U-dual fluxes and Generalized Geometry

We perform a systematic analysis of generic string flux compactifications, making use of Exceptional Generalized Geometry (EGG) as an organizing principle. In particular, we establish the precise map between fluxes, gaugings of maximal 4d supergravity and EGG, identifying the complete set of gaugings that admit an uplift to 10d heterotic or type IIB supegravity backgrounds. Our results reveal a rich structure, involving new deformations of 10d supergravity backgrounds, such as the RR counterparts of the $\beta$-deformation. These new deformations are expected to provide the natural extension of the $\beta$-deformation to full-fledged F-theory backgrounds. Our analysis also provides some clues on the 10d origin of some of the particularly less understood gaugings of 4d supergravity. Finally, we derive the explicit expression for the effective superpotential in arbitrary N = 1 heterotic or type IIB orientifold compactifications, for all the allowed fluxes.


Introduction
T-duality is a distinctive symmetry of String Theory, in the sense that stringy aspects come into play. In its simplest form it identifies a theory compactified on a circle of a given radius with a theory compactified on a circle of inverse radius and exchanges compact momenta and winding modes, which simply do not exist in a field theory of particles.
When compactification on a more general background with d isometries is considered, T-duality action is enhanced to an O(d, d, Z) group that, among other features, mixes the metric modes with the antisymmetric NSNS B 2 field components. In a 2d sigma model approach, Buscher rules [1] indicate the precise way in which target space fields transform under T-duality, highlighting the fact that different backgrounds lead to the same CFT.
When fluxes are turned on [2], the situation becomes richer and more intricate. A The last T-duality transformation in eq.(1.1) is more obscure. In fact, it has been conjectured [3] from the 4d effective action viewpoint that if a T-duality transformation along a direction p is performed in a background with Q mn p fluxes, we would arrive to a "truly non-geometric" flux R mnp for which not even a local 10d supergravity description is available. In this sense, the resulting 4d effective supergravity should not be thought of as a dimensional reduction of a 10d effective supergravity background, but as a 4d theory that incorporates information of the full string theory.
From the above chain of T-dualities a "stringy" generalization of the concept of Riemannian geometry seems to emerge, on which symmetries of the B 2 field are taken now on equal footing than diffeomorphisms. Thus, in a series of papers starting from [8,9], the concept of Generalized Geometry [10] (see also [11] for an introduction to the topic) was proposed as a natural framework to describe string compactifications with fluxes. In besides diffeomorphisms, an O(6, 6) action involving, in particular, the B 2 field. 2 1 Strictly speaking, the usual derivation of Buscher rules is actually not allowed in these cases since the isometry is not globally well defined. However, following these rules even in these "obstructed" cases seems to be meaningful. For instance, to some of these solutions an interpretation in terms of asymmetric orbifolds can be given [6] (see also [7]). 2 It is worth noticing that, in this description, fields do depend on six dimensional coordinates of the as backgrounds for the RR and NSNS field-strengths of 10d supergravity, metric fluxes or β-deformed backgrounds, our analysis also reveals new deformations related to the RR counterparts of the β-deformation. These provide the natural extension of the latter to full-fledged F-theory backgrounds (see also [23]). Moreover, our analysis also sheds light on the 10d origin of some of the particularly less understood gaugings of 4d N = 4 supergravity, such as the ones transforming in the vector representation of O(6, 6) [25] (see also [26,27,24] for related work).
In the second part of the paper then we consider aspects related to the global SU (3) structure of N = 1 compactifications. More precisely, making use of the tools of EGG, we write explicitly the effective superpotential for all the U-dual fluxes allowed in general N = 1 heterotic or type IIB orientifold compactifications.
A more detailed outline of the paper is as follows. In Section 2 we introduce the basic concepts of Generalized Geometry. In particular we discuss the action of different generators of O(6, 6) structure group in connection with diffeomorphisms, B 2 gauge transformations and the extra β transformation, associated to non globally defined geometries.
The interpretation of NSNS field strength fluxes associated to such transformations and how they combine to fulfill an O(d, d) representation is also presented. In Section 3 we address the inclusion of RR fields into geometry, defining EGG. The structure group is thus enlarged to the full E 7 U-duality group and the generalized tangent space becomes 56 dimensional. U-dual field strengths are discussed in Section 4. In Section 5 we consider the simplest compactifications, that is toroidal compactifications, in the presence of general dual fluxes. We discuss the algebra and global constraints that U-dual fluxes must satisfy, and derive the flux induced effective superpotential in these compactifications.
The relation with 4d N = 8 and N = 4 gauged supergravity is also discussed in that Section. Section 6 is devoted to the formulation of N = 1 backgrounds in EGG and to the construction of N = 1 untwisted sector superpotentials for general SU(3) structure compactifications. Section 7 provides some final comments. Some notation and useful results related to the E 7 algebra are summarized in the Appendices. At the intersection of two patches U α and U β , the generalized vectors X = x+ξ (where x is a vector and ξ a one-form), are identified as follows where a ∈ GL(d, R) (and a −T ≡ (a −1 ) T ) is a conventional diffeomorphism, b αβ is a twoform and ι x ′ β means a contraction along x ′ β ≡ a αβ x β . This patching corresponds to the following transformation h ∈ O(d, d) The presence of the 2-form b 2 in the patching implies that the generalized tangent bundle is not just the sum T M ⊕ T * M but is an extension of the tangent bundle by the cotangent one where b αβ describes how T * M is fibered over T M. b 2 is required to be locally exact, i.e.
b αβ = dΛ αβ and (in an analogous way as one patches a U(1) bundle) the local 1-forms Λ αβ should satisfy the cocycle condition The gauge parameters Λ αβ allow to define a local 2-form gauge field B α whose patching is characterized by a bi-vector β 2 . The "β-transform" action on a generalized vector is One could a priori allow for patchings involving these transformations as well. This is perfectly fine as long as β 2 is a globally defined bi-vector. If that is not the case, there is no gauge transformation like (2.5) that allows to define it patchwise. Roughly speaking, this is because the derivative in (2.5) has an index down, and even when combined with a vector it cannot define a bi-vector. This is related to the fact that the manifold is still d-dimensional, even if we have defined an extended 2d-dimensional tangent bundle on it.
Hence, whenever β 2 is not globally well-defined, there is no "gerbe-like" construction such as that outlined above for B 2 .
The 36-dimensional space of metric and B 2 field parameterize the coset O(6,6) O(6)×O (6) , or in other words g and B 2 combine to define an O(6) × O(6) structure on E. This structure can also be defined by the splitting of the generalized tangent bundle into two orthogonal 6-dimensional sub-bundles E = C + ⊕ C − such that the natural O(6, 6)-invariant metric decomposes into a positive-definite metric on C + and a negative-definite metric on C − .
The sub-group of O(6, 6) that preserves each metric separately is O(6) × O(6). One can now define a positive definite generalized metric A generic element of C + or C − cannot be purely a vector or a one-form, since these are null with respect to η. We can therefore write X + ∈ C + as x + M x for some matrix M. Taking M = B 2 + g, the patching condition (2.1) implies g α = g β , while B 2 is patched according to (2.5). Orthogonality between C + and C − implies that an element X − ∈ C − can be written as and parameterizes the space of scalar fields of the resulting 4d effective supergravity. The generalized metric can be acted by O(6, 6) transformations. The effect of diffeomorphisms is changing the metric and B 2 field accordingly (namely (g, One can therefore think about a background with a given B 2 field B as the b 2 -transform with b = −B of a background with no B 2 field. A β-transformation leads to a more complicated action so that it does not seem possible to tell apart the new g and the new B 2 . However, it is always allowed to perform an O(6) × O(6) transformation in the stabilizer of g + B 2 (i.e., a transformation that does not change the metric and B 2 field) such that the β-transform rotates into a diffeomorphism and a b-transform. In other words, given a metric and B 2 field at a point on the manifold, the full O(6,6) O(6)×O(6) orbit can be reached by acting only with the geometric subgroup of O(6, 6) that implies that β-transforms can be locally "gauged away". We will come back to this point in the next Section.
The transformations b 2 and β 2 are related by T-duality. In fact, there is another basis for O(d, d) given by diffeomorphisms, b-transforms and T-dualities along any two directions on the tangent space. A T-duality along the first two basis vectors is realized by the following O(6, 6) transformation (2.10) where I 2,4 are the 2 × 2 and 4 × 4 identity matrices. Hence, T-duality transforms each It is not hard to check that the action of T-duality along a fiber f on a generic element (split among its base and fiber components) in the Lie algebra o(d, d) is Here the index A = 1, ..., 2d is split into d indices up and d indices down, and each of them is further split into fiber and base indices. We have also definedâ ≡ −a T and b 2 and We see that the effect of T-duality is, roughly speaking, to raise and lower f -indices and move the building block to the corresponding new place according to the structure of the indices. Hence, transformations purely on the base are not touched by T-duality. A fiber-base (base-fiber) diffeomorphism is exchanged with a fiber-base (base-fiber) b 2 (β 2 )-field, while b 2 and β 2 purely along the fiber are interchanged. A β-transform with one or two legs along the fiber can therefore be generated by T-dualizing a diffeomorphism or a b 2 field with respectively one and two legs along the fiber.  [12,13]. Even though we will stick to a six-dimensional manifold, from a purely group-theoretic point of view we shall introduce a "derivative up" ∂ m , T-dual to the standard derivative ∂ m , such that it combines with the latter to form a 12-dimensional 1-form ∂ A . The 220

T-dual field strengths
representation of the fluxes is therefore obtained by 3 Exceptional Generalized Geometry and U-dual gauge fields Following [29], in this section we incorporate also the RR fields to the geometry, similarly to what we did for B 2 in the previous section. With that aim we extend the structure group of the generalized tangent bundle to E 7 .

The 56 representation of the gauge parameters
In order to geometrize the RR fields at the same time as the B 2 field, one needs to extend the generalized tangent bundle to one whose structure group is the full U-duality group  5 Examples of this are given in [28].
where i = 1, 2, and a minus denotes a sum of odd forms on the internal space. 6 The 56 degrees of freedom can be accounted for by gauge parameters. Six come from gauge transformations of the B 2 field, given by one-forms, and six from vectors pointing in the directions of the diffeomorphisms. Those build the fundamental 12 representation of O(6, 6). We have to add to them gauge transformations of the RR fields, given by odd forms in the 32' representation (in this paper we will concentrate on type IIB, where the RR potentials are even, and their gauge transformations are given by odd forms).
However, in order to fill out the 56 representation (or in other words, to have a closed set under U-duality), we need another 12 parameters. These are the magnetic duals in the NS sector, namely a 5-form, corresponding to gauge transformations of B 6 (the dual of B 2 ), and the duals of the diffeomorphism vectors, given by elements of T * M ⊗ Λ 6 T * M (whose corresponding gauge charges are the Kaluza-Klein monopoles). The 56dimensional exceptional generalized tangent space is therefore given by The embedding of GL(6) ⊂ O(6, 6) × SL(2, R) picks out a vector that breaks the 2 of SL(2, R) into 1 + 1 such that in one direction that we will call v i , the 12 representation of O(6, 6) is built out of a vector and a 1-form, while in the other (called ω i ) it contains a 5-form and a 1-form tensor a volume form. This is the result of an uneven assignment of GL(6) weights (see Appendix A of [29] for details). Without loss of generality one can There are 4d gauge fields associated to each of the above gauge parameters. These come from 10d gauge fields with one space-time index. In terms of the notation in eq.(3.1), the 4d vectors are whereb m µ ≡ 1 5! ǫ mnopqr b nopqrµ (i.e. it is the Hodge dual of the 6-form b 6 with one external and five internal indices),k mµ ≡ k mµ 123456 is the magnetic dual of the vector a µ associated with diffeomorphisms, andc − µ ≡ p=2n+1 1 p! ǫ i 1 ...i 6 c µi 1 ...ip are a sum of odd multi-vectors corresponding to the Hodge dual of the RR potentials with one space-time index.

The 133 representation of the gauge fields
The analogue of the O(6, 6) action on the generalized tangent space is an E 7 action on E.
The adjoint 133 representation of E 7 decomposes under O(6, 6) × SL(2, R) as Under the GL(6) ⊂ O(6, 6) × SL(2, R) embedding, these further decompose into [29] A b 6 and c + transformations form the geometric subgroup G geom ⊂ E 7 used to patch the exceptional generalized tangent space (see more details in [18,29]). Using the notation in (3.5) we embed the generators of the geometric subgroup of E 7 in the following way [29] A geom = where c + = c 0 + c 2 + c 4 + c 6 and v i = ǫ ij v j .
As in the generalized geometric case, the geometric subgroup is a semi-direct product where G A is the nilpotent subgroup corresponding to the shift symmetries A p → A p + a p for the p-form gauge fields B 2 , B 6 and C + . The remaining elements of E 7 play an analogous role to the β-transformation in the O(6, 6) case: they can locally be gauged away by SU(8) ⊂ E 7 transformations that do not change the metric, dilaton, B and C-fields (which together define an SU(8) structure), but in the case where they are not globally well defined, they might carry some topologically non-trivial flux that signals a non-geometric background. Besides the bi-vector β 2 , the non-geometric gauge fields are a sum of even vectors that we will call γ + (which are SL(2, R)-dual to the RR gauge fields), a scalarĉ 0 , and a six-vector β 6 that together with b 6 form the triplet of SL(2, R). The full 133 representation arranges in the following way with respect to the To make contact between E 7 and Generalized Geometry, we have decomposed the fundamental and adjoint representations of E 7 into representations of O(6, 6) × SL(2, R).
In this decomposition, the O(6, 6) subgroup is the one of Generalized Geometry, whereas the SL(2, R) factor corresponds to fractional linear transformations of the complex axiondilaton which appears in heterotic string compactifications, The embedding of O(6, 6) × SL(2, R) into E 7 is, however, not unique. For instance, while T-duality (in the adjoint of O(6, 6)) acts straightforwardly on eq.(3.8), type IIB S-duality seems more complicated. The latter should exchange b 2 and c 2 , which are contained in different representations of O(6, 6) × SL(2, R). This implies that S-duality is not in the SL(2, R) piece, but it is a combination of generators in both SL(2, R) and O(6, 6).
We can therefore select another decomposition O(6, 6) × SL(2, R)| B ⊂ E 7 for which the SL(2, R) subgroup is the one that contains type IIB S-duality, which acts on by fractional linear transformations. In this basis, eq.(3.8) is reexpressed as, Note that type IIB S-duality now corresponds to the exchangeω →ṽ,ṽ → −ω. Thus, from eq.(3.11) we observe that γ 2 is the S-dual of β 2 (which is itself T-dual of b 2 ), while β 6 and γ 6 are also related by S-duality.
In what follows, we will refer to the above two decompositions as O(6, 6) × SL(2, R)| H and O(6, 6) × SL(2, R)| B . To relate fields in one decomposition to fields in the other, it is useful to assign every gauge parameter in the 56 representation and every gauge field in the 133 a weight vector, in the same spirit than e.g. [20].
Given that choice, the assignment of weights is unique in order to reproduce the algebra satisfied by (3.8), given in eq.(A.4). We summarize the weight of each gauge field in Table   1 below, where we have made use of the shorthand notation ± ≡ ± 1 2 . Weight vectors for the O(6, 6) × SL(2, R)| H and O(6, 6) × SL(2, R)| B decompositions of E 7 are related by a change of basis. Given the assignment of roots in the O(6, 6) × SL(2, R)| H basis, in order to find such change, it is enough to require that b 2 and c 2 form a doublet of S-duality in the new basis, and that diffeomorphisms are mapped to themselves. For convenience, we take the case on which a → a T . Callingẽ a , a = 1, ..., 8, the orthonormal basis vectors in the type IIB basis, we get the following dictionary between the two bases (3.14) Thus, for instance decomposition, which has the structure given in eq. (3.11). We present also in Table 1 the assignment of weights in this type IIB basis.
field  From Table 1 we observe that (in the O(6, 6) × SL(2, R)| H basis) G geom contains all positive roots, the Cartans and a few negative roots (corresponding to a i j , i < j) which are minus a simple root. 7 This is referred to as a parabolic subgroup. The further subalgebra containing just the shift symmetries b 2 , b 6 and c + is its unipotent radical. The geometric subgroup G geom contains in particular the Borel subgroup of E 7 , generated by all positive roots and Cartans. As it has been commented in the previous subsection, it is possible to locally gauge away all the transformations which lie outside the Borel subgroup by means 7 In the O(6, 6) × SL(2, R)| B basis one can change conventions such that the same thing happens.
of SU(8) transformations. The 70 dimensional space of fields in the Borel subgroup (the dilaton, the metric and the B and C + fields) define a SU(8) structure on the generalized tangent bundle and parameterize the coset E 7 /SU (8). This is also the space of scalar fields of the resulting 4d effective supergravity (7 fundamental scalars corresponding to the Cartans, 48 axions and 15 diffeomorphisms corresponding to the positive roots, which transform non-linearly under E 7 /SU(8) [30,31]). After orientifolding only half of the supersymmetries are preserved, and E 7 is broken to the U-duality group of the resulting 4d N = 4 supergravity, O(6, 6) × SL(2, R). In this context, the change of basis vectors in eqs.(3.13)-(3.14) corresponds to T-dualizing along all the coordinates of the internal T 6 and applying type I -SO(32) heterotic duality, thus dualizing from a type IIB compactification with O3-planes to a heterotic compactification.
By comparing the assignment of weights for the two basis in Table 1 (or alternatively eqs.(3.8) and (3.11)) we can see how gauge transformations are mapped under this type IIB -heterotic duality. For instance, the heterotic b 2 is mapped to the type IIB c 4 , whereas the heterotic b 6 is mapped to the type IIB c 0 . We will see more details on toroidal compactifications with general fluxes in Section 5.
One may easily check that states transforming in spinorial representations of O(6, 6) (e.g. 32, 32 ′ , 352, etc.) have negative parity under the orientifold action, whereas the remaining states have positive parity. In particular, note that the gauge fields surviving 8 Note that we are making a rather general use of the term 'orientifolding', referring also to the nonperturbative description of an orientifold. In this general sense, the heterotic string would be for instance considered an orientifold of type IIB String Theory through the duality chain type IIB → type I → heterotic.
the orientifold projection are precisely those in the adjoint of O(6, 6) × SL(2), for both decompositions. In terms of weight vectors, we can therefore introduce an operator P such that it acts on a given state k as This operator is identified in heterotic compactifications with the space-time fermionic number for left-movers (−1) F L , whereas in the type IIB basis corresponds to the orientifold where Ω P is the worldsheet parity and σ an orientifold involution which reverses all coordinates of T 6 . Acting with P on the states in Table 1 we see indeed that and γ 4 are kept in the O(6, 6) × SL(2, R)| B basis. We have stated in Table 1 the parity for the different fields under (−1) F L and Ω P obtained in this way.

U-dual field strengths
The discussion of field strengths associated to U-duality covariant gauge potentials follows closely the one for T-dual field strengths in Section 2.  , where the traceless condition on the 352 representation (which corresponds to a traceless vector-spinor) is encoded in the absence of a 0-form in the first line, the absence of a 6-form in the second line, and the extra condition P ′m mi 2 i 3 i 4 = 0 that has to be imposed on this flux. Note that as defined in eq.(4.3), f A+ does not satisfy Γ A f A+ = 0, but it has the same number of degrees of freedom of a traceless vector-spinor. We summarize in Table   6 of Appendix B the assignment of E 7 weights for the field strengths in both basis and their parity under (−1) F L and Ω P . 10 To shed light on the 10d supergravity uplift of these field strengths we can compute their explicit expression in terms of derivatives of the fields in Table 1, similarly to what we did in eqs. (2.12) for T-dual field strengths. In the language of representation theory, this is equivalent to computing the Clebsh-Gordan coefficients for the 912 representation expressed in terms of elements of the 56 and 133. 9 Our notation for some of the fluxes is slightly different from that of [5]. In particular, H ′ ↔ F ′ and P ′ ↔ Q ′ are exchanged. We find this notation more suited to the (−1) FL charges of the fluxes. Moreover, Q → −Q and ω → −ω with respect to [5].
10 Depending on the context, we represent the tensor structure of the fluxes in slightly different ways by making use of the 6d antisymmetric tensor. Thus, for instance, The notation has been chosen in such a way that there is not possible ambiguity.
With that aim, we take the highest weight of the 912 representation expressed as a linear combination of weights belonging to the tensor product 56 × 133, The numerical coefficients have been determined in such as way that the r.h.s. of the equation vanishes when acted with any positive root of E 7 , as corresponds to the highest weight of a representation. We refer the reader to Appendix A for further details on the E 7 algebra and root system.
In terms of the elements in Tables 1 and 6, eq.(4.4) reads, where, for convenience, we have introduced a generalized exterior derivative in the 56 The assignment of weights is presented in Table 2.
Note that the first term in the r.h.s. of eq.   The procedure can be systematized with the aid of the computer. 11 In terms of where Γ M are the O(6, 6) Gamma matrices, which act on forms by 12 There are some subtleties that have to be taken into account, however. In particular, notice that there can be independent sequences of negative roots which result in the same weight of 912. This is the origin of the multiple copies of the same weight appearing in the (220, 2) and (12,2), or in the (352, 1) . In order to disentangle weights appearing in various representations of O(6, 6) × SL(2, R) it is important to stress that in the conventions of Appendix A, E −αi , i = 1, . . . , 6 is also a basis of negative roots of O(6, 6), whereas the negative root of SL(2, R) is given by the combination where the sequence of subindices is 7, 6, 4, 5, 3, 4, 6, 7, 2, 3, 4, 6, 5, 4, 3, 2, 1. 12 ι m means a contraction along m, which acts like ι m and the bracket denotes the Mukai pairing defined by  Table 1 (i.e. we set to zero all the exotic derivatives in eq.(4.6)), and which therefore admit an uplift to locally geometric solutions of 10d type IIB supergravity. We can organize them as follows: 1. NSNS fluxes: 2. RR fluxes: 3. Metric fluxes: 5. γ-deformations (RR): Any vacuum of the 4d theory involving only these backgrounds should admit a consistent, locally geometric, description in terms of 10d type IIB supergravity. Indeed, the uplift to 10d is almost automatic for NSNS, RR and metric fluxes, as these field strengths are in one to one correspondence with field strengths of type IIB supergravity. From the algebraic point of view they correspond to derivatives of elements in G geom . On the other hand, elements which lie outside the Borel subalgebra (β-and γ-deformations) in general require also local SU(8) transformations to be described as a locally geometric 10d background.
Backgrounds of 10d supergravity which involve β-deformations have been considered in the recent literature [34,35], resulting important in the context of the AdS/CFT correspondence for understanding some of the marginal deformations of N = 4 super Yang-Mills [36]. Backgrounds of 10d supergravity involving γ-deformations, on the other hand, have been much less studied (see, however, [23] for some partial results). As it has been commented, γ-deformations provide the RR counterpart of the β-deformation.
The orientifold projection selects field strengths which lie in the (220, 2) + (12, 2) representation. These can be read directly from Table 6, for both type IIB with O3-planes where S B is the type IIB complex dilaton defined in eq.(3.10). As we will see in Section 6, N = 1 supersymmetry equations require the above combination to be an anti-holomorphic (0, 2) complex bi-vector. We postpone further comments on this type of backgrounds to that Section.
It is also interesting to stress that some of the field strengths in eqs. Hence, according to Table 4, most of the field strengths in the (12, 2) representation admit a priori an uplift to 10d locally geometric type IIB or heterotic supergravity backgrounds. These are related to non-traceless metric fluxes, β-and γ-deformations. These field strengths will turn out particularly relevant in the context of 4d N = 4 gauged supergravity, as it will become manifest in next Section.
weight IIB with D3/D7 heterotic Components between parenthesis do not admit a 10d locally geometric description.

Toroidal compactifications and dual fluxes
Up to here we have been discussing various aspects related to the local E 7 structure of the generalized tangent bundle, or the O(6, 6)×SL(2, R) subgroup which survives after taking the orientifold projection. We have in particular introduced a set of gauge parameters, gauge fields and field strengths which are covariant under the structure group.
In the present (and forthcoming) Section we consider the inclusion of topologically non-trivial fluxes for these field strengths. The global structure of the tangent bundle therefore becomes relevant also. In that regard, here we consider the simplest possible situation, namely compactifications on manifolds of trivial structure (tori, or "twisted tori"). Since tori are parallelizable manifolds, global and local structure groups in this case coincide. The more interesting case of (not necessarily toroidal) compactifications with global SU(3) structure will be treated in Section 6.
The correspond to the embedding tensor of 4d N = 4 gauged supergravity [25].
In this context, it results particularly interesting the observation found in Section 4 that all gaugings in the (12, 2) representation admit an uplift to locally geometric solutions of 10d supergravity. From the 4d point of view, some of these gaugings are known to be related to twists by an axionic rescaling symmetry [26]. Their higher dimensional origin, however, has been a more obscure and longstanding problem. Some of them have been identified as arising on particular Scherk-Schwarz reductions of 10d heterotic supergravity [27]. This is consistent with Table 4, where we observe that gaugings in the first row admit an uplift to heterotic compactifications with non-traceless metric fluxes. Also, more recently it has been shown that some of these gaugings admit an uplift to type IIA orientifold compactifications with dilaton fluxes [24]. In this regard, the results of Section 4 reveal that all gaugings in the (12, 2) representation can be also understood as originating from type IIB orientifold compactifications with non-traceless β-and γdeformations.

U-dual gauge algebra and constraints
The different fluxes, grouped in the 912 representation of the U-duality group E 7 , are expected to obey diverse constraints. In fact, from the 10d point of view, Bianchi identities as well as tadpole cancelation equations must be satisfied. In [23] it was shown, for a particular type IIB orientifold, that these Jacobi identities are actually identified with the Bianchi identities and tadpole equations of the higher dimensional theory. Therefore, once the dictionary between structure constants and fluxes is known we should be able to read the flux constraints directly from the Jacobi identities of the 4d gauge algebra.
In a generic situation (see for instance [16]), the function F P AB expressed in terms of the fluxes is not antisymmetric in the subindices A, B and therefore does not define a con-sistent algebra. 13 In order to ensure antisymmetry of the commutators, extra constraints must be imposed. Namely, the symmetric part of the structure constants, contracted with generators must vanish, F P (AB) X P = 0. Jacobi identities are therefore satisfied when a contraction with a third generator is considered. In terms of structure constants this new condition reads In what follows we sketch a possible way to obtain the expression of the structure constants in terms of fluxes. The idea is to follow similar steps as in Ref. [23]. In other words, by starting with a previously known sector of the algebra (5.1), we can apply E 7 generators to it in order to construct the entire algebra. Similarly, the generators of the algebra are decomposed accordingly to eq.(3.1), where S − is a weight belonging to the 32' representation. Alternatively, X ′ S − can be expressed in terms of a set of multi-vectors, X ′ S − = X ′ m ⊕ X ′ mno ⊕ X ′ mnopq . 13 In the general situation where F P (AB) = 0, in order to covariantize the Lagrangian 4d gauge bosons have to be supplemented with the 2-forms which result from dualizing the scalars of the theory [38].
The general structure of the algebra, written in terms of O(6, 6) × SL(2, R) content hence reads where A = (Ai, S − ) and B = (Bj, S ′− ). In order to start with, we choose the subsector given by the fluxes transforming in the (12, 2) + (220, 2) representation of O(6, 6) ×

SL(2, R)
[X Ai , X Bj ] = F P k AiBj X P k (5.9) In this case the explicit expression for structure constants is known to be [25] F kP with f Ai and f ABCi given in eq.(4.2), while the antisymmetry constraint is Notice that performing a similar analysis as above in terms of representations, the left member of eq.(5.9) leads to For the sake of clarity let us present a concrete example and choose i = 1, j = 2 and A = 1 + 6, B = 2 + 6. Then, in terms of type IIB fluxes, we read from eq.(5.9) where, in order to avoid confusion with the notation we have defined X m ≡ X (m+6)1 , X m ≡ X (m+6)2 , m = 1, . . . , 6 (5.14) to denote the two elements of the SL(2) doublet in (12,2), with weights (0, 1, 0, 0, 0, 0, ∓, ±) (where 1 is in the m-th entry). The first row of (5.13) contains fluxes in the (220, 2) representation, summarized in Table 3, while the second row corresponds to fluxes in the (12,2), presented in Table 4.
As mentioned, by applying E 7 generators the full algebra can be reconstructed. For instance, by acting with E α 7 , the generator corresponding to the positive simple root and a spinor in the (32 ′ , 1). Hence, acting in the left hand side of (5.13) and taking into Here, X ′m is the generator corresponding to the weight (−, −, −, +, −, −; 0, 0) of the 32' representation, with + in the m-th position.
In the same way we can obtain the terms on the right hand side. Thus, by acting with E α 7 on the fluxes we see that whereas the action on the other terms which appear in the right hand side of eq.(5. 13) vanishes. Putting all together, and proceeding in a similar way for the other [X m , X n ] commutators, we get [X ′m ,X n ] = ω ′mn pX p + (P ′p,mn + 1 √ 2 P mnp )X p + 2ω ′mX n + ω ′nX m (5.18)

Type IIB orientifold compactifications
We focus here on type IIB compactifications on T 6 /[Ω P (−1) F L σ ×Γ], where the orientifold involution σ reverses all coordinates of T 6 . In particular, σ(J) = J and σ(Ω) = −Ω. There are O3-planes spanning the space-time directions and, depending on the particular choice of Γ, there can be also O7-planes wrapping complex 4-cycles within the T 6 . Consistently with this, D3 and/or D7-branes may be also required in the compactification to cancel the total RR charge.
Following [39] we can introduce a basis for 3-forms in the covering T 6 as, In terms of these, the holomorphic 3-form can be expanded as, where, The scalars U mq correspond to moduli of the complex structure defined by Ω in T 6 , Similarly, the moduli parameterizing deformations of the Kähler structure of T 6 can be extracted from the complexified 4-form, 14 where ω lm is a basis of integer 4-forms even under the orientifold involution, 15 The complex axion-dilaton S B has been defined in eq.  Table 5.
The metric of this moduli space is given in terms of the Kähler potential [40], 14 We have taken C 4 → −C 4 and J c → −J c with respect to [5,23] in order to match the usual conventions in Generalized Complex Geometry. 15 Notice that before applying the orbifold projection Γ, the first of the equations in (6.1) below is generically not satisfied, and it is only after the projection that J and Ω define an SU(3) structure.  An explicit expression of the second integral in terms of the complex structure moduli for a general T 6 can be found, for instance, in [39]. The superpotential (5.27) is at the core of many of the recent phenomenological approaches to string theory, where moduli stabilization is a must. However, in Section 4 it was recast that F 3 and H 3 are only a piece of the (220, 2) representation of O(6, 6) × SL(2, R)| B . In Table 3 we summarized all the elements in the representation.
We can therefore generalize eq.(5.27) to incorporate all the remaining fluxes in the (220, 2) representation. Following the same strategy than in [3,5,23], we act with the O(6, 6) × SL(2, R)| B generators on (5.27), or rather, on the invariant superpotential, G = log|W O3 | 2 +K O3 , withK O3 given in (5.26). For that we observe that W O3 transforms with weight (1, 1, 1, 0, 0, 0; − 1 2 , 1 2 ) and moduli transform with weights summarized in Table  5. We can build O(6, 6)/[O(6)×O(6)] covariant 3-forms linear in the fluxes by considering contractions of the fluxes in Table 3  Based on the above observations, it is possible to show then that the superpotential involving the full (220, 2) representation of fluxes is given by, where, and similarly for P · J c , Q ′ · J 2 c and F ′ · J 3 c . The reader may check that all terms in this equation indeed transform with weight (1, 1, 1, 0, 0, 0; − 1 2 , 1 2 ), as desired. Of course, one has to bear in mind that in concrete models many of the flux components which appear in (5.28) are projected out by the orbifold action Γ. In particular, one may check that (5.28) reproduces the results derived by similar arguments in [5,23] for the case of Γ = Z 2 × Z 2 and G = SL(2, R) 7 . 16 In addition, the quadratic constraints 16 For completeness we present here the embedding of SL(2, R) 7 ⊂ E 7 selected by the Z 2 × Z 2 orbifold. This is given by 7 coordinate vectors, σ i , one per SL(2, R) factor. In the conventions of [5] for SL(2, R) 7 , these are given by, withẽ i the orthonormal vectors of the B basis, introduced in Section 3.3.
derived in Section 5.1 require in general further components to vanish. Hence, many of the vacua which result from (5.28) are actually related by 4d electric-magnetic duality.
Systematic analysis of the the vacuum structure induced by the first two pieces in the above superpotential for Γ = Z 2 × Z 2 have been carried out recently in [42].
We could consider also fluxes transforming in the (12, 2) representation of O(6, 6) × SL(2, R)| B , since they survive to the orientifold projection too, by acting with E 7 generators on eq.(5.28). We postpone however the discussion of these fluxes to Section 6.3, where the complete flux induced effective superpotential is derived in the broader context of general SU(3) structure compactifications.

Heterotic orbifold compactifications
The To get the full superpotential, the starting point is the known effective superpotential for heterotic compactifications with 3-form NSNS flux and torsion [43,44,45,46] W het. = Ω ∧ (H 3 + idJ) (5.32) We can now proceed as before in order to extend this superpotential to the full set of fluxes transforming in the (220, 2) representation of O(6, 6)×SL(2, R)| H . These are summarized in the last column of Table 3. After some algebra we arrive to the expression, where the contractions are defined as, and similarly for Q ′ , ω ′ and H ′ (c.f. footnote 10).
It is illuminating to express this superpotential in terms of the tensors f ABCi and f Ai , defined in eqs. (4.3), which in the present context are just the embedding tensors of the parent N = 4 gauged supergravity theory [25]. We find the compact expression where we have included also fluxes transforming in the (12, 2) representation whose superpotential will be derived in Section 6.3. G 3 and G 1 are defined as where a subindex 2 (1) refers to the direction v i (ω i ), and the contraction · is the O(6, 6) action f ABCi Γ ABC and f Ai Γ A , with the gamma matrices acting as in eq.(4.12). In what follows we first review the O(6, 6) and E 7 covariant formulations of N = 1 backgrounds, following respectively [14,15] and [29]. Then we address in Section 6.3 the computation of the effective superpotential in general SU(3) structure compactifications with arbitrary fluxes.
6.1 O(6, 6) formulation of N = 1 backgrounds Four-dimensional N = 1 supersymmetry requires the existence of nowhere vanishing internal spinors η 1 and η 2 such that the 4d supersymmetry parameter ε is obtained from the two 10d ones ǫ 1,2 by We have chosen the chirality of the 10d spinors to be negative. There is no requirement a priori on the relative orientation of the spinors η 1,2 . Each of them is invariant under an SU(3) structure on the 6-dimensional space. If the two spinors are the same, they give rise to a single SU(3) structure. Whenever the spinors do not coincide, the two SU (3) structures intersect into an SU (2). If there are points on the manifold where the spinors are parallel, there is no global SU(2) structure, but only a local one.
The two spinors can be combined to form complex pure spinors of O(6, 6) Making use of Fierz identities, it is possible to express Φ ± 0 as sums of spinor bilinears, namely By chirality, only even (odd) p contribute to Φ + (Φ − ) and these can equivalently be thought of as sums of even or odd forms in the 32 and 32' representations of O(6, 6). In the special case where the two spinors coincide, i.e. η 1 = η 2 , the pure spinors read where J and Ω are those of eqs.(6.1).
By construction Φ ± 0 are pure, i.e. each of them is annihilated by half of the twelve O(6, 6) gamma matrices Γ A , splitting the generalized tangent bundle into a 6-dimensional complex holomorphic bundle (given by i ± = 1, ..., 6 such that Γ i ± Φ ± 0 = 0) and its complex conjugate antiholomorphic bundle. 17 In other words, Φ ± 0 define each a generalized (almost) complex structure J ± 0 , satisfying (J ± 0 ) 2 = −I, that can be obtained from the spinors by where O(6, 6) gamma matrices act as in (4.12), and the bracket is the Mukai pairing defined in eq.(4.13), which is the natural bilinear on O(6, 6) spinors.
Two compatible pure spinors define a generalized (positive definite) metric on the generalized tangent space. Using the generalized almost complex structures associated to the pure spinors, the generalized metric is given by This is the same metric as in eq.(2.9). For the pure spinors (6.5), it gives a block diagonal matrix (corresponding to B 2 = 0), where g is obtained in the standard way from an SU (3) structure (in complex coordinates g i = −iJ i ).
In order to obtain a generalized metric that contains a B 2 field, one needs to Btransform the pure spinors, i.e. to apply an O(6, 6) transformation corresponding to a B 2 field to eq.(6.5) (or more generally to eq.(6.3)). Making use of eq.(A.4) and the representation for gamma matrices in eq.(4.12), it is not hard to see that the B-transform on the pure spinors amounts to a wedge product of B 2 and the component forms. Exponentiating the action we get Φ ± = e B 2 Φ ± 0 which for the case η 1 = η 2 , corresponding to a single SU(3) structure, implies In terms of the pure spinors, the Kähler potential for the N = 1 theory in the context of type IIB compactifications reads [15] (6.9) 17 Furthermore, Φ ± 0 are by construction compatible, which implies that the two 6-dimensional holomorphic bundles have a 3-dimensional intersection.
It is easy to see that this is equivalent to (5.26). Note that the contribution of B 2 drops out, as it should be since the Mukai pairing is an O(6, 6) invariant, and the B 2 field is an adjoint O(6, 6) action.
The second term in the Kähler potential should be written in terms of the N = 1 variables J C and S B . These can be read off from a combination of Φ + , the dilaton and the RR potentials into the following complex form [15] where θ is an angle that defines the N = 1 orientifold projection. For O3/O7 planes θ = 0, while for O5/O9, θ = π/2. In the case of type IIB compactifications with O3- . Hence the 0-form component of Φ + C is the complex axion-dilaton S B , while the 4-form component is precisely J C defined in eq.(5.24). Note that even if B 2 , C 2 and C 6 are projected out of the spectrum in this case, we keep them in Φ + C as their fluxes are not. In terms of Φ + C the O(6, 6)-covariant superpotential is given by Here the fluxes F 3 and H 3 are encoded in their local definition in terms of the gauge fields, i.e. F 3 = dC 2 , H 3 = dB 2 . It is not hard to check that this reproduces eq.(5.27).
The term in eq.(5.28) involving the locally geometric NSNS flux Q ij k can also be easily encoded in (6.11). For that, note that in general Φ + C = e B 2 (Φ + C ) 0 , where (Φ + C ) 0 is built out of C + and the pure spinor Φ + 0 . If instead of a B-transform one acts by a β-transformation, the action of d on β 2 gives the Q · J C term in eq.(5.28).
The remaining NSNS fluxes Q ′ and H ′ in the 220 representation are locally nongeometric and their contribution to the superpotential cannot be obtained from (6.11), not even promoting d to an element in the 12 of O(6, 6). The reason for this is that, as explained in Section 4, these fluxes require extending the derivatives even further, going beyond the fundamental representations of O(6, 6) to the 56 representation of E 7 .
Similarly, non-geometric RR fluxes should also only appear when promoting the O(6, 6) covariance to a full E 7 one.

E 7 formulation of N = 1 backgrounds
We review now the E 7 covariant formulation of N = 1 backgrounds that descend from N = 2 ones, following [29]. We work in the O(6, 6) × SL(2, R)| H basis, where the O (6,6) subgroup is that of Generalized Geometry.
The first step is to embed Φ − 0 and Φ + 0 into representations of E 7 . The easiest way is to embed them in the 32' and 32 representations of O(6, 6) that appear respectively in the decomposition of the 56 and 133 representations of E 7 (c.f. eqs.(3.1) and (3.5)). This assignment is also consistent with the degrees of freedom in the N = 2 theory: those in Φ − are the scalars of vector multiplets, while the deformations of Φ + build up, together with the dilaton and the RR axions, the hypermultiplets.
The moduli space is a direct product of these two moduli spaces, which implies in particular that Φ − 0 should be a singlet under SL(2, R)| H . The pure spinor Φ − 0 is therefore embedded as follows where we have used the notation in eq.(3.1).
The other spinor, Φ + 0 , combines with the dilaton to form a doublet of SL(2, R)| H . It should also combine with the RR fields and transform non-trivially under the SU(2) R Rsymmetry. The way to realize all these conditions is to promote Φ + 0 to an SU(2) R triplet of elements (K 0 1 , K 0 2 , K 0 3 ), with K 0 a ∈ 133, satisfying the real su(2) algebra [K 0 a , K 0 b ] = 2κǫ abc K 0 c , with κ = e −2φ vol 6 . Using the notation in eq.(3.5), we define K 0 where v i , ω i are those introduced in Section 3.1 and S H is the heterotic complex dilaton defined in eq.(3.9). K 0 − is just the complex conjugate of K 0 + , while K 0 3 , obtained by demanding the su(2) commutation relations, is given by with u ≡ (v i S H + ω i ), uū = (S −S) H = 2ie −2φ vol 6 and J +A 0 B is the generalized almost complex structure defined by Φ + 0 in eq.(6.6).
In an analogous way as for the b-transform of Φ ± , we can define b 2 , b 6 and c + transformed objects of λ 0 and K 0 as Hence, defining the N = 1 field as we have that the spinor component along ω i in the (2,32) piece of K C is indeed the N = 1 chiral field in eq.(6.10) for type II compactifications.

General U-duality covariant superpotential
We are now ready to compute with the above tools the U-duality covariant superpotential for general SU(3) structure compactifications. In terms of the geometric objects λ and K C the superpotential in the E 7 covariant formulation is given by [29] W = S(λ, DK C ) . (6.18) Here D is the generalized derivative defined in eq.(4.6), transforming in the 56 representation, K C = gK 0 C and λ = gλ 0 , with g = e A a generic group element of E 7 . S is the symplectic invariant in the 56 representation, whose O(6, 6) × SL(2, R) decomposition is given in eq.(A.5) of the Appendix and z a is the vector introduced in (6.16). For consistency DK a , the generalized derivative of K a , is projected onto the 56 representation.
Extracting the group elements g and making use of the symplectic invariance, the superpotential above can be recast as where the derivative is now acting on the bare objects with a superindex 0, while derivatives of the gauge fields (the fluxes) have been encoded in the generalized connection [29,47], with While in Ref. [29] the derivative was restricted to the standard derivative, D = (∂ m v i , 0), and at the same time the gauge fields were taken to be in the geometric subgroup, here we consider the full D defined in eq.(4.6) and the full set of 133 gauge fields. Hence, the tensor F AB C ∈ 912 involves all the field strengths introduced in Section 4. The generalized connection applied to K CD in the 133 and projected onto the 56 gives  (6.24) This is almost exactly the same expression than the one that we derived for toroidal heterotic orbifold compactifications, eq.(5.33), except that we are missing the complexification of J. A bare gauge field like B 2 appears because the connection F AB C in (6.21) is actually defined in terms of generalized derivatives of the group elements g ∈ This factor can be actually absorbed by a Kähler transformation of (6.9), leading to the canonical Kähler metric for heterotic string compactifications, given in eq.(5.31).
Hence, taking all these observations into account, we have that the flux induced superpotential, derived from eq.(6.18), for general heterotic compactifications with global SU(3) structure is given by, where contractions were defined in eqs. (5.34), (6.26) and similarly for ω ′ · J 2 c . In a similar way we can compute the superpotential for type IIB compactifications with O3-planes. This requires a little more work. The orientifold projection sets in this 18 We have canceled the vol 6 factor in S H with inverse volume factors in the ω i components of the fluxes (see eq.(4.3)).
The fluxes surviving the projection are in the (2,12) and (2, 220) representations of Table 4). In the notation of eq.(4.2), they read f iABC =ṽ i F abc + Q ab c + P ′a bc ǫ 123456 + H ′abc ǫ 123456 +ω i H abc + P bc a + Q ′a bc ǫ 123456 + F ′abc ǫ 123456 . (6.29) Making use of vol 6 = J 3 /6 and the fact that (J + ) mn and (J + ) mn are respectively proportional to J mn and J mn , we obtain where the contractions are as in eqs.(5.34) for the (220, 2) fluxes, while Q m and P a act by a single contraction, as in (6.26).
We proceed now to express (6.30) in terms of the N = 1 variables S B and J c defined in (3.10) and (5.24) respectively. For that aim, we observe that J and J 3 , which build up Im(Φ + 0 ), are given in terms of Re(Φ + 0 ) by the derivative of the Hitchin functional [10,14].
This leads to the useful relations Note that eq.(6.30) is also missing the contribution from the RR axions C 0 and C 4 .
The reason is the same than for the heterotic superpotential. For instance, notice that when considering higher orders in the exponential e A , eq.(A.4) tells us that the group element corresponding to the 2-form part alongṽ i (whose generator is c 2 ) gets a shift Taking into account these two facts, it is not hard to show that the flux induced superpotential derived from eq.(6.18) for general type IIB orientifold compactifications with O3-planes and global SU(3) structure is, Let us stress that expressions (6.25) and (6.32) have been derived in the context of general SU(3) structure compactifications, and therefore not only apply to toroidal orbifold compactifications, but also to general N = 1 compactifications of string theory.
It is also interesting to comment on the possible uplift of superpotential (6.32) to Ftheory. Indeed, superpotential (5.27) has a well-known interpretation in terms of F-theory compactifications on elliptically fibered Calabi-Yau 4-folds [41]. From this point of view, NSNS and RR 3-form fluxes correspond to different components of the M-theory/F-theory 4-form, where θ i is a basis of integral 1-forms in the elliptic fiber, which we have complexified in the r.h.s. of this expression. In terms of G 4 , the superpotential (5.27) becomes, where Ω 4 is the holomorphic 4-form, whose existence and uniqueness is guaranteed by the SU(4) holonomy of the Calabi-Yau 4-fold.
Similarly, the second piece in eq.(6.32) also admits a natural uplift to F-theory. This term is related to a background for the field strength of the complex parameter (4.19).
As it has been noted in [23], topologically non-trivial β 2 and γ 2 -deformations (that is, Q ij k and P ij k fluxes) correspond to locally geometric type IIB backgrounds with a holomorphic complex dilaton on which there is a deficit of (p, q) 7-brane charge. Thus, generic transition functions gluing different patches contain not only diffeomorphisms and shifts of the B and C fields, but also T-dualities and/or SL(2)| B rotations.
The form of (6.32) suggest that β 2 and γ 2 are different components of a 1-form along the F-theory fiber, so that the first two terms in eq.(5.28) are recast as, Note also that taking ∂ U mq W = W = 0 (in absence of 3-form fluxes), for all complex structure moduli, requires (Q − S B P ) · J c to be a primitive (2, 1)-form. This fact, together with the condition obtained from imposing ∂ T mq W = 0 for all Kähler moduli, implies that (4.19) has to be an anti-holomorphic (0, 2) bi-vector in order to satisfy the N = 1 supersymmetry equations.

Conclusions
We have performed a detailed analysis of the relation between Exceptional Generalized Geometry and 4d gauged supergravities, providing an organizing principle for generic string flux compactifications. The U-duality group, E 7 , which turns out to be also the structure group of the generalized bundle in EGG, encodes much information on the 10d origin of 4d gaugings. In particular, we have established a precise dictionary between weights of the 912, 133 and 56 representations of E 7 on one side, and fluxes, gauge fields and gauge parameters, respectively, on the other. Moreover, from different ways of decomposing E 7 into O(6, 6)×SL(2, R), we have identified different orientifold projections in 10d, generalizing the results of [20].
The Clebsh-Gordan decomposition of the 912 representation onto the tensor product 56 × 133 has provided us also with explicit local expressions for the field strengths in terms of the gauge potentials. This in particular allows to disentangle fluxes which admit an uplift to 10d supergravity backgrounds, from locally non-geometric fluxes, whose field strengths involve exotic exterior derivatives and which should be understood as 4d gaugings resulting from String Theory backgrounds which do not admit a higher dimensional supergravity limit. Hence, the uplift of these backgrounds to 10d should be thought directly in terms of String Theory compactifications. In addition, the E 7 group structure allows us also to distinguish globally geometric from globally non-geometric compactifications.
In this way, we have identified systematically all gaugings of 4d supergravity which admit an uplift to backgrounds of 10d supergravity. Some of these backgrounds were already known, corresponding to fluxes of the NSNS and RR field-strengths, metric fluxes or β-deformed backgrounds. Our analysis, however, reveals other types of 10d supergravity backgrounds apart from the above ones. These turn out to be related to the RR counterparts of the β-deformation, which we have dubbed γ-deformations. We have The above analysis gives also some clues on the possible 10d origin of some of the particularly less understood gaugings of 4d supergravity. In particular, we have shown that 4d gaugings of N = 4 supergravity transforming in the (12, 2) representation of O(6, 6) × SL(2, Z) can be uplifted to backgrounds of 10d type IIB supergravity, corresponding to β and/or γ-deformed backgrounds. Our results are also consistent with [27], where it was noticed that a small subset of these gaugings can be also interpreted as particular Scherk-Schwarz reductions (metric fluxes) of heterotic supergravity.
Whereas the above results directly apply to toroidal orbifold or orientifold compactifications, where in absence of fluxes the bulk preserves N = 8 or N = 4 supersymmetry, they are also relevant in more generic setups. The reason is that even if in generic N = 1 compactifications the global structure group is reduced to SU(3), the structure group of the generalized tangent bundle is still given by O(6, 6) × SL(2, R). Hence, in the last part of this work we have made use of the tools of EGG to derive the effective superpotential induced by the fluxes in general N = 1 heterotic compactifications or type IIB compactifications with O3 and/or O7-planes. These superpotentials, which we have derived here in a 10d context, allow for the study of the F-term conditions associated to general backgrounds, in particular those which involve β and γ-deformations apart from standard fluxes. We hope to come back to this point in future work.
A Summary of group theory results on E 7 A.1 E 7 root and weight system E 7 has rank seven (7 Cartan generators) and dimension 133. In the Cartan-Weyl basis, the commutation relations of the algebra read, where H I , I = 1 . . . 7, denote matrix representations of the I-th Cartan of E 7 , and similarly, E α denote matrix representations of the generator associated to the root α ∈ ∆, with ∆ the root lattice of E 7 . Even if ǫ α,β = ±1 can be explicitly computed, we will not need them in our computations. Representations are constructed by systematic application of generators (associated to simple roots) to states, without anticommuting them.

A.2 Some relevant formulas
The action of the adjoint representation (with parameter A) on the fundamental representation, decomposed in elements of O(6, 6) × SL(2, R) is The adjoint action on the 133 representation (with parameter A ′ ) is given by δA = [A ′ , A] where The E 7 symplectic invariant is S(λ, λ ′ ) = ǫ ij λ iA λ ′jB η AB + λ + , λ ′ + , (A.5) The projection on the 56 in the tensor product 912 × 133 is given by where n 1 ...n 9 are some constant coefficients which we leave undetermined.

B Tensor structure of U-dual fluxes
In the following tables we summarize the tensor structure of fluxes transforming in the whereas Q ′[i,jklm] , Q ij j , ω i im and ω ′i ij (Q ′[i,jklm] , Q ij j , P ij j and P ′[i,jklm] ) transform in the (220, 2). Similarly, P i , P ijk and P ijklm (ω i , ω ′i and P ijk ) transform in the (32 ′ Table 6: Field strengths transforming in the 912 of E 7 . We use the shorthand notation ± ≡ ± 1 2 .