Generalized N = 1 and N = 2 structures in M-theory and type II orientifolds

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ate bundle between the tangent and the generalized tangent bundle that admits the action of SL(8) ⊂ E 7 (7) . This might indicate the existence of a twelve-dimensional theory (similar to F-theory) in which some of the charges will be geometrized. SU (7) and SU (6) structures are related to Spin (7) and SU(4) structures in SL (8).
After we briefly review the E 7(7) -covariant formalism of exceptional generalized geometry in section 2, we will discuss SU (7) and SU (6) structures in M-theory in section 3. In particular, we will discuss all classical couplings in these backgrounds. Subsequently, we will relate these results to the equivalent structures in type IIA string theory in section 4. While SU (6) structures descend in a straight-forward way, SU(7)-structures are related to O6-orientifold backgrounds in type IIA. Moreover, SU (6) and SU(7) structures should be related by involutions that project out half of the supersymmetry, such as orbifolding and orientifolding in M-theory and type II string theory. In section 5 we will show how an SU(7) structure is obtained from an SU(6) structure via such involutions. In particular, we will determine the N = 1 couplings in terms of the N = 2 parent theory. Finally, we will identify the involutions given by standard orbifolding and orientifolding in M-theory and type II string theory and discuss a few new involutions, for instance an involution creating objects with tension and charge opposite to those of NS5-branes. We conclude with a summary of the results and some outlook. Appendix A contains all the relevant formuli concerning E 7 (7) representations, and appendix B presents the details of some of the calculations done along the paper.

Exceptional Generalized Geometry (EGG)
In this section we review the basic concepts of Exceptional Generalized Geometry (EGG), emphasizing the role of the eight-dimensional intermediate tangent bundle T 8 . The idea of EGG is to covariantize the U-duality group E 7 (7) in M-theory and type II compactifications to four dimensions. Though only torus compactifications admit globally the action of the U-duality group, locally any background admits it, as the tangent space is isomorphic to R 7 (R 6 ). In EGG the internal seven-(or six-)dimensional tangent bundle of an M-theory (type II) compactification to four dimensions is enlarged to a 56-dimensional exceptional generalized tangent bundle such that the U-duality symmetry group E 7(7) acts linearly on it. Thereby, the U-duality group promotes to a geometric action on this bundle. As the U-duality group maps all bosonic supergravity degrees of freedom into each other, EGG gathers them all in a metric on this exceptional generalized tangent bundle. The patching of the exceptional generalized tangent bundle [4] resembles the global aspects of the compactification. More details can be found in [3][4][5][6].

An eight-dimensional tangent space T 8
In the case of compactifications of type II, the exceptional tangent bundle combines the string internal momentum and winding charges (6+6 elements), their magnetic duals (another 6+6) as well as all the D-brane charges (32 elements). These together form the fundamental 56 representation of E 7 . In M-theory, it is a result of combining momentum and its dual (Kaluza-Klein monopole charge) (7+7) together with M2 and M5-brane JHEP03(2013)145 charges (21+21). These charges can be combined into SL (8, R) representations. We can think of this group as acting on an 8-dimensional tangent bundle T 8 , which will be split into 7 + 1 for M-theory, and further split into 6 + 1 + 1 for type IIA. Of course there is a priori no eight-dimensional manifold with a tangent bundle T 8 appearing in M-theory or type IIA. Therefore, T 8 should be seen as some kind of generalized tangent bundle. In terms of SL (8, R) representations, the fundamental of E 7 decomposes as E = Λ 2 T 8 ⊕ Λ 2 T * 8 , 56 = 28 ⊕ 28 ′ . (2.1) Similarly, for the adjoint we have where the subscript 0 denotes traceless. We will also need the 912 representation, which splits according to where S 2 denotes symmetric two-tensors. When we later consider spinors it is also useful to use the maximal compact subgroups of the groups that are involved. The maximal compact subgroup of E 7 is SU (8), and the group-theoretical decompositions are completely analogous to the SL(8, R) case and are given by (2.1)- (2.3). Note though that the SU(8) that transforms the spinors is not the compact subgroup SU(8) of SL(8) that acts on T 8 . Nevertheless the two SU (8) subgroups are related by some E 7 (7) transformation and the decomposition of E 7 (7) representations is the same in both cases. More details can be found in appendix A.2. When we then consider SL(8, R), spinors transform under the corresponding spin group Spin(8) and its maximal compact subgroup SO (8). Note that for Spin(8), we can impose a Majorana-Weyl condition on the spinor. The Weyl-spinors are in one-to-one correspondence to Spin(7) spinors that are considered in the M-theory compactification.

M-theory and GL(7) decompositions
For compactifications of M-theory on seven-dimensional manifolds, we should decompose further the SL(8) into GL (7) representations, or in other words split the 8-dimensional dual vector bundle T * 8 into a 7-dimensional one T * 7 , plus a scalar piece. Choosing an overall power of the 7-dimensional volume form to get the correct embedding in SL(8) (see more details in [4]), we get
In turn, the adjoint (2.2) decomposes into We recognize here the GL(7) adjoint (first term), and the shifts of the M-theory 3-form potential A 3 (last term) and its 6-form dual A 6 (fourth term). These build up the geometric subgroup of transformations that are used to patch the exceptional tangent bundle. The other pieces correspond to "hidden symmetries", very much like the β-transformations in generalized geometry. Note that A 6 and A 3 come respectively from the 63 and 70 representations of SL(8) in (2.2), i.e. they embed into the SL(8) representations in (2.2) as where we defined the four-form with meaning the full contraction of A 6 with (vol 7 ) −1 .

Supersymmetric reductions of M-theory in EGG
In this section we review reductions of M-theory preserving N = 2 and N = 1 supersymmetry in the language of Exceptional Generalized Geometry following [4,5]. In the N = 2 case, we show how an SU(4) structure on T 8 emerges, and write the EGG structures in terms of the complex and symplectic structures on this space. To get a supersymmetric effective four-dimensional theory, there should exist nowherevanishing spinors η i on the seven-dimensional internal space such that the eleven dimensional spinor can be decomposed as (for spinor conventions, see [4]) where ξ i ± are four-dimensional Weyl spinors, η i are complex Spin(7) spinor, the superscript c denotes charge conjugation and i = 1, . . . , N determines the amount of 4D supersymmetry. When we embed Spin(7) into Spin(8), we can choose η i to be of positive chirality with JHEP03(2013)145 respect to Spin (8), which means that we can take η i to transform in the 8 of SU (8). Note that there exists a Majorana condition for Spin(7) (and also for Spin (8)) so that actually the real and imaginary parts of η i are independent spinors. Both the real and imaginary part of η i are stabilized by a Spin(7) subgroup in SL(8, R), respectively. Therefore, inside SL(8, R) each complex spinor η i defines a pair of Spin (7) structures. Inside E 7(7) though, each η transforms in the fundamental of SU(8) and its real and imaginary parts are not independent any more, as they transform into each other under SU (8). Consequently, a single η defines an SU(7) structure [4], and in general N non-mutually parallel spinors define an SU(8 − N ) ⊂ E 7(7) structure.
3.1 N = 2 reductions and SU(6) × SU(6) structures For reductions with N = 2 supersymmetry in four-dimensions, there should be a pair of globally defined no-where vanishing (and nowhere parallel) SU(8) spinors η 1 , η 2 . As explained above, each of these spinors can be complex so that Re η i and Im η i each define an SU(6) structure. Without loss of generality, we can take them to be orthogonal and having the same norm, namelyη where i, j = 1, 2 is an SU(2) R index, and we introduced an arbitrary normalization factor e −K L /2 which we will further discuss below. On the other hand, we have in general the following inner products for some complex 2 × 2 matrix C = (c ij ). 2 The scalar degrees of freedom of N = 2 theories, coming from vector and hypermultiplets, are encoded respectively in an SU(2) R singlet L (0) and an SU(2) R triplet K (0) of bispinors [5]. The former embeds in the 28 representation of SU (8), which appears in the fundamental 56 representation of E 7 , and in terms of the decomposition (2.1) readŝ where for later convenience we have defined two real bispinors in the 28 and28 that are related to L (0) by L (0) =L (0) + iL (0) . FromL (0) we can define also the almost complex structure J (0) L that relates real and imaginary parts of L (0) , given by where ½ is the identity matrix with Spin(8) spinorial indices, δ α β , × denotes the product of two objects in the fundamental 56 representation into the adjoint 133 representation of E 7 , cf. (A.8) and in the last equality we have used (3.2).
Furthermore, the SU(2) R triplet K (0) a transforms in the 63 adjoint representation of SU (8), which is embedded in the 133 adjoint representation of E 7 , and reads in terms of the decomposition (2.2) where σ a are the Pauli sigma matrices and we introduced another normalization factor κ.
Note that the product of L and K in the 56, given in (A.5), vanishes, i.e.
which means that the stabilizers of L (0) (E 6(2) ⊂ E 7(7) ) and K (0) a (SO * (12) ⊂ E 7 (7) ) intersect in SU (6). General L and K a are then constructed by acting with the shift matrix A shifts in the adjoint of E 7(7) on them, i.e.
The Kähler potential for the space of structures L and the hyper-Kähler potential hyper-Kähler cone over the space of K a has been given in [5]. The Kähler potential for L is given by the moment map for the rotation of L by a phase (generated by J L ) and can be expressed by the logarithm of the quartic invariant ofL [5], cf. (A.9), where (·, ·) refers to the trace in the adjoint, given in (A.7). The hyper-Kähler potential of the hyper-Kähler cone over the moduli space of K a is determined by [5] It determines the normalization of the K a by where ν is an arbitrary element in the 56 representation. The supersymmetric couplings between the two objects L and K a are given by the Killing prepotentials [5,9] P a = ǫ abc (D L K b , K c ) , (3.12) and D L = L, D − L × D is the Dorfman derivative [6] along L, with D being the standard differential operator. 3 Inserting this, we get [9] P a = ǫ abc ( L, D K b , K c ) + 4κ(L, DK a ) . In the last section we defined a general SU(6) structure with an arbitrary product (3.3). Note that in general, the spinors η i are complex, and real and imaginary part of each spinor, if they are never parallel, already define an SU(6) structure, so in general we would have "SU(6) × SU(6)" structures (and secretly, a theory with more supersymmetries). In such a generic case, it is difficult to be more explicit as typically all SL(8) components are present JHEP03(2013)145 in L and K a . The case of real (i.e. Majorana) spinors η i simplifies the form of L and K a and gives rise to a natural interpretation in terms of the bundle T 8 . The relation (3.3) reduces for real spinors to the normalization condition (3.2) so that (3.14) We will express now both L (0) and K (0) a in (3.4) and (3.6) in terms of objects in SL(8, R) representations. For this we define a pure spinor out of the real spinors η i . 4 We then find for L (0) and K where we made the following definitions In terms of the x m we find ) The four-form Ω

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In terms of these objects, the structure L (0) has the following SL(8, R) decomposition (cf. (2.1)) 6 where we have used (A.21) and vol −1 8 means contraction with the eight-dimensional volume form, i.e. with the numeric epsilon tensor (the 8-dimensional space has no volume modulus). We see that L (0) can be understood as the shift of the bi-vector −ρ 8 ∧v by some imaginary four-form i ρ 8 ∧ v (0) ∧ J. 7 The almost complex structure relating real and imaginary part of L (0) given in (3.5) can be computed to be Similarly, using (3.16) and (A.22), we can determine K (0) a for the decomposition (2.2) to be We also have ιv(0)J = ιρ 8 J = 0, and 1 Furthermore, the compatibility condition (3.7) of L (0) and K (0) a implies ιv(0)Ω = 0 and J ∧ Ω = 0, or in other words J and Ω 4 define an SU(4) structure on the eight-dimensional space T 8 , i.e. inside SL(8, R). 8 From the compatibility condition (3.7) it also follows that g (3.29) 6 For the generic case of complex spinors ηi, the form of L (0) is also given by eq. (3.23), but in this case and similarly for J and v (0) . 7 Note that this shift is not an E 7(7) transformation because the four-form is not real. 8 As stated above, these objects define an SU(6) structure inside E 7 (7) .

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is a symmetric object. Furthermore, explicit computation shows that g (0) 8 is of the form Similar to (3.29) we also find for the inverse metric g General elements L, K a in the orbit can be achieved from L (0) , K (0) a by the action of E 7(7) , cf. (3.8). The degrees of freedom that can modify the above embeddings are the remaining massless fields of M-theory, namely the three-form gauge field A 3 and its magnetic dual six-form field A 6 . Their action in SL(8) language is shown in (2.7). This gives and where A 4 is defined in (2.8) and the shifted J and Ω 4 are The symplectic and complex structure J 8 and Ω 4 are also compatible, which means J 8 ∧ Ω 4 = 0. Finally, the almost complex structure I We see that the description of an SU(6) ⊂ E 7 structure in M-theory is completely analogous to the type II case discussed in [5], namely given by one object in the fundamental representation and a triplet in the adjoint. These are in turn described respectively by J 8 and Ω 4 , which define an SU(4) ⊂ SL(8) structure, in a form that very much resembles the pure spinors e − i J 8 , Ω 4 of two generalized almost complex structures. We will come back to this in section 4.
We turn now to the Kähler potentials and prepotentials for the space of structures L and (the hyper-Kähler cone over) K a . They both have the expected form in terms of the SU(4) structure objects J 8 and Ω 4 . The Kähler potential for L can be easily computed from (3.9)to be

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The hyper-Kähler potential of the hyper-Kähler cone over the moduli space of K a is given by (3.10) and reads in terms of Ω 4 as Finally, the Killing prepotentials (3.13) are computed in appendix B.2, using the differential operator D in the 56 representation that is given by the embedding The result of the computation is given by where Lv is the Lie derivative along v.

N = 1 reductions
For reductions with N = 1 supersymmetry, there is a single Spin(8) internal spinor η, which defines an SU (7) structure in E 7 (7) . 9 This structure can be encoded in a nowhere-vanishing object φ in the 912 representation of E 7(7) [4]. The 912 decomposes in SU (8) representations in the same way as for SL (8), given in (2.3). In terms of this decomposition, we have 8 is the metric on T 8 (3.30), the complex constant c is defined by and φ Note that at points where |c| = e −K φ /2 real and imaginary part of η actually define a (local) SU(6)-structure. In terms of the split into 7 + 1 in (2.4), this can be written as where α 3 is a complex three-form that defines a G 2 × G 2 -structure which reduces to a real three-form in the G 2 -structure case (i.e. when η is Majorana). Note that φ (0) 4 is self-dual as η is of positive chirality in Spin (8).
A general element φ is obtained by where A 4 , defined in (2.8), acts by the E 7 adjoint action (A.12), while A 6 shifts the metric (3.30) by and the 4-form Note that the volume form vol 8 of the metric g 8 in (3.47) is still given by the eightdimensional epsilon tensor (with entries ±1 and 0), as there is no R + factor in E 7 (7) corresponding to the eight-dimensional volume. We will therefore sometimes abuse notation and make no distinction between an eight-form and a scalar. The stabilizer of φ turns out to be indeed SU(7) [4]. Therefore, the existence of φ is completely equivalent to η. We will denote the real and imaginary parts of φ byφ andφ. The product 912 × 912 → 133 gives us the generator J φ for the almost complex structure e J φ related to φ. More precisely, we have The Kähler potential K φ in the orbit of φ is given by the quartic invariant in the 912 and turns out to be where in the last step we used (ιÂ * 7 α 3 ) ∧ * 7 α 3 = 0. Therefore, the Kähler potential does not depend on A 6 and A 4 , in agreement with E 7 (7) invariance. Note that this expression reduces to the known one in the G 2 -structure case.

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As has been derived in [4], the superpotential W is given through the eigenvalue equation where D is defined in (3.38). We present the detailed computation of the superpotential in appendix B.1. We get which is exactly the result of [4]. A similar result for manifolds of G 2 structure has been obtained in [11].
3.4 N = 2 in the language of N = 1 To finish this section, we note that the SU(6) structure of N = 2 reductions can also be understood as a triple of SU (7) structures, by using the product 133 × 56 → 912 given in (A.11). We find the objectŝ which are non-zero and define an SU(2) triplet of SU (7) structures of the form given in (3.40). The index a here labels the various symmetric combinations of the η i . A spinor η = α i η i corresponds then to the SU(7) structure defined by a . For an SU(6) structure the φ (0) a can be computed in terms of the SU(4) forms J 8 and Ω 4 on the 8-dimensional bundle. For this, we use (3.29) and (3.31). In terms of the decomposition (2.3), we then find 8 is given in (3.30). As before, the general form of φ a is obtained by acting with A 6 and A 3 on the above expressions. The action of A 6 turns g into Ω 4 (see (3.34)), while A 3 acts as usual by the 4-form shift e ρ 8 ∧A 3 . One can therefore understand the SU(6) structure in E 7 relevant for N = 2 compactifications as a triple of SU(7) structures. When we perform a projection on the SU(6) structure, for instance by orbifolding or orientifolding, a single combination of these SU(7) structures will survive, giving the N = 1 description one expects. We will discuss such projections in section 5.

Type IIA and GL(6) decompositions
To descend to type IIA, we further split the 7-dimensional tangent space into six plus one-dimensional pieces, i.e.
2) where from the last expression we recognize the type IIA charges, namely momentum, winding, their magnetic duals and the p-brane charges.

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The embedding of the gauge fields B 2 , B 6 and C ∓ (that we will call collectively A shifts ) in type IIA and type IIB is the following [5] A shifts = 0 We can now write explicitly the one-form v and gauge fields A 3 (or A 4 ), A 6 of the previous sections in terms of their type IIA counterparts v (0) = dy + C 1 , Inspecting (2.7), we can write the E 7(7) embedding of the gauge fields in SL(8, R) decomposition (cf. (2.2)) as follows whereĈ 5 is the vector associated to C 5 , i.e. As in M-theory, reductions to 4D with N = n supersymmetry require n internal SU(8) spinors η i , which show up in the decomposition of the two ten-dimensional spinors ǫ 1,2 as follows where η i are SU (8) spinors that combine the SU(4) ∼ = O(6) spinors that build up ǫ 1 and ǫ 2 . Given η 1,2 and η for the case of N = 2 and N = 1, the reductions work exactly as in M-theory, namely one builds the structures L (0) , K (0) a as in (3.4), (3.6) in the case of N = 2 [5], and φ (0) as in (3.40) in the case of N = 1. Their full orbit is obtained by the E 7(7) action of the gauge fields (4.9).
Let us concentrate on the N = 2 case now. The form of L resembles very much the pure spinor counterpart e B+i J for an SU(3) structure. Indeed, we get Note however that J here is a 2-form constructed as a bilinear of two different SU(8) spinors, and furthermore it is complex. To make the comparison with GCG more straightforward, one can parameterize them as 11 (4.13)
5 From N = 2 to N = 1: orientifolding and orbifolding in EGG As we will show in detail in section 5.2.4, orientifolds break the E 7(7) covariance into that of the subgroup O(6, 6) × SL(2, R) O , where the "O" makes it explicit that this is a different subgroup from that of T and S-duality. Furthermore, as we will see, each orientifold projection gives rise to a different O(6, 6) × SL(2, R) subgroup. The splitting of the fundamental and adjoint representations of E 7 (7) has been given in (4.5) and (4.6).

Orbifold action on T 8 and its reduction to M-theory
The M-theory uplift of type IIA O6-orientifolds are a geometric involution on the 7dimensional space. Such an involution can in turn be uplifted to an orbifold actionσ * on the 8-dimensional space T 8 such that the SL(8) bundle decomposes at the locus of the action into a positive and a negative eigenbundle and such that both subspaces are four-dimensional. The adjoint of E 7(7) (2.2) decomposes under (5.1) as where T 0 is the element of the adjoint that acts as ±1 on T ± 8 . By comparing to (4.6), we can see how E 7 is broken into O(6, 6) × SL(2, R): the first line builds up the adjoint of O(6, 6) × SL(2, R) O , which is even under the orbifold action, while the terms in the second line are odd and form the (2, 32 ′ ) representation. Similarly, the fundamental of E 7(7) , given in (2.1) in terms of SL(8) representations, decomposes as where the first four terms are even and form the (12, 2) representation of O(6, 6)×SL(2, R) O (see eq. (4.5)), while the last two, odd terms, form the (32, 1).

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To descend to M-theory, we require the orbifold action to have positive eigenvalue when acting onρ 8 . Eq. (2.4) tells us then that T + 7 is three-dimensional, while T − 7 is fourdimensional. Finally, to recover the type IIA orbifold action that gives rise to O6 planes, we requirev to have negative eigenvalue. In summary We will come back to the full orientifold projection later in section 5.2.1 Now let us see how the orbifold acts on the N = 2 structures defined in the previous sections. In type IIA, an involutive symmetry σ that can be used to mod out the theory should be anti-holomorphic if N = 1 supersymmetry is to be preserved [12]. This means that for an SU(3) structure defined by J and Ω, it should act as σ * J = −J, σ * Ω =Ω. This is easy to uplift to an action on J 8 , Ω 4 defining the SU(4) structure on T 8 , namely we require σ to act asσ This implies that the action induced byσ * on K a and L in the 133 and the 56 representation, which define an SU(6) ⊂ E 7 structure, should bẽ Therefore, L and K 2/3 are not well-defined any more in the presence of fixed points. However, their products are well-defined under the orbifold action. More precisely, their product in the 912 representation is defined by 13 where the pre-factor √ κ appears due to the different normalizations in (3.2) and (3.41). This φ in the 912 defines in turn an SU(7) structure following section 3.3, corresponding to the single spinor that survives the orbifolding. φ defines the metric g and a four-form φ 4 , cf. (3.46). From (5.7) we find We find the second expression using c = √ κe −K L /2 . The orbifold projection selects an N = 1 special Kähler subspace inside the N = 2 Kähler and quaternionic spaces. The generator of its complex structure is given in (3.50), which can in turn be written in terms of the SU(4)-structure objects J 8 and Ω 4 in L and K respectively as Therefore, the complex structure on the N = 1 Kähler space e J φ is the tensor product of the complex structures e K 1 with e J L . If the orbifold singularities were blown-up (see comments below), e J φ would not be block-diagonal any more. As long as the singularities are

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not blown up, the Kähler potential (3.53) simplifies to the sum the two Kähler potentials for L and K, i.e.
The superpotential is given in (3.55). On the other hand, we have the Killing prepotentials (3.39) that should descend to the N = 1 description. Comparing both expressions we find the relation The above formulas are valid for the orbifold of an SU(6) structure. If we blow-up the singularities resulting from the orbifolding, we switch on additional modes in φ that alter its form from the one given in (5.7). More precisely, the objects K 2/3 and L are not well-defined on the blown-up manifold, while φ still defines the geometry. The blow-up should lead to new modes that enter φ 4 as extra four-forms.

O6 orientifolds
To recover O6-orientifolds, the orbifold involutionσ should have negative eigenvalue on T y , i.e. act on T 7 as diag(σ, −1) where σ is an involution on the 6-dimensional space whose action is given above (5.5). Furthermore, the O6 projection mods out by the action of σΩ p (−1) F L , where Ω p is the worldsheet parity, and (−1) F L gives an additional minus sign on the RR sector. The uplift of this combination is the purely geometric involutionσ.
The combined operation Ω p (−1) F L has a different action on the different O(6, 6) × SL(2, R) components of L and K a . On O(6, 6) bispinors, such as the RR potentials or the pure spinors of generalized complex geometry Φ ± , which are tensor products of a left and a right-moving spinor, it acts in the following way [13] where λ is the following action on forms This can be understood since worldsheet parity exchanges the left and right-moving sectors, and on the bispinors, which are tensor products of left and right moving spinors, it acts by transposition. Since the orientifold projection keeps states which are even under the action of σ(−1) F L Ω p , one requires the involution to satisfy We want to define an analogous "λ-operation" as an action on fundamental SL(2) and O(6, 6) indices. The following operator acting respectively on the 12 of O(6, 6), 2 of SL(2, R) and 32 of O(6, 6) does the job

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For higher representationsλ just acts on all indices. 14 Therefore, on the fundamental 56 representation, which decomposes into O(6, 6) × SL(2, R) as in (4.5), this action reads On the adjoint representation, whose O(6, 6) × SL(2, R) decomposition is given in (4.6), we getλ The claim is thatλ acts like Ω p (−1) F L on K 1 , while on L and K 2/3 it is −λ that does the job. The fields that will survive the orientifold projection are therefore those for which σ acts in the following way In the language of (4.13), we see from (3.4)-(3.6) that the action ofλ corresponds to the exchange of the two spinors η i . For the corresponding vector fields L µ = (L iA µ , L + µ ) in the 56 representation we have Here, L Ai µ are the electric and dual magnetic vectors coming from the off-diagonal components fo the metric and the B-field, while L + µ collects the Ramond-Ramond fields with one external leg, i.e. L + µ = (C 1 ) µ + (C 3 ) µ + (C 5 ) µ + (C 7 ) µ .

Type IIB orientifolds
For completeness (and because we will use later O9 as an illustration) we give the action for type IIB orientifolds. 15 There, the theory is modded out by σΩ p (−1) F L for O3/O7 projection, and σΩ p for O5/O9. This means that the latter projection has an extra minus sign on the O(6, 6) spinors with respect to the type IIA case, i.e. we definẽ where the plus sign is for O3/O7 projections, while the minus applies to O5/O9. This contributes to a ± sign in the last components of (5.16) and (5.17). We then require (5.18) and (5.19), withλ replaced byλ IIB .

New Z 2 projections
In general, new orientifold actions can be found by conjugating known orientifold actions with elements in E 7(7) (Z). For all these new orientifolds our discussion applies. A simple example of a new Z 2 action is the NS5-projection that is related to O5-orientifolding in type IIB by S-duality in ten dimensions. Concerning the involution, S-duality only exchanges the roles of the B field and the field C 2 . Therefore, the NS-NS (R-R) sector is even (odd) under the resulting involution, which thus can be written as (−1) F L σ, where F L is the left-moving fermion number on the world-sheet and σ is an involution of the internal space that inverts four of the internal directions, satisfying the following on the pure spinors of GCG and the RR fields Correspondingly, the involutionλ NS5 is Subsequently, the roles of K 1 and K 3 are exchanged with respect to (5.6). S-duality implies that the fixed points of this action are negative tension objects with negative NS5-brane charge.
Note that this action can be defined in a completely analogous way in type IIA, and it can be uplifted to M-theory to find the orientifolding for M5-branes, with The analogous D4-orientifolding lifts to the same expression, but without any involution of dy. Though we can uplift these involutions to M-theory, even with these assignments forσ M5 , the M5-involution does not become an orbifold action. More precisely, it acts on the adjoint representation as σ * M5 but on the exceptional generalized tangent bundle with an extra minus sign, i.e. with −σ * M5 . Therefore, the projection on L and K a is given bỹ in contrast to the orbifold action given in (5.6). Thus, no SU(7) structure surviving the involution can be defined. As a consequence the fixed points of σ M5 cannot be resolved in a geometric way within M-theory, in contrast to the orbifold fixed points of section 5.1.

Kähler subspaces
Here we show how the orientifold projection selects the N = 1 special Kähler subspaces inside the N = 2 Kähler and quaternionic ones. Before we analyze how a Kähler space emerges from the projection on the hypermultiplets, we first want to understand the reduction of E 7(7) under the orientifolding. Let us first start with an orientifolding to O9-planes, i.e. σ is the identity (and we are modding out the theory just by the action of Ω p ). As the representations of E 7(7) split into the even and odd parts underλ, they form representations of a subgroup of E 7(7) that is the subgroup of even transformations. Therefore, we

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analyze the action ofλ on the adjoint of E 7 (7) , split into representations of the subgroup O(6, 6) × SL(2, R) corresponding to T-and S-duality subgroups, cf. eq. (4.6). 16 On the adjoint of O(6, 6),λ acts as in the first component of (5.17). The 66 representation therefore gets projected to Thus we find O(6, 6) → Gl(6, R) = SL(6, R) × R + (and the subindex denotes the charge under this R + , corresponding to the volume). Similarly and thus SL(2, R) → R + (and the subindex denotes the charge under this R, corresponding to the dilaton). Finally, on the (32 ′ , 2) representation we get (recall that for O9,λ IIB acts as −λ on the spinor) On the first line we stated the SL(6, R) representations and denoted the charges under the two R + factors coming from the volume and the dilaton, and on the second line the superscript denotes the SL(2) component, while the number in parenthesis in the subscript denotes the degree of the form. We see that the diagonal R + factor together with the two scalars in (5.27) forms an SL(2, R) group, while the two 15s together with the non-diagonal R + factor enhance the SL(6, R) to O(6, 6). Thus, the new covariance group is O(6, 6) × SL(2, R). Since this is a different O(6, 6) × SL(2, R) from the original one associated with S-and T-duality (see Footnote 16), we call this O(6, 6) × SL(2, R) O . Thus we get where in the last line we have used the O(6, 6) × SL(2, R) embedding of the B and C-fields given in (4.7). If σ is not the identity, the situation is slightly more involved. The orientifolding in general maps different points onto each other. Only at the locus of the O-planes the JHEP03(2013)145 covariance group can be really projected to a subgroup. Let us consider the case of an O(3 + p)-plane. At the O(3 + p)-plane we can split the tangent space where the supraindex in parenthesis indicates the dimensions of each space. The involution σ * acts as +1 on T and as −1 on T ⊥ . Therefore, the combination σ * λ projects the geometric group Furthermore, σ * λ projects . Hence, we see that for all orientifold actions, the covariance group projects to O(6, 6) × SL(2, R) O(p+3) (as we saw this subgroup is different for each type of orientifold). Now let us consider the projection (5.18) on the vector and hypermultiplet sectors. The vector fields that survive the orientifold projection are those that are even under σ * λ (see (5.19)). For O9, where σ is the identity andλ acts as −λ in the spinor part, we get that the surviving vector fields are the Kaluza-Klein vectors as well as the vectors associated with the internal one-form (C 2 ) µ (and their magnetic duals). As for the N = 1 chiral fields that descend from N = 2 vector multiplets, we keep from L only the pieces that are invariant under −σ * λ . Again, for an O9 and an SU(3) structure, we have that the projection onto states that are invariant underλ gives that all degrees of freedom in the three-form Ω is kept.

Discussion
In this work we derived the form of the couplings for general SU(7) and SU(6) structures in M-theory and type IIA, which correspond to (off-shell) N = 1 and N = 2 supersymmetric compactifications to four dimensions, building on and extending the work of [4,5]. Using EGG we could reformulate all degrees of freedom in such backgrounds by a set of fundamental objects in E 7 (7) representations. Moreover, the effective couplings are easily determined as singlets that are tensor products of the fundamental objects and their first derivatives in JHEP03(2013)145 E 7 (7) . In particular, N = 1 backgrounds are determined by an SU(7) structure φ in the 912 representation. Its quartic invariant gives the Kähler potential, while the superpotential is determined by an eigenvalue equation. In contrast, N = 2 backgrounds admit two sectors: vector-and hyper-multiplets. The former is described by one object L in the (fundamental) 56 representation whose quartic invariant gives the Kähler potential. The hypermultiplets are described by an SU(2) subalgebra spanned by a triplet of structures K a in the adjoint representation. The normalization of the SU(2) commutator relations gives the hyper-Kähler potential of the hyper-Kähler cone over this quaternionic Kähler space. L and K a together define an SU(6) structure. The couplings of the two sectors, i.e. the prepotentials, are given by a triple tensor product of these two objects with the derivative operator.
Furthermore, we discussed involutions in EGG that are supposed to project an N = 2 background to an N = 1 one. Examples of these involutions are orbifoldings in M-theory or orientifoldings in type II. We found the explicit map between the original SU(6) structure and its SU(7) descendant. In particular, while L as well as K 2 + i K 3 are projected out, their tensor product produces φ, which defines the SU(7) structure and survives the blow-up to a smooth geometry. The N = 1 Kähler potential and superpotential are then naturally determined by the N = 2 Kähler and hyper-Kähler potentials, and the prepotentials. We also determined the projection that creates negative tension objects with negative M5-brane charge in M-theory and observed that no SU(7) structure can be defined in that case, i.e. one cannot describe the resolution of singularities from involutions other than orbifoldings in EGG. In particular, the orientifold singularities related to D6-branes in type IIA cannot be resolved, but the corresponding M-theory orbifold fixed points can. In other words, the pure existence of an extra coordinate enables to resolve the singularities of D6-branes and O6-planes. It seems in order to describe D-branes in EGG one needs to introduce extra coordinates. For instance, while NS5-branes cannot be described in generalized geometry, they could in principle be described in doubled geometry. It would be interesting to understand the resulting doubled geometries in the presence of NS5-branes and their negative tension counterparts further. Even more challenging would be the realization of a 56-dimensional space that covariantizes E 7 (7) and could describe all branes in type II string theory or M-theory.
We pointed moreover out that there exists an intermediate generalized tangent bundle T 8 in M-theory that transforms under an SL(8) subgroup of E 7 (7) . From the type IIA point of view, this SL(8) group contains the geometric transformations SL(6) and the group SL(2) transforming the four-dimensional axiodilaton τ = B 6 + i e −φ . In this language, N = 1 backgrounds are described by a four-form on T 8 , N = 2 backgrounds by a real two-form and a complex four-form, i.e. they correspond to Spin(7) and SU(4) structures in eight dimensions. This suggests that there should exist a lift to an eight-dimensional space M 8 on which T 8 is the tangent bundle, similar to F-theory, whose volume is normalized everywhere. In the fashion described above, M 8 would not only geometrize D6-branes, but also some kind of exotic branes (as described in [14,15]) that form a set of (p, q)-branes for the four-dimensional axiodilaton, similar to F-theory. It would be very interesting to understand such geometries further.
In the following we give relevant formulas for products of representations in the SL(8) decomposition and their translation into the SU(8) decomposition of spinors, following and extending [7].
A.1 E 7(7) group theory in terms of SL (8) representations The E 7 (7) representations of interest are the fundamental 56, the adjoint 133 and the 912, whose decompositions under SL(8) are given in (2.1), (2.2) and (2.3), respectively. In the following we denote objects in the fundamental representation by α, β, etc. and write them in terms of SL(8) representations as α = (α ab , α ab ) . (A.1) Similarly, the adjoint representation decomposes as The 912 representation finally is given by We will use the following notation for the product of representations: ( where rep is any representation of E 7 (7) . The action of the adjoint on the fundamental representation, in other words the product 133 × 56 → 56, is given by The symplectic invariant on the 56 reads α, β = α ab β ab − α ab β ab . (A.6)

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where we defined γ ab βα = C βγ (γ ab ) γ α and C αβ is the matrix that induces transposition on spinors. In the 133, if only the 63 adjoint representation of SU (8)  In this appendix we give the computation of the superpotential given in (3.55). We start from (3.54) where the differential operator is given by (3.38). The form of φ is given in (3.46)-(3.48). In order to compute (3.54), we consider first where we used Sp(56) indices. 17 Now we can translate this back into E 7(7) indices, rewriting the above equation as where we used the notation of eq. (A.4) and in the last term the differential operator D acts on φ. If we use the form A 4 = (0, ρ 8 ∧ A 3 ) and the form of the differential operator (3.38), we find that D · A 4 = 0 , A 4 · D = 0 . The last equation actually puts all further terms in (B.4) to zero. From this together with (3.54), the superpotential can be computed to be In order to make the dependence on A 6 explicit, we use Formula (B.1) but now for A = (ρ 8 ⊗Â, 0) and note that D BÂA B = 0 andÂ D C D C = 0. Therefore, we find Furthermore, we compute D ⊗Â = (0, 0, (ρ 8 ∧ dA 6 )ρ 8 ⊗ ρ 8 , 0) , (B.8) giving (B.9) From self-duality of φ (0) 4 we find that the first two terms actually agree. This means that we finally have e K φ /2 W = ι ρ 8 φ In this appendix we compute the Killing prepotentials of the N = 2 theory given in (3.39). This computation is very similar to the one in appendix B.1. We start with the second term in (3.13) where the differential operator is given by (3.38). The objects K a and L defining the SU(6) structure are given in (3.33) and (3.32), respectively. In order to compute (3.13), we consider first where we again used Sp(56) indices. Translating this back into E 7(7) indices, we find where in the last term the differential operator D acts on K. The last two terms vanish due to A 4 = (0, where the coefficients are given in (B.5). We can treat the transformationsÂ = (ρ 8 ⊗Â, 0) using the same equations and (B.8), so that DK = e A shifts D(e −A shifts K) − e A shifts D ⊗ A 4 ⊙ (e −A shifts K) − e A shifts D ⊗Â ⊙ (e −A shifts K) . (B.14) For the first term in (3.13) we have essentially to determine the scalar derivative (L, D). Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.