Mariana Graña
email: mariana.grana@cea.fr

Hagen Triendl
email: hagen.triendl@cea.fr

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

) and SU(6) structures on this bundle and thereby derive the effective four-dimensional N = 1 and N = 2 couplings, in particular we compute the Kähler and hyper-Kähler potentials as well as the triplet of Killing prepotentials (or the superpotential in the N = 1 case). These structures and couplings can be described in terms of forms on an eightdimensional tangent space where SL(8) ⊂ E 7 acts, which might indicate a description in terms of an eight-dimensional internal space, similar to F-theory. We finally discuss an orbifold action in M-theory and its reduction to O6 orientifolds, and show how the projection on the N = 2 structures selects the N = 1 ones. We briefly comment on new orientifold projections, U-dual to the standard ones.

Introduction

It has been a long-standing problem to understand more general flux backgrounds in string theory. The most successful tool for this task has been supersymmetry, since it reduces the second-order equations of motion to first-order supersymmetry equations that are much simpler to solve and provides non-renormalization theorems for certain sets of couplings. The simplest string backgrounds are Ricci-flat geometries that admit a number of covariantly constant spinors. However, in the presence of fluxes, the first-order supersymmetry equations become more involved, as all bosonic fields of string theory appear in these equations. The global bosonic symmetries of the massless string spectrum then can help to write these first-order equations in a simpler form. This is the approach of generalized geometry. The introduction of a generalized tangent bundle covariantizes these symmetries by combining the diffeomorphisms and gauge transformations of the gauge fields into a single vector bundle. The metric on this bundle then includes, besides the metric, the other massless bosonic fields. The idea was first applied to the T-duality group that transforms the metric and the B-field into each other. This description goes by the name of Generalized Complex Geometry [START_REF] Hitchin | Generalized Calabi-Yau manifolds[END_REF].

More recently, the formalism was extended to the U-duality group of type II string theory to also include its Ramond-Ramond fields and to M-theory, covariantizing its three-form gauge field [START_REF] Hull | Generalised Geometry for M-Theory[END_REF][START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF][START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF][START_REF] Coimbra | E d(d) × R + Generalised Geometry, Connections and M Theory[END_REF]. In both cases the covariantized symmetry group relevant to compactify to four dimensions is E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] , and the generalized tangent bundle of type IIA is recovered from the one of M-theory by dimensional reduction. All possible supersymmetric backgrounds of type II and M-theory can then be understood as (generalized) G-structure backgrounds, where G is, depending on the number of supercharges, a subgroup of SU(8) (the maximal compact subgroup of E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] ). 1 The first-order supersymmetry equations can then be written in a covariant way [START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF][START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF][START_REF] Waldram | [END_REF]. Furthermore, G-structure backgrounds that solve the equations of motion but not the first-order supersymmetry conditions can usually be understood as backgrounds with spontaneously broken supersymmetry. In this sense, generalized geometry serves as an off-shell supersymmetric formulation for general G-structure backgrounds.

Generalized G structures are characterized by fundamental objects (defined by bispinors) in certain representations of E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] such that their common stabilizer is exactly G. These fundamental objects determine the generalized metric (which is stabilized by SU [START_REF] Waldram | [END_REF]) and thereby the entire bosonic supergravity background. The first-order supersymmetry conditions are resembled by first-order differential equations on them. Moreover, the couplings of the off-shell supersymmetric formulations are given in terms of tensorial combinations of these fundamental objects into E 7(7) singlets.

A major aim of this work is to understand SU [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] and SU [START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF] structures in M-theory and type II, as they correspond to backgrounds with the off-shell structure of fourdimensional N = 1 and N = 2 supergravity. SU [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] structures in M-theory have already been discussed to some extend in [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF], SU [START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF] structures in type IIA have already been discussed in [START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF]. Of particular interest in our discussion will be the existence of an eight-dimensional intermediate bundle between the tangent and the generalized tangent bundle that admits the action of SL( 8) ⊂ E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] . This might indicate the existence of a twelve-dimensional theory (similar to F-theory) in which some of the charges will be geometrized. SU [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] and SU [START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF] structures are related to Spin [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] and SU(4) structures in SL [START_REF] Waldram | [END_REF].

After we briefly review the E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] -covariant formalism of exceptional generalized geometry in Section 2, we will discuss SU [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] and SU [START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF] 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 [START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF] and SU [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF] resembles the global aspects of the compactification. More details can be found in [START_REF] Hull | Generalised Geometry for M-Theory[END_REF][START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF][START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF][START_REF] Coimbra | E d(d) × R + Generalised Geometry, Connections and M Theory[END_REF].

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 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

A =(T 8 ⊗ T * 8 ) 0 ⊕ Λ 4 T * 8 , 133 =63 ⊕ 70 , (2.2)
where the subscript 0 denotes traceless. We will also need the 912 representation, which splits according to

N =S 2 T 8 ⊕ (Λ 3 T 8 ⊗ T * 8 ) 0 ⊕ S 2 T * 8 ⊕ (Λ 3 T * 8 ⊗ T 8 ) 0 , 912 =36 ⊕ 420 ⊕ 36 ′ ⊕ 420 ′ , (2.3)
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 [START_REF] Waldram | [END_REF], 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 [START_REF] Waldram | [END_REF] subgroups are related by some E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] transformation and the decomposition of E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Waldram | [END_REF]. Note that for Spin [START_REF] Waldram | [END_REF], we can impose a Majorana-Weyl condition on the spinor. The Weyl-spinors are in one-to-one correspondence to Spin [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Waldram | [END_REF] (see more details in [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF]), we get

T * 8 = (Λ 7 T * 7 ) -1/4 ⊗ (T * 7 ⊕ Λ 7 T * 7 ) 8 = 7 ⊕ 1 ρ a = (ρ m , ρ 8 ) (2.4)
where ρ is some one-form, and a = 1, ..., 8, m = 1, ..., 7. Note that the eight-form ρ 1 ∧ • • • ∧ ρ 8 is just one on T 8 . This fits nicely with the fact that only SL(8) acts on this bundle. This implies that the fundamental 56 representation (2.1) decomposes as

E M = (Λ 7 T 7 ) 1/2 ⊗ (T 7 ⊕ Λ 2 T * 7 ⊕ Λ 5 T * 7 ⊕ (T * 7 ⊗ Λ 7 T * 7 )) , 56 = 7 ⊕ 21 ⊕ 21 ⊕ 7 (2.5)
corresponding respectively to momentum, M2-and M5-brane charge, and KK monopole charge.

In turn, the adjoint (2.2) decomposes into

A = T 7 ⊗ T * 7 ⊕ Λ 6 T 7 ⊕ Λ 6 T * 7 ⊕ Λ 3 T 7 ⊕ Λ 3 T * 7 , 133 = 48 ⊕ 1 ⊕ 7 ⊕ 7 ⊕ 35 ⊕ 35 (2.6)
We recognize here the GL( 7) adjoint (first term), and the shifts of the M-theory 3form 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 3 and A 6 come respectively from the 70 and 63 representations of SL( 8) in (2.2), i.e. they embed into the SL(8) representations in (2.2) as

A shifts = Â ⊗ ρ 8 , A 4 , (2.7)
where we defined the four-form

A 4 ≡ A 3 ∧ ρ 8 , (2.8)
and the vector  ≡ (vol 7 ) -1 A 6 , (2.9)

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 [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF][START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF]. 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 [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF])

ǫ = ξ i + ⊗ η c i + ξ i -⊗ η i , (3.1)
where ξ i ± are four-dimensional Weyl spinors, η i are complex Spin [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] spinor and i = 1, .., N determines the amount of 4D supersymmetry. When we embed Spin [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] into Spin(8), we can choose η i to be of positive chirality with respect to Spin [START_REF] Waldram | [END_REF], which means that we can take η i to transform in the 8 of SU [START_REF] Waldram | [END_REF]. Note that there exists a Majorana condition for Spin [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] (and also for Spin [START_REF] Waldram | [END_REF]) 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Waldram | [END_REF]. Consequently, a single η defines an SU(7) structure [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF], 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, namely

ηi η j = e -K L /2 δ ij , (3.2)
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

η T i η j = c ij , (3.3)
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 and an SU(2) R triplet of bispinors [START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF]. 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) reads

L(0) = 1 2 (ǫ ij η i ⊗ η j , ǫ ij ηi ⊗ ηj ) , L(0) = 1 2 (-i ǫ ij η i ⊗ η j , i ǫ ij ηi ⊗ ηj ) , (3.4)
where for later convenience we have defined two real bispinors in the 28 and 28 that are related to L by L = L(0) + i L(0) . From L(0) we can define also the almost complex structure J L that relates real and imaginary parts of L, given by

J L = 2 L × L = L × L = (e -K L /2 δ ij η i ⊗ ηj -1 4 e -K L 1, 0) . (3.5)
where 1 is the identity matrix with Spin(8) spinorial indices, δ α β . Furthermore, the SU(2) R triplet transforms in the 63 adjoint representation of SU [START_REF] Waldram | [END_REF], which is embedded in the 133 adjoint representation of E 7 , and reads in terms of the decomposition (2.2)

K (0) a = (κ e K L /2 σ i a k δ kj η i ⊗ ηj , 0) (3.6)
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 vanishes, i.e.

K (0) a • L (0) = 0 , (3.7)
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 [START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF]. 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.

L = e A shifts L (0) , J L = e A shifts J (0) L , K a = e A shifts K (0) a . (3.8)
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 [START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF]. 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 of L [START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF]

K L = -log -1 4 (J L , J L ) = -log -q( L) = -log -i 8 L (0) , L(0) . (3.9)
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 [START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF]

κ = -1 4 (K a , K a ) . (3.10)
It determines the normalization of the K a by

K a • (K b • ν) = -1 2 κ 2 δ ab ν + κ ǫ abc K c • ν , (3.11)
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 [START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF][START_REF] Waldram | [END_REF]]

P a = ǫ abc (D L K b , K c ) , (3.12)
and D L = L, D -L × D is the Dorfman derivative [START_REF] Coimbra | E d(d) × R + Generalised Geometry, Connections and M Theory[END_REF] along L, with D being the standard differential operator. 3 Inserting this, we get [8]

P a = ǫ abc ( L, D K b , K c ) + 4κ(L, DK a ) . (3.13)
3.2 N = 2 reductions and SU (4) structure on T 8

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 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

c ij = e -K L /2 δ ij . (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

χ = η 1 + i η 2 (3.15)
out of the real spinors η i . 4 We then find for L (0) and K (0)

a that K (0) 3 ± i K (0) 1 =(χ ⊗ χ T , 0) , J (0)
L ± i K (0) 2 =(χ ⊗ χ -1 4 e K L /2 1, 0) . (3.16)
On the other hand, the bispinors Φ 1 = χ ⊗ χ and Φ 2 = χ ⊗ χ T are a pair of compatible O(8, 8) pure spinors that define an SU(4) structure on T 8 , given by [1]

Φ 1 = 1 4 e -K L /2 exp(-i e K L /2 J (0) 8 ) , Φ 2 = Ω (0) 4 , (3.17)
where we made the following definitions

J (0) 8 = -i χγ ab χ dx a ∧ dx b = ǫ ij ηi γ ab η j dx a ∧ dx b , Ω (0) 4 = -1 4 κ e K L /2 χ T γ abcd χ dx a ∧ dx b ∧ dx c ∧ dx d = -1 4 κ e K L /2 (η 1 + i η2 )γ abcd (η 1 + i η 2 )dx a ∧ dx b ∧ dx c ∧ dx d . (3.18)
In terms of the x m we find

J (0) 8 = J + v (0) ∧ ρ 8 , J = ǫ ij ηi γ mn η j dx m ∧ dx n , (3.19)
Ω (0) 4 = (ρ 8 + i v (0) ) ∧ Ω , Ω = ι ρ8 Ω (0) 4 , (3.20)
where the one-form v (0) is defined by 5

v (0) = ι ρ8 ǫ ij ηi γ ab η j dx a ∧ dx b . (3.21)
The four-form Ω (0) 4 defines a complex structure

I (0) 8 by (I (0) 8 ) m n = 1 κ 2 ( * 8 Im Ω (0) 4 ) mpqr (Re Ω (0) 4 ) npqr = 1 2κ 2 ((v (0) ) m (ρ 8 ) n -ρm 8 v (0) n + (Im Ω) mpq (Re Ω) npq -(Re Ω) mpq (Im Ω) npq ) .
(3.22) In terms of these objects, the structure L (0) has the following SL(8, R) decomposition (cf. (2.1)) 6

L (0) = L(0) + i L(0) =(-ρ 8 ∧ v(0) + e K L vol -1 8 ( 1 2 ρ 8 ∧ v (0) ∧ J ∧ J), i(ρ 8 ∧ v (0) -J)) =(e K L vol -1 8 ( 1 3! J (0) 8 ∧ J (0) 8 ∧ J (0) 8 ), -i J (0) 8 ) =e i ρ 8 ∧v (0) ∧J • (-ρ 8 ∧ v(0) , 0) , (3.23) 4 A complex Spin(8) spinor χ is called pure if χ T χ = 0.
5 The hat denotes the vector dual to a one-form. In the special case that v is the one-form dy of a circle, one can reduce the theory to type IIA, where v m γ m8 then plays the role of chirality. 6 For the generic case of complex spinors η i , the form of L (0) is also given by Eq. (3.23), but in this case J (0) 8 is complex, given by J (0) 8 = ǫ ij η T i γ ab η j dx a ∧ dx b , and similarly for J and v (0) .

where we have used (A.20) 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

J L (0) = (0, J 8 ∧ J 8 ) .
(3.24)

Similarly, using (3.16) and (A.21), we can determine

K (0)
a for the decomposition (2.2) to be K

(0) 1 =(0, Im Ω (0) 4 ) , K (0)
2 =(κI

(0) 8 , 0) , K (0)
3 =(0, Re Ω (0) 4 ) .

(3.25)

From the Fierz identities (or from the SU(2) algebra) it follows that Ω (0) 4 transforms holomorphically under I (0) 8 , and obeys

Ω (0) 4 ∧ Ω (0) 4 = 0 , Ω (0) 4 ∧ Ω(0) 4 = 2κ 2 vol 8 . (3.26)
We also have ι v(0) J = ι ρ8 J = 0, and 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 (0) 8 defined by g

1 3! v ∧ J ∧ J ∧ J = e -2K L vol 7 or in other words 1 4! J (0) 8 ∧ J (0) 8 ∧ J (0) 8 ∧ J (0) 8 = 1 3! J ∧ J ∧ J ∧ v ∧ ρ 8 = e -2K L vol 8 . ( 3
(0) 8 = e K L /2 J (0) 8 • I (0) 8 , (3.29)
is a symmetric object. Furthermore, explicit computation shows that g (0)

8 is of the form

g (0) 8 = (detg 7 ) -1/4 g 7 0 0 detg 7 . (3.30)
Similar to (3.29) we also find for the inverse metric g

(0) -1 8
the expression

g (0) -1 8 = e 3K L /2 I (0) 8 • (vol -1 8 ( 1 3! J (0) 8 ∧ J (0) 8 ∧ J (0) 8 )) . (3.31)
General elements L, K a in the orbit can be achieved from L (0) , K (0) a by the action of E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] , 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

L =e A 4 (vol -1 8 ( 1 3! J 8 ∧ J 8 ∧ J 8 ), -i J 8 ) , J L = -e A 4 (0, J 8 ∧ J 8 ) , (3.32) and K 1 =e A 4 (0, Im Ω 4 ) , K 2 =e A 4 (κI 8 , 0) , K 3 =e A 4 (0, Re Ω 4 ) , (3.33)
where A 4 is defined in (2.8) and the shifted J and Ω 4 are

J 8 = J + v ∧ ρ 8 , Ω 4 = (1 + i a v 6 )ρ 8 + i v (0) ∧ Ω -i ρ 8 ∧ v (0) ∧ ι ÃΩ (3.34)
with  defined in (2.9), and

v = v (0) -ι ÂJ , a v 6 = vol -1 7 • (v (0) ∧ A 6 ) . (3.35)
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 (0) 8 is shifted so that Ω (0) 4 is replaced by Ω 4 in (3.22).
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 [START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF], 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

K L = -1 2 log( 1 4! J 8 ∧ J 8 ∧ J 8 ∧ J 8 ) . (3.36)
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

κ = 1 2 Ω 4 ∧ Ω4 . (3.37)
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

D = (0, ρ 8 ∧ d) . (3.38)
The result of the computation is given by

P 3 + i P 1 = -4κ i J ∧ ρ 8 ∧ dΩ (0) 4 + ι v Ω (0) 4 ∧ ρ 8 ∧ dA 3 , P 2 = e -K L Im (L v Ω(0) 4 ) ∧ Ω (0) 4 -2κ 2 e K L ρ 8 ∧ J ∧ J ∧ v (0) ∧ dv (0) + 4κρ 8 ∧ dA 6 -(A 3 + i κv (0) ∧ J) ∧ dA 3 , (3.39)
where L v 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] structure in E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] . 9 This structure can be encoded in a nowherevanishing object φ in the 912 representation of E 7(7) [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF]. The 912 decomposes in SU [START_REF] Waldram | [END_REF] representations in the same way as for SL [START_REF] Waldram | [END_REF], given in (2.3). In terms of this decomposition, we have

φ (0) = (2η ⊗ η, 0, 0, 0) , φ(0) = (0, 0, 2η ⊗ η, 0) , (3.40)
with the normalization ηη = e -K φ /2 . (3.41) Using (A.22) we can write this in terms of SL( 8) representations (2.3)

φ (0) = -(c(g (0) 8 ) -1 , g (0)
8

• (vol -1 8 φ (0) 4 ), i cg (0) 8 , i(g (0) 8 ) -1 • φ (0) 4 ) (3.42)
where g

(0)
8 is the metric on T 8 (3.30), the complex constant c is defined by

η T η = c , (3.43)
and φ

(0) abcd = 3 8 η T γ abcd η . (3.44)
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

φ (0) 4 = ρ 8 ∧ α 3 + * 7 α 3 , (3.45)
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 [START_REF] Waldram | [END_REF].

A general element φ is obtained by

φ = e A shifts φ (0) = -e A 4 (cg -1 8 , g 8 • (vol -1 8 φ 4 ), i cg 8 , i g -1 8 • φ 4 ) , (3.46)
where A 4 , defined in (2.8), acts by the E 7 adjoint action (A.11), while A 6 shifts the metric (3.30) by

g 8 = (detg 7 ) -1/4 g 7 + ( * 7 A 6 ) 2 (detg 7 ) 1/2 * 7 A 6 (detg 7 ) 1/2 * 7 A 6 detg 7 , (3.47)
and the 4-form

φ 4 = φ (0) 4 + ρ 8 ∧ ι Â * 7 α 3 . (3.48)
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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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) [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF]. Therefore, the existence of φ is completely equivalent to η. We will denote the real and imaginary parts of φ by φ and φ. 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

J φ = 2 φ × φ , (3.49)
which in terms of the SL(8) decomposition (2.2) is The Kähler potential K φ in the orbit of φ is given by the quartic invariant in the 912 and turns out to be

J φ = e A 4 • (I φ , φ r 4 )) , (3.50
K φ = -log -q( φ) = -log -1 4 (J φ , J φ ) = -log -1 4 tr(I 2 φ ) -1 2 log 1 4! φ r 4 ∧ φ r 4 = -log -1 4 tr((I (0) φ ) 2 ) -1 2 log 1 4! φ r (0) 4 ∧ φ r (0) 4 , (3.53)
where in the last step we used (ι Â * 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] invariance. Note that this expression reduces to the known one in the G 2 -structure case.

As has been derived in [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF], the superpotential W is given through the eigenvalue equation (Dφ)

• φ = (e K φ /2 W )φ , (3.54)
where D is defined in (3.38). We present the detailed computation of the superpotential in Appendix B.1. We get

e K φ /2 W = ι ρ 8 φ (0) 4 ∧ dφ (0) 4 + ρ 8 ∧ dA 6 -2 i φ 4 ∧ ι ρ 8 dA 4 + ι ρ 8 A 4 ∧ dA 4 . (3.55)
which is exactly the result of [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF]. A similar result for manifolds of G 2 structure has been obtained in [START_REF] House | M-Theory compactifications on manifolds with G(2) structure[END_REF].

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.10). We find the objects

φ(0) a = L(0) ⊗ K (0) a = 2κ(ǫ ik (σ a ) k j η i ⊗ η j , 0, ǫ ik (σ a ) k j ηi ⊗ ηj , 0) , φ(0) a = L(0) ⊗ K (0) a = 2κ(-i ǫ ik (σ a ) k j η i ⊗ η j , 0, i(ǫ ik (σ a ) k j ηi ⊗ ηj , 0) , (3.56)
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 φ = (α i ǫ ik (σ a ) k j α j )φ a , where φ (0) a = φ(0) a + i φ(0) 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

φ (0) 1 = -2e -K L /2 (0, i g (0) 8 • (vol -1 8
Re Ω (0) 4 ), 0, (g

(0) 8 ) -1 • Re Ω (0) 4 ) , φ (0) 2 = -2e -K L /2 κ((g (0) 8 ) -1 , e K L g (0) 8 • (vol -1 8 (J (0) 8 ∧ J (0) 8 )), i g (0) 8 , i e K L (g (0) 8 ) -1 • (J (0) 8 ∧ J (0) 8 )) , φ (0) 3 = -2e -K L /2 (0, -i g (0) 8 • (vol -1 8 Im Ω (0) 4 ), 0, (g (0) 8 ) -1 • Im Ω (0) 4 ) , (3.57)
where g (0) 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 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 reductions 4.1 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.

T * 7 = T * 6 ⊕ T * y (4.1) ρ m = (ρ m, ρ y )
Choosing T y to have no GL(6) weight, we get that the exceptional tangent bundle (2.5) decomposes as 56 = 6 ⊕ 6 ⊕ 6 ⊕ 6 ⊕ 32, i.e.

E IIA =(Λ 6 T 6 ) 1/2 ⊗ (T 6 ⊕ Λ 0 T * 6 ⊕ Λ 2 T * 6 ⊕ T * 6 ⊕ Λ 5 T * 6 ⊕ Λ 4 T * 6 ⊕ (T * 6 ⊗ Λ 6 T * ) ⊕ Λ 6 T * =(Λ 6 T 6 ) 1/2 ⊗ (T 6 ⊕ T * 6 ⊕ Λ 5 T * 6 ⊕ (T * 6 ⊗ Λ 6 T * ) ⊕ Λ even T * 6 , (4.
2) where from the last expression we recognize the type IIA charges, namely momentum, winding, their magnetic duals and the p-brane charges.

The adjoint decomposes as

A IIA =[λ ⊕ (T 6 ⊗ T * 6 ) 0 ⊕ (T y ⊗ Λ 2 T 6 ) ⊕ (T * y ⊗ Λ 2 T * 6 )] ⊕ [ϕ ⊕ Λ 6 T 6 ⊕ Λ 6 T * 6 ] ⊕ [(T y ⊗ T * 6 ⊕ Λ 3 T * 6 ⊕ (T * y ⊗ Λ 5 T * 6 )] ⊕ [(T * y ⊗ T 6 ⊕ Λ 3 T 6 ⊕ (T y ⊗ Λ 5 T 6 )] = (T 6 ⊗ T * 6 ) ⊕ Λ 2 T 6 ⊕ Λ 2 T * 6 ⊕ R ⊕ Λ 6 T * 6 ⊕ Λ 6 T 6 ⊕ Λ odd T * 6 ⊕ Λ odd T 6 (4.3) where λ = dy ⊗ ∂ y , ϕ = 2dy ⊗ ∂ y + dx m ⊗ ∂ m , (4.4)
are linear combinations of the generators for shifts in the volumes of the six-dimensional part and the eleventh dimension. From the last equality we recognize the O(6, 6) adjoint, where the GL( 6) and B 2 -field transformations live, the SL( 2) adjoint where the dual B 6 of B 2 is, and the sum of odd forms corresponding to the shifts of the RR potentials.

More precisely, under the breaking of the U-duality group E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] into the product of T-duality and S-duality group SO(6, 6) × SL(2) ,the fundamental representation splits according to 1056 = (12, 2) ⊕ (32, 1) ,

L A = (L Ai , L ± ) (4.5)
where A = 1, ..., 12 (i = 1, 2) is a fundamental O(6, 6) (SL( 2)) index, and ± represents a positive or negative chirality O(6, 6) spinor (the plus is relevant for type IIA and Mtheory, while for type IIB we need the negative chirality representation). The adjoint decomposes as the adjoint of each group, plus a spinor contribution

133 = (66, 1) ⊕ (1, 3) ⊕ (32 ′ , 2) , K = ( K A B , K i j , K ∓i ) . (4.6)
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 [4]

A shifts = 0 B 2 0 0 , 0 B 6 0 0 , C ∓ 0 . (4.7)
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 , C 5 = ι ∂y A 6 , A 4 = ρ 8 ∧ (C 3 -dy ∧ B 2 ) , ι ∂y C 3 = 0 . (4.8)
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

A shifts = (C 1 ⊗ ∂ y + ρ 8 ⊗ Ĉ5 , ρ 8 ∧ C 3 -ρ 8 ∧ dy ∧ B) (4.9)
where Ĉ5 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

Ĉ5 = vol

ǫ 1 ǫ 2 = ξ i -⊗ η i + c.c. (4.11)
where η i are SU [START_REF] Waldram | [END_REF] 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

a as in (3.4), (3.6) in the case of N = 2 [START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF], 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

L = e (C 1 ⊗∂y+ρ 8 ⊗ Ĉ5 ,ρ 8 ∧C 3 ) e ρ 8 ∧dy∧(-B-i J) • (ρ 8 ∧ v, 0) . (4.12)
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 as11

η 1 = θ 1 + θ1 - , η 2 = θ2 + θ 2 - . (4.13)
For the special ansatz θi = 0, L and K have particularly nice forms in in terms of the O(6, 6) pure spinors12 Φ ± = e B (θ 1 + ⊗ θ2 ± ) (4.14) the SL(2, R) vielbein u i related to the four-dimensional axiodilaton, i = 1, 2, and the RR-spinor

C = C 1 + C 3 + C 5 as [4] L = e C (0, Φ + ) , K 3 + i K 1 = e C (0, 0, u i Φ -) , (4.15)
where this is written in the O(6, 6) × SL(2, R) decomposition of E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] , given explicitly below in (4.5), (4.6), and K 2 is just the commutator of K 3 and K 1 .

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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] has been given in (4.5) and (4.6).

product in the 912 representation is defined by13

L ⊗ (K 2 + i K 3 ) = √ κφ , (5.7)
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

φ 4 = √ κ -1 e -K L /2 Im Ω 4 + √ κe K L /2 J 8 ∧ J 8 φ r 4 =e -K L Im Ω 4 + κJ 8 ∧ J 8 . (5.8)
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

J φ = e A 4 • (0, e -K L Im Ω 4 + κJ 8 ∧ J 8 ) = e -K L K 1 + κJ L .
(5.9)

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 not blown up, the Kähler potential (3.53) simplifies to the sum the two Kähler potentials for L and K, i.e.

K φ = -1 2 log κ + K L + 1 2 log 2 , (5.10)
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 e K φ /2 W = 1 2 κ -1 (P 2 + i P 3 ) .

(

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.

Further descent to type IIA

O6 orientifolds

To recover O6-orientifolds, the orbifold involution σ 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 involution σ.

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 [START_REF] Benmachiche | Generalized N = 1 orientifold compactifications and the Hitchin functionals[END_REF] (-1)

F L Ω p Φ + = λ(Φ + ) , (-1) F L Ω p Φ -= -λ( Φ-) , (-1) F L Ω p C = λ(C) (5.12)
where λ is the following action on forms λ(α 2p ) = (-1) p α 2p , λ(α 2p-1 ) = (-1) p α 2p-1 .

(5.13)

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

σ * Φ + = λ(Φ + ) , σ * Φ -= -λ( Φ-) , σ * C = λ(C) . (5.14)
We want to define an analogous "λ-operation" as an action on fundamental SL( 2 (5.15)

For higher representations λ 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

λ(L) = L m1 -L m1 -L m2 L m2
, -λ(L + ) .

(5.16)

On the adjoint representation, whose O(6, 6) × SL(2, R) decomposition is given in (4.6), we get

λ(K) = K m n -K mn -K mn K m n , K 1 1 -K 1 2 -K 2 1 K 2 2 , -λ(K -1 )
λ(K -2 ) (5.17) (where by construction K 2 2 = -K 1 1 and K m n = -K m n). The claim is that λ 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

σ * L = -λ(L) , σ * K 1 = λ(K 1 ) , σ * K 2/3 = -λ(K 2/3
) .

(5.18)

In the language of (4.13), we see from (3.4)-(3.6) that the action of λ 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

σ * L µ = λ(L µ ) . (5.19)
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 define

λIIB = 1 6×6 0 0 -1 6×6 ⊗ 1 0 0 -1 ⊗ ±λ, (5.20)
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), with λ replaced by λIIB .

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

σ * Φ ± = Φ ± , σ * C = -C . (5.21) Correspondingly, the involution λNS5 is λNS5 = 1 12×12 ⊗ 1 2×2 ⊗ (-1) 32 . (5.22)
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.

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 under λ gives that all degrees of freedom in the three-form Ω is kept.

In the hypermultiplet sector, since E 7( 7) is projected onto O(6, 6) × SL(2, R) O(p+3) , we know from (5.18) that both K 2 and K 3 will be in the (32 ′ , 2) representation of O(6, 6) × SL(2, R) O(p+3) , which is odd under the projection. Since [K 2 , K 3 ] ∼ K 1 and each K 2 and K 3 determine each other, K 2 + i K 3 defines a pure O(6, 6) spinor tensored with a doublet of SL(2, R). Therefore, K 2 + i K 3 parametrizes a special Kähler space. On the other hand, K 1 is in the (66, 1) ⊕ (1, 3) of O(6, 6) × SL(2, R) O(p+3) . It can be understood as a generalized almost complex structure on the orbit of K 2 + i K 3 .

Discussion

In this work we derived the form of the couplings for general SU [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Pacheco | M-theory, exceptional generalised geometry and superpotentials[END_REF][START_REF] Grana | E7(7) formulation of N=2 backgrounds[END_REF]. Using EGG we could reformulate all degrees of freedom in such backgrounds by a set of fundamental objects in E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] representations. Moreover, the effective couplings are easily determined as singlets that are tensor products of the fundamental objects and their first derivatives in E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] . In particular, N = 1 backgrounds are determined by an SU [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] . 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] and SU(4) structures in eight dimensions. This suggests that there should exist a lift to an eightdimensional 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 [START_REF] Bergshoeff | Brane orbits[END_REF][START_REF] De Boer | Exotic Branes in String Theory[END_REF]) 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.

  .27) Furthermore, the compatibility condition (3.7) of L (0) and K (0) a implies ι v(0) Ω = 0 and J ∧ Ω = 0, or in other words J

8 Re φ 4 )

 84 acde (Im φ 4 ) bcde (3.51) and φ r 4 = 14 Re(cφ 4 ) = 1 2 e K φ /2 ηγ abcd η dx a ∧ dx b ∧ dx c ∧ dx d . (3.52)

into g 8

 8 (cf. Eq. (3.47)), J (0) 8 into J 8 and Ω (0) 4 into Ω 4 (see (3.34)), while A 3 acts as usual by the 4-form shift e ρ 8 ∧A 3 .

-1 6 •

 6 C 5 . (4.10) 4.2 N = 2 and N = 1 reductions

  ) and O[START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF][START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF] indices. The following operator acting respectively on the 12 of O(6, 6), 2 of SL(2, R) and 32 of O(6,

3 ( 8 µ) cdeb -φ b cde (vol - 1 8µ-(φ acd e ψ bcd e + φ bcd e ψ acd e ) -1 8 A. 2

 38182 The adjoint action 133 × 912 → 912 on the 912 is given by(µ • φ) ab =µ a c φ cb + µ b c φ ac + 2 φ) ab =µ c a φ cbµ c b φ ac + 2 3 (µ cde(a φ cde b) ) , (µ • φ) abc d = -3µ e [a φ bc]e d + µ d e φ abc e + µ abce φ ed + µ ef [ab φ ef d c]µ ef g[a φ ef g b δ d c] .(A.11)The product 912 × 133 → 56 reads(φ ⊙ µ) ab = -(φ ac µ b cφ bc µ a c ) -2φ abc d µ ) cdea ) , (φ ⊙ µ) ab =(φ ac µ c bφ bc µ c a ) -2φ abc d µ c d -2 3 (φ cde a µ cdebφ cde b µ cdea ) . (A.12)Similarly, the product 912 ⊗ 56 → 133 reads(φ • α) a b =α ca φ cb + α cb φ ca + α cd φ cda bα cd φ cdb a , (φ • α) abcd = -4(φ [abc e α d]e -1 4! ǫ abcdef gh φ ef g i α hi ) . (A.13)The product 912 ⊗ 912 → 133 reads in the SL(8) decomposition(φ × ψ) a b =φ ac ψ cb + φ bc ψ ca + 1 3 (φ cde a ψ cde b + φ cde b φ cde a ) δ a b (φ cd ψ cd + φ cd ψ cd -2 3 φ cde f ψ cde f -2 3 φ cde f ψ cde f ) , (φ × ψ) abcd =4(φ e[a ψ bcd] e + φ [bcd e ψ a]e + 1 4! ǫ abcdef gh (φ ei ψ f gh i + φ f gh i ψ ei )) + 2(φ f [ab e ψ f cd]e + 1 4! ǫ abcdef gh φ ef j i ψ ghi j ) .(A.14) The relation to SU (8) representations The SU(8) representation is spanned by anti-symmetric products γ ab of the matrices γ a that obey the Clifford algebra {γ a , γ b } α β = 2g ab δ α β . (A.15) Furthermore, the gamma matrices γ a fulfill (γ a ) α β (γ a ) γ δ = δ α δ δ γ β . (A.16) Under SU(8), the 56 decomposes according to α = (α αβ , ᾱαβ ) , 56 = 28 ⊕ 28 , (A.17)

Often these are called generalized G-structures as they are structures on the generalized tangent space, as opposed to usual G-structures on the tangent space.

The matrix C is complex symmetric and in general not diagonalizable.

L, D is the symplectic invariant (A.6), and (L×D) is the projection onto the adjoint in the product of two 56 elements, given in (A.8).

Note that this shift is not an E 7[START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] transformation because the four-form is not real.

As stated above, these objects define an SU (6) structure inside E 7[START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] .

This spinor can have a real and imaginary part and thereby actually defines an SU (7) × SU (7) structure.

S-duality here does not refer to the type IIB S-duality, but as the group that acts by fractional linear transformations on τ = B 6 + i e -φ .

In standard Calabi-Yau compactifications, one takes θ 1 = θ 2 = θ, θi = 0. Note that in this case η i as defined in (4.13) are not Majorana, but the combinations η 1 + η 2 and i(η 1η 2 ) are.

Note that for this ansatz, J (Ω) contains the bilinears between the two θ's involving two (three) Spin(6) gamma matrices.

Note that the 56 component of their tensor product is forced to be zero by the compatibility condition.

Note that the action on T ⊕ T * then induces the action λ in (5.13) on the spinor representation (isomorphic to a sum of forms).

Note that in type IIB the roles of Φ + and Φ -are exchanged.

Here S-duality does not refer to the type IIB S-duality acting on the RR axion C 0 and the dilaton, but the S-duality within the NS sector, that acts on τ = B 6 + i e -φ .

Note that E 7(7) ⊂ Sp(56).

Acknowledgments

We would like to thank Diego Marqués, Ruben Minasian, Eran Palti and Daniel Waldram for useful discussions. This work was supported in part by the ANR grant 08-JCJC-0001-0 and the ERC Starting Grants 259133 -ObservableString and 240210 -String-QCD-BH.

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 action σ * on the 8-dimensional space T 8 such that the SL [START_REF] Waldram | [END_REF] 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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] (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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] , given in (2.1) in terms of SL [START_REF] Waldram | [END_REF] 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).

To descend to M-theory, we require the orbifold action to have positive eigenvalue when acting on ρ8 . Eq. (2.4) tells us then that T + 7 is three-dimensional, while T - 7 is four-dimensional. Finally, to recover the type IIA orbifold action that gives rise to O6 planes, we require v 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 [START_REF] Acharya | Orientifolds, mirror symmetry and superpotentials[END_REF]. 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 as σ * J 8 = -J 8 , σ * Ω 4 = -Ω4 .

(5.5)

This implies that the action induced by σ * on K a and L in the 133 and the 56 representation, which define an SU(6) ⊂ E 7 structure, should be

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

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 σ * M5 dy = -dy , σ * M5 ρ 8 = -ρ 8 .

(5.23)

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 for σM5 , 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 by σ *

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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] split into the even and odd parts under λ, they form representations of a subgroup of E 7(7) that is the subgroup of even transformations. Therefore, we analyze the action of λ on the adjoint of E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] , 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 66 → 35 0 ⊕ 1 0 , (5.25)

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, λIIB 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 [START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF][START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF]. 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 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 Gl(6, R) → Gl (p, R) × Gl ⊥ (6p, R) .

(5.29) Furthermore, σ * λ projects

and these give each the (p, 6p) representations that enhance Gl (p, R) × Gl ⊥ (6p, R) to Gl(6, R) O(3+p) . Thus, as for the case of O9-planes, we find O(6, 6) → Gl(6, R), but now to a different Gl(6, R) indicated by the subindex. Under the breaking O(6, 6) → Gl(6, R) O (3+p) , the (2, 32 ′ ) representation projects as in (5.27). Here, the two surviving singlets are Λ p T * and Λ (6-p) T * ⊥ , which form singlets under the emerging Gl(6, R) O(3+p) . 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 and λ 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

Appendix A E 7(7) group theory

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 [START_REF] Grana | N=1 vacua in Exceptional Generalized Geometry[END_REF].

A.1 E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] group theory in terms of SL [START_REF] Waldram | [END_REF] 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 φ = (φ ab , φ abc d , φ ab , φ abc d ) .

(A.3)

We will use the following notation for the product of representations:

where rep is any representation of E 7 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] .

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 abα ab β ab .

(A.6)

The trace in the adjoint is

The 56 × 56 → 133 reads

The action of the adjoint onto itself, i.e. the 133 × 133 → 133 is given by

(A.9)

The product 56 × 133 → 912 is given by

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) is nonzero, i.e. if µ αβγδ = 0, one recovers the following SL(8, R) components

where µ ba = -µ ab and * 8 µ abcd = µ abcd (the symmetric and anti-self-dual pieces are obtained from the 70 representation µ αβγδ ) and µ ab = g ac µ c b . Similarly, if only the 36 and the 36 components are non-zero in the SU(8) decomposition of the 912, we get the SL(8, R) components 36 and 420 in the following way

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 [START_REF] Grana | N=2 vacua in Generalized Geometry[END_REF] indices, rewriting the above equation as

where we used the notation of Eq. (A.

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  = (ρ 8 ⊗ Â, 0) and note that D B ÂA B = 0 and ÂD C D C = 0. Therefore, we find

Furthermore, we compute

From self-duality of φ (0)

4 we find that the first two terms actually agree. This means that we finally have

(B.10)

B.2 The N = 2 prepotentials

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 From this together with (3.13), the Killing prepotentials can be computed to be