AdS5 Backgrounds with 24 Supersymmetries

We prove a non-existence theorem for smooth AdS5 solutions with connected, compact without boundary internal space that preserve strictly 24 supersymmetries. In particular, we show that D=11 supergravity does not admit such solutions, and that all such solutions of IIB supergravity are locally isometric to the AdS_5 * S^5 maximally supersymmetric background. Furthermore, we prove that (massive) IIA supergravity also does not admit such solutions, provided that the homogeneity conjecture for massive IIA supergravity is valid. In the context of AdS/CFT these results imply that if strictly N=3 superconformal theories in 4-dimensions exist, their gravitational dual backgrounds are either singular or their internal spaces are not compact.


Introduction
Warped AdS 5 backgrounds of 10-and 11-dimensional supergravity theories are of particular interest within the AdS/CFT correspondence as they are dual to 4-dimensional superconformal theories, see [1] for a review. The most celebrated example of such correspondence is the statement that IIB superstring theory on the maximally supersymmetric AdS 5 × S 5 background is dual to N = 4 supersymmetric gauge theory in four dimensions [2]. AdS spaces have been used in supergravity compactifications, for a review see [3].
To establish such dualities to other 4-dimensional superconformal theories with less than maximal supersymmetry requires the construction of AdS 5 supergravity backgrounds preserving less than 32 supersymmetries. Recently it has been shown in [4,5,6] that AdS 5 backgrounds preserve 8,16,24 or 32 supersymmetries in type II 10-dimensional supergravities and in 11-dimensional supergravity 1 . The maximally supersymmetric AdS 5 backgrounds have been classified in [7] where it has been shown that no such backgrounds exist in either 11-dimensional or (massive) IIA supergravities, and all maximally supersymmetric AdS 5 backgrounds in IIB supergravity are locally isometric to the previously known AdS 5 × S 5 solution of the theory, see [8] and reference within. To our knowledge there is no classification of AdS 5 backgrounds preserving 16 or 24 supersymmetries. The geometry of AdS 5 solutions preserving 8 supersymmetries has been investigated in [11,12], after assuming that the fields are invariant under the so(4, 2) symmetry of AdS 5 together with some additional restrictions 2 on the form of Killing spinors. Moreover, many AdS 5 solutions have been found, see for example [13]- [22]. In [4,5,6] a different approach to investigate the geometry of AdS backgrounds was proposed, which was based on earlier work on black hole near horizon geometries [9] which has the advantage that all additional restrictions are removed and the only assumption that remains is the requirement for the fields to be invariant under the so(4, 2) symmetry of AdS 5 .
In this paper, we shall demonstrate, under the assumptions we describe in detail below, that there are no AdS 5 solutions in 11-dimensional and (massive) IIA supergravities that preserve 24 supersymmetries. Furthermore we shall show that all AdS 5 solutions of IIB supersgravity that preserve 24 supersymmetries are locally isometric to the maximally supersymmetric AdS 5 × S 5 background.
One application of our results is in AdS/CFT and in particular on the existence of gravitational duals for strictly N = 3 superconformal theories in four dimensions. It is known for sometime that the field content and component actions of N = 3 and N = 4 superconformal theories with rigid supersymmetry are the same. As a result N = 3 superconformal symmetry classically enhances to N = 4. Quantum mechanically, the picture is more involved as the quantization of these theories with manifest N = 3 and N = 4 supersymmetry will require the use techniques like harmonic superspace [23,24], and these are different for these two theories. Nevertheless the interpretation of 1 There are no supersymmetric AdS 5 backgrounds in either heterotic or type I supergravities. 2 Typically it is assumed that the Killing spinors factorize as Killing spinors on AdS and Killing spinors along the internal space. A factorization of this type has been investigated in [4,5,6] and it was found that it imposes more restrictions on the backgrounds than those required for invariance under the isometries of AdS. Therefore the generality of the factorization approaches must be re-investigated on a case by case basis.
the equivalence of the classical actions is that perturbatively the two quantum theories are indistinguishable 3 . Therefore if a theory exists with strictly N = 3 superconformal symmetry, it must be intrinsically non-perturbative. The properties of such N = 3 superconformal theories have been investigated in [25] and an F-theory construction for such a theory has been proposed in [26]. In this context our results imply that, unlike the N = 4 superconformal theories, there are no smooth gravitational duals, with compact without boundary internal spaces, for strictly N = 3 superconformal theories in four dimensions.
The proof of the above result utilizes the near horizon approach of [4,5,6] for solving the Killing spinor equations (KSEs) of supergravity theories for AdS backgrounds as well as a technique developed for the proof of the homogeneity conjecture in [10]. Furthermore, to prove our results we make certain smoothness and global assumptions. In particular apart from implementing the so(4, 2) symmetry on the fields, we also assume that the warped AdS 5 × w M D−5 backgrounds, for D = 10 or D = 11, satisfy the following conditions 4 : (i) All the fields are smooth, and (ii) the internal space M D−5 is connected 5 and compact without boundary. Both these additional restrictions, apart from the connectness of M D−5 , can be replaced with the assertion that the data are such that the Hopf maximum principle applies. These assumptions are essential as otherwise there are for example AdS 5 backgrounds in 11-dimensions which preserve more than 16 supersymmetries, see also section 2.
This paper is organized as follows. In section 2, we prove the non-existence of AdS 5 backgrounds preserving 24 supersymmetries in 11-dimensional supergravity, and in section 3, we demonstrate the same result for both standard and massive IIA supergravities. In section 4, we show that the AdS 5 backgrounds that preserve 24 supersymmetries in IIB supergravity are locally isometric to the maximally supersymmetric background AdS 5 × S 5 . In section 5, we give our conclusions and explore an application to AdS/CFT. Furthermore, in appendix A we briefly summarize some of our conventions, and in appendix B for completeness we present a technique we use to derive our results which has been adapted from the proof of the homogeneity conjecture.

2
AdS 5 × w M 6 Solutions in D=11 We begin by briefly summarizing the general structure of warped AdS 5 solutions in 11dimensional supergravity, as determined in [4], whose conventions we shall follow throughout this section. Then we shall present the proof that there are no such solutions preserv-3 In fact it may be possible to prove this by demonstrating via Ward identity techniques like those in [27] that N = 3 superconformal symmetry quantum mechanically always enhances to N = 4. We would like to thank Paul Howe for suggesting this. 4 We also assume the validity of the homogeneity conjecture for massive IIA supergravity. This has not been proven as yet but it is expected to hold. 5 In fact M D−5 is required to be path connected but all manifolds are path connected if they are connected since they are locally path connected. From now on, we shall assume that M D−5 is always connected.
ing 24 supersymmetries. The metric and 4-form are given by where we have written the solution as a near-horizon geometry [9], with (u, r, z, x 1 , x 2 ) are the coordinates of the AdS 5 space, A is the warp factor which is a function on M 6 , and X is a closed 4-form on M 6 . A and X depend only on the coordinates of M 6 , ℓ is the radius of AdS 5 . The 11-dimensional Einstein equation implies that where D is the Levi-Civita connection on M 6 . The remaining components of the Einstein and gauge field equations are listed in [4], however we shall only require (2.3) for the analysis of the N = 24 solutions. In particular, (2.3) implies that A is everywhere nonvanishing on M 6 , on assuming that M 6 is connected and all fields are smooth. We adopt the following frame conventions; e i is an orthonormal frame for M 6 , and e + = du , e − = dr + rh , e z = Adz , e a = Ae z/ℓ dx a . (2.4) We use this frame in the investigation of KSEs below.

The Killing spinors
The Killing spinors of AdS 5 backgrounds are given by where we have used the light-cone projections and σ ± and τ ± are 16-component spinors that depend only on the coordinates of M 6 . We do not assume that the Killing spinors factorize as Killing spinors on AdS 5 and Killing spinors on M 6 . The remaining independent Killing spinor equations (KSEs) are: and In particular algebraic KSEs (2.8) imply that σ + and τ + cannot be linearly dependent. For our Clifford algebra conventions see also appendix A.

Counting the Killing Spinors
In order to count the number of supersymmetries, note that if σ + is a solution of the σ + KSEs, then so is Γ 12 σ + . Furthermore, τ + = Γ z Γ 1 σ + and τ + = Γ z Γ 2 σ + are solutions to the τ + KSEs. The spinors σ + , Γ 12 σ + , Γ z Γ 1 σ + , Γ z Γ 2 σ + are linearly independent. The positive chirality spinors also generate negative chirality spinors σ − , τ − which satisfy the appropriate KSEs. This is because if σ + , τ + is a solution, then so is and also conversely, if σ − , τ − is a solution, then so is So for a generic AdS 5 × w M 6 solution, all of the Killing spinors are generated by the σ + spinors, each of which gives rise to 8 linearly independent spinors via the mechanism described here. The solutions therefore preserve 8k supersymmetries, where k is equal to the number of σ + spinors.

Non-existence of N = 24 AdS 5 solutions in D=11
To consider the AdS 5 solutions preserving 24 supersymmetries, we begin by setting and defining Then (2.7) implies that so W is an isometry of M 6 . In addition, the algebraic conditions (2.8) imply that Also, (2.7) implies that Combining (2.15), and (2.16) we have In addition (2.7) implies that Hence, on taking the divergence of (2.17), we find A maximum principle argument then implies that Λ 2 is constant. Substituting these conditions back into (2.16), we find the condition and ⋆ 6 denotes the Hodge dual on M 6 . To prove a non-existence theorem for N = 24 solutions, we consider spinors of the type For a N = 24 solution, there are 12 linearly independent spinors of this type, because of the algebraic conditions (2.8). Next, consider the condition (2.20). This implies that where W is the isometry generated by Λ as defined in (2.13). A straightforward modification of the reasoning used in [10], which we describe in Appendix B, implies that for N = 24 solutions, the vector fields dual to the 1-form bilinears W generated by the Λ spinors span the tangent space of M 6 . Then the condition i W dA = 0 implies that A is constant, and furthermore, (2.20) implies that i W H = 0, which also implies that H = 0, and so X = 0.
However, the Einstein equation (2.3) admits no AdS 5 solutions for which dA = 0 and X = 0, so there can be no N = 24 AdS 5 solutions.
We should remark that the two assumptions we have made on the fields to derive this result are essential. This is because any AdS d+1 background can locally be written as a warped product ds 2 (AdS d+1 ) = dy 2 + A 2 (y)ds 2 (AdS d ) for some function A which has been determined in [29]. For d = 2, this has previously been established in [28]. As a result the maximally supersymmetric AdS 7 × S 4 solution of 11-dimensional supergravity can be seen as a warped AdS 5 background. This appears to be a contradiction to our result. However, the internal space M 6 in this case is non-compact and so it does not satisfy the two assumptions we have made.
3 AdS 5 × w M 5 solutions in (massive) IIA supergravity As in the 11-dimensional supergravity investigated in the previous sections, there are no N = 24 AdS5 backgrounds in (massive) IIA supergravity. We shall use the formalism and follow the conventions of [6] in the analysis that follows. Imposing invariance of the background under the symmetries of AdS 5 all the fluxes are magnetic, ie their components along AdS 5 vanish. In particular the most general AdS 5 background is where we have denoted the 10-dimensional fluxes and their components along M 5 with the same symbol, A is the warp factor, Φ is the dilaton and S is the cosmological constant dressed with the dilaton. A, S and Φ are functions of M 5 , while G , H and F are the 4-form, 3-form and a 2-form fluxes, respectively, which have support only on M 5 . The coordinates of AdS 5 are (u, r, z, x a ) and we introduce the frame (e + , e − , e z , e a ) as in (2.4).
The fields satisfy a number of field equations and Bianchi identities which can be found in [6]. Those relevant for the analysis that follows are the field equation for the dilaton and the field equation for G respectively, and the Einstein equations both along AdS 5 and M 5 respectively, where D is the Levi-Civita connection of M 5 and R (5) ij is the Ricci tensor of M 5 . The former is seen as the field equation for the warp factor A.

Killing spinor equations
The killing spinors of IIA AdS 5 backgrounds are given as in (2.5) where now σ ± and τ ± are 16-component spinors that depend only on the coordinates of M 5 . The remaining independent conditions are the gravitino KSEs the dilatino KSEs and the algebraic KSEs and where D is the spin connection on M 5 and see appendix A for our Clifford algebra conventions. The counting of supersymmetries is exactly the same as in the D=11 supergravity described in the previous sections.

N = 24 AdS 5 solutions in (massive) IIA supergravity
Before we proceed with the analysis, the homogeneity conjecture 6 [10] together with the results [30] on the classification of (massive) IIA backgrounds imply that both Φ and S are constant functions over the whole spacetime which we shall assume from now on. Next let us set and define Then (3.6) implies that so W is an isometry of M 5 . After some straightforward computation using the gravitino KSEs, one finds On the other hand (3.8) gives Using this, (3.14) can be written as Furthermore using (3.6), one can show that Taking the covariant derivative of (3.16) and using the above equation, one finds that This in turn implies after using the maximum principle that Λ 2 is constant.
Using the constancy of Λ 2 , (3.14) and (3.16) imply that and Next taking the difference of the two identities below and upon using (3.20), we find ie τ + and σ + are orthogonal.
To continue, multiply Ξ + Λ = −ℓ −1 τ + with Γ xy , and using the fact Γ xy τ + is again a type τ + Killing spinor, and the equation above, one obtains that As straightforward modification of the argument used in [10] to prove the homogeneity conjecture, see also appendix B, one can show that the vector fields W span the tangent spaces of M 5 . As a result, the above equation implies that A is constant. Next using the dilatino KSE (3.7) to eliminate the G-dependent term in (3.19) and that A = const, one finds In what follows, we shall investigate the standard and massive IIA supergravities separately.

Massive IIA supergravity with S = 0
On writing G = ⋆ 5 X, where X is a 1-form on M 5 , the condition which is derived from the dilatino KSE (3.7), can be rewritten as Furthermore, the G field equation implies that dX = 0, and we assume 7 that L W G = 0 which implies L W X = 0. This condition, together with dX = 0, gives that i W X is constant. Then it follows from (3.28) that Λ, Γ 11 Λ is also constant.
On differentiating the condition Λ, Γ 11 Λ = const using the gravitino KSEs, we obtain the condition and hence However, using an argument directly analogous to that used to show that the vector fields W span the tangent space of M 5 , it follows that the vectors Λ, Γ j Λ ∂ j also span the tangent space of M 5 , see appendix B. Therefore, Next, act on the right-hand-side of the dilatino equation (3.7) with / XΓ 11 and take the inner product with Λ. On making use of i X F = 0, we find the condition Λ, X ℓ 1 H ℓ 2 ℓ 3 ℓ 4 Γ ℓ 1 ℓ 2 ℓ 3 ℓ 4 Λ = 0 , (3.32) and hence Again, as the vectors Λ, Γ 11 Γ xyz Γ j Λ ∂ j span the tangent space of M 5 , this condition implies that Another useful condition is to note that L W X = 0 implies that and as the W span the tangent space of M 5 , it follows that D i X i must be constant. However the integral of D i X i over M 5 vanishes, and hence it follows that ie X is co-closed. As it is also closed, X and so G are harmonic. This condition, together with dX = 0, imply that one can write On using the Einstein equation (3.5), together with the conditions i X F = 0, X ∧ H = 0, we find which can be written as on using the dilaton equation (3.3) to eliminate the G 2 term. As the right-hand-side of this expression is a sum of non-negative terms, an application of the maximum principle implies that X 2 is constant 8 and for all Killing spinors Λ. However, this is a contradiction. 8 The condition X 2 = const also follows from L W X 2 = 0 together with homogeneity.
To see this, let the 12-dimensional vector space spanned by the Killing spinors Λ be denoted by K. Then the above condition implies that The dimension of space of all Majorana Spin(9, 1) spinors ζ satisfying the lightcone projection Γ + ζ = 0 is 16. As K has dimension 12, K ⊥ has dimension 4. As Γ 11 K is 12dimensional it cannot be included in K ⊥ as required by the assumption (3.41). Therefore there are no AdS 5 solutions in massive IIA supergravity which preserve 24 supersymmetries.
We would like to remark that the proof of this result is considerably simpler if M 5 is simply connected. As has already been proven, G is harmonic. On a simply connected M 5 , G vanishes. In such a case, (3.27) again implies (3.41). Then the non-existence of such AdS 5 backgrounds follows from the argument produced above that (3.41) cannot hold for all Killing spinors.

4
AdS 5 × w M 5 solutions in IIB supergravity The active fields of AdS 5 × w M 5 IIB backgrounds as well as the relevant field and KSEs have been determined in [5]. In particular, in the conventions of [5], the metric and other form field strengths are where again we have written the background as a near-horizon geometry [9], with

2)
A is the warp factor which is a smooth function on M 5 , G is the complex 3-form, P encodes the (complexified) axion/dilaton gradients, F is the real self-dual 5-form and Y is a real scalar. The AdS 5 coordinates are (u, r, z, x a ) and we introduce the frame (e + , e − , e z , e a ) as in (2.4).
For the analysis that follows, we shall use the Bianchi identities and the 10-dimensional Einstein equation along AdS 5 which gives the field equation for the warp factor A. The remaining Bianchi identities and bosonic field equations, which are not necessary for the investigation of N = 24 solutions, can be found in [5]. We also assume the same regularity assumptions as for the eleven dimensional solutions, and remark that (4.4) implies that A is nowhere vanishing on M 5 .

The Killing spinors
Solving the KSEs of IIB supergravity for AdS 5 × w M 5 backgrounds along AdS 5 , one finds that the Killing spinors can be written as in (2.5), where now σ ± and τ ± are Weyl Spin(9, 1) spinors which depend only on the coordinates of M 5 that obey in addition the lightcone projections Γ ± σ ± = Γ ± τ ± = 0. The remaining independent KSEs are the gravitino parallel transport equations together with the dilatino KSEs and some additional algebraic conditions which arise from the integration of the KSEs along the AdS 5 subspace where and C is the charge conjugation matrix. Again, we have not made any assumptions on the form of the Killing spinors. The counting of the Killing spinors, and the way in which one can construct the σ ± , τ ± spinors from each other proceeds in exactly in the same way as for the D = 11 AdS 5 solutions. So, again, for a generic AdS 5 × w M 5 solution, all of the Killing spinors are generated by the σ + spinors, each of which gives rise to 8 linearly independent spinors. The solutions therefore preserve 8k supersymmetries, where k is equal to the number of σ + spinors.

N = 24 AdS 5 solutions in IIB
To proceed with the analysis first note that as a consequence of the homogeneity conjecture proven in [10] is that the solutions with 24 supersymmetries must be locally homogeneous, with ξ = 0 . (4.10) Then, we set and define Then (4.5) implies that so W is an isometry of M 5 . Next, using (4.5), we find (4.14) Furthermore, the algebraic condition (4.8) implies that On substituting this condition back into (4.14) we find However, (4.5) also implies that So combining this condition with (4.16), we find A maximum principle argument then implies that Λ 2 is constant. Then (4.14) and (4.16) imply 19) or, equivalently Next, we shall show that the spinors σ + , τ + are orthogonal with respect to the inner product Re <, >. To see this, note that (4.8) implies that On expanding out, and subtracting these two identities, one finds that the real and imaginary parts of the resulting expression imply and Y Re τ + , Γ xy σ + + 1 48 Im For N = 24 solutions there are 6 linearly independent σ + spinors, and 6 linearly independent τ + spinors, hence the spinors of the type Λ = σ + + τ + span a 12 dimensional vector space over R, which we shall denote by K.
It is also particularly useful to note that the algebraic condition (4.8) implies On taking the real part of this expression, one finds On expanding this expression out, and adding it to (4.29), one obtains the condition or equivalently Again, as the W isometries span the tangent space of M 5 , one obtains So either H = 0, or C * Λ, Λ = 0 for all Λ ∈ K. We shall prove that C * Λ, Λ = 0 cannot be satisfied for all Λ. Indeed, suppose that C * Λ, Λ = 0 for all Λ ∈ K. We remark that C * Λ 1 , Λ 2 is symmetric in Λ 1 , Λ 2 , and so C * Λ, Λ = 0 for all Λ ∈ K implies that C * Λ 1 , Λ 2 = 0 , then the condition (4.36) implies thatK ⊂ K ⊥ . However, this is not possible, becauseK is 12 dimensional, whereas K ⊥ is 4-dimensional. So, one cannot have C * Λ, Λ = 0 for all Λ ∈ K. It follows that and hence the spinors Λ satisfy as a consequence of (4.4). The integrability condition of (4.39) implies that where we have used the Bianchi identity dQ = 0. Then (4.41) gives that or equivalently As the isometries W span the tangent space of M 5 , it follows that and hence M 5 is locally isometric to the round S 5 . It follows that all (sufficiently regular) AdS 5 solutions with N = 24 supersymmetries are locally isometric to AdS 5 × S 5 , with constant axion and dilaton, and G = 0. This establishes that there are no distinct local geometries for IIB AdS 5 × M 5 backgrounds that preserve strictly 24 supersymmetries.

Concluding remarks
We have proven, under some assumptions, a non-existence theorem for AdS 5 × w M D−5 , D = 10, 11, backgrounds that preserve strictly 24 supersymmetries in all 10-and 11dimensional supergravity theories. In particular we have demonstrated that such backgrounds cannot exist in 11-dimensional and (massive) IIA supergravities, and all such IIB backgrounds must be locally isometric to the maximally supersymmetric AdS 5 × S 5 solution of the theory.
Our assumptions are that the fields must be smooth and the internal space M D−5 , D = 6, 5, must be connected, compact and without boundary. Alternatively, these assumptions can be summarized by saying that the data are such that the maximum principle applies. It turns out that these assumptions are required to establish our results. It is known that if the compactness assumption for M 6 is removed, then the maximally supersymmetric AdS 7 × S 4 solution of 11-dimensional supergravity can be written locally as a warped AdS 5 × w M 6 solution. This would appear to be a contradiction to our result for eleven dimensions, but for such a solution M 6 is not compact [29]. Because of this, it is not apparent that the smoothness and global assumptions on M D−5 can be removed. This in particular leaves open the possibility that there are AdS 5 × w M D−5 backgrounds in 10-and 11-dimensional supergravities but such backgrounds would either be singular or M D−5 will not be compact and without boundary. Another possibility of constructing AdS 5 backgrounds in IIB with 24 supersymmetries is to take appropriate orbifolds of the maximally supersymmetric AdS 5 × S 5 solution of the theory. Though such a possibility cannot be ruled out, it is unlikely. It is also supported by the results of [25], that there are no relevant N = 3 deformations of N = 4 theory.
The existence of a smooth AdS 5 background with compact without boundary internal space in a 10-or 11-dimensional supergravity theory with distinct local geometry from that of maximally supersymmetric backgrounds would have raised the expectation that it should have been the AdS/CFT dual to a N = 3 4-dimensional superconformal theory. This would have been in parallel with the well known duality that string theory on AdS 5 × S 5 is AdS/CFT dual to N = 4 U(N) gauge theory. Because both N = 3 and N = 4 theories have the same classical action, it is believed that in perturbation theory the two theories are indistinguishable. Though such a proof is not known, it may be possible to demonstrate this by proving that quantum mechanically N = 3 Ward identities imply, using for example techniques similar to [27], that the symmetry enhances to N = 4. In any case assuming that perturbatively the two theories cannot be distinguished, the possibility that remains is that if a theory exists with strictly N = 3 superconformal symmetry, it has to be intrinsically non-perturbative. The properties of such a theory have been investigated in [25] and an F-theory construction has been proposed in [26]. Our results prove that the gravitational duals of strictly N = 3 superconformal theories cannot be smooth with compact without boundary internal spaces. This is unlike the gravitational duals that preserve more than 16 supersymmetries of other superconformal theories. The quantum mechanical consistency of N = 3 superconformal theories requires further investigation.

Appendix B Homogeneity of Internal Spaces
In this Appendix, we prove that for the N = 24 AdS 5 solutions in eleven-dimensional supergravity, the isometries on M 6 generated by the Λ spinors via span the tangent space of M 6 . The proof for this is a straightforward adaptation of a similar result used in the proof of the homogeneity conjecture [10]. To begin, let K denote the 12-dimensional vector space spanned by the Killing spinors Λ. Define the map ϕ : K ⊗ K → T M 6 by As ϕ(Λ 1 , Λ 2 ) = ϕ(Λ 2 , Λ 1 ), it follows that the W span T (M 6 ) iff ϕ is surjective. However, ϕ is surjective iff the only vector V ∈ T (M 6 ) satisfying for all Λ 1 , Λ 2 ∈ K is V = 0, i.e. the perpendicular complement of the image of ϕ is trivial. Suppose, for a contradiction, that the perpendicular complement of the image of ϕ is not trivial. Then there exists nonzero V ∈ T (M 6 ) such that Observe that K⊕K ⊥ is a 16-dimensional vector space spanned by the Majorana Spin(10, 1) spinors ζ that satisfy the lightcone projection Γ + ζ = 0. Thus as K is 12-dimensional, K ⊥ is a 4-dimensional subspace. As V = 0, the kernel of the map V i Γ i Γ xyz : K → K ⊥ is zero and so it is injective. However this is not possible as the image V i Γ i Γ xyz (K) is 12-dimensional while K ⊥ is 4-dimensional. Thus the hypothesis that V = 0 is not valid and ϕ is surjective, and so the vectors W span the tangent space of M 6 .
The argument for the AdS 5 backgrounds of massive IIA supergravity is the same as that described above upon replacing M 6 with M 5 . It also generalizes for the AdS 5 solutions in IIB supergravity, after replacing the norm <, > with Re <, >, and M 6 with M 5 throughout.