A second look at gauged supergravities from fluxes in M-theory

We investigate reductions of M-theory beyond twisted tori by allowing the presence of KK6 monopoles (KKO6-planes) compatible with N=4 supersymmetry in four dimensions. The presence of KKO6-planes proves crucial to achieve full moduli stabilisation as they generate new universal moduli powers in the scalar potential. The resulting gauged supergravities turn out to be compatible with a weak G2 holonomy at N=1 as well as at some non-supersymmetric AdS4 vacua. The M-theory flux vacua we present here cannot be obtained from ordinary type IIA orientifold reductions including background fluxes, D6-branes (O6-planes) and/or KK5 (KKO5) sources. However, from a four-dimensional point of view, they still admit a description in terms of so-called non-geometric fluxes. In this sense we provide the M-theory interpretation for such non-geometric type IIA flux vacua.

How to get masses from extra dimensions [1] has captured the attention of theoretical physicists during the last thirty five years. How massless theories in higher dimensions lead to massive theories in lower dimensions remains at the core of the connection between strings or M-theory and the real world. One may think of two approaches. The first is the top-down approach where a higher-dimensional theory like strings or M-theory is selected and then a lower-dimensional effective model is derived from the choice of a compactification scheme.
In this way the dynamics in lower dimensions follows from the reduction prescription. The higher-dimensional interpretation of such effective models is clear but, as a downside, one often engineers classes of compactifications which do not produce satisfactory physics. Alternatively, the bottom-up approach begins with an effective field theory (EFT) in lower dimensions selected using low-energy dynamical or phenomenological criteria. Only then can one try to relate such well-motivated models to more fundamental theories in higher dimensions like strings or M-theory. This may be feasible if some guiding principles are respected in the selection of the EFT.
A crucial ingredient in the construction of an EFT expected to describe classes of strings/M-theory compactifications is the number of preserved or broken supersymmetries. This is even more relevant than the space-time dimension since the existence of supercharges severely restricts the field content and the structure of the effective action in all dimensions.
For the cases of 32 (maximal) and 16 (half-maximal) supercharges in four dimensions (4d), the guiding principle which governs the structure of the EFT is the embedding tensor (ET) formalism. This framework allows for a systematic exploration of N = 8 (maximal) [2] and N = 4 (half-maximal) [3] effective supergravity models -in the form of gauged supergravities -which, on the other hand, are of special interest due to their plausible realisation in higher dimensions as maximally supersymmetric and 1/2-BPS backgrounds.
However the identification between parameters in the embedding tensor formalism and quantities in a higher-dimensional theory turns out to be a subtle task and has occasionally led to some confusion in the literature. This has been for instance the case for the effective STU-models of ref. [4] arising from massive type IIA orientifold reductions including background fluxes, D6-branes and O6-planes. These were the first string constructions featuring full moduli stabilisation in a vacuum without requiring non-perturbative effects, such as Euclidean brane instantons or gaugino condensation, to stabilise the Kähler moduli [5]. In ref. [4] an N = 1 flux-induced superpotential W IIA was presented and the fluxes (couplings in W IIA ) were related to N = 4 gauging parameters, thus establishing a correspondence between flux compactifications and (N = 1 truncations of) N = 4 gauged supergravity in the context of type IIA orientifold reductions (16 supercharges). The string vacuum of ref. [4] was reconsidered in ref. [6] and found to actually require the presence of KK5 monopoles due to a relation of the form ω ω = 0 ⇒ Net charge of KK5 (KKO5) sources (1.1) involving the Scherk-Schwarz metric ω-flux along the six-dimensional internal space X 6 . This result indicated the necessity to extend the twisted tori picture of ref. [1], which demands ω ω = 0 as a consistency relation. However, and only after the advent of the embedding tensor formalism, a thorough study of type IIA orientifold reductions [7] showed that ω ω = 0 violates the consistency conditions of N = 4 gauged supergravity [3]. As a consequence, the string vacuum of ref. [4] is not a solution of N = 4 gauged supergravity although it still is a perfectly acceptable solution of the N = 1 supergravity specified by the superpotential W IIA .
Nevertheless various type IIA orientifold models actually corresponding to N = 4 gauged supergravities, i.e. satisfying ω ω = 0 , have been worked out afterwards on the basis of the ET formalism [7,8]. In all the cases where full moduli stabilisation occurred, the massive version [9] of the type IIA theory was needed.
Gauged supergravities related to M-theory reductions to four dimensions have been much less explored [10][11][12] than their type IIA relatives. Ref. [10]  between the N = 1 superpotentials of the M-theory models (32 supercharges) and of the type IIA orientifold models (16 supercharges) which can be summarised as One of our main results is that full moduli stabilisation can be achieved in M-theory scenarios provided that the fluxes (c 3 , d 0 ) are activated. The minimally setup requires an N = 8 → N = 4 breaking of supersymmetries (from 32 supercharges to 16) in the effective STU-models. Using the embedding tensor formalism as an organising principlefor this we will derive a precise ET/flux dictionary in M-theory -we will show that the set of N = 4 consistency relations is compatible with a relaxation of the Scherk-Schwarz conditions ω ω = 0 involving the metric ω-flux in M-theory, in contrast to what happened in the type IIA case. Along the lines of ref. [6], we will introduce the corresponding KK6 monopoles entering the relation which now involves the Scherk-Schwarz metric ω-flux along the seven-dimensional internal space X 7 , and discuss their compatibility with preserving N = 4 supersymmetry in the effective action. The aim of this work is to extend the study of type IIA/M-theory reductions initiated in ref. [10] by exploiting the power of the embedding tensor formalism used to systematically analyse maximal and half-maximal gauged supergravitites in four dimensions.
The paper is organised as follows. In section 2 we review the reductions of M-theory on G 2 -manifolds with fluxes [10] and their interpretation as type IIA orientifold constructions in order to introduce the effective STU-models considered in the rest of the paper. In section 3 we establish the precise correspondence between STU-models and (half-)maximal gauged supergravities in four dimensions. We present the flux/ET dictionary, discuss the interplay between supersymmetry and Scherk-Schwarz conditions as well as the relation to the absence/presence of KK6 monopoles and finally characterise the effective supergravity in terms of the universal moduli powers appearing in the scalar potential. In section 4 we exhaustively classify the structure of 4d flux vacua by making a combined use of duality transformations in the STU-models and algebraic geometry techniques in order to solve the extremum conditions of the scalar potential and the consistency relations imposed by supersymmetry. A systematic analysis of the critical points identifying the required sources as well as the underlying N = 4 gauging is performed. We conclude with section 5 and present some relevant data associated to the M-theory flux vacua in the two appendices.

M-theory on G -manifolds with fluxes
Our starting point is the Scherk-Schwarz reduction of M-theory on G 2 -manifolds with fluxes derived in ref. [10]. It is an orbifold reduction on X 7 = T 7 Z 2 ×Z 2 ×Z 2 including G (4) and G (7) background fluxes for the A (3) and A (6) gauge potentials of 11d supergravity, as well as a metric ω-flux associated to a twist along the internal space X 7 . We will re-derive the four dimensional effective theory of ref. [10] in order to establish the set of conventions we are using in this work.
Before introducing the twist, the G 2 -holonomy of the orbifold is encoded in a G 2 invariant three-form and its 7d dual four-form ϕ G 2 = dy 127 + dy 347 + dy 567 + dy 135 − dy 146 − dy 362 − dy 524 , 7 ϕ G 2 = dy 3456 + dy 1256 + dy 1234 − dy 2467 + dy 2357 + dy 4517 + dy 6137 , .., 7 in the above expressions. The metric of the internal space is simply the flat metric of the ambient T 7 , i.e. ds 2 7 = (η A ) 2 , where η A = R A dy A and R A=1,...,7 denote the radii of the seven internal circles. We denote Φ (3) Consequently the internal component of the gauge potential A (3) has a similar expansion and both can be combined into a G 2 invariant complexified three-form where the ω A (y) entering the above expansion are the seven basis elements of H 3 (X 7 ). The seven coefficients T A (x) represent moduli fields in the four-dimensional effective action.
After a twist is turned on by means of a metric flux 5 , i.e. ω BC A = 0, the G 2 -holonomy of the original orbifold is replaced by a G 2 -structure. The set of left invariant twisted forms η A along the internal space then satisfy the Maurer-Cartan equations and can be used to build the set H p (X 7 ) of cohomology classes of X 7 .
The preserved G 2 -structure ensures N = 1 supersymmetry in the reduced theory. The Kähler potential for the seven moduli fields T A in the expansion (2.3) is given by [10,16]  corresponding to a scalar manifold M scalar = [SU(1, 1)/U(1)] 7 . In addition a scalar potential also emerges upon reduction -see refs [16][17][18] for reductions with G 2 -holonomy and ref. [10] for weak G 2 -holonomy and cocalibrated G 2 -structures -. This potential can be derived from the flux-induced superpotential [10] using the standard N = 1 supergravity formula where K AB is the inverse of the Kähler metric is the Kähler derivative. The exterior derivative entering the last term in (2.6) corresponds to the twisted operator d = ∂ + ω that incorporates the metric ω-flux in the internal space The superpotential (2.6) consists of three pieces: The first piece is induced by G (7) an produces a constant term. The second piece is induced by G (4) and gives rise to linear couplings for the seven moduli. The third piece is induced by the metric ω-flux and produces quadratic terms T A T B (with A = B) in the superpotential.

M-theory flux-induced superpotential
Let us now derive the form of the M-theory superpotential (2.6) in the case of a reduction on X 7 = T 7 /(Z 2 × Z 2 × Z 2 ) which has untwisted Betti numbers b 1 (X 7 ) = b 2 (X 7 ) = 0 and b 3 (X 7 ) = 7. The geometry of the orbifold is encoded in its sets of invariant forms. Splitting the basis of left invariant twisted 1-forms as with a = 1, 3, 5 and i = 2, 4, 6 , then the seven basis elements of H 3 (X 7 ) are given by (2.9) The complementary elements spanning H 4 (X 7 ) are then obtained by 7d Hodge duality and (2.10) The cohomology basis then satisfies the orthogonality conditions with I, J = 1, 2, 3 and where the volume of X 7 is defined as V 7 = X 7 η 1234567 .
Using the above set of invariant forms, it is possible to turn on background fluxes for G (4) and G (7) as well as for the metric ω-flux. In terms of the elements in (2.10), the G (4) background flux can be expanded as The expansion of the background for G (7) is simply In addition to the gauge fluxes (2.12) and (2.13), there are 21 metric ω-fluxes compatible with the orbifold symmetries. The entire set of M-theory fluxes is summarised in Table 1.
In terms of the basis elements (2.9), the expansion of the complex three-form in (2.3) can be rewritten as where S , T I and U I have the type IIA interpretation of dilaton, complex structure and Kähler moduli, respectively 6 . Moreover we also find where we have defined the quantities 7 and where C 1 is the flux matrix introduced in ref. [19] By plugging (2.12)-(2.15) into the flux-induced superpotential (2.6) and using the orthogonality conditions (2.11), one finds the M-theory superpotential With this we conclude the re-derivation of the effective supergravities coming from twisted reductions of M-theory on an X 7 = T 7 /(Z 2 × Z 2 × Z 2 ) orbifold with fluxes and set up the scenario we will analyse later.

G 2 -structure of the M-theory reduction
The geometry of the twisted X 7 = T 7 /(Z 2 × Z 2 × Z 2 ) orbifold we are considering determines the set of G 2 -structure relations

19)
6 Notice the somehow unconventional names for the type IIA moduli fields. We have made this choice in order to exactly reproduce the generalised superpotential of ref. [8] derived in the context of type IIB compactifications and further connected to the embedding tensor framework for N = 4 supergravity. 7 In the expressions (2.16) the I = J = K assignments have to be understood in a cyclic manner, namely (I, J, K) = (1, 2, 3) , (2, 3, 1) , (3, 1, 2). For instance one has P 1 = a in the particular case of A (3) = 0 and R A = 1 for the seven radii in X 7 . This requires to evaluate the expression (2.14) at the point S = T I = U I = i so that the twisted versions of the G 2 invariant forms in (2.1) are recovered. These are with the non-standard normalisation 1 7 After some algebra we obtain a one-parameter family -the non-trivial condition in (2.19) is linear -of torsion classes satisfying (2.19). It is given by where we have introduced the flux-dependent quantities The coefficients in the expansion of W 27 also depend on the flux parameters and read 8 In order to recover the standard G 2 relations for the properly normalised Φ (3) and 7 Φ (3) forms [21,22] one must set the parameter κ = −5/7 in (2.21). Up to an overall 1 16 factor coming from the normalisation of (2.6), this is consistent with the relation [22] between the potential energy induced by the metric ω-flux and the Ricci scalar of X 7 (2.25) 8 As in (2.16), the I = J = K assignments are understood in a cyclic manner also in (2.23). This time ) and similarly for the others.
Generic M-theory flux vacua will activate the two torsion classes W 1 and W 27 thus specifying a cocalibrated G 2 -structure. However under certain circumstances -for instance at N = 1 supersymmetric AdS 4 solutions [10] -one might have W 27 = 0 determining a weak G 2 -holonomy or even W 1 = W 27 = 0 restoring a G 2 -holonomy. We will investigate this issue for the set of M-theory flux vacua we will obtain in section 4.
whereas the expansion of the NS-NS flux can be taken as Importantly, the F (0) = −a 3 flux parameter in (2.31) corresponds to the Romans mass in massive type IIA supergravity [9] and does not directly descend from M-theory. The full set of type IIA fluxes including also metric fluxes is summarised in Table 2.
In type IIA orientifold compactifications including O6-planes and D6-branes, the fluxinduced superpotential takes the form [4,23] where the (complexified) Kähler two-form J c and the holomorphic three-form Ω c can be read off from (2.14) by requiring 1 Using the type IIA metric ω-fluxes in X 6 displayed in Table 2 one finds and an explicit computation of the superpotential (2.33) yields (2.36) As noticed in refs [10,13], the ordinary type IIA orientifold reductions including gauge plus metric fluxes miss the c and d (I) 0 which induce the last two terms in (2.18). The situation can be described as follows We will elaborate more on the consequences of turning on these type IIA non-geometric fluxes c (I)  3 and d (I) 0 as well as on the interpretation of the corresponding flux-induced vacua as backgrounds containing KK monopoles, thus going beyond twisted tori as suggested in ref. [10] (see discussion in section 5.2 therein). Our approach here will be completely fourdimensional as we will be using the effective theory of N = 4 gauged supergravity [3] as the theoretical framework in which to describe the backgrounds.

Cyclic symmetry and STU-models
In order to simplify the setup as much as possible we will further restrict to the isotropic scenario in which a cyclic SO(3) symmetry I → J → K is imposed [4]. This simplification is compatible with an Ansatz for the four-dimensional moduli fields. The Kähler potential in (2.5) then reduces to the isotropic form for the M-theory metric ω-fluxes in Table 1 and similarly for the gauge fluxes The above content of fields and fluxes has been shown to be part of the SO(3) invariant sector of the maximal and half-maximal supergravities in four dimensions, the latter being coupled to six vector multiplets [8,24]. We will exploit this fact later on in the paper to investigate the effect of introducing M-theory monopoles in the compactification scheme.
In the isotropic limit, the expression (2.18) of the M-theory flux-induced superpotential takes the form whereas the type IIA superpotential in (2.36) reduces to [4] (2.43) These are the M-theory and type IIA superpotentials we will consider during the rest of the paper. Notice that the relation (2.37) still holds in its isotropic version with the flux-dependent coefficients in W 27 given by Constraining the torsion classes, e.g. demanding W 27 = 0 to have weak G 2 -holonomy, imposes linear relations on the background fluxes that simplify the resulting STU-models.
In this section we investigate the connection between the consistency conditions in Scherk-Schwarz reductions of M-theory (top-down) and the consistency conditions in effective N = 4 and N = 8 gauged supergravities (bottom-up). We will link such conditions to the absence/presence of KK6 monopoles in the M-theory background and characterise the resulting scalar potential in the effective supergravity action.

Scherk-Schwarz reductions and BI
The M-theory fluxes are restricted by a set of quadratic constraints coming from the consistency of the reduction down to four dimensions [1,10,25,26]. In an ordinary Scherk-Schwarz reduction of M-theory these are coming respectively from the nilpotency (d 2 = 0) of the twisted derivative operator d = ∂ + ω as well as from the twisted Bianchi identity (BI) dG (4) = 0 along the internal space X 7 .
Moreover the symmetries of the (compact X 7 with no boundary), thus implying a well-defined Lagrangian upon reduction [1].
The first quadratic constraint in (3.1) gives rise to a set of 6 + 6 + 3 + 1 + 3 + 6 + 3 = 28 conditions of the form 10 whereas the second quadratic constraint in (3.1) is automatically satisfied due to the orbifold symmetries. This can be straightforwardly verified using the M-theory fluxes in Table 1. the set of quadratic constraints in (3.2) to only 7 conditions. These are given by We will investigate the connection between the set of quadratic constraints in (3.3) and those required if demanding N = 8 or N = 4 supersymmetry in the effective action. We will discuss it in the framework of the embedding tensor [3].

Extended supersymmetry and gaugings
The M-theory superpotential in (2.18)   f aj as the invariant metric to raise and lower SO (6,6) indices. If we further split the index M as M = ( a , i , a , i ) , then the fluxes/ET dictionary is presented in Table 3. Notice the presence of electric (α = +) as well as magnetic (α = −) components within the embedding tensor f αM N P = f αM N Q η QP . Both are simultaneously required in order to avoid a runaway behaviour for the dilaton modulus [27].
The consistency of a gauging in N = 4 supergravity [3] imposes a set of quadratic constraints on the embedding tensor f αM N P . These are given by where αβ = αβ with +− = − −+ = 1 is used to raise and lower the SL(2) index α. In order to make contact with the Scherk-Schwarz conditions in (3.1) for M-theory reductions, we have to set the Romans mass to zero, i.e. a 3 = 0, among the fluxes in Table 3 as it corresponds to a non-geometric flux in M-theory. The explicit computation of the constraints in (3.6) produces the following conditions As a result, the quadratic constraints of N = 4 supergravity (3.6) fail to reproduce the two additional conditions iv) and vii) in (3.2). Therefore, demanding N = 4 in the effective theory is less restrictive than demanding a higher-dimensional interpretation as an ordinary Scherk-Schwarz reduction of M-theory.
In ref. [28] it was shown that the N = 4 constraints (3.6) must be supplemented with two additional ones in order to have an N = 4 → N = 8 supersymmetry enhancement in the effective action.
The label SD in the first constraint in (3.8) restricts it to the self-dual part of the SO (6,6) six-form αβ f α[M N P f βQRS] . Once more, an explicit computation of these two constraints hence completing the set of conditions in (3.2). In other words, there is a one-to-one correspondence between the N = 8 quadratic constraints and the conditions required by an ordinary Scherk-Schwarz reduction of M-theory.

KK6 monopoles and N = 8 → N = 4 breaking
In the previous section we have seen that requiring an N = 4 description of the effective supergravity allows for a relaxation of the conditions iv) and vii) in (3.2). However these still have to be imposed in any ordinary Scherk-Schwarz reduction of M-theory establishing the link to N = 8 supergravity.
On the other hand, a violation of some of the ω ω = 0 conditions in (3.1) has been connected to the presence of KK6 monopoles in the compactification scheme, thus going beyond twisted tori [6]. From the effective field theory point of view, we will refer to the would-be companion sources carrying negative charge as KKO6-planes following a similar terminology to that of ref. [6]. Schematically, where ψ refers to the S 1 direction along which the KK6 is fibered and [• • •] specifies the 3-form dual to the 7-cycle filled by the KK6 and the S 1 fiber. The KK6 monopoles will induce a positive contribution to the scalar potential whereas the one coming from the KKO6-planes will be negative [6].
In the case of  Table 2, thus corresponding to non-geometric type IIA flux backgrounds. For the set of conditions in (3.2), the corresponding types of KK6 monopoles are summarised in Table 4.
Our last concern is that of supersymmetry breaking in the presence of KK6 monopoles. then N = 4 supersymmetry is still preserved but one goes beyond Scherk-Schwarz reductions of M-theory due to the violation of (3.1). We will exhaustively explore these two types of effective theories in the next section.

Universal IIA moduli, KK6 monopoles and scalar potential
A way of understanding the effect of including M-theory sources in the background is to analyse the moduli powers appearing in the scalar potential. In order to make contact with previous results in the literature [29][30][31][32][33] we will reinterpret the M-theory potential from a type IIA point of view. To this end, let us introduce the three universal IIA moduli fields (τ, ρ, σ) entering the 10d metric 11) which are related to the STU fields as τ = Im(S) 1/4 Im(T ) 3/4 , ρ = Im(U ) , σ = Im(S) −1/6 Im(T ) 1/6 . The functions A n (ρ, σ) that determine the different terms V n = A n (ρ, σ) τ −n in the potential take the following form: (3.14) Let us discuss the V n terms in the M-theory scalar potential (3.13) when adopting a type IIA point of view using the M-theory/type IIA dictionary in Table 3. In the previous section we saw that including a net charge for those KK6 sources in Mtheory which correspond to KK5 and KK5 monopoles in the IIA picture -types i) and ii) in Table 4 -was not compatible with preserving N = 4 supersymmetry in the effective action.
The reason was that the associated conditions i) and ii) in (3.3) still hold after relaxing N = 8 → N = 4. The effect of adding such monopoles has been investigated in refs [30,31] and found to induce an extra piece V KK5 ∝ τ −2 ρ −1 in the potential supplementing the one already induced by the IIA metric flux V ω IIA ∝ τ −2 ρ −1 with the same moduli powers. More importantly, this extra piece V KK5 turned out to help in finding de Sitter solutions [30,31].
Even though we cannot include such KK5 and KK5 monopoles when demanding N = 4 supersymmetry, the M-theory fluxes (c 3 , d 0 ) will potentially induce the desired τ −2 ρ −1 extra piece within V 2 (third line in A 2 ) 11 . Despite this promising fact, only Anti-de Sitter (AdS 4 ) solutions will happen to exist in these N = 4 STU-models. As we already discussed, demanding isotropy imposes an additional "plane exchange" SO (3) cyclic symmetry among the three two-tori in T 6 = T 2 × T 2 × T 2 (inside X 7 ). This additional symmetry can be interpreted as an enhancement of the truncation from a Z 2 × Z 2 truncation to an SO (3)  on the STU-model is that of rescaling and shifting the moduli fields as with λ S,T,U and ∆ S,T,U being real parameters. By using the tranformations (4.1), any moduli configuration corresponding to a critical point of the scalar potential can be brought to the origin of the moduli space defined as The above set of equations in (4.3) can be completely solved -with or without relaxing iv) and vii) -by using algebraic geometry tools included in the computational package singular [35]. In particular, we have used the GTZ built-in algorithm for primary decomposition of ideals. The outcome is that (4.3) contains several prime factors, each of which corresponds to a physically different content of KK6 monopoles in the M-theory background.
We will discuss them in detail later on.
Some other advantages of bringing the moduli configurations to the origin are : i) closed expressions for the particle mass spectra at a critical point of the scalar potential have been worked out [36] ii) the fermion mass terms get a much simpler form. The fermion masses can be viewed as "dressing up" the embedding tensor with the moduli dependence [3]. When evaluated at the origin, the N = 4 gravitini mass matrix acquires the very simple form with the two independent entries given by M-Theory at the origin of the moduli space as it has to in order to identify the N = 1 gravitino mass with |W   Table 3), so the IIA solutions in ref. [24] will not appear in an M-theory context. Finally the M-theory ω-twist corresponds to G ω = Solv 6 U(1) in agreement with the analysis of twist groups performed in ref. [10].

Backgrounds with KK6 (KKO6)
We have rederived the result that there is no moduli stabilisation (without flat directions) in the absence of KK6 (KKO6) monopoles [10]. Next step is then to remove the conditions  corresponding to a no-scale STU-model analogous to that in (4.7) upon exchanging T ↔ U .
The associated vacuum -we will refer to it as "vac 0" from now on -is a non-supersymmetric Minkowski vacuum with non-vanishing W 1 and W 27 torsion classes in (2.19).
Using the mass formula in ref. [36], the scalar mass spectrum is given by so it does not contain tachyons but presents thirteen flat directions, i.e., zero-mass modes not associated to Goldstone bosons. The spectrum of vector masses reads for the R-R gauge potential C (7) . The BI for F (2) along the internal space X 6 reads thus demanding O6 orientifold planes lifting to KKO6-planes in M-theory (see ref. [6] and references therein for a discussion of the lifting).

Including only KK6 (KKO6) → KK6 (KKO6 ) sources
The second prime factor in the decomposition of the algebraic system (4.3) is compatible  Table 5. In all the solutions the net charge of KK6 (KKO6 ) sources is

Including both types of KK6 (KKO6) sources
The third prime factor in the decomposition of the system (4.3) demands to simultaneously relax the two condtions This last prime factor is also of dimension one and therefore can be solved explicitly. We find a rich structure of AdS 4 critical points they all compatible with 2 c 1 =c 1 = λ , (4.19) so that the U T term in the superpotential (2.42) is absent. Up to discrete multiplicities, we now find nine inequivalent M-theory flux vacua labelled as "vac 9" to "vac 17". As in the previous case the phenomenological consequences are very diverse and we have moved a detailed discussion of these M-theory backgrounds to Appendix B. A summary of the main features of these vacua is also included in Table 5. For these solutions the net charge of localised sources are requiring the presence of O6 -and KKO6 -planes in the backgrounds. Moduli fields can also be fully stabilised at supersymmetric and non-supersymmetric vacua for some of these M-theory backgrounds but, generically, instabilities happen to occur more often.

Monopoles and duality orbits of N = 4 gaugings
The results in the previous section have shown that M-theory backgrounds including KK6 (KKO6) monopoles lead to moduli stabilisation in the effective STU-models. Now we will investigate the N = 4 gaugings underlying such M-theory backgrounds with sources.
Let us start by recalling the [SU(1, 1)/U(1)] 3 duality transformation in (4.1) needed to bring a given moduli configuration to the origin, namely, The action of applying (4.21) to the M-theory superpotential in (2.42) has the effect of redefining the flux background (moduli couplings) in the following way where we have identified one duality orbit with one supergravity model. The N = 4 gauging underlying each of the four inequivalent STU-models can be computed by looking at any of the orbit representatives. We decide to select the first representative in each of the models.
They are given by together with (4.16) for Model 1 and 2 as well as (4.19) for Model 3 and 4. A detailed description of these M-theory backgrounds is collected in the appendices. Remarkably they require a 1 = b 0 = c 0 = 0 implying a vanishing flux G (4) = 0 (see Table 1) as well as a non-vanishing G (7) = 0 . The result is then an M-theory superpotential (2.6) of the form uniquely induced by metric ω and G (7) background fluxes. The underlying N = 4 gauging turns out to have a very simple algebra structure as we will investigate now.
The four inequivalent STU-models can be simultaneously explored by considering the gauge algebra G 0 ⊂ SL(2) × SO(6, 6) induced by the set of seven fluxes Using this decomposition, the antisymmetry of the brackets in (3.4) determines the magnetic generators in terms of the electric ones so that only an independent twelve-dimensional algebra is gauged. One obtains According to the Levi decomposition of finite dimensional real lie algebras, we find that G 0 = G semi G solv with a 3-dimensional semisimple piece and a 9-dimensional solvable piece.
The generators {Z i , X i ; Z a } span the solvable (actually nilpotent) ideal with non-vanishing brackets which can be understood as G solv = U (1) 6 U (1) 3 . The mixed brackets between the solvable and the semisimple pieces read whereas the non-vanishing commutators between generators X a in the semisimple piece are given by The full 12-dimensional N = 4 gauging thus corresponds to a gauge group G 0 = SO(3) Nil 9 (2) , (4.30) with Nil 9 (2) being a 9-dimensional U(1) 6 U(1) 3 nilpotent ideal of order two (three steps) with lower central series The different values of the fluxes in the four disconnected STU-models only determine how the semisimple products are specifically realised but do not modify the identification of the full group as G 0 = SO(3) Nil 9 (2). This gauge group (with a different realisation in terms of brackets) has also appeared in twisted reductions of massive type IIA strings [7,8].
However, as we already emphasised, those massive type IIA backgrounds cannot be obtained from our M-theory reductions due to the lack of the Romans mass parameter 13 .  Table 5. This scenario was discussed in detail in section 4.3.1.
In the first part of the paper we investigated the interplay between the conditions (3.1) required by an ordinary Scherk-Schwarz reduction and the consistency relations (3.6) and In the second part of the paper we performed a systematic and exhaustive study of M-theory flux vacua by combining the use of dualities in the STU-models with algebraic geometry tools available in the computer algebra system singular. We proved that full moduli stabilisation can be achieved in N = 4 flux models coming from M-theory provided KKO6-planes are included as background sources (see Table 5 and section 4.5 for a summary of the results). The underlying N = 4 gauging is unique and identified with G 0 = SO(3) Nil 9 (2). Moreover, we also showed that these models correspond to genuine M-theory backgrounds which do not admit an interpretation in terms of regular type IIA orientifold constructions. In the latter, moduli stabilisation seems to demand a non-vanishing Romans mass parameter [9] and therefore a deformation already in higher dimensions [7,8].
This deformation parameter does not appear from M-theory upon ordinary dimensional reduction so, in the M-theory backgrounds we have found here, full moduli stabilisation is achieved from a massless theory in higher dimension. Moreover a background flux for the G (7) form -a 0 parameter in (2.42) -seems to be mandatory in this case, thus playing a similar role as the Romans mass for moduli stabilisation in a type IIA context but having a neat "field strength" interpretation in higher dimensions as the dual of a purely external G (4) flux [10,39].
Finally we want to stress once more the four-dimensional, bottom-up approach adopted in this paper which justifies to adopt the ET formalism to analyse the effect of the partic- additional motivation for pursuing this goal. We hope to come back to this in the future.

Acknowledgments
We want to thank Ulf Danielsson, Giuseppe Dibitetto and Diederik Roest for interesting discussions and specially Giuseppe Dibitetto for the careful reading of an early draft of this paper. The work of the authors is supported by the Swiss National Science Foundation.
14 Four-dimensional gauged supergravities have been shown to capture the dynamics of asymmetric orbifold constructions for which a ten-dimensional "geometry" does not even exist due to the difference between the left X L and right X R sectors. Further interesting connections to non-geometric flux backgrounds have also been established in refs [40][41][42]. shows that supersymmetry is completely broken at this solution.
Vacuum 3 This solution is compatible with the family of flux parameters so this time the superpotential is given by    (B.32) and is also non-supersymmetric.