Exceptional Flux Compactifications

We consider type II (non-)geometric flux backgrounds in the absence of brane sources, and construct their explicit embedding into maximal gauged D=4 supergravity. This enables one to investigate the critical points, mass spectra and gauge groups of such backgrounds. We focus on a class of type IIA geometric vacua and find a novel, non-supersymmetric and stable AdS vacuum in maximal supergravity with a non-semisimple gauge group. Our construction relies on a non-trivial mapping between SL(2) x SO(6,6) fluxes, SU(8) mass spectra and gaugings of E7(7) subgroups.


Introduction
In the last decade a lot of progress has been made in understanding flux compactifications in type II string theory [1]. The introduction of background fluxes is needed in order to perturbatively generate a potential for the moduli fields and stabilise them into a minimum.
Generically, a given flux background contains a number of local sources such as branes and orientifold planes that can break different amounts of supersymmetry, thus yielding an effective four-dimensional (4D) description with lower supersymmetry.
Flux backgrounds compatible with minimal supersymmetry have received a lot of attention in the literature, with a particular focus on the mechanism of inducing an effective superpotential from fluxes [2], e.g. in the context of the so-called ST U -models in four dimensions [3][4][5]. However, the task of finding de Sitter (dS) solutions for cosmological purposes in this set-up turns out to be quite hard; the only known dS solutions in geometric backgrounds are presented in refs [6]. The situation gets dramatically worse if one specialises to backgrounds preserving extended supersymmetry; in particular, in N = 4 [7,8], geometric fluxes have been found to be insufficient to obtain dS solutions [9,10].
A crucial ingredient for dS extrema is therefore given by non-geometric fluxes [11], whose existence was first conjectured in order for the low energy effective theory to be duality covariant. This duality is correctly encoded in the global symmetry of the underlying gauged supergravity in four dimensions [12]. In this sense, T-duality singles out the important role of half-maximal supergravities, whereas, in order to supplement it with non-perturbative dualities to generate the full U-duality group, one has to consider maximal supergravity.
Concentrating in particular on T-duality, different ways have been investigated in the literature in order to implement T-duality covariance at a more fundamental level in order to gain a better understanding of how non-geometric fluxes change the compactification prescription, thus appearing in the effective theory [13]. One direction to follow is generalised geometry [14], in which the internal manifold is given a particular bundle structure in which the gauge fields now span the full O (6,6) group. Another possibility is that of doubling the internal coordinates [15,16] by supplementing them with the corresponding duals to winding modes and viewing a non-geometric flux background as something created by means of a twisted double torus compactification [17].
Recently, this second approach has been further developed into the so-called Double Field Theory (DFT) [18], which aims to promote T-duality to a fundamental symmetry even independently of whether spacetime directions are compact or not [19]. This theory in 10+10 dimensions is formulated in terms of a generalised metric, whose action can be constructed to be fully O (10,10) invariant. Since it has been shown how to obtain gaugings of N = 4 supergravity from DFT [20], this might provide a higher-dimensional origin for non-geometric flux backgrounds, even though the concrete construction leading to the most general background still needs to be accomplished.
In a parallel development, progress has also been made on classifying the landscape of such theories. In ref. [10], an exhaustive analysis of the landscape of isotropic type II geometric vacua has been carried out within the context of half-maximal gauged supergravity. By making use of the dictionary between fluxes and embedding tensor components, the most general geometric flux background in type IIA and IIB has been studied, all the vacua found and the underlying gaugings identified. The crucial ingredient to obtain these results turned out to be the fact of dealing with sets of embedding tensor components which were invariant under non-compact duality transformations. This allows one to restrict the search for critical points to the origin without loss of generality, where all the equations of motion for the scalars simply reduce to quadratic conditions in the fluxes, which can be treated algebraically.
The set of geometric vacua in type IIB, where only gauge fluxes are allowed, turned out to only contain the Minkowski solutions found in ref. [2]. Much more interesting is the vacua structure of type IIA N = 4 compactifications, where metric flux is allowed as well as gauge fluxes. There turn out to be two inequivalent theories (related by a Z 2 ) with gauge group ISO(3) U(1) 6 , having 4 inequivalent Anti-de Sitter (AdS) critical points each.
Surprisingly, two of these break all supersymmetry and nevertheless has non-negative masses, indicating perturbative stability. Furthermore, all critical points of this theory turned out to be compatible with the absence of supersymmetry-breaking sources, hence admitting an uplift to maximal supergravity. Again, we would like to stress that these solutions were obtained by moving to the origin all the SL(2) × SO (6,6) scalars; for this procedure to be fully general, our set of fluxes needed to be invariant under non-compact S-and T-duality transformations. Indeed this is the case for geometric fluxes in these theories.
The combination of these recent developments makes it interesting to further investigate the structure of extended gauged supergravities in order to better understand which role T-and U-dualities play in the context of flux compactifications. The embedding of half-maximal into maximal supergravity [21,22] allows one to study flux backgrounds that preserve maximal supersymmetry. An interesting fact is that the completion of half-maximal supergravity deformations to maximal is given by objects which behave as spinors under Tduality. Following the generalised geometry approach, a further extension has been proposed called exceptional generalised geometry [23,24] in which the internal manifold gains more structure by including R-R gauge fields which extend O(6, 6) to E 7 (7) .
From a purely supergravity point of view, a number of critical points of maximal theories have been found. In particular, the SO(8)-gauged theory has been studied in detail and a classification of AdS critical points based on their residual symmetry group G has been carried out [25]. These critical points correspond to configurations in which only G-invariant scalars can acquire a vacuum expectation value and hence they are to be found as solutions of the G-truncated theory, in which only the G-invariant sector of the theory is retained.
Amongst the solutions with SO(3) × SO(3) invariance, there is the first example of a nonsupersymmetric and nevertheless stable vacuum in a theory of gauged supergravity with the maximal amount of supersymmetry [26]. Note that this critical point relies on the Breitenlohner-Freedman bound to be perturbatively stable. More recently, this classification has been extended with a number of critical points with smaller or trivial invariance groups, which have been obtained with a numerical procedure [27].
Moving beyond SO (8), analytic continuations such as SO(p, q) and their contracted versions have been investigated for critical points [28]. Moreover, by restricting to a subset of all embedding tensor irrep's and by applying the method developed in ref. [10], critical points were found in theories with non-semisimple gaugings [29]. In terms of the Levi decomposition these gauge groups can be written as the semi-direct product of a semisimple group with an Abelian ideal. In this case, the subset of embedding tensor irrep's is not closed under non-compact duality transformations. Therefore one cannot employ all such transformations; in other words, one has to restrict to a submanifold of the full moduli space.
The aforementioned results provide a strong indication that the issue of stability without supersymmetry in extended supergravity is much more involved than what was generally expected in the past. In particular, the presence of non-supersymmetric yet stable Antide Sitter critical points came somewhat as a surprise. Whether this is also possible for Our goal in this work will be to elaborate on the results of [10] and explicitly show how these IIA geometric flux backgrounds and any other type II background can be embedded in maximal supergravity. To this end we will therefore need to relate different formulations of N = 8 gauged supergravity. The embedding tensor formalism provides an E 7(7) covariant formulation of maximal gauged supergravity in D = 4. However, in order to make contact with flux compactifications, we need a rewriting of this theory in terms of irrep's of the Sand T-duality groups, i.e. SL(2) × SO(6, 6), following the philosophy of ref. [24]. Finally, in order to study the physical properties of scalars, such as equations of motions and the mass matrix, SU(8) is the correct group rearranging all the 70 scalar physical degrees of freedom into an irrep. This can be summarised as ↔ Fluxes SL(2) × SO(6, 6) ↔ Mass spectra SU (8) Employing this mapping, we will derive the mass spectrum and the gauge group of such "exceptional" cases of flux backgrounds without branes.
Interestingly, we will find that the mass spectrum of one of the geometric IIA solutions remains non-negative in the maximal theory, providing the first example of a nonsupersymmetric critical point with this property. Similarly, the gauge group will turn out to be a semi-direct product between a semisimple group and a nilpotent ideal. Again this provides the first such example in maximal supergravity. A point to note is that once this set of N = 4 fluxes will be embedded in N = 8 , it will not be a closed set under all non-compact E 7(7) transformations; these fluxes will be part of a bigger U-duality orbit of configurations. Therefore our results leave open the possibility of including fluxes which are odd under the orientifold involution (i.e. the spinorial embedding tensor components), which would fill out the U-duality orbits. In a further line of research we hope to investigate backgrounds with non-zero spinorial fluxes in order to study the landscape of geometric type II compactifications within N = 8 .
The paper is organised as follows. In section 2, we first review maximal supergravity in four dimensions in its formulation in terms of SU (8)

Maximal supergravity in four dimensions
Maximal supergravity appears when reducing type II ten-dimensional supergravities on a torus down to four dimensions. In recent years, a unified framework where to systematically investigate maximal supergravities has been developed, the so-called embedding tensor formalism [31]. This formalism relies on the gauging procedure, i.e., promoting to local a part of the E 7(7) global symmetry of the 4D theory. After applying a gauging, a non-Abelian gauge symmetry is realised in a way compatible with still keeping N = 8 supersymmetry in four dimensions. Moreover, a non-trivial potential V for the 70 (physical) scalar fields in the lower-dimensional theory (a.k.a moduli fields) is also generated, hence opening the possibility for them to get stabilised, i.e., to acquire a mass, due to the gauging. The aim of this paper is to explore the interplay between gaugings and moduli stabilisation in the context of maximal supergravity.  . This translates into the following set of linear constraints On the other hand, the tensor X MNP must also satisfy a set of quadratic constraints coming from the consistency of the gauge algebra in (2.1). These quadratic constraints sit in the (133 × 133) a = 133 + 8645 irrep's of E 7 (7) and are given by The above set of linear (2.2) and quadratic (2.3) constraints guarantee the consistency of the maximal gauged supergravity in four dimensions [31].
Switching on a gauging has strong implications for the scalar sector of the four-dimensional theory. It consists of 133 scalars out of which only 70 are physical degrees of freedom -the remaining 63 can be removed from the theory after gauge fixing -and parameterise an . Because of the gauging, a non-trivial scalar potential appears which is invariant under the linear action of E 7 (7) transformations. This scalar potential might contain a reach structure of critical points where to stabilise all the moduli fields in the four-dimensional theory.

Fermionic mass terms and the SU(8) formulation
The Lagrangian of maximal supergravity in four dimensions can be unambiguously written in terms of SU(8) tensors, since SU(8) is one of the maximal subgroups of E 7 (7) . More concretely, it is its maximal compact subgroup and is identified with the R-symmetry group under which the eight gravitini of the theory get rotated amongst themselves.

Bosonic field content
Under its SU (8) In the SU(8) formulation, the Sp(56, R) invariant (skew-symmetric) matrix Ω MN takes the form Fermionic mass terms and scalar potential The A IJ and A I J KL tensors play a central role in the SU(8) formulation of maximal supergravity. They determine the fermionic mass terms for the gravitini ψ I µ and the dilatini χ IJ K in the four-dimensional Lagrangian [31] (where in this formula µ, ν are understood as space-time indices) PQR . The number of supersymmetries preserved by an AdS (V 0 < 0) or Minkowski (V 0 = 0) solution of the theory is related to the number of spinors satisfying the Killing equations The scalar potential in (2.4) can also be rewritten in terms of the fermionic mass terms as where |A 1 | 2 = A IJ A IJ and |A 2 | 2 = A I J KL A I J KL . This potential will possess a structure of critical points satisfying ∂V ∂φ IJ KL φ IJ KL = 0 , (2.10) where φ IJ KL denotes the VEV for the 70 physical scalar fields. Provided V IJ = 0 for the vector fields, maximally symmetric solutions of the theory are obtained by solving the equations of motion [32] of the physical scalars At these solutions, the mass matrix for the physical scalars [30,32] reads (2.12) Defining the normalised mass as (mass 2 ) norm = 1 |V 0 | (mass 2 ) , then the Breitenlohner-Freedman (B.F.) bound for the stability of an AdS solution is given by where m 2 denotes the lowest eigenvalue of the normalised mass matrix at the AdS extremum with energy V 0 < 0 . We will make extensive use of (2.12) and (2.13) in the last part of the work when discussing stability of solutions in specific maximal supergravity models arising from flux compactifications of type II strings.

Quadratic constraints
The set of quadratic constraints in (  The SO(8) gauging [33] as well as the CSO(p, q, r) with p + q + r = 8 contractions thereof [29,34] are either simple gauge groups or straightforward contractions thereof. For this reason it has been relatively easy to explore these from a pure supergravity viewpoint, irrespective of their realisation in string theory. However, as the relation between gauged supergravities and flux compactifications of string theory became better understood [12], more complicated non-semisimple gauge groups other than the CSO gaugings have gained interest both in maximal [35] and half-maximal [8-10, 36, 37] supergravity. The reason is that, as we will show later, the CSO gaugings turn out to correspond to non-geometric flux backgrounds for which an origin in string theory remains unknown 2 , whereas gaugings corresponding to flux backgrounds with a higher-dimensional origin are in general not CSO.  This group of transformations corresponds with the global symmetry group of half-maximal N = 4 supergravity in four dimensions [38]. The relation between half-maximal supergravity and string compactifications with fluxes has been explored in refs [7,9,10]. As a speculative remark -and up to quantum requirements such as the discrete nature of the gaugings when understood as fluxes -, by covering the different SL(2) × SO(6, 6) orbits of half-maximal supergravities, one might have access to intrinsically stringy effects involving winding modes and/or dyonic backgrounds, even though it is formulated as a supersymmetric field theory of point-like particles.
Both S-duality and T-duality belong to a larger U-duality group, the E 7(7) global sym-metry group of maximal supergravity in four dimensions. Consequently, in order to go from half-maximal to maximal supergravity [22], one has to enlarge the field content of the theory, i.e. vectors, scalars and embedding tensor components, to complete irrep's of E 7 (7) .
It is at this point where an alternative formulation of maximal supergravity in terms of SL(2) × SO(6, 6) tensors becomes mandatory in order to understand the relation between flux compactifications of string theory and maximal supergravity.

Bosonic field content
Complementary to the SU(8) formulation of the previous section, maximal supergravity can also be unambiguously expressed in terms of SL(2) × SO(6, 6) tensors since that is a maximal subgroup of E 7(7) as well. Under SL(2) × SO(6, 6) , we now have the following branching for the E 7(7) representations associated to vectors, scalars and embedding tensor respectively.
ii) The scalars satisfy However, in contrast to the SU (8)  In order to avoid confusion, we will adopt the convention of B µν = 1 32 when referring to the origin of the moduli space, hence being compatible with the natural choice of as the origin of field space.
iii) The different pieces of the embedding tensor satisfy In the SL(2) × SO(6, 6) formulation, the Sp(56, R) skew-symmetric invariant matrix Ω MN becomes block-diagonal and reads where αβ is the Levi-Civita SL(2)-invariant tensor (normalised as +− = 1 ) and where η M N and C µν are the metric and the charge conjugation matrix of SO(6, 6), respectively.
We have summarised our conventions for spinorial representations, gamma matrices, etc. of SO (6,6) in the appendix B.

Fluxes and the embedding tensor
The decomposition of the 56 of E 7(7) under SL(2) × SO(6, 6) translates into the index splitting M = αM ⊕ µ . When expressed in terms of the different pieces of the embedding tensor, the tensor X MNP entering the gauge brackets in (2.1) can then be split into components involving an even number of fermionic indices which turn out to be sourced by f αM N P and ξ αM , together with those involving an odd number of them The brackets of the gauge algebra in (2.1) involving the X M = X αM ⊕ X µ generators in maximal supergravity then takes the form where, if looking at the part only involving the X αM generators, namely we rediscover the gauge algebra of half-maximal supergravity [38] where the X µ generators have been projected out of the theory.
Moving to explicit string constructions, the f αM N P and ξ αM embedding tensor pieces have been related to different background fluxes, e.g. to gauge, geometric and non-geometric fluxes, in compactifications of type II and Heterotic strings producing half-maximal supergravities [7][8][9]. These fluxes restore the invariance of the four-dimensional supergravity under T-and S-duality, i.e. under SL(2) × SO(6, 6) transformations. The F Mμ and Ξ αβµ embedding tensor pieces are related to additional background fluxes which restore the invariance of the theory under U-duality, i.e. under E 7(7) transformations [24]. Nevertheless, the identification between embedding tensor components and fluxes strongly depends on the string theory under consideration. For instance, a component of the embedding tensor corresponding to a metric flux ω in a type IIA construction might correspond to a non-geometric Q flux in a type IIB one and vice versa (see tables 3 and 4 in appendix E). We will take this fact into account in the last section when analysing specific type II flux models.

Quadratic constraints
Plugging the expression for the components of the tensor X MNP in ( as an organising principle for the quadratic constraints. After a straightforward but tedious computation, one finds the following set of quadratic constraints: Let us comment a bit more about the above set of quadratic constraints. If we refer to the embedding tensor pieces f αM N P and ξ αM as "bosonic" and to F Mμ and Ξ αβµ as "fermionic", then the first seven conditions can be understood as (bos × bos) + (fermi × fermi) = 0 quadratic constraints whereas the last three are of the form (bos × fermi) = 0 . As a check of consistency, the first seven conditions reduce to those of the form (bos × bos) = 0 in ref. [22] by setting F Mμ = Ξ αβµ = 0 , namely, by switching off fluxes associated to SO(6, 6)fermi irrep's of the embedding tensor. In this case, the last three conditions are trivially satisfied.
As mentioned in the introduction, looking for a higher-dimensional origin of dual fluxes is becoming a very exciting line of research. As far as fluxes related to the f αM N P components of the embedding tensor are concerned, only purely electric (or equivalently magnetic) SO(6, 6) gaugings have been so far addressed by Double Field Theory [20]. However, the explicit twelve-dimensional twist matrices producing such gaugings have only been built in some particular cases [16,17]. In order to firstly extend to SL(2) × SO(6, 6) gaugings including fluxes related to ξ αM and secondly to E 7(7) gaugings involving also fluxes related to the F Mμ and Ξ αβµ components (such as R-R gauge fluxes amongst others), a generalisation to a 56-dimensional "twisted megatorus" reduction has been proposed [17].

Connecting different formulations
In order to relate the SL(2) × SO(6, 6) and the SU (8) formulations of maximal supergravity, it is mandatory to derive the expression of the X MNP tensor entering the brackets in (2.23) as a function of the fermionic mass terms in (2.7). This can be done in a two-step procedure as follows: 1) By using the tensors A IJ and A I J KL , we can build the so-called T -tensor [31,32]. The components of this T -tensor take the form 2) The constant X MNP tensor in the SU(8) formulation, let us denote it X MNP to avoid confusion with that in the SL(2) × SO(6, 6) formulation, can then be obtained by removing the dependence of the T MNP tensor on the scalar fields (see footnote 1) is the E 7(7) /SU(8) vielbein in the SU(8) formulation [31]. After removing the scalar dependence, the X MNP and X MNP constant tensors in the SU (8)  Schematically, the connection between the two formulations of maximal supergravity works in the following way By inverting the above chain 4 we are able to relate a flux background given in terms of f αM N P , ξ αM , F Mμ and Ξ αβµ to certain fermionic mass terms A IJ and A I J KL . This amounts to know the relations The F Mμ = Ξ αβµ = 0 case Let us derive the relations (2.45) between fermionic mass terms and embedding tensor components when F Mμ = Ξ αβµ = 0 . In the string theory side, this means that fluxes related to fermionic components of the embedding tensor are set to zero, so that (2.46) Before presenting the explicit form of the relations in (2.46), we want to point out an issue that appears during the computation, the way to overcome it and the corresponding price to pay: i) In the SL(2)×SO(6, 6) formulation of maximal supergravity, the scalar fields split into "bosonic" {φ αβ , φ M N } and "fermionic" φ αμ ones. While the former enter the vielbein V M N in a simple way, the latter do it in a very complicated way. In the derivation of the relations (2.46), we will set φ αμ = 0 which means that all the "fermionic" scalars are fixed to their values at the origin of the moduli space. Therefore, the relation between fluxes and fermionic mass terms that we present here is only valid in the submanifold of the moduli space where φ αμ = 0 .
ii) Being tight to the submanifold with φ αμ = 0 is perfectly consistent with embedding N = 4 flux compactifications (and truncations thereof) inside N = 8 supergravity, since "fermionic" scalars are projected out (set to zero) when truncating from maximal to half-maximal supergravity in four dimension [22]. A special point in this submanifold is the origin of the moduli space defined in (2.18), where both "bosonic" and "fermionic" scalars are set to zero.
iii) One of the main consequences of taking F Mμ = Ξ αβµ = 0 as well as φ αμ = 0 is that the method introduced in ref. [10] (and further exploited in ref. [29]) for charting critical points of the scalar potential becomes more subtle. This method relies on the fact that the manifold spanned by the scalars, i.e. E 7(7) /SU (8) in the case of maximal supergravity, is homogeneous so any critical point can be brought back to the origin of the moduli space by applying an E 7 (7) transformation. However, neither F Mμ = Ξ αβµ = 0 nor φ αμ = 0 are U-duality covariant conditions: E 7(7) transformations will mix "fermionic" and "bosonic" embedding tensor components and scalars, hence rendering the relations in (2.46) no longer valid. We will be back to this point in the last section when discussing specific flux backgrounds yielding maximal supergravities.
Taking Considering this reduced set of vielbein components, we can build the explicit mapping between fermionic mass terms and fluxes by following the prescription in (2.44). It will be useful to define the tensors which reproduce the fermionic mass terms in N = 4 supergravity [38], together with their counterparts which complete the N = 8 theory. In terms of these, the relation between fluxes and fermionic mass terms is given by for the components inside A IJ and In the next section we present a series of consistent truncations of maximal supergravity yielding simpler theories with a smaller set of fields and embedding tensor components.
Later, in the last section, we will investigate the lifting of (solutions of) these truncations to maximal supergravity making use of the explicit correspondence between flux backgrounds and fermionic mass terms derived here. Due to the presence of vectors, this theory might have interesting applications in holographic superconductivity as well as in Cosmology as far as the existence of de Sitter solutions via D-terms uplifting is concerned. We hope to come back to these two issues in the near future.

A web of group-theoretical truncations
• Step from -2-to -3-: the truncation is now with respect to an H = Z 3 discrete subgroup of the G = SL(2) T × G 2(2) global symmetry of the previous N = 2 theory via the chain More concretely we mod-out the different fields in the theory by a Z 3 element of the form e i 2π 3 q , where q mod(3) denotes the charge of the fields with respect to the U(1) U factor in (3.4). The field content inside the box -3-follows from the G 2(2) irrep decompositions where the subindex in n (q) refers to the U(1) U charge q of the SU(2) irrep n . The truncated theory has an SL(2) T × SU(2, 1) global symmetry and still keeps N = 2 supersymmetry. This fact can be seen by obtaining the theory directly from an SU (3) truncation of maximal supergravity without any intermediate step, as we see next. • Step from -1-to -3-: truncating maximal supergravity with respect to a compact H = SU(3) subgroup of its G = E 7(7) global symmetry via the chain associated to one vector multiplet and one hyper multiplet respectively. This theory can therefore be seen as a truncation of that in box -2-where one of the hyper multiplets is projected out after modding out by the Z 3 discrete subgroup previously introduced.
The same truncation was explored in ref. [39] and further investigated in refs [40] as gravity dual of non-relativistic field theories. • Step from -1-to -4-: this is the truncation connecting maximal supergravity and half-maximal supergravity coupled to six vector multiplets. It can be seen as a This N = 1 supergravity theory has been extensively investigated because of its direct connection to string theory via type II orientifold compactifications with fluxes [3][4][5].
The resulting supergravity models are referred to as ST U -models and different background fluxes in the string theory side correspond with different embedding tensor configurations in the supergravity side. However, not all the embedding tensor configurations in the supergravity side have a higher-dimensional interpretation since most of them corresponds to non-geometric flux backgrounds for which an origin in string theory, if possible, remains to be found.
It is worth noticing here that this theory can be lifted to that in box -2-by completing it with the fermionic irrep's removed by the Z 2 truncation taking from the box -1-to the box -4-in figure 1. • Step from -5-to -6-: this truncation is with respect to an H = Z 3 discrete subgroup of the G = SL(2) S × SL(2) T × SL(2) U global symmetry in -5-via the chain As happened when truncating from -2-to -3-before, we mod-out again the different fields in the theory by a Z 3 element of the form e i 2π 3 q , with q mod(3) being this time the charge of the fields with respect to the U(1) U factor in (3.13). Now, the relevant branchings in order to derive the field content inside the box -6-are where, as before, the subindex in 1 (q) refers to the U(1) U charge q of the state. The truncated theory still has N = 1 supersymmetry since the gravitino in the parent theory was already a singlet with respect to both U(1) T and U(1) U .
The scalars in the truncated theory span the scalar manifold (3.14) It can be parameterised by two complex scalars S and T associated to the SL(2)/SO (2) factors plus an extra real scalar associated to the Cartan generator (rescalings) inside the SL(2) U factor in the global symmetry group of the parent theory. This scalar relates to the R U factor in the scalar manifold (3.14). As summarised inside the box -6-in figure 1, the embedding tensor consists of two pieces sitting in the same irrep of the global symmetry group of the theory.
In the next section we concentrate on the N = 1 theory inside box -5-which can be seen as a truncation of the N = 4 theory inside box -4-. We will investigate the lifting of some vacuum solutions to N = 8 supergravity (box -1-) making use of the relations

Exceptional flux backgrounds
When compactifying type II ten-dimensional supergravities down to four dimensions, background fluxes threading the internal space can be switched on during the compactification procedure giving rise to gauged maximal supergravity models. As introduced in section 2.3, flux backgrounds on the string side correspond to deformation parameters related to the f αM N P , ξ αM , F Mμ and Ξ αβµ pieces of the embedding tensor on the supergravity side. For the sake of simplicity, we will restrict our study to the case this is, to string backgrounds not including fluxes associated to SO(6, 6) fermionic irrep's of the embedding tensor. However, even though the remaining f αM N P and ξ αM pieces reproduce those of half-maximal supergravity, the set of quadratic constraints they are restricted by will be that of maximal supergravity derived in section 2.3. Setting to zero spinorial fluxes as in (4.1) does not amount to modding out maximal supergravity by a Z 2 symmetry.
While the former does not affect other fields in the theory (as scalars and vectors), the latter projects out some of them in order to truncate from maximal to half-maximal supergravity.
On the string theory side, modding out by this Z 2 symmetry is commonly referred to as applying an orientifold projection.

String theory embedding vs moduli stabilisation
Thus far, we have discussed in detail the correspondence between maximal gauged supergravities and type II flux compactifications. However, one might also be interested in the interplay between gaugings, fluxes and moduli stabilisation: in short, fluxes were introduced in order to achieve moduli stabilisation. Sketchily, the picture in this respect seems to be the following Semisimple gaugings are likely to produce critical points and moduli stabilisation [27][28][29], but we will show that their embedding as type II flux compactifications involves highly nongeometric backgrounds. On the other hand, nilpotent gaugings can be obtained from type II compactifications including gauge fluxes [35], but they seem not to be enough to get moduli stabilisation. Intermediate gaugings containing a semisimple part and an Abelian part have recently been found in ref. [29], although their embedding into string theory/M-theory has not been explored yet.
Here we will present a novel intermediate gauging consisting of a semisimple and a nilpotent part which allows for moduli stabilisation and can be embedded into string theory as a type IIA flux compactification including gauge and metric fluxes. We will reduce our search of critical points to the origin of the moduli space and discuss the issues of stability and supersymmetry at those solutions. However, due to the restriction gauge groups with p + q + r = 8 . By applying the Z 2 orientifold projection truncating from maximal to half-maximal supergravity, the CSO(p, q, r) gauge groups get broken to the direct product of two smaller CSO ± groups as with p ± + q ± + r ± = 4 . As explained in ref. [34], each of the CSO ± factors in the r.h.s of (4.3) can be parameterised in terms of two real symmetric 4×4 matrices M ± andM ± which determine their embedding into an SO(3, 3) ± group, respectively. In terms of generalised flux components, these matrices read

Setting up the flux models
together with where the concrete identification between flux entries in M ± andM ± and embedding tensor

Type IIA with gauge and metric fluxes
Now we investigate specific flux backgrounds having a higher-dimensional interpretation in terms of type IIA string compactifications including geometric fluxes: these are R-R F 0,2,4,6 and NS-NS H 3 gauge fluxes together with a metric flux ω associated to the spin connection of the internal space.
By using again the fluxes/embedding tensor correspondence of  [42], they can be seen as the uplifting to maximal supergravity of half-maximal supergravity solutions compatible with the total absence of sources 9 .
With the fermionic mass terms A IJ and A I J KL at our disposal, we can now compute the different values of the cosmological constant (2.9) at the above set type IIA solutions.   These are given by In addition, we can also obtain the complete mass spectrum for the 70 physical scalars by using the mass formula (2.12) and check stability as well as the amount of supersymmetry preserved. The mass spectrum at the critical points in table 2 turns out to be the following: • At the solution 1 (s 1 ,s 2 ) , the normalised scalar field masses and their multiplicities are given by (×3) , for the 38 scalars surviving the truncation from maximal to half-maximal supergravity, together with 1 3 4 ± √ 6 (×3) , 6 (×3) , T-duality transformations, hence being physically equivalent. 8 We are setting g = 1 2 in analogy to ref. [10]. 9 The presence of sources as O6-planes and D6-branes in these type IIA scenarios modifies the set of N = 8 quadratic constraints.
for the additional 32 scalars in the maximal theory. There are two tachyons in the spectrum both with the same normalised mass m 2 = − 2 3 , so this AdS solution is completely stable since it satisfies the B.F. bound in (2.13).
• At the solution 2 (s 1 ,s 2 ) , the values of the normalised scalar masses and their multiplic- At this point, the nature of the two Z 2 factors labelled by (s 1 , s 2 ) becomes clear. The first one, as already pointed out in ref. [10], is a symmetry of the N = 4 theory and hence it does not really label different solutions, whereas, at this level, the second Z 2 seems to appear as an accidental symmetry forcing the value of the energy and the mass spectra of inequivalent critical points to be identical. When lifting these solutions to maximal supergravity, the second Z 2 becomes a symmetry as well: it corresponds precisely to the SU(8) element interchanging SU(4) time-like with SU(4) space-like , thus relating equivalent solutions. As a consequence, the number of inequivalent critical points reduces to four and they can be seen as different solutions of the same maximal gauged supergravity.
Let us now identify the gauge group underlying these type IIA geometric backgrounds in a maximal gauged supergravity context. Since we set F Mμ = Ξ αβµ = 0 , the X MNP components in (2.22) do vanish. Then, the brackets (2.1) of the gauge group G 0 take the simpler form (4.11) The 12-dimensional subgroup G bos ⊂ G 0 spanned by the linearly independent 10 X αM bosonic generators in (4.13) turns out to be As an aside remark, we have taken the real realisation of gamma matrices (see appendix B) when building the structure constants of the gauge algebra in (4.11). Otherwise, if taking the SU(4) × SU(4) covariant realisation, the structure constants turn out to be complex an so the gauge generators in the adjoint representation. Thus, one still would have to impose a reality condition upon vectors when it comes to identify the gaugings.
Because of all the aforementioned, we conclude that the gauge group in (4.13) gives rise to N = 1 supersymmetric and non-supersymmetric AdS stable solutions of maximal supergravity at the origin of the moduli space which can be embedded in string theory as type IIA flux compactifications in the presence of geometric fluxes.

Conclusions
We have presented the SL (2)  Building on these results, we have constructed the uplift to maximal supergravity of a set of four AdS vacua of type IIA geometric flux compactifications. These solutions can be seen as different critical points of the same theory with gauge group G 0 = SO(4) Nil (22) .
One of those is supersymmetric, whereas the other three are non-supersymmetric. However, amongst the non-supersymmetric solutions, we find that one is not only perturbatively stable, but it even turns out to be a minimum.
Additionally, we have studied the uplift of a particular class of type IIB non-geometric flux backgrounds by requiring any brane sources to be absent. We have compared the results with the list of critical points of maximal supergravity which are already present in the literature and we have analysed their mass spectra. Our results are in perfect agreement with ref. [29].
This translates into the following rules and the consistency of the two rules implies where the indices µ andμ respectively denote left-and right-handed M-W spinors. Accordingly to this decomposition, the Γ-matrices split into 32 × 32 blocks as follows and the charge conjugation and conjugation matrices become In terms of these 32 × 32 gamma matrices, the relations (B.1) and (B.4) can be respectively written as Antisymmetrised products of two gamma matrices can be defined both for left-and right-handed M-W representations as and further extended to antisymmetrised products of an even number of gamma matrices.
However, only those up to degree six are linearly independent since higher-degree ones (from 7 to 12) are related to them by Hodge duality 11 . After defining all the products of gamma matrices, one can make use of C µν , Cμν and their inverse transpose in order to rise and lower indices. As a result, antisymmetrised products of two and six gamma matrices are symmetric, whereas the ones with four are antisymmetric.  When written in terms of SU(4) ∼ SO(6) invariant tensors, the antisymmetric charge conjugation matrices in (B.8) take the block off-diagonal form

SU(4) × SU(4) covariant formulation of M-W spinors
whereas the gamma matrices in (B.7) split into a set of time-like matrices with a blockdiagonal structure and a set of space-like ones together with the self-dual G a ones Notice that they are complex matrices and then will lead to a complex representation of gamma matrices in (B.15)-(B.18). This is related to the fact that SO (6) producing a non-standard definition of the origin of the moduli space, as discussed in detail in the main text.

Real formulation of M-W spinors
In addition to the SU(4) × SU(4) covariant formulation of M-W spinors described above, we can adopt another realisation such that: i) it is a real realisation of M-W spinors ii) it is compatible with the standard choice of (2.19) as the origin of the moduli space.
We build our real 64 × 64 Γ-matrices in a Majorana representation out of the 2 × 2 Pauli matrices σ 1,2,3 in the following way where we decide to use a set of Pauli matrices satisfying [σ i , σ j ] = 2 i ijk σ k . This corresponds to the choice Building the 64 × 64 charge conjugation matrix as We will use this real representation of M-W spinors when it comes to identify gaugings associated to critical points at the origin of the moduli space.
C X MNP in the SL(2) × SO (6,6) formulation In this appendix we derive the explicit form of the components of the X MNP tensor given in (2.21) and (2.22). Let us first start by giving the explicit form of the E 7(7) symmetric generators [t A ] MN in the fundamental representation following the conventions in ref. [22]. which were already derived in ref. [22], together with a set of additional ones involving an odd number of fermionic indices. The most general ansatz for the latter according to the symmetry is given by we must go to a common basis where to simultaneously describe SL(2) × SO(6, 6) and SU (8) indices. This common basis turns out to be SO(2) × SO (6)  In what follows we will make an extensive use of two different decompositions of an E 7 (7) fundamental index: i) the decomposition with respect to SL(2) × SO(6, 6) whereas those of the inverse vielbein V M N read couplings SO(6, 6) SO(2, 2) type IIB type IIA fluxes Table 4: Mapping between primed fluxes, embedding tensor components and couplings in the flux-induced superpotential. We have made the index splitting M = {a, i,ā,ī} for SO (6,6) light-cone coordinates, not to be confused with the same indices appearing in appendix A.
theory when it comes to associate embedding tensor components to fluxes. In this picture, the superpotential in (E.2) contains flux-induced polynomials depending on both electric and magnetic pairs -schematically (e, m) -of gauge (F 3 , H 3 ) fluxes and non-geometric (Q, P ) fluxes, as well as those induced by their less known primed counterparts (F 3 , H 3 ) and (Q , P ) fluxes, P F = a 3 + 3 a 2 U + 3 a 1 U 2 + a 0 U 3 ,

(E.4)
For the sake of clarity, we have introduced the flux combinations C i ≡ 2 c i −c i , D i ≡ 2 d i −d i , C i ≡ 2 c i −c i and D i ≡ 2 d i −d i entering the superpotential (E.2), and hence also the scalar potential.