Geometric non-geometry

We consider a class of (orbifolds of) M-theory compactifications on $S^{d} \times T^{7-d}$ with gauge fluxes yielding minimally supersymmetric STU-models in 4D. We present a group-theoretical derivation of the corresponding flux-induced superpotentials and argue that the aforementioned backgrounds provide a (globally) geometric origin for 4D theories that only look locally geometric from the perspective of twisted tori. In particular, we show that Q-flux can be used to generate compactifications on $S^{4} \times T^{3}$. We thus conclude that the effect of turning on non-geometric fluxes, at least when the section condition is solved, may be recovered by considering reductions on different topologies other than toroidal.


Introduction
The issue of studying compactifications of string theory producing satisfactory phenomenology has always been of utmost importance from several different perspectives. In particular, dimensional reductions of type IIA string theory and the possibility of generating a perturbative moduli potential induced by gauge fluxes and geometry has been widely explored in the literature over the last decade.
More specifically, type II reductions on twisted tori with gauge fluxes have received a lot of attention over the years owing to the possibility of analysing them in terms of their underlying lower-dimensional supergravity descriptions. In this context, a central role is played by those string backgrounds that can be described by a class of minimal supergravity theories a.k.a. STU-models in four dimensions due to their remarkable simplicity.
However, the search for (meta)stable de Sitter (dS) extrema within the above class of STU-models has turned out to be unsuccessful [1][2][3][4]. A possible further development of this research line includes the possibility of taking some strongly-coupled effects into account. Therefore, a very natural framework is that of M-theory compactifications. The corresponding flux-induced superpotentials present a complete set of quadratic couplings induced by the curvature [5]. Still, in such a context, reductions on twisted tori are known not to allow for any dS solutions either [6].
Within those STU-models describing M-theory on twisted tori, all the couplings higher than quadratic are still judged as non-geometric [7], i.e. they do not admit any elevendimensional origin. Nevertheless, by moving to topologies other than toroidal, it is actually possible to find examples of flux superpotentials with homogenous degree higher than two.
1 A particularly enlightening case is that of reduction on S 7 yielding maximal SO (8) gauged supergravity in four dimensions admitting a truncation to an STU-model featuring quartic superpotential couplings. Analytic continuations thereof describe non-compact gaugings exhibiting unstable dS extrema where, however, the internal space is non-compact [8].
The goal of our work is to investigate which STU-models containing non-geometric fluxes can be understood as M-theory reductions on internal spaces with non-trivial topologies. It is worth mentioning that, by construction, all our models will admit a locally geometric description in the sense that they rely on an eleven-dimensional formulation correctly satisfying the section condition [9] in the language of Exceptional Field Theory (EFT) [10][11][12]. This is in the spirit of ref. [13] and does not lead to non-geometric duality orbits in the sense of ref. [14]. However, such a formulation will in general only be equivalent to the traditional one up to total derivative terms [15] that might play an important role upon reductions on non-toroidal topologies.
Even though our present work aims at shedding further light on the meaning of nongeometric fluxes, one cannot conclude anything about those non-geometric STU-models that were found to allow for stable dS critical points [16][17][18]. Whether it is possible to find novel examples of stable dS vacua satisfying the section condition still remains to be seen. Even so we expect that there will be compactness issues due to the no-go result in ref. [19].
The paper is organised as follows. We first present a group-theoretical truncation of maximal supergravity in four dimensions leading to isotropic STU-models with three complex scalars. We then employ some group theory arguments applied to the embedding tensor formalism in order to derive the flux-induced superpotentials describing M-theory compactifications on a twisted T 7 , S 7 and S 4 × T 3 . The result of this procedure will be a quadratic, quartic and cubic superpotential, respectively. We then discuss our results as well as some possible implications and future research directions. Finally, we collect some technical details concerning group theory in appendix A.

M-theory on Different Geometries and Topologies
The low-energy M-theory action in its democratic formulation reads .., 10. We choose the following reduction Ansatz where ρ represents the volume of the internal space X 7 and τ is suitably determined, such that the Ansatz in (2.2) yield a 4D Lagrangian in the Einstein frame.
In the second part of this section we will be considering different choices for X 7 within the class of S d × T 7−d leading to STU-models within N = 1 supergravity in 4D. We will start out revisiting the case of a twisted T 7 , and we will derive the flux-induced superpotential for this class of compactifications through group-theoretical considerations. This will help us construct our working conventions, which will be used in the analogous derivations carried out for different choices of X 7 other than twisted tori. Before we do this, we first need to introduce a particular group-theoretical truncation of maximal supergravity in 4D leading to the isotropic STU-models that we are interested in.  (7) and retaining only the G 0 -singlets. Such a truncation is guaranteed to be mathematically consistent due the E 7(7) covariance of the eom's of maximal supergravity.
In the second step, we perform a truncation to the SO ( The physical scalar fields span the coset space involving three SL (2) The Kähler potential of the theory reads In addition, the embedding tensor of the theory contains 40 independent components (coming this time from the decomposition of the 912 of E 7 (7) according to the chain in (2.4)) which can be viewed as the superpotential couplings 3 representing a complete duality-inviariant set of generalised fluxes [7]. This yields the following duality-covariant flux-induced superpoten- involving the three complex moduli S, T and U surviving the SO(3)-truncation introduced ealier in this section.
as well as those induced by their primed counterparts (F , H ) and (Q , P ) fluxes [24], (2.10) 3 The connection between the N = 1 and N = 4 theory was extensively investigated in refs [22,23].
However, the explicit agreement between the scalar potentials up to quadratic constraints was first shown in ref. [4].
For the sake of simplicity, we have introduced the flux combinations (2.8), and hence also the scalar potential.
In order to relate our 4D deformed supergravity models to M-theory reductions on different geometries and topologies, one needs to fix some conventions for assigning a Z 2 parity to the seven physical coordinates on X 7 . We adopt a set of conventions that is inherited from the link with type IIA compactifications with O6-planes [25], where such a parity transformation can be viewed as orientifold involution.
realise the compact geometry of X 7 . Retaining only even fields and fluxes w.r.t. the action of the above Z 2 will automatically restrict the supergravity theory obtained through an M-theory reduction to the framework of N = 1 STU-models.
The metric (2.2) splits accordingly into The relationship between the STU scalars and the above geometric moduli reads

Compactifications on a twisted T 7
The seven compact coordinates of the torus transform in the fundamental representation of the SL (7) subgroup of E 7 (7) , which can be viewed as the group of diffeomorphisms on T 7 with twist. The relevant chain of decompositions is and finally down to  where one should, furthermore, only restrict to isotropic objects. Following the philosophy of ref. [25], one can match the naïve scaling behaviour coming from dimensional reductions of the various terms in the action (2.1) with the correct STU-charges by using the relations in (2.13). This results in the following mapping between the group-theoretical R + charges obtained from the above decomposition and the STU-charges realised in N = 1 supergravity. Such a mapping allows one to derive a dictionary between fluxes and superpotential couplings.
From the decomposition of the fundamental representation of E 7(7) (see appendix A for the details)  (3) i . As far as the fluxes are concerned, one has to decompose the embedding tensor of maximal supergravity E 7(7) ⊃ SL (8) , to find where we have used the decomposition in (A.1). By using the dictionary (2.14), we were able to reproduce all the correct STU scalings of the fluxes on a twisted T 7 . The results of this procedure are collected in table 1 and agree with those already found earlier in refs [5,6]. As a consequence, the flux-induced superpotential in this case reads

Compactifications on S 7
Let us now consider the compactification of M-theory on S 7 . In refs [28,29] it was already noted that such a compactification is described by an SO(8) gauging in 4D maximal supergravity. The components of the embedding tensor are parametrised by a symmetric 8 × 8 matrix Θ AB transforming 4 in the 36 of SL(8). 4 We adopt the following conventions X A ≡ X a , X i , X 7 , X 8 for the fundamental representation of SL (8).  In the relevant S 7 example, the embedding tensor reads We will now interpret this theory as an STU-model and rederive the corresponding fluxinduced superpotential by means of group theory arguments. In this case the relevant decomposition is still the same as the one in the twisted T 7 case

(2.24)
This gives the STU-couplings collected in table 2 upon using the dictionary (2.14). The associated flux-induced superpotential is given by which matches what was found in refs [27,32] in the context of STU-models. The N = 1 scalar potential computed from (2.25) coincides with (2.23) upon using the correct identification of the STU scalars inside the coset representative M AB .

Compactifications on S 4 × T 3
We have seen how for S 7 the superpotential contains only the constant part and some quartic parts. We will now analyse the flux-induced superpotential for S 4 × T 3 to find that cubic terms will appear, thus mimicking the effect of the presence of Q-flux.
Given the natural factorisation that X 7 has in this case, the relevant branching one should analyse goes through and finally down to where, as usual, only isotropic objects should be retained within our STU-model. By following the new branching of the fundamental representation of E 7(7) 26) and demanding that the physical derivative operators identified in (2.16) be the same, one as a new dictionary between STU-scaling weights and group-theoretical R + Q × R + 1 × R + 2 charges. Note that this procedure of identifying the seven physical derivative operators 9 corresponds to choosing the relevant solution to the section condition in the EFT sense. In this case, several a priori different choices are possible but they all yield a superpotential that is unique up to modular transformations.
As far as the fluxes are concerned, in total analogy with the S 7 case, now we expect to be able to describe the S 4 geometry with G (4) flux by turning on embedding tensor deformations transforming in the 15 of SL(5), i.e. a symmetric 5 × 5 matrix Θ IJ . This would in itself lead to a maximal SO(5)-gauged supergravity in 7D [33].
However, these deformations can be supplemented with G (7) flux wrapping the whole Out of these Model 1 is the only choice that is compatible with SO(3)-invariance thus yielding an isotropic STU-model. This is the model we will focus on, and for which we will provide details. Dealing with Model 2 and Model 3 requires further breaking SL(3) a × SL(3) i symmetry down to SL(2) a × SL(2) i × R + a × R + i , this giving rise to non-isotropic STUmodels. We will only sketchily show that such non-isotropic superpotentials will still be cubic.
• Model 1: In this case our decomposition contains the following relevant pieces The results of this procedure are collected and shown in table 3. The associated fluxinduced superpotential is given by STU couplings M-theory fluxes Flux labels R +  One should note that the geometry of the twisted T 3 sits in the 36. The explicit way the corrsponding ω-flux is embedded inΘ is given bỹ where now the indices a, b, c, . . . label the legs of the T 3 .
Thus, in contrast with the S 7 case, such a background lies in the non-semisimple branch of solutions to the constraints (2.20) where three suitable and uniquely determined linear combinations of the five R + charges introduced above represent the STU-charges. As already anticipated earlier, the resulting superpotentials are non-isotropic and their explicit form is beyond the present scope. Nevertheless, the qualitative analysis of the STU couplings induced by fluxes in these models are respectively collected in tables 4 and 5.

Discussion
In this paper we have considered compactifications of M-theory on manifolds with non-trivial topologies. After reviewing the twisted T 7 and the S 7 cases, we also analysed the S 4 ×T 3 case.
By means of a group-theoretical approach, we have derived the flux-induced superpotentials in all the different cases in question. While the twisted T 7 superpotential contains terms that are at most quadratic in the complex scalars, the S 7 and S 4 × T 3 , contain some quartic and cubic terms, respectively.
The appearence of the aforementioned higher-degree superpotential couplings may be  which would correspond to turning on the Romans' mass after reduction on a circle down to type IIA. This is in line with the statement that the mass parameter cannot be written as a derivative of any gauge field in a generalised geometry language [35].
Nevertheless, whether or not such models actually contain new physics needs to be checked case by case. The origin of this open question is to be found in the local equivalence between different reformulations of eleven-dimensional supergravity that relate inequivalent solutions to the section condition. At a global level, the different and locally equivalent Lagrangians may differ by total derivative terms that might become important upon dimensional reduction. Such a fact has been already investigated in ten dimensions in the context of the so-called β-supergravity [36,37].
In particular, as far as (meta)stable dS extrema are concerned, all the examples known so far generically violate the section condition, thus being genuinely non-geometric. A possible future issue to be addressed is the existence of such (meta)stable dS vacua within locally geometric backgrounds that can be made globally geometric by following our approach.
Another possible line of research left open by our analysis is the possible relevance of exotic differentiable structures on spheres. The idea that our approach might capture some information about those comes from observing that the expression of the scalar potential associated with the S 7 reduction contains volume scaling behaviours that are beyond the ones predicted by its ordinary Riemannian structure. In ref. [29], it was already observed that M-13 theory solutions on the S 7 seem to require other parallelisable differentiable structures beyond the Riemannian one. This might be seen as an evidence that maximal supersymmetry and 11D supergravity are in fact sensitive to exotic differentiable structures on S 7 . If this turns out to be the case, one could imagine using M-theory reductions, and their underlying lowerdimensional gauged supergravity descriptions, to test the presence of exotic differentiable structures in other cases of special mathematical interest like, e.g., S 4 .