A non-existence theorem for N > 16 supersymmetric AdS 3 backgrounds

: We show that there are no smooth warped AdS 3 solutions in 10- and 11-dimensional supergravities which preserve strictly more than 16 supersymmetries and have internal space a compact without boundary manifold.


Introduction
The main purpose of this paper is to complete the classification of AdS backgrounds that preserve strictly more than 16 supersymmetries in 10-and 11-dimensional supergravities. Such backgrounds have found many applications originally in supergravity compactifications and more recently in the AdS/CFT correspondence, for reviews see [1,2]. In the JHEP07(2018)178 latter case AdS backgrounds that preserve N > 16 supersymmetries are associated with the best understood examples of the correspondence [3,4].
The maximally supersymmetric AdS backgrounds 1 have been classified in [6] and it has been found that they are locally isometric to the AdS 4 ×S 7 [7] and AdS 7 ×S 4 [8] solutions of 11-dimensional supergravity, and to the AdS 5 × S 5 solution of IIB supergravity, see [9] and comment within. There are no AdS 7 backgrounds that preserve 16 < N < 32 supersymmetries [10][11][12] and no smooth AdS 6 backgrounds that preserve N > 16 supersymmetries with compact without boundary internal space [13]. More recently, it has been demonstrated under the same assumptions on the internal space that there are no smooth AdS 5 backgrounds that preserve 16 < N < 32 supersymmetries [14]; see [15][16][17] for applications to AdS/CFT. It has also been shown in [18] that the only smooth AdS 4 solution with compact without boundary internal space that preserves 16 < N < 32 supersymmetries is locally isometric to the N = 24 IIA solution AdS 4 ×CP 3 of [19]. Moreover it has been shown in [20] that there are no smooth AdS 2 backgrounds in 10-and 11-dimensional supergravities with compact without boundary internal space that preserve N > 16 supersymmetries. Product solutions AdS n × M D−n with M D−n a symmetric space have been classified in [21]- [24]. Furthermore the geometry of all heterotic AdS 3 backgrounds have been investigated in [25] and it has been found that there are no solutions that preserve N > 8 supersymmetries.
The only class of backgrounds that remains to be investigated are the warped AdS 3 backgrounds with the most general allowed fluxes in 10-dimensional type II and 11dimensional supergravity theories that preserve N > 16 supersymmetries. For these, we shall demonstrate a non-existence theorem provided that the solutions are smooth and their internal spaces are compact manifolds without boundary. It suffices to establish the non-existence theorem up to local isometries. The more general result follows as there are no new geometries that can be constructed by taking quotients by discrete groups.
The method used to establish this result relies on a number of recent developments. One of them is the integration of both field equations and Killing spinor equations (KSEs) of 10-and 11-dimensional supergravities over the AdS n subspace for all warped AdS backgrounds, AdS n × w M D−n , without making any other assumptions on the fields and Killing spinors apart from imposing the symmetries of the AdS subspace on the fields [10][11][12]. This integration reduces the field equations and KSEs of 10-and 11-dimensional supergravity theories to a system of independent equations on the internal space M D−n . To solve these, we use another key development which is the homogeneity theorem in [26]. This states that all 10-and 11-dimensional supergravity backgrounds that preserve N > 16 supersymmetries are Lorentzian homogeneous spaces and all the fields are invariant tensors. Another ingredient for the proof is the classification of Killing superalgebras of all warped AdS backgrounds in [27]. For AdS backgrounds that preserve N > 16 supersymmetries this enables us to identify all the Lorentzian algebras that act transitively and (almost) effectively 2 on the spacetime. The requirement for a transitive action is a consequence JHEP07(2018)178 of the homogeneity theorem while that of the (almost) effective action is needed for the super-Jacobi identities of the superalgebra to be satisfied.
The strategy of the proof is as follows. First one establishes that for all AdS 3 × w M D−3 backgrounds that preserve N > 16 supersymmetry, the warp factor is constant. Therefore the geometry is a product AdS 3 × M D−3 . To show this, either one uses that the solutions are smooth and the internal space is compact without boundary as well as techniques from the proof of the homogeneity theorem or that the even subalgebra g 0 of the Killing superalgebra g of AdS 3 × w M D−3 decomposes as g 0 = iso(AdS 3 ) ⊕ iso(M D−3 ), where iso(AdS 3 ) is an isometry algebra of AdS 3 and iso(M D−3 ) = t 0 is an isometry algebra of the internal space M D−3 .
Having established that the N > 16 AdS 3 backgrounds are products, AdS 3 × M D−3 and that g 0 = iso(AdS 3 ) ⊕ t 0 , where t 0 is an algebra of isometries on M D−3 , we obtain as a consequence of the homogeneity theorem that the internal space is a homogeneous space G/H with Lie G = t 0 . In addition, the theorem requires that all fields are invariant under the left action of G on G/H.
The final part of the proof involves the identification of all homogeneous spaces 3 in seven and eight dimensions that admit a transitive and an almost effective action of a group G with Lie algebra t 0 . For t 0 semisimple, one can identify the relevant homogeneous spaces using the classification results of (simply connected) 7-and 8-dimensional homogeneous manifolds in [28]- [32]; for a concise description see [29]. There is also the need of a procedure on homogeneous spaces which tests whether a t 0 can act effectively on a given G/H. In section 2, we refer to it as "modification" of a homogeneous space. A similar approach can be used for the case that t 0 is not semisimple. After identifying all the suitable homogeneous spaces, a substitution of the geometric data into the field equations and KSEs of supergravity theories in 10-and 11-dimensions establishes our non-existence theorem.
Before we proceed with the proof, let us investigate the need for the assumptions we have made. First one can establish that if AdS 3 × w M D−3 is smooth and M D−3 is compact without boundary, then the even subalgebra of the Killing superalgebra of AdS 3 × w M D−3 will decompose as g 0 = iso(AdS 3 ) ⊕ t 0 [27]. The requirement that M D−3 must be compact without boundary may be weakened but not completely removed. If it is removed, then g 0 may not decompose as stated above. In addition the warp factor of AdS 3 × w M D−3 backgrounds with N > 16 supersymmetries may not be constant and there exist AdS 3 backgrounds that preserve N > 16 supersymmetries, see [35] for a detailed exposition. In particular the maximally supersymmetric AdS 4 × S 7 and AdS 7 × S 4 backgrounds of 11-dimensional supergravity can be viewed as warped AdS 3 backgrounds but the internal spaces are not compact without boundary.
The paper is organized as follows. In section 2, we describe the Killing superalgebras of AdS 3 backgrounds and introduce the notion of a modification of a homogeneous space which allows us to test whether an algebra can act effectively on it. In sections 3, 4 and 5, we prove the main result of our paper for 11-dimensional, IIA and IIB supergravities, respectively. In appendix A, we describe our conventions. In appendix B, we present some JHEP07(2018)178 aspects of the structure of homogeneous spaces admitting a transitive action by a compact but not semisimple Lie group that are useful in the proof of our results, and in appendix C, we describe the geometry of the N k,l homogeneous space.
2 Symmetries of AdS 3 backgrounds 2.1 Killing superalgebras of AdS 3 backgrounds As AdS 3 is locally a group manifold, the Killing superalgebras of warped AdS 3 backgrounds with the most general allowed fluxes decompose as g = g L ⊕ g R , where g L and g R are associated with the left and right actions. The left and right Killing superalgebras g L and g R have been identified in [27]. This has been done under the assumptions that either the internal space is compact without boundary or that the even subalgebra decomposes as (g L ) 0 = sl(2, R) L ⊕ (t L ) 0 and similarly for g R . As is isomorphic to either sl(2, R) L or sl(2, R) R if the background has only either left or right supersymmetries, respectively, or iso(AdS 3 ) = sl(2, R) L ⊕ sl(2, R) R if the background has both left and right supersymmetries.
It has been shown in [27] that for AdS 3 backgrounds t 0 may not be semisimple and in addition may admit central terms c which commute with all other generators of the superalgebra. We shall show below that in all cases but one c L = {0}. If c L = {0}, it will have at most dimension 3. The left and right superalgebras are isomorphic and so it suffices to present only the left ones. These are tabulated in table 4 1.

Central terms
We shall focus on g L as the description that follows below also applies to g R . It has been observed in [27] that the Killing superalgebras of AdS 3 backgrounds may exhibit central terms. Such terms may occur in all cases apart from osp(n|2) and D(2, 1, α). However it has been shown in [27] that both f(4) and g(3) do not exhibit such terms. Though sl(2|2)/1 4×4 can exhibit up to three central terms. This is because sl(2|2)/1 4×4 arises as a special case of D(2, 1, α) at special values of the parameter α. At those values three of the R-symmetry generators of D(2, 1, α) span the R-symmetry algebra so(3) of sl(2|2)/1 4×4 and the other three become central.
It can also be shown that sl(n|2), n > 2 and osp(4|2n), n > 1, do not exhibit central terms either. This can be seen after an analysis of the condition of [27], whereṼ rs = −Ṽ sr are the generators of (t L ) 0 and α is described in [27]. For sl(n|2), n > 2, the central terms that can occur are (2,0) and (0,2) components of theṼ . However one can show that these do not satisfy (2.1) unless they vanish. Thus c = {0}. It remains to investigate the superalgebra with (t L ) 0 /c L = sp(n) ⊕ sp(1). The central generators that can occur are theṼ which lie in the complement of sp(n) ⊕ sp(1) in so(4n). Taking the trace of (2.1) with one of the three complex structures that are associated with sp(n) ⊕ sp (1), one can demonstrate that all such generatorsṼ must also vanish. Thus again c = {0}. Therefore apart from sl(2|2)/1 4×4 , all the other superalgebras in table 1 do not exhibit central terms.

On the G/H structure of internal spaces
We shall demonstrate later that the spacetime of all AdS 3 backgrounds that preserve > 16 supersymmetries in 10-and 11-dimensional supergravities is a product AdS 3 × M D−3 and that M D−3 is a homogeneous space M D−3 = G/H such that LieG = t 0 . Of course G acts transitively on M D−3 . In addition it is required to act "almost effectively" on M D−3 . This means that the map of Lie G into the space of Killing vector fields of M D−3 is an inclusion, ie for every generator of Lie G there is an associated non-vanishing Killing vector field on M D−3 . We shall also refer to this property as Lie G acting "effectively" on M D−3 . This latter property is essential as otherwise the super-Jacobi identities of the AdS Killing superalgebra will not be satisfied. It is also essential for the identification of the manifolds that can arise as internal spaces of all AdS, and in particular AdS 3 , backgrounds preserving some supersymmetry.
For AdS 3 backgrounds, there are two cases to consider. The first case arises whenever t 0 is a simple Lie algebra. Then the internal spaces can be identified, up a to factoring with a finite group, using the classification of the simply connected 7-and 8-dimensional homogeneous spaces in [28]- [32]. This is sufficient to identify the internal spaces of all such AdS 3 backgrounds that preserve N > 16 supersymmetries.
However for most AdS 3 backgrounds t 0 is not simple. Typically it is the sum of two Lie algebras, t 0 = (t L ) 0 ⊕ (t R ) 0 , one arising from the left sector and another from the right sector. In addition, it may not be semisimple. For example, we have seen that t 0 = u(3) = su(3) ⊕ u(1) for the sl(3|2) Killing superalgebra and t 0 = su(2) ⊕ c for the sl(2|2)/1 4×4 superalgebra with a central term c. Furthermore, t 0 is not semisimple for all AdS 3 backgrounds that exhibit either N L = 4 or N R = 4 supersymmetries. Given that t 0 may not be JHEP07(2018)178 simple, the question then arises how one can decide given a G ′ /H ′ space chosen from the classification results of [28]- [32] whether t 0 acts both transitively and effectively on G ′ /H ′ .
Let us illustrate this with examples. It is known that both U(n) and SU(n) act transitively and effectively on S 2n−1 . Thus S 2n−1 = U(n)/U(n − 1) and S 2n−1 = SU(n)/SU(n−1). However for n > 2, it is u(n) which appears as a subalgebra of sl(n|2) and so u(n) is expected to act transitively and effectively on the internal spaces instead of su(n). From this perspective U(n)/U(n−1) can arise as a potential internal space of an AdS 3 background whereas SU(n)/SU(n − 1) should be discarded. As in the classification results for homogeneous spaces it is not apparent which description is used for a given homogeneous space but essential for the classification of AdS 3 backgrounds, let us investigate the above paradigm further. To see how S 2n−1 = SU(n)/SU(n − 1) can be modified to be written as a U(n)/U(n − 1), consider the group homomorphism i from SU(n − 1) × U(1) into SU(n) as In fact i has kernel Z n−1 and so factors to U(n − 1). Next consider SU(n) × U(1) and the group homomorphism j of SU(n − 1) × U(1) into SU(n) × U(1) as Again j has kernel Z n−1 and so factors to U(n−1). Then SU(n)×U(1)/j(SU(n−1)×U(1)) = S 2n−1 with SU(n) × U(1) acting almost effectively on S 2n−1 . Furthermore one can verify that U(n) = (SU(n) × U(1))/Z n acts effectively on S 2n−1 as expected.
The key point of the modification described above is the existence of U(1) ⊂ SU(n) such that SU(n − 1) × U(1) ⊂ SU(n) and that this U(1) acts on both SU(n) and the U(1) subgroup of SU(n) × U(1). Observe that after the modification the isotropy group is larger and so the invariant geometry of S 2n−1 as a U(n)/U(n − 1) homogeneous space is more restrictive than that of S 2n−1 = SU(n)/SU(n − 1).

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One finds that Sp(2) × Sp(1)/j(Sp(1) × Sp(1)) is diffeomorphic to S 7 , with Sp(2) × Sp(1) acting almost effectively and descending to an effective action for Sp(2) · Sp(1). Again the additional Sp(1) introduced in the isotropy group acts both on Sp(2) and the additional Sp(1) introduced in the transitive group. The geometry of the homogeneous space S 7 = Sp(2) · Sp(1)/Sp(1) · Sp(1) is more restrictive than that of S 7 = Sp(2)/Sp (1). In fact the former is a special case of the latter. As a final example SU(2) × SU(2)/SU(2) can be seen as a modification of the homogeneous space SU(2)/{e}. From now on we shall refer to such constructions as "modifications" of a homogeneous space. On the level of Lie algebras the modifications can be viewed as follows. Suppose t 0 decomposes as t 0 = k ⊕ e, where k and e are Lie algebras, and that there is a homogeneous space K/L with Lie(K) = k. To see whether K/L can be modified to admit an effective action of the whole t 0 algebra, it is first required that l ⊕ e is a subalgebra of k, where LieL = l. Then, up to possible discrete identifications, K/L can be modified to K×E/L×E, where now E with Lie E = e acts on both the K and E subgroups of the transitive group.
All 7-and 8-dimensional K/L homogeneous spaces with K semisimple are known up to possible modifications. Because of this for t 0 semisimple, one can systematically search for all modifications to K/L homogeneous spaces to find whether a Lie algebra t 0 can act transitively and effectively on a modified homogenous space. If t 0 is not semisimple, we have argued in appendix B that up to discrete identifications one can construct all the homogeneous spaces G/H with Lie G = t 0 as product of a modification of a homogeneous space K/L with K semisimple with the abelian group × k U(1).
As we shall see the modifications of homogeneous spaces are necessary to identify all possible internal spaces of AdS 3 backgrounds that can preserve some supersymmetry. For such modifications to exist for K/L a necessary condition is that the rank of L must be strictly smaller than that of K. It turns out that this is rather restrictive in the analysis that follows.
Let us now turn to investigate the homogeneous geometry of a modification K×E/L×E of the homogenous K/L space. One can show that this can be explored as a special case of that of K/L. Indeed suppose that k = l ⊕ m. Then observe that one can choose the generators of Lie(K × E) such that Lie(K × E) = j(l ⊕ e) ⊕ m, where j : l ⊕ e → k ⊕ e is the inclusion of the modification. Therefore the tangent space at the origin of the original K/H space and that of the modification K ×E/L×E can be identified with the same vector space m. The only difference is that m as the tangent space at the origin of K/L is the module of a representation of l while after the modification m is the module of a representation of l ⊕ e. Thus all the local homogeneous geometry of the modification K × E/L × E is that of K/L which in addition is invariant under the representation of e on m. fields of 11-dimensional supergravity are a metric ds 2 and a 4-form field strength F . Following the description of AdS 3 × w M 8 backgrounds presented in [10], these can be written as where (u, r, z) are the coordinates of AdS 3 , ℓ is the AdS 3 radius, A is the warp factor which is a function of M 8 , and Q and X are a 1-form and 4-form on M 8 , respectively. The dependence of the fields on the AdS 3 coordinates (u, r, z) is explicit while ds 2 (M 8 ), A, Q, X depend only on the coordinates y I of M 8 . Next we define a null-orthonormal frame as The field equations for F give that and where our Hodge duality conventions can be found in appendix A. Similarly, the Einstein equation along AdS 3 gives rise to a field equation for the warp factor A and the Einstein equation along M 8 reads where R (8) ij is the Ricci tensor of the internal manifold M 8 . Note in particular that (3.7) implies that A is nowhere vanishing, provided that A and all other fields are smooth.

The Killing spinors
Here we summarize the solution of the gravitino KSE of 11-dimensional supergravity of [10] for warped AdS 3 × w M 8 backgrounds. In this approach, the KSE of 11-dimensional supergravity is first solved along the AdS 3 subspace and then the remaining independent KSEs along the internal space M 8 are identified. The Killing spinors can be expressed 5 as The gamma matrices are always taken with respect to the null-orthonormal frame (3.3).

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where the dependence on the AdS 3 coordinates is explicit and σ ± and τ ± are Majorana Spin(10, 1) spinors that depend only on the coordinates of M 8 and satisfy the light-cone projections The remaining independent KSEs on M 8 are and 14) The conditions (3.11) can be thought of as the restriction of the gravitino KSE of 11dimensional supergravity on M 8 while (3.12) arises from the integration of the gravitino KSE along the AdS 3 subspace.
To make a connection with the terminology used to describe the Killing superalgebras of AdS 3 backgrounds in section 2, the Killing spinors ǫ that depend only on the σ ± type of spinors are in the left sector while those that depend on τ ± spinors are in the right sector. The existence of unrelated 6 σ ± and τ ± types of spinors is the reason that the Killing superalgebra g of AdS 3 decomposes as g = g L ⊕ g R . Furthermore, it has been noted in [10] that if σ + and τ + solve the KSEs (3.11) and (3.12), so do

.1 Factorization of Killing vectors
It has been shown in [27] that for compact without boundary internal spaces M 8 , the even part of the Killing superalgebra g 0 decomposes into the algebra of symmetries of AdS 3 and those of the internal space M 8 . This together with the homogeneity theorem of [26] can be used to show that the internal space M 8 is homogeneous for N > 16 backgrounds. For AdS 3 backgrounds, the condition [27] for

A is constant and M 8 is homogeneous
Let us define the spinor bilinear where χ either stands for σ + or τ + . The gravitino KSE (3.11) then implies ie W is a Killing vector 7 on M 8 . From (3.16) it follows that the only non-vanishing Killing vector fields W are those that are constructed as bilinears of either σ + or τ + spinors. As a consequence of the algebraic KSEs (3.12), one has Im σ 1 + , Ξ (+) σ 2 + = 0 and Im τ 1 + , (Ξ (+) + ℓ −1 )τ 2 + = 0. Expanding these, one finds that where W is a bilinear of either σ + or τ + spinors. As it has been mentioned, (3.16) implies that the only non-vanishing Killing vectors W on M 8 are those constructed from either σ + or τ + spinors. Therefore (3.25) will be valid for all non-vanishing Killing vectors W on M 8 . Suppose now that N > 16. A similar argument to that used for the proof of the homogeneity theorem in [26] implies that the set of all Killing vectors W span the tangent space of M 8 . Thus A is constant and M 8 is homogeneous. It should be noted that if (3.16) is not valid, then the vector fields W in (3.25) may not span all the Killing vectors on M 8 .

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Therefore we conclude that all N > 16 supersymmetric AdS 3 backgrounds are products AdS 3 × M 8 , where M 8 is a homogeneous space. In the analysis that follows, which includes that of AdS 3 backgrounds in type II 10-dimensional supergravities, we shall focus only on such product spaces.

Electric solutions do not preserve 16 < N < 32 supersymmetries
A consequence of the constancy of the warp factor is that it rules out the existence of electric solutions that preserve 16 < N < 32 supersymmetries. Indeed for electric solutions X = 0. The algebraic KSE (3.12) on σ + reduces to which implies the integrability condition 1 9 On the other hand the field equation for the warp factor (3.7) yields 1 6 Q 2 = 1 ℓ 2 which is a contradiction as the radius of AdS 3 does not vanish, ℓ = 0.

N > 16 solutions with left only supersymmetry
Suppose first that the solutions only have left-hand supersymmetry. In such a case, the Lie algebras that must act transitively and effectively on the internal spaces are so(n) L , n = 9, · · · , 15 , (N = 2n) ; 28) where N < 32 as there are no AdS 3 solutions which preserve maximal supersymmetry. Furthermore solutions that preserve N = 30 supersymmetries have already been excluded in [36]. An inspection of the list of homogeneous spaces reveals that the only possibility that can occur is S 8 = Spin(9)/Spin(8) which can preserve 18 supersymmetries. However S 8 is a symmetric space and there are no invariant 1-and 4-forms. Thus Q = X = 0 which in turn implies F = 0. This leads to a contradiction as the field equation for the warp factor cannot be satisfied.

N > 16 solutions with N R = 2
For N R = 2 there are no right isometries and so all the symmetries of the internal space are generated by (t L ) 0 . The Lie algebras (t L ) 0 that act transitively and effectively on the internal spaces are where the last case is associated with the Killing superalgebra f(4). An inspection of the 8-dimensional homogeneous spaces in table 2 reveals that there are only two possibilities that can occur , Table 3. Decomposition of (3.34) into eigenspaces.
Observe that G 2 (C 4 ) = SU(4)/S(U(2) × U(2)) could have been included as a potential internal space of an AdS 3 background with N = 18 supersymmetries provided that it admitted an effective u(4) action. However this is not the case as the rank of the isotropy group S(U(2) × U(2)) is the same as that of SU(4) and so it cannot admit a modification such that U(4) acts almost effectively on G 2 (C 4 ). For confirmation, we have also excluded this case with an explicit calculation which we shall not present here. In addition AdS 3 × S 8 can also be excluded as a solution with an identical argument to the one we produced in the previous case with no right-handed supersymmetries. The remaining case is investigated below.
The closure and co-closure of X give a relation between α 1 and α 2 , and between α 3 and α 4 , but they are not essential here. Also X ∧ X = 0 implies that α 1 α 3 = 0.
On the other hand the algebraic KSE (3.12) can be written as

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where J 1 = Γ 1212 , J 2 = Γ 3412 and J 3 = Γ 7812 . We have chosen the orientation such that Γ z σ + = −J 1 J 2 J 3 σ + . The decomposition of the algebraic KSE into the eigenspaces of J 1 , J 2 and J 3 as well as the relations implied amongst the fluxes for each eigenspace can be found in table 3.
As each common eigenspace of J 1 , J 2 and J 3 has dimension two for solutions with N > 16 supersymmetries one has always to consider either one of the eigenspinors |+, −, ± and |−, +, ± or all the eigenspinors |±, ±, + and |±, ±, − . In the former case, we have that 1 36 where we have chosen without loss of generality the eigenvalue +1 of J 3 . Taking the difference of the equation above with the warp factor field equation we find that α 1 = α 2 = α 3 = α 4 = 0, and so X = 0, which is a contradiction. In the latter case we have that α 1 = α 2 = α 4 = 0 and 1 Comparing this with the warp factor field equation above again leads to a contradiction. There are no solutions with internal space Sp (2) The only right superalgebra that gives rise to 4 supersymmetries is osp(2|2) which in turn leads to an so(2) R right-handed symmetry. Therefore the Lie algebras that act both transitively and effectively on the internal spaces M 8 are so(n) L ⊕ so(2) R , n = 7, · · · , 13 , (N = 2n + 4) ; u(n) L ⊕ so(2) R , n = 4, 5, 6 , (N = 4n + 4) ; (sp(n) ⊕ sp(1)) L ⊕ so(2) R , n = 2, 3 , (N = 8n + 4) ; Up to a finite cover, the allowed homogeneous spaces are Observe that all the cases that arise, up to discrete identifications, are products of 7dimensional homogeneous spaces with S 1 . This is because it is not possible to modify 8-dimensional homogeneous spaces which admit an effective and transitive action of the (t L ) 0 Lie algebras in (3.37) to homogeneous spaces which admit an effective and transitive JHEP07(2018)178 |P 1 , P 2 , P 3 relations for the fluxes  (2) R . This is due to the fact that for all candidate homogeneous spaces that can occur the rank of the isotropy group is the same as the rank of (t L ) 0 . However a modification has been used to include the homogeneous space (1)) × S 1 . This is because an AdS 3 solution with internal space Sp(2)/Sp(1) × S 1 , is expected to preserve N = N L + N R = 10 + 4 = 14 < 16 supersymmetries as sp(2) = so(5) and so should be discarded, while with internal space (1)) × S 1 is expected to preserve N = 20 supersymmetries as it is associated to the (sp(n) ⊕ sp(1)) L ⊕ so(2) R subalgebra in (3.37) and therefore has been included. A modification has also been used to include The coset space Spin(8)/Spin(7) × S 1 can immediately be excluded as the 4-form field strength F is electric and we have shown there are no electric solutions which preserve 16 < N ≤ 32 supersymmetries. It remains to investigate the rest of the cases.
The metric on the homogeneous space Spin(7)/G 2 × S 1 can be chosen as where a description of Spin(7)/G 2 can be found in [18] whose conventions we follow, a, b > 0 are constants and ℓ 8 is an invariant frame on S 1 , dℓ 8 = 0. The most general invariant fluxes are where * 7 ϕ and ϕ are the fundamental G 2 forms and α, β, γ are constants. Furthermore, the Bianchi identity dX = 0 implies that β = 0. It is straightforward to observe that the investigation of the number of supersymmetries preserved by the algebraic KSE is exactly the same as that for the AdS 4 backgrounds with internal space Spin(7)/G 2 in [18], where instead of Γ x we have Γ 8 . In particular, the algebraic KSE can be written as

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To preserve N > 16 supersymmetries, it is required to consider the subspace in table 4 with 7 eigenspinors. The integrability condition of the remaining algebraic KSE gives while the warp factor field equation implies (3.43) Clearly, these are mutually inconsistent. So there are no AdS 3 solutions that preserve N > 16 supersymmetries with internal space Spin(7)/G 2 × S 1 .
Let us briefly summarize the homogeneous geometry of S 7 = U(4)/U(3) which is useful for our investigation of other cases below as well. There is a left-invariant frame (ℓ r , ℓ 7 ), r = 1, · · · , 6, on U(4)/U(3) such that the invariant metric can be written as where a, b > 0 are constants. The invariant forms on U(4)/U(3) are generated by the invariant 1-form ℓ 7 and the 2-form ω which can be chosen as For more details see eg [18], where the homogeneous geometry of SU(4)/SU(3) is also described.
Turning to the investigation at hand, the metric on U(4)/U(3) × S 1 can be written as where ds 2 (U(4)/U (3)) is as in (3.44), ℓ 8 is the invariant frame on S 1 , dℓ 8 = 0, and c > 0 is constant. The most general invariant fluxes Q and X that satisfy the Bianchi identities (3.4), dX = dQ = 0, are where α, β are constants. Next consider the Einstein equation along S 1 . As X does not have non-vanishing components along S 1 and the metric factorizes into that of U(4)/U(3) and S 1 , we have where R (8) 88 is the Ricci tensor along S 1 . This must vanish, R

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and ℓ 8 is an invariant frame on S 1 , dℓ 8 = 0. Note that ℓ r can be chosen up to an SO (3) transformation. This can be used to choose b without loss of generality to be diagonal. The most general invariant fluxes are As the Bianchi identities require that dQ = 0, one finds that γ r = 0. Set β = β 1 . As Q is also co-closed, we have that X ∧ X = 0 which in turn gives α 1 α 2 = 0. Suppose first that α 1 = 0. In that case, the algebraic KSE can be written as 1 6  The only right-handed superalgebra with 6 odd generators is osp(3|2). This gives rise to an so(3) R action on the internal space. Therefore the symmetry algebras that act transitively and effectively on the internal spaces are so(n) L ⊕ so(3) R , n = 6, · · · , 12 , (N = 2n + 6) ; u(n) L ⊕ so (3)   The homogeneous spaces S 6 × S 2 = Spin(7)/Spin(6) × SU(2)/U(1) and S 5 × S 3 = Spin(6)/Spin(5) × SU(2) can immediately be excluded as giving potential solutions. For S 6 × S 2 , X = Q = 0 and so the warp factor field equation cannot be satisfied. The same is the case for S 5 × S 3 after applying the Bianchi identity dQ = 0 to show that Q = 0. This homogeneous space is considered as an internal space because su(4) = so(6) and so it may give rise to a solution which preserves 18 supersymmetries. The most general invariant metric in the conventions of [18] is where (ℓ i ), i = 7, 8, is a left-invariant frame on S 2 and (ℓ r , ℓr) r,r = 1, 2, 3, is a left-invariant frame on CP 3 and a, b > 0 are constants. The invariant forms are generated by the volume form on The Bianchi identities are trivially satisfied but the field equation for Q gives the condition  Table 5. Decomposition of (3.68) KSE into eigenspaces. β = 0. For β = 0, the flux X is simply X = 1 2 α ω ∧ ω. Going to an orthonormal frame, in which the Kähler form is ω = e 12 + e 34 + e 56 , we find for the algebraic KSE (3.12) where J 1 = Γ 1234 and J 2 = Γ 1256 are mutually commuting Clifford algebra operators with eigenvalues ±1. The decomposition in terms of the common eigenspaces is summarised in table 5. A similar analysis applies to τ + , except that the right-hand side is −1/(ℓA).
To find solutions that preserve N > 16 supersymmetries, one has to choose spinors from the eigenspaces (1) in table 5. In such a case, the integrability condition of the remaining Γ z projection on the spinors is whereas the field equation for the warp factor (3.7) requires where the Clifford algebra operators J are defined as and The decomposition of the algebraic KSE (3.71) into the eigenpaces of these mutually commuting Clifford algebra operators is illustrated in table 6. A similar analysis applies to the τ + spinors with the right-hand side replaced by − 1 ℓA . For solutions to preserve N > 16 supersymmetries, we need to consider the eigenspinors given in row (1) of table 6. This gives  Table 6. Decomposition of (3.71) KSE into eigenspaces.
The geometry on S 5 as a U(3)/U(2) homogeneous space can be described in a similar way as that for S 7 = U(4)/U(3) which can be found in section 3.7.2. In particular the metric is The most general invariant 1-form Q can be written as The Bianchi identity, dQ = 0, in (3.4) implies that α = β r = 0. So, we have Q = 0. Furthermore, the Bianchi identities (3.4) also imply that dX = 0, and as Q = 0 the field equation (3.5) also implies that d * X = 0. Thus X is harmonic and represents a class in H 4 (S 5 × S 3 ). However H 4 (S 5 × S 3 ) = 0 and so X = 0. This leads to a contradiction as the field equation for the warp factor (3.7) cannot be satisfied.
Note that the above calculation rules out the existence of AdS 3 solutions with internal space S 5 × S 3 = Spin(6)/Spin(5) × SU(2) as this is a special case of the background examined above. The most general invariant metric is

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where a, b, c > 0 are constants, ℓ i , i = 1, 2, 3, 4, is a left-invariant frame on S 4 viewed as the symmetric space Spin(5)/Spin(4), and (ℓ 5 , ℓ 6 ) and (ℓ 7 , ℓ 8 ) are left-invariant frames on the two S 2 's, respectively. As there are no invariant 1-forms Q = 0. Moreover X can be written as As X ∧ X = 0, which follows from the field equation of Q, we have that α 1 α 2 = 0. If α 2 = 0, then the integrability condition of the algebraic KSE will give 1 36 (3.81) Comparing this with the field equation of the warp factor leads to a contradiction. This is also the case if instead α 1 = 0. There are no supersymmetric AdS 3 solutions with internal space S 4 × S 2 × S 2 .

S
The existence of AdS 3 solutions with G 2 /SU(3) × SU(2)/U(1) internal space can be ruled out by a cohomological argument. Observe that su(3) acts on m with the [3] R = 3 ⊕3 representation. Using this, one concludes that there are no invariant 1-forms on M 8 and so Q = 0. In such a case X is both closed and co-closed and so harmonic. However, H 4 (M 8 ) = 0 as M 8 = S 6 × S 2 and so X = 0. This in turn leads to a contradiction as the field equation for the warp factor cannot be satisfied.

N > 16 solutions with N R = 8
The right-handed superalgebras with 8 supercharges are osp(4|2), D(2, 1, α) and sl(2|2)/1 4×4 . These give rise to right-handed isometries with Lie algebras so(4) R , (so(3) ⊕ so(3)) R and su(2) R , respectively. In the latter case there can also be up to three additional central generators. As so(4) R = (so(3) ⊕ so(3)) R , it suffices to consider (so(3) ⊕ so(3)) R and su(2) R , and in the latter case include up to 3 central generators. Furthermore as N > 16, one has N L > 8. Collecting the above and using the results of table 1, the allowed algebras that act transitively and effectively on the internal space are the following.
so(n) L ⊕ (t R ) 0 , n = 5, . . . , 11 , (N = 2n + 8) ; where (t R ) 0 is either (so(3) ⊕ so(3)) R or su(2) R ⊕ c R with c R spanned by up to 3 central generators. The homogeneous spaces that can admit a transitive and an effective action by the above Lie algebras have been tabulated in table 7. A detailed examination of the homogeneous spaces that may give rise to supersymmetric AdS 3 solutions with N R = 8 reveals that the only cases that have not been investigated where α and β are constant parameters and ψ = ℓ 1234 , σ = ℓ 56 and ρ = ℓ 78 . Furthermore where γ are constants. As Q is parallel, the field equation for Q, (3.6), gives X ∧ X = 0 and so we obtain the condition that either α = 0 or β = 0. Let us proceed to investigate JHEP07(2018)178 α = 0, as the case for β = 0 can be dealt with in complete analogy. As X = βψ, the algebraic KSE (3.12) becomes The integrability condition of this is (3.87) On the other hand the warp factor field equation (3.7) gives (3.88) The last two equations are incompatible and so there are no supersymmetric solutions.

M
Here we shall consider two cases that with and where a, b, c, f > 0 are constants. The invariant forms are generated by ℓ 5 , ℓ 8 , ω = ℓ 12 +ℓ 34 and σ = ℓ 67 . The independent differential relations between the invariant forms are where we have used the description of the geometry on S 5 as in section 3.8.2. As dQ = 0, we have that Q = γℓ 8 . Furthermore after imposing dX = 0 the most general flux X is  Table 8. Decomposition of (3.94) into eigenspaces.
Squaring this, we find 1 36 where J 1 = Γ 1267 and J 2 = Γ 3467 . The decomposition of this condition on σ + into eigenspaces of J 1 and J 2 is given in table 8. Each common eigenspace of J 1 and J 2 has dimension 4. So to find solutions with N > 16 supersymmetries, we have to consider at least two of these eigenspaces. Hence this would necessarily involve either one of the eigenspinors |+, − and |−, + or both eigenspinors |±, ± . In the former case taking the difference of the condition that arises on the fluxes with the warp factor field equation 1 12 one finds that α = β = γ = 0 which is a contradiction. In the latter case, we find that αβ = 0. Using this and comparing the condition on the fluxes in table 8 with the warp factor field equation above again leads to a contradiction. There are no AdS 3 solutions that preserve N > 16 supersymmetries with internal space S 5 × S 2 × S 1 .
We have also performed the calculation for S 5 = SU(3)/SU(2) which gives rise to an X flux with additional terms to those in (3.92) because of the presence of an invariant complex (2,0) form. After some investigation, we find that again there are no solutions with N > 16 supersymmetry.

SU(3) k,l
In this context SU(3) is viewed, up to a discrete identification, as a homogeneous space with isotropy group SU(2) × U(1) and almost effective transitive group SU As we have mentioned the geometry of such cosets is more restrictive than that of SU(3) viewed as the homogeneous space SU(3)/{e}. Thus it suffices to investigate whether SU(3) is a solution. As SU(3) does not admit closed 1-forms, Q = 0. In such case X is harmonic. However H 4 (SU(3), R) = 0 and so X = 0. This leads to a contradiction as the warp factor field equation cannot be satisfied.

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3.9.4 N k,l,m × S 1 = SU(2)×SU(3)×U(1) ∆ k,l,m (U(1) 2 )·(1×SU(2)) × S 1 Let us denote the left-invariant frame along S 1 with ℓ 8 , dℓ 8 = 0. N k,l,m can be thought of as a modification of N k,l and so for the analysis that follows we can use the description of the geometry of N k,l in appendix C. In particular, the most general Q flux is (3.97) As dQ = 0, we deduce that γ 2 = 0 and set γ 1 = γ. The most general invariant metric is where (ℓ r ,l r , ℓ 5 , ℓ 6 , ℓ 7 ), r, s = 1, 2, is a left-invariant frame on N k,l , ℓ 8 is a left-invariant frame on S 1 and a, b, c, f > 0 are constants. Next X can be chosen as where α 1 , α 2 , α 3 , α 4 are constants. As dX = 0, one deduces that α 3 = α 4 = 0. Choosing an orthonormal frame as the algebraic KSE can be written as The form of this KSE is the same as that in (3.93). A similar analysis again reveals that there are no solutions that preserve N > 16 supersymmetries.

N > 16 solutions with N R = 10
The only superalgebra that gives rise to ten right-handed supersymmetries is osp(5|2) with (t R ) 0 = so(5). As we are investigating backgrounds with N > 16 and we have chosen that N L ≥ N R , we conclude that 10 ≤ N L < 22. Using this and the results of table 1, the allowed algebras that can act transitively and effectively on the internal spaces are so(n) L ⊕ so (5)   where α, β are constants. The field equation for Q, (3.6), yields the condition that either α = 0 or β = 0. Without loss of generality, we take β = 0. Substituting X into the algebraic KSE (3.12), one finds and hence obtains as an integrability condition. However, the warp factor field equation (3.7) implies that (3.108) Thus there is a contradiction and there are no such supersymmetric solutions.

N > 16 solutions with N R ≥ 12
Imposing the restriction that N L ≥ N R , it is easy to see that there are no homogeneous spaces that admit a transitive and effective t 0 action. This follows from a detailed examination of the classification results of [28]- [32] as well as their modifications.

Field equations and Bianchi identities for N > 16
The bosonic fields of (massive) IIA supergravity are the metric ds 2 , a 4-form field strength G, a 3-form field strength H, a 2-form field strength F , the dilaton Φ and the cosmological constant dressed with the dilaton S. Following the description of warped AdS 3 backgrounds in [12], we write the fields as

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where we have used a null-orthonormal frame (e + , e − , e i ), i = 1, . . . , 7, defined as in (3.3) and ds 2 (M 7 ) = δ ij e i e j . The fields Φ, W, S and the warp factor A are functions, Y is a 1-form, F is a 2-form, Z is a 3-form and X is a 4-form on M 7 . As the 2-form field strength F is purely magnetic we have denoted the field and its component on M 7 by the same symbol. This is also the case for Φ and S. The dependence of the fields on the AdS 3 coordinates is hidden in the definition of the frame e + , e − and e z . The components of the fields in this frame depend only on the coordinates of M 7 .
As we have demonstrated in 11-dimensional supergravity, the description for the fields simplifies considerably for AdS 3 backgrounds preserving N > 16 supersymmetries. In particular, a similar argument to the one presented for 11-dimensional backgrounds gives that the warp factor A is constant. The proof of this is very similar to that given in 11dimensions and so we shall not repeat the analysis. Furthermore it is a consequence of the homogeneity theorem and Bianchi identities of the theory that the scalars Φ, S and W are constant.
To focus the analysis on the IIA AdS 3 backgrounds that preserve N > 16 supersymmetries, we shall impose these conditions on the Bianchi identities, field equations and Killing spinor equations. The general formulae can be found in [12]. In particular taking A, W, Φ and S to be constant the Bianchi identities can be simplified as A consequence of this is that either S = 0 or W = 0. Furthermore, the field equations of the form fluxes can be written as respectively. As M 7 is compact without boundary observe that d * 7 Y = −Z∧X implies that To see this, first observe that homogeneity implies that * 7 (Z ∧ X) is constant. On the other hand the integral of Z ∧ X over M 8 is the constant * 7 (Z ∧ X) times the volume of M 8 . As the integral of Z ∧ X is zero, this constant must vanish giving (4.4). The dilaton field equation is The Einstein equation along AdS 3 and M 7 implies

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where ∇ and R (7) ij denote the Levi-Civita connection and the Ricci tensor of M 7 , respectively. The former condition is the warp factor field equation.

The Killing spinor equations
The solutions to the KSEs of (massive) IIA along AdS 3 may be written as in (3.9), although now σ ± and τ ± are Spin(9, 1) Majorana spinors which satisfy the lightcone projections Γ ± σ ± = Γ ± τ ± = 0 and only depend on the coordinates of M 7 . These are subject to the gravitino KSEs the dilatino KSEs (4.8) and the algebraic KSEs If M 7 is compact without boundary, one can demonstrate that σ + = const , τ + = const , σ + , τ + = 0 , τ + , Γ iz σ + = 0 . (4.11) As in eleven dimensions, the last condition is essential to establish that the warp factor A is constant for IIA AdS 3 backgrounds preserving N > 16 supersymmetries with compact without boundary internal space M 7 .

N > 16 solutions with left only supersymmetry
AdS 3 backgrounds admit the same Killing superalgebras in 11-dimensional, IIA and IIB supergravities. As a result the Lie algebras t 0 that must act transitively and effectively on the internal spaces of IIA and IIB AdS 3 backgrounds can be read off those found in the 11-dimensional analysis. So for N R = 0, these are given in (3.28). An inspection of the 7-dimensional homogeneous spaces in table 9 reveals that there are no N > 16 supersymmetric AdS 3 backgrounds with N R = 0. (2) Sp(1) diffeomorphic to S 7 (5)

S 7 = U(4)/U(3)
The geometry of S 7 = U(4)/U(3) has been summarized in the beginning of section 3.7.2. The metric is given in (3.44). The invariant forms are generated by the 1-form ℓ 7 and 2-form JHEP07(2018)178 ω as in (3.45), dℓ 7 = ω. Given these data, the most general invariant fluxes can be chosen as As the Bianchi identities require that dZ = 0, we have β = 0. Furthermore the remaining Bianchi identities imply 14) and the field equations for the fluxes give Suppose first that S = 0. Then W = 0 which in turn gives α = γ = δ = 0. As both Z = Y = 0, the dilaton field equation in (4.5) implies that the rest of the fluxes vanish which in turn leads to a contradiction as the warp factor field equation in (4.6) cannot be satisfied.
Next suppose that S = 0. Then α γ = 0. Take that W = 0 otherwise there will be a contradiction as described for S = 0 above. If γ = 0, this will imply that δ = 0 and so again the dilaton field equation in (4.5) will imply that the rest of the fluxes must vanish.
It remains to investigate the case α = 0. The dilatino KSE (4.8) and algebraic KSE (4.9) become Eliminating the flux F , one finds The integrability condition gives (4.18) Comparing this with the field equation for the warp factor (4.6) leads to a contradiction. There are no supersymmetric solutions.
The geometry of S 7 = (Sp(2) × Sp(1))/Sp(1) × Sp(1) has been described in section 3. The metric on the internal space can be chosen as ds 2 (M 7 ) = ds 2 (S 4 ) + ds 2 (S 3 ) = a δ ij ℓ i ℓ j + b rs ℓ r ℓ s , (4.19) where (ℓ i ), i = 4, . . . , 7, is a left-invariant frame on S 4 and (ℓ r ), r = 1, 2, 3 is a left-invariant frame on S 3 = SU(2), a > 0 is a constant and (b rs ) a positive definite 3 × 3 symmetric matrix. Note that where α, β, γ r and δ r , r = 1, 2, 3, are constants. First observe that Z ∧ X = 0 implies that αβ = 0. Next suppose that S = 0. It follows from the Bianchi identities (4.2) that Z = 0 as dF = 0. In addition, the Bianchi identities (4.2) give Next the dilaton field equation in (4.5) implies that S = X = F = 0 which is a contradiction to the assumption that S = 0. So let us now consider that S = 0. Again αβ = 0 and so either Z = 0 or X = 0. Let us first take Z = 0 and X = 0. In such a case the field equation for X in (4.3) gives W = 0. If W = 0, the Bianchi identities (4.2) will imply that Y = 0. This in turn leads to a contradiction as the field equation for the dilaton in (4.5) implies that X = 0.
Suppose now that both Z = X = 0. As S = 0 as well, the dilatino and algebraic KSEs can be re-written as in (4.16). This in turn gives (4.17) that leads to the integrability condition (4.18). Substituting this into the field equation for the warp factor in (4.6) and after eliminating Y 2 , one finds a contradiction.
It remains to investigate the case that Z = 0 and X = 0. First the Bianchi identity for Y (4.2) implies that Then field equation for F , d * 7 F = −W * 7 Y , together with (4.23) imply that Next turn to the Einstein equation along S 4 . As X = 0 and the fields Z, F and Y have non-vanishing components only along S 3 , we find that

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Using (4.24), one can show that R ij = 0. This is a contradiction as R (7) ij is the Ricci tensor of the S 4 subspace which is required to be strictly positive. Therefore we conclude that there are no supersymmetric IIA AdS 3 solutions with internal space S 4 × S 3 .

M
This homogeneous space admits invariant 3-and 4-forms which are the fundamental G 2 forms ϕ and * 7 ϕ. However the 3-form ϕ is not closed and so Z = 0. As there are no invariant 1-forms and 2-forms Y = F = 0. In such a case the dilaton field equation in (4.5) implies that W = S = X = 0. In turn, the warp factor field equation in (4.6) becomes inconsistent.

N > 16 solutions with N R = 4
The Lie algebras that must act both effectively and transitively on the internal space M 7 are the same as those found in D = 11 supergravity and given in (3.37). After an inspection of the 7-dimensional homogeneous spaces those admitting an effective and transitive action by such Lie algebras are the following  The only non-vanishing k-form flux, k > 0, allowed is Y = αℓ 7 , where ℓ 7 is a left-invariant frame along S 1 and α is constant. The dilatino KSE (4.8) can be re-written as which leads to the integrability condition As a result W = S = Y = 0. This leads to an inconsistency as the warp field equation (4.6) cannot be satisfied. There are no supersymmetric IIA AdS 3 backgrounds with Spin(7)/Spin(6) × S 1 internal space.
The differential algebra of a left-invariant frame on M 7 modulo terms in su(3) ∧ m which involve the canonical connection is where λ r is a complex frame,λ r = λr, on S 6 and ℓ 7 is a left-invariant frame on S 1 . The invariant forms on S 6 are the 2-form The most general invariant metric on M 7 is where a, b > 0 are constants. Moreover the most general invariant fluxes are where α's, β, γ and δ are constants and we have used that dZ = 0. As Z ∧ X = 0, we have that α 2 β = 0. Furthermore as dF = SZ, we have that Let us first consider the case that S = 0. This implies that W = 0. As either α 2 = 0 or Z = 0, let us investigate first the case that Z = 0. In such a case the Bianchi identities (4.2) and the field equations (4.3) imply that X is harmonic and as H 4 (S 6 × S 1 ) = 0, we have X = 0. Using this, we also find that F is harmonic and so as H 2 (S 6 × S 1 ) = 0, F = 0. Next the dilatino KSE (4.8) becomes (5S + / Y Γ z )σ + = 0 which in turn implies that 25S 2 + Y 2 = 0. This is a contradiction as S = 0. Thus there are no such supersymmetric AdS 3 backgrounds.
Next suppose that α 2 = 0. The field equation d * 7 X = Z ∧ Y gives Thus again S = 0 which is a contradiction. So to find solutions, we have to set S = 0. The Bianchi identity dF = 0 gives F = 0. Furthermore the field equation d * 7 Z = 0 gives Z = 0. We also have from the field equations (4.3) that W * 7 Y = 0. If we choose Y = 0 and as Z vanishes as well, Z = 0 , the dilaton field equation in (4.5) implies that the rest of the fields vanish which contradicts the warp factor field equation. So let us take W = 0. In such a case X is harmonic and so X = 0. This is also the case for Z and so Z = 0. In turn the dilatino KSE implies that / Y σ + = 0 which gives Y = 0. Thus all the fields vanish leading to a contradiction with the warp factor field equation. There are no supersymmetric AdS 3 solutions with internal space The geometry of the homogeneous space S 5 = U(3)/U(2) has been described in section 3.8.2. Using this, the most general invariant metric on M 7 can be written as The most general invariant fluxes are The Bianchi identity dZ = 0 implies that β 1 = β 2 = 0. So Z = 0. The remaining Bianchi identities imply that To continue first take S = 0. In such a case W = 0 and so δ = 0. As both Y = Z = 0, the dilaton field equation implies that S = X = F = 0. This is a contradiction to the assumption that S = 0.
Therefore we have to set S = 0. Furthermore W = 0 as otherwise Y = Z = 0 and the dilaton field equation will imply that all other fluxes must vanish. This in turn leads to a contradiction as the warp factor field equation cannot be satisfied. As W = 0, we have γ 2 = 0. Then the field equation for the fluxes (4.3) give * 7 X ∧ F = 0 which in turn implies that α 2 γ 1 = 0 , γ 1 α 1 = 0 . (4.44)
As we have established that Z = X = 0, we can follow the analysis of the KSEs in However this is the Ricci tensor of S 1 and hence vanishes. This in turn gives S = 0 which is a contradiction to our assumption that S = 0.

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Thus we have to set S = 0. The Bianchi identities (4.2) and the field equation (4.4) give that αγ = 0 , W β = 0 , (4.50) and the field equations (4.3) of the form field strengths imply that Therefore if W = 0, we will have F = Y = X = 0. Furthermore as R 77 = 0, the Einstein equation reveals that Z = 0. Then the dilaton field equation implies that W = 0 which is a contradiction to our assumption that W = 0.
It remains to investigate solutions with W = S = 0. Notice that we should have that Z = 0, or equivalently γ = 0, as otherwise the Einstein equation R (7) 77 = 0 will imply that X = Y = F = 0 and so the warp factor field equation cannot be satisfied leading to a contradiction. Thus Z = 0 and as αγ = 0, we have that X = 0. Inserting X = S = W = 0 into the dilatino and algebraic KSEs we find that they can be rewritten as where J 1 = Γ 56 Γ z Γ 11 and J 2 = Γ 7 . As each common eigenspace of J 1 and J 2 has dimension 4 to find solutions with N > 16 supersymmetries we have to choose at least two of these eigenspaces. One can after some calculation verify that for all possible pairs of eigenspaces As N k,l,m is a modification of N k,l , see [33,34], we can use the local description of the geometry of the latter in appendix C to describe the former. In particular the metric can be written as where (ℓ r ,l r , ℓ 5 , ℓ 6 , ℓ 7 ) is a left-invariant frame and a, b, c > 0 constants. From the results of appendix C, one can deduce that there are no closed 3-forms and so Z = 0. The remaining invariant form field strengths are where α 1 , α 2 , γ 1 , γ 2 , δ are constants. The Bianchi identities (4.2) imply that Clearly from (4.55) either S = 0 or W = 0. Suppose that S = 0. Then W = 0 and from the rest of the conditions arising in the Bianchi identities Y = 0. As both Y = Z = 0, the dilaton field equation implies that S = F = X = W = 0 which is a contradiction to our assumption that S = 0. Therefore we set S = 0. We also take W = 0 as otherwise the same argument presented above leads again into a contradiction. As S = 0, the last condition in (4.56) implies that α 2 γ 1 = 0. However γ 1 cannot vanish. Indeed if γ 1 = 0, then (4.55) will lead to Y = 0. Since Y = Z = 0, the dilaton field equation in (4.5) will imply that the rest of the fields vanish. In turn the warp factor field equation (4.6) cannot be satisfied. Thus we have to set γ 1 = 0. In such case α 2 = 0 and the first equation in (4.56) gives α 1 = 0. As both α 1 = α 2 = 0, X = 0.
We have shown that the remaining non-vanishing fields are W , Y and F . To continue consider the dilatino and algebraic KSEs. These can be written as in (4.16). Then a similar argument as that presented in section 4.4.1 leads to a contradiction. There are no supersymmetric AdS 3 solutions with internal space N k,l,m . The bosonic fields of IIB supergravity are a metric ds 2 , a complex 1-form field strength P , a complex 3-form field strength G and a real self-dual 5-form field strength F . For the investigation of IIB AdS 3 × w M 7 backgrounds that follows, we shall use the analysis presented in [11] where all the necessary formulae can be found. As we are focusing on backgrounds that preserve N > 16 supersymmetries, the homogeneity theorem implies that the scalars are constants and so P = 0. We shall use this from the beginning to simplify the relevant field equations, Bianchi identities and KSEs. Imposing the symmetries of the AdS 3 subspace on the fields, one finds that the non-vanishing fields are where a null-orthonormal frame (e + , e − , e z , e i ), i = 1, . . . , 7, is defined as (3.3) and ds 2 (M 7 ) = δ ij e i e j . Y is a real 2-form, X is a complex function and H a complex 3-form on M 7 . The dependence of the fields on AdS 3 coordinates is hidden in the definition of the frame (e + , e − , e z ). All the components of the fields in this frame depend on the coordinates of M 7 .

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and C is the charge conjugation matrix followed by complex conjugation. In the expressions above we have used that P = 0 and that A is constant. As in 11-dimensional and IIA supergravities, the IIB AdS 3 backgrounds preserve an even number of supersymmetries.
5.3 N > 16 solutions with N R = 0 and N R = 2 The existence of solutions that preserve strictly 28 and 30 supersymmetries has already been excluded in [37]. As in the IIA case, IIB N > 16 supersymmetric AdS 3 solutions with N R = 0 can also be ruled out because there are no 7-dimensional homogeneous manifolds that admit a transitive and effective action of the t 0 subalgebra of the expected symmetry superalgebra of such backgrounds. So we shall begin with backgrounds with N R = 2. The homogeneous spaces are as those in IIA and are given in (4.12). The homogeneous space S 7 = Spin(8)/Spin (7) can be ruled out immediately. This symmetric space does not admit invariant 2-and 3-forms. Therefore Y = H = 0. Then a field equation in (5.3) implies that X = 0 as well and so the warp factor field equation in (5.4) cannot be satisfied.
Similarly S 7 = Spin(7)/G 2 can also be ruled out as it does not admit an invariant closed 3-form and so H = 0. Also it does not admit an invariant 2-form either and so Y = 0. Then because of the field equations in (5.3), one deduces that X = 0 and so the warp factor field equation in (5.4) becomes inconsistent.

S 7 = U(4)/U(3)
Following the description for the geometry of the homogeneous space U(4)/U(3) as in section 3.7.2, the most general allowed fluxes are Y = α ω , H = β ℓ 7 ∧ ω . (5.9) The Bianchi identity dH = 0 requires that β = 0. In turn a field equation in (5.3) implies that X = 0. Substituting this back into the Bianchi identities (5.2), one finds that Y is harmonic and so it must vanish. As all fluxes vanish, the warp factor field equation in (5.4) becomes inconsistent. There are no AdS 3 solutions with internal space U(4)/U(3).

S
The geometry of this homogeneous space described in section 3.7.3 reveals that there are no invariant 2-forms and closed 3-forms. As a result Y = H = 0. The field equations (5.3) imply that X = 0 as well. Therefore there are no solutions as the warp factor field equation cannot be satisfied.

S 4 × S 3 = Spin(5)/Spin(4) × SU(2)
The geometry of this homogeneous space has been described in section 4.4.3. The most general fluxes can be chosen as

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where r, s, t = 1, 2, 3. As Y is both closed and co-closed and H 2 (S 4 × S 3 ) = 0, we deduce that Y = 0 and so a Bianchi identity in (5.2) implies that Xβ − Xβ = 0 . The homogeneous internal spaces are given in (4.26). It is straightforward to show that S 6 × S 1 = Spin(7)Spin(6) × S 1 is not a solution as Y = H = X = 0 which contradicts the warp factor field equation.
In the notation of section 4.5.2 the metric can be chosen as ds 2 = aδ rs λ r λs + b(ℓ 7 ) 2 and the most general invariant Y and H forms are The Bianchi identity dH = 0 implies that β 2 = β 3 = 0. In what follows set β 1 = β. It follows that the Bianchi identity for Y in (5.2) gives Next turn to the dilatino KSE. Setting λ r = ℓ 2r−1 + iℓ 2r , it can be written as β a 3 2 where J 1 = Γ z Γ 136 , J 2 = Γ z Γ 235 and J 3 = Γ z Γ 246 . These are commuting hermitian Clifford algebra operators with eigenvalues ±1. For all choices of eigenspaces either X = 0 or X = ±4β/a 3 2 . Substituting this into the first field equation in (5.3), we find that β = 0. Therefore X = 0 as well. Then (5.13) and (5.14) imply that Y = H = 0. Thus the warp factor field equation in (5.4) cannot be satisfied. Therefore there are no supersymmetric AdS 3 solutions with internal space G 2 /SU(3) × S 1 .
The geometry of this homogeneous space has already been described in section 4.6.1 and the metric is given in (4.39). The most general fluxes can be chosen as Y = α 1 ω + α 2 σ , H = β 1 ℓ 5 ∧ ω + β 2 ℓ 5 ∧ σ . The allowed homogeneous internal spaces are given in (4.45). All the cases have already been investigated apart from those with internal space S 4 × S 2 × S 1 = Spin(5)/Spin(4) × SU(2)/U(1) × S 1 and N k,l,m which we shall examine next. which together with a field equations in (5.3) imply that H = X = 0. The only nonvanishing field is Y . However as in the previous case after evaluating the Einstein equation along S 2 , one finds a similar relation to (5.19). This is a contradiction as the Ricci tensor of S 2 is strictly positive and so there are no AdS 3 solutions with internal space Spin (5) Table 10. Decomposition of (5.24) into eigenspaces.
5.6.2 N k,l,m = (SU(2) × SU(3) × U(1))/∆ k,l,m (U(1) × U(1)) · (1 × SU (2)) The metric can be chosen as in (4.53). From the results of appendix C, one can deduce that there are no closed invariant 3-forms and so H = 0. The field equations (5.3) imply that X = 0 as well. The most general 2-form Y is The Bianchi identities imply that Y must be harmonic. Observe that dY = 0. The co-closure condition implies that where dvol = 1 2 ω 2 1 ∧ ω 2 ∧ ℓ 7 . Next the algebraic KSE (5.7) can be written as As the common eigenspaces of J 1 , J 2 , J 3 have dimension 2 to find solutions preserving N > 16 supersymmetries, one needs to choose at least three such eigenspaces. The eigenspaces that lead to the relation α 2 c = ± 1 ℓA for the fluxes can be ruled out because of the warp factor field equation. Therefore we have to choose three eigenspaces from the remaining cases in table 10. For every choice of a pair of relations either α 1 or α 2 vanishes. Then the co-closure condition (5.23) implies that Y = 0. There are no AdS 3 supersymmetric solutions preserving N > 16 supersymmetries.

B Coset spaces with non-semisimple transitive groups
We have already argued that if t 0 is simple then the internal spaces G/H, with Lie G = t 0 can be identified from the classification results of [28]- [32]. This is also the case for t 0 semisimple provided that in addition one considers modifications to the homogeneous spaces as described in section 2.3.
Here, we shall describe the structure of homogeneous G/H spaces for G a compact but not semisimple Lie group. As we consider homogeneous spaces up to discrete identifications, we shall perform the calculation in terms of Lie algebras. The Lie algebra of G can be written as Lie G = p ⊕ a, where p is semisimple and a is abelian. Suppose now that we have a G/H coset space, where H is a compact subgroup of G. Then Lie (H) = q ⊕ b, where q is a semisimple subalgebra and b is abelian. Let us now focus on the inclusion i : Lie (H) → Lie G. Consider the projections p 1 : Lie G → p and p 2 : Lie G → a. Then we have that p 2 • i| q = 0 as there are no non-trivial Lie algebra homomorphisms from a semisimple Lie algebra into an abelian one. Thus p 1 • i| q is an inclusion. Furthermore p 1 • i| Ker(p 2 •i| b ) is also an inclusion. Therefore q ⊕ Ker(p 2 • i| b ) is a subalgebra of p. As p is semisimple and for applications here dim G/H ≤ 8, there is a classification of all coset spaces P/T with Lie P = p and Lie (T ) = q ⊕ Ker p 2 • i| b .
Next consider b 1 = b/Ker (p 2 • i| b ). Suppose first that i(b 1 ) is contained in both p and a, then up to a discrete identification a coset space W/X with Lie W = p ⊕ i(b) and

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Lie X = q⊕b is a modification of P/T with an abelian group which has Lie algebra p 2 •i(b). Furthermore the generators of a/i(b) commute with Lie X and of course are not in the image of i. As a result G/H up to a discrete identification can be written as W/X ×T k , where k = dim a/i(b), ie up to a discrete identification G/H is the product of an abelian modification of a coset space with semisimple transitive group and of an abelian group. On the other hand if i(b) is all contained in a, then G/H up to a discrete identification is a product P/T ×T m , where m = dim i(a). In the classification of AdS 3 backgrounds, we use the above results to describe the geometry of the internal spaces whenever t 0 is not a semisimple Lie algebra.

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