N = 2 vacua in generalized geometry

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Introduction
The study of four-dimensional configurations with reduced supersymmetries is crucial to connect string theory with phenomenology.Even if in many physically interesting situations supersymmetry is expected to be broken, the scale of supersymmetry breaking can be much lower than the compactification scale, and studying supersymmetric compactifications is a first step towards understanding the non-supersymmetric setups.In particular, supersymmetry has been shown to constrain the allowed internal geometries to certain specific classes.When no fluxes are turned on, supersymmetric backgrounds of type II supergravity of the form M 1,9 = R 1,3 × M 6 require the internal manifold M 6 to be Calabi-Yau [1].Such manifolds satisfy an algebraic condition, namely the existence of a global section on the spinor bundle over T M 6 (i.e.there should be a globally defined nowhere vanishing internal spinor), and a differential one, that the spinor is covariantly constant.The algebraic condition is necessary in order to recover a supersymmetric (N = 2) effective theory in four dimensions, while the differential one is required in order to have a supersymmetric vacuum.Turning on fluxes on the internal manifold is phenomenologically and mathematically interesting in many respects.They were primarily motivated in light of their potential to solve the problem of moduli stabilization [2].Their presence also leads to warped spaces, which are of interest in the Randall-Sundrum scenario [3], giving a stringy origin to the hierarchy of scales [2].From the mathematical point of view, while they leave the algebraic constrain intact, vacua with fluxes are possible on manifolds which have weaker differential properties.Rephrasing these constraints in a similar language as those for fluxless solutions was very much guided by the framework of generalized complex geometry developed by Hitchin [4].Generalized complex geometry was used in [5,6] to characterize N = 1 vacua.In analogy with the fluxless case, off-shell supersymmetry requires an algebraic condition to hold, namely the existence of a pair of pure spinors on the spinor bundle over the generalized tangent bundle T M 6 ⊕ T * M 6 .These pure spinors geometrize the entire NS-NS content of type II string theories, as they determine the metric, B-field and dilaton.To describe a vacuum, the pair of pure spinors should also satisfy specific differential conditions [6], namely the pure spinor that has the same parity as the R-R fluxes should be closed (and thus the manifold is said to be generalized Calabi-Yau), while the non closure of the second pure spinor is due to the R-R fluxes.Alternatively [7,8], these conditions can be obtained from the F and D-terms of the effective four-dimensional gauged supergravity [9,10].It has been also proven that the pure spinor equations can be deduced from a generalized calibration condition for D-branes [11,12].The R-R fields are not geometrized in the language of generalized complex geometry.Including them in some geometric structure necessarily demands enlarging the generalized tangent bundle, so that it includes the extra charges carried by D-branes.The natural generalization appears to be Exceptional (or Extended) Generalized Geometry (EGG) [13,14,15], its name alluding to the covariance under the exceptional groups appearing in U-duality.The algebraic conditions to have N = 2 supersymmetry in four-dimensions have been worked out in the language of EGG in [16].Very much in analogy to the generalized complex geometric case, they require the existence of two algebraic structures on the exceptional generalized tangent bundle (in fact one of them, rather than a single struc-ture, is actually a triplet satisfying an SU(2) R algebra), which are built by tensoring the internal SU(8) spinors.The SU(2) R -singlet structure, that we call L, describes the vector multiplet moduli space, while the triplet of structures (named K a ) describes the hypermultiplets.In type IIA (IIB) the structure L contains a difference of two even (odd) O (6,6) pure spinors, plus extra vectorial degrees of freedom, while the structures K a contain their odd (even) chirality counterparts, plus an additional bivector, two-form and a couple of scalars.Differential conditions in order to have an N = 1 vacuum in this language have been studied in [17] 1 , where it was found that N = 1 supersymmetry requires on one hand closure of both L and r a K a , where r a is a vector pointing in the direction of the N = 1 supersymmetry preserved.On the other hand, the structure along the complex orthogonal direction2 is closed upon projecting onto the holomorphic sub-bundle defined by L. The aim of this paper is to investigate the differential conditions on these structures required by N = 2 supersymmetric vacua on four-dimensional Minkowski space, and their corresponding expression in terms of the O(6, 6) pure spinors that they contain.A generic N = 2 theory possess an SU(2) R R-symmetry, which must left be unbroken in the N = 2 compactification, and therefore the conditions should be the same for the three K a .We expect that N = 2 supersymmetry should be translated into integrability of the structures L, K a .We show in this paper that all but one component of the derivative of L and K a are required to vanish.The two components (one in the derivative of L and one in the derivative of K) that do not vanish, involve representations that should be projected out in order to obtain a standard N = 2 effective four-dimensional supergravity description (i.e. a description without massive gravitini multiplets), but are there in the ten-dimensional formulation.We work in type IIA, though we expect the same equations to hold in type IIB 3 .We also write these equations in the language of GCG, i.e. we find the equations governing the two pairs of pure spinors that build up L and K.These equations involve the twisted differential d − H∧, and the R-R fluxes appear on the right hand side only when we consider the extra degrees of freedom.Conditions for unbroken N = 2 supersymmetry for type IIB compactifications on conformal Calabi-Yau manifolds were obtained in [18], by further restricting the N = 1 requirements found in [19] 4 .On more general manifolds and using the generalized geometric language, our current understanding of the conditions for N = 2 vacua amounts to checking whether there are two pairs of pure spinors giving the same metric, B-field and dilaton, which separately satisfy the N = 1 conditions5 .This is how N = 2 solutions have been obtained in [20] (for their description in terms of N = 2 gauged supergravity see [21]) and [22].On the other hand, a detailed analysis of the supersymmetric conditions leading to N = 2 AdS 4 or Minkowski vacua from a gauged supergravity point of view is done in [23,24], which provide concrete examples, some of which in the context of flux compactifications of M-theory.We will make contact with these works in the discussion.The study of N = 2 vacua is interesting also for the applications of the AdS/CFT correspondence in settings with reduced number of supersymmetries.This has for instance been investigated in the circle reduction of M 1,1,1 giving a massive deformation of the AdS 4 × M 6 backgrounds [25], as well as for a first order perturbative expansion in the Romans mass [26] .The paper is organized as follows: in Section 2 we introduce the necessary concepts of generalized complex geometry.In Section 3 we show the main features of the extended or exceptional version of generalized geometry.In Section 4 we present the differential conditions on the algebraic structures required by N = 2 supersymmetry on-shell.In Section 5 we write the equations for vacua in terms of pure spinors, and we finish by a discussion in Section 6. Appendix A reviews the generalized complex and exceptional geometric formulation of N = 1 vacua.Appendix B shows the different components of the algebraic structures in terms of pure spinors.Appendix C contains the tensor product formulae needed in computing the derivatives of the algebraic structures.Appendix D gives the equations on the SU(8) spinors obtained from the ten-dimensional supersymmetry transformations, and Appendix E includes the details of the derivation of the Eqs.presented in Sections 4 and 5.

Generalized complex geometry
In Generalized (Complex) Geometry [4], one constructs algebraic structures on the generalized tangent bundle T M ⊕ T * M.These structures appear in compactifications of type II theories as they are constructed from the tensor product of two internal spinors.We will concentrate on compactifications of type IIA to four-dimensional warped Minkowski space, i.e. the ten-dimensional metric is In order to recover an N = 2 effective action in four-dimensions, the following splitting of the ten-dimensional spinors should be globally well-defined where the minus (plus) sign on the chirality of η 2 is for type IIA (IIB).We will later see that this is not the most general ansatz for the four-six splitting, but in terms of the effective 4D theory, as well as to study N = 1 vacua, one can always make a redefinition such that the splitting has this form.This is not true though when we study N = 2 vacua.We will come back to this point several times in the text.
Tensoring the two internal spinors, one can build Weyl pure spinors 6 of O(6, 6), namely where the plus and minus refer to spinor chirality, and φ is the dilaton, which defines the isomorphism between the spinor bundle and the bundle of forms.Using Fierz identities, these can be expanded as Using the isomorphism between the spinor bundle and the bundle of differential forms (often referred to as Clifford map): the spinor bilinears (2.4) can be mapped to sums of forms.Under this isomorphism, the inner product of spinors Φχ is mapped to the following action on forms, called the Mukai pairing Φ, χ = (Φ ∧ s(χ)) 6 , where s(χ) = (−) and the subindex 6 means the six-form part of the wedge product.
For Weyl O(6, 6) spinors, the corresponding forms are only even (odd) for a positive (negative) chirality O(6, 6) spinor.In the special case where η 1 = η 2 ≡ η, familiar from the case of Calabi-Yau compactifications, we get where J, Ω are respectively the symplectic and complex structures of the manifold.Pure spinors can be "rotated" by means of O(6, 6) transformations.Of particular interest is the nilpotent subgroup of O(6, 6) defined by the generator with B an antisymmetric 6×6 matrix, or equivalently a two-form.On spinors it amounts to the exponential action We will refer to Φ as naked pure spinor, while Φ D will be called dressed pure spinor.The pair (Φ + D , Φ − D ) defines a positive definite metric on the generalized tangent space, which in turn defines a positive metric and a two-form (the B field) on the six-dimensional manifold.In an analogous way as an O(6) spinor defines an SU(3) structure (i.e., it is invariant under an SU(3) subgroup of O(6)), a pure O (6,6) spinor defines an SU(3, 3) ⊂ O(6, 6) structure.Its 32 degrees of freedom minus one corresponding to the norm parameterize the coset O(6, 6)/SU (3,3).Furthermore, two O(6) spinors which are never parallel, define an SU(2) structure, which is the intersection of the two SU(3) structures.Similarly, two pure O(6, 6) spinors, whenever they satisfy the following compatibility condition define an SU(3) × SU(3) structure.Pure spinors which are tensor products of O( 6) spinors as defined in (2.3) are automatically compatible.We finish this section by mentioning that the 6d annihilator space of an O(6, 6) pure spinor can be thought as the holomorphic bundle of a generalized almost complex structure (GACS) J , which is a map from T M ⊕ T * M to itself such that it satisfies the hermiticity condition (J t ηJ = η) and J 2 = −1 12 .Therefore there is a one-to-one correspondence between a pure spinor of O(6, 6) and a GACS.The GACS can be obtained from the pure spinor by [9]7

Exceptional Generalized Geometry
To incorporate the R-R fields to the geometry, in exceptional generalized geometry (EGG) [13,14] one extends the tangent space (or rather the generalized tangent space T ⊕ T * ) such that there is a natural action of the U-duality group on it.In this paper we will be interested in compactifications of type II theories (and in particular we will work with type IIA8 ) on six-dimensional manifolds, where the relevant exceptional group is E 7 (7) .Shifts of the B-field a well as shifts of the sum of internal R-R fields to particular E 7 (7) adjoint actions.To form a set of gauge fields that is closed under U-duality, we also have to consider the shift of the six-form dual of B 2 , which we will call B. 9In what follows, we will mainly use the decomposition of E 7 (7) under SL(8, R).This subgroup contains the product SL(2, R) × GL(6, R), and allows to make contact with SU(8)/Z 2 , the maximal compact subgroup of E 7 (7) .The latter is the group under which the spinors transform, and therefore the natural language to formulate supersymmetry using the Killing spinor equations.
The adjoint has the following decomposition where the first piece corresponds to the adjoint of SL(8, R), and we have µ a a = 0 and µ abcd is fully antisymmetric.We also define µ abcd ≡ 1 4! ǫ abcdef gh µ abcd = (⋆µ) abcd10 .For the 912 we have d and φ abc c = 0.The fundamental representation is where the charges live.Momentum and winding charges are embedded respectively in ν ′2m and ν 1m , while D0, D2, D4 and D6-brane charges in ν ′12 , ν mn , ν ′mn and ν 12 (for more details, see [17]).The gauge fields live in the adjoint representation.Their embedding in terms of the ) Finally, the fluxes live in the 912, and are embedded as11

E 7(7) structures as spinor bilinears
The supersymmetry parameters transform under the maximal compact subgroup of the duality group, which in the case at hand is SU (8).The action of this group on the spinors 12 is manifest if we combine the two ten-dimensional supersymmetry ǫ 1 , ǫ 2 as follows where ζ 1,2 − are four-dimensional spinors of negative chirality, and θ 1,2 are never parallel, and can be parameterized as The notation is chosen so that the standard ansatz for the ten-dimensional spinors (2.2) (used for example in [9]) corresponds to η1 = η2 = 0. Introducing ηI gives us the most general ansatz for four-dimensional N = 2 supersymmetry.A nowhere vanishing spinor θ defines an SU(7) ⊂ SU(8) structure.The pair (θ 1 , θ 2 ) defines an SU(6) structure 13 .Without loss of generality, we can choose a basis where the spinors are orthonormal, namely θI θ J = δ I J . (3.9) where The two spinors can be combined into the following SU(2) R singlet and triplet structures, which replace the pure spinors of GCG, and parameterize respectively the scalars from vector multiplets and hypermultiplets The triplet K a satisfies the su(2) algebra with a scaling given by the dilaton, i.e.
L and K a are the E 7 (7) structures that play the role of the generalized almost complex structures Φ + and Φ − .They belong respectively to the 28 and 63 representations of SU (8), which are in turn part of the 56 and 133 representations of E 7 (7) .We will use the SL(8, R) decomposition of L and K a .The former is obtained from the SU(8) object in (3.10) by where λ and λ ′ are respectively the 28 and 28 ′ (real) components of L, and α, β = 1, ..., 8 are Spin(8) spinor indices.As for K a , given that it is in the 63 representation of SU(8), we get that its SL(8, R) components are where K ba = −K ab (and K ab = K a c ĝcb ) and ⋆K abcd = −K abcd (the symmetric and selfdual pieces would be obtained from the 70 representation K αβγδ , which is not there).We give in Appendix B the different SL(8, R) components of L and K a in terms of bilinears of the 6d spinors η I , ηI in (3.8) that build up θ I .The structures L and K a can be dressed by the action of the gauge fields B, B and C − , i.e. we define where the action of C, B, B on L and K is given respectively by (C.3) and (C.4) and we have to use their embedding in E 7 (7) , given in (3.5).They span orbits in E 7 (7) which are respectively Special Kähler and quaternionic.As shown in [16], the structure L D is stabilized by E 6(2) , and the corresponding local Special Kähler space is . The triplet K aD is stabilized by an SO * (12) subgroup of E 7 (7) , and the corresponding orbit is the quaternionic space E 7 (7) SO * (12)×SU (2) , where the SU(2) factor corresponds to rotations of the triplet.The SO * (12) and E 6(2) structures intersect on an SU(6) structure if L and K a satisfy the compatibility condition where we have to apply the projection on the 56 on the product 56×133.This condition is automatically satisfied for the structures (3.10) built up as spinor bilinears.
4 Conditions for N = 2 flux vacua In this section we determine the equations for the structures (3.10) required by N = 2 supersymmetric compactifications on warped Minkowski space, i.e.where the tendimensional metric has the form (2.1).From the point of view of the effective fourdimensional N = 2 action, these come from setting to zero the triplet of Killing prepotentials P a , along with its variations.In the language of EGG, the triplet of Killing prepotentials reads [16] where S is the symplectic invariant on the 56 representation, given in (C.1) and D is the derivative twisted by the fluxes constructed as explained below.From demanding that this is zero under variations of L and K a , one expects that N = 2 supersymmetry requires that both L and the whole triplet K a are closed under D. We will see that this is roughly the case, though some subtleties arise.But before presenting the equations on L and K a , we will explain very briefly, following [17], how the twisted derivative is built and how it acts.We define where the derivative D is in the 56 representation, and its SL(8, R) decomposition (see (all otheer components are zero), and the fluxes F are in the 912 representation, and are given in terms of SL(8, R) in (3.6) 14 .
The equations involve the twisted derivatives of L and K projected onto specific representations, respectively the 133 and 56.We therefore have to use the following tensor products ).The powers of the dilaton in this metric explain the different powers of the dilaton appearing in these equations, as we will show later.
We will now briefly show how we obtained these equations, leaving the full details to Appendix E.

Explicit form of the twisted derivative
The twisted derivative defined in (4.2), applied to L and projected onto the 133 representation as in (4.4) (where the tensor products needed are given in (C.2) and (C.6) in terms of SL(8, R) decompositions), gives the following components 16   (DL) On the other hand, for K we use (4.5), and the tensor products in (C.3) and (C.7) and get the following SL(8, R) components ) 16 Here we are giving the equations for a complex 28 object as defined in (3.12).

Comparing to equations coming from Killing spinors
As shown in detail in Appendix D, the supersymmetry variations of the internal and external gravitino and the dilatino give algebraic and differential conditions on the tendimensional spinors ε 1 , ε 2 .Using the splitting into four and six-dimensional spinors corresponding to N = 2 supersymmetry, given in (3.8), these turn into conditions on the spinors θ I .We give these conditions in (D.5)-(D.7).Multiplying these by θ J (or θJ ) we get conditions on L and K that we give in Appendices E.1 and E.2 respectively.We show for instance how supersymmetry implies that Eq. (4.13) should vanish (this condition is written on the second line in (4.6)): multiplying (E.1) by Γ 12 , as well as l d times (E.3) by i Γ m , and tracing over spinor indices, we recover where for the third equality we have taken l d = 1.The calculations for the other components of the derivative of L are given in Appendix E.1.
We now show how one of the conditions on the derivative of K (defined in (4.8)), namely the second one in (4.7), can be recovered by using supersymmetry.Taking (E.21) multiplied by Γ p1 and summing over internal indices, together with n e times (E.22) and n d times (E.23) multiplied by −iΓ 2 , and tracing the overall sum over the spinor indices, we get For the mn components, a similar argument holds, while for the other equations a slightly more involved calculation is needed.We show in Appendix E.2 how to obtain the rest of the conditions for DK .We note that the equations that have an explicit power of the dilaton in (4.6) are those that involve a derivative of a component of L with one internal and one SL(2.R) index.
For instance (DL) 1  1 is proportional to ∇ p L p2 .According to the metric in (B.1), this component of L transforms as e −φ ⊗ T M (see (B.3)), and this power of the dilaton is compensated by the explicit e φ factor appearing in the first equation in (4.6).On DK, pieces that involve a derivative of a component of K that transforms with a power of e −φ (such as K mnp2 , for example, as shown in B.4)) do not carry explicit dilaton factors, while otheer powers are compensated by explicit powers of the dilaton.For example (DK) m1 contains ∇ m K 2 1 , which transforms as e −2φ , and this is compensated by the explicit e φ on the fourth line of (4.7).

Conditions for N = 2 vacua in GCG
Using the splitting of θ I in terms of SU(4) spinors η I , ηI as in (3.8), we can obtain L and K + = K 1 + iK 2 in terms of O(6, 6) pure spinors, namely where here Φ+ is defined in an analogous way as Φ + , Eq. (2.3), but using η; the superscript T denotes the transpose of the bispinor and we have defined (5. 2) The normalization condition (3.9) implies Note that the second condition is equivalent to Λ + 0 + Λ′ + 0 = 0, where the subindex 0 denotes the zero-form component.The structures L and K + contain two pure spinors Φ and Φ of positive and negative chirality respectively, plus extra degrees of freedom involving bilinears between η and η (which are zero in the "standard N = 2 ansatz" introduced in 2.2).In the case of K + , the two pure spinors Φ − and Φ− appear as independent degrees of freedom (unlike Φ + and Φ+ in L).In order to get the SL(8, R) components of L and K we use (3.12), (3.13) and the decomposition of the Gamma matrices in (B.2).The result is given in (B.3) and (B.4).We can see clearly that the extra degrees of freedom in L are "vectorial" type (i.e., in 6 representations of O(6), or in terms of the O(6, 6) × SL(2, R) subgroup of E 7 (7) they are in the (2,12)), while the extra degrees of freedom in K are in the adjoint of SL(2) and the adjoint of O (6,6).It is useful to define the polyforms Using the explicit form of the twisted derivatives in (4.9)-(4.24)we get that conditions (4.6) on L imply the following equation on δΦ +18 where d H = d − H∧ and we have defined the polyform and the Clifford action i.e. in the n + 1-form equation in (5.5), the real part of Λ 1 acts as a vector contracted on F n+2 , while the imaginary part is a one-form wedged on F n .From equations (4.7) specialized to K + , we get 19d H (e A−φ δΦ − ) = −2e A−φ Λ 0 F . (5.7) Before writing the additional equations on the other degrees of freedom that appear in L and K, we note that these equations involve sums and differences between Φ and Φ.While this is expected in the equations coming from L, since Φ + and Φ+ are not independent degrees of freedom, we expect more equation on Φ − and Φ− .Indeed, supersymmetry constraints the derivative of, for instance, K mnp1 , which does not appear in (4.17)- (4.24).Using the extra equations that we present in Appendix E.2.1, which involve the combination of Φ − and Φ− with an opposite sign as that of (5.4), we get the following set of equations where we have defined δΛ + (±) = Λ + ± Λ′+ (5.12) Let us make a few comments before we go on to the discussion.First, note that the equations do not look exactly like a pair of N = 1 equations of the form (A.4)-(A.6).This is because the N = 2 EGG formulation selects the pure spinors Φ and Φ, instead of Λ 1 , Λ 2 , defined in (B.5), which would be the natural ones from the N = 1 point of view.
In other words, the present equations are the natural ones when one thinks of N = 2 backgrounds in terms of an SU(6) structure on the exceptional generalized tangent space, and not in terms of a pair of SU( 7) structures.Then, we notice that the equations for the pure spinors involve the H-twisted differential, while the R-R fluxes appear on the RHS only when Λ 0 , Λ 1 are not-zero, or in other words when (at least one of) the spinors ηI is not zero.In the case ηI = 0, we get that Φ + and e 2A Φ − are d H closed, that A = −φ and the R-R fluxes should obey certain algebraic constraints.One solution within this class is the generalized Kähler solution [4] (previously called "bihermitian geometry" [28]), where F = A = φ = 0, and the two pure spinors are H-twisted closed.

Discussion
We have found the conditions on the twisted derivative of the structures L and K a required by compactifications to four-dimensional Minkowski vacua preserving N = 2 supersymmetry.As expected from doing variations on the triplet of Killing prepotentials in (4.1), N = 2 supersymmetry requires these structures to be twisted closed.Two subtleties arise, though.The first one is that there is one component of DL and one component of DK which are not zero.Massaging these two equations as much as possible, we were able to write the obstruction to twisted closure in terms respectively of a single R-R and NS-NS flux contracted with an appropriate L and K.These combinations are not set to zero by the other equations.The fact that these components of the twisted derivatives of L and K a are not zero is surprising, but does not contradict with the expectation coming from four-dimensional supergravity, since they involve derivatives of components of L and K a that need to be projected out in order to obtain a standard N = 2 off-shell formulation (see [16] for more details).The second subtlety is that there are explicit powers of the dilaton appearing in certain equations, though we could make sense of them considering how the dilaton appears when embedding GL(6, R) into SL(8, R).Furthermore, these powers of the dilaton appear uniformly in the GCG counterpart equations.
In the language of gauged supergravity, N = 2 conditions arise from requiring that the matrices S, W and N, appearing respectively in the susy variations of the gravitini, gaugini and hyperini vanish.The conditions obtained in [23]- [24] from setting to zero S and W , should be equivalent to our conditions on DK a , while the ones coming from setting N = 0, should translate into our conditions on DL.It would be nice to have an explicit check of this, as was done in [7], [8] in the case of N = 1 vacua and the equations on the pure spinors of generalized complex geometry.By parameterizing the SU( 8) spinors in terms of SU( 4) ones, we decomposed L and K + into O(6, 6) pure spinors.The structure L contains the difference between two even pure spinors Φ and Φ, while K + contains their odd counterparts, and they each have additional degrees of freedom.The N = 2 equations for Φ and Φ involve the H-twisted differential d − H∧, while the R-R fluxes appear on the right hand side, multiplying the extra degrees of freedom.These equations simplify considerably in the "standard N = 2 ansatz", where L and K + contain just Φ + and Φ − respectively.In this case the R-R fluxes completely decouple (and should obey some algebraic constraints), while the pure spinors are twisted integrable, i.e. they describe a generalized Kähler structure.
B SL(8, R) components of L and K a in terms of pure spinors To obtain the SL(8, R) components of L and K a , we use (3.12) and (3.13).Then we want to split the SL(8, R) index into a GL(6, R) and an SL(2, R) index.For that, we use the embedding of GL(6, R) into SL(8, R) given by the following metric (for more details see [17]) as well as the following decomposition for Cliff (8) gamma matrices in terms of the Cliff( 6) ones γ m This gives, for the 12 components of L for example where the subscript 0 denotes the zero-form piece of the polyform corresponding to the O(6, 6) spinor through the Clifford map (2.5), and we have used the fact that L transforms in the 56 representation weighted by a power of g 1/4 ≃ (Λ 6 T * M) 1/2 .We now note that given the factor of g 1/2 in L 12 , this transforms as a six-form, namely the Hodge star of the zero-form.Using additionally that the pure spinors are imaginary anti self-dual, i.e. * Φ ± = −iΦ ± , we can write . We proceed similarly for the other components of L and get where ǫ is a numeric totally antisymmetric tensor (i.e. with values ±1, 0), such that L mn , for example, transforms as a 4-form.
For K + , weighting by a factor g 1/2 , we get the following components where we have multiplied the whole 133 representation by a factor g 1/2 .To obtain the components of K 3 , we first write it in terms of pure spinors as where we have defined Then, for the SL(8, R) components of K 3 we just need to make the following replacements in (B.4) where the operation s on forms, which corresponds to minus (plus) the transposed of the bispinors, was defined in (2.6).

D Supersymmetric variations for the N = 2 spinor anstaz
The supersymmetry transformations of the fermionic fields of type IIA, namely the gravitino ψ and dilatino λ read, in the democratic formulation [27] δψ (5 − n)/ F (10)   n where ψ, λ and ǫ are a doublet of spinors of opposite chirality, as in (3.8), and P = −σ 3 , P n = (−σ 3 ) n/2 σ 1 act on the doublet.We use the standard decomposition of ten-dimensional gamma matrices γ (10) while from the external gravitino variation, we get and from the dilatino variation In these equations we have defined in terms of the "hermitean" and "antihermitean" pieces of F , namely and a slash means  We will also need the "transposed" of the equation coming from internal gravitino, namely Given L and product of gamma matrices Γ a 1 ...a i we will make use of the following type of combinations (E.7) and similarly for the anticommutator.Multiplying these equations by appropriate combinations of gamma matrices, we recover the combinations involved in (4.9)-(4.16).Unless otherwise specified, we will take in the very last step of the following equations.We start from (4.13) Similarly, we have Choosing this time l d = 2 we recover (E.13) We are then left with two equations.Using only the internal gravitino constraint we get For (DL) n m we have on one hand and on the other hand we can use By comparing the two, one recovers the following constraint Consider then the following combination using the commutator defined in (E.7) The following choice for l d and l e makes the equation look as simple a possible for which we obtain

E.2 DK
We need the hermitean conjugate of Eq. (D.5), namely Multiplying (D.5) by σ a θJ , and (E.20) by θ I σ a , we get the following condition (for any a) Using a similar trick on the external gravitino and dilatino equations (D.6) and (D.7), we also get and their "transposed" versions We sketch in the following how conditions (4.7) arise from supersymmetry.We first look at the mn components.We have