Metastable vacua from torsion and machine learning

By implementing an error function on a Machine Learning algorithm we look for minimal conditions to construct stable Anti de Sitter and de Sitter vacua from dimensional type IIB String theory compactifcation on K\"ahler manifolds with torsion. This allows to have contributions to the scalar potential from the five-form flux and from D-branes wrapping torsional cycles, interpreted as non-BPS states. The former implies the possibility to construct stable AdS vacua while the later constitutes a mechanism to uplift AdS to dS vacua. Particularly we consider $\hat{D5}$ non-BPS states to uplift the stable AdS vacua to an (apparently) stable dS minimum. Both results -- the generation of an AdS vacuum and the corresponding uplifting to a dS one -- are restricted to certain type of configurations, specifically with the number of O3 orientifolds bounded from below by the number of D3-branes and fluxes. Under these conditions we report over 170 dS (classical) stable vacua. In all of them, the uplifted effective potential becomes very flat indicating the presence of possible source of instabilities. We comment about their relation with the Swampland Conjectures.


Introduction
The Swampland Program has received a lot of attention over the last few years. Its importance relies on the establishment of some criteria to separate effective quantum field theories −considered as consistent with Quantum Gravity, a.k.a. String Theory− from those which are not. The program focuses in different proposals commonly referred as Conjectures which appear to rule out some of the string model engineering constructions so far presented in the literature. Some of those conjectures are involved in our work, such as the instability of non-SUSY Anti de Sitter (AdS) vacua, the AdS scale separation and the Refined de Sitter conjecture, which in turn seem to be interconnected [1][2][3]).
The refined dS conjecture establishes that the minima of the scalar potential coming from the dimensional reduction of the low energy theory in string theory have to be AdS otherwise they are tachyonic or not consistent to Quantum Gravity, at least in the asymptotic regions of moduli space [4][5][6][7][8]. Even more restrictive, the AdS conjecture establishes that the scale of the lightest moduli is not parametrically separated from the AdS scale, and thus any attempt to uplift an AdS to a dS vacuum shall result in their destabilization 1 .
Recently it has been argued that the use of non-BPS states, classified by K-theory, shall be an interesting corner to evade these restrictions [10,11], unless the total K-theory charge must cancel as pointed out in [12] and related to the cobordism conjecture in [13]. The use of non-BPS states, typically constructed from a pair of stable branes and anti-branes in the presence of an orientifold plane, emulates the role played by non-perturbative contributions in KKLT scenarios by breaking the non-scale structure of the N = 1 superpotential and providing a nice mechanism to stabilize all the moduli. However, their inclusion is not suffice to guarantee the presence of apparently stable dS vacua but contributions to the effective scalar potential coming from the RR 5-form are necessary.
We are interested in two main aspects. First, in constructing a (meta)-stable dS vacuum by identifying the minimal set of ingredients the effective scalar potential must posses in the spirit of [14,15] and also to find possible compactification scenarios where such conditions might be present. Second, in case we can construct a classical stable dS vacuum we want to look for possible sources of instabilities which in turn can be taken as evidence (or not) of the realization of the above referred Swampland Conjectures .
In this work we consider a compactification on a Kähler manifold admiting torsion, upon which there is a contribution of the torsional part of F 5 to the scalar potential allowing to find AdS vacua (but not dS). For that to happen it is necessary that the number of orientifold fixed points be greater than the number of D3-branes such that their contribution to the tadpole be negative, i.e., N 3 < 0. Under this context it is then possible to wrap D5-branes on torsional 2-cycles which we claim are preciselyD5 non-BPS states and that contribute with a positive amount of energy such that uplift the AdS vacuum to a dS one.
As in the case of the AdS vacua, the realization of dS minima require some extra conditions, namely that there are fluxes in RR and NS-NS sectors supported in more than two 3-cycles and that the number of orientifold 3-planes has a lower bound given by where A H 3 , A F 3 and A 3 are the contributions −upon dimensional reduction− of 3-form fluxes and 3-dimensional sources as D3-branes and O3 − -planes, while N flux is the usual flux number entering into the D3-brane charge tadpole contribution.
These conditions were inferred after implementing a Machine Learning (ML) algorithm specifically designed to look for dS vacua. The use of ML algorithms and tools has been proved to be prolific (and a more systematic way) to explore the vacua in string theory compactifications (see for instance [16][17][18][19][20][21][22][23][24][25]). For that we implemented a hybrid algorithm to explore the minima of a scalar potential of the form 2 V eff = V eff (H 3 , F 3 , F 5 , D5) subject to the constraints of 1) having a positive value at the minimum, 2) zero value of its derivative with respect to each of the moduli, 3) positive definiteness of the mass matrix, and 4) positiveness of the contribution of the D5-brane. In the context of ML, these restrictions can be implemented through an objective function written as where each of the error i contributions takes into account every single restriction above mentioned with the α parameter a real value giving a weight to each error contribution. In the present work we employ the hybrid algorithm including the Simulated Annealing (SA) as well as the Gradient Descent (GD). The SA algorithm is a heuristic method for solving optimization problems which, inspired by the annealing procedure of metal working, is able to look for an approximate solution to the optimization problem. The GD algorithm on the other hand is a second order iterative optimization algorithm designed to find local minima provided that the first derivative is known. Thus, at a first step, the SA algorithm shall look for interesting points in the error function whereas the GD shall improve the solution guaranteeing the zero value of the first derivative of the scalar potential. We describe in detail these algorithms in Appendix A.
Our work is then organized as follows: In section 2 we present the most usual conditions for a type IIB compactification and specify the notation we use along the paper. In section 3 we show that it is possible to construct AdS vacua by compactification of type IIB string Theory on a Kähler manifold with torsion, such that the RR five-form has a torsional contribution to the effective scalar. For that we implement a ML algorithm through the presence of an error function which allows to easily find a large number of stable and unstable vacua. In this case we report 389 different AdS vacua which existence relies upon the requirement that the number of D3-branes be less than one-half of the number of orientifold O3 − -planes. However no dS vacua were found under these conditions. In section 4, once we take a compactification on a manifold with torsion, we also consider D5-branes wrapping torsional 2-cycles while fulfilling the aforementioned conditions on fluxes and the orientifold bound. Extra assumptions were taken, such as the non existence of torsional components of all 3-form fluxes. For this case, we report over 170 different dS stable vacua. In section 5 we discuss the conditions upon which the AdS vacua can be lifted to dS ones and comment about the implications with respect to the Swampland Conjectures. In section 6 we present our conclusions, while in the Appendix we describe some useful technical information in relation with the Machine Learning algorithm to be implemented in our search, particularly about the incorporation of the above mentioned two algorithms: the Simulated Annealing and the Conjugate Gradient.

Contribution to the scalar potential
Let us review the standard dimensional reduction procedure to construct the effective scalar potential. Consider the type IIB superstring compactified on a manifold X 6 in the presence of 3-form fluxes and 3-dimensional local sources. We are not including 7-branes or orientifold 7-planes. As usual, the action for the massless modes in the string frame is where in terms of the string length l s , T 3 is the D3-brane tension, N 3 = N D3 − 1 2 N O3 counts the number of D3-branes minus the number of orientifold planes O3 − with µ 3 = T 3 = 2π l 4 s . We consider the DBI action at leading order in α for D3-branes and O3 − -planes along the extended coordinates, where the RR fluxes arê Thus, the action S F 5 (before self-duality is imposed) can be written as Due to the action of the orientifold planes O3 − , the RR and NS-NS potentials C 2 and B 2 are projected out and the equations of motion from δS/δC 4 = 0 give us the tadpole condition for the 3-dimensional sources (2.11) Therefore, the contribution from S F 5 + S CS + S 3 to effective the scalar potential −in a compactification on a CY manifold− vanishes. As we shall see we are going to depart from a CY compactification into a more general setup such that S F 5 does have a contribution.
In order to construct the effective scalar potential V eff , we specify the ten-dimensional metric as where e −2Ω is the conformal factor fixed as to change into the Einstein frame, with V 6 = d 6 y √ h 6 . Notice we are not taking into account warping effects on the internal metric.
In terms of the axionic moduli fields τ and s are given by (2.14) so, the contributions for the action terms S G 3 and S DBI are given by 16) where A F 3 , A F 3 and A 3 are the corresponding contributions not depending on τ and s where S = C 0 + is, and T = C 4 + iτ. (2.17) On the above we have assumed that complex structure moduli z i are fixed through 3-form fluxes, by D z i W = 0, where as usual but D S W = 0. Therefore SUSY is broken at least by the axio-dilaton modulo S, and the fluxes we are turning on, have not (1, 2)-components. Together with the Kähler potential of the form the flux contribution to the scalar potential reduces to Comparing with expression (2.15), .
(2.22) As known, by exploring different values for A F 3 , A H 3 and A 3 we find that no stable vacuum is obtained. More ingredients are required.

Stable non SUSY AdS vacua from torsion
As suggested in the literature (see [14,15] and [27][28][29][30][31][32][33][34][35][36]), it is possible to find stable vacua by turning on different contributions to the scalar potential. Here we are interested in a non-vanishing contribution from S F 5 to V eff . For that, we shall take into account the presence of torsion in the internal manifold X 6 which, as we shall argue, naturally comes into play in the presence of orientifold planes [37,38]. This implies that the Kähler 2-form J 2 is no longer closed, i.e., dJ 2 = 0 pointing out the necessity to compactify on generalized CY manifolds. By using the ML algorithm described in the Appendix A, we find that AdS stable vacua are obtained under some specific conditions we shall describe in detail.

Effective scalar potential from Torsion
Let us start by writing the action component S F 5 in (2.10) as where As said, in generic compactifications on X 6 with orientifold planes O3 − , 2-forms are divided on odd or even according to the orientifold action on them [39]. Since 2-form RR and NS-NS potentials are odd under an O3 − action, and the fluxes F 3 , H 3 are even and it follows that Therefore, for a generic CY manifold, ω 5 does not contribute to V eff . Also notice that in the presence of orientifold O3 − -planes, the RR potential C 6 is projected out and it is not possible to have stable BPS D5-branes. The effective 4-dimensional scalar potential only receives contributions from the rest of the terms in the action S and from the Dirac-Born-Infeld action of extended objects wrapping internal cycles on X 6 , as D3-branes and orientifold planes O3 − .
However, in the presence of orientifold planes, it is natural and expected to have torsional cycles. For instance, in a IIB toroidal orientifold, the quotient space T 6 /Z 2 contains torsional cycles of different dimension (dual to torsional fluxes), meaning that there are cycles that after wrapping them a certain number of times, one ends up with a subspace of T 6 with boundary. Since we are considering the presence of orientifold planes, we shall assume the existence of torsional cycles in generic Kähler manifolds. Under this context we shall study whether or not ω 5 contributes to V eff via torsional cycles.
The pth-cohomology group of a six-dimensional Kähler manifold is written as where b p is the Betti number for H p (X 6 , Z) and k i ∈ Z. Let us consider the case for p = 3. A 3-form in the torsional part can be decomposed as with i = 1, . . . , n according to (3.4) and λ i ∈ Z. In the case in which the set of integers λ i has a greatest common divisor (gcd) κ, there exists a non-closed 2-formω 2 such that dω 2 = κπ tor 3 , i.e., π tor 3 ∈ Z k . The set of such 2-forms is denotedΩ 2 (X 6 ). If λ i = κ i k i only for some i, then there existŝ ω i ∈Ω 2 i (X 6 , Z) such that dω i = k i π tor 3,i . In this scenario, generic RR and NS-NS potentials are given by where ω a ∈ H 2 − (X 6 , Z),ω i ∈Ω 2 (X 6 , Z) with a = 1 . . . h 1,1 − (X 6 ) and i = 1, . . . n. The presence of 2-formsω i implies that the Kähler form J 2 can also be written as from which dJ 2 = k it i π tor 3,i . Hence fort i = τ i /k i , dJ 2 is non trivial in H 3 (X 6 , Z) and X 6 is not a CY manifold but at least a Kähler manifold modulo k i .
If now we restrict the compactification over a Kähler manifold with torsion as above, the contribution from S F 5 is not longer zero, but with A(y) the warping factor in Eq.(2.12). Therefore, the contribution of F 5 -form to the scalar potential, in the Einstein frame, is given by where A 5 = A mod k i for some A.

Conditions for finding stable AdS vacua
The above contribution to V eff from S F 5 together with the contributions from 3-form fluxes, D3-branes and O3 − -planes, lead us to a scalar potential of the form which actually has some stable AdS minima if there is at least one negative contribution from the above terms. However, since the flux contribution A G 3 is positive definite 3 and A F 5 is defined modulo an integer, the only option left is that, from the contribution of 3-dimensional sources, N 3 must be negative.
By restricting the flux configurations and local sources to satisfy that N 3 < 0, the number of O3 − -planes must be larger than the number of D3-branes, implying that at some points in the internal space, there must be isolated orientifold planes, or in other words that there are no D3-branes of top of some of the O3 − -planes. This follows from the usual assumption that orientifold planes are immovable and from the fact that there is an attraction between D3-branes and O3 − -planes due to the RR D3-brane charge they carry. For instance, the most simple configuration involving the presence of D3-branes with N 3 < 0 is to have 4 orientifold fixed points and a single D3-brane sitting at one of those points. In such case, N 3 = −1 (see Figure 1 for a schematic representation of this configuration).  Under these conditions we implemented our ML algorithm described in Appendix A. With it, we were able to find 389 different stable AdS vacua. However, in spite of designing our algorithm such that finding dS vacua was favored over AdS, no dS one was found. Our results are shown in Figure 2 where all found vacua, stable or not, are represented by black squares. × Figure 2. Plot of the critical points found by the hybrid algorithm. The black squares correspond to the cases where F 5 contributions were taken into account without non-BPS states, whereas the blue crosses consider the presence ofD5 non-BPS states. On the second image, we present a zoom of the stable cases.

Stable dS vacua from Non-BPS states
The presence of torsion opens up the possibility to consider wrapping D-branes on torsional cycles. The existence of torsional cycles follows from the dual maps between homology and cohomology, where This last assertion means that the homology group H 2 (X 6 , R) also has a torsion component, i.e., Σ i,tor 2 ∈ Tor H 2 (X 6 , Z) andΠ i 3 ∈Ω 3 (X 6 , Z). It follows then that Tor H 2 (X 6 , Z) ∼ Tor H 3 (X 6 , Z). We shall follow the argument in which these states −D-branes wrapped on torsional cycles− are in fact related to the well-known non-BPS states constructed from K-theory [40].
The existence of non-BPS states in the presence of an orientifold plane O3 − can be inferred by applying T-duality on the corresponding coordinates on a torus compactification of Type I string theory, which actually have non-BPS branes asD7,D8,D0 andD(−1). Hence, by taking for instance a non-BPSD7-brane spanned on 4 coordinates on T 6 immersed in an O9 − -plane and applying T-duality on the compact coordinates, we get an extended O3 − -plane and a 5-brane wrapping a 2dimensional space in the covering space. We expect this object to carry a topological Z 2 charge as its T-dual partner. Indeed, by computing the 2nd-homology group of T 6 /Z 2 we see that there are torsional 2-cycles. Wrapping D5-branes of type IIB theory on such cycles seem to be the way to construct the aforementioned non-BPS states. Moreover, by computing the corresponding T-dual Ktheory group one sees that stable non-BPS states are present, carrying discrete topological charge Z 2 with three extended coordinates while the other are wrapped on the compact space.
For a more general compactification, one must compute the K-theory groups of intersecting sources, i.e., of configurations of branes intersecting orientifold planes wrapping cycles on a compact manifold. This is indeed a difficult task. However, ignoring the compact component of the space, it is possible to classify intersecting branes with orientifolds by the use of the Kasparov KK-theory [41,42]. Since the formulation is quite technical and it is beyond the scope of this work 4 , we just present the KK-theory group which classifies 5-branes fully intersecting an O3 − -plane, i.e., with 2 transversal coordinates to the orientifold plane and its relation to orthogonal K-theory group. This is: as expected.
Based on these results we are taking as valid the construction of stable non-BPS states by wrapping D-branes on torsional cycles of a Kähler manifold X 6 . In particular, we can construct a non-BPŜ D5-brane by wrapping a D5-brane on a torsional 2-cycle Σ tor 2 ∈ H tor 2 (X 6 , Z), where Σ tor 2 is the cycle where the 2-formω 2 is supported as in Eq.(4.1).
Summarizing, a compactification on a Kähler manifold X 6 with torsional components in (co)homology, leads us to the possibility to include D-branes wrapping torsional cycles. Here we shall 4 The KK-theory group classifying Dd-branes on top of an Op − -plane, with p = 3 mod 4 and d > p is given by [42]  consider the contribution to the effective scalar potential from non-BPSD5-branes. However, before that we must discuss possible sources of instability on a configuration constructed with fluxes, D3branes, O3 − -planes and non-BPS states.

Consistency by adding non-BPSD5-branes
As it is known [40], the non-BPS braneD7 in type I theory can be constructed by a pair of a D7 andD7-branes, where the tachyon on the open sector string connecting the two branes is projected out by the orientifold O9 − . However, since in type I theory there are 32 D9-branes, there is also a tachyon from the open string between D9-branes and D7-branes, making the non-BPSD7-brane to be unstable [43].
In a T-dual version, upon compactification on a six-dimensional torus, the above configuration is mapped into D3-branes and O3 − -planes sitting at different points on T 6 and D5-branes wrapping torsional 2-cycles on the compact space, corresponding to the non-BPS statesD5. Therefore, by Tduality, it is expected that in a given fixed point in the internal space, aD5-brane coinciding with at least one D3-brane, would be unstable to decay into a field configuration while preserving its topological charge Z 2 . This instability is not present (at least locally) if at the given fixed point, there are not D3-branes, a configuration we can have if there are more orientifolds than D3-branes, i.e., if In order to cancel the D3-brane charge tadpole, we then require a positive contribution from fluxes. These two characteristics, N 3 < 0 and N flux > 0 are essential to guarantee the stability of adding non-BPSD5-branes. Notice that N 3 < 0 is one of the conditions to assure the existence of stable AdS vacua without adding non-BPS states.
Under the above circumstances, we shall take a D5-brane and wrap it on a torsional 2-cycle Σ tor 2 ∈ Tor H 2 (X 6 , Z). Following [44], we argue that such a state is classified by the corresponding K-theory group on X 6 . Also, we shall consider the contribution of this non-BPSD5-brane to the effective scalar from the DBI term. However, it is important to notice that its contribution must be measured as mod 2 since a pair of non-BPS branes with topological charge Z 2 anihillate to each other. This means that if the total discrete charge vanishes, the effective contribution from non-BPS branes is null [10,13]. Another important fact we must have in mind is that we are ignoring torsional components for 3-form fluxes, although there is no restriction for their presence 5 .
Hence, the effective contribution of a non-BPS braneD5 at leading order in α is given by the DBI action, where g 6 is the determinant of the induced metric on theD5-brane worldvolume. Therefore, the 5 In [11] some consequences of turning on torsion components of fluxes are commented.
corresponding effective scalar potential in the Einstein frame reads where 2nAD 5 = 0 for n ∈ Z.

Stable dS vacua with non-BPS states
In order to look for dS minima we shall employ an hybrid method which consists in applying a stochastic method known as Simulated Snnealing followed by the gradient descent algorithm (see Appendix A). The effective scalar potential constructed by contributions from 3-form fluxes, 3dimensional sources, a torsional component of F 5 and non-BPSD5-branes is As discussed in [15] (see also [14]), it is expected that this anzats evades the no-go theorems and increases the possibility to find some stable dS vacua.
In Figure 2 it is shown by blue crosses, the critical points found by the above-mentioned algorithm. Notice the presence of many stable dS vacua. In Table 4.2 we present the explicit values of the scalar potential contributions for some of these vacua.

Uplifting conditions by non-BPS states
In this section we are interested in discussing the uplifiting of AdS stable vacua to dS by the presence of non-BPS states as theD5-branes. As previously observed, a dimensional reduction in the presence of 3-form fluxes H 3 and F 3 , as well as 3-dimensional sources as D3-branes and O3 − -planes together with a torsional F 5 form, leads us to the possibility to construct AdS stable vacua. For A D5 = 0, the minimum for V eff is located at Notice that in the case we are turning on a single flux G 3 , meaning that we are considering a contribution to the superpotential along one single period, ∆ reduces to zero due to the tadpole cancellation. Therefore, it is necessary to consider more than one flux in order to uplift the AdS vacua while keeping |A 3 N 3 | > 2(A H 3 A F 3 ) 1/2 such that τ 0 > 0. Therefore we require that two specific conditions must be taken: 1. W = G 3 ∧ Ω must be constructed from more than just one period.
We shall restrict the rest of our analysis to such a case.
The minima of the AdS can be written in function of the vacuum expectation value (vev) of the Kähler modulus as thus, the larger τ 0 , the smaller value for the AdS vacua, which is compatible with the KKLT scenario.
The eigenvalues can be written in terms of the vev's as and we see that for large values of τ 0 , the smallest eigenvalue is always in the τ direction. Now, to uplift from stable AdS to dS vacua it is necessary to add energy associated to the non-BPS statesD5 as in Eq.(4.6), which change the vevs of the moduli shifting its numerical values to greater values. In this case the Kähler modulus modify to which in the limit of AD 5 1 can be written as Notice from this and from Eq.(5.1) that for ∆ > 0 this branch of solution takes real values. In this context, one also can express the effective potential at leading terms in AD 5 as where the uplifting from AdS to dS depends on how deep is the AdS vacuum.
However it is important to analyze whether the uplifting would be stable or not. For that we shall study under which conditions there are tachyons. Let us start by establishing the required stability criteria for the AdS vacua. Since we are interested only in their presence, we shall take the the mass matrix as with i, j = s, τ . The eigenvalues λ AdS are given by AdS . According to the Silverster's criterium, a stable minimum exists always that tr M 2 AdS > 0 and α be real. Notice that large values for the eigenvalues λ AdS indicate that it is difficult to destabilize the minimum. On the contrary, small values of λ AdS correspond to very flat potentials from which it is easy to escape from. Following this line of reasoning, we want to show that by adding non-BPS statesD5 the eigenvalues related to an AdS vacuum become smaller.
For that, let us consider adding the contribution from non-BPS states VD 5 , such that One realizes that the eigenvalues for each of the moduli decreases as we add the AD 5 term. To clearly show this, lets us split tr M 2 and det M 2 in terms of the contributions of AD 5 as where f (AD 5 ) and g(AD 5 ) are positive definite homogeneous functions of degree 1 on AD 5 . If the added potential is of the form with n, m > 0, which indeed is our case. Thus, by adding the AD 5 terms, there is a contribution δλ to the eigenvalues λ AdS as λ = λ AdS + δλ. (5.12) In this context, we say that if δλ < 0, the eigenvalues of the mass matrix decrease due to the contribution of the non-BPS states. Indeed, the change of the eigenvalues can be written explicitly as (5.14) Since f and g are positive functions and α < tr M 2 , then γ is positive definite. In consequence the term 1 − √ 1 + γ shall be negative. This in general implies that Finally, putting f and α in terms of the determinant and trace of the mass matrix we find that and δλ < 0.
Adding non-BPS states drives two important features in the effective potential. In one hand, uplifts the value of V AdS to a dS one, but in the other hand, since the contribution to the energy at the minimum is positive, the scalar potential becomes very flat increasing the probabilities for this vacuum to be destabilized. We show this behavior, for one case, in Figure 3

Comments about some Swampland conjectures
We have described a way to construct a dS vacuum by adding the contribution to the scalar potential from a non-BPSD5-brane to a non-SUSY AdS vacuum (D S W = 0). However, as recently studied, there are some constraints around the construction of both states. First of all, it has been argued that a non-supersymmetric AdS vacuum is at most metastable in the context of the Swampland program [45,46]. Second of all, it is expected a constraint on the AdS scale with respect to the lightest moduli mass, and finally, in case of uplifting the non-SUSY vacuum to a dS one, the final vacuum is at most, metastable. Let us comment about these three points and how they are manifested in our setup.
As mentioned, one way to assure the construction of an AdS vacuum by considering the contribution of F 5 in a manifold with torsion, implies the stabilization of the complex structure by D U W = 0 while keeping D S W = 0. Therefore, the AdS vacuum is non-SUSY. According to the Swampland conjectures, such an AdS vacuum must be at most metastable. In our case, the source for instabilities could come from two places: first, from our assumption of not considering torsional components of 3-form fluxes, which usually drives some topological transitions as pointed out in [11]. Second, since the contribution from F 5 is based on the existence of torsional cycles, it is possible that the total discrete charge must vanish following the recent proposal about having zero global charges in Quantum Gravity and its relation to K-theory by cobordisms as proposed in [13]. We believe that both aspects are in fact related.
The second point concerns the AdS scale which it is also conjectured to satisfy a relation of the form where c ∼ 1 and R AdS ∼ |V 0 | 1/2 in order to keep a robust realization of a dS vaccum. Recent studies argue that effective models which support such a parametric hierarchy are in fact in the Swampland. Again, in our case the above two factors can be expressed in terms of each of the contributions to the scalar potential, for which we obtain that As all the constants A i for i = {H 3 , F 3 , D3, O3} are of the same order, the energy added by F 5 , for a Z k discrete torsion, vanishes up to a multiple of k. Hence, unless k is too large, the quotient (5.18) is slightly larger than order 1, and by taking k = 2, m mod R AdS c .
In this context, it is possible to add energy for the uplifting in such a way we stay in a region where stability can be (parametrically) controlled. Indeed, in our model, the AdS vacua do not contain tachyons neither in the axio-dilaton nor along Kähler directions. Besides, the scale of the AdS is smaller that the energy coming from the lightest moduli violating the AdS conjecture. Thus, by adding a non-BPS states which its energy contribution scales a s −1/2 τ −5/2 generates a flattering effect accordingly.
Finally, according to the Swampland conjectures, a source of instabilities is expected to affect the uplifted dS vacuum. They could come from the fact that the pairD5 − D3 (dual to theD7 -D9) is unstable [47,48] and although a decay into a final state does not dilute the discrete charge, it is canceled out by requiring a vanishing K-theory charge [12,43]. However, in our caseD5-branes come from D5-branes wrapping torsional 2-cycles around an O3 − -plane with no D3-branes. Hence, at least locally, there are no instabilities at such points. Thus, the non-BPS states are stable and the only decay channel is through tunneling leading to the decompactification limit [49] probably described by a topological transition driven by torsional 3-form fluxes, as suggested in [11]. A detail study about this process is reserved for a future work.

Conclusions and Final comments
As expected, the incorporation of F 5 fluxes by contribution to the effective scalar potential V eff seems to be fundamental to find classical stable dS vacua in an orientifolded flux compactification of string theory. However, since in a Calabi-Yau manifold F 5 does not contribute to V eff , we need to consider other internal manifolds, such as the considered in [50,51].
As shown in [38] a Kähler manifold admitting torsion is a suitable example in which F 5 contributes to V eff . Moreover, these type of manifolds allow to wrap D5-branes on torsional cycles, by which one can construct non-BPS states actually classified by K-theory, with a non-zero contribution to V eff .
Under these circumstances and by implementing a novel ML algorithm we were able to find more than 200 dS critical points for V eff out of which 170 are stable.
We also find that there are certain specific conditions that our configurations of branes and fluxes must fulfill in order to generate a stable dS vacuum by uplifting an AdS one. First, to obtain a stable AdS it is necessary to turn on the torsional part of F 5 and to have a configuration of branes and orientifolds such that the number of O3-planes or fixed points are larger than the number of D3branes implying that N f lux > 0. Second, for these vacua to be uplifted to dS by incorporating the non-BPS statesD5 we ougth to have that: 1. The RR and NS-NS 3-form fluxes are supported in more than a single cycle, where A H 3 and A F 3 are the contributions to V eff (independent of moduli) from the fluxes H 3 and F 3 , while N 3 A 3 is the corresponding from 3-dimensional sources, with N 3 = N D3 − 1 2 N O3 . Under these conditions, it is possible to obtain that all mass eigenvalues are positive under the uplifting by non-BPS states. We observe, that 1. Getting a small positive value for V min seems to be a natural consequence by uplifting AdS vacua with a small deep. There are two consequences of this: the resulting uplifted potential is very flat while the probability for a destabilization of the dS vacua increases since at the limit for large volume, the potential goes to zero, indicating the presence of a barrier potential between the dS local vacuum and the the run away region for the Kähler moduli.
2. We believe that this possibility of the scalar potential to become unstable could be generated by extra mechanisms or topological transitions driven by torsional components on the 3-form fluxes as suggested in [11] and in consequence, establishing a rich scenario where Swampland conjectures can be tasted.

Acknowledgments
We

A Machine Learning algorithm
In this section we want to show with some detail the characteristics of our Machine Learning algorithms and how they can help us to find stable vacua in a flux string compactification. We make use of two specific algorithms called the Simulated Annealing (SA) and the Conjugate Gradient (CG).

A.1 Simulated Annealing
The SA algorithm is one of the most preferred heuristic methods for solving the optimization problems. SA was introduced by inspiring the annealing procedure of the metal working. In general manner, SA algorithm adopts an iterative movement according to a variable parameter which imitates the annealing transaction of the metals. Thus, by taking the objective function as "Error", the SA takes the probability distribution with support ∆Error used to replace a new solution as for ∆Error the change in the error function which depends on an arbitrary number of parameters such as moduli vev's and numerical coefficients that depends on fluxes as well as the non-BPS states, and i the current iteration. Thus as ∆Error or the iteration i becomes large the probability to replace a new solution decreases. The SA takes an initial value φ 1 and check if if true, φ best is replaced by φ 1 , otherwise it is replaced with a probability P [∆Error]. A schematic picture of the SA algorithm is shown in Figure A.1.

A.2 Conjugate gradient
Conjugate gradient (CG) is a second-order iterative optimization algorithm designed to find a local minimum provided that the first derivative is known (other alternatives which give us a similar result is the Powells algorithm). The main idea consist in to take repeated steps in conjugate directions of the scalar potential at a given point of the moduli space, since this is the direction of steepest  . In a schematic view, the SA algorithm starts at an arbitrary point in the parameter space of the Error function, then it lets to find in a random manner a best solution leading to a local minima. Once the local minima is reached, the algorithm shall find for alternative paths that finds a better minima by perturbing the solution with a probability P [∆Error] avoiding to get stuck in a local minima.
descent. Conversely, stepping in the direction of the gradient which leads to a local minima of the scalar potential. Thus, if the Error function near to the global minima is approximated by the residual defined as implies that ∂ a Error(φ i+1 ) = −r i a vanishes at an extremum. Now, in order to move to the minima of the error function, the changes in the gradient have to follow the direction along This implies that the directions ∂ a Error(φ i ) and ∂ b Error(φ i ) have to be conjugated. Thus, the CG moves through a conjugate direction leading to a local minima for convex problems. By starting with an initial vector φ 0 a the conjugate gradient method find two sequences of vectors as where A ab ∂ a Error i ∂ b Error j = 0 for j < i, s is an small parameters and is chosen in order to guarantee that the gradients in successive iteration steeps are conjugated.

A.3 Error functions
The objetive function can be written as for α i ∈ R and in a range of 0, 10 4 6 . This parameters is employed to give a penalty to regions on the moduli space that are not of interest. For instance, if we are looking for dS vacua, the error induced by finding a AdS is weighted by this factor, forcing the algorithm to look for another direction. Thus, in the present work we are interested in finding dS vacua free of negative mass square moduli. Thus our penalty functions shall require 1) to avoid tachyons, 2) to avoid AdS vacua, 3) require that A D5 to be positive. In order to penalize this constraints the following errors are employed • As we are looking for extrema of the scalar potential, the first error contribution is related to the derivative of the scalar potential. This is applied as • The second contribution of the errors is defined by proposing a continuous function that penalize the error function each time that the parameter space is in a AdS vacua. This is, • The third contribution to the error function is proposed in order to avoid tachyons in the spectrum. For the simple case of two real moduli, the positive mass square moduli require that tr m 2 ij > 0 as well as det m 2 − 1 4 (tr m) 2 > 0. Thus the third contribution of the error is defined as as well as • The fifth contribution to the error is associated to a penalization of the error function each time that the algorithm moves into the region of A D5 < 0. This requirement is implemented in the algorithm as (A.13) 6 Notice that this way to implement penalty functions is known as regularization in machine learning and is equivalent to implement Lagrange multipliers in an approximate manner, this is inside the bounds of the convergence criteria.

B More generic vacua
The implementation of our ML algorithms allows to look for stable vacua in more generic conditions. Here we present numerical results by considering extra terms in the scalar potential without wondering wether they can be constructed or not in a consistent scenario. Specifically, we incorporate the contributions to the scalar potential from O5 and O7 planes and the internal curvature R 6 besides the usual 3-form fluxes, the O3-plane , D3-branes and the non-BPSD5-branes. Our results are shown in Figure 5 where we have plotted each vacua in function of the energy value at the extreme point and the value of the minimal mass eigenvalue. We observe that the majority of the vacua are unstable but some of them are actually dS and stable.
All the cases explored in this landscape contain a contribution of the curvature R 6 of the internal space. In order to check the landscape of critical points we employ different configurations with a different content of fluxes and O-planes. Some particular comments for each case follows: • For the case with F 3 , H 3 , O3 and O5, the algorithm is able to find stable dS minima. However, almost all the critical points are unstable.
• For the case of F 3 , H 3 , O5 and O7, the algorithm was able to find a few stable dS minima.
• For the case of F 3 , H 3 , O3 and O5 the algorithm was not able to find any dS minima.
• For the case of F 3 , H 3 , F 5 , O3 andD5 the algorithm was able to find several dS minima. In particular, it is observed an abundance of dS superior to all the other cases. Besides, as theD5 contribution is removed (black squares), the algorithm was not able to find any dS minima. This suggest that non-BPS states play an important role in stabilizing the vacua.
Finally and just for sake of comparison, we want to show that the implementation of a penalty constraints in the ML algorithms really impacts on the number of stable vacua we find. Let us look for critical point with the same algorithm and by considering the same content the fluxes as in the body of the paper, i.e., F 3 , H 3 , and F 5 , as well as O3-planes and non-BPSD5-branes (no curvature term). In this case we realize that • As we remove the constraints the algorithm finds a lot of critical points but just 6 stable dS against 529 stable AdS. This case is similar to the one obtained by employing the GA+NN classification of our previous work.
• As we implement the penalty functions, the algorithm is able to find 203 stable dS and 170 stable AdS. This is shown in Figure 6. Figure 5. Landscape found by the hybrid method. The red points are found by using a scalar potential with O3, the blue dots are the vacua found by using O7, the green circles are the critical points found by using O5, the cyan circles are found by employing RR F 5 fluxes andD5 and the black squares are found by using RR F 5 fluxes but notD5.