For simplicity we will assume that all elements of the two-form expansion basis {ω a } are odd under the action of R, see e.g.[27] for the more general case.Similarly, the complex structure deformations are defined via the expansion where (α κ , β λ ) is a symplectic basis of three-forms. 1 The orientifold projection decomposes this basis into R-even (α K , β Λ ) and R-odd (β K , α Λ ) 3-forms, and eliminates half of the degrees of freedom of the original complex periods of Ω.Then one defines the complexified 3-form Ω c as where C ≡ e −ϕ e 1 2 (Kcs−K K ) [40].Here ϕ is the 10d dilaton, K cs ≡ − log iℓ −6 s X 6 Ω ∧ Ω and K K is defined as in (2.5).Finally, the moduli including the complex structure are defined as: For simplicity in the following we will collect both sets of moduli {N K , U Λ } into U µ , introducing a unified basis {α µ } = {α K , −β Λ } and {β µ } = {β K , α Λ } and modifying the fluxes accordingly.
In the regime of diluted fluxes, the kinetic terms of all these bulk fields are encoded in a Kähler potential of the form K ≡ K K + K Q , where and (2.6) 1 For Calabi-Yau metrics the set of p-forms {ℓ −2 s ωa, ℓ −3 s ακ, ℓ −3 s β λ , ℓ −4 s ωb }, with ℓs = 2π √ α ′ , corresponds to a basis of harmonic forms that are integrally quantised, and chosen such that ωa ∧ ωb = ℓ 6 s δ b a and ακ ∧ β λ = ℓ 6 s δ λ κ .For the more general SU(3)-structure case this set may include non-harmonic forms as well, as in the general framework for dimensional reduction on SU(3)-structure manifold developed in [37,38].There the definition of quantised p-form becomes more subtle, but it can be made precise in terms of smeared delta forms [39].
Here K abc ≡ −ℓ −6 s X 6 ω a ∧ ω b ∧ ω c are triple intersection numbers, K ≡ K abc t a t b t c = 6Vol X 6 and ϕ 4 = ϕ − 1 2 log Vol X 6 is the 4d dilaton.An interesting setup is the case where the smeared metric for X 6 is Calabi-Yau as then, by including RR and NSNS three-form fluxes, one can apply the analysis of [9,17] to generate infinite families of AdS 4 vacua.We work with the democratic formulation of type IIA supergravity [41], in which all RR potentials are grouped in a polyform C = C 1 + C 3 + C 5 + C 7 + C 9 , and so are their gauge invariant field strengths while W NS contains the complex structure fields and H-flux quanta Here e 0 , e a , m a , m, h µ are all integers, defined following the same conventions as in [20] (2.10) These coefficients of the superpotential are redefined when taking into account curvature corrections, see e.g.[20].
Let us now focus on the case where X 6 = (T 2 ) 3 /Γ with Γ ⊂ SU (3) a discrete orbifold group, where one can apply the results of [9,10] (see also [11,12]) to provide explicit expressions for the untwisted moduli vevs in terms the flux quanta.For instance, in the case where Γ = Z 2 × Z 2 the Kähler potentials for the untwisted sector take the simplified form while the corresponding superpotentials take the form (2.8) and (2.9) with K 123 = 2 and K iij = 0, ∀i, j.In this case, the vacua analysis of the 4d F-term scalar potential yields the following relation between Kähler moduli and fluxes: , where êa ≡ e a − 1 2 for n ∈ N, which corresponds to an overall rescaling of the volume of X 6 .This is the scaling used in [9,10] to construct an infinite family of scale-separated vacua, which can be implemented in a general Calabi-Yau.In the present setup one can however consider richer scalings, that depend differently on each flux quanta.For instance: For r = s = 1, one recovers (2.12), but other values for r, s lead to other branches of vacua.One may then generalise the analysis of [9,10,17] to arrive at the following scaling dependence for the remaining moduli To address the separation of scale, we define the AdS radius R AdS as where Λ the 4d vacuum energy in 4d Planck mass units M P .In the current setup we have [17] |Λ| ∼ (t 1 t 2 t 3 ) This last scaling is to be compared with that of the KK or winding mode scale M KK and M w (2.17) We thus find obtaining that if r > 0 and r + s > 0 there is an asymptotic separation of scales.This is true even when s < 0, which is a possibility as long as the flux m 1 vanishes.Choosing s < 0 takes us to a small radius limit in the torus factor (T 2 ) 1 associated with t 1 .In that case, it is more sensible to describe the compactification in a T-dual frame, for instance after two T-dualities on (T 2 ) 1 , which flips the sign of s in (2.13).This double T-duality is precisely the frame considered in [22][23][24] to describe a family of vacua in a massless type IIA compactification, which explains the similarity of the scalings above with those obtained in [24]. 2 The following sections aim to generalise such scalings for non-toroidal setups.
A simple approach for such a generalisation is to describe the two T-dualities at the level of the 4d effective theory.This can be implemented by performing the following change of variables in terms of which the Kähler potential transforms as where we have used that the 4d dilaton is invariant under T-duality.The shift in the Kähler potential can be absorbed in a Kähler transformation with F ≡ log T 1 .After performing it, we recover the previous Kähler potential and a superpotential of the form One can interpret the new superpotential in terms of the set of fluxes that one obtains after two T-dualities.Notice that the would-be cubic term m 1 T 1 T 2 T 3 is missing due to consistency with an exact scaling of the form (2.13) for s < 0. The lack of cubic term in (2.22a) signals the absence of Romans mass G 0 in the new T-dual frame, while the bilinear structure of (2.22b) indicates that the former H-flux has been traded for a metric flux [10][11][12].
At the perturbative level, the F-term potential derived from (2.22) is equivalent to the previous one with superpotential (2.8) + (2.9).Therefore the 4d vacua analysis leading to the scalings (2.18) works mutatis mutandis as before.However, if in the current frame we want to stay in the regime of large compactification volumes where the 10d supergravity approximation can be trusted, one is prompted to take s < 0 in the original setting (2.13), which means that the standard DGKT-CFI scaling (2.12) cannot be implemented.Still, a negative s allows for parametric scale separation in (2.18), which translates into an infinite family of scale separated vacua at large volume in the T-dual setup.The existence of such a family was pointed out in [24], precisely in this toroidal setup.
Achieving scale separation with the superpotential (2.22) brings us to the question of how this family of vacua compares to those obtained in [27], where a general analysis of type IIA AdS 4 vacua with p-form and metric fluxes was performed, assuming a specific F-term pattern and obtaining no parametric scale separation.As we will see in the next section, the AdS 4 vacua analysed in [23,24] belong to a branch of vacua which was missed in the classification performed in [27], but that one can characterise using the same techniques.This characterisation will allow us to generalise the construction in [22][23][24] to more involved settings, unveiling new branches of AdS 4 orientifold vacua that in principle can realise parametric scale separation.Since these new constructions necessarily feature metric fluxes, it is not so obvious how to identify the set of compact manifolds X 6 that they correspond to.However, the above discussion gives us a hint on the simplest geometries where similar scalings to the above should be realised: Calabi-Yaus with an SL(2, Z) perturbative duality on a Kähler modulus.These geometries include toroidal orbifolds and smooth elliptic fibrations.In section 5 we will see how to use the latter to propose new families of AdS 4 vacua with parametric scale separation.

General class of 4d vacua
In this section, we review the 4d approach taken in [27] to classify type IIA AdS 4 orientifold vacua, and point out branches of solutions that were missed in the original classification.As we will show, the supersymmetric family of AdS 4 vacua with scale separation discussed in [24] is part of these new branches, as well as its non-supersymmetric counterpart.The framework of [27] allows us to describe the new branches in a general manner, without specifying the underlying SU(3)-structure manifold, but providing certain constraints for its torsion classes which we discuss in the (smeared) 10d interpretation of these vacua.In subsequent sections we apply this general description to propose new families of scale-separated vacua.

4d analysis
Let us briefly review the results of [27], to see how new branches of vacua arise from the 4d analysis developed therein.The working assumption of [27, section 4] is a 4d Kähler potential where f aµ ∈ Z represent metric fluxes.In practice, they are defined in terms of the action of the twisted external derivative over the basis introduced in the previous section in agreement with the general scheme for dimensional reduction in SU(3)-structure manifolds [37,38].We refer to [27, Appendix A] for more details regarding the action of the metric fluxes.
When working with the smeared approximation in this setup, the Bianchi identity for the RR two-form flux threading X 6 reads where δ O6/D6 are smeared delta forms of the three-cycles wrapped by D6-branes and O6-planes, see [39] for a precise definition.Therefore, the tadpole cancellation condition puts constraints on the available values of the flux quanta m, m a , h µ and f aµ , but not on the rest.
From the above superpotential and Kähler potential, it follows that the F-terms read where we have made use of the shift symmetries of the Kähler potential to define . From here one obtains the F-term potential where the relevant field space metrics are Upper indices denote their inverses and cµν ≡ c µν − 4u µ u ν .Finally, we have defined the flux-axion polynomials ) ) where ξ µ ≡ Re U µ .Notice that, up to now, we have not made any assumption on the background fluxes, except that they correspond to a geometric compactification.As before, we have assumed that there are no two-forms ϖ α that are even under the orientifold action, which in particular prevents a D-term contribution V D to the potential.This assumption will not change our results, since as shown in [27] V D vanishes at the class of vacua that we will consider.
The flux potential (3.5) has the bilinear structure [44][45][46], where the matrix entries Z AB only depend on the saxions {t a , n µ }, while the ρ A displayed in (3.7) are gauge-invariant combinations of flux quanta and the axions {b a , ξ µ }.This allows to implement the techniques used in [17,[47][48][49] to perform a systematic search for vacua.This approach was applied in [27], together with the assumption of on-shell F-terms of the form with λ K ℓ s , λ Q ℓ s ∈ C, which includes all supersymmetric vacua.Translated to the language of flux-axion polynomials, this F-term constraint results in the following on-shell relations ) where P, Q, M, N are real functions of the 4d light fields.The resulting F-terms are given by The functions P, Q, M, N are constrained by the extrema conditions resulting from the flux potential (3.5), with which they must be compatible.In particular, plugging (3.9) into the equations of motion for the axions {ξ µ , b a } one obtains which must be satisfied on-shell.Similarly, the equations of motion for the saxions {n µ , t a } are As it was argued in [27], generically (3.11b) and (3.12b) imply the constraint which was the additional ingredient in the Ansatz used in [27] for a systematic search of vacua.
It turns out that this condition leads to a particular set of SU(3)-structure manifolds, nearly-Kähler manifolds, which is a restricted set of solutions with respect to the class of supersymmetric SU(3)-structure backgrounds obtained in [50].In particular, it does not contain the AdS 4 vacua constructed in [23,24], which feature half-flat manifolds.
This suggests that there must be branches of vacua compatible with (3.8) for which (3.13) is not satisfied, as it is indeed the case.In order to move away from this proportionality relation, one needs to demand that the two pairs of brackets in (3.11b) and (3.12b) vanish simultaneously, which severely constrains the parameters of the Ansatz (3.9).The two possible options are summarised in the two rows of table 1, where we have defined for clarity the new quantity Table 1: Parameter configurations of (3.9) for which the brackets in (3.11b) and (3.12b) vanish independently.
At this point there are two branches: one supersymmetric (SUSY) and one non-supersymmetric (non-SUSY).

Branch
Parameters find three new non-SUSY families of solutions.From the 10d perspective, addressed in section 3.2, these four branches describe half-flat manifolds, which contrasts with the nearly-Kähler geometry arising when (3.13) is satisfied.
Motivated by the T-duality considerations of section 2, from now on we focus on the solutions with vanishing Romans mass, which leads us take P = M = 0 for the SUSY branch.We also focus on the non-SUSY branch that features P = M = 0 since it shares a lot of similarities with its SUSY partner.From table 2 we thus read the condition and that the SUSY sub-branch and its non-SUSY counterpart are described by: SUSY: Both branches display a negative vacuum energy given by3 which is independent of the choice of sign in (3.16).This sign choice is however physically relevant, since it distinguishes between supersymmetric and non-supersymmetric vacua, as we argued before.In the non-supersymmetric case, the F-term structure corresponds to (3.8) with 4 which indeed reproduces (3.17).In other words, we find two branches of vacua related by a sign flip, which preserves the vacuum energy but breaks supersymmetry.This is reminiscent of the structure found in DGKT-CFI vacua (see also [17]) where the sign flip is implemented in the (smeared) RR four-form flux background G 4 .This is a first hint that these new branches may contain the T-duals of DGKT-CFI vacua discussed in the section 2. Further evidence can be gathered by providing a 10d interpretation of these 4d solutions, as we now proceed to do.

10d interpretation
As emphasised in [44][45][46], the flux-axion polynomials {ρ 0 , ρ a , ρa , ρ} represent the type IIA gauge invariant RR fluxes G 2p threading the compactification manifold X 6 with p = 3, 2, 1, 0 respectively, as opposed to the flux quanta {e 0 , e a , m a , m}.Additionally, one may connect with the results of [23,24,50] by interpreting the metric fluxes ρ aµ in terms of the intrinsic torsion classes of the SU(3)-structure manifold X 6 .In order to proceed, we make the assumption that the presence of fluxes causes a small deformation of the internal geometry such that the underlying Calabi-Yau structure can still be used as a reference, as in [34][35][36][37][38].In particular, we have a well-defined Kähler potential that satisfies the standard relations Using them, the definition (3.2) and the Ansatz (3.9) and (3.16), we have that Then, in order to derive the torsion classes we only need to decompose the flux G 2 in its primitive G P 2 and non-primitive 4 Again here we abuse notations and W is now in Einstein frame and expressed in four-dimensional Planck units.
with G P 2 ∧ J 2 = 0.The Ansatz (3.9) tells us that ℓ s K a ρa = Q, which implies From the above discussion we deduce that when the localised sources are smeared (so that the Bianchi identities amount to the tadpole condition (3.3), already taken into account by our analysis), the branches in table 2 correspond to the following 10-dimensional flux configurations and SU(3)-structure relations where Φ 6 is the normalised volume 6-form and ρ 0 takes the value − 3 2 Q in the SUSY branch and − 1 2 Q(1 − 12/S) in the non-SUSY branches.The two relations in (3.25) translate into the following SU (3) torsion classes Therefore, in terms of an internal SU(3)-structure manifold, our vacua correspond to a subfamily of half-flat compactifications with W 3 = 0, like the ones considered in [23,24,50,51].This complements the picture found in [27], where it was observed that flux configurations satisfying while G 2 and G 6 are present in the compactification, together with metric fluxes.This is indeed what we expect to find after applying two T-dualities to a DGKT-CFI vacuum, at least in the approximation of smeared localised sources.The absence of Romans mass and H-flux was already pointed out in section 2, while the absence of G 4 in the smearing approximation and the presence of G 2 and G 6 can be motivated by looking at the solutions found in [23,24].
We close this section by recalling that the current discussion does not mean that the internal metric of X 6 corresponds to an SU(3)-structure.As in the 10d uplift of the 4d supersymmetric Calabi-Yau vacua [9], recently analysed in [15,16], one expects that upon localisation of sources, the 10d background displays an SU(3)×SU(3)-structure that is simplified to an SU(3)-structure in the smearing approximation.An example that motivates this picture in the context of half-flat manifolds are the twisted-tori solutions with localised sources obtained in [24].

Moduli stabilisation and scaling symmetries
In this section we analyse the vacua equations for the two branches obtained in (3.16), in order to work out explicit expressions for the fixed moduli vevs in terms of flux quanta.We show how, in simple cases, one recovers scaling symmetries of the form (2.13), that lead to an infinite family of vacua with the scale-separation relations (2.18).Two key assumptions to find such scaling symmetries are a metric-flux matrix of rank one and a compactification manifold of the form , where X 4 = T 4 or K3, Γ is an orbifold group and × indicates that metric fluxes fibre T 2 over X 4 .These examples are directly related to the ones considered in [23,24] for and in [50] for X 4 = K3.In section 5 we extend our 4d analysis to manifolds that are more general elliptic fibrations, and show that for them the scaling symmetry is no longer exact.
Nonetheless, we will argue that such geometries can also lead to an infinite family of vacua with a parametric scale separation similar to the cases discussed in here.

A rank-one metric flux Ansatz
Let us start by rewriting the on-shell relations (3.15) and (3.16) as: ) ) ) where a priori Q can be function of the moduli.The negative sign corresponds to the supersymmetric branch of vacua (3.16), and the positive to the non-supersymmetric one.If these vacua are obtained by applying two T-dualities to a DGKT-CFI vacuum, one expects a metric flux matrix f aµ of rank one, as observed in (2.22b).Let us then take this as a working assumption, 5and write with σ a , σ µ ∈ Z.Then, in order to satisfy the vacuum condition ρ µ = 0 from (3.7e), we necessarily need to impose the following condition on the fluxes with σ ∈ Z, which partially fixes some axion directions.To proceed we consider the matrix J ab ≡ K abc m c and impose a second working assumption: Namely, that the vector σ a is in the image of the symmetric matrix J.This assumption is automatically satisfied whenever J is invertible, which will be the case in all of our examples.
Under the assumption (4.4) and the condition ρ = 0, the vacuum equation ρ a = 0 in (3.7b) can only be satisfied if also e a = J ab ζ b for some which implies that those Kähler axions in the kernel of J are left unfixed.Note also that only the complex structure axions that enter the linear combination σ µ ξ µ are fixed by the vacuum equations, although these axions are rendered massive via Stückelberg couplings involving space-time filling D6-branes [10].Using both (4.3) and (4.5), we find that the vev for such a combination is where in the second equation we have assumed that J is invertible, with inverse J ab .Plugging this expression into the definition of ρ 0 in (3.7a) and using (4.1a) we finally find J ab e a e b + 1 2 and the corresponding expression in terms of (χ a , ζ b ) when J is not invertible.That is, we have found explicit formulas for all the stabilised axion vevs and for Q in terms of fluxes quanta.
Recall that this last value essentially fixes the vacuum energy via (3.17).In simple examples, like for instance the backgrounds considered in [24], we have that e a = h µ = 0, and then Q 2 = 4 9 e 2 0 .Let us now consider the vacuum equations involving the saxions, namely (4.1b) and (4.1c).
We first have that (4.1c) implies the on-shell relations which when plugged back into (4.1b)leads to The first of these last relations represents a non-linear equation on the Kähler saxions which, when solved, can be plugged back into (4.8) to obtain the vev of the complex structure saxions.

Scaling symmetries in twisted factorised geometries
Solving the saxion vacuum equations (4.8) and (4.9) explicitly is in general non-trivial, and it is also not clear how the dependence of the saxion vevs on the fluxes can lead to a scaling symmetry of the form (2.13) that generates an infinite family of solutions.At this stage the only general information that we have is that the fluxes with a non-trivial contribution to the Bianchi identity of G 2 cannot scale, which in this case corresponds to the product m a f aµ = m a σ a σ µ .
This means that one can scale arbitrarily Q and the flux quanta m a that do not enter in m a σ a .
The simplest way to generate an infinite family of vacua from such a flux scaling is whenever there is a scaling symmetry in the vacuum equations.Looking at (4.9) this looks difficult to implement unless the triple intersection numbers of X 6 factorise as with η ab = η ba and a σ a η ab = 0. Note that we do not include any normalisation factor in our definition of the symmetrisation of the indices.This structure simplifies (4.9) to where we have dubbed t L ≡ t a σ a .We can then implement the scalings and which reproduce (2.13) and (2.14) except for a sign flip on s expected from T-duality.With these scalings one can check that and so scale separation is obtained whenever r > s ≥ 0.
To illustrate this picture it is useful to consider a particular geometry for X 6 .Clearly, the structure (4.10) for the triple intersection numbers occurs for factorised geometries like T 2 × X 4 .
However, these may not be acceptable geometries to solve the Bianchi identities.Indeed, recall that in the smeared approximation the exterior derivative for the RR two-form flux threading X 6 is given by (3.3) with ρ µ = 0. Due to (4.8) we have that m a f aµ β µ ∝ Re Ω, and so one needs an O6-plane content such that δ O6 ∝ Re Ω.This cannot be achieved in factorised geometries, unless the compactification is modded out by an appropriate orbifold group Γ, as done in [23] for the case of nilmanfolds.Therefore in the following we will focus on geometries of the form X 6 = (T 2 ×X 4 )/Γ, where X 4 = T 2 × T 2 or K3, the symbol × means that we are twisting the product by introducing a rank-one metric flux f aµ that fibres the T 2 over X 4 , and Γ is a orbifold quotient such that δ O6 ∝ Re Ω.Finally, in order to recover a structure of the form (4.10), we will focus our attention on the orbifold untwisted sector of the theory.Notice in this setting there is no general scheme to stabilise the orbifold twisted sector away from the orbifold point, and as a result the final compactification will typically display orbifold singularities.
For both cases of X 4 we can split the Kähler index as a = {L, A}, where L refers to T 2 and A runs over the Kähler moduli of X 4 .We have that where κ ∈ N and η AB is the intersection matrix of X 4 .This implies where we have defined η ≡ η AB t A t B .The structure (4.10) is recovered if we take σ a = σ L δ aL with σ L = κ.Then, eq.(4.8) amounts to In addition we have that where we have defined M ≡ η AB m A m B .Since J ab is invertible we can rewrite (4.9) as the more explicit expressions ) where η A ≡ η AB t B .This system is solved by which is in agreement with (4.11) and realises the expected scalings (4.13) via the following rescaling of flux quanta This also reproduces the results presented in [24], for the choice of scaling parameters a = 2r and b = c = r − s defined therein.In order to maintain the consistency of the scaling symmetry across the remaining equations of motion (4.5), (4.6) and (4.7) we also take Given that e −K Q is a homogeneous function of degree four and using u µ ∼ n 2r−s we deduce which in turn leads to the following scaling for the 10d string coupling Consequently in order to stay in the weak coupling limit one further needs to require r > 3 2 s, in agreement with the results of [24].
The Kaluza-Klein scale is approximated by the largest compact direction, which is contained in X 4 .Therefore the corresponding scaling reads where in the last step we have expressed the result in units of the Planck mass M P .The AdS scale can be obtained from the scalar potential (3.17) Therefore we conclude which means that this new branch of solutions provides scale separated vacua whenever r > s ≥ 0 in the choice of flux rescaling (4.22), as predicted in (4.14).Finally, let us verify that higher order flux corrections are under control in the regime in which scale separation arises.
Following [23], we compute the norm |g s G p | 2 of the flux density of the fields that are present in our setup, i.e.G 2 and G 6 .Applying the scalings described above we have and therefore we conclude that this corrections are suppressed in the regime of large n whenever 2r > s ≥ 0, which is compatible the scenario required to achieve scale separation.

Elliptic fibrations
We now generalise the previous results to the more involved case of smooth elliptically fibered geometries X 6 .The possibility to extend the results of [24] and obtain similar scalings in this context has already been anticipated at the end of section 2, and the general idea goes as follows.
Let us consider the DGKT-like set of vacua based on a smooth elliptically fibered Calabi-Yau.
One can then single out a four-form flux quantum dual to the class of the elliptic fibre and scale it like ê1 in (2.13), while all the remaining four-form fluxes scale like êi .As in the toroidal case, for negative values of s one expects to go to a small fibre area limit.Therefore, by performing a double T-duality along the fibre, or in other words a Fourier-Mukai transformation, one should be taken to a large-volume type IIA compactification on a different elliptic fibration X 6 .From the Fourier-Mukai action on D-brane charges (see e.g.[52]) one deduces that at the 4d EFT level this duality must be implemented by transformations of the form (2. 19) and (2.20), and that most of the features of the toroidal setting should also apply to this case.In particular, the Nevertheless, in this setup the analogy with toroidal compactification is not exact, due to the more involved structure of the triple intersection numbers, that no longer factorises.It is therefore a natural question whether or not one may achieve an infinite family of vacua with parametric scale separation via similar flux scalings.To address this question we will first describe an exact scaling symmetry of the vacuum equations to see how we one can reproduce the scalings (2.13) and (2.14) in this more involved setting.It turns out that such a scaling symmetry is however only formal, with little hope to be implemented in practice, since one would need to scale topological data of X 6 and as a result an infinite family of vacua cannot be generated.In contrast, one can generate such an infinite family by implementing an approximate scaling symmetry that reduces to (2.13) and (2.14) at leading order in a perturbative expansion.
This expansion leads to a perturbative approach to solve the saxionic vacuum equations around the saxionic vevs of the trivially fibered case analysed in the previous section, which is a more and more accurate approximation as we move along the infinite family of vacua.We will eventually refer to numerical results obtained in specific examples and displayed in Appendix A, to confirm the robustness of the perturbative analysis and the existence of this new infinite families of scaleseparated vacua on smooth elliptically fibrations.

An exact scaling symmetry
Let us assume that one can build a DGKT-like family of vacua on a smooth elliptically fibered Calabi-Yau W 6 with base X 4 .As in section 4.2, we split the Kähler index as a = {L, A}, where L refers to the elliptic fibre and A runs over the Kähler moduli of the base X 4 .Expanding the first Chern class of the base like c 1 (X 4 ) ≡ c A ω A , the triple intersection numbers read which implies (5.2) We now perform two T-dualities along the fibre or, equivalently, a Fourier-Mukai transform.
Following the reasoning of section 3 and the results of [36,53], the former H-flux H ∝ Re Ω should be translated into a more involved elliptic fibration that leads to an SU(3)-structure manifold X 6 , such that it has same structure of triple intersection numbers as W 6 , but now with non-trivial intrinsic torsion classes.This last feature should be captured by a rank-one metric flux matrix of the form (4.2), and more precisely such that σ a = σ L δ aL .It is in this manifold X 6 where we are going to look for solutions to the 4d vacua equations (3.15) and (3.16).
The discussion parallels the one in section 4.2, except that now we have the more involved triple intersection numbers (5.1).This implies that the matrix J ab and its inverse read with N ≡ M + η AB c A m B m L .Notice that we recover (4.18) and (4.19) with κ = 1 when the Chern class of the base vanishes.Moreover, with this choice of metric flux eq.(4.8) becomes while eq.(4.9) multiplied by J −1 yields Splitting this relation for the index L and indices A we get ) where x t ηy ≡ η AB x A y B .In particular, M = m t ηm and η = t t ηt.
From the system of equations above, we can derive an exact scaling symmetry of the form which gives exactly the desired scalings (2.13) and (2.14) for the saxions to obtain scale separation.This symmetry is however only formal since it involves scaling the first Chern class c A of the base of the elliptic fibration, which cannot be done in general.Nevertheless, the observation that the Chern class needs to scale with a positive power of n (recall that we take r > s) in order to have an exact symmetry hints at the fact that the terms breaking the symmetry when we keep X 4 fixed are subleading in the limit of large n.This is supported by the fact that each Chern class term in (5.7) is multiplied by additional factors {m L , t L } in detriment of {m A , t A }.
Therefore, if the scalings of section 4.2 were to be taken, the terms associated with the first Chern class of the base would grow slower than the ones associated to a factorised geometry with a damping power n s−r .In the following, we will consider this scenario in more detail and perform a perturbative expansion around the factorised limit.

An approximate scaling symmetry through a perturbative expansion
Let us now implement the scaling of the flux quanta (that are not involved in the tadpole) while keeping the fibration base X 4 fixed.In this case the scaling symmetry (5.8) is broken, and so it is not clear that one can generate an infinite family of scale-separated vacua.To argue that this is the case, we resort to a perturbative expansion of the 4d equations of motion, related to an approximate scaling symmetry.Indeed, recall that the scaling m A ∼ n r−s leads to an exact symmetry either when c A vanishes or it is rescaled as in (5.8).When it is non-vanishing and fixed, one can define the expansion parameters ϵ A to be which are all equally suppressed for large values of n and r > s ≥ 0.
One can now solve the 4d vacua equations order by order in ϵ.In particular, inserting the vev solutions (4.20a) and (4.20b) of the factorised case into the relations (5.7) turns out to cancel the leading order terms.We therefore write the perturbative expansion around this solution as where we have defined t L (0) to be the vev of t L in the factorised case.For future reference, we also define the leading-order vev of t A (0) as well as the next-to-leading-order corrected vevs t L and t A (1) .We thus have where Q scales like in the factorised case: Q ∼ n 2r .For the expansion (5.10) to make sense, we need the corrections to behave like ∆ L = O(ϵ) ∼ n s−r and ∆ A = O(1) ∼ const.Inserting the expansion inside the equations (5.7) we find which indeed scale consistently.
This perturbative analysis thus suggests a mechanism to generate an infinite family of scaleseparated vacua with scalings (2.13) and (2.14) asymptotically at large n, thanks to the choice which this time only involves flux quanta allowed to scale, exactly as in the factorised case.
Appendix A performs a numerical analysis of the 4d vacua equations for two specific elliptically fibered models, and confirms the existence of the infinite family of vacua as well as the robustness of the analytical results derived in this section.

Stability
To study the perturbative stability of the vacua described in this paper, we follow the same approach as in [17,27].That is, we compute the Hessian matrix from the scalar F-term potential (3.5) in our branches of solutions (3.16) under consideration.From there, to obtain the canonically normalised masses, we make use of the decomposition of the Kähler metric into a primitive and non-primitive part [17,27].
We will first display the generic expression of the Hessian, valid for any SU(3)-structure compactification within the branches of vacua (3.15) and (3.16).With this general expression it is not possible to extract the eigenvalues in full generality.This task is however doable for the twisted factorised geometries In both cases the masses are the same and do not induce any perturbative instability.For the case of more general elliptic fibrations, we expect to recover a similar mass spectrum at large scaling parameter n where the perturbative expansion described in sect.5.2 converges towards the factorised setup.We will refer to numerical results given in Appendix A to confirm this expectation and conclude that our new infinite family of scale-separated vacua is perturbatively stable.We will eventually discuss aspects of stability at the non-perturbative level in sect.6.4.

Generic Hessian matrix
The strategy to express the Hessian mirrors the one implemented in [27, Appendix C].We thus proceed with the following steps: • From the generic scalar potential (3.5) translated to Einstein frame and expressed in fourdimensional Planck units, one can compute double derivatives with respect to all the fields and this defines the Hessian matrix.
• One then particularises these expressions to the branches of solutions under consideration by implementing the Ansatz (3.9) and the definition of the branches (3.16).It is already useful at this point to decompose the Kähler metric into its primitive and non-primitive parts, both for the Kähler and complex-structure sectors.We have where the subscripts and superscripts P and NP stand for the primitive and non-primitive parts of the metrics and their inverses, At this point, one can realise that the matrix is block diagonal with the saxions decoupled from the axions.
• For the eigenvalues of the matrix to be mapped to the canonically normalised masses, the Hessian should be expressed in an orthonormal basis which splits into vectors along the (one dimensional) non-primitive subspaces of the Kähler metrics and the (h 1,1 − -and h 2,1dimensional) primitive subspaces [17,27].The unit-norm vectors along the non-primitive spaces are denoted ξ, b, û and t for the axions and saxions respectively.On the other hand, the unit-norm vectors spanning the primitive spaces are written ξμ , bα , ûμ and tα with μ ∈ {1, . . ., h 2,1 − 1} and α ∈ {1, . . ., h The implementation of all these steps is a rather involved process and we will simply display the result.We denote A the axionic submatrix and S the saxionic one.We find where Σ ≡ e −K K σ a K ab σ b and similarly J ab ≡ e −K K J ac K cd J db .The quantities with hatted indices are obtained by contractions with ta α.For example, J αb ≡ J ab ta α and the single-hattedindex matrices J α and J α are obtained by further contraction with t b .The quantity δ is 0 or 1 respectively at vacua that are supersymmetric or not.Notice that the saxionic mass matrix does not depend on δ and it is thus identical in the SUSY and non-SUSY branches.The third row/column associated to ξμ and ûα is understood to be h 2,1 -dimensional and diagonal.
Note that these expressions for the Hessian are only implicit in the sense that it is not possible to have knowledge about the canonically normalised basis in full generality.We can however uncover a universal spectrum for simple geometries as we will see now.

Twisted factorised geometries
Let us investigate the Hessian for the twisted factorised geometries X 6 = (T 2 ×X 4 )/Γ considered in sect.4.2 with X 4 = T 2 × T 2 or K3.We first focus on the orbifold untwisted sector for each choice of compactification, and then comment on the orbifold twisted sector.As we argue, we do not expect perturbative instabilities to arise in such a twisted sector, although this will in general depend on the specific choice of Γ.
T 6 with metric fluxes Let us consider the case X 4 = T 2 × T 2 that corresponds to a factorised T 6 with metric fluxes, and an abelian orbifold Γ such that h 1,1 untw (T 6 /Γ) = 3.Then one can explicitly express the three basis vectors ( t, tα ) (the vectors ( b, bα ) are then the same, acting on the proper axionic subspace).
The only non-vanishing intersection number is K 123 such that the Kähler sector metric reads from which we deduce the following orthonormal basis: With this explicit basis, with σ a = (1, 0, 0) to match the structure (4.10) and with the saxionic vevs given by (4.21), we find the following universal behaviour for the Hessian: To get the masses, we need to divide the eigenvalues by a factor two and it is also useful to normalise the spectrum with the Breitenlohner-Freedman bound m BF .In Planck units, it is given by m 2 BF ≡ −3/4|V | which in our setup translates into m 2 BF = −9e K Q 2 .We obtain the following masses m 2 s for the saxions and m 2 a for the axions: The saxionic tachyon has multiplicity h 2,1 untw (T 6 /Γ) ≤ 3 and is just above the BF bound.On the other hand, the axionic massless mode has the same multiplicity and the tachyon in the non-SUSY branch is also above the BF bound.There is thus no instability induced at the perturbative level.Note that this spectrum matches the universal one found in the T-dual DGKT-CFI setup [17].We interpret this exact matching as a consequence of the factorised structure for the triple intersection numbers, which allows us to implement T-duality on a Kähler modulus that does not mix kinematically with the rest of the (untwisted) Kähler sector.
T 2 × K3 with metric fluxes Now we consider the slightly more involved example that arises when we take X 6 = (T 2 ×X 4 )/Γ with X 4 a K3 manifold.One may attempt to work directly with (6.3a) and (6.3b), however, there is a simpler way to build the canonically normalised masses and recover a universal behaviour of the Hessian by taking advantage of the factorised structure of metric and the fact that the metric fluxes run along the fibre.We thus treat separately the direction of the fibre and only split the base sector of the fibration in its primitive and non primitive components.
The metric in this case is of the form where A, B go from 1 to h 1,1 − (X 4 ) and η AB are the intersection numbers of the X 4 base.We can split the base part of the metric in its primitive and non-primitive sectors, obtaining We introduce a convenient basis ( tL , t, tα ), with tL defined along the fibre, t in the nonprimitive sector of the base and tα in the primitive sector of the base.Since we want all of them to be orthonormal we choose tL = 2(t L , 0, . . . ) , t = √ 2(0, t 1 , . . . ) , (6.12) and leave the conditions for the primitive sector of the base implicit.In the basis (û, ûμ , tL , t, tα ) the Hessian becomes where the entries associated to the second component of the basis have multiplicity h 2,1 untw (T 2 × K3/Γ) and the ones associated to the fifth component have multiplicity h 1,1 − (X 4 ) − 1.In particular the entry (5, 5) corresponds to a diagonal matrix with shape (h 1,1 − (X 4 ) − 1) × (h 1,1 − (X 4 ) − 1).To get the masses, we divide the eigenvalues by a factor two and normalise the spectrum with the Breitenlohner-Freedman bound m 2 BF = −9e K Q 2 as in the previous example.We obtain the following masses m 2 s for the saxions and m 2 a for the axions: where the multiplicities of the masses are (1, h 1,1 − (X 4 ) − 1, 1, 1, h 2,1 untw (T 2 × K3/Γ)).As we can see, the mass spectrum is the same as in (6.8), up to multiplicity.Again, this is to be expected from the relatively simple setting, that displays a factorisation of the triple intersection numbers.These allow us to link our setup to DGKT-like vacua via T-duality on a Kähler modulus kinematically decoupled from the rest of the untwisted Kähler sector, which presumably explains the exact matching with the universal spectrum found in [17].

Orbifold twisted sector
Let us now turn to potential instabilities in the closed-string twisted sector localised at fixed loci of the orbifold group Γ.By assumption, such fields have vanishing vev and do not enter the flux superpotential, so we can describe them as a perturbation of the untwisted Kähler potential, as usually done for the chiral sector of a compactification [54]: Here H m = {T a , U µ } represent the untwisted sector, and C α the closed string fields of the twisted sector.For many orbifolds quotients the Kähler metrics of the twisted sector are diagonal Kα β (H m , Hm ) = δ α β Kα (H m , Hm ) , (6.17) implying that we can use the following formula for the flux-induced mass on each twisted field [54], expressed in Planck units: with the rhs evaluated at the vacuum.For the supersymmetric branch of vacua, where F-terms vanish, we find matching the mass spectrum of the untwisted complex structure fields that do not enter the superpotential.For the non-supersymmetric branch (3.16b) we need to make use of the F-term structure (3.8), (3.18).From here we obtain that the universal contribution to (6.18) reads which is positive.To generate a perturbative instability in the twisted sector, the non-universal contribution to (6.18) should therefore be negative and large enough to violate the BF bound.
To evaluate this second contribution one needs to specify the choice of Γ.To get an estimate, let us consider the Kähler metrics that arise in toroidal orbifolds of the form T 6 /Γ, with Γ an Abelian orbifold: where the rational numbers n a α , l µ α are the modular weights of the orbifold singularity [55,56].
From here one can deduce that the non-universal contribution to the twisted masses is The number L α is typically negative, lowering the mass for the twisted field, but as long as it takes values above −22/9, as is often the case, no instability will be induced.Finally, notice that this result should be reconsidered if the twisted fields enter the flux superpotential, as further contributions to the mass terms would then be induced by a Giudice-Masiero mechanism.

Elliptic fibrations
For non-trivial elliptic fibrations, one can neither explicitly express the primitive/non-primitive basis as in the factorised torus case nor exploit the block diagonal aspect of the metric between the fibre and the base components as in the T 2 ×X 4 setup.Moreover, unlike in our previous analysis, now we observe deviations from the universal DGKT-like mass spectrum (6.8).This could be interpreted from the fact that, in the T-dual DGKT-like picture, there is a Kähler modulus that is small and that mixes kinematically with all the remaining Kähler moduli.As such, to fully account for T-duality at the level of the 4d EFT one should incorporate to the Kähler potential (2.5) the associated leading curvature corrections, which were neglected in the large-volume setup of [17].However, we still expect from the perturbative analysis developed in sect.5.2 to recover the universal mass spectrum asymptotically at large n, since in this limit the said Kähler modulus decouples kinematically from the rest.As such, our infinite family of scale separated vacua is safe from perturbative instabilities.
To check these expectations we explored numerically two specific examples of non-trivial elliptic fibrations.We generated vacua along our infinite family with increasing n and computed numerically, for each one of them, the Hessian matrix from the scalar potential (3.5).At a given vacuum with completely determined numerical vevs, it is then easy to perform the change of basis in order to get the canonically normalised masses.Alternatively, instead of starting from the potential, we could have evaluated directly the expressions (6.3a) and (6.3b).However, the former strategy does not rely on the heavy computations required to obtain the analytical of AdS 4 parallel to its boundary, in the present context.Using the results of section 3 we may do so at the level of smeared-source backgrounds, focusing for concreteness on the sub-branch (3.15).From (3.17 where ρ 0 ≡ −ι 3 2 Q and ι Q ≡ sign Q.Such four-form fluxes support 4d membranes that come from (anti-)D2-branes point-like in X 6 and from (anti-)D6-branes wrapping calibrated four-cycles of X 6 , respectively.Following [20], one can see that upon choosing the appropriate orientation one recovers and these objects are marginally stable, both for the SUSY and non-SUSY branches in (3.16).This is to be expected, as upon two T-dualities these objects are mapped to (anti-)D4-branes wrapping holomorphic two-cycles in DGKT-like vacua. 7It would be interesting to check whether this equality is still maintained when including the first-order corrections to the 10d smeared background, as it happens for DGKT-like vacua.masses m 2 s normalised by the BF bound in Planck units m 2 BF ≡ −3/4|V | are shown in fig.2(a) with respect to ϵ = 1/n on the x-axis.These masses are identical in both the supersymmetric and non-supersymmetric branches of vacua.We observe that at small ϵ the masses converge towards the values derived in the factorised torus or trivial K3×T 2 fibration cases, while being above the BF bound for the whole investigated ϵ range.Figs.2(b) and 2(c) display the normalised axionic masses m 2 a for the supersymmetric and non-supersymmetric branches respectively.Again, these masses converge towards the expected values at small ϵ and do not introduce any instability.Denoting with the index L the direction along the fibre and with A, B ∈ {1, 2} the two others, the triple intersection numbers read Similarly to the two-parameter model studied above, fig.3(a) shows the exact solutions t L and t i of the vacuum equations (5.7) as well as the leading order and next-to-leading order approximations t L (0) , t 1 (0) , t i (1) and t i (1) with respect to n in the range [2,200].We again observe that the leading order expressions are good approximations of the actual solution and that the next-to-leading order ones are better.The relative errors are displayed in fig.3(b) and they again scale as predicted by the perturbative expansion.(1,1,2,8,12) model and their leading-order and next-to-leading order approximations.The green dotted curve coincides with the orange dotted one while t 1 and t 2 converge to the same asymptotic value because of our choice to have the same unit coefficient in front of the scalings for m 1 and m 2 .That would not be the case with other scalings.(b) The relative error of these approximations with respect to the actual solution.As in the two-parameter case, the plots have been generated in the supersymmetric branch but similar behaviours are observed for non-supersymmetric vacua.
For each vacua, we compute the Hessian from the scalar potential (3.5) and express it in the primitive/non-primitive basis defined by (6.2).We compared again our numerical results with a direct implementation of the analytical expressions (6.3a) and (6.3b) and found again a perfect agreement in this more involved model.This shows the robustness of the analytical derivation of the mass matrix.The saxionic squared masses m 2 s normalised by the BF bound defined like in the previous example are shown in fig.4(a), while figs.4(b) and 4(c) show the axionic masses m 2 a for the supersymmetric and non-supersymmetric branches respectively.Again, all these masses converge towards expected values at small ϵ and do not introduce any instability.(c) The axionic masses in the non-supersymmetric branch with a tachyon identical to the saxionic one.

( 3 .
13) describe nearly-Kähler compactifications.From this framework, one can interpret the vacuum constraint(3.15)as the absence of certain internal fluxes: transformed background contains no Romans mass, and the H-flux must be traded by metric fluxes.The corresponding flux vacua should then lie within the branches (3.15) and (3.16).
) one first deduces the following relation for the AdS 4 radius R measured in the 10d string frame ℓ s the smeared internal fluxes threading X 6 one deduces the following fluxes that translate into 4d four-form fluxes upon dimensional reduction:

Figure 2 : 4 ( 1 , 1 , 1 , 6 , 9 )
Figure 2: The squared masses normalised by the BF bound depicted with the black dashed line for the twoparameter P 4 (1,1,1,6,9) model.(a) The saxionic masses, identical in both supersymmetric and non-supersymmetric branches.The tachyon has degeneracy h 2,1 and is slightly above the BF bound.(b) The axionic masses in the supersymmetric branch.It is now the massless mode that has degeneracy h 2,1 .(c) The axionic masses in the non-supersymmetric branch.In this family, a single tachyon arises in this sector, with same mass as the saxionic tachyons.

1 Figure 3 :
Figure 3: (a) Vacuum expectation values of the saxions t L t 1 and t 2 for the three-parameter P 4(1,1,2,8,12) model and their leading-order and next-to-leading order approximations.The green dotted curve coincides with the

Figure 4 :
Figure 4: The squared masses normalised by the BF bound depicted with the black dashed line for the threeparameter P 4 (1,1,2,8,12) model.(a) The saxionic masses, identical in both supersymmetric and non-supersymmetric branches and with degenerate tachyons slightly above the BF bound.(b) The axionic masses in the SUSY branch.
.7) with H the three-form NSNS flux, d H ≡ (d − H∧) is the H-twisted differential and Ḡ a formal sum of closed p-forms on X 6 .The fluxes enter the perturbative superpotential W ≡ W RR + W NS , where W RR involves .14)By construction, the parameters of table 1 satisfy the equations of motion of the Kähler sector.Demanding that this restricted Ansatz also solves the equations of the complex structure sector further constrains the non-SUSY branch by imposing the vanishing of the Romans mass.
As a result, we arrive at the four branches of solutions displayed in table2.By evaluating the above solutions in the F-terms equations (3.10) one can check that the first row of table 2 corresponds to a supersymmetric branch of vacua.Actually, these results generalise the SUSY branch found in[27, table 2]beyond the case (3.13).In addition, we also Branch Parameters ℓ s ρ 0

Table 2 :
Complete set of branches of solutions to the equation of motion under the Ansatz (3.9) such that the brackets in (3.11b) and (3.12b) vanish simultaneously.The SUSY branch has two independent parameters P and N while the non-SUSY branches only have one, i.e.N .