Leaving the Swampland: Non-geometric fluxes and the Distance Conjecture

We present an analytical solution of toroidal compactification of Type IIB string theory with non-geometric fluxes. Under the assumption of a mass hierarchy among the moduli fields, we obtain a scalar potential with a runaway direction. We show that the existence of an upper bound in field space for this non-stabilized field follows from the presence of hierarchy on the moduli fields and in consequence from a particular type of flux configuration. In this context, the Swampland distance conjecture is satisfied by consistency in the selection of fluxes. This suggests the possibility to leave or enter the Swampland by a parametric control of the fluxes. This is achieved upon allowing the non-geometric fluxes to take fractional values. In the process we are able to compute the cut-off scale below which the theory is valid, completely depending on the flux configuration. We also report on the appearance of a discrete spectrum of values for the string coupling at the level of the effective theory.


Introduction
The Swampland program [1][2][3][4][5] has received a great attention in the last few years (for reviews see e.g. [6,7]). The proposed criteria to discern wether an effective theory is compatible with a quantum gravity theory leads us to the possibility to understand important issues concerning the construction of effective theories directly from string theory or inspired by it [8]. The possible microscopic origin of the criteria is also an opportunity to question wether our knowledge about dimensional reduction and compactification in generic scenarios is complete [9,10].
One interesting question constitutes the validity of the Swampland criteria as a way to have a well-defined boundary in field theory, separating those compatible effective theories with string theory to those which are not. If true, one should be able to cross it in both directions [11] (see [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] for many different tests). Entering the Swampland seems to be easy by departing from an effective theory constructed from string theory restricted to many of the self-consistent aspects of string theory and high energies and by adding many extra assumptions.
One question arises however, about the meaning of leaving the Swampland. Let us say that one has an effective theory constructed by implementing some set of assumptions, inspired on a string construction. The number of assumptions could lead to a model violating some of the Swampland criteria. This is reflected in the apparent possibility to extend the model to transplanckian scales or to scales below the validity of the field theory. In this case the model is not just incomplete from the quantum gravity perspective but inconsistent at the level of field theory. However, suppose it is possible to identify some set of assumptions whose removal allows to fulfill the Swampland criteria and enter the Landscape. Is the final model compatible with string theory? Are the removed assumptions a way to trace back consistent models? If all the above is true, one can establish a way to identify those assumptions that can be relaxed in an effective theory by entering the Landscape. In the process one would learn more about the UV completions of the effective theory.
Of particular interest becomes the construction of effective theories inspired by string theory, or as coined in [7], string-inspired models. On those, the direct construction of a model from a concrete string theory is not completely known. In consequence the set of assumptions is an arbitrary election in their construction, playing in some cases, an important role in the consistency of the theory.
The presence of many assumptions could lead us to an effective theory violating some consistency checks as the Swampland criteria, pushing the model directly into the Swampland. As proposed, if one of the Swampland criteria is violated the effective model cannot be completed in the UV pointing out the presence of a model incompatible with some extension to quantum gravity, or as in the case, to string theory. A second case could just leads us to a model valid untill some scale Λ SW above which some corrections or removal of some taken assumptions need to be implemented in order to have a model compatible with string theory. See Figure 1.
Consider the case of a scalar potential with a runaway direction. In this case the refined dS conjecture is fulfilled over a finite range on the modulus field, such that infinite trajectories are limited by the distance conjecture. The relationship between these two Swampland criteria has been intensively studied recently [28][29][30][31][32][33] indicating a link between them in terms of modular symmetries [34] and the presence of non-perturbative objects as instantons [35,36]. Those works establish important advances in the search of the microscopic origin of the Swampland criteria (see also [37][38][39]   By taking a string-inspired model resulting from a non-geometric flux compactification on an isotropic torus and by introducing strong constraints (represented by the arrowed trajectory 1 in the field space) we can end up with an effective theory in the Swampland in which the energy scale factor is unlimited (typically according to the value of some model-dependent parameters) below and above the effective field theory scale Λ eff . The more separated the trajectory from the line center, the more number of taken assumptions. By removing some set of assumptions (in principle different from those taken in trajectory 1) one can re-enter into the Landscape (trajectory 2), making possible to construct an effective theory valid till some scale Λ SW > Λ eff (trajectory 3).
In this paper we explore the relation between the existence of a finite distance in field space and specific flux configurations by studying a simple string-inspired model consisting on a toroidal compactification of type IIB in the presence of non-geometric fluxes [40][41][42]. These fluxes have been considered in the construction of effective models in order to generate a superpotential depending on all moduli fields, including Kähler moduli (see [43] for a review) by assuming the presence of T-duality at the level of the effective theory. Despite some significant success, mainly in the search for stable vacua with stabilized moduli [44][45][46][47][48][49], nongeometric fluxes lack for a complete global construction from the ten-dimensional perspective [48,50] (see however [51] for a proposal based on double field theory). Their incorporation into compactification models usually requires the imposition of some set of plausible constraints, making these type of construction perfect examples of the so called string-inspired models. The most usual assumptions involve the extension of tadpole and Bianchi identities to the corresponding T-dual fluxes, quantization of non-geometric fluxes and a null back-reaction by non-geometric fluxes on the internal geometry implying the assumption of a well-defined internal volume.
Non-geometric fluxes have also been studied in the context of the flux-scale scenario in order to have some parametrical control to generate almost flat directions on moduli space, testing wether inflationary directions and stabilization of all moduli come along [48,49,52,53]. This approach was followed in the context of F-term axion monodromy inflation [54,55] (see [56][57][58][59] for relations among the flux-scaling scenario, hierarchy on moduli mass and the Swampland). So far all results seem to enforce an intriguing idea: for inflationary directions to be present and to have a parametrical control over the different scales, fractional fluxes are required. In the context of the Swampland criteria, the above could be interpreted as a way to enter the Swampland, i.e. by adding an extra assumption concerning the presence of non-integer fluxes.
Within this context, we study a scenario of non-geometric flux compactification on an isotropic T 6 with orientifold three-planes [60] as an string-inspired effective model. Besides the inherent assumption about the validity of T-duality in four-dimensions, we also assume a hierarchy on the masses for the complex structure and the axio-dilaton against the mass of the Kähler modulus. We concentrate on a particular solution for which the superpotential component(depending on the Kähler moduli and the vevs for the complex structure and the axio-dilaton) vanishes [61]. By this considerable increment on the number of assumptions, we find an analytical solution in which the scalar potential exhibits a runaway direction on the real component of the Kähler modulus (τ ) resembling some characteristics of KKLT-scenario 1 (before the inclusion of antibranes). There are some important results in our model we want to stress out here: 1. Compatibility with the Hierarchy Assumption on moduli fields forces the existence of a range in the field space for τ as suggested in [61]. Since such a hierarchy comes from an appropriate selection of fluxes we conclude that in this case, the constraints on the flux configuration can be interpreted as the microscopic origin of the distance conjecture constraint. Notice as well that this provides the model with an essential feature since the volume is restricted to a range as expected to the geometric back-reaction of nongeometric fluxes.
2. The above condition defines a scale of energy Λ SW up to which the model is valid and it depends purely on non-geometric fluxes.
3. Due to the number of constraints, the effective scalar potential depends only on 2 nongeometric fluxes.
4. For different flux configurations (eight thousand) we find numerical evidence that the string coupling s 0 always acquires discrete values (see Figure 3).

5.
Taking integer values for all fluxes leads us to an incompatible effective theory with Λ SW > M s and with an internal volume smaller than 1/M 6 s , with M s the string scale.
6. Only by considering fractional values for non-geometric fluxes, the model is consistent and the distance conjecture is satisfied for a scale Λ SW below M s . Similarly, the scale's hierarchies M s > M KK > M U,S are also accomplished and more important, it is possible to have a parametrical control by fluxes. We comment about fractional fluxes in our conclusions.
By all the above we show a specific procedure which takes an effective model out of the Swampland by removing some, in principle, essential assumptions. Wether this mechanism is an available and general option to construct consistent effective models within the Swampland project is something we want to discuss.
This work is organized as follows: In Section 2 we review some toy models on which we based our proposal, stressing out important characteristics of having a runaway direction to check for consistency with the Swampland conjectures, specifically the refined dS and the distance conjectures. In Section 3 we describe the consistency conditions of the model by assuming a hierarchy on the moduli fields. The main part of our work is described in Section 4, where we present an analytical solution for fixing vevs of some moduli fields. This establishes a way to construct a scalar potential with a runaway direction which allows us to leave the Swampland by relaxing the condition of having integer-valued fluxes. We discuss the consistency of the model, the Swampland criteria and the implications in the flux scaling scenario. Additionally we present numerical evidence supporting our assertions and a simple example in which the field τ acquires a small range in the field space parametrically controlled by the flux configuration. Finally we present our conclusions. In Appendix A we write the conditions for the scalar potential extrema in terms of the superpotential covariant derivates and Appendix B is devoted to an exhaustive discussion of the notation used throughout the paper.

Swampland criteria in type IIB toroidal compactifications
In this Section we review some of the main features that effective scalar field theories possess when constructed directly from a ten-dimensional string theory. Effective models with a runaway direction constructed by a toroidal compactification of Type IIB string theory, constitute interesting models for which the Swampland refined de Sitter and Distance criteria are satisfied. Those are usually driven by the real part of Kähler modulus.

Toy model 1: GKP and the Swampland criteria
Let us start by considering the Gukov-Vafa-Witten (GVW) superpotential W derived from a 6-dimensional isotropic toroidal compactification of Type IIB string theory. The superpotential depends only on two moduli, the complex structure U = u + iv and the complex axio-dilaton S = s + ic, where f 's and h's refer to RR and NS-NS fluxes. See Appendix B for more details on this notation. Being a no-scale superpotential for the Kähler modulus, the minimum for the scalar potential V, supersymmetric or not, is constrained to be positive or null. Since the dependence of V on the Kähler modulus T = τ + iθ only comes from the Kähler potential K it contains a flat direction on θ and a runaway direction on τ . The moduli vacuum expectation values U 0 and S 0 are fixed by the equations ∂ U V = ∂ S V = 0. Since the potential is of the form with F(U 0 , S 0 ) a positive real function. The canonically normalized field t 1 = √ 3 √ 2 ln(T +T ) serves to define the first refined-dS criterium as: This is fulfilled, notice that a SUSY solution implies F = 0. Similarly, since we have a flat direction on θ, the second criterion reads is violated. However it is necessary to restrict the range of values for τ in modulus space according to the Distance Conjecture (DC). This simple no-scale model is out of the Swampland for every value of τ subject to an upper bound.

Toy model 2: non-SUSYà la KKLT
In this Section we review the scalar potential for a type IIB string theory toroidal compactification. The superpotential dependence on the Kähler modulus arises from D7 branes. This model is also a case that satisfies the refined Swampland constraints.
Let us consider a toroidal compactification with a superpotential given by where the contribution depending on T coming from gaugino condensation or instanton contributions from D7-branes as in KKLT, is given by W (T ) = Ae −λT with λ > 0. Assume as well a hierarchy on the moduli such that U and S are fixed independently of T . This is obtained by solving the equations D U W = D S W = 0. Since D U W and D S W are exponentially suppressed by τ , the hierarchy assumption is valid and D U W = D S W ∼ 0 for large values of τ . In this scenario the scalar potential is given by with γ(θ) = Re(W 0 e iλθ ) and θ 0 = arg(W 0 ) with W 0 = W(U 0 , S 0 ). The potential, as known, exhibits an AdS vacuum or a run-away direction on τ depending to the values on the involved constants. In the case of a runaway τ -direction it has been pointed out in [5] that the distance conjecture allows the potential to fulfill de refined dS criteria since large values on moduli space for τ would imply the apperance of extra light modes. The effective model is then consistent for large values of τ , with values limited by the appearance of extra light KK modes. Such that an upper bound on τ must be imposed by hand.
Now, as stated in [27] corrections to the superpotential on the Kähler modulus adds extra terms on the scalar potential. According to the refined dS conjecture, whether these corrections lead us to the swampland or not, will be an indicative of compatibility with a quantum gravity theory such as string theory. In that context we shall explore under which conditions the inclusion of non-geometric fluxes satisfies the above bounds. This implies considering only tree-level corrections of the superpotential trough the inclusion of a linear term W (U 0 , T ) = iT P 3 (U 0 ) in the superpotential which introduces an interaction of the Kähler modulus with the complex structure modulus.

Inclusion of non-geometric fluxes and the Moduli Hierarchy Assumption
We consider type IIB string theory compactified on an isotropic six-dimensional torus with fluxes and orientifold 3-planes. A generic scheme involves 6 real moduli fields and 8 different integer fluxes 2 . The corresponding superpotential reads 3 with P 3 being also a cubic polynomial on U given by with b's and β's corresponding to non-geometric fluxes (see Appendix B for notation). The scalar potential V depends on all moduli U, S, T and some extrema are expected in a general flux configuration.
However we are interested in studying effective models with runaway directions, particularly on the Kähler field τ . For that we present an analytical solution for fixing the vevs of U and S such that a kind of "no-scale" behavior is present on the effective model. The first assumption for such a goal is the presence of hierarchies on the moduli. Therefore we proceed to clearly describe this assumption.

On the Moduli Hierarchy Assumption
In this Subsection a moduli hierarchy is discussed in the scalar potential. We write conditions to obtain a separation between the scales of certain moduli. Based on the above toy models, an interesting scenario is that the complex-structure moduli and the axio-dilaton are stabilized in a first step and the Kähler moduli are stabilized in a second step.
Consider a superpotential W = W (φ I ) depending on N moduli fields φ I with I = 1 . . . N and let us assume that the vacuum expectation values for some of the moduli fields φ a are (almost) fixed independently of the rest of the moduli denoted as φ i with i = a. We shall refer to this assumption as the Moduli Hierarchy Assumption. Under this scheme the vacuum expectation values of φ a are assumed to be barely modified by the dynamics of the rest of moduli φ i , implying strong constraints on flux configurations as we shall see. In terms of the scalar potential, the moduli hierarchy assumption implies that the fields φ i are fixed at a minimum of the potential where the fields φ a are fixed at their vevs denoted (φ 0 ) a .
The implications of the hierarchy moduli assumption strongly depend on the form of the superpotential. In this case we are thinking on a superpotential W consisting of a component W which depends on the moduli φ a and a second component W being a function of φ i and containing interactions between φ a and φ i . Those would be obtained by compactification or dimensional reduction of string theory. The complete superpotential is of the form To clearly specify the constraints followed by our assumption it is important to observe that the scalar potential can be written as [67] V According to the anzatz, (φ 0 ) a is a solution of ∂ a V| (φ 0 )a = 0 (see Appendix A for the specific case of the isotropic torus).
On the other hand, the vacuum expectation values for the moduli fields φ i , denoted (φ 0 ) i should be determined by the system for each i. The assumed hierarchy for the fields φ a would be consistent with the above method of computing the vevs and mass of the rest of moduli if The moduli mass are given, as usual, by the eigenvalues of the mass matrix M 2 IJ = ∂ IJ V . However, according to the assumed hierarchy, the following constraints must be fulfilled such that with the corresponding eigenvalues M 2 a and M 2 i satisfying A mass hierarchy of order 10 can be obtained by F-terms in the presence of fluxes as described in the flux-scaling scenario [48]. Our goal is to elucidate how the hierarchy moduli assumption determines an effective theory is in or out the Swampland.

Hierarchy and the Swampland Distance Conjecture
As stated before, we shall consider the GVW superpotential with tree-level corrections depending on the Kähler modulus. For that we assume that the complex structure modulus U and the dilaton S are fixed independently of T implying that non-geometric fluxes contributions are sub-leading. In the following we shall use of the notation introduced in the last section. To start with, we consider the GVW superpotential written in the form W = P 1 (U ) − iSP 2 (U ). Then we have a no-scale scalar potential of the form Fixing U independently of T implies finding a solution of ∂ U V = 0 which involves a dependence on S. If we assume that U = U 0 is also fixed independently of S, the solution of ∂ U V = 0 is the same as the equation D U W = 0 (see Appendix A were we derive extremum scalar potential conditions with the covariant derivatives of the superpotential). We are therefore considering a model in which Being a no-scale model, the last assertion also implies that at the minimum V 0 = 0 where supersymmetry could or could not be broken by T . However, it is straightforward to see that a supersymmetric solution leads us to trivial solutions for U 0 . Henceforth we consider the case in which D T W = 0.
Generically we know that a solution to the above equations implies turning on a G 3 form of type (2, 1) and (0, 3) [68]. However, here we are interested in expressing the fluxes in terms of their symplectic components (f I , f I , h I , h I ). Hence, stabilization of S is obtained from D S W = 0 from which we obtain In this scheme the stabilized value of S depends on U .

A particular solution
In this Subsection we present an analytic solution to the conditions D S W = D U W = 0. Only RR and NSNS fluxes are on, generating a superpotential W(U, S) giving a setup where complex structure and axio-dilaton are stabilized in a first step. As we shall see, a consistent solution will require a relation between fluxes.
A generic solution would imply to substitute the dilaton as a function on the complex structure in D U W = 0. From this last equation with P 1 (U ) and P 2 (U ) different from zero in order to have W = 0 at the minimum of the potential and P i (U ) being the derivative of P i with respect to U . A general analytical solution seems difficult to obtain since we also have to satisfy the tadpole condition. We proceed to consider a particular solution by solving (4.4) written as: A common solution for U in the above pair of equations will also stabilize the value of the dilaton S. Indeed, there is a common root for the equations (U + U * )D U P 1 (U ) = 0 and (U + U * )D U P 2 (U ) = 0 if there is a relation among RR and NS-NS fluxes. In order to see that, observe that the above equations are also represented by a quadratic polynomial on Re(U ) and cubic in Im(U ). As solutions we have 4 for i = 1, 2, where the RR fluxes A 1 , B 1 , C 1 and the NS-NS fluxes A 2 , B 2 , C 2 are given by 4 One additional solution has Re(U0) = 0. We shall not consider such unphysical case. and We observe that one way for both equations to be simultaneously fulfilled, is that fluxes satisfy the relations while assuring a non-zero tadpole contribution from fluxes f and h 5 . Notice that for both solutions the magnitude of U 0 is independent of fluxes B. The value of P 1 at the minimum reads In terms of the above fluxes, the dilaton vev is given by from which the string coupling s 0 = e −φ = 1/g s reads (4.12) Using the tadpole condition (B.11) and the constraints (4.9), s 0 reduces to Therefore, any physical solution implies a new flux constraint of the form (4.14) Some comments are in order. First of all observe that s 0 only depends on 4 NS-NS fluxes h. In principle we start with 8 fluxes h and f , but only 5 of them are free once we take the flux constraints (4.9) together with the tadpole cancellation condition. In that context we select 4 NS-NS fluxes h and one RR f corresponding to the 5 degrees of freedom. Second, since not all terms are positive in the denominator, it is possible to have some flux configurations leading us outside the physical region for which s 0 is smaller than unity.
We have fixed U and S independently of the Kähler modulus T , which has to be incorporated. Therefore our next step is to consider such tree level correction on the superpotential, looking for the required conditions such that our hierarchy assumption is consistent.

Non-geometric fluxes
In this Subsection we discuss the implications of the solutions for U and S presented in (4.6) and (4.11) for RR and NSNS fluxes satisfying (4.9). Now we incorporate non-geometric fluxes having a dependence on the Kähler modulus.
Consider now the whole superpotential of the form with a scalar potential V (T ) = V (U 0 , S 0 , T ). The values U 0 , S 0 are the previously computed vevs 6 . However, these values turn out to constraint the polynomial P 3 since a root U 0 of the polynomial (U + U * )P 2 (U ) − 3P 2 (U ) = 0 is also a root of P 3 (U ) for an isotropic set of fluxes.
This follows from the use of Jacobi Identities for the non-geometric fluxes, Q·H = Q·Q = 0 from which it is possible to establish a set of relations among non-geometric and NS-NS fluxes. Before discussing the implications, let us first show that indeed P 3 (U 0 ) = 0. For the isotropic case there is a particular solution for Jacobi (B.22) and Bianchi identities (B.23) given by allowing us to express four non-geometric fluxes in terms of just two of them, namely b * and b * . Notice the difficulty on having integer fluxes satisfying the above constraints. Then the polynomial P 3 takes the form We observe that P 3 depends only on 2 non-geometric fluxes through Q(U ) while q 3 (U 0 ) = 0. Observe that U 0 depends only on NS-NS fluxes and in fact, it is the only solution of Here we have shown that P 3 (U 0 ) = 0, now let us discuss the physical implications of our solution.

Physical viability
Once we have shown that P 3 (U 0 ) = 0 we must check if our solution is compatible with the moduli hierarchy assumption and proceed to study the implications on the scalar potential properties. First, notice from (3.6) that the scalar potential is given by Hence we see that V (T ) reaches a minimum at θ 0 = 0 in the θ-direction and has a runaway direction on τ . Since P 3 (U 0 ) = 0 we also have The covariant derivatives are evaluated on the vevs U 0 and S 0 . From these expressions we see that in order for the approximations (3.8) to be valid and the moduli fields U and S to be fixed independently of T it is necessary that which guarantee that U 's and S's vev's are approximately kept at the values U 0 and S 0 respectively. Therefore our task now is to asure the viability of vanishing of P 3 (U 0 ) or of the above two constraints. Let us start by checking wether P 3 (U 0 ) can vanish or not. First of all, in terms of fluxes Vanishing of |P 3 (U 0 )| implies either that δ 2 = 0, or that However, non-geometric fluxes b * and b * are real while δ 2 , besides being real, must also be different from zero (see (4.6)), implying that |P 3 (U 0 )| cannot vanish. Therefore the only option to be consistent with our hierarchy assumption is to tune on the non-geometric fluxes such that constraints (4.23) and (4.24) hold. As we shall see, this is deeply connected to the Swampland Distance Conjecture.

Implications on the flux scaling-scenario
Before discussing our model's consistency with the moduli hierarchy assumption and the fixing of the cut-off scale by a proper selection of non-geometric fluxes, it is important to analyze the implications on the flux-scale scenario [48]. We will analyze the hierarchy of physical scales.
The following hierarchy of scales is expected where m 3/2 is the gravitino mass and V is the volume of T 6 in the Einstein frame. We have used the relation Since s 0 , τ > 2 for a supergravity description to be valid in a physical region, it follows that M pl > M s > M KK in concordance with (4.27). Notice that all mass scales as well as the moduli masses, are unfixed and depend inversely on τ , while the relevant ratios are determined by Observe that this behavior is the same as the models with frozen complex structure studied in the flux-scale scenario [48], implying that τ > s 0 . Therefore, for consistency and taking θ = θ 0 = 0, The scale of supersymmetry breaking is determined by the non-vanishing F-term evaluated at S 0 and U 0 and given by .

(4.33)
Notice that the scale at which SUSY is broken is determined by fluxes f and h where nongeometric ones are not playing a role since the above ratio does not depend on τ . Finally, the moduli mass eigenvalues depend on τ as following that where i = U, S, T . Since τ is not fixed, there is only a range of values in moduli space in which M KK > M i , actually for τ > 1. It is then important to check the bounds for τ .

Moduli Hierarchy and the Swampland Distance Conjecture
Up to here we have presented a model in which the presence of non-geometric fluxes have not altered the runaway profile of the scalar potential on the τ -direction but have stabilized θ.
Since any scalar potential with a dependence on τ n for any integer value of n satisfies it automatically satisfies one of the refined dS bounds. The same occurs to the potential (4.19) even for θ = 0 since with H = M 2 pl |P 3 (U 0 )| 2 /48πu 0 s 0 . According to the refined dS conjecture such potentials can be considered out of the Swampland. Therefore, the moduli hierarchy assumption allows us to have a model with tree-level corrections on the superpotential depending on the Kähler modulus by the presence of non-geometric fluxes which is actually compatible with the refined dS criteria. Such a model is therefore an effective theory compatible with a quantum gravity theory such as string theory. Notice however that although S and U are stabilized, all moduli masses still depend on τ for which they remain unfixed unless there is some criteria to constraint the value of τ .
In the following we shall use the constraints on our moduli hierarchy assumption to derive some bounds on τ . Even more we shall show they are compatible with the distance conjecture, allowing us to establish a cutoff scale at which the effective model is valid.
From constraints (4.23) and (4.24), we obtain that at θ = 0 which is an available range of τ if Notice that this is a restriction on non-geometric fluxes since NS-NS and RR fluxes have been already fixed at higher scales. The above range of viable displacement on τ is a direct consequence of the Moduli Hierarchy Assumption and allows us to estimate the range of scales at which our effective model is valid. First of all, a supergravity description of the 10-dimensional model requires the internal volume V 6D > 1 with with V being the volume of T 6 in the Einstein frame. Therefore it follows that τ > 1/2 implying that |P 3 (U 0 )| < 2.
Second of all, from the bounded value for τ in (4.38) we have that the gravitino mass is constrained to the values , indicating that for the gravitino mass to have an available range of values, |P 3 (U 0 )| must be less than unit.
A third important consequence of (4.38) is the following: it has been conjectured that moduli fields can not take large displacements otherwise massive fields interacting with the moduli must be taken into account. In such context and by taking string theory as the quantum gravity theory, the displacements are argued to be of the form [30] ∆τ whereτ is the canonical normalized Kähler modulus and λ has been typically taken of order 1 7 . In our case, given the Kähler potential, λ = 2/ √ 3 and implying that Λ SW is fixed by |P 3 (U 0 )|,. Once f and h fluxes have been chosen, it depends only on non-geometric fluxes b * and b * . Therefore Notice that small values for |P 3 (U 0 )| would fix the scale Λ SW below string mass. This means that the canonical normalized fieldτ have a non-zero range in which the model is consistent. 7 See Reference [30] for a discussion on the scale of λ τ Otherwise, for values of |P 3 (U 0 )| greater than unity,τ has a zero range of consistent values.
Therefore, by all the above implications, the range of viability for τ is fixed as 8 Hence, the smaller the value for |P 3 (U 0 )|, the larger the allowed range of displacement for τ . For |P 3 (U 0 )| = max(2, (12u 0 s 0 ) 1/3 ) = τ 0 , τ in principle is fixed to a single value although the model is not consistent. Notice that with large values for |P 3 (U 0 )| (which implies a better approximation consistent with the hierarchy assumption) the range for τ diminishes making more difficult to satisfy the Swampland Distance Conjecture. However all we need is to have a non-zero range of viability for τ defined far away from the minimum and maximum values established by the hierarchy assumption while having τ > 1 for SUGRA to be a valid approach. See Figure 2. The question is if one can have such scenarios for concrete flux configurations.

A numerical analysis
From all the above it seems there exists a link between the Moduli Hierarchy Assumption and the distance conjecture, establishing a range of viability for τ . Taking into account all the constraints, our model possesses 7 degrees of freedom (4 NS-NS, 1 R-R and 2 non-geometric fluxes) implying the necessity of a numerical analysis. Using only even integer fluxes we find 8000 flux configurations fulfilling all restrictions. The results are plotted in Figure 3. Interestingly, we find that for integer fluxes, non of the cases contains a small value for |P 3 (U 0 )| indicating that the model is not consistent. This means that the upper bound on τ is smaller than unity, destroying the supergravity approach. However, if one consider fractional non-geometric fluxes b * and b * , the values for |P 3 (U 0 )| become less than one, and in turn it allows a consistent range for τ .
A particularly surprising issue that becomes evident from this plot is the discreetness of the values of s 0 (which is independent of the non-geometric fluxes). The numerical evidence points out a maximum value for s 0 of 2 (although by considering odd-fluxes this can increase). This feature was also noticed in [31] and it is probably related to the high number of constraints (as shown in the expression (4.13) for s 0 in terms of fluxes which only depend on 4 NS-NS fluxes. Discrete values for the string coupling seem to be related to strong constraints after compactification, as fulfilling the tadpole condition or directly related to the topology of the internal space.
Similarly, it is possible to have a simple argument to scketch how fractional fluxes are linked to small values of |P 3 (U 0 )|. By assuming quantization of NS-NS and R-R fluxes one could assume its extension to non-geometry fluxes by imposing quantization on the action of non-geometric fluxes on (p + 1)-forms, as where Σ p is an internal p-cycle and ω p+1 a (p + 1)-form extended on internal coordinates. At some effective level, below Λ SW , one can write where M S the string mass scale. From (4.44) it follows that Hence, for |P 3 (U 0 )| less than one, |P 3 (U 0 )| p−1 can be approximated as 1/k with integer k > 1. Therefore, up to Λ SW The p-form Q · ω p+1 has fractional values in an effective theory allowing for b * and b * to be fractional. The presence of non-integers and non-constant fluxes have been already considered in literature as fluxes sourcing punctures on a sphere [69][70][71] or fractional fluxes arising as a consequence of the topology of the internal manifold [72,73]. Fractional fluxes can be considered as the result of the backreaction of the metric by the presence of non-geometric fluxes or equivalently by assuming T-duality on the internal manifold threaded with NS-NS fluxes. Under this perspective, Dirac quantization is a feature compatible with string theory in its ten dimensional version which can be modified by an specific compactification setup. This is consistent with our theory as soon as the quantization be reinforced at high scales but weakened at lower energies. We have shown that this is indeed our case.

A toy example: Quintessence and the swampland
In this Subsection we particularize our solution to a set of fluxes satisfying (4.9), Bianchi identities and Tadpole cancellation conditions. In this scheme we analyze the implications for the Swampland constraints.
Let us focus on an example which satisfies all the constraints, namely: thus one can see that in order to get even integer fluxes the NS fluxes are highly constrained. Indeed the only values allowed for the NS are ±4 and ±2 in Planck units. The U and S moduli are fixed at which is a solution of the scaling type and a physical solution implies that h 0 and h * have opposite signs. Thus, to stay in the perturbative regime it is required that |h * | = 2 (otherwise s 0 < 0) which is compatible with the flux quantization condition. As is stated by [31], it is possible to evade the Dine-Seiberg problem and to keep the theory in the perturbative regime just by fluxes, if the dilaton is stabilized at a value that it is not exponentially large. The mass hierarchy is controlled by the value of |P 3 (U 0 )|, which in terms of the fluxes is written as and it has to be as small as possible. Now, since h 0 and h * have oposite signs, the magnitude of P 3 (U 0 ) lies in a circle of radius of order O (10) violating the hierarchy condition. The hierarchies are preserved if we consider fractional non-geometric fluxes, which apparently violates the Dirac quantization condition. However, the cohomology group at which the nongeometric fluxes belong has to be determined. We let this subtle question for future work and we shall proceed with the approach of considering non-geometric fluxes with magnitude less than 1, preserving a parametric control on the mass hierarchies. Fixing the values of the NS fluxes as h 0 = −h * = −2, the scalar potential takes the form which implies a potential for a quintessence scalar rolling to positive values. Since the quintessence field is represented by the Kähler modulus it can potentially lead to fifthforces through its coupling with SM fields. However, since the rolling of the scalar field is parametrized by the non-geometric fluxes, it could be slow enough to effectively fix the couplings to SM fields avoiding fifth-forces.
This model, breaks SUSY spontaneously though F terms, with the sgoldsitno direction pointing mainly in the complex structure direction as τ becomes larger as where by construction the S directions is set to zero, and the swampland criteria is satisfied. As noticed in [74], this swampland criteria is not parametrically controlled by fluxes, instead it is possible to get a numerical control by a suitable choice of the scalar potential. The Moduli Hierarchy Assumption implies that Eq. (4.38) must hold, which for this particular solution can be written as thus, for h 0 = −h * , the allowed non-geometric fluxes lie in a circle of radius 2 3h * 2/3 which is lower than 1. Thus, together all the swampland criteria are satisfied if the non-geometric fluxes take fractional values less than 1. In this way the field range allowed by the distance conjecture can be parametrically controlled.

Final comments
In this work we proposed a particular path in the field space to leave the Swampland by starting with a string-inspired model. In the following see Figure 1. Generically we consider a string-inspired theory at some scale Λ N G , meaning that we construct a model in which some set of assumptions -denoted A-, and inspired by string theory, have been taken into consideration. Set A contains expected assumptions derived directly from string theory to give consistency at scales below the string scale but above the scale of validity of the effective model we are constructing. Also A contains all those assumptions we expect could be needed and extended at the effective level.
Hence, one can construct an effective model starting with a string-inspired model in which we have taken some assumptions contained in A, i.e., the assumptions we take to construct the effective model belongs to a subset A ⊂ A. At this step, the model could be out or in the Swampland. Consider now a different set of assumptions denoted B, containing those assumptions that are not related to symmetries or constraints from string theory. By adding some extra assumptions from set B one can end up with a model in the Swampland. However, if we retrieve some assumptions in A it is possible that our model leaves the Swampland as ilistrated by the arrowed trajectory in Figure 1. If that is the case, one could wonder about the physical meaning of the retrieved assumptions.
Leaving the Swampland means that we have constructed a consistent theory till some scale Λ SW < Λ N G < M s . If the Swampland conjectures are indeed a way to divide the field space into consistent and non-consistent theories with compatible extensions to quantum gravity, it is necessary to explain how is that the removed assumptions in A are not essential as first thought. If correct, this undoubtedly could establish a method to discern fundamental assumptions at different scales of energy.
In this work we present a specific example of the above described proposal by starting with a compactification on an isotropic six-dimensional torus with non-geometric fluxes, orientifold 3-planes and no D-branes. The assumptions taken at this scale Λ N G are the validity of T-duality in the effective four-dimensional theory −from which non-geometric fluxes have been introduced− non-geometric flux quantization, extension of tadpole and Bianchi identities. All of them constitute self-consistent assumptions derived directly from string theory or constructed by consistency at some level below the string scale −as the compactification scale−. Notice that by assuming T-duality, non-geometric fluxes interaction with Kähler moduli is introduced, suggesting the existence of a shift-symmetry on Kähler moduli derived from a symmetry on non-geometric fluxes.
At this point we take our first extra assumption (from set B): by some proper selection of fluxes it is possible to give a vev to U and S independently of T and in consequence of the non-geometric fluxes. We called it the Moduli Hierarchy Assumption and we look for solutions to stabilize the complex structure and the axio-dilaton in this form. For that we take our second assumption: a particular solution which relates NS-NS and R-R fluxes as shown in Eqs. (4.9). Together with the extension of Bianchi Identities and Tadpole conditions it is possible to show that such assumption leads us to two important consequences. First, that there are only 2 unconstrained non-geometric fluxes 9 (b * and b * ). Second, that the superpotential component depending on the Kähler moduli vanishes once evaluated at the vevs of U and S. These two results restrict the effective model to have a runaway direction along the real part of the complex Kähler modulus τ .
However, the effective model must be consistent with the hierarchy assumption on moduli. Due to the property of our solution, the superpotential written as W (U, S, T ) = W(U, S) + W (U, T ).
This implies that W (U 0 , T ) = iT P 3 (U 0 ) = 0 and that the effective scalar potential must be very small for the hierarchy assumption to be valid. The bound established by the hierarchy assumption is reflected on the range of values that τ can take by impossing an upper bound value depending on |∂ U W |. In other words we suggest that at least for this particular model, the distance conjecture is a consequence −manifested in the four-dimensional theory− of a particular selection of flux configurations which in turn enable the hierarchy assumption on moduli to be valid. Moreover, all different scales, (depending on τ ) show a hierarchy as expected in models as the flux-scaling scenario. In this context, it is possible to compute the scale Λ SW at which the effective theory is valid turning out to be established solely by |∂ U W | which only depend on the two non-geometric fluxes b * and b * .
Finding concrete examples with integer values for the non-geometric fluxes was impossible after a numerical computation of near three thousand different configurations, showing that by taking integer non-geometric fluxes implies that our model is inconsistent (there are no values for τ satisfying the bound constraints). Integrality of non-geometric fluxes is reflected in large values for |∂ UW |. Therefore, the only option to have a consistent effective theory is to have small values for |P 3 (U 0 )| which can be gathered by selecting fractional non-geometric fluxes.
According to our previous description, this consideration refers to the removal of some taken assumption in set A, i.e., based on their implications we suggest that incorporation of fractional non-geometric fluxes at this stage allows the model to fulfill all Swampland criteria making the effective theory to be safely in the Landscape. This is based on all the results obtained and enumerated in the introduction. Namely that a viable range for τ is acquired consistently with the Swampland distance conjecture and that all scales obey a hierarchy as expected in these kind of compactifications. Moreover, we can have a parametric control on both, the field distance and the hierarchy in a flux-scaling scenario. It is also interesting to stress out that due to tadpole constraints, the values for the string coupling shows a discrete pattern as suggested in [31].
Fractional fluxes can arise at the compactification scale due to the internal manifold topology. In our case could be a consequence of the backreaction of the internal metric by assuming T-duality. This issue, although very well known by the community, has been ignored in order to stabilize the internal volume and make estimations on the KK scale. Since the distance conjecture is deeply connected with the internal volume by having a runaway direction on τ , it is expected to have definite range in the τ field space. We also show how by fixing the scale Λ SW one can argue that fractional values for non-geometric fluxes are expected.
In summary we have presented what we consider is a reliable method to construct effective models based on non-geometric fluxes. In the way, we have elucidated the necessity to remove some assumptions, as the quantization of non-geometric fluxes, which allows to re-enter into the Landscape. If the Swampland criteria indeed divides the field space into two types of effective models, the requirements for some of them to be in the Landscape could mean a way to understand implications of a quantum theory of gravity in four-dimensional effective theories.

A Hierarchies
The extrema of the scalar potential can be recast in terms of the covariant derivatives of the superpotential. In the simpler case where only an U dependence of the superpotential exists, one obtains that the condition ∂ U V = 0 is equivalent to D U W = 0. When only a dependence on U, S is on the conditions ∂ U V = ∂ S V = 0 can be satisfied simultaneously if D U W = D S W = 0. There is however another solution, that is ∂ S V = 0 is satisfied by D U W = ∂ S D U W = 0; and ∂ U V = 0 is satisfied by D S W = ∂ U D S W = 0. When the three moduli are on the conditions ∂ U V = ∂ S V = ∂ T V = 0 can be satisfied simultaneously for D U W = D S W = D T W = 0. But there are other simple cases which we summarize in the table 1. Table 1. Particular extrema of the scalar potential in terms of superpotential covariant derivatives.

B Fixing the notation
We consider a compactification on a six-dimensional torus in the presence of NS-NS and RR three-form fluxes. The corresponding superpotential is given by W(U, S) = G 3 ∧ Ω, (B.1) where G 3 = F 3 − iSH 3 . In terms of the 3-form cohomology symplectic basis (α I , β I ), we have that with G 3 = F 3 − iSH 3 = g I α I − g I β I , where g I = f I − iSh I , The symplectic basis is given by In the considered model the complex structure is identical for the three tori T 2 and it is determined by the complex coordinate for each T 2 i , z i = x i +iU y i . Thus the (3,0) holomorphic form reads Ω = dz 1 ∧ dz 2 ∧ dz 3 , We have considered a unique U for the isotropic T 6 and use the following notation: from which the superpotential can be written as In terms of the NS-NS and RR fluxes, the superpotential reads where P i (U ) are cubic polynomials on U shown in expressions (2.2) and (2.3). Finally, for a compactification on an isotropic T 6 in the presence of O3-planes, the tadpole condition reads: where N O3 measures the contribution of an O3 − -plane to the internal D3-brane charge. In terms of the fluxes f and h the above expression reduces to

B.1 Superpotential with non-geometric fluxes
Now we shall turn on non-geometric fluxes, meaning that we are considering a superpotential of the form [60] W (U, S) = W(U, S) + 1 κ 2 (Q · J c ) ∧ Ω, (B.12) with the 3-form Q · J c = iT (b I α I − b I β I ). It is useful to rearrange all 24 non-geometric fluxes in the following matrices: with i, j = 1, 2, 3. With this notation, Eq. (B.15) recasts the form given in (3.2).
In terms of the matrices given in Eq. can be written as and similarly for Q · Q = 0. For the isotropic torus the above identities become while the tadpole condition on the non-geometric fluxes takes the form for the flux conditions (4.9) and the relations (4.16), meaning that seven branes are absent in our model.