On supersymmetric AdS6 solutions in 10 and 11 dimensions

We prove a non-existence theorem for smooth, supersymmetric, warped AdS6 solutions with connected, compact without boundary internal space in D = 11 and (massive) IIA supergravities. In IIB supergravity we show that if such AdS6 solutions exist, then the NSNS and RR 3-form fluxes must be linearly independent and certain spinor bilinears must be appropriately restricted. Moreover we demonstrate that the internal space admits an so3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{so}(3) $$\end{document} action which leaves all the fields invariant and for smooth solutions the principal orbits must have co-dimension two. We also describe the topology and geometry of internal spaces that admit such a so3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{so}(3) $$\end{document} action and show that there are no solutions for which the internal space has topology F × S2, where F is an oriented surface.


Introduction
AdS spaces have found widespread applications first as compactifications of supergravity theories and more recently as a tool to explore superconformal field theories in the context of the AdS/CFT correspondence, for reviews see [1,2]. As AdS/CFT provides a correspondence between AdS n backgrounds of 10 and 11-dimensional supergravity theories with conformal field theories in (n − 1)-dimensions, properties of superconformal theories can be investigated in the context of supergravity theories. This has given a new impetus to understanding the AdS backgrounds that preserve a fraction of the spacetime supersymmetry.
In this context, and in particular for AdS 6 backgrounds, several solutions have been found in [3][4][5][6][7][8][9] and explored in the context of AdS 6 /CFT 5 , see eg [10][11][12][13][14]. Furthermore it has been shown in [15][16][17] that AdS 6 backgrounds preserve either 16 or 32 supersymmetries in all 10-and 11-dimensional supergravity theories. Moreover it is known for sometime that these theories do not admit maximally supersymmetric backgrounds [18] which are strictly locally isometric to AdS 6 . Some additional non-existence results have been established in [6,19] for smooth AdS 6 solutions preserving 16 supersymmetries in 11-dimensional and IIA supergravities under the assumption that the Killing spinors factorize into Killing spinors of AdS and Killing spinors on the internal space and some additional restrictions on the internal spaces.
It has been demonstrated in [20] that the Killing spinors of AdS backgrounds do not factorize into Killing spinors of AdS and Killing spinors on the internal space, and so requiring factorization is an additional assumption. Related to this some care is required in establishing no-go theorems for AdS backgrounds as AdS n spaces can be written as warped products of AdS k , k < n and so AdS n backgrounds can be re-interpreted as warped AdS k solutions [21]- [24]. For example the AdS 7 maximally supersymmetric background of 11dimensional supergravity can be re-interpreted as a warped maximally supersymmetric AdS 6 solution. To exclude such a scenario, we put some global assumptions on the internal spaces of AdS 6 backgrounds that we shall describe below.
In this paper, we shall prove a non-existence theorem for smooth warped AdS 6 backgrounds in 11-dimensional and (massive) IIA imposing only as assumptions 1 that the internal space is closed, 2 ie it is compact and without boundary. As this theorem for maximally supersymmetric backgrounds has already been demonstrated, the main focus is to establish the result for AdS 6 solutions preserving 16 supersymmetries.
Furthermore we shall demonstrate some non-existence results for smooth closed AdS 6 backgrounds in IIB supergravity provided some additional assumptions are made. In turn these assumptions can be viewed as necessary conditions for the existence of IIB AdS 6 solutions. In particular, we demonstrate that if smooth closed AdS 6 IIB backgrounds exist, the NSNS and RR 3-form fluxes must be linearly independent. This rules out the existence of smooth closed solutions with only NSNS or RR 3-form fluxes and all their JHEP12(2017)009 SL(2, R) duals. We also find that this linear independence condition on the 3-form fluxes is met provided some spinor bilinears are appropriately restricted. In particular, a certain (twisted) scalar bilinear must vanish, another (twisted) scalar bilinear must be somewhere vanishing on the internal space and a 2-form bilinear must not be zero, and must have rank at most two. The full set of conditions is summarized in section 4.4. Another necessary condition is that closed AdS 6 IIB backgrounds must always have active scalars.
Next we show that the internal spaces of all AdS 6 IIB backgrounds admit a non-trivial 3 so(3) action which leaves all the fields invariant. If the infinitesimal so(3) action can be integrated to an effective SU (2) or SO(3) action and AdS 6 IIB backgrounds are smooth and closed, then the principal orbits must be of co-dimension 2. This rules out all solutions for which so(3) acts on the internal space with co-dimension 1 principal orbits. The diffeomorphic type of internal manifolds in the oriented case can be specified by utilizing the classification results of [25]. These include the spin manifolds where M (F ) is the unique spin oriented 2-sphere bundle over a surface F , see also appendix C. In particular if F is oriented, then M (F ) = F × S 2 . Next we demonstrate with a partial integration argument that there are no smooth closed AdS 6 solutions that have internal spaces with topology F × S 2 .
The methodology that will be followed has been developed in [15][16][17] for investigating all AdS backgrounds. In particular, it is assumed that the metric and fluxes are invariant under the isometries of the AdS space but otherwise there are no additional assumptions like an ansatz on the form of the Killing spinors. Then the supergravity KSEs are integrated along the AdS space which gives the dependence of the Killing spinor on the AdS coordinates as well as a set of KSEs along the internal space. These are associated with the gravitino and other algebraic KSEs of the original supergravity theory, and in addition there is an algebraic Killing spinor equation which arises as an integrability condition associated with the solution of the KSEs along the AdS. The existence of smooth closed solutions is then explored by applying techniques, like that of the Hopf maximum principle, and taking into account the assumptions made on the topology of the internal space. It turns out that this methodology is sufficient to prove the non-existence of supersymmetric AdS 6 backgrounds in 11-dimensional and (massive) IIA supergravities. However in IIB, we have not been able to establish such a result. Instead, we have given some necessary conditions for the existence of smooth closed AdS 6 solutions, and specified their diffeomorphic type using classification results of [25] for 4-dimensional manifolds admitting an effective This paper has been organized as follows. In sections 2 and 3, we establish the nonexistence of smooth closed AdS 6 backgrounds for 11-dimensional and (massive) IIA supergravities, respectively. In section 4, we investigate the existence of smooth closed AdS 6 solutions in IIB supergravity. In appendix A, we investigate the isometries of the internal space and state our conventions. In appendix B, we present some Fierz identities that JHEP12(2017)009 have been used in our derivations. In appendix C, we summarize the results of [25] on the structure of 4-manifolds admitting a non-abelian group action and in appendix D, we present various formulae for the spinor bilinears of IIB AdS 6 backgrounds.

Warped AdS 6 backgrounds in D=11
We begin by briefly summarizing the general structure of warped AdS 6 solutions in 11dimensional supergravity, as determined in [15], whose conventions we shall follow throughout this section. The metric and 4-form which are invariant under the isometries of AdS 6 are given by where ds 2 (M 5 ) is the metric on the internal space M 5 , and we have written the solution as a near-horizon geometry [26], with The coordinates (u, r, z, x 1 , x 2 , x 3 ) are those of the AdS 6 space, A is the warp factor which is a function on M 5 , and X is a closed 4-form on M 5 . The metric ds 2 (M ), A and X depend only on the coordinates of M 5 , ℓ is the radius of AdS 6 . The 11-dimensional Einstein equation implies that the warp factor satisfies the equation where D is the Levi-Civita connection on M 5 . The remaining components of the Einstein and gauge field equations are listed in [15], however we shall only require (2.3) for the analysis that follows. In particular, (2.3) implies that A is everywhere non-vanishing on M 5 , on assuming that M 5 is connected and all fields are smooth.

The Killing spinors
The KSEs of 11-dimensional supergravity can be solved along AdS 6 yielding where the spinors σ ± and τ ± are Majorana Spin(10, 1) spinors that depend only on the coordinates y I of M 5 and satisfy the light-cone projections

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The gamma matrices have been adapted to the spacetime frame e + = du , e − = dr + rh , e z = Adz , e a = Ae z/ℓ dx a , e i = e i I dy I , (2.6) where ds 2 (M 5 ) = δ ij e i I e j J dy I dy J . The expression for the Killing spinor ǫ (2.4) is derived after intergrating the KSEs along AdS 6 . In particular notice that we do not assume that the Killing spinor ǫ factorizes as a Killing spinor on AdS 6 and a Killing spinor on M 5 which is an additional assumption on the form of the Killing spinors [15,20].
The remaining independent Killing spinor equations (KSEs) are and The (2.7) KSEs are a suitable restriction of the gravitino KSE of 11-dimensional supergravity on M 5 while the (2.8) conditions arise during the integration process of the KSEs along AdS 6 . Notice that the algebraic KSEs (2.8) imply that σ + and τ + cannot be linearly dependent. In fact, we shall later demonstrate that they must be orthogonal. For our Clifford algebra conventions see [15].

Counting the Killing spinors
The counting of supersymmetries of warped AdS 6 backgrounds will be given in 11dimensional supergravity. A similar counting applies to (massive) IIA and IIB supergravities and so it will not be repeated below for these theories.
As a result the number of Killing spinors of AdS 6 backgrounds is a multiple of 16. Thus if there are AdS 6 solutions of supergravity theories, they will either preserve 16 or 32 supersymmetries. It has been shown sometime ago that 11-dimensional, (massive) IIA and IIB supergravities do not admit maximally supersymmetric AdS 6 solutions [18]. As a result, it remains to investigate the AdS 6 backgrounds that preserve 16 supersymmetries.

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2.3 Proof of the main theorem in D=11 supergravity 2.3.1 Orthogonality of τ + and σ + spinors Before we proceed with the proof of the main theorem, we shall first establish the orthogonality of τ + and σ + Killing spinors. It will be convenient to define where⋆ denotes the Hodge dual on M 5 , so W is a 1-form. To proceed, we set and using (2.7), we find where Γ (4) denotes the product of the gamma matrices in the 4 directions of AdS 6 spanned by e z and e a , with the convention that Using the algebraic KSE (2.8), (2.12) can be rewritten as and the gravitino KSE (2.7) implies that On taking the divergence of (2.14), and utilizing (2.15), we find A maximum principle argument then implies that Note that for the application of the maximum principle it is sufficient to assume that the backgrounds are smooth and M 5 is closed. As Λ 2 is constant, (2.12) and (2.14) imply and respectively.

A non-existence theorem
Next, note that as a consequence of how the σ + , τ + spinors are generated from each other as described in the previous subsection, it follows that Γ (4) σ + = τ ′ + , and hence (2.21) implies that So, on substituting Λ = σ + into (2.18), we find and so the warp factor is constant. The gravitino KSE (2.7) also implies that 24) and hence this expression together with (2.22) also give that and hence X = 0. So, we have proven that for AdS 6 solutions, one has However, as A is constant and X = 0, the field equation (2.3) for the warp factor A does not admit a solution. This proves the theorem. To summarize, combining the result 4 of [18] with the proof described above, one concludes that there are no smooth closed warped AdS 6 solutions of 11-dimensional supergravity.
One question that arises is whether some of the assumptions we have made can be lifted. First our result can be generalized somewhat. For this notice that we need the smoothness assumption as well as the restrictions on the internal space M 5 to demonstrate JHEP12(2017)009 that the only solution of (2.16) is that Λ is of constant length (2.17). Thus we can replace all these assumptions with (2.17).
The theorem is not valid if one removes all conditions on the fields and the internal space. This is because locally AdS n+1 spaces can be written as warped products of AdS n spaces. 5 As a result the AdS 7 × S 4 maximally supersymmetric solution of 11-dimensional supergravity can be written locally as AdS 6 × w (R × S 4 ) and so it can be interpreted as an AdS 6 solution. This demonstrates that 11-dimensional supergravity admits maximally supersymmetric AdS 6 solutions. However such solutions are not a contradiction to our theorem as their internal space is not compact, see also [20].

Warped AdS 6 backgrounds in IIA supergravity
The non-vanishing fields of (massive) IIA supergravity for warped AdS 6 × w M 4 backgrounds in the conventions of [17] are where A is the warp factor, Φ is the dilaton, S is related to the cosmological constant, F and G are the 2-form and 4-form R-R field strengths correspondingly, and H is a 3-form the NS-NS 3-form field strength. A, Φ and S are functions on the internal space M 4 , while F , G and H are 2-, 4-and 3-forms on M 4 ; all of them depend only on the coordinates of M 4 as well as the metric of the internal space ds 2 (M 4 ). We have also introduced the frame (e + , e − , e z , e a , e i ) as in (2.6) and ℓ is the radius of AdS 6 . It will be convenient to define where W is now a 1-form on M 4 and X is a function on M 4 . The Bianchi identities of the (massive) IIA supergravity give where ⋆ denotes the Hodge dual on M 4 . Furthermore, the field equations of (massive) IIA supergravity give

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which is interpreted as the field equation for the warp factor. The Bianchi identity dS = SdΦ implies that if S is smooth, then either S is nowhere vanishing on M 4 or S ≡ 0 everywhere on M 4 . In what follows, we shall use the conventions and methodology of [17] for the investigation of AdS spaces where more details can be found.

The Killing spinors
The solution of the KSEs of (massive) IIA supergravity along AdS 6 can be expressed as in (2.4), where σ ± and τ ± depend only on the coordinates of M 4 and are Majorana Spin(9, 1) spinors that satisfy the lightcone projections Γ ± σ ± = Γ ± τ ± = 0. The remaining KSEs have been stated in [17]. For the analysis which follows, it suffices to consider those acting on χ = σ + and χ = τ + . The IIA gravitino KSE implies that Furthermore the IIA dilatino KSE implies that and there is a further algebraic KSE which arises during the integration of the KSEs along AdS 6 given by where c = 1 for χ = σ + and c = −1 for χ = τ + . This is the analogue of the (2.8) in D = 11 supergravity. Note that (3.9) implies that σ + and τ + are linearly independent. The counting of supersymmetries of IIA AdS 6 backgrounds proceeds as in the D = 11 case and so IIA AdS 6 backgrounds preserve either 16 or 32 supersymmetries. As there are not maximally supersymmetric AdS 6 backgrounds in IIA supergravity [18], it remains to investigate the AdS 6 backgrounds preserving 16 supersymmetries.
3.2 Proof of the main theorem 3.2.1 Orthogonality of σ + and τ + spinors As in D = 11 supergravity, we proceed by setting Furthermore, (3.6) also implies that On taking the divergence of (3.11) and using (3.12), we find An application of the maximum principle implies that For this, it is sufficient to require that the solutions are smooth and the internal space M 4 is closed. Hence (3.11) implies that Returning to the algebraic conditions (3.9); on taking inner products and making use of (3.15), we obtain On subtracting these expressions, one obtains This establishes the orthogonality of σ + and τ + spinors.

Additional properties of KSEs
To establish some additional properties of the KSEs observe that if σ + is a Killing spinor, then Γ (4) σ + solves the KSEs as a τ + Killing spinor. This follows from the relation between σ + and τ + Killing spinors as explained in the context of 11-dimensional supergravity theory, section 2.2, that also applies in IIA supergravity. Then it follows from the orthogonality condition (3.17) of σ + and τ + spinors that To proceed further, on eliminating the F terms between the algebraic KSEs (3.8) and (3.9), and using (3.18), one obtains the condition

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The KSEs also imply that which together with (3.18) gives There are then two cases to consider depending on whether S ≡ 0 or S ≡ 0.

A non-existence theorem for standard IIA
In the special case for which S vanishes, S ≡ 0, (3.19) gives that .22) and (3.21) implies that The dilaton field equation (3.4) then becomes On integrating this expression over M 4 , one finds and also where again we have used that M 4 is closed. However, the warp factor equation (3.5) then admits no solution which establishes the non-existence theorem.

A non-existence theorem for massive IIA
As we have already mentioned a consequence of the Bianchi identity dS = SdΦ is that S is nowhere vanishing. Furthermore, left-multiplying (3.8) withΓ (4) , taking χ = σ + , and then taking the inner product with σ + , implies that Also, taking the inner product of (3.8), with χ = σ + , and using (3.19), (3.21) and (3.27) to eliminate the spinor bilinear terms, one finds

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Then, using (3.28) to eliminate the S 2 term, the dilaton field equation (3.4) can be rewritten as Integrating this expression over M 4 and using the assumption that M 4 is closed yields the conditions and (3.29) then simplifies to The maximum principle then implies that The algebraic KSE (3.9) then gives that and on squaring this expression we find However, this equation admits no regular solution. Hence there are no smooth closed supersymmetric warped AdS 6 solutions in massive IIA supergravity. The assumptions mentioned in the description of the non-existence theorem are essential. First observe that the maximum principle has been used to establish that the length of the Killing spinor Λ is constant as in the 11-dimensional case. In addition, further partial integration arguments are required to establish the result, which require topological restrictions on the internal space. These assumptions can possibly be weakened, but not entirely removed. This is because one can reduce the AdS 7 × S 4 solution of 11-dimensional supergravity, which has been interpreted as an AdS 6 solution in section (2.3.2), along Killing directions of S 4 to find AdS 6 solutions in IIA supergravity. However the existence of such solutions will not be a contradiction, as they do not satisfy the conditions of our theorem. In addition, the solutions of [3] are singular and therefore do not satisfy our regularity assumptions.

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4 AdS 6 × w M 4 solutions in IIB supergravity The non-vanishing form fluxes of IIB AdS 6 backgrounds have support on the internal space M 4 . In particular as the 5-form R-R field strength F is self-dual, it vanishes. The rest of the fields can be written as where G is a twisted 6 complex 3-form which includes the R-R and the 3-form NS-NS 3form field strengths, and P is the twisted 1-form field strength of the dilaton and axion of the theory. Thus H and ξ are twisted complex 3-and 1-forms on the internal space M 4 , respectively. We also introduce the frame (e + , e − , e z , e a , e i ) as in (2.6) and ℓ is the radius of AdS 6 . We follow the conventions, notation and methodology of [16] for investigating IIB AdS backgrounds.
To continue it is convenient to define where W is a complex (twisted) 1-form on M 4 . In such a case the Bianchi identities and the field equations that we shall use below can be expressed as and respectively, where Q is the connection of the U(1) bundle on the upper-half plane pulled back on M 4 .

The Killing spinors
The KSEs of IIB supergravity can be solved along AdS 6 [16] to yield an expression for the Killing spinor as in (2.4), where now σ ± and τ ± are complex Weyl Spin(9, 1) spinors that depend only on the coordinates of M 4 and satisfy the projections Γ ± σ ± = Γ ± τ ± = 0. Next as in the IIA case define

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with the convention that Γ (4)Γ(4) σ ± = ±σ ± , (4. 6) and similarly for τ ± , as these spinors. Using these conventions, the remaining independent KSEs are and and ∇ is the Levi-Civita connection on M 4 . The (4.7) KSEs are a suitable restriction of the gravitino and dilatino KSEs of IIB supergravity on τ ± and σ ± while the (4.8) KSEs arise as integrability conditions of the solution of IIB KSEs along AdS 6 . Observe also that these KSEs imply that the σ + and τ + Killing spinors are linearly independent. The counting of Killing spinors is similar as that presented in more detail for AdS 6 solutions of 11-dimensional supergravity and so IIB AdS 6 backgrounds preserve either 16 or 32 supersymmetries. It is known for sometime that there are no maximally supersymmetric IIB AdS 6 backgrounds [18]. Therefore it remains to explore the IIB AdS 6 backgrounds that preserve 16 supersymmetries.

The orthogonality of τ + and σ + Killing spinors
As in previous cases setting Λ = σ + + τ + and upon using the gravitino KSE, one finds Next the algebraic KSE (4.8) gives Thus, one finds

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Furthermore observe that as a consequence of the gravitino KSE in (4.7), one has ∇ i (ARe τ + , Γ iz σ + ) = 0 . (4.14) Taking the divergence of (4.13) and using the above equation, one deduces that The maximum principle then gives that Λ is constant and in turn where it is sufficient to assume the smoothness of the fields and Killing spinors, and that M 4 is closed. Furthermore the algebraic KSEs (4.8) give that This establishes the orthogonality of τ + and σ + spinors.

A formula for the warp factor
Observe that as a consequence of the first equation in (4.16) as well as the orthogonality condition in (4.17) that where we have used thatΓ (4) σ + is a τ + Killing spinor because of (4.6) and the relation between τ + and σ + spinors. Taking the derivative of the first equations in (4.16), we obtain for Λ = σ + that Moreover the dilatino KSE in (4.7) can be used to give Eliminating the term that contains ξ in the above two equations, we find This is the key equation that we shall explore in what follows to establish non-existence theorems. Observe that apart from the last term, it has the required form to apply the maximum principle on the warp factor A.
and substituting into As the last term in (4.21) vanishes an application of the maximum principle reveals that the warp factor is constant and W = 0. This in turn makes the field equation for the warp factor A (4.4) inconsistent and so such backgrounds do not exist. Furthermore, it is required that smooth closed AdS 6 solutions must have active scalars. Indeed if ξ = 0, then the algebraic KSE in (4.7) implies that which can be used to eliminate the last term in (4.21). Then the maximum principle can apply to yield that A is constant and W = 0 which in turn are inconsistent with the field equation for the warp factor A (4.4).

4.2.4
A non-existence theorem for σ + , C * σ + = 0 To demonstrate this non-existence theorem for smooth closed IIB AdS 6 solutions, we shall first show that either the bilinear σ + , C * σ + vanishes identically or it does not vanish anywhere on the internal space. To begin, we evaluate the derivative of this bilinear and after using the gravitino in (4.7) and algebraic (4.8) KSEs, to find that This is a parallel transport equation which establishes the statement as A does not vanish anywhere on the internal manifold. Next observe that the second equation in (4.18) can be written as This condition is as that in (4.22) with f = σ + , C * σ + . Therefore if σ + , C * σ + does not vanish identically, then the non-existence theorem follows from the results of the previous section. Observe that σ + , C * σ + satisfies the requirements on the function f .

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Incidentally observe that if the bi-linear α = σ + ,Γ (4) C * σ + does not vanish anywhere on M 4 , then there are no smooth closed solutions. Indeed from (4.16), one finds that Substituting this into (4.21), the latter can be rearranged such that the maximum principle applies leading to W = 0 which together with the field equations for the warp factor A establishes the statement.

A non-existence theorem for det ω = 0
So far we have seen that smooth closed IIB AdS 6 backgrounds exist provided that and α vanishes somewhere on the internal space. However, there are additional restrictions on the spinor bilinears required for the existence of such AdS 6 backgrounds. To find them, let us explore further the geometry of the internal space. A more systematic investigation of the spinor bilinears can be found in appendices B and D.
To continue define Observe that σ + , Γ ijΓ(4) σ + = − 1 2 ǫ ij kl σ + , Γ kl σ + . An application of the gravitino KSE reveals that These in particular imply that ie A * ω is a Killing-Yano tensor. Furthermore, it is easy to show using the equation (4.31) that Thus A 2 ω ij * ω ij = const. If this constant does not vanish, then ω has rank 4 everywhere on the internal manifold M 4 .
On the other hand using (4.29), one can show from the dilatino KSE that Thus if A 2 ω ij * ω ij = 0, and so both * ω and ω are invertible, then W will vanish everywhere on M 4 . Again in such a case there are no smooth closed AdS 6 solutions. This establishes our theorem. Therefore if smooth closed AdS 6 solutions exist, they require that ω has rank 2 or less everywhere on the internal space.

Symmetries and topology of internal manifold
Further restrictions on the internal space of smooth closed IIB AdS 6 backgrounds can be found by exploring the isometries of AdS 6 backgrounds. We have established in appendix A that the internal space of AdS 6 backgrounds admits an so(3) action generated by three 7 vector fields It is then a consequence of the KSEs that this action leaves all the fields invariant. First let us demonstrate that this action is non-trivial. For this we shall show that the Z Q cannot vanish identically for smooth closed solutions. To see this first note that the Fierz identities in appendix B give where α = σ + ,Γ (4) C * σ + as in previous sections and we have used that σ + , C * σ + = σ + ,Γ (4) σ + = 0. Now suppose that there are smooth closed solutions for which Z r vanish identically on the internal space. Then as a consequence of the Fierz identities ω 2 = 0 and so ω = 0. On the other hand if ω vanishes, then an application of the maximum principle on (4.21) implies that there are no smooth closed AdS 6 solutions which is a contradiction. This establishes that for smooth closed solutions so(3) acts non-trivially on the internal space.
The groups with Lie algebra so(3) act on 4-dimensional manifolds and the principal 8 orbits are either of co-dimension 1 or 2. To continue, let us suppose that the principal orbits of so(3) on M 4 have co-dimension 1. As all fluxes are invariant up to a possible gauge U(1) transformation and Z i Q W i = 0 (A.2), W ∧W descends on the space of orbits. As the space of orbits is 1-dimensional W ∧W vanishes. Then we can apply the maximum principle on (4.21) to conclude that the warp factor is constant and W = 0 which lead to an inconsistency in the field equation for the warp factor. Thus, there are no smooth closed AdS 6 backgrounds with principal so(3) orbits on the internal space of co-dimension 1.
It remains to investigate the case that for which the principal orbits of so(3) action on M 4 have co-dimension 2. For this let us assume that the so(3) action on M 4 can be integrated to an effective SU(2) or SO(3) on M 4 , and that M 4 is oriented. Oriented 4dimensional manifolds admitting an effective non-abelian group action have been classified in [25] and references within, see also [27]. The principal orbits are 2-spheres. These can degenerate to either points or to the 2-dimensional real projective space RP 2 . The space of orbits is a 2-dimensional surface F with boundary which is a union of circles. The principal orbits degenerate at the boundary circles. Spin oriented 4-dimensional manifolds JHEP12(2017)009 M 4 which admit an effective SU(2) or SO(3) with principal co-dimension 2 orbits include 9 those presented in (1.1). For M (F ), the surface F has no boundary and the principal orbits do not degenerate. If F is oriented, then M (F ) = F × S 2 . However we shall demonstrate later that there are no smooth closed AdS 6 backgrounds with internal space M 4 that has topology F × S 2 .

Aspects of geometry
To explore the geometry of the internal space of IIB AdS 6 backgrounds, let us assume that σ + , C * σ + = σ + ,Γ (4) σ + = 0 and det ω = 0. Observe that (4.34) can be rewritten as and the algebraic and dilatino KSEs give The 2-form Aω descends on the space of orbits F as L Z Q Aω = 0 and i Z Q ω = 0. The latter follows from (B.3). Note that ω gives rise to a Hermitian structure on F away from the fixed points. This can be seen from the Fierz identity (B.5) which can be re-written as If there are points for which |α| = σ + 2 , ω is not a Hermitian form on F . Incidentally, the points for which |α| = σ + 2 are the fixed points of the group action as can be seen from (B.4).
In addition, setting as before α = σ + ,Γ (4) C * σ + , one finds that Using the algebraic KSE Ξ to eliminate the last term, we find that This condition can be solved for W to determine the flux in terms of the geometry of the internal space.

A non-existence theorem for
We shall now demonstrate that there are no smooth closed AdS 6 solutions for which the internal space has topology 10 M 4 = F × S 2 , where F is an oriented surface. To see this, observe that in this case the action of so ( In turn the first equation in (4.36) implies that ω is nowhere vanishing. As A is nowhere vanishing, the 2-form A n ω is nowhere vanishing as well for n ∈ Z. Furthermore A n ω descends on F and as A n ω is nowhere vanishing, one has On the other hand, an application of the gravitino KSE (4.7) reveals that for a = 1, 2, 3, z. Using this and the KSE (4.8) we obtain where ζ i = σ + , Γ zi σ + . Pulling this equation back on F with a global section of the trivial fibration F × S 2 , we find upon application of Stoke's theorem that This is in contradiction to (4.43) for n = 2. We therefore conclude that there are no smooth closed AdS 6 solutions with internal manifolds that have topology F × S 2 .

Conditions for the existence of AdS 6 IIB solutions
We have seen that the existence of smooth closed IIB AdS 6 backgrounds is rather restricted. However, we have not been able to establish as strong a result as that for D = 11 and (massive) IIA backgrounds. If smooth closed IIB AdS 6 backgrounds exist, the statements of non-existence theorems that we have established can turn to necessary conditions that such backgrounds must satisfy. Of course the issue of existence of smooth closed IIB AdS 6 backgrounds can be settled with an example. However to our knowledge all solutions that have been found in the literature [3]- [9] so far are singular. In particular those found in [9] are singular because the warp factor A of the AdS 6 subspace vanishes at some points on the internal space. It has been shown in [16] that irrespective of the frame chosen, A is related to the length of a Killing spinor [16], σ − 2 = c 2 A 2 , where c is a constant. For smooth closed AdS 6 backgrounds, the Killing spinors must be nowhere vanishing, and so A = 0 everywhere on the internal space. Here we shall summarize the necessary conditions for existence of smooth closed IIB AdS 6 backgrounds.

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We have seen that the bilinears of the Killing spinor σ + must satisfy the following conditions Violation of either of the first two conditions leads to a non-existence result while the third condition is a consequence of the orthogonality of σ + and τ + Killing spinors.
Furthermore the bilinear σ + ,Γ (4) C * σ + , (4.48) must vanish at some points of the internal space M 4 . Otherwise again a non-existence theorem for smooth closed AdS 6 backgrounds can be established.
In addition, it has been shown in appendix D that for smooth closed AdS 6 backgrounds, the following spinor bilinears also vanish σ + , Γ 12 σ + = σ + , Γ 13 σ + = σ + , Γ 23 σ + = 0 , (4.49) otherwise such backgrounds cannot exist. This is equivalent to the statement that all σtype Killing spinors are orthogonal over the complex numbers. This is despite the fact that the KSEs of IIB supergravity are linear over the real numbers.
Another necessary condition for the existence of smooth closed IIB AdS 6 backgrounds is that they must have active scalars. If the IIB scalars are constant, then one can demonstrate that there is a non-existence theorem for such solutions. This follows from the dilatino KSE on setting ξ = 0, and making use of (4.21). As demonstrated in section 4.2.3, another necessary condition for the existence of such backgrounds is that the RR and NSNS 3-form fluxes must be linearly independent.
Moreover we have demonstrated that if smooth closed IIB AdS 6 solutions exist, the internal space has principal orbits of so(3) of co-dimension 2. Otherwise again a nonexistence theorem can be established. Combining this with the classification results of [25], it is possible to determine the diffeomeorphic type of the internal space that includes manifolds as those in (1.1). Using these classification results, we also excluded the existence of smooth closed IIB AdS 6 solutions with internal space which has topology F × S 2 , where F is an oriented surface. This restricts further the possible topologies of smooth closed IIB AdS 6 backgrounds to S 4 , p(S 1 × S 3 )#q(S 1 × RP 3 ) and M (F ), where F is a nonoriented surface.
Amongst these, an interesting topology for an internal space is p(S 1 ×S 3 )#q(S 1 ×RP 3 ). The space of orbits is a surface F with boundary ∂F which is a union of circles. If n is the number of circles that S 2 degenerates to a point, then q = rankH 0 (∂F ) − n and p = rankH 1 (F ) − q. For q = 0 and p = 1, F is a cylinder as expected and n = 2, while for q = 1 and p = 0 again F is a cylinder but now n = 1. The internal spaces of examples explored in [8,9] may have such topology.

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We also define where the Γ i are the Dirac gamma matrices. The inner product ·, · we use on the space of spinors is that for which space-like gamma matrices are Hermitian while time-like gamma matrices are anti-hermitian, ie the Dirac spin-invariant inner product is Γ 0 ·, · . For more details on our conventions see [15][16][17].

B IIB Fierz identities
In the investigation of the geometry of the internal spaces of IIB AdS 6 backgrounds, we have used some Fierz identities. These are listed below without assuming the vanishing of any of the spinor bilinears which are necessary for the existence of smooth closed IIB AdS 6 backgrounds. So the identities below are valid in general. However, we shall comment on their consequences in the special cases of smooth closed IIB AdS 6 backgrounds.
The first Fierz identity gives ω 2 in terms of scalar bilinears as Observe that the length of ω can vanish for smooth closed IIB AdS 6 backgrounds whenever Γ (4) C * σ + = σ + as a consequence of the Cauchy-Schwarz inequality. Next, one finds that For smooth closed IIB AdS 6 solutions, this implies that ω must have rank strictly less than 4. This was derived already using a different reasoning. The Fierz identity is instrumental to show that A ω descends on the space of orbits of so(3) action on the internal space M 4 for smooth closed backgrounds. The Fierz identity

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has been instrumental in showing that the so(3) acts non-trivially on the internal space of smooth closed AdS 6 backgrounds. The Fierz identity has been used to investigate the existence of a Hermitian structure on the space of orbits of the so(3) action on the internal space M 4 .

C 4-manifolds with large symmetry groups
It has been known for sometime [25] that closed oriented 4-dimensional manifolds that support an effective action of SO (3)  From these those collected in (1.1) are spin, closed, and SO(3) or SU(2) have codimension 2 principal orbits. The SO(3) or SU(2) group action on M (F ) have orbits 2-spheres S 2 and are all principal. M (F ) is uniquely characterized as the 2-sphere bundle over a surface F that admits a spin structure. M (F ) = F × S 2 , if F is oriented. The spaces p(S 1 × S 3 )#q(S 1 × RP 3 ) arise in the case that the principal orbit S 2 degenerates to several other special orbits. For oriented 4-manifolds S 2 either collapses to a point or degenerates to RP 2 . In both cases, the special orbits are not isolated but rather lie on a circle. In particular in S 1 ×S 3 the S 2 fibre collapses to a point at two different circles, while in S 1 × RP 3 the S 2 fibre collapses to point at a circle and becomes RP 2 at another. There are some more oriented 4-manifolds admitting co-dimension 2 principal orbits. An example is P (F ) for F a 2-dimensional surface for which at each boundary circle the principal S 2 orbit degenerates to RP 2 . If F is a two 2-disk B 2 , then P (F ) = M (RP 2 ). For other 2dimensional surfaces F with boundaries, P (F ) gives another class of oriented 4-dimensional manifolds which exhibit co-dimension 2 principal orbits. However, we have not been able to establish whether P (F ) are spin for F = B 2 .
Note that in IIB, the Killing spinors have a Spin c structure rather than a Spin structure. In particular the spinors are twisted with an additional U(1) bundle which is the pull back on the spacetime of canonical U(1) bundle that arises after viewing the target space of IIB scalars as a SU(1, 1)/U(1) coset space. As this coset space is the upper half plane, it is JHEP12(2017)009 contractible unless one considers further identifications which we shall not investigate here. As a result the U(1) bundle is topologically trivial and so this Spin c structure is equivalent to a standard Spin structure.
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