N=1 vacua in Exceptional Generalized Geometry

We study N=1 Minkowski vacua in compactifications of type II string theory in the language of exceptional generalized geometry (EGG). We find the differential equations governing the EGG analogues of the pure spinors of generalized complex geometry, namely the structures which parameterize the vector and hypermultiplet moduli spaces of the effective four-dimensional N=2 supergravity obtained after compactification. In order to do so, we identify a twisted differential operator that contains NS and RR fluxes and transforms covariantly under the U-duality group, E7(7). We show that the conditions for N=1 vacua correspond to a subset of the structures being closed under the twisted derivative.


Introduction
Since the seminal paper of Candelas, Horowitz, Strominger and Witten [1], the geometrical perspective in compactifications of string theory from ten to four dimensions had great insights. Supersymmetry conditions have been shown to constrain the allowed internal manifolds to certain specific classes. When there are no fluxes, the internal spaces should be Calabi-Yau. Such manifolds satisfy an algebraic condition, namely the existence of a globally defined, nowhere vanishing, internal spinor, and a differential one, that the spinor is covariantly constant. The algebraic condition is necessary in order to recover a supersymmetric (N = 2) effective theory in four dimensions, while the differential one is required in order to have supersymmetric vacua. In the presence of fluxes, the algebraic condition stays intact (i.e., in order to have N = 2 supersymmetry off-shell, a globally defined internal spinor is needed), but the differential one becomes more intricate.
The role of fluxes in string theory, combined with the warped nature of the compactification, has become of primary interest mainly for the possibility of fixing moduli and providing a hierarchy of scales [2]. This motivated the search for a geometric description of backgrounds with fluxes, which was very much guided by the framework of generalized geometry developed by Hitchin [3,4]. In rough terms, generalized complex geometry is complex geometry applied to the generalized tangent bundle of the space, consisting of the sum of tangent and cotangent bundles. The parameters encoding the symmetries of the metric plus the B-field, namely diffeomorphisms plus gauge transformations of B, live on this bundle. This formulation has therefore a natural action of T-duality, which exchanges these two. On the generalized tangent bundle one can define (generalized) almost complex structures, and study their integrability (integrable generalized complex structures allow to integrate the one-forms dZ i and find global complex coordinates). Generalized almost complex structures are in one-to-one correspondence with pure spinors, which are built by tensoring the internal spinor with itself and with its charge conjugate. Spinors on the generalized tangent bundle are isomorphic to sums of forms on the cotangent bundle, and the integrability condition for the structure is nicely recast into closure of the pure spinor under the exterior derivative twisted by H 1 .
Generalized complex geometry was used in [5,6] to characterize N = 1 vacua. In analogy with the fluxless case, off-shell supersymmetry requires an algebraic condition, namely the existence of two pure spinors on the generalized tangent bundle. N = 1 vacua require one of the pure spinors to be closed (and therefore the generalized almost complex structure associated to be integrable), while RR fluxes act as a defect for integrability of the other structure. In order to geometrize the RR fields as well, and give a purely algebraic geometrical characterization of the vacua (which would allow, for example, to study their deformations, i.e. their moduli spaces, in a model-independent manner), one needs to extend the generalized tangent bundle such that it includes the extra symmetries corresponding to gauge transformations of the RR fields. Such extension has been worked out in [7,8], and was termed exceptional generalized geometry, alluding to the exceptional groups arising in U-duality. In this paper we study compactifications of type II down to four-dimensions, where the relevant group is E 7 (7) .
The algebraic conditions to have off-shell N = 2 supersymmetry in four-dimensions 1 Integrability condition is actually weaker, it requires (d − H∧)Φ = XΦ for some generalized tangent vector X, where Φ is the pure spinor corresponding to the generalized almost complex structure.
have been worked out in [9]. Very much in analogy to the generalized complex geometric case, they require the existence of two algebraic structures on the exceptional generalized tangent bundle (in fact one of them, rather than a single structure, is actually a triplet, satisfying an SU(2) R algebra), which are built by tensoring the internal spinors. The SU(2) R -singlet structure, that we call L, describes the vector multiplet moduli space, while the triplet of structures (named K a ) describes the hypermultiplets. The N = 1 preserved supersymmetry breaks the SU(2) R into U(1) R , selecting a vector r a along this U(1), and a complex orthogonal vector z a . The complex combination z a K a describes the N = 1 chiral multiplets contained in the hypermultiplets 2 .
In this paper we obtain the differential conditions on the algebraic structures L, K a required by N = 1 on-shell supersymmetry 3 . The first step is to identify the appropriate twisted derivative that generalizes d − H∧ to include the RR fluxes, or in other words to identify the right generalized connection. Such connection is obtained as in standard differential geometry by the operation g −1 Dg, where g are the E 7 -adjoint elements corresponding to the shift symmetries (the so-called "B-and C-transforms"), and the derivative operator D is embedded in the fundamental representation of E 7 [9]. The key point is that this connection, which a priori transforms as a generic tensor product of adjoint and fundamental representations, should only belong to a particular irreducible representation in this tensor product, which in the case at hand is the 912. Having identified the appropriate connection, we rewrite supersymmetry conditions in terms of closure of the structures. The equations we get are given in (5.12)-(5.16). We find that N = 1 supersymmetry requires on one hand closure of L, as conjectured in [9]. On the other hand, the components of the twisted derivative of r a K a with an even number of internal indices have to vanish, while those with an odd number are proportional to derivatives of the warp factor. A similar thing happens with z a K a , except that this time closure occurs upon projecting onto the holomorphic sub-bundle defined by L.
The paper is organized as follows: in section 2 we review the basic features of generalized geometry and its extension achieved by exceptional generalized geometry. In section 3 we present the relevant algebraic structures for compactifications with off-shell N = 2 supersymmetry. In section 4 we review the constrains on the (traditional and generalized complex) structures imposed by on-shell supersymmetry. In section 5 we study supersymmetric vacua in the framework of exceptional generalized geometry. In particular, we introduce the twisted derivative operator in 5.1, we present the N = 1 equations in 5.2, and finally in section 5.3 we outline the proof that supersymmetry requires those equations. Various technical details, as well as the full derivation of the equations, are left to Appendices A to G.

Generalized Complex Geometry
In this section we present the basic concepts of Generalized Complex Geometry (GCG) in six-dimensions (we will restrict to the six-dimensional case, though most of what follows can be generalized to any dimension), which will be used as a mathematical tool for describing flux vacua.
In Generalized (Complex) Geometry, the algebraic structures are not defined on the usual tangent bundle T M but on T M ⊕ T * M, on which there is a natural metric η η = 0 1 6 Following the language of usual complex geometry, a generalized almost complex structure (GACS for short) J is a map from T M ⊕ T * M to itself such that it satisfies the hermiticity condition (J t ηJ = η) and J 2 = −1 12 . One can define projectors Π ± for the complexified generalized tangent bundle as which can be used to define a maximally isotropic sub-bundle (six-dimensional) of T M ⊕ T * M as the i-eigenbundle of J There is a one-to-one correspondence between a GACS and a "pure spinor" Φ of O(6, 6). A spinor is said to be pure if its annihilator space is maximal (here · refers to the Clifford action X · Φ = X A Γ A Φ). The one-to-one correspondence is then One can construct the GACS from the spinor by Weyl pure spinors of O(6, 6) can be built by tensoring two O(6) spinors (η 1 , η 2 ) as follows where the plus and minus refers to chirality, and φ is the dilaton, which defines the isomorphism between the spinor bundle and the bundle of forms. Using Fierz identities, these can be expanded as Using the isomorphism between the spinor bundle and the bundle of differential forms (often referred to as Clifford map): the spinor bilinears (2.8) can be mapped to a sum of forms. Under this isomorphism, the inner product of spinors Φχ is mapped to the following action on forms, called the Mukai pairing and the subindex 6 means the six-form part of the wedge product.
For Weyl spinors, the corresponding forms are only even (odd) for a positive (negative) chirality O(6, 6) spinor. In the special case where η 1 = η 2 ≡ η, familiar from Calabi-Yau compactifications, we get where J, Ω are respectively the symplectic and complex structures of the manifold (more details are given in section 4.1.1).
Pure spinors can be "rotated" by means of O(6, 6) transformations. Of particular interest is the nilpotent subgroup of O(6, 6) defined by the generator with B an antisymmetric 6×6 matrix, or equivalently a two-form. On spinors, it amounts to the exponential action where on the polyform associated to the spinor, the action is e −B Φ = (1 − B ∧ + 1 2 B ∧ B ∧ +...)Φ. We will refer to Φ as naked pure spinor, while Φ D will be called dressed pure spinor. The pair (Φ + D , Φ − D ) defines a positive definite metric on the generalized tangent space, which in turn defines a positive metric and a two-form (the B field) on the six-dimensional manifold.

Exceptional Generalized Geometry
Exceptional generalized geometry (EGG) [7,8] is an extension of the O(6, 6) (T-duality) covariant formalism of generalized geometry to an E 7(7) (U-duality) covariant one, such that the RR fields are incorporated into the geometry.
We saw in the previous section that there is a particular O(6, 6) adjoint action (2.12) corresponding to shifts of the B-field. In EGG, shifts of the B-field as well as shifts of the sum of internal RR fields C − = C 1 + C 3 + C 5 5 , which transforms as a chiral O(6, 6) spinor, correspond to particular E 7 adjoint actions. To form a set of gauge fields that is closed under U-duality, we also have to consider the shift of the six-form dual to B 2 , which we will callB. 6 Decomposing the adjoint 133 representation of E 7 (7) under O(6, 6) × SL(2, R), we have 133 = (3, 1) + (1, 66) + (2, 32 ′ ) (2.14) , raised and lowered with ǫ ij , and the O(6, 6) fundamental indices A, B = 1, ..., 12 are raised and lowered with the metric η in (2.1). The B-transform action (2.12) is part of µ A B , while the C-transformations are naturally embedded in one of the two 32 ′ representations. Let us call v i the SL(2, R) vector pointing in the direction of the C-field, which we can take without loss of generality to be The GL(6) assignments of the different components shown in Appendix C, indicate that the shift symmetries are given by the following sum of generators where v i = ǫ ij v j . Using (A.4) it is not hard to show that given this embedding we recover the right commutation relations where the first term on the rhs is a six-form and therefore corresponds to aB transformation and the other two, to an RR shift. It combines all the gauge transformations: vectors plus one-forms correspond to diffeomorphisms and gauge transformations of the B-field. Their SL(2, R) duals 7 are gauge transformations of B 6 (given by a five-form, or analogously a vector) and diffeomorphisms for the dual vielbein (sourced by KK monopoles), given by a one-form tensored a six-form. Gauge transformations of the RR fields combine forming again a spinor representation, this time with positive chirality. The generalized tangent bundle T ⊕ T * is therefore extended to the exceptional tangent bundle (EGT) E In what follows, we will mostly use the decomposition of E 7 under SL(8, R). The fundamental representation decomposes as 56 = 28 + 28 ′ (2.20) ν = (ν ab ,ν ab ) 7 The SL(2, R) here is the "heterotic S-duality", where the complex field that transforms by fractional linear transformations is S =B + i.e. −2φ . For the connection between this and type IIB S-duality, see where a, b = 1, ..., 8 and ν ab = −ν ba . The adjoint decomposes as where µ a a = 0 and µ abcd is fully antisymmetric. In order to identify the embedding of the gauge fields (2.16) in SL(8, R), we use the GL(6, R) properties of the different components of the adjoint representation given in (C.4). We get 8 22) or in other words where the factors and signs are chosen in order to match the supergravity conventions.
Here and in the following, * refers to a six-dimensional Hodge dual, while we use ⋆ for the eight-dimensional one.

E 7(7) algebraic structures
In this section we present the algebraic structures in E 7 constructed in [9] that play the role of the O(6, 6) pure spinors Φ ± . We start by building the analogous of the naked pure spinors, and then discuss their orbits under the action of the gauge fields A in (2.16), (2.22).
Spinors transform under the maximal compact subgroup of the duality group. In the GCG case, this subgroup is O(6) × O(6), which acts on the pair (η 1 , η 2 ). In EGG, the relevant group is SU (8). We can combine the two ten-dimensional supersymmetry parameters such that the SU(8) transformation of their internal piece is manifest. The most general ten-dimensional spinor ansatz relevant to four-dimensional N = 2 theories is where ζ 1,2 − are four-dimensional spinors of negative chirality, and θ 1,2 are never parallel. In this paper we will be dealing with equations for N = 1 vacua, where there is a relation between ζ 1 and ζ 2 . In that case, we can use the special parameterization A nowhere vanishing spinor θ defines an SU(7) ⊂ SU(8) structure. The pair (θ 1 , θ 2 ) defines an SU(6) structure 9 . We can take the SU(4) spinors to be normalized to 1. In that case the SU(8) spinors are orthonormal, namelȳ where I = 1, 2 is a fundamental SU(2) R index (for conventions on the conjugate spinors, see Appendix B). The two spinors can be combined into the following SU(2) R singlet and triplet combinations where we have introduced K 0 for future convenience. The triplet K a satisfies the su(2) algebra with a scaling given by the dilaton, i.e.
[K a , K b ] = 2ie −φ ǫ abc K c To make contact with the pure spinors of GCG, we note that using the parameterization (3.2), we get where the operation s is introduced in (2.10).
Using (3.2), we get for K ± = K 1 ± iK 2 while for K 3 we get We see that L contains the pure spinor Φ + , which spans the vector multiplets in type IIA (see (2.11)), while K + is built from the pure spinor Φ − , which is part of the hypermultiplets. K 3 contains on the contrary the even-form bilinears of the same SU(4) spinor, or in other terms the symplectic structures defined by each spinor (see (2.11)).
To get the SL(8) components of L and K a , we use (B.8). Using the decomposition of the gamma matrices given in (B.9), we get that the only non-zero components of L and K a are L : (3.10) where L 12 and L mn involve the zero and two-form pieces of Φ + , K mi + , K mnpi + contain the one and three-form pieces of Φ + (where the difference between the two SL(2) components is a different GL(6) weight), while K 3 contains the different components of Φ + 1 and Φ + 2 . In an analogous way as for the pure spinors, the structures L and K a can be dressed by the action of the gauge fields B,B and C − in (2.16), (2.23), i.e. we define In the GCG case, the B-field twisted pure spinors span the orbit O(6,6) is the stabilizer of the pure spinor and the R + factor corresponds to the norm. Quotenting by the C * action Φ D → cΦ D , we get the space O(6,6) U (3,3) which is local Special Kähler. Similarly, our EGG structures L D and K aD span orbits in E 7 which are respectively Special Kähler and Quaternionic-Kähler. As shown in [9], the structure L D is stabilized by E 6(2) , and the corresponding local Special Kähler space is E 7 E 6(2) × U(1). The triplet K aD is stabilized by an SO * (12) subgroup of E 7 , and the corresponding orbit is the quaternionic space (2) , where the SU(2) factor corresponds to rotations of the triplet. The SO * (12) and E 6(2) structures intersect on an SU(6) structure if L and K a satisfy the compatibility condition where we have to apply the projection on the 56 on the product 56×133. This condition is automatically satisfied for the structures (3.4) built as spinor bilinears.

String vacua and integrability conditions
In the previous sections we have presented the relevant algebraic structures that are used to describe an off-shell N = 2 four-dimensional effective action. We now turn to the differential conditions imposed by requiring on-shell supersymmetry, or in other words, by demanding that the vacua are supersymmetric. As we will show, these translate into integrability of some of the algebraic structures.

Warm up: fluxless case
It will be useful for the following to recall the conditions for supersymmetric vacua in the absence of fluxes. We start by reviewing the integrability conditions in ordinary complex geometry, and then re-express them in the language of GCG.

Conditions for the structures on T M
In the absence of fluxes, inserting the N = 2 spinor ansatz (3.2) in the supersymmetry condition δψ m = 0 (see (F.2)), we get When there is only one globally defined spinor η, we take η 1 = η 2 ≡ η, and (4.1) reduces to the familiar Calabi-Yau condition which implies that the SU(3) structure defined by η is integrable, or in other words that the manifold has SU(3) holonomy [10]. The holonomy is defined as the group generated by parallel transporting an arbitrary spinor around a closed loop. Riemaniann geometries can be classified by specifying the holonomy of the Levi-Civita connection.
A general Riemaniann six-dimensional space has holonomy SO(6) ≃ SU(4). However if the manifold admits one (or more) Killing spinors, the group is reduced: it lies within the stabilizer group. In six dimensions, the existence of a globally defined, nowhere vanishing, covariantly constant spinor implies that the holonomy is reduced to SU(3) ⊂ SU(4).
Integrability of an SU(3) structure can also be recast in terms of integrability of two seemingly very different algebraic structures that intersect on an SU(3), namely a complex and a symplectic one. The existence of a globally defined nowhere vanishing spinor is equivalent to the existence of an almost symplectic 2-form J (which defines an almost symplectic Sp(6, R) structure) and a 3-form Ω (which defines an almost complex GL(3, C) structure). These two structures intersect on an SU(3). If the structures are integrable, i.e. if they satisfy for any one-form ξ, one can define local complex and local symplectic coordinates which can be "integrated" (i.e. there exist local complex coordinates z i and symplectic ones (x i , yî) (i,î = 1, 2, 3) such that the local complex and symplectic one forms dz i , (dx i , dyî) are indeed their differentials). If additionally ξ = 0, then the canonical bundle is holomorphically trivial and the manifold is Calabi-Yau. Since J and Ω can be written as bilinears of the spinor η, the supersymmetry requirement (4.2) is equivalent to the conditions (4.3) and the additional requirement ξ = 0.
Note that for an almost complex structure, there are many equivalent ways to check its integrability. Instead of the second requirement in (4.3), one can find conditions on the corresponding map I : and [ , ] denotes the Lie bracket. As we will see, either requirement (4.3) and (4.4) will have its analogue in generalized complex geometry. In exceptional generalized geometry, we will only deal with conditions of the form (4.3).

Conditions for the structures on T M ⊕ T * M
As shown in section 2.1, almost complex and symplectic structures on the tangent bundle are expressed on the same footing in terms of generalized almost complex structures on T M ⊕ T * M. Furthermore, a generic GACS reduces on the tangent bundle to a structure that is locally a product of lower dimensional complex and symplectic structures.
As in the case of ordinary complex structures, Eq.(4.4), a GACS is integrable if its ieigenbundle is closed under an extension of the Lie bracket to where the projectors Π ± are defined in (2.2) and the bracket is the Courant bracket with L the Lie derivative. Again, in a similar fashion to ordinary complex structures, the integrability condition (4.5) is equivalent to requiring that the pure spinor Φ associated to J satisfies dΦ = X · Φ (4.7) for some generalized vector X = x + ξ, and where · is the Clifford product, whose action on forms is The N = 2 supersymmetry requirement (4.1) that arises in the absence of fluxes, translates into dΦ which means that both GACS are integrable (and both canonical bundles are trivial), or in other words that the SU(3) × SU(3) structure is integrable. In the case η 1 = η 2 = η, this reduces to the Calabi-Yau conditions (4.3) with ξ = 0. Manifolds satisfying (4.9) have been termed "generalized Calabi-Yau metric geometries" in [4] 11 . They are more general than Calabi-Yau's in the sense that the pure spinors need not be purely complex or pure symplectic, as happens when η 1 = η 2 , but can correspond to (integrable) hybrid complex-symplectic structures.

Flux case in CGC
In this section we review the results of [11] (in the language of GCG, as in [12]) and [6] where the conditions for respectively N = 2 supersymmetry with NS flux only, and N = 1 with NS and RR fluxes were found.

Vacua with NS fluxes
In section 2.1 we saw how GCG incorporates the B-field, in particular by means of the Btwisted pure spinors (2.13). When B is not globally well-defined, i.e. when NS fluxes are switched on, the B-twisted pure spinors are not global sections of T M ⊕ T * M, but they are rather sections of a particular fibration of T * M over T M involving the B-field. For reasons that will become clear later, in this paper we choose the alternative "untwisted picture" as in [4], where pure spinors are naked (or dressed by just a closed B field), and the H-flux is introduced explicitly in, e.g. the integrability conditions 12 .
A closed B field is an automorphism of the Courant bracket, while in the presence of The H-twisted Courant bracket is defined by adding this last term to (4.6).
If a GACS is "twisted integrable", then the corresponding pure spinor satisfies where the H-twisted differential is Note the equivalence between the twisted and untwisted picture. If a naked pure spinor is twisted closed, then the dressed pure spinor is closed under the ordinary exterior derivative, i.e.
This shows how to construct the twisted exterior derivative from the ordinary one, and the action of the B-field d H = e B de −B (4.14) which will be extended in section (5.1) to include the RR fluxes.
Supersymmetry conditions in the presence of H-flux amount precisely to H-twisting the generalized Calabi-Yau metric condition (4.9). More precisely, vacua preserving fourdimensional N = 2 supersymmetry in the presence of NS fluxes should satisfy [12] i.e. they require H-twisted generalized Calabi-Yau metric structures.
Vacua with N = 1 supersymmetry in the presence of NS fluxes were obtained in [13], and reinterpreted in the language of G-structures in [14]. They read where Φ ± are those for an SU(3) structure, (2.11). Note that in the second equation H does not enter as a twisting in the standard way, and therefore the even pure spinor is not twisted integrable. It would be interesting to get the right GCG description of N = 1 vacua with NS fluxes.

Vacua with NS and RR fluxes
Compactifications on Minkowski space preserving N = 1 supersymmetry in the presence of NS and RR fluxes require the spacetime to be a warped product, i.e.
The preserved spinor can be parameterized within the N = 2 spinor ansatz (3.2) by a doublet n I = (a,b) such that the supersymmetry preserved is given by ǫ = n I ǫ I , i.e.
and we take |η 1 | 2 = |η 2 | 2 = 1 (while |a| and |b| are related to the warp factor, as we will see). The vector n I distinguishes a U(1) R ⊂ SU(2) R such that any triplet can be written in terms of a U(1) complex doublet and a U(1) singlet by means of the vectors Using these vectors, one can extract respectively an N = 1 superpotential and D-term from the triplet of Killing prepotentials P a that give the potential in the N = 2 theory, by For type IIA compactifications, the triplet P a reads [15] The conditions for flux vacua have been obtained in the language of GCG either using the ten-dimensional gravitino and dilatino variations [6], or by extremizing the superpotential of the four-dimensional N = 1 theory and setting the D-term to zero [16,17]. For the case |a| = |b|, which arises when sources are present, they read Finally, N = 1 supersymmetry requires The susy condition (4.22) says that the GACS corresponding J + is twisted integrable, and furthermore that the canonical bundle is trivial, and therefore the required manifold is a twisted Generalized Calabi-Yau (see footnote 11). The other GACS appearing in (4.23)-(4.24) is "half integrable", i.e. its real part is closed, while the non-integrability of the imaginary part is due to the RR fluxes. In the EGG formulation, RR fluxes are also encoded in the twisting of the differential operator, and therefore we expect to rephrase these equations purely in terms of integrability of the structures defined on the EGT space. Note that in the limit of RR fluxes going to zero, Eqs. in the absence of sources, while in the presence of D-branes or orientifold planes, the right hand sides get modified by the appropriate charge densities.

Flux vacua in Exceptional Generalized Geometry
In this section we discuss the conditions for N = 1 vacua in the language of EGG. The putative conditions for supersymmetric vacua come from variations of the (where we are using again v i = ǫ ij v j = (0, −1)), DK a in (5.1) is an element in the 56 × 133, which is projected to the 56 by the symplectic product. In the second equality in (5.1) we have used the E 7 invariance of the symplectic product to untwist the structures L D and K aD and express the Killing prepotentials in terms of naked structures, and a twisted derivative. We will now see how to properly define this twisted derivative, needed to get the equations for vacua.

Twisted derivative and generalized connection
For the gauge fields A and the derivative operator D A , A = 1, ..., 56, one can define a connection φ AB C ∈ 56 × 133 by the following twisting of the Levi-Civita one The connection φ contains derivatives of the gauge fields. The key point is that in the tensor product 56 × 133 = 56 + 912 + 6480 (5.5) only the terms in the 912 representation involve exterior derivatives of the gauge potentials [18], while the other representations contain non-gauge invariant terms (like divergences of potentials). We therefore define the twisted derivative as The fact that the fluxes lie purely in the 912 is consistent with the supersymmetry requirement that the embedding tensor of the resulting four-dimensional gauge supergravity be in the 912 [19].
The 912 decomposes in the following O(6, 6) × SL(2, R) representations where Γ A F A− = 0 and F iABC is fully antisymmetric in ABC. The only nonzero components of the connection (5.6) are (see Appendix D for details) where F + = e B dC − .
In the SL(8, R) decomposition, the generalized connection decomposes in the following representations 912 = 36 + 420 + 36 ′ + 420 ′ (5.8) where F ba = F ab and F abc c = 0 and similarly for the objects with a tilde. The NS and RR fluxes give the following non-zero components In applying the twisted derivative to the algebraic structures L and K, the following tensor products appear If we think of the vacua equations as coming from variations of the Killing prepotentials (5.1), out of these tensor products of representations, the equations should lie in the 133 representation for DL, and in the 56 in DK. We give in (E.1)-(E.16) the full expression for the twisted derivative of an element in the 56 and an element in the 133. In section 5.3 we rewrite the only components that are non-zero in the case of N = 1 vacua, i.e. for L and K whose only non-zero components are those in (3.10).

Equations for N = 1 vacua
By following the same reasoning that leads from the superpotential to the equations for N = 1 vacua in the GCG case, a set of three equations were conjectured in [9] to be the EGG analogue of (4. Here, we show that the conjectured equations do work if we introduce two modifications: first, instead of using dressed bispinors and an untwisted derivative, we use undressed bispinors and a twisted derivative, appropriately projected onto the 912. This gets rid of the non gauge invariant terms arising in the vector parts of the equations conjectured in [9]. Second, we add a right hand side to the equations with a single internal index, proportional to the derivative of the warp factor or the dilaton.
The equations are written in terms of L and K a using the following parameterisation for the spinors With this parameterisation, the combinations that are relevant for N = 1 supersymmetry are In the language of EGG, N = 1 supersymmetry requires for L ′ , for DK ′ 1 | 56 (DK ′ 1 ) m2 = 0, (DK ′ 1 ) m1 = 0 , 13 We are using the notation in (2.20), where a tilde denotes the component in the 28 ′ representation and for DK ′ + | 56 The remaining components of DK (all with one internal index) are proportional to derivatives of the dilaton and warp factor as follows The equations for L, K ′ 3 and K ′ + in (5.12)-(5.14) are respectively the EGG version of (4.22), (4.23) and (4.24). The vectorial equations are a combination of (4.22)-(4.24) plus (4.26).

From SUSY conditions to EGG equations
We will sketch here the proof that N = 1 supersymmetry requires (5.12)-(5.14) and leave the details, as well the proof of Eqs (5.15), (5.16), to Appendix G.
Using (3.10) in (E.1)-(E.8), we get that the only nontrivial components of Eq. (5.12) are where we used (B.7), while for K ′ 1 we get (5.25) and for K ′ where we should keep in mind that the components of K + with an odd (even) number of internal indices are proportional to K 2 (K 3 ) (see (3.10)).
We now show that supersymmetry requires (5.12), in particular the components appearing in (5.17) and (5.18). The proof for the rest of the components is in Appendix G.1.
It is not hard to show that exactly the same combination of RR fluxes appearing on the right hand side of (5.17) is obtained by multiplying Eq. (G.5), coming from the external gravitino variation, by Γ 2 , and tracing over the spinor indices, namely where in the second equality the term proportional to the derivative of the warp factor goes away by symmetry, and we have used (B.7) to relate the SU(8) and SL(8) components of L. Supersymmetry requires therefore (DL ′ ) 1 2 = 0. For the equations that involve a covariant derivative of L ab , we use (G.1) coming from the internal gravitino variation, multiplied by Γ ab and we trace over the spinor indices (see Eq. (B.7)). For ab = 12, for example, this gives where π ′ is defined in (G.2) and (G.3). Now we use Eqs. (G.4) and (G.6) multiplied by Γ m and traced over the spinor indices to cancel the terms containing derivatives of the dilaton and warp factor. In doing this, the term involving H and F fluxes completely cancel, i.e.
We show in Appendix G.1 how supersymmetry requires the remaining equations, (5.19) and (5.20), to vanish.
The equations for K work similarly. For example, to show that (5.21) should vanish, we use (G.11) coming from internal gravitino, in the following way We combine this with external gravitino equations (G.14), (G.28) and dilatino equations (G.15), (G.17) to get (see more details in Appendix G.2) where we have used the notation in (G.29).
We give the details about the rest of the components of the twisted derivative of K ′ 1 and K ′ + in Appendix G.2. We will now connect the equations found to their generalized complex geometric counterparts, Eqs.  )). The mn and 12 components of the EGG equations for K ′ 1 and K ′ + combine to build up respectively (4.23) and (4.24). Interestingly, Eq. (4.26), which is not part of the pure spinor equations but has to be added by hand in the GCG language, becomes one of the EGG equations, namely the one on the second line of (5.16). This can be seen by using (5.29) and the fact that K ′ The other vectorial components of DK involve for example terms of the form F, Γ A Φ − , which making use of (4.22)-(4.24), can be shown to be proportional to derivatives of the warp factor. Since (4.22)-(4.26) were shown in [20] to be equivalent to supersymmetry conditions, we conclude that the EGG equations (5.12)-(5.16) are completely equivalent to requiring N = 1 supersymmetry, i.e., supersymmetry requires (5.12)-(5.16), and (5.12)-(5.16) implies supersymmetry.
As mentioned in section 3, L defines an E 6(2) structure in E 7 . We have shown here that N = 1 supersymmetry requires this structure to be twisted closed, upon projection to the 133. It would be very nice to show that this is equivalent to the structue being integrable 14 . For constant warp factor and dilaton, also K ′ 1 is twisted closed. Most of the components of K ′ + are also twisted closed after projection onto holmorphic indices in the 56. The vectorial components of DK are proportional to derivatives of the warp factor and dilaton, except the second equation in (5.16), which does not seem to be expressible in terms of such derivatives.

Acknowledgements
We would like to thank Diego Marqués, Hagen Triendl and especially Daniel Waldram for many useful discussions. This work is supported by the DSM CEA-Saclay and by the ERC Starting Independent Researcher Grant 259133 -ObservableString.
A E 7(7) basics and tensor products of representations E 7 (7) can be defined as the subgroup of Sp(56, R) which in addition to preserve the symplectic structure S(λ, λ ′ ), preserves also a totally symmetric quartic invariant. We exploit the decomposition of E 7(7) representations under two subgroups 1. SL(2, R) × O(6, 6) is the physical subgroup appearing as the factorization of ("heterotic") S-duality and the T-duality group that emerges in the framework of generalized geometry 2. SL(8, R). This subgroup contains the product SL(2, R) × GL(6, R), and allows to make contact with SU(8)/Z 2 , the maximal compact subgroup of E 7 (7) . The latter is the group under which the spinors transform, and therefore the natural language to formulate supersymmetry via the Killing spinor equations. where Γ A Φ A− = 0 and φ iABC is fully antisymmetric in ABC.
There are various tensor products projected on some particular representation that are used throughout the paper. These are: 56 × 56 1 (i.e. the symplectic invariant) the adjoint action on the adjoint, i.e. 133 × 133 133 ;

A.2 SL(8, R)
The decomposition of the E 7 representations we use in terms of SL(8, R) are the following. with µ a a = 0, and µ abcd fully antisymmetric. For the 912 we have d and φ abc c = 0 and similarly for the tided objects. The SL(8, R) decomposition of the tensor products is the following. The adjoint action on the fundamental, 56 × 133 56 is 15 .
The symplectic invariant 56 × 56 1 reads The 56 × 56 133 reads where ⋆µ is the 8-dimensional Hodge dual, while the adjoint action on the adjoint 133 × 133 133 gives The 56 × 133 912 is The 912 × 56 133 gives  15 Note tht this convention differs by a sign in the ⋆µ term than the one used in [8,21]. This choice is correlated with the choice in (E.17), and affects a few signs in the equations that follow (those in the terms involving ⋆µ). and finally 912 × 133 56 is The spinor θ α transforms in the fundamental representation of SU (8). The standard interwining relations allow to define the conjugate spinors Under SU (8) where µ α α = 0 andμ αβγδ = ⋆µ αβγδ . Note that these are very similar to the SL(8, R) decompositions (A.6), (A.7). To go from one to the other, we use for the 56 [8] In the main text we use a complex 28 object, defined from its real pieces λ ab ,λ ab in the obvious way From the 63 adjoint representation of SU(8) (i.e. taking µ αβγδ = 0) one recovers the following SL(8, R) components where µ ba = −µ ab and ⋆µ abcd = −µ abcd (the symmetric and self-dual pieces are obtained from the 70 representation µ αβγδ ) and µ ab = g ac µ c b (at this point there is a metric since SL(8) ∩ SU(8) = SO(8)).
When breaking SU(8) → SU(4) × SU(2), the spinor index decomposes in a pair of indices α =αI, whereα is an SU(4) spinor index. For the Cliff (8, 0) gamma matrices, we have used the following basis in terms of Cliff (6, 0) and Pauli sigma-matrices The intertwiners A, C, D also split into Cliff (6)⊗Cliff (2) intertwiners. In particular, C splits as whereĈ is the intertwiner We get that C αβ =Ĉ ⊗ σ 1 (B.12) We will use a basis for the Cliff (6, 0) gamma matrices in whichÂ =Ĉ =D = I, and therefore the SU(4) conjugate spinors are just and η − ≡ η * + . In this basis, the SU(8) spinors in (3.2) have conjugates θ 1t = (0, η 1T − ) (B.14) The GL(6, R) weights of the different O(6, 6) × SL(2, R) representations is worked out in [9]. It turns out that the two components of an SL(2, R) doublet have different GL(6, R) weights. To find the GL(6, R) weight in the SL(8, R) decomposition, we use that SL ( where M ∈ SL(8, R), a ∈ GL(6, R), and we have added explicit factors of the dilaton that are needed in order to get the right transformation properties of the connection. Since a six-form transforms by a factor (detg) 1/2 (or equivalently 1/deta), we can write the 8-dimensional metric aŝ The different SL(8, R) components of 56 representation ν = (ν ab ,ν ab ) transform therefore according tõ where we have introduced a trivial real line bundle L with sections e −φ ∈ L to account for factors of the dilaton. The adjoint µ = (µ a b , µ abcd ) has the following GL(6, R) and dilaton assignments (C.4) Finally, the 912 multiplied by L⊗(Λ 6 T * M) −1/2 (a T-duality invariant factor), transforms as

D Computing the twisted derivative
We show in the following how to obtain the connection from twisting the Levi-Civita covariant derivative (5.2) by the gauge fields B,B and C − in the 133 representation.
Using the Hadamard formula we get for any element A in the adjoint We now promote the Levi-Civita connection ∇ to an element carrying a fundamental 56 index, as in (5.2): D A = (v i ∇ A , 0) and ∇ A = (0, ∇ m ). Finally, we project to the 912 representation using the tensor product 56 × 133 912 for the subgroup SL(2, R) × O (6,6) given in (A.5). We recover the simple result where F + = e B dC − , and all the other components are zero.
One can alternatively express the connection in terms of the SL(8, R) subgroup. The derivative D A is given in this case by while all other components are zero. Applying this to the gauge fields in (2.23), and projecting onto the 912 using (A.13), we find the following non-vanishing components (D.4) Notice that the mass parameter F (0) cannot be obtained this way, and should be added by hand. Using (C.5), we note that the component φ 11 transforms as a scalar, and we therefore assign E Twisted derivative of L and K Inserting the SL(8, R) decomposition of the derivative and of the fluxes given respectively in (5.3) and (5.9), and the corresponding SL(8, R) components of the tensor products given in (A.11) and (A.14), we get the following expressions for the twisted derivative of λ = (λ ab ,λ ab ), projected onto the 133 To get the twisted derivative of K projected on the 56, we use the tensor products (A.9) and (A.15). We find where we have used that which is a consequence of fact that K is purely in the 63 of SU(8).
We use the N = 1 spinor ansatz (4.18), parameterised using two internal spinors, namely θ = θ 1 + θ 2 , where θ 1 , θ 2 given in (5.10), we get from the internal components of the gravitino variation that N = 1 supersymmetry requires and the same exchaging 1 ↔ 2, where we have defined in terms of the "hermitean" and "antihermitean" pieces of F , namely and finally We will also need the equations involvingθ, which is From the external gravitino variation, we get that N = 1 vacua should satisfy / F e θ 2 = 0, (F.10) and similarly exchanging 1 and 2, where The hermitean conjugate equation reads From the dilatino variation, we get where we have defined The hermitean conjugate equation reads where we have defined We will also need the SL(8) object π abcd , which we define to be Multiplying (F.10) (coming from external gravitino variation on θ 1 ) by θ 2 , and substracting to the equation with θ 1 and θ 2 exchanged, we get the following equation If instead we multiply (F.10) by θ 1 and substract the corresponding equation for θ 2 multiplied by θ 2 , we get Doing the same on the dilatino (F.14) we get and a similar equations with L and π exchanged, that will not be used.
We show here how supersymmetry requires equations (5.18)-(5.20) to vanish. For each of them, we use (G.1) plus l e times (G.4) and l d times (G.6), and take in the one to last step l e = −2 , l d = 1 .
We show that susy requires Eq. (5.18) to vanish by To get (5.19) we do while for (5.20) we use

G.2 DK
We define the following quantities Combining (F.5) multiplied byθ with (F.9) multiplied by θ, we obtain where the factors of the dilaton inside the covariant derivatives are there to cancel the explicit dilaton dependence of K (see (3.6)).
Multiplying the external gavitino or dilatino equation, Eqs. (F.10) and (F.14) byθ 2 on the right, and adding it to the same equation with θ 1 and θ 2 exchanged, we get We can also use the complex conjugate equations (F.13), (F.16) multiplied on the left by θ 2 . This gives We will also need the corresponding equations mixing K 3 and K 2 and the following ones involving K 0 and K 1 Given a generic K and product of gamma matrices Γ a 1 ...a i we will make use of the following type of combinations (G.29) and similarly for the anticommutator.
We want to show that susy requires (5.13) and (5.15). We recall that as shown in (3.10), K 1 has only nonzero components with an odd number of internal indices.
The idea is to reconstruct the twisted derivative of the corresponding K ′ appearing in each of the equations by summing an equation coming from internal gravitino (which gives a covariant derivative of K with no dilaton or warp factors) together with equations coming from external gravitino plus dilatino, which contribute the required derivatives of dilaton and warp factor.
We start by showing that susy requires (5.21) to vanish. We use the following combination of equations: (G.11) coming from internal gravitino, (G.14) and (G.28) from external gravitino, and (G.15), (G.17) from dilatino (the last four multiplied by arbitrary coefficients n e and n d , that will be set to n e = 1, n d = −1).
where in the third equality we have used the values n e = 1, n d = −1.
To show that (5.22) vanishes, we use − H pq[m K ′ where we have used K 2 = K 1 Γ 12 and K 0 = −iK 3 Γ 12 , and in the last line we have used (5.24). For the last equation (5.25) we use (G.10) and (G.19) where in the second equality we have used again K 0 = −iK 3 Γ 12 , and in the third equality we have used (5.16) (which will be shown to hold below).

+
The other set of equations involves From (3.10), we see that K + with an odd number of internal indices is proportional to iK 2 , while for an even number of internal indices, K + is proportional to K 3 .
We are left with the vectorial components. The last equation in (5.14) is trivial (see (E.13)). To show the m1 component, we use For the ( DK ′ ) m2 equation, we first note that supersymmetry requires their RR pieces to vanish by itself, namely 0 = Tr (∆ m K ′ 3 ) = e φ ( * F 6 )(K ′ + ) m2 + F mp K ′1p + + ( * F 4 ) pq (K ′ + ) 1pqm = F RR m2 , while in the m1 equation, the RR piece is proportional to a derivative of the warp factor, i.e.