AdS4 backgrounds with N>16 supersymmetries in 10 and 11 dimensions

We explore all warped $AdS_4\times_w M^{D-4}$ backgrounds with the most general allowed fluxes that preserve more than 16 supersymmetries in $D=10$- and $11$-dimensional supergravities. After imposing the assumption that either the internal space $M^{D-4}$ is compact without boundary or the isometry algebra of the background decomposes into that of AdS$_4$ and that of $M^{D-4}$, we find that there are no such backgrounds in IIB supergravity. Similarly in IIA supergravity, there is a unique such background with 24 supersymmetries locally isometric to $AdS_4\times \mathbb{CP}^3$, and in $D=11$ supergravity all such backgrounds are locally isometric to the maximally supersymmetric $AdS_4\times S^7$ solution.


Introduction
AdS backgrounds in 10 and 11 dimensions that preserve N supersymmetries with N > 16 have found widespread applications both in supergravity compactifications and in AdS/CFT correspondence, for reviews see [1,2] and references therein. One of the features of such backgrounds in AdS/CFT [3] is that the CFT R-symmetry group acts transitively on the internal space of the solution and this can be used to establish the dictionary between some of the operators of the CFT and spacetime Kaluza-Klein fields [4]. Therefore the question arises whether it is possible to find all such AdS solutions. Despite the progress that has been made during the years, a complete description of all AdS solutions that preserve N > 16 supersymmetries remains an open problem.
Recently however, there have been several developments which facilitate progress in this direction for a large class of warped flux AdS solutions. In [5][6][7], the Killing spinor equations (KSEs) of supergravity theories have been solved in all generality and the fractions of supersymmetry preserved by all warped flux AdS backgrounds have been identified. Furthermore global analysis techniques have also been introduced in the investigation of AdS backgrounds which can be used to a priori impose properties like the compactness of the internal space and the smoothness of the fields. Another key development is the proof of the homogeneity theorem [8] which for the special case of AdS backgrounds states that all such backgrounds that preserve N > 16 supersymmetries are Lorentzian homogeneous spaces.
So far it is known that the warped flux AdS n , n ≥ 6, backgrounds preserve either 16 or 32 supersymmetries and those that preserve 32 supersymmetries have been classified in [9]. In addition, it has been shown that there are no N > 16 AdS 5 backgrounds in D = 11 and (massive) IIA supergravities while in IIB supergravity all such backgrounds are locally isometric to the maximally supersymmetric AdS 5 × S 5 solution [10]. In particular the existence of a IIB AdS 5 solution that preserves 24 supersymmetries has been excluded. Moreover the AdS n × M D−n solutions with M D−n a symmetric coset space have been classified in [11][12][13][14]. Furthermore heterotic supergravity does not admit AdS solutions that preserve more than 8 supersymmetries [15].
The main task of this paper is to describe all warped AdS 4 backgrounds that admit the most general fluxes in 10 and 11 dimensions and preserve more than 16 supersymmetries. It has been shown in [5][6][7] that such backgrounds preserve 4k supersymmetries. Therefore, we shall investigate the backgrounds preserving 20, 24 and 28 as those with 32 supersymmetries have already been classified in [9]. In particular, we find that JHEP01(2018)087 • IIB and massive IIA supergravity do not admit AdS 4 solutions with N > 16 supersymmetries.
• Standard IIA supergravity admits a unique solution up to an overall scale preserving 24 supersymmetries locally isometric to the AdS 4 × CP 3 background of [16].
These results have been established under certain assumptions. 1 We begin with a spacetime which is a warped product AdS 4  It has been shown in [21] that for all AdS backgrounds, the first assumption implies the second. In addition for N > 16 AdS 4 backgrounds, 2 the second assumption implies the first. This is because t 0 is the Lie algebra of a compact group and all internal spaces are compact without boundaries. Smoothness also follows as a consequence of considering only invariant solutions. The proof of the main statement of our paper is based first on the results of [5][6][7] that the number of supersymmetries preserved by AdS 4 backgrounds are 4k and so the solutions under consideration preserve 20, 24, 28 and 32 supersymmetries. Then the homogeneity theorem of [8] implies that all such backgrounds are Lorentzian homogeneous spaces. Moreover, it has been shown in [21] under the assumptions mentioned above that the Killing superalgebra of warped AdS 4 backgrounds that preserve N = 4k supersymmetries is isomorphic to osp(N/4|4), see also [22], and that the even subalgebra osp(N/4|4) 0 = so(3, 2) ⊕ so(N/4) acts effectively on the spacetime with t 0 = so(N/4) acting on the internal space. Thus together with the homogeneity theorem osp(N/4|4) 0 acts both transitively and effectively on the spacetime. Then we demonstrate in all cases that the warp factor A is constant. As a result all N > 16 AdS 4 backgrounds are product spaces AdS 4 × M D−4 . So the internal space M D−4 is a homogeneous space, M D−4 = G/H, and Lie G = so(N/4). Therefore, we have demonstrated the following, • The internal spaces of AdS 4 backgrounds that preserve N > 16 supersymmetries are homogeneous spaces that admit a transitive and effective action of a group G with Lie G = so(N/4). 1 Some assumptions are necessary to exclude the possibility that a warped AdS4 background is not locally isometric to an AdSn background with n > 4. This has been observed in [19] and explored in the context of KSEs in [20]. 2 In what follows, we use "N > 16 AdS backgrounds" instead of "AdS backgrounds that preserve N > 16 supersymmetries" for short.

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Having established this, one can use the classification of [23][24][25][26] to identify all the 6and 7-dimensional homogeneous spaces that can occur as internal spaces for N > 16 AdS 4 backgrounds, see also tables 3 1 and 3. Incidentally, this also means that if N > 16 backgrounds were to exist, the R-symmetry group of the dual CFT would have to act transitively on the internal space of the solution.
A direct observation of the classification of 6-dimensional homogeneous spaces G/H in table 1 reveals that those that can occur as internal spaces of AdS 4 backgrounds with N > 16 in 10 dimensions are Spin(7)/Spin (6)  where N denotes the expected number of supersymmetries that can be preserved by the background and we always take G to be simply connected. Observe that there are no maximally supersymmetric AdS 4 solutions in 10-dimensional supergravities in agreement with the results of [9]. The proof of our result in IIB supergravity is based on a cohomological argument and does not use details of the 6-dimensional homogeneous spaces involved. However in (massive) IIA supergravity, one has to consider details of the geometry of these coset spaces. Solutions with strictly N = 28 and N = 20 supersymmetries are ruled out after a detailed analysis of the KSEs and dilaton field equation. In the standard IIA supergravity there is a solution with 24 supersymmetry and internal space locally isometric to the symmetric space SU(4)/S(U(1) × U(3)) = CP 3 . This solution has already been found in [16]. The homogeneous space Sp(2)/Sp(1) × U(1), which is diffeomorphic to CP 3 , gives also a solution at a special region of the moduli space of parameters. This solution admits 24 supersymmetries and is locally isometric to that with internal space SU(4)/S(U(1) × U(3)). The classification of 7-dimensional homogeneous spaces G/H in table 3  where Sp(1) max and ∆(Sp(1)) denote the maximal and diagonal embeddings of Sp(1) in Sp (2), respectively, and G is chosen to be simply connected. It is known that there is a maximally supersymmetric solution AdS 4 ×S 7 with internal space S 7 = Spin(8)/Spin(7) [17,18]. After a detailed investigation of the geometry of the above homogeneous spaces, the solutions of the KSEs and the warp factor field equation, one can also show that the rest of the coset spaces do not give solutions with strictly 20, 24 and 28 supersymmetries. However as the homogeneous spaces Spin(7)/G 2 , SU(4)/SU(3) and Sp(2)/Sp(1) are diffeomorphic 3 These tables list the simply connected homogeneous spaces. This suffices for our purpose because we are investigating the geometry of the backgrounds up to local isometries. As so(N/4) is simple the universal cover of G/H with Lie(G) = so(N/4) is compact and homogeneous, see eg [27]. So the internal space can be identified with the universal coverG/H of G/H for whichG can be chosen to be simply connected.

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to S 7 , there is a region in the moduli space of their parameters which yields the maximally supersymmetric AdS 4 × S 7 solution. The paper is organized as follows. In section 2, we show that there are no IIB N > 16 AdS 4 × w M 6 solutions. In section 3, we show that there is an up to an over scale unique solution of IIA supersgravity that preserves 24 supersymmetries. In section 4, we demonstrate that all N > 16 AdS 4 backgrounds of 11-dimensional supergravity are locally isometric to the maximally supersymmetric AdS 4 × S 7 solution. In section 5 we state our conclusions. In appendix A, we explain our conventions, and in appendix B we summarize some aspects of the geometry of homogeneous spaces that is used throughout the paper. In appendices C, D and E, we present some formulae for the homogeneous spaces that admit a transitive action of a group with Lie algebra su(k) or so(5) = sp(2).
2 N > 16 AdS 4 × w M 6 solutions in IIB To investigate the IIB AdS 4 backgrounds, we shall use the approach and notation of [6] where Bianchi identities, field equations and KSEs are first solved along the AdS 4 subspace of AdS 4 × w M 6 and then the remaining independent conditions along the internal space M 6 are identified. The bosonic fields of IIB supergravity are the metric, a complex 1-form field strength P , a complex 3-form field strength G and a real self-dual 5-form F . Imposing the symmetry of AdS 4 on the fields, one finds that the metric and form field strengths are given by where the metric has been written as a near-horizon geometry [30] with The warp factor A is a function on the internal manifold M 6 , H is the complex 3-form on M 6 , ξ is a complex 1-form on M 6 and Y is a real 1-form on M 6 . The AdS 4 coordinates are (u, r, z, x) and we introduce the null-ortho-normal frame where ds 2 (M 6 ) = δ ij e i e j . All gamma matrices are taken with respect to this null orthonormal frame. The Bianchi identities along M 6 which are useful in the analysis that follows are

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where Q is the pull-back of the canonical connection of the upper-half plane on the spacetime with respect to the dilaton and axion scalars of IIB supergravity. Similarly, the field equations of the warp factor is and those of the scalar and 3-form fluxes are The full set of Bianchi identities and field equations can be found in [6]. Note in particular that (2.5) implies that if A and the other fields are smooth, then A is nowhere vanishing on M 6 .

The Killing spinors
After solving the KSEs along AdS 4 , the Killing spinors of the background can be written as where we have used the light-cone projections and σ ± and τ ± are Spin(9, 1) Weyl spinors depending only on the coordinates of M 6 . The remaining independent KSEs are  13) and C * is the charge conjugation matrix followed by standard complex conjugation. For some explanation of the notation see appendix A. ( As Q is the pull-back of the canonical connection of the upper half plane with respect to the scalars and these are constant, Q = 0 as well. Setting Λ = σ + + τ + and after using the gravitino KSE (2.9), we find Next, observe that the algebraic KSE (2.11) implies which, when substituted back into (2.16), yields However, the gravitino KSE (2.9) also implies that Thus, in conjunction with (2.18), we obtain

The warp factor is constant and the 5-form flux vanishes
AdS 4 backgrounds preserving 4k supersymmetries admit k linearly independent Killing spinors σ + . For every pair of such spinors σ 1 + and σ 2 + define the bilinear Then the gravitino KSE (2.9) implies that Therefore W is a Killing vector on M 6 . Next consider the algebraic KSE (2.11) and take the real part of σ 1 + , Ξ (+) σ 2 + − σ 2 + , Ξ (+) σ 1 + = 0 to find that where we have used (2.22). Similarly, taking the real part of the difference σ 1 + , Γ zx Ξ (+) σ 2 + − σ 2 + , Γ zx Ξ (+) σ 1 + = 0 and after using the condition (2.23), we find The conditions (2.26) and (2.27) are valid for all IIB AdS 4 backgrounds. However if the solution preserves more than 16 supersymmetries, an argument similar to that used for the proof of the homogeneity theorem in [8] implies that the Killing vectors W span the tangent spaces of M 6 at each point. As a result, we conclude that dA = Y = 0 . (2.28) Therefore the warp factor A is constant and the 5-form flux F vanishes. So the background is a product AdS 4 × M 6 , and as it has been explained in the introduction M 6 is one of the homogeneous spaces in (1.1).

Proof of the main statement
To begin, it has been shown in [31] that all IIB AdS backgrounds that preserve N ≥ 28 supersymmetries are locally isometric to the maximally supersymmetric ones. As there is not a maximally supersymmetric AdS 4 background in IIB, we conclude that there does not exist a AdS 4 solution which preserves N ≥ 28 supersymmetries.
To investigate the N = 20 and N = 24 cases, substitute (2.28) into the Bianchi identities and field equations to find that H is harmonic and If H were real, this condition would have implied H = 0 and in turn would have led to a contradiction. This is because the field equation for the warp factor (2.5) cannot be satisfied. Thus we can already exclude the existence of such backgrounds. Otherwise for solutions to exist, M 6 must be a compact, homogeneous, 6-dimensional Riemannian manifold whose de-Rham cohomology H 3 (M 6 ) has at least two generators and which admits a transitive and effective action of a group with Lie algebra isomorphic to either so(6) or so (5) for N = 24 and N = 20, respectively [21]. The homogeneous spaces that admit a transitive and effective action of so(6) or so(5) = sp(2) have already been listed in (1.1) and none of them satisfies these cohomology criteria. All compact homogeneous 6-manifolds have been classified in [25] and the complete list of the simply connected ones relevant here is given in table 1. Therefore, we conclude that there do not exist AdS 4 backgrounds preserving N > 16 supersymmetries in IIB supergravity. 4 3 N > 16 AdS 4 × w M 6 solutions in (massive) IIA To begin, let us summarize the solution of Bianchi identities, field equations and KSEs for (massive) IIA AdS 4 × w M 6 backgrounds as presented in [7] whose notation we follow. The bosonic fields of (massive) IIA supergravity are the metric, a 4-form field strength G, a 3-form field strength H, a 2-form field strength F , the dilaton Φ and the mass parameter S of massive IIA dressed with the dilaton. Imposing the symmetries of AdS 4 on the fields, one finds that (1) Spin(7) Spin(6) = S 6 , symmetric space (2) equations after solving along the AdS 4 subspace can be found in [7]. Relevant to our analysis that follows are the Bianchi identities and the field equations for the fluxes where ∇ and R

The Killing spinor equations
The solution of KSEs of (massive) IIA supergravity along the AdS 4 subspace can again be written as (2.7), where now σ ± and τ ± are spin(9, 1) Majorana spinors that satisfy the lightcone projections Γ ± σ ± = Γ ± τ ± = 0 and depend only on the coordinates of M 6 . After the lightcone projections are imposed, σ ± and τ ± have 16 independent components. These satisfy the gravitino KSEs and the algebraic KSEs The first two equations arise from the naive restriction of the gravitino and dilatino KSEs of the theory on σ ± and τ ± , respectively, while the last algebraic equation is an integrability condition that arises from the integration of the IIA KSEs on AdS 4 . As in the IIB case, the solutions of the above KSEs are related as in (2.14) and so such backgrounds preserve 4k supersymmetries.

Conditions on spinor bilinears
The methodology to establish conditions on the Killing spinor bilinears which follow from our assumption that either the solutions are smooth and the internal space is compact without boundary or that the even subalgebra of the Killing superalgebra decomposes as stated in the introduction is the same as that presented for IIB. However, the formulae are somewhat different. Setting Λ = σ + +τ + and upon using the gravitino KSE (3.5), one finds After multiplying the algebraic KSE (3.7) with Γ iz on the other hand, one gets

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Using this, one can rewrite (3.9) as On the other hand the gravitino KSE (3.5) gives Therefore taking the divergence of (3.11), one finds An application of the Hopf maximum principle gives that Λ 2 is constant, which when inserted back into (3.9) and (3.11) yields and respectively. The above condition can also be expressed as σ 1 + , Γ ix σ 2 + = 0 for any two solutions σ 1 + and σ 2 + of the KSEs. As in IIB, the algebraic KSE (3.7) implies that τ + , This together with (3.15) give that σ + , τ + = 0 and so the τ + and σ + Killing spinors are orthogonal.

The warp factor is constant
To begin, for every pair of solutions σ 1 + and σ 2 + of the KSEs we define the 1-form bilinear Then the gravitino KSE (3.5) implies that therefore W is an Killing vector on M 6 . Next the difference σ 1 where we have used (3.15). So far we have not used that the solutions preserve N > 16 supersymmetries. However if this is assumed, then (3.18) implies that the warp factor A is constant. This is a consequence of an adaptation of the homogeneity theorem on M 6 . The homogeneity theorem also implies that Φ and S are constant. X is also constant as a consequence of the Bianchi JHEP01(2018)087 identity (3.2). Therefore we have established that if the backgrounds preserve N > 16 supersymmetries, then As the warp factor is constant, all backgrounds that preserve N > 16 supersymmetries are products, AdS 4 × M 6 . In addition as it has been explained in the introduction, M 6 is a homogeneous space admitting a transitive and effective action of a group G with Lie algebra so(N/4). These homogeneous spaces have been listed in (1.1). In what follows, we shall explore all these 6-dimensional homogeneous spaces to search for IIA solutions that preserve N > 16 supersymmetries.

N = 28
There are no maximally supersymmetric AdS 4 backgrounds in (massive) IIA supergravity [9]. So the next case to be investigated is that with 28 supersymmetries. In such a case M 6 admits a transitive and effective action of a group with Lie algebra so (7).
Amongst the homogeneous spaces presented in (1.1), the only one with this property is Spin(7)/Spin(6) = S 6 . As Spin (7)/Spin(6) = S 6 is a symmetric space, all left-invariant forms are parallel with respect to the Levi-Civita connection and so represent classes in the de-Rham cohomology.
As it is the sum of two commuting terms one Hermitian and the other anti-Hermitian, the existence of solutions requires that both must vanish separately. As a result S = X = 0. Therefore all fluxes must vanish. This in turn leads to a contradiction as the field equation of the warp factor (3.4) cannot admit any solutions. Thus there are no (massive) IIA AdS 4 backgrounds preserving 28 supersymmetries.

N = 24
The internal space of AdS 4 backgrounds that preserve 24 supersymmetries admits a transitive and effective action of a group with Lie algebra so(6) = su(4). The only space in (1.1) compatible with such an action is SU(4)/S(U(1) × U(3)) = CP 3 . Again this is a symmetric space and so all invariant forms are parallel with respect to the Levi-Civita connection. In turn they represent classes in the de-Rham cohomology. As H odd (CP 3 ) = 0, this implies that H = 0. It is well-known that this homogeneous space is a Kähler manifold and the left-invariant metric is given by the standard Fubini-Study metric on CP 3 . The even cohomology ring of CP 3 is generated by the Kähler form ω. As a result the 2-and 4-form fluxes can be written as for some real constants α and β to be determined.

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To determine α and β, let us first consider the dilatino KSE (3.6) which after imposing (3.19) reads The Hermitian and anti-Hermitian terms in this equation commute and so they can be separately imposed. Notice that the only non-trivial commutator to check is [ / F , / Y ] which vanishes because F is proportional to the Kähler form while Y is a (2,2)-form with respect to the associated complex structure. Thus we have and Inserting these into the algebraic KSE (3.7) simplifies to The integrability condition of this yields Next let us focus on (3.23) and (3.24). Choosing without loss of generality Γ 11 = Γ +− Γ zx Γ 123456 , (3.23) can be rewritten as and similarly (3.24) as where we have chosen an ortho-normal frame for which ω = e 12 + e 34 + e 56 . To solve (3.27) and (3.28), we decompose σ + into eigenspaces of J 1 = Γ 3456 and J 2 = Γ 1256 and find that this leads to the relations for the eigenspaces |+, + , |+, − , |−, + , and for the eigenspace |−, − . Before we proceed to investigate the KSEs further, let us focus on the field equations for the fluxes and the warp factor. Observe that α = 0. Indeed if α = 0, then the KSEs would JHEP01(2018)087 have implied that X = 0. As H = X = 0, the dilaton field equation in (3.3) implies that all fluxes vanish. In such a case, the warp factor field equation in (3.4) cannot be satisfied.
Thus α = 0. Then the field equation for the 3-form flux in (3.3) becomes α(S +4β) = 0 and so this implies that β = −1/4 S. This contradicts the results from KSEs in (3.29) and (3.30) above unless β = S = 0. Setting S = Y = 0 in the dilaton field equation in (3.3), it is easy to see that it is satisfied if and only if α = −1/3X and so σ + lies in the eigenspaces |+, + , |+, − and |−, + . As S = 0, (3.26) implies that X = ±3 −1 A −1 and so α = ∓ −1 A −1 . The algebraic KSE (3.25) now reads Γ x σ + = ∓σ + . As α = −1/3X, the common eigenspace of Γ x , Γ 3456 and Γ 1256 on σ + spinors has dimension 6. Thus the number of supersymmetries that the background can preserve is 24. To establish that (3.31) preserves 24 supersymmetries, it remains to investigate the gravitino KSE (3.5). As CP 3 is simply connected it is sufficient to investigate the integrability condition Then a substitution of this and the rest of the fluxes into the integrability condition reveals that it is satisfied without further conditions. In a similar manner, one can check that the Einstein equation along M 6 is also satisfied. This is the IIA N = 24 solution of [16,28].

N = 20
The internal space of AdS 4 backgrounds that preserve 20 supersymmetries admits an effective and transitive action of a group which has Lie algebra so(5) = sp (2). An inspection of the homogeneous spaces in table 1 reveals that there are two candidate internal spaces namely the symmetric space Sp(2)/U(2) and the homogeneous space The symmetric space is the space of complex structures on H 2 which are compatible with the quaternionic inner product while the homogeneous space is identified with the coset space of the spherexx +ȳy = 1, x, y ∈ H, with respect to the action (x, y) → (ax, ay), a ∈ U(1). The latter is diffeomorphic to CP 3 .

Sp(2)/U(2)
The geometry and algebraic properties of this symmetric space are described in appendix E. The most general left-invariant metric is ds 2 = a δ rs δ ab ra sb = δ rs δ ab e ra e sb , (3.34)

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where a > 0 is a constant and ra , and e ra = √ a ra are the left-invariant and orthonormal frames, respectively, and where r, s = 1, 2, 3 and a, b = 4, 5. The invariant forms are generated by the 2-form is a Kähler manifold with respect to the pair (ds 2 , ω).
To continue we choose the metric on the internal manifold as (3.34) and the fluxes as in the SU(4)/S(U(1) × U(3)) case, i.e.
but now ω is given in (3.35), where α and β are constants. Since there are no invariant 3-forms on Sp(2)/U(2), this implies H = 0. Performing a similar analysis to that in section 3.4, we find that β = S = 0, α = ∓ −1 A −1 and X = ±3 −1 A −1 , and σ + to satisfy the same Clifford algebra projections as in e.g. (3.27). This requires an appropriate re-labeling of the indices of the ortho-normal frame e ra so that the left-invariant tensors take the same canonical form as those of SU(4)/S(U(1) × U (3)) expressed in terms of the ortho-normal frame e i . As a result, there are 24 spinors that solve the KSEs so far. It remains to investigate the solutions of the gravitino KSE (3.5). As in the SU(4)/S(U(1) × U(3)) case in section 3.4, we shall investigate the integrability condition instead. This is again given as in (3.32). The curvature of the metric of this symmetric space is presented in (E.7). Using this the integrability condition (3.32) is written as Contracting with δ ab , one finds that there are solutions which preserve more than 8 supersymmetries provided a = 2 A 2 . Then taking the trace of (3.37) with ab δ rs , we find that which is in contradiction to the condition (3.23) arising from the dilatino KSE. The symmetric space Sp(2)/U(2) does not yield 5 AdS 4 solutions that preserve 20 supersymmetries.
r , the most general left-invariant metric is

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where we have introduced the ortho-normal frame e a = √ a a , e r = √ b r , and where r = 1, 2 and a, b = 1, . . . , 4. The invariant forms are generated by and their duals, where The matrices (I (±) r ) ab are a basis in the space of self-dual and anti-self dual 2-forms in R 4 and are defined in (E.10). Imposing the Bianchi identities (3.2), one finds the relation and that the fluxes can be written as where α, β, h, γ and δ are constants. The dilatino KSE (3.6) is the sum of hermitian and anti-hermitian Clifford algebra elements which commute and thus lead to the two independent conditions Using this to simplify the algebraic KSE (3.7), one finds If we then insert the fluxes (3.43) into the above KSEs and set J 1 = Γ 241 Γ 11 , J 2 = Γ 131 Γ 11 and J 3 = Γ 232 Γ 11 , we obtain As J 1 , J 2 , J 3 are commuting Hermitian Clifford algebra operators with eigenvalues ±1, the KSE (3.45) can be decomposed along the common eigenspaces as described in table 2.
From the results of table 2, there are two possibilities to choose five σ + Killing spinors, namely those in eigenspaces (1) and (3) and those in eigenspaces (1) and (4). For both of these choices, the Bianchi identity (3.42) and the dilaton field equation give  Let us first summarize some of the properties of AdS 4 × w M 7 backgrounds in 11-dimensional supergravity as described in [5] that we shall use later. The bosonic fields are given as

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where the null ortho-normal frame (e + , e − , e z , e x , e i ) is as in (2.3), but now i, j = 1, . . . , 7, and the metric on the internal space M 7 is ds 2 (M 7 ) = δ ij e i e j . X and Y are a function and 4-form on M 7 , respectively. The Bianchi identities of the 11-dimensional supergravity evaluated on the AdS 4 × w M 7 background yield Similarly, the field equations give and where ∇ is the Levi-Civita connection on M 7 .

The Killing spinors
The solution of the KSEs of D = 11 supergravity along the AdS 4 subspace of AdS 4 × w M 7 given in [5] can be expressed as in (2.7) but now σ ± and τ ± are spin(10, 1) Majorana spinors that depend on the coordinates of M 7 . Again they satisfy the lightcone projections Γ ± σ ± = Γ ± τ ± = 0. The remaining independent KSEs are and where The former KSE is the restriction of the gravitino KSE on σ ± and τ ± while the latter arises as an integrability condition as a result of integrating the gravitino KSE of 11-dimensional supergravity over the AdS 4 subspace of AdS 4 × w M 7 .

Conditions on spinor bilinears
The conditions that arise from the assumption that M 7 be compact without boundary and the solutions be smooth are similar to those presented in the (massive) IIA case. In particular, one finds The proof follows the same steps as in the (massive) IIA case and so we shall not repeat it here.

The warp factor is constant
Using arguments similar to those presented in the (massive) IIA case, one finds that W i = A Im σ 1 + , Γ iz σ 2 + are Killing vectors on M 7 for any pair of Killing spinors σ 1 + and σ 2 + and that i W dA = 0. Next, let us suppose that the backgrounds preserve N > 16 supersymmetries. In such a case a similar argument to that presented for the proof of the homogeneity conjecture implies that the W vector fields span the tangent space of M 7 at every point and so A is constant. From the Bianchi identity (4.2) it then follows that X is constant as well. Thus we have established that As a result, the space time is a product AdS 4 × M 7 , where M 7 is a homogeneous space. Further progress requires the investigation of individual homogeneous spaces of dimension 7 which have been classified in [26,27] and they are presented in table 3. Requiring in addition that the homogeneous spaces which can occur as internal spaces of N > 16 AdS 4 backgrounds must admit an effective and transitive action of a group that has Lie algebra so(N/4), one arrives at the homogeneous spaces presented in (1.2). In what follows, we shall investigate in detail the geometry of these homogeneous spaces to search for N > 16 AdS 4 backgrounds in 11-dimensional supergravity.

N = 28, Spin(7)/G 2
The maximally supersymmetric solutions have been classified before [9] where it has been shown that all are locally isometric to AdS 4 ×S 7 with S 7 = Spin(8)/Spin (7). The only solution that may preserve N = 28 supersymmetries is associated with the homogeneous space Spin(7)/G 2 , see (1.2). The Lie algebra spin(7) = so (7) is again spanned by matrices M ij as in (E.1) satisfying the commutation relations (E.2) where now i, j = 1, 2, . . . , 7. Let us denote the generators of g 2 subalgebra of spin(7) and those of the module m, spin(7) = g 2 ⊕ m, with G and A, respectively. These are defined as (2) Spin(7) Sp (2) Sp(1)max , Berger space SU (3) ∆ k,l (U(1)) = W k,l k, l coprime, Aloff-Wallach space ∆ k,l (U(1))·(1×SU(2)) = N k,l k, l coprime ∆p,q,r(U(1) 2 ) = Q p,q,r p, q, r coprime where ϕ is the fundamental G 2 3-form, * 7 ϕ is its dual and * 7 is the duality operation along the 7-dimensional internal space. The non-vanishing components of ϕ and * 7 ϕ can be chosen as and we have raised the indices above using the flat metric. We have used the conventions for ϕ and * 7 ϕ of [29], where also several useful identities satisfied by ϕ and * 7 ϕ are presented. In particular observe that ϕ i jk G jk = 0. The spin (7) generators can be written as and using this we obtain Clearly, Spin(7)/G 2 is a homogeneous space. As  we may choose an ortho-normal frame e i such that where a > 0 is a constant. The left-invariant forms are and its dual * 7 ϕ. So the Y flux can be chosen as Using this the algebraic KSE (4.7) can be expressed as However, the field equation for the warp factor A (4.4) gives These two equations imply that α = 0 and so Y = 0. As Y = 0, the algebraic KSE is simplified to and so σ + lies in one of the 8-dimensional eigenspaces of Γ x provided that X = ± 3 A . Thus instead of preserving 28 supersymmetries, the solution can be maximally supersymmetric.
Indeed this is the case as we shall now demonstrate. The integrability condition of the gravitino KSE (4.6) becomes To investigate whether this can yield a new condition on σ + , we find after a direct computation using the results of appendix B that the Riemann tensor in the ortho-normal frame is given by So S 7 = Spin(7)/G 2 is equipped with the round metric. For supersymmetric solutions, one must set a −1 = 1 81 X 2 = 1 9 2 A 2 . In such a case, the integrability condition of the gravitino KSE is automatically satisfied and so the solution preserves 32 supersymmetries. This solution is locally isometric to the maximally supersymmetric AdS 4 × S 7 solution.

N = 24, SU(4)/SU(3)
As so(6) = su(4), it follows from (1.2) that the internal space of an AdS 4 solution with 24 supersymmetries is the 7-dimensional homogeneous manifold SU(4)/SU(3). The geometry of this homogeneous space is described in appendix C. The left-invariant metric can be rewritten as with eigenvalues ±1, where tan θ = γ/β. Then upon inserting Y into the algebraic KSE (4.7) and using the above Clifford algebra operators, we obtain  Table 5. Decomposition of (4.29) KSE into eigenspaces.
The algebraic KSE (4.7) can then be decomposed into the eigenspaces of J 1 , J 2 and J 3 . The different relations on the fluxes for all possible sets of eigenvalues of these operators are listed in table 5.
The only possibility to obtain solutions with N > 16 supersymmetries is to choose the first set of eigenspinors in table 5. This leads to the integrability condition from the remaining KSE. This together with the warp factor field equation (4.4) and so α = β = γ = 0. Therefore Y = 0 and the solution is electric. As a result, the algebraic KSE (4.4) becomes and so for X = ±3 −1 A −1 it admits 8 linearly independent σ + solutions. So potentially, the background is maximally supersymmetric. It remains to investigate the gravitino KSE. First of all, we observe that for Y = 0 the Einstein equation (4.5) along the internal space becomes Therefore, the internal space is Einstein. After some computation using the results in appendix C, one finds that the homogeneous space SU(4)/SU(3) is Einstein provided that b = 9 4 a. In that case, the curvature of the metric in the ortho-normal frame becomes

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and so the internal space is locally isometric to the round 7-sphere. As expected from this, the integrability condition of the gravitino KSE (4.6)

N = 20
As mentioned in the introduction, the internal space of AdS 4 backgrounds that preserve 20 supersymmetries admits an effective and transitive action of a group which has Lie algebra so(5) = sp (2). The field equation for Y (4.3) is As X is constant, note that for generic 4-forms Y this defines a nearly-parallel G 2 -structure on M 7 , see e.g. [33] for homogeneous G 2 structures. However, in what follows we shall not assume that Y is generic. In fact in many cases, it vanishes. Amongst the 7-dimensional compact homogeneous spaces of (1.2), there are three candidate internal spaces. These are the Berger space B 7 = Sp(2)/Sp(1) max , V 2 (R 5 ) = Sp(2)/∆(Sp(1)), and J 7 = Sp(2)/Sp(1), corresponding to the three inequivalent embeddings of Sp(1) into Sp(2). We will in the following examine each case separately, starting with the Berger space Sp(2)/Sp(1) max .

Sp(2)/Sp(1) max
The description of the Berger space B 7 = Sp(2)/Sp(1) max as a homogeneous manifold is summarized in appendix D. B 7 is diffeomeorphic to the total space of an S 3 bundle over S 4 with Euler class ∓10 and first Pontryagin class ∓16 [34]. As a result H 4 (B 7 , Z) = Z 10 and B 7 is a rational homology 7-sphere. As sp(2) = so(5) and sp(1) = so(3), one writes so(5) = so(3) ⊕ m and the subalgebra so(3) acts irreducibly on m with the 7 representation. So B 7 admits a unique invariant metric up to a scale and it is Einstein. As the embedding of so(3) into so(7) factors through g 2 , it also admits an invariant 3-form ϕ given in (4.13) which is unique up to a scale. Because there is a unique invariant 3-form ϕ, dϕ ∝ * 7 ϕ and B 7 is a nearly parallel G 2 manifold. Using these, we find that the invariant fields of the theory are where we have introduced the ortho-normal frame e i = √ a i , * 7 ϕ is given in (4.13) and a, α are constants with a > 0.

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As the pair (ds 2 , Y ) exhibits the same algebraic relations as that of the Spin(7)/G 2 case, the algebraic KSE (4.19) can be solved in the same way yielding the results of table 4. To find N > 16 AdS 4 solutions, one should consider the first set of eigenspinors of the table which in turn imply the relation (4.20) amongst the fluxes. This together with the field equation of the warp factor (4.21) leads again to the conclusion that α = 0 and so Y = 0.
As a result of the analysis of the algebraic KSE, so far the background can admit up to 32 supersymmetries. It remains to investigate the solutions of the gravitino KSE. The curvature of B 7 is given by where k α is given in appendix D. The integrability condition of the gravitino KSE for Y = 0 is given in (4.23). To solve this condition, we decompose the expression into the 7 and 14 representations of g 2 using the projectors (4.40) and noting that k α as 2-forms are in the 14 representation. The integrability condition along the 7 representation gives X 2 = 81 5 a −1 while along the 14 representation gives that the Killing spinors must be invariant under g 2 . It turns out that there are two such σ + spinors however taking into account the remaining projection arising from the algebraic KSE, see (4.22), we deduce that the solution preserves 4 supersymmetries in total. This solution has already been derived in [23].

Sp(2)/∆(Sp(1))
The decomposition of the Lie algebra sp(2) = so(5) suitable to describe this homogeneous space can be found in appendix E. Writing A m A = ra M ra + 7 T 7 for the left-invariant frame, r = 1, 2, 3 and a = 4, 5, the most general left-invariant metric is ds 2 = δ rs g ab ra sb + a 4 ( 7 ) 2 , (4.41) where g ab is a positive definite symmetric 2 × 2-matrix, a > 0 a constant, and the leftinvariant forms are generated by To simplify the analysis of the geometry that follows, we note that without loss of generality the matrix (g ab ) can chosen to be diagonal. To see this, perform an orthogonal transformation O ∈ SO(2) to bring (g ab ) into a diagonal form. Such a transformation can be compensated with a frame rotation Demanding that A m A is invariant implies that M ra has to transform as M ra → O b a M rb . However, it is straight forward to observe that such a transformation is an automorphism of so(5) that preserves the decomposition (E.5), i.e. the structure constants of the Lie algebra remain the same. As a result, we can diagonalize the metric and at the same time use the same structure constants to calculate the geometric quantities of the homogeneous space. Under these orthogonal transformations the first two left-invariant forms are invariant while there is a change of basis in the space of left-invariant 3-forms.
To continue take 6 (g ab ) = diag(a 1 , a 2 ). Then introduce the ortho-normal frame e 7 = √ a 4 7 , e r4 = √ a 1 r4 and e r5 = √ a 2 r5 . In this frame the most general left-invariant metric and Y flux can be written as ds 2 = δ ab δ rs e ra e sb + (e 7 ) 2 , Y = β 1 e 7 ∧ χ 444 + β 2 e 7 ∧ χ 445 + β 3 e 7 ∧ χ 455 + β 4 e 7 ∧ χ 555 + β 5 ψ , (4.44) where β 1 , β 2 , · · · , β 5 are constants, and ω = 1 2 δ rs ab e ra ∧ e sb . (4.46) The Bianchi identity for Y is automatically satisfied. On the other hand the field equation for Y in (4.3) yields the conditions where we have chosen the top form on M 7 as dvol = e 7 ∧ χ 444 ∧ χ 555 . Before we proceed to investigate the various cases which arise from solving the linear system (4.47), let us consider first the case in which F is electric, i.e. it is proportional to the volume form of AdS 4 . In such a case β 1 = · · · = β 5 = 0. The algebraic KSE then gives (4.48) and the field equations along M 7 imply that and so M 7 is Einstein. The Einstein condition on the metric of M 7 requires that a 1 = a 2 , a 4 = 3 2 a 1 .
(4.50) 6 We have performed the analysis that follows also without taking (g ab ) to be diagonal producing the same conclusions.

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To investigate whether there are solutions preserving 20 supersymmetries, it remains to consider the integrability condition of the gravitino KSE (4.36). Indeed using the expressions (E.18) and (E. 19) for the curvature of this homogeneous space, the integrability condition along the directions 7 and ra gives X 2 = (27/8)a −1 1 while along the ra and sb directions requires additional projections. For example after taking the trace with δ ab and setting r = 1 and s = 2, the condition is Γ 1245 σ + = σ + which leads to solutions that preserve 16 supersymmetries or less, where the gamma matrices are in the ortho-normal frame and Γ r4 = Γ r , Γ r5 = Γ 3+r . Hence there are no N > 16 AdS 4 solutions.
Next let us turn to investigate the solutions of the linear system (4.47). The last condition implies that (4.51) To continue consider first the case that β 5 = 0.
Substituting the second equation in (4.51) into the linear system (4.47), one finds that Now there are several cases to consider. First suppose that the parameters of the metric a 1 , a 2 , a 4 are such that the only solutions of the linear system above are β 1 = β 2 = β 3 = β 4 = 0. In such case Y = β 5 ψ and Y has the same algebraic properties as that of the SU(4)/SU(3) case with β = γ = 0 and α = β 5 . As a result, the algebraic KSE together with the Einstein equation for the warp factor imply that β 5 = 0 as well and so Y = 0. This violates our assumption that β 5 = 0. In any case, the 4-form flux F is electric which we have already investigated above and have found that such a configuration does not admit solutions with N > 16 supersymmetries. Next suppose that the parameters of the metric are chosen such that either β 1 = β 3 = 0 , or β 2 = β 4 = 0 . with 3 4 a 2 1 − a 4 (a 4 − a 2 ) = 0. Setting P 1 = Γ 7156 , P 2 = Γ 7345 and P 3 = Γ 7264 , the algebraic KSE can be written as  Table 6. Decomposition of (4.55) KSE into eigenspaces.
As P 1 , P 2 , P 3 are commuting and have eigenvalues ±1, the above algebraic equation decomposes into eigenspaces as tabulated in table 6.
To find solutions with 20 supersymmetries or more, we can either choose one of the two eigenspaces with 3 linearly independent eigenspinors and both eigenspaces with a single eigenspinor or both eigenspaces with 3 linearly independent eigenspinors. In the former case the algebraic KSE will admit 20 Killing spinors and in the latter 24 Killing spinors.
Let us first consider the case with 20 Killing spinors. In such a case, we find that and where we have considered the second eigenspace with 3 eigenspinors in table 6. The case where the first such eigenspace with 3 eigenspinors is chosen can be treated in a similar way. The condition (4.57) follows as an integrability condition to the remaining algebraic KSE involving Γ z and Γ x . On the other hand, the field equation of the warp factor (4.4) implies that which together with (4.57) gives β 1 = 0 and so Y = 0. The solution cannot preserve N > 16 supersymmetries. Next consider the case with 24 Killing spinors. In this case, we find that and the integrability of the remaining algebraic KSE gives  Table 7. Decomposition of (4.66) KSE into eigenspaces.
Comparing this with (4.60), one finds that the β's vanish and so Y = 0. Thus there are no solutions with N > 16 for either β 1 , β 3 or β 2 , β 4 non-vanishing. It remains to investigate the case that all β 1 , . . . , β 5 = 0. This requires that the determinant of the coefficients of the linear system (4.52) must vanish, i.e.
Taking the difference of the two equations, we find that either a 1 = a 2 , or a 4 = 3 4 (a 1 + a 2 ) . (4.63) Substituting a 4 above into (4.62), we find that a 1 = a 2 . So without loss of generality, we set a 1 = a 2 = a. Then the linear system (4.52) can be solved to yield the algebraic KSE (4.7) can be rewritten as where tan θ = β 3 /β 2 and α = β 2 2 + β 2 3 . As these Clifford algebra operations commute and have eigenvalues ±1, the restrictions of this equation to the eigenspaces of P 1 , P 2 and P 3 are given in table 7.
To find solutions with 20 supersymmetries, one needs to consider the eigenspace in table 7 with 6 eigenspinors. In such a case the integrability of the remaining KSE requires that 1 36 (4.67)

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Comparing this with the field equation of the warp factor 1 9 we find that all β's must vanish and so Y = 0. Thus the flux F is electric and as we have demonstrated such background does not admit N > 16 AdS 4 supersymmetries.
β 5 = 0. Since the backgrounds with electric flux F cannot preserve N > 16 supersymmetries, we have to assume that at least one of the pairs (β 1 , β 3 ) and (β 2 , β 4 ) do not vanish. If either the pair (β 1 , β 3 ) or (β 2 , β 4 ) is non-vanishing, the investigation of the algebraic KSE proceeds as in the previous case with β 5 = 0. In particular, we find that the algebraic KSE (4.7) together with the field equation for the warp factor imply that all β's vanish and the flux F is electric. So there are no solutions preserving N > 16 supersymmetries. It remains to investigate the case that β 1 , β 2 , β 3 , β 4 = 0. If this is the case, the determinant of the linear system (4.47) must vanish which in turn implies that The solution of these equations is Substituting the latter equation into (4.69), one again finds that a 1 = a 2 . So without loss of generality we take a 1 = a 2 in which case For the latter case, the linear system (4.47) gives After setting β 5 = 0, the investigation of the algebraic KSE can be carried out as that described in table 7. As a result after comparing with the field equation for the warp factor, the β's vanish and F is electric. Thus there are no solutions preserving N > 16 supersymmetries. It remains to investigate the case that X = 1/(2 √ a 4 ) in (4.71). In this case, the linear system (4.47) gives Using the P 1 , P 2 and P 3 as in (4.65), the algebraic KSE (4.7) becomes and the solutions in the eigenspaces of P 1 , P 2 and P 3 are described in table 8. To preserve N > 16 supersymmetries, one has to consider either one of the eigenspaces with 3 eigenspinors and the eigenspace with 2 eigenspinors or both of the eigenspaces with 3 eigenspinors. In either case, one finds that all β's vanish and so Y = 0. Then F is electric and such solutions do not preserve N > 16 supersymmetries. Therefore we conclude that the homogenous space Sp(2)/∆(Sp(1)) does not give rise to AdS 4 backgrounds with N > 16.

Sp(2)/Sp(1)
The geometry of this homogeneous space is described in appendix E where the definition of the generators of the algebra and expressions for the curvature and invariant forms can be found. A left-invariant frame is A m A = a W a + r T (+) r , where a = 1, . . . , 4 and r = 1, 2, 3. Then the most general left-invariant metric is where a > 0 is a constant and (g rs ) is any constant 3×3 positive definite symmetric matrix.
To simplify the computations that follow, it is convenient to use the covariant properties of the decomposition of sp(2) = so(5) as described in (E.9) to restrict the number of parameter in the metric. In particular, observe that the decomposition (E.9) remains invariant under the transformation of the generators s . The orthogonal rotations O act on the matrix (g rs ) as g → OgO −1 . As (O, U ) is an automorphism of so(5) which leaves the decomposition (E.9) invariant, we can use O to put the matrix (g rs ) into diagonal form. So from now on without loss of generality, we set (g rs ) = diag(b 1 , b 2 , b 3 ) with b 1 , b 2 , b 3 > 0, see also [35].
The left-invariant 4-forms are generated by where where there is no summation over the indices r and s on the left-hand side of the last equation and β = δ rs β rs . Before we proceed to investigate the solutions of the linear system, notice that if β rs = 0, then α = 0 and so F is electric. The supersymmetry preserved by these solutions will be investigated later. As we shall demonstrate such solutions cannot preserve more than 16 supersymmetries.
Furthermore writing Y = αψ + Y β , where Y β = β rs ρ rs , the field equation of the warp factor in (4.4) can be written as As we shall demonstrate, the compatibility of this field equation with the algebraic KSE rules out the existence of N > 16 backgrounds.
Returning to the solutions of (4.82), let us focus on β rs with r = s. There are several cases to consider.
Either β rs = 0 for all r = s or β rs = 0 for all r = s. If β rs , r = s, are all non-vanishing, the last equation in (4.82) implies that As a result, the metric is invariant under SO(3) and this can be used to bring β rs into diagonal form. Of course (β rs ) is also diagonal if β rs = 0 for all r = s. So without loss of generality, we can assume that (β rs ) is diagonal. Setting where all gamma matrices are in the ortho-normal basis and {Γ i } = {Γ a , Γ 4+r }, the algebraic KSE can be written as  Table 9. Decomposition of (4.86) KSE into eigenspaces.
The decomposition of the algebraic KSE into the eigenspaces of the commuting Clifford algebra operators J 1 , J 2 , J 3 is illustrated in table 9.
To construct N > 16 solutions, we have to include the eigenspace with four eigenspinors. The integrability condition of the remaining KSE described in table 9 gives Comparing (4.87) with the field equation for the warp factor (4.83), we find that α = β rs = 0. Therefore Y = 0 and so F is electric.
β 12 , β 13 = 0 and β 23 = 0. As the other two cases for which either β 13 = 0 or β 12 = 0 with the rest of the components non-vanishing can be treated in a similar way, we take without loss of generality that β 23 = 0 and β 12 , β 13 = 0. In such a case, the last condition in (4.82) gives The metric is invariant under an SO(2) ⊂ SO(3) symmetry which acts with the vector representation on the vector (β 12 , β 13 ) and leaves the form of (β rs ) invariant. As a result up to an SO(2) transformation, we can set β 13 = 0 as well. Furthermore, if b 1 = b 2 , the diagonal terms in the last condition in (4.82) give On the other hand if b 1 = b 2 the analysis reduces to that of the previous case. Therefore for b 1 = b 2 , Y can be written as  where tan θ = β 12 /β 11 , the algebraic KSE can be written as The decomposition of the algebraic KSE into the eigenspaces of the commuting Clifford algebra operators J 1 , J 2 , J 3 is illustrated in table 10.
To construct solutions preserving more than 16 supersymmetries, we have to include the eigenspace with four eigenspinors leading again to the integrability condition (4.87). Comparing again with the field equations of the warp factor (4.83), we deduce that F is electric. β 13 = β 23 = 0 but β 12 = 0. All three cases for which only one of the three off-diagonal components of (β rs ) is non-zero can be treated symmetrically. So without loss of generality, one can take β 13 = β 23 = 0 but β 12 = 0. In this case, the last equation in (4.82) has four branches of solutions depending on the choice of the b 1 , b 2 and b 3 components of the metric.
The last equation in (4.82) then implies X = 2/ √ b and the aforementioned residual SO(3) symmetry can be used to put β rs to be diagonal.
The last equation in (4.82) then implies X = 2/ √ b 3 and β 33 = 0. The aforementioned residual SO(2) symmetry can be used to put β rs to be diagonal.
The last equation in (4.82) then implies X = (b 1 + b 2 )/ √ b 1 b 2 b 3 and β 11 = β 22 = 0. In such a case, Y reads Y = αψ + β 33 ρ 33 + β 12 (ρ 12 + ρ 21 ) .  Table 11. Decomposition of (4.95) KSE into eigenspaces. Choosing the algebraic KSE can be written as The decomposition of the algebraic KSE into the eigenspaces of J 1 , J 2 , J 3 is illustrated in table 11. Again the eigenspace with four eigenspinors has to be included in the construction of N > 16 backgrounds. As a result, this leads to the integrability condition (4.87) which together with the warp factor field equation (4.83) imply that F is electric.
The last equation in (4.82) then implies In such a case, Y reads With the choice of commuting Clifford algebra operators as in (4.91), the algebraic KSE can be written as  It remains to investigate the number of supersymmetries preserved by the solutions for which F is electric. For this, one has to investigate the integrability condition of the gravitino KSE (4.36). Using the expression for the curvature of metric in (E.25)-(E. 28) and requiring that the solution preserves N > 16, we find that

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As X 2 = 9 −2 A −2 , we have b = 2 A 2 and a = (1/2) 2 A 2 . The rest of the integrability condition is satisfied without further conditions. So every solutions that preserves N > 16 supersymmetries is maximally supersymmetric and so locally isometric to AdS 4 × S 7 . One can confirm this result by investigating the Einstein equation (4.5). As all solutions with electric F are Einstein R (7) ij = (1/6)X 2 δ ij , it suffices to identify the left-invariant metrics on Sp(2)/Sp(1) that are Einstein. There are two Einstein metrics [35,36] on Sp(2)/Sp(1) given by and where the first one is the round metric on S 7 , see also [37]. The second one does not give N > 16 supersymmetric solutions.

Conclusions
We have classified up to local isometries all warped AdS 4 backgrounds with the most general allowed fluxes in 10-and 11-dimensional supergravities that preserve N > 16 supersymmetries. We have demonstrated that up to an overall scale, the only solutions that arise are the maximally supersymmetric solution AdS 4 × S 7 of 11-dimensional supergravity [17,18] and the N = 24 solution AdS 4 × CP 3 of IIA supergravity [16]. These two solutions are related via dimensional reduction along the fibre of the Hopf fibration S 1 → S 7 → CP 3 .
The assumption we have made to prove these results is that either the solutions are smooth and the internal space is compact without boundary or that the even part g 0 of the Killing superalgebra of the backgrounds decomposes as g 0 = so(3, 2) ⊕ t 0 . In fact these two assumptions are equivalent for N > 16 AdS 4 backgrounds. It may be possible to weaken these assumptions but they cannot be removed altogether. This is because in such a case additional solutions will exist. For example the maximally supersymmetric AdS 7 ×S 4 solution of 11-dimensional supergravity [38] can be re-interpreted as a maximally supersymmetric warped AdS 4 solution. However in such case the "internal" 7-dimensional manifold M 7 is not compact and the even subalgebra of the Killing superalgebra g 0 does not decompose as so(3, 2) ⊕ t 0 .
We have identified all AdS 4 backgrounds up to a local isometry. Therefore, we have specified all the local geometries of the internal spaces G/H of these solutions. However the possibility remains that there are more solutions which arise via additional discrete identifications Z\G/H, where Z is a discrete subgroup of Z ⊂ G. The AdS 4 × Z\G/H solutions will preserve at most as many supersymmetries as the AdS 4 × G/H solutions.  [39]. A systematic investigation of all possible N > 16 AdS 4 × Z\G/H backgrounds will involve the identification of all discrete subgroups of G. The relevant groups here are SU(4) and Spin (8), see e.g. [46] for an exposition of discrete subgroups of SU(4) and references therein.
It is clear from our results on AdS 4 backgrounds that supersymmetric AdS solutions which preserve N > 16 supersymmetries in 10-and 11-dimensions are severely restricted. Consequently there are few gravitational duals for superconformal theories with a large number of supersymmetries which have distinct local geometries. For example, the superconformal theories of [40][41][42] have gravitational duals which are locally isometric to the AdS 5 × S 5 maximally supersymmetric background as there are no distinct local AdS 5 geometries that preserve strictly 24 supersymmetries [10]. In general our results also suggest that there may not be a large number of backgrounds that preserve N > 16 supersymmetries in 10-and 11-dimensional supergravities. So it is likely that all these solutions can be found in the future.

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Consider the left coset space M = G/H, where G is a compact connected semisimple Lie group G which acts effectively from the left on M = G/H and H is a closed Lie subgroup of G. Let us denote the Lie algebras of G and H with g and h, respectively. As there is always an invariant inner product on g, it can be used to take the orthogonal complement of h in g and so Denote the generators of h with h α , α = 1, 2, . . . , dim h and a basis in m as m A , A = 1, . . . , dim g − dim h. In this basis, the brackets of the Lie algebra g take the following form If f AB C = 0, that is [m, m] ⊂ h, the space is symmetric.
Let g : U ⊂ G/H → G be a local section of the coset. The decomposition of the Maurer-Cartan form in components along h and m is which defines a local left-invariant frame A and a canonical left-invariant connection Ω α on G/H. The curvature and torsion of the canonical connection are The latter condition is required for invariance under the right action of H on G. All left-invariant forms are parallel with respect to the canonical connection. It remains to describe the metrics of G/H which are left-invariant. These are written as where the components g AB are constant and satisfy f αA C g BC + f αB C g AC = 0 . (B.8)

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For symmetric spaces, the canonical connection coincides with the Levi-Civita connection of invariant metrics. So all non-vanishing left-invariant forms are harmonic and represent non-trivial elements in the de Rham cohomology of G/H. However if G/H is strictly homogeneous this is not the case since the canonical connection has non-vanishing torsion. Suppose G/H is homogeneous and equipped with an invariant metric g. To describe the results of the paper, it is required to find the Levi-Civita connection of g and its curvature. Let Φ be the Levi-Civita connection in the left-invariant frame. As the difference of two connections is a tensor, we set As Φ is metric and torsion free, we have These equations can be solved for Q to find that In turn the Riemann curvature 2-form R A B is This is required for the investigation of the gravitino KSE. Note that the expression for Φ A B is considerably simplified whenever the coset space is naturally reductive because the structure constants f ABC = f AB E g CE are then skew symmetric.

C su(k)
Here we shall collect some formulae that are useful in understanding the homogeneous spaces that admit a transitive action of a group with Lie algebra su(k). A basis over the reals of anti-hermitian k × k traceless complex matrices is where ν(ab) is a normalization factor and a, b, c, d = 1, . . . , k. The trace of these matrices yields an invariant inner product on su(k). In particular the non-vanishing traces are tr(M ab M a b ) = − 1 2 (δ aa δ bb − δ ab δ ba ) , tr(N ab N a b ) = − ν(ab)ν(a b ) 2 δ aa δ bb + δ ab δ ba − 2 k δ ab δ a b . (C.2)

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It is customary to choose the normalization factors ν such that all generators have the same length. In such a case, they will depend on k. However in what follows, it is more convenient to choose ν = 1. Then re-write each basis element using the 5 representation so (3)  where O ∈ so(3) and S ∈ m. Using this one can proceed to describe the homogeneous space B 7 . However, this decomposition does not automatically reveal the G 2 structure which is necessary in the analysis of the supersymmetric solutions. Instead, we shall follow an adaptation [34] of the description in [23] and [45, appendix A.1]. For this use the inner product W ab , W a b = − 1 2 tr(W ab W a b ) , (D.4) which is so(5) invariant and the basis W ab , a < b, is ortho-normal. In this basis, the structure constants of so(5) are skew-symmetric. Then identify the so(3) subalgebra of so (5) with the span of the ortho-normal vectors