KK-monopoles and G-structures in M-theory/type IIA reductions

We argue that M-theory/massive IIA backgrounds including KK-monopoles are suitably described in the language of G-structures and their intrinsic torsion. To this end, we study classes of minimal supergravity models that admit an interpretation as twisted reductions in which the twist parameters are not restricted to satisfy the Jacobi constraints $\omega\, \omega=0$ required by an ordinary Scherk-Schwarz reduction. We first derive the correspondence between four-dimensional data and torsion classes of the internal space and, then, check the one-to-one correspondence between higher-dimensional and four-dimensional equations of motion. Remarkably, the whole construction holds regardless of the Jacobi constraints, thus shedding light upon the string/M-theory interpretation of (smeared) KK-monopoles.

would easily allow for their relaxation. Such a possibility has been considered in ref. [20], where it was argued that (smeared) KK-monopoles can be viewed as sources to the r.h.s. of the Jacobi constraints ω ω = 0 ⇒ KK-monopoles .
(1.2) Some work in this direction has been done in refs [21,22], where M-theory backgrounds with KK-monopoles and their relation to type IIA are analysed with several complementary motivations.
The aim of this work is to interpret twisted T 6 /Z 2 2 and T 7 /Z 3 2 reductions as reductions on SU(3)-and G 2 -structure manifolds, respectively, and establish a solid correspondence between twist parameters (a.k.a. metric fluxes) and torsion classes of the corresponding G-structure. We want to make use of the above correspondence in order to construct an explicit uplift of the STU-models to M-theory or type IIA and provide an interpretation in terms of G-structure compactifications, regardless of whether or not the Jacobi constraints are satisfied. This will justify and corroborate the validity of the bottom-up approach, at least as far as STU-models are concerned. In addition, it will also shed new light on the nature of (smeared) KK-monopoles by giving them a natural and geometric interpretation: unlike for M-branes or D-branes, the KK-monopoles are secretly built-in within the bulk action of eleven-dimensional and massive type IIA supergravity. As a result, they can be nicely described using the framework of G-structures and their intrinsic torsion.
The paper is organised as follows. In section 2, we review relevant facts regarding Mtheory reductions on manifolds with G 2 -structure as well as massive type IIA reductions on manifolds with SU(3)-structure. In section 3 we revisit the STU-models obtained as particular twisted orbifold reductions of either M-theory or massive type IIA string theory and establish the connection to G 2 -and SU(3)-structures and their intrinsic torsion. We compare the two types of reductions in the context of SU(3)-structures of 7d vs 6d manifolds, comment on the non-geometric type IIA interpretation of some M-theory reductions and discuss the relaxation of the Jacobi constraints and how this fact is interpreted as adding KKmonopoles as sources. In section 4 we construct the massive type IIA/M-theory uplift of the corresponding STU-models by studying the ten-/eleven-dimensional equations of motion and showing their one-to-one correspondence with the four-dimensional supergravity equations of motion coming from varying the effective scalar potential of the STU-model. Interestingly, such a correspondence works regardless of the Jacobi constraints, namely, whether or not smeared KK-monopoles are included in the construction. We conclude with a discussion of the results and pose some remaining issues which might open up new possible developments.
A summary of conventions concerning the geometry and topology of the twisted T 6 /Z 2 2 and T 7 /Z 3 2 orbifolds is presented in the appendix A. 3

M-theory/Type IIA on G-structure manifolds
In this section we review a class of orbifold reductions of M-theory and massive type IIA strings on twisted tori with gauge fluxes and their corresponding four dimensional (4d) supergravity effective descriptions as STU-models. Furthermore, we will respectively connect them to reductions on seven dimensional (7d) G 2 -structure and six dimensional (6d) SU(3)structure manifolds.

M-theory on a G 2 -manifolds X 7 with fluxes
We start with a discussion of the 4d effective supergravities coming from reductions of 11d supergravity on G 2 -structure manifolds with fluxes.

G 2 -structure manifolds
A seven-dimensional manifold X 7 with a G 2 -structure [23,24] is specified in terms of a G 2 invariant three-form Φ (3) or, equivalently, in terms of a covariantly constant spinor η such that Φ ABC ∝ η † γ ABC η with A =, 1, ..., 7. The presence of such G 2 invariant objects can be inferred from the decomposition of the corresponding SO (7) representations under the maximal G 2 ⊂ SO (7) subgroup (2.1) The invariant three-form Φ (3) thus corresponds to the singlet appearing in the decomposition of the 35. The failure in the closure of Φ (3) is understood as the presence of a non-vanishing torsion T AB C ∈ Λ 2 (X 7 ) ⊗ Λ 1 (X 7 ) = (7 ⊕¨r r 14 which splits into a set of torsion classes, namely different G 2 representations, given by As anticipated, the above torsion classes act as sources in the r.h.s of the closure relations for the invariant three-form Φ (3) and its 7d dual four-form 7d Φ (3) . These are given by (2.5) Later on we will restrict to the case of vanishing τ 1 = τ 2 = 0 which corresponds to cocalibrated G 2 structures. These include the case of the X 7 = T 7 /Z 3 2 orbifold we will investigate in this work.

Ricci scalar and scalar potential
The 7d metric g AB of the G 2 -manifold can be constructed from the invariant form Φ (3) using the standard formula g (7) The associated Ricci scalar can be expressed in terms of the torsion classes entering (2.5).
The result is given by where |τ 2 p | ≡ 1 p! τ A 1 ...Ap τ A 1 ...Ap and where seven-dimensional indices are raised using the inverse of the metric g (7) AB introduced in (2.6). The 7d Ricci scalar (2.7) becomes (part of) the scalar potential upon reduction of the 11d Ricci scalar S 11d ⊃ d 11 x g (11) R (11) . (2.8) Taking the 11d metric to be of the form ds 2 (11) = τ −2 ds 2 (4) + ds 2 (7) , (2.9) requires the four-dimensional dilaton to be identified as τ 2 = g (7) in order to recover the Einstein frame in four dimensions. This is then compatible with a four-dimensional action of the form which results in the appearance of a scalar potential due to the internal geometry of the form (2.11) We will verify the above relation latter on for the case of the the X 7 = T 7 /Z 3 2 orbifold for which τ 1 = τ 2 = 0.

N = 1 effective action and flux-induced superpotential
Because of the singlet in the decomposition of the 8 in (2.1), reductions of M-theory on G 2 -manifolds with fluxes produce N = 1 effective supergravities in 4d. The M-theory fluxinduced superpotential is given by [13,17] where, for the twisted orbifold reductions we will consider in this work, d is the 7d twisted derivative operator d = ∂ + ω acting on a generic p-form T (p) as with ω AB C being the 7d twist parameters (metric fluxes). C (3) is the three-form gauge potential of 11d supergravity and G (4) its associated background flux along the internal directions. In addition, G (7) corresponds to the dual of a background flux entirely along the external directions, i.e. a Freund-Rubin parameter [25]. Having non-vanishing G (7) = 0 proved to be a necessary ingredient to fully stabilise moduli in the X 7 = T 7 /Z 3 2 reductions of refs [16,18].

Massive IIA on an SU(3)-manifold X 6 with fluxes
Let us now discuss the 4d effective supergravities arising upon reduction of massive type IIA supergravity on SU(3)-structure manifolds with fluxes.

SU(3)-structure manifolds
A six-dimensional manifold X 6 with SU(3)-structure is characterised by the presence of two globally defined and SU(3)-invariant fundamental forms -a real 2-form J and a holomorphic 3-form Ω -defining an interpolation between a complex and a symplectic structure. By decomposing the 2-and (anti-)self-dual 3-form representations of SO(6) w.r.t. its SU (3) maximal subgroup, one indeed finds 6 featuring the three singlets corresponding to J, Ω andΩ respectively. Besides the above topological constraint, the SU(3)-structure (together with supersymmetry) requires a set of special differential conditions which select only some allowed SU(3) irrep's inside the expression of the exterior derivatives of the above fundamental forms. Such irrep's are usually referred to as torsion classes and can be obtained by decomposing the most general metric connection T mn p into SU(3) pieces where the contribution coming from the adjoint representation of SU(3) has been crossed out since it drops out whenever acting on invaraint forms like J and Ω [26]. This procedure where W 1 is a complex 0-form, W 2 is a complex primitive 2-form, i.e. such that and, finally, W 4 and W 5 are real 1-forms. The full expression of the exterior derivatives of the fundamental forms in terms of the torsion classes reads We shall in the following restrict ourselves to the case W 4 = W 5 = 0, which certainly includes the example of X 6 = T 6 /Z 2 2 that we want to make contact with, as well as any other manifold X 6 without 1-and 5-cycles.

Ricci scalar and scalar potential
In terms of the fundamental forms, one can subsequently introduce a metric on X 6 . The intermediate step is defining the quantity [8] where Ω R ≡ Re(Ω) , Ω I ≡ Im(Ω) and λ is a moduli-dependent quantity fixing the correct normalisation of I to I 2 ! = −1 6 . As a consequence, the metric is defined as The Ricci scalar R (6) for such six-dimensional SU(3)-structure manifolds is then expressed as a function of the torsion classes via [27]  As happened before, the 6d Ricci scalar (2.22) becomes (part of) the scalar potential upon reduction of the 10d Ricci scalar in the string frame S 10d ⊃ d 10 x g (10) e −2φ R (10) . (2.23) Taking the 10d metric to be of the form ds 2 (10) = τ −2 ds 2 (4) + ds 2 (6) , (2.24) requires the four-dimensional dilaton φ 4 to be identified as τ 2 = e −2φ g (6) ≡ e −2φ 4 in order to recover the Einstein frame in four dimensions. This is then compatible with a four-dimensional action of the form which results in the appearance of a scalar potential due to the internal geometry of the g (6) R (6) . (2.26) We will verify the above relation latter on for the case of the the X 6 = T 6 /Z 2 2 orbifold for which W 4 = W 5 = 0.

N = 1 orientifolds and flux-induced superpotential
The presence of two singlets in the decomposition of the 8 in (2.14) indicates that reductions of type IIA supergravity in SU(3)-manifolds produce N = 2 effective supergravities in 4d.
Further applying an orientifold projection, the resulting N = 1 supergravity is specified in terms of the flux-induced superpotential [15,28] (2.27) In the case of twisted orbifold reductions, the operator d is the 6d twisted derivative d = ∂ +ω acting on a generic p-form T (p) as with ω mn p being the 6d twist parameters (metric fluxes). J c is the complexified Käler form

Twisted orbifolds
The aim of this section is to provide explicit examples of X 7 and X 6 manifolds with G 2and SU(3)-structure respectively and investigate their connection to flux compactifications of M-theory/Type IIA.

STU-models from M-theory/Type IIA
Twisted orbifolds provide simple examples of manifolds with G 2 -and SU(3)-structure which are easy to handle. In particular we will focus on the X 7 = T 7 /Z 3 2 orbifold in the case of M-theory reductions and X 6 = T 6 /Z 2 2 for orientifolds of type IIA reductions. Twisting an orbifold amounts to introduce a constant metric ω-flux such that the left-invariant forms η A (η m ) globally defined in X 7 (X 6 ) satisfy the Maurer-Cartan equations Twisted X 7 :   (2.27). For the twisted orbifolds X 7 = T 7 /Z 3 2 and X 6 = T 6 /Z 2 2 we are considering in this work, the reduction gives rise to an N = 1 supergravity 2 , more concretely, to a so-called STU-model. The M-theory/Type IIA flux content compatible with the orbifold symmetries is summarised in Table 1.
The reductions on such geometries with fluxes have been carried out in ref [15] (for type IIA on X 6 ) and refs [17,18] (for M-theory on X 7 ). This paper follows the conventions of 2 The discrete orbifold action reduces the amount of supersymmetry in the effective action to four supercharges (N = 1) in the M-theory case and eight supercharges (N = 2) in the case of Type IIA reductions.
Modding out the latter by an extra Z 2 orientifold action further reduces to four supercharges (N = 1).
ref. [18] and we refer the reader to the original literature in order to follow the details of the reduction procedure. Upon reduction, the scalar sector of the N = 1 effective action contains seven complex fields 3 , a.k.a moduli, which serve as coordinates in the coset space (SL(2)/SO(2)) 7 . We denote them T A = (S , T I , U I ) with A = 1, ..., 7 and I = 1, 2, 3 . The set of moduli T A is the natural one to describe M-theory reductions on X 7 where one has the expansion for the complexified three-form entering the superpotential (2.12). On the other hand, splitting the moduli as S, T I and U I makes the connection to the Type IIA forms entering the superpotential (2.27) more transparent. These are given by  Table 1, one finds the M-theory result as well as the Type IIA result The orbifold X 7 = T 7 /Z 3 2 has non-vanishing (untwisted) Betti numbers b 3 (X 7 ) = 7. In the case of X 6 = T 6 /Z 2 2 , one has (untwisted) Betti numbers b − 2 (X 6 ) = 3 and b + 3 (X 6 ) = 4 with the appropriate parity behaviour under the orientifold action σ O6 : previously derived in refs [17,18]. Notice that both superpotentials only differ in the fluxinduced terms appearing in the last lines: the (c Finally the kinetic Lagrangian for the moduli fields -we use conventions where axions are associated to Re(T A ) and dilatons to Im(T A ) -follows from the Kähler potential (3.6) Using (3.4) and (3.5) as well as (3.6), the M-theory/Type IIA scalar potential can be computed from the standard N = 1 supergravity formula where K AB is the inverse Kähler metric and D A W = ∂ A W + (∂ A K)W denotes the Kähler derivative. 12 3.2 Co-calibrated G 2 -structure from M-theory on In this section we work out the co-calibrated G 2 -structure associated to the orbifold space 3 2 in terms of the flux-induced torsion classes. We will show that the co-calibrated G 2 -structure holds regardless of the Jacobi constraints for the metric fluxes, namely, irrespective of the introduction of KK6-monopoles. This motivates the use of the G-structure as a powerful tool to uplift backgrounds with sources. In the last section, we will present the lifting to 11d of backgrounds with an arbitrary configuration of KK6-monopoles.

M-theory metric fluxes and torsion classes
Let us start by introducing the G 2 invariant forms for the orbifold X 7 = T 7 /Z 3 2 . For this particular geometry, they can be written as in terms of the seven-dimensional real forms v (one-form) and J (two-form) and the complex with κ = k 1 k 2 k 3 τ 1 τ 2 τ 3 , and are manifestly invariant under SU(3) ⊂ G 2 ⊂ SO (7). As a result, the forms v, J and Ω specify an SU(3)-structure in X 7 . However this SU(3)-structure is restricted in the sense that it is liftable to a G 2 -structure 4 .
Using the standard expression (2.6) to obtain the 7d metric g AB in terms of Φ (3) in (3.8), one finds so the 7d metric takes the form of a circle fibration over a 6d metric The fibration in (3.10) ensures that the metric on the 6d base is in the string frame when moving to a type IIA description of the backgrounds [31,32]. 4 An SU(3)-structure in a 7d manifold will in general not be liftable to a G 2 -structure.

13
The orbifold X 7 = T 7 /Z 3 2 has (untwisted) Betti numbers b 1 (X 7 ) = b 5 (X 7 ) = 0 which translates into a vanishing of the torsion classes τ 1 = τ 2 = 0. The G 2 -structure specified by the relations (2.5) is then called co-calibrated and takes the simple form (3.12) The above relations can be inverted to obtain the torsion classes τ 0 and τ 3 :  14) and is sourced by all the M-theory metric fluxes in Table 1 with different e φ -weights. The second torsion class τ 3 is a three-form which has an expansion providing again different e φ -weights to different fluxes. The three components associated wirth the β I basis elements can be written in a compact form as Finally, the three components associated to the ω I basis elements in (3.15) can also be collectively given as Notice that also the triplets τ 3 (I) and τ (I) 3 come out with different e φ -weights for different fluxes. In summary, the above set of torsion components completely codifies the G 2 -structure induced by an M-theory metric flux ω BC A = 0.

The matching of the scalar potentials
Equipped with the torsion classes computed in the previous section we can move to compute the Ricci scalar using (2.7), which, in the case of a co-calibrated G 2 -structure, simplifies to We are not displaying the expression for R (7) after plugging in the results for τ 0 and τ 3 since we do not gain any additional understanding on the M-theory reduction. However, let us discuss in more detail the connection to the scalar potential derived from the the N = 1 superpotential in (3.4). More concretely, we are interested in the relation (2.11) reading where we have used the expression for 7d metric g (7) in (3.10).
Considering only the terms coming from the twist ω BC A in the M-theory superpotential (3.4) -these are the quadratic coupling in the second and third lines -it is straightforward to compute their contribution to the full M-theory scalar potential. We will denote the purely metric-flux-induced contribution to the potential . (3.21) In order to check whether the two potentials (3.20) and (3.21) do match, a precise identification between the N = 1 chiral moduli fields in (3.4) and the geometric moduli entering the G 2 invariant forms in (3.9) is required 5 . This identification is given by [15,17,28] Im(S) = e −φ κ , Im(T I ) = e −φ κ τ J τ K (I = J = K) , Im(U I ) = k I , (3.22) with I, J, K = 1, 2, 3 and where κ = k 1 k 2 k 3 τ 1 τ 2 τ 3 was already introduced in (3.9). After an exhaustive term-by-term check of the two potentials (3.20) and (3.21) one finds a perfect matching of the form where the factor 1/16 comes from the overall normalisation of the superpotential in (2.12).
At first sight, the perfect matching (3.23) between the potentials (3.20) and (3.21) might appear as something to be expected from the consistency of the M-theory reduction. However, in order to have a standard twisted torus interpretation of the reduction, one has to impose the Jacobi constraints which are satisfied in a group manifold reduction [19]. Remarkably, the matching ( Reductions of type IIA strings on a twisted T 6 /Z 2 2 orbifold with fluxes and one single O6plane (orientifold ) have been extensively studied in the literature. Such orientifold planes split the space-time coordinates into transverse and parallel directions as follows O6 -plane : and can be located at the fixed points of the Z 2 involution The six-dimensional coordinates y m on X 6 split into orientifold-even y a (a = 1, 3, 5) and orientifold-odd y i (i = 2, 4, 6) sets under (3.25), as introduced in Table 1.
We will show that the N = 1 effective STU-models arising from type IIA orientifolds of X 6 = T 6 /Z 2 2 nicely fit within the framework of half-flat SU(3)-structure manifolds regardless of the Jacobi constraints for the metric fluxes. As for the previous M-theory case, what we will eventually find is a linear relation between type IIA metric flux components dressed up with the moduli and torsion classes. This will be shown explicitly in the case of vanishing axions. In this case Ω R and Ω I acquire a definite parity under the orientifold involution σ O6 such that Ω σ →Ω.

Type IIA metric fluxes and torsion classes
The symmetries of the X 6 = T 6 /Z 2 2 orbifold naturally induce an SU(3)-structure on X 6 specified by an invariant two-form J and a three-form Ω given by , where κ = k 1 k 2 k 3 τ 1 τ 2 τ 3 as previously introduced in (3.9). Notice that J and Ω in (3.26) correspond to two-and three-forms in six dimensions, unlike in (3.9) where they were understood as forms in seven dimensions. It is immediate to check that they satisfy the orthogonality now take the simpler form thus determining dJ and dΩ purely in terms of a real W 1 and W 2 . Such an SU(3)-structure is usually referred to as half-flat structure [33]. The above set of relations (3.28) can again be inverted to obtain the torsion classes as a function of J and Ω. This process gives Using the basis of left-invariant two-and three-forms H 2 (X 6 ) and H 3 (X 6 ) given in (A. 6) and (A.9), one finds the following expansions for the torsion classes where now, due to half-flatness, all the components in (3.30) are real. Although again quite tedious, the explicit computation of the torsion classes (3.29) is performed without surprises.
It results in the following expressions for the metric-flux-induced torsion classes. The torsion class W 1 reads in agreement with the structure found in the first line of (3.14). The three components with ω K in the expansion (3.30) of the W 2 torsion class are collectively given by also in agreement with the structure in the first line of (3.18). Finally the coefficients associated with the singlet β 0 and the triplet α K of basis elements in the expansion (3.30) of W 3 take the form

The matching of the scalar potentials
The set of torsion classes we obtained in the previous section can be used to compute to Ricci scalar (2.22). In this case, it has the simpler form Using the 6d metric in (3.11), which is compatible with (2.21) if setting λ −1 = 24 k 1 k 2 k 3 in (2.20), one finds the relation We are again interested in the relation between the purely metric-flux-induced contribution to the scalar potential coming from the superpotential (3.5), namely required to have a standard twisted torus interpretation of the reduction [19]. Therefore, the SU(3)-structure can potentially be used to lift background also including KK5-monopoles.
These sources were used to build simple de Sitter vacua in refs [34,35].

Beyond twisted tori
We have argued that the framework of G-structures is able to accommodate twisted reductions of M-theory/type IIA regardless of the Jacobi constraints on the twist parameters, namely, In the M-theory case of X 7 = T 7 /Z 3 2 , the first set of conditions in (3.39) amounts to require the 4d effective action to preserve all the 32 supercharges (N = 8) of the 11d theory [17,18].
However, in the type IIA orientifold case of X 6 = T 6 /Z 2 2 , the second set of conditions in (3.39) is not enough to guarantee the 16 supercharges (N = 4) of the orientifolded theory and additional metric-gauge flux conditions -in the form of tadpole conditionshave to be supplemented to ensure a vanishing net charge of O6/D6 sources [16,29]. These sources generically reduce the amount of supersymmetry in the effective action down to 4 supercharges (N = 1) and are secretly taken into account by the IIA superpotential (3.5) [28].
On the other hand, a non-vanishing r.h.s. in (3.39) amounts to having KK6-monopoles (M-theory) or KK5-monopoles (type IIA) in the background [20]. Upon an 11d → 10d reduction, KK6-monopoles give rise to KK5-monopoles as well as to O6-planes/D6-branes and more exotic sources associated to non-geometric type IIA fluxes [18,20,21]. We will discuss the higher-dimensional description of these sources later on in the paper. Now we will introduce a framework where to compare both M-theory and type IIA reductions with generic background fluxes and sources going beyond the twisted tori picture, i.e. not restricted by the conditions (3.39).

SU(3)-structures in six and seven dimension
Manifolds with SU(3)-structure in seven and six dimensions represent the natural framework to compare M-theory reductions on X 7 and type IIA orientifolds on X 6 . Expressing the G 2 -structure of X 7 "a la SU(3)" [11,17,36,37] will help us to understand what is the role played by the metric fluxes in M-theory that correspond to a R-R two-form flux F (2) [11,31,32,38] and to non-geometric fluxes in the type IIA picture [39,40].
Let us derive the SU(3)-structure of the seven-dimensional manifold X 7 = T 7 /Z 3 2 . It is specified in terms of the seven-dimensional invariant forms v (one-form), J (two-form) and Ω (three-form) introduced in (3.9). The failure of the closure of v, J and Ω is again identified with the presence of non-trivial torsion classes in the seven-dimensional manifold X 7 . An explicit computation reveals where W 1 , W 2 and W 3 were respectively given in (3.31), (3.32) and (3.33). We will concentrate on the contributions R 1 , R 2 , R 3 and R 4 in (3.40) as they parameterise how much does the seven-dimensional SU(3)-structure deviate from being understandable as a sixdimensional one. The piece R 1 has an expansion in terms of H 2 (X 6 ) given by and is induced by the M-theory fluxes corresponding to the R-R two-form flux F (2) in the type IIA picture. It is then easy to show that R 2 = 0 due the orbifold symmetries. The third piece R 3 has an expansion in terms of the basis elements of H 3 (X 6 ) given by where the coefficients read .

(3.43)
Finally, the last piece R 4 has an expansion in terms of the basis elements of H 3 (X 6 ) given this time by with coefficients ) which correspond to non-geometric fluxes in the type IIA picture (see Table 1). As a result, these types of metric fluxes in seven dimensions induce deformations in the geometry that cannot be retrieved in a six-dimensional setup and therefore look like non-geometric ingredients from a six-dimensional viewpoint.

Lifting STU-models to higher dimensions
Following the philosophy of refs [5,8], one can use the fundamental forms and torsion classes of G-structure manifolds in order to rewrite the M-theory (type IIA) background fluxes in Table 1 in a way that produces by construction a well-behaved stress-energy tensor with respect to the G-structure underlying the geometry. In this section we will show explicitly how this rewriting works and subsequently revisit some known supergravity solutions inspired by M-theory [18] and type IIA [29] compactifications (in a bottom-up sense) and reinterpret them as G-structure reductions.

Isotropic STU-models
The M-theory/type IIA four-dimensional minimal supergravities in refs [18,29] that we will uplift to 11d/10d correspond to so-called isotropic STU-models which are further invariant under a plane-exchange-symmetry [15] identifying the chiral moduli as  After the simplifications in (4.1) and (4.2), the isotropic M-theory superpotential in (3.4) takes the simpler form [18] W (iso) whereas the type IIA superpotential in (3.5) gets simplified to [15,29]

(4.4)
Based on the scalar potential derived from the N = 1 superpotentials (4.3) and (4.4), we will uplift to eleven and ten dimensions the entire set of maximally symmetric solutions found in ref. [18] and ref. [29], respectively.
Finally, the plane-exchange symmetry (4.1) of the STU-models amounts to identify the geometric moduli in (3.22) as These moduli determine the invariant forms of the underlying G-structure of X 7 and X 6 .
Combined with the simplification on the fluxes (4.2), the isotropy restriction will notably reduce the expressions for the G-structure intrinsic torsion.

Isotropic torsion classes and Ricci tensor of X 7
The Ricci tensor of the 7d manifold X 7 with co-calibrated G 2 -structure can be expressed in terms of the skew-symmetric three-form torsion [41] T = 1 The torsion class τ 0 in an isotropic background is given by whereas τ 3 still has the decomposition into a singlet and two triplets in (3.15). The singlet component reads (4.12) the democratic formulation of M-theory in a 4 + 7 splitting and restricting to those components which are invariant under the orbifold symmetries, the above action (4.6) is compatible with takingĈ (3)A1A2A3 as the dynamical gauge potential as well asĜ (4)µA1A2A3 ,Ĝ (4)A1A2A3A4 andĜ (7)A1···A7 as curvatures [17].
Upon the isotropic restriction, the three components of the first triplet τ 3 (I) become equal and take the form whereas the three components in the second triplet τ (I) 3 get also identified and read (4.14) The expression for the Ricci tensor reads Taking (4.10) and plugging in the torsion classes computed above, one finds a diagonal Ricci tensor R AB = diag ( r x , r y , r x , r y , r x , r y , r 7 ) , (4. 16) where the three functions r x , r y and r 7 depend both on the geometric moduli (k, τ, φ) and on the M-theory metric fluxes. The first one is given by whereas the second one reads The last function takes the form Let us discuss the set of components of the Ricci tensor of X 7 . It is worth noticing that by switching off the M-theory metric fluxes without counterparts in Table 1 as type IIA metric fluxes, namely a 2 = 0 and (d 0 , c 3 ) = (0, 0), one finds that r 7 = 0 and that the expressions for r x and r y get reduced to the first line in (4.17) and (4.18). Additionally turning on the M-theory metric flux a 2 that corresponds to a R-R flux F (2) in type IIA activates the second line in (4.17) and (4.18) as well as the first one in (4.19). Finally, the M-theory metric fluxes (d 0 , c 3 ) which are non-geometric in the type IIA picture are responsible for the last line in (4.17), (4.18) and (4.19). Computing the Ricci scalar R (7) by contracting with the (inverse) metric in (3.10) reproduces the result in (3.19).

Isotropic gauge backgrounds
Recalling the M-theory gauge fluxes in Table 1 with vol 7 = g (7) andŴ 3 ≡ W 3 /|W 3 | being the normalised version of the W 3 torsion class (3.33), and where It is worth noticing here that (4.21) is just a rewritting of the original expansion (4.20) in which we have used G-structure data as the basis for expanding forms instead of leftinvariant forms. The form of the G (4) flux in (4.21) was found in ref. [17] to be the one connecting M-theory to type IIA backgrounds upon reduction along the 11th space-time dimension. The g 1 function then maps to a R-R four-form flux F (4) = g 1 (k, τ, φ) J ∧ J whereas the quantity between parenthesis in e − 2 3 φ (...) ∧ v does it to a NS-NS three-form flux H (3) = g 2 (k, τ, φ) Ω I + g 3 (k, τ, φ)Ŵ 3 . This is also in perfect agreement with the universal Ansatz for type IIA reductions on SU(3)-structure manifolds investigated in refs [5,6,8]. We will elaborate more on type IIA reductions in the next section.

Matching between 11d and 4d EOM
Equipped with the results for the Ricci tensor of X 7 and the gauge backgrounds forĜ (4) andĜ (7) , it is now possible (and tedious) to check the eleven-dimensional equations of motion. Focusing on backgrounds with vanishing axions and constant geometric moduli, i.e. maximally symmetric solutions, we find six independent equations descending from (4.7) : three of them coming from the EOM ofĈ (3) and the other three coming from the Einstein equations. The BI forĈ (3) is automatically satisfied due to the orbifold symmetries [17].
From the EOM ofĈ (3) we obtain the equations whereas the Einstein equations reduce to As a summary, we have provided the eleven-dimensional uplift of any maximally symmetric solution (at vanishing axions) of the N = 1 superpotential (4.3) coming from twisted reductions of M-theory on an X 7 = T 7 /Z 3 2 orbifold with background fluxes. We have shown that X 7 corresponds to a G 2 -structure manifold whether or not the Jacobi constraints in (3.39) hold. For those cases in which they don't, the underlying G 2 -structure geometry accounts for KK6-monopoles in the background and supersymmetry is generically broken by the sources from N = 8 down to N = 1 [18,20].

Type IIA uplift of STU-models
Let us introduce the bosonic part of massive type IIA supergravity in ten dimensions. In the string frame, it is given by and with M = 0, ..., 9.
The above action contains a topological term of the form 27) and an extra piece accounting for the localised O6/D6 sources with j (3) representing the 3-form current associated with local D6/O6 sources. Taking these sources in the smeared limit, the contribution from the local sources can be rewritten as with a 3-form current, in the isotropic case, of the form The quantities N 6 = N O6 −N D6 and N ⊥ 6 = N O6 ⊥ −N D6 ⊥ respectively count the number of O6/D6 sources parallel and orthogonal to the orientifold directions and the functions j 1 (k, τ ) and j 2 (k, τ ) read From the above action (4.26), and in the smeared limit, one derives the following set of ten-dimensional EOM and BI for backgrounds with a constant dilaton φ. The EOM's for B (2) ,Ĉ (1) andĈ ( whereas the one for the dilaton φ reads Using purely internal background fluxes, which is more convenient to match results in the flux compactification literature, we will rewrite the first and third equations for the gauge potential in (4.32) as (4.34) We will use this last form of the equations when deriving the 10d/4d correspondence of the type IIA flux models.
The ten-dimensional Einstein equations take the standard form where the symmetric energy-momentum tensor T M N is defined as M N T loc. , (4.36) with the last term representing the contribution coming from sources. Focusing on the purely internal part, the contribution from the sources can be written as In addition to the above EOM, the set of BI for theB (2) ,Ĉ (1) andĈ (3) gauge potentials takes the form Plugging the set of type IIA fluxes in Table 1 into the second equation in (4.38) and using (4.31) one recovers the standard flux-induced tadpoles for the local sources O6 /D6 and O6 ⊥ /D6 ⊥ [39,40]. In the isotropic case, the number of such sources is then given by (4.39) The above combinations of fluxes in (4.39) are the relevant ones when uplifting four-dimensional backgrounds to ten dimensions. As we will see now, the isotropic restriction also simplifies the torsion classes W 1 , W 2 and W 3 specifying the half-flat SU(3)-structure in (3.28).
Isotropic torsion classes and Ricci tensor of X 6 Imposing the isotropic restriction on geometric moduli as well as on the type IIA fluxes, one finds that the torsion classes W 1 and W 2 read (4.40) The torsion class W 3 decomposing into a singlet and a triplet of components in (3.30) is given by (4.41) The Ricci tensor of the 6d manifold X 6 with half-flat SU(3)-structure can be recovered by setting a 2 = 0 and (d 0 , c 3 ) = (0, 0) in the expressions (4.17) and (4.18). This is turning off the M-theory fluxes corresponding to F (2) and non-geometric fluxes in the type IIA picture.
In this way one finds that the expression for the Ricci tensor is R mn = diag ( r x , r y , r x , r y , r x , r y ) , (4.42) where the functions r x and r y depending on the geometric moduli (k, τ ) and on the type IIA metric fluxes are given by Notice the identifications with the first lines in (4.17) and (4.18), respectively. As a check of consistency, the computation of the Ricci scalar R (6) using the (inverse) metric in (3.11) matches the result in (3.34).

Isotropic gauge backgrounds
By considering massive type IIA gauge fluxes in Table 1 with vol 6 = g (6) andŴ 3 ≡ W 3 /|W 3 | being again the normalised version of the W 3 torsion class (3.33). This form of gauge flux backgrounds was originally proposed in ref. [6], where it was denominated universal flux Ansatz due to its nice feature of automatically producing a well-behaved stress-energy tensor appearing in the Einstein equations, i.e. respecting the underlying SU(3)-structure. In our particular X 6 = T 6 /Z 2 2 isotropic orbifold, the functions entering (4.45) read and depend on the flux background parameters and the four-dimensional geometric moduli.
Matching between 10d and 4d EOM By combining the results for the Ricci tensor of X 6 with the flux Ansatz (4.45), we were able to mimic the calculation done in ref. [8] and check the ten-dimensional equations of motion.
First of all, let us take a look at the BI. The one forĈ 1 gives rise to the relations (4.31) defining the current j (3) as a funcion of the number of O6/D6 sources in (4.39). All the other BI are automatically satisfied because of the orbifold symmetries.
Focusing again on backgrounds with vanishing axions and constant geometric moduli, i.e. maximally symmetric solutions, we find six independent equations descending from the ten-dimensional EOM: one of them coming from the EOM ofB (2) , two of them from the EOM ofĈ (3) , one from the EOM of φ and the last two coming from the Einstein equations.
From the EOM ofB (2) we obtain the equation whereas from the EOM ofĈ (3) we obtain the equations The EOM of the ten-dimensional dilaton φ reads and the Einstein equations reduce to  Similarly to the M-theory case, we have now provided the ten-dimensional uplift of any maximally symmetric solution (at vanishing axions) of the N = 1 superpotential (4.4) coming from twisted reductions of massive type IIA on an X 6 = T 6 /Z 2 2 orbifold with background fluxes. We have shown that X 6 corresponds to a SU(3)-structure manifold whether or not the Jacobi constraints in (3.39) hold. For those cases in which they don't, the underlying SU(3)-structure geometry accounts for KK5-monopoles in the background and supersymmetry is generically broken by the sources from N = 8 down to N = 1 [18,20]. 31 We have investigated massive type IIA/M-theory reductions in the presence of background fluxes using the framework of G-structures and their intrinsic torsion. Taking a twisted orbifold as the internal space for the reduction -the X 6 = T 6 /Z 2 2 orbifold for type IIA and the X 7 = T 7 /Z 3 2 orbifold for M-theory -we have established a precise correspondence between four dimensional data, namely, twist parameters ω (metric fluxes) and moduli fields, and the torsion classes of the flux-induced G-structure underlying the reduction.
These types of twisted orbifold reductions produce classes of minimal supergravities dubbed STU-models which are specified in terms of N = 1 flux-induced superpotentials.
Remarkably, we observed that the Ricci scalar of the internal G-structure manifold computed from the flux-induced torsion classes exactly reproduces the scalar potential obtained from the flux-induced superpotentials without having to impose any extra condition on the twist parameters ω . In other words, we found a perfect matching regardless of whether or not the usual Jabobi constraints ω ω = 0 on the twist parameters are satisfied. These Jacobi constraints are required in an ordinary Scherk-Schwarz reduction, thus taking our G-structure reductions beyond the standard twisted tori picture.
Relaxing the Jacobi constraints has been connected to the presence of (smeared) KKmonopoles in the background [20]. Therefore, it becomes natural to propose the framework of G-structures and their intrinsic torsion as the natural playground where to describe KKmonopoles in a geometric manner. With this motivation, we have studied two different classes of STU-models and have provided their uplift to type IIA string theory with an internal SU(3)-structure manifold and to M-theory with an internal G 2 -structure manifold. In particular, we showed that any maximally symmetric solution to the four-dimensional equations is automatically a solution also to the equations of motion in ten or eleven dimensions.
The uplift nicely works without requiring the Jacobi constraints at any time in the computations. As a consequence, we can accommodate (smeared) objects such as KK5-monopoles in type IIA string theory and KK6-monopoles in M-theory and provide a higher-dimensional description of the AdS 4 vacua presented in refs [18,29]. However, even though we can map STU-models to explicit SU(3)-or G 2 -structures, it still does not mean that we in general know the actual geometry that realises the torsion classes. This is a problem that needs to be examined on a case by case basis.
Our uplifting formulas assume the sources to be smeared. From the point of view of type IIA string theory in ten dimensions this can be criticised. In general, at least when supersymmetry is broken, one can expect considerable differences between localised and smeared solutions. With localisation there are warp factors to be taken into account that will modify the effective four-dimensional dynamics. It is not even clear whether the solutions of the smeared models will have anything to do with solutions of the localised ones. The same criticism cannot be voiced against the M-theory scenario. There, everything is geometry and all of the 4d physics can be realised in the form of smooth configurations of geometry and gauge fields. The way KK-monopoles are captured by our construction is a nice illustration of how this may happen. In this context it is interesting to speculate that the smeared solutions in type IIA string theory actually do have a physical meaning even though one must go beyond IIA supergravity and use an M-theory perspective to make sense of them.
The relation between string theory and M-theory would be like the one between classical physics and quantum mechanics. It would be interesting to investigate this perspective further.