Mink$_3\times S^3$ solutions of type II supergravity

We initiate the classification of supersymmetric solutions of type II supergravity on $\mathbb{R}^{1,2} \times S^3 \times M_4$. We find explicit local expressions for all backgrounds with either a single Killing spinor or two of equal norm, up to PDE's. We show that the only type II AdS$_4\times S^3$ solution is the known $\mathcal{N}=4$ AdS$_4$ background obtained from the near-horizon limit of intersecting D2-D6 branes. Various known branes and intersecting brane systems are recovered, and we obtain a novel class of $\mathbb{R}^{1,2} \times S^2\times S^3$ solutions in IIA.


Introduction
The advent of the AdS-CFT correspondence has led to significant interest in the construction of Anti-de Sitter string backgrounds in various dimensions and with various amounts of supersymmetry. One of the most famous AdS 4 backgrounds is the AdS 4 × CP 3 solution. The discovery of this solution far pre-dates the correspondence [1], however it was not realised how it fit into the holographic paradigm until the works of [2] and [3]. A plethora of other such AdS 4 classes and explicit examples have been found using (in some cases vastly) different methods and exhibiting different amounts of supersymmetry: consider the very incomplete list of [4][5][6][7][8][9][10][11][12][13][14] for N = 1, [15][16][17][18][19][20][21] for N = 2, [22] for N = 3 and [23][24][25][26] for N = 4. Solutions with N > 4 where recently classified in [27], they are very restricted.
One of the more prominent methods of finding AdS d backgrounds is to find bosonic solutions with an AdS d factor to the supersymmetry constraints, which also satisfy the Bianchi identities. As a consequence of various integrability theorems, such solutions automatically solve the equations of motions. The Killing spinor equations reduce to constraints on the internal manifold, which can then be solved by means of G-structure and generalised geometrical techniques. The literature usually approaches this problem by assuming an AdS d from the start. However we are also interested in solutions of relevance to flux compactifications and the broader definition of holography that includes non conformal solutions. As such we shall consider assume Minkowski factor, in this case Mink 3 , so that our results are more broadly applicable.
Finding Minkowski solutions using G-structure techniques [28][29][30][31] or otherwise is by now quite a mature program, see [32][33][34][35] for some recent examples. Usually the aim is to preserve minimal or even no supersymmetry for phenomenological reason which makes the problem in general quite hard. We shall take inspiration from [36] and assume the existence of an S 3 factor in the metric. This will necessarily mean that we are dealing with at least N = 2 which is of less phenomenological interest, however with these solutions classified it should then be possible to systematically break some (or even all) of this symmetry by deforming the S 3 .
In this paper we classify all supersymmetric solutions of Type II supergravity on R 1,2 ×S 3 ×M 4 , under the assumption that the seven-dimensional internal Killing spinors have equal norms and that the physical fields of the solution respect the ISO(1, 2) × SO(4) isometry subgroup. Our classification is quite detailed, going as far as to give explicit local expressions the metric, fluxes and dilaton in terms of simple (Laplace-like) PDE's. As we shall see, solutions in this class are generically N = 2, from the Minkowski perspective, and support a SU (2) R-symmetry realised geometrically as one factor of the SO(4) SU (2) + × SU (2) − isometry group manifold of S 3 -the remaining SU (2) factor is a "flavour" under which the Killing spinors are uncharged. 1 This may sound strange as there is no 3d superconformal algebra with SU (2) R , but this only matters for solutions where R 1,2 is part of a AdS 4 factor so that SO(2, 3) is realised. Ultimately our results end up side stepping this issue as in general their is either an enhancement of the R-symmetry to SO(4) via the emergence of an additional S 2,3 factor, or an enhancement of the Minkowski factor to dimensions where SU (2) R is a necessary part of the superconformal algebra.
The classification recovers various well-known intersecting brane systems listed in [40] and some of their U-duals and some of their S-duals. New classes we find include a pure NS R 1,2 × S 3 × S 2 × R vacuum, its U-dual in IIB, the cone over R 1,2 × S 3 × S 3 , and a novel class of R 1,2 × S 2 × S 3 × Σ 2 solutions in massless and massive IIA.
One of our main results is that the only compact AdS 4 × S 3 × M 3 solution of type II supergravity is the known N = 4 solution of type IIA on a foliation of AdS 4 × S 3 × S 2 over an interval which is the near-horizon limit of the an D2-D6 brane system. The required SO(4) R is realised with one SU (2) from each sphere, and not the S 3 alone. Indeed this is to be expected as if the 3-sphere does realise two SU (2) R-symmetries there would be two sets of N = 2 spinors transforming in the (2,1) and (1,2) of SO(4) -there is no N = 4 super-conformal algebra in 3d with Q-generators that transform in this fashion. So it seems likely that the only avenue left open for holographic duals of N = 4 is to seek AdS 4 × S 2 × S 2 solution like [23,24], but in massive IIA.
Our other main result is the discovery of a new class of N = 4 solutions on R 1,2 × S 2 × S 3 × Σ 2 preserving an SO(4) R-symmetry but no AdS 4 . These generically have all possible IIA fluxes turned on and can be divided into cases either in massless or massive IIA at which point solutions are in one to one correspondence with a single PDE on Σ 2 . In particular the massless solutions are governed by a 3d cylindrical Laplace equation with axial symmetry. These classes look very promising both for finding compact Mink 3 solutions, but also possibly solutions that asymptote to AdS.
Let us now describe the outline of the paper: in order to solve the supersymmetry constraints, we will make use of the reformulation of the Killing spinor equations in terms of so-called pure spinor equations. Such pure spinor equations were first used for backgrounds of the form M 10 = R 1,3 × M 6 , where it was shown that they are related to integrability constraints of generalised almost complex structures on the internal space M 6 [30]. For backgrounds of the form M 10 = R 1,2 × M 7 , the pure spinor equations were constructed in [31] (see also [37]). Next, we decompose M 7 = S 3 × M 4 , leading to pure spinor equations on the internal M 4 . We explain this setup in detail in section 2. The resulting supersymmetry constraints vary significantly, depending on whether the theory at hand is type IIA or type IIB. We will solve the supersymmetry constraints as well as the Bianchi identities for IIB backgrounds in section 3 and for IIA backgrounds in section 4. In section 5, we then show that there is a unique solution with a warped AdS 4 factor, obtained from the D2-D6 system . In addition to the case where the internal Killing spinors have equivalent norm, in section 6 we examine all backgrounds in the case where one of the Killing spinors vanishes, i.e., 2 = 0. In this case, there is no need to distinguish between IIA and IIB; we demonstrate that all such backgrounds are pure NSNS and give the solutions. In the appendix, we discuss conventions and identities used, a mild extension of the 3+7 pure spinor equation construction (including the non-equivalent norm case), and a discussion on similar backgrounds from an M-theory perspective.

Mink 3 with an S 3 factor
We are interested in solutions to type II with at least a three-dimensional external Minkowski component, with the fluxes respecting the three-dimensional Poincaré invariance: where the RR flux f is a polyform on M 7 and the warp factor A and the dilaton Φ are functions on M 7 . 2 Moreover, we take the NSNS 3-form H to be internal as well. The Killing spinors for N = 1 supersymmetric solutions decompose as where ζ is a Majorana spinor of Spin(1, 2) and χ 1,2 are Majorana spinors of Spin (7) and where the upper (lower) signs are taken in IIA (IIB). Following [31], we define two real seven-dimensional bispinors Φ ± in terms of χ 1,2 : where the subscript +/− refers to the even/odd forms in the decomposition of the polyform. The conditions for unbroken N = 1 supersymmetry are equivalent to as long as the norms of spinors χ 1,2 are equal 3 , which leads to The assumption of equal norm is a global requirement for AdS 4 (see footnote 6 in section 5), and a local requirement for the existence of calibrated D-branes or O-planes (see section 6), however this is not a requirement in general -rather we view this as a well-motivated simplifying assumption.
Next, we require that the internal space can be decomposed locally as M 7 = S 3 × M 4 , and in order to ensure that compactification leads to an SO(4) global symmetry we insist that the fluxes respect the SO(4) isometry. As a result, the metric and fluxes decompose further as We decompose the 7d spinors in the same fashion in terms of a single 4 pseudoreal (i.e., (ξ c ) c = −ξ) Killing spinor ξ on S 3 , and two pseudoreal spinors η 1,2 on M 4 : In [31], [37] an additional constraint that was imposed in order to derive (2.4) was that the external component of the NSNS 3-form flux is trivial; unlike in four dimensions, this is not enforced by Poincaré invariance. It turns out that this second assumption is redundant though, as is shown in appendix D: if |χ 1 | 2 = |χ 2 | 2 and spacetime does not admit a cosmological constant, then supersymmetry enforces that the external NSNS flux vanishes. 4 As explained in Appendix C, there are two independent types of Killing spinors on S 3 , ξ + and ξ − -however they cannot be mapped to each other using the SO(4) invariants of the fluxes or the Killing spinor equations. This is all that appears when one decompose M 7 = S 3 × M 4 , so if one were to include terms like ξ + ⊗ η + and ξ − ⊗ η − then reduced the 7d spinor conditions to 4d ones you would find that η ± never mix. So setting one of η ± to zero excludes no solutions in our analysis.
which is the most general parameterisation consistant with an S 3 ×M 4 product and the Majorana condtion. 5 Note that we do not restrict the Spin(4) spinors η i to be chiral and we normalise η † 1,2 η 1,2 = 1 . The Killing spinors on S 3 satisfy the Killing spinor equation which preserves two supercharges for each of ν = ±1. We will not make a choice of ν so we can establish whether any solutions are independent of this choice -the S 3 of such a solution would preserve 4 supercharges. As explained in Appendix C a spinor on S 3 defines a doublet which is charged under one SU (2) factor of SO(4) = SU (2) + × SU (2) − , depending on the sign of ν -ξ a is a singlet under the action of the second SU (2). As such, a generic solution with Mink 3 × S 3 will have an R-symmetry SU (2) R and an additional global flavour symmetry SU (2) F . Such solutions preserve at least N = 2 supersymmetry from the 3d perspective, so 4 real supercharges -indeed, the 10d Killing spinors may be written as where ζ a is a doublet of Killing spinors on R 1,2 , that allow the 10d spinors to be invariant under SU (2) R transformations. However we only need to solve an N = 1 sub-sector, because the part of the Killing spinor which couples to ζ 1 is mapped to the part coupling to ζ 2 under the action of SU (2) R -so if you solve one part, the other is guaranteed. If a solution ends up being independent of ν then there is a copy of (2.10) for each sign and supersymmetry is doubled to N = 4 -there are two SU (2) R-symmetries, but they do not appear as a product so do not form SO(4) R -as we shall see, this only happen in a small number of special cases.
Using the gamma matrix decomposition (A.2), the seven-dimensional bispinor (2.3) decomposes as Here,γ is the four-dimensional chirality matrix and the ± subscripts again refer to even and odd form components. We see that the components are in fact matrices and that the sevendimensional bispinor is constructed as the trace of the product of the components.
The S 3 component leads to the bispinor matrix where K i is a vielbein defining a trivial structure on S 3 (see appendix A).
The M 4 component leads to the bispinor matrix where Since the matrix entries are somewhat involved, we refer to appendix B for details. Plugging both components (2.11), (2.12) into the seven-dimensional bispinors (2.10), it follows that At this point, the IIA and IIB supersymmetry equations diverge, and we shall relegate their explicit form to the relevant sections.
With our set up, a solution to the supersymmetry equations is a solution to the equation of motion if and only it satisfies the Bianchi identites [4] [38] [39]. These are given by d H F = dH = 0 away from localised sources. By definition, a localised (magnetic) source manifests itself in the Bianchi identity of some field strength F as dF = Qδ n (x) and hence in such cases F is discontinuous. Loosely speaking, a localised source corresponds physically to an extended object (such as a brane) located at a submanifold of the ten-dimensional spacetime S ⊂ M 10 which is pointlike in some of the local coordinates. The standard approach to obtaining backgrounds, which we follow as well, is to first solve the supersymmetry equations by introducing local coordinates, and then afterwards determine the physically sensible range of these local coordinates by examining the obtained geometry and fluxes. The presence of localised sources is signified by discontinuities of not just the fluxes, but of the spacetime geometry as well, precisely at the location of the sources. Therefore, it is possible to obtain solutions with localised sources even when making use of the Bianchi identities with no sources: one examines possible discontinuities in the geometry and fluxes and determines whether or not such discontinuities are associated with localised sources or not by comparing them with the divergent behaviour of known extended objects.
Making use of the flux decomposition (2.1), (2.6), the Bianchi identies thus reduce to (2.15) This is after imposing H 0 = 0, which turns out to be a requirement for every solution to the supersymmetry equations that we obtain.

Summary of obtained backgrounds
As the rest of the paper is somewhat technical, let us summarise our results here. We find a number of well-known backgrounds, as well as some new ones.
3. A generalization of the D5-brane generated by U-duality. The metric is given by (3.43), the fluxes by (3.41), scalar field constraints by (3.42).

4.
A new background on the cone over R 1,2 × S 3 × S 3 sq , with S 3 sq a generically squashed threesphere admitting an SU (2) × U (1) isometry group. For the unsquashed limit, the metric is given by (3.59), the fluxes by (3.57), the scalar field constraints by (3.58). In the generic squashed case, the metric and dilaton are given by (3.75), the fluxes by (3.74). We note that the more general squashed case can be obtained from the unsquashed case by a duality chain.
3. A generalization of the D4-D8 system generated by U-duality. The metric is given by (4.30), the fluxes by (4.29), and scalar constraints by (4.27).

4.
A class of new backgrounds. The metric contains an R 1,2 × S 3 × S 2 factor, with various warpings, and is given by (4.42). The warp factors are constrained by various PDE, given in (4.43). In general, all fluxes are turned on and are given by (4.44). This new class of backgrounds contains a subset with a U (1) isometry. In this case, T-dualising along the isometry direction leads to the new IIB backgrounds outlined above, with generic squashing.
In addition, we find two more backgrounds when setting 2 = 0. These backgrounds are pure NS, and as such, can be found in both type IIA and type IIB. We find: 1. The NS5-brane, with metric (6.18), flux (6.16) and the scalar constraints (6.17).

A pure NS background on
The metric is given by (6.22), the flux by (6.21). All scalars are determined up to constant factors.

Mink with an S factor in IIB
The type IIB supersymmetry equations are obtained by plugging the decomposed seven-dimensional bispinors (2.14) into the seven-dimensional supersymmetry constraints (2.4). This leads to the following constraints on the four-dimensional bispinors while the fluxes are determined by and must additionally satisfy the pairing equation In order to solve these, we will first examine the 0-form conditions, These are given by We solve the first two of these in Appendix B, which leads to a spinor ansatz depending on 6 real functions with support on M 4 α, a 1 , b 1 , λ 1 , λ 2 , λ 3 (3.5) subject to the constraint The third 0-form constraint, which is unique to IIB, still needs to dealt with. After making use of (B.12), it reduces to H 0 (a 1 cos Here, as well as in IIA, the solutions depend drastically on the behaviour of α. We can distinguish between three different cases: α = 0, α = 1 2 π, and generic α ∈ (0, π), α = 1 2 π. Let us reiterate that we introduced α in (B.5) by defining where η is a locally defined non-chiral spinor, where the chiral components are normalised. Note that the non-chirality is crucial: it ensures that η can be used to define the local trivial structure (i.e., the vielbein) via (B.3). In the case that α = 0, the 4d internal Killing spinors η 1 = η are such that the chiral components of η 1 have equal norm. In the case that α = π/2, we see that η 1 becomes chiral. It turns out that we can treat this case together with α = 0, but find no such solutions. Thus we seperate our solutions into two branches.

(3.13)
Making use of these, one finds that the supersymmetry equations imply cos β(e C cos δdθ + 2ν sin δv 1 ) = cos β(e C cos δ sin θdφ − 2ν sin δw 2 ) = 0 , which is not a compete list. The first thing to establish is how to solve (3.14a) -if we set sin β = 0, one needs to set cos δ = 0 to solve (3.14c), but since ν = ±1, (3.14b) leads to a contradiction. The next conditions we consider are (3.14d). For cos β = 0 we see that either sin δ = dθ = dφ = 0, or 0 < sin δ < π 2 in which case (θ, φ) define local coordinates on a 2-sphere. We are ignoring cos β = 0 because, as should be clear from, (3.9), this is a subcase of sin δ = dθ = dφ = 0. Let us now prove that 0 < sin δ < π 2 is not possible: Since H 0 = 0 we can solve (3.14b)-(3.14c) by introducing local coordinates x and ρ = e 2A+2C−Φ cos β cos δ such that We can also rewrite (3.14e) as The key point here is that v 2 , w 1 only have legs in (ρ, x) while v 1 , w 2 sit orthogonal to this with legs in (θ, φ) only. This means that the equation above, cannot be solved as there is a Vol(S 2 ) term whose coefficient is non-vanishing. Thus we can conclude in general that sin δ = dθ = dφ = 0. Plugging this back into (4.1), one finds that nothing depends on the specific values these paramaters take so we can set without loss of generality, leaving one undetermined function β.
We are now ready to write the supersymemtry conditions that follow when α = 0, however we find it helpful to perform a second rotation of the canonical vielbein by considering to ease presentation. The necessary and sufficient conditions for supersymmetry in the α = 0 branch are We can simplify this system further, but not without making assumptions about β. We now proceed to study the systems that follow from different values of β, we find that the physical interpretation is quite different in each case.
3.1.1 Sub case: β = 0 Upon setting β = 0 in (3.1.2) one can show that the supersymmetry conditions reduce to We can solve (3.18b) by using it to define a vielbein in terms of local coordinates ψ, x 1 , x 2 and (3.20) . We now have enough information to calculate the fluxes. First we find (3.21) We can then use coordinate dependence of the physical fields and local expression for the vielbein (3.20) to take the Hodge dual in (3.18c) arriving at Plugging this into (3.18d) we find (∂ x 2 f 1 − ∂ x 1 f 2 ) = 0 which mean that V is closed and so we can locally fix with a shift ψ → ψ − η for dη = V , without loss of generality. Taking this into account the ten-dimensional fluxes are The final thing we need to do is impose the Bianchi identities, which away from localised sources rise to the PDEs The local form of the metric is then This corresponds to the intersecting D3-D7 brane system, where the D3-branes are embedded in the D7-branes [40].

Sub case
Setting β = π 2 in leads to the following nessesary and sufficient condtions for unbroken supersymmetry Here we have introduced the notation both to ease notation and to stress that the vielbeine u i obey a cyclic property. Exploiting this property will be very helpful in solving this system and other systems we shall encounter which mirror this behaviour, so we will be very explicit in our derivation here, but less so elsewhere. The first thing to note is that the combination (3.28c This implies that the 1-form in large brackets is zero. This can be seen by writing it as X j u j +X 0 v 2 for some functions X j , noting that the vielbeine u 1,2,3 are independent, and then considering the resulting constraints for i = 1, 2, 3. Next by examining (3.28c which implies that d(e −A u i ) has no leg in u i , it is then not hard to see that since (3.28b) has no ijk u j ∧ u k term it is in fact zero. So we can conclude without loss of generality that which imply (3.28b)-(3.28c) without further constants. We can then solve these conditions by using them to define the vielbeine in terms of local coordinates as We can now solve (3.28a), which in fact just tells us that up to rescaling g s and that e 2A is a function of ρ only, making ∂ x i all isometry directions parameterising either R 3 or T 3 locally. The only non-trivial flux is the RR 3-form and its Bianchi identity, dF 3 = 0, imposes that This is the warp factor of a D5-brane or O5-hole, depending on the sign of c 2 (see for example [41]). Indeed the metric locally takes the form As we will see, this is a subcase of the solution in the next section.

Sub case: Generic β
For generic 0 < β < π 2 we are free to divide by the trigonometric functions in (3.1.2). Using sin β = 0 it is possible to show that supersymmetry requires by following the same line of reasoning as in the previous subsection. First we solve (3.38b) by using it to define the vielbeine on M 4 locally where we have used the first of (3.38a) to simplify these somewhat. Next (3.38a) is solved when . As a result, ∂ x i are isometries. We then use (3.39) to take the Hodge dual of (3.38d), (3.38e) arriving at the fluxes which solve (3.38e) without restriction. The Bianchi identities impose that which is again the warp factor of a D5-brane or O5-hole. However, in this case the metric takes the local form where T 2 is spanned by (x 2 , x 3 ). This generalises the solution in the previous section by introducing an additional warping factor for a T 2 submanifold, thus breaking SO(1, 5) Lorentz symmetry and leading to more general fluxes. In fact, this solution can be generated from the D5-brane solution of the previous section via "G-structure rotation" [42] which is formally a U-duality [43].

Branch II: α non-zero solutions
For the second branch with 0 < α < π, we begin by studying the lower form conditions that follow from (3.1). Here we find it useful to rotate the canonical frame of ( We then find the following necessary, but not sufficient, conditions for supersymmetry First we note that if either θ or φ become constant or if cos α = 0 then (3.46b), (3.46c) require that sin β = cos δ = 0 which makes (θ, φ) drop out of (3.11) entirely and the final line of (3.46f) vanishes (setting sin θ = 0 leads to the same conclusion). In this case we can conclude that we can set without loss of generality, which we study in section 3.2.1. If we assume sin θ and cos α don't vanish, then (θ, φ) are local coordinates on a 2-sphere and we can take ρ = e 2A+2C−Φ cos α sin β as a local coordinate. We can then use (3.46b)-(3.46d) to rewrite (3.46f) as (3.48) which we can then use to fix some of the free functions. First we note that if we solve (3.46a) with sin δ = 0, then (3.48) fixes cos β = 0 -this is because sin β sin αv 2 − cos β sin αw 2 is parallel to dρ in this limit and so cannot generate the Vol(S 2 ) factor that comes from v 1 ∧ w 1 . Next, for H 0 = 0 and for generic values of (α, β, δ) we can use (3.46b)-(3.46e) to locally define the vielbein on M 4 by introducing another local coordinate such that dx = e 2A+3C−Φ (sin α sin βw 2 + sin α−δ cos βv 2 ), but then we must once more set the v 1 ∧ w 1 term in (3.48) to zero which fixes either cos β = 0 or cos(α − δ) = 0. Thus, for H 0 = 0 and a priori generic (θ, φ, α, β, δ), we end up with just two cases. Firstly cos β = 0, which solves (3.46a) and makes the δ dependence of (3.11) drop out such that we can set without loss of generality We shall examine this case in detail in section 3.2.2 where we find that it contains no solution.
Secondly cos(α − δ) = 0, such that we can set without loss of generality which we shall study in section 3.2.3, finding a new class of solution.
There is one final option one can consider for H 0 = 0, by taking both cos α and sin θ non vanishing-one can tune the values of (α, δ, β) such that (sin α sin βw 2 +sin(α−δ) cos βv 2 )) becomes parallel to dρ and so can no longer be used to introduce a local coordinate. This requires fixing It will then be (3.48) that will be used to define the final vielbein direction, which will necessarily be fibred over S 2 . We shall examine this possibility in section 3.2.4.

Sub case β = 0
For β = 0 the supersymmetry conditions reduce to where as usual u = (v 1 , w 1 , w 2 ). Note that this system reduces to that of section 3.1.2 when sin α = 0, and that (3.52b) imposes that cos α = 0, so we can take 0 < 2α < π. By adding linear combinations of wedge products of (3.52a), (3.52b) and the vielbein to (3.52b), it is then possible to derive enough independent 2-form conditions to establish that This is sufficient to establish that e 2A , e 2C and α are functions of a single local coordinate ρ, which v 2 is parallel to -specifically The final condition in (3.53) implies that d(e −A sec αu 1 ) ∝ u 2 ∧ u 3 and cyclic permutations, however plugging this into (3.52b) and (3.52b) we realise we can without loss of generality take whereK i are necessarily SU (2) invariant forms which furnish a frame for a round S 3 . We have now without loss of generality determined the vielbein on M 4 , which is a foliation of S 3 over an interval, and (3.52a)-(3.52c) are solved when The only non-trivial 10d flux can be extracted from (3.52d) and is given by The pairing equation (3.52e) is equivalent to the Bianchi identity at this point; either one implies The metric is of the form This solution has both an SO(4) R-symmetry and SO(4) flavour symmetry, and is S-dual to the one that we find in section 6, as will be explained in that section.

Sub case
Here one can show that supersymmetry implies There is no solution to this set of constraints, as we will now show. First, we note that H 0 = 0 is imposed by (3.60d). Let ρ = e 2A+2C−Φ cos α. Due to (3.60a), we have w 2 ∼ dx, v 2 ∼ dρ. It is then possible to rewrite (3.60d), (3.60e) as The first equation implies tan α = ρ −1/2 f (x), which is incompatible with the second equationthus this putative class contains no solutions.

Sub case
As explained below (3.46f), here we necessarily have β > 0 and 0 < α < π 2 -for this reason it will turn out the case contains no solutions. As the proof is similar to that of the previous section we shall be brief, this time only quoting sufficient supersymmetry conditions to prove this. In addition to the rotation of (B.3) we find it useful to send v 1 + iw 1 → e −iβ (v 1 + iw 1 ), then a set necessary (but insufficient) conditions for supersymmetry are As elsewhere we can take (3.62c)-(3.62d) as a local definition of the vielbein without loss of generality -v 1 , w 1 are clearly the local vielbeine of a round S 2 in terms of the local coordinates (θ, φ). For v 2 , w 2 we introduce local coordinates x and ρ = e A+C− 1 2 Φ √ cos α sin β such that We can then use (3.62e) to define H 3 without loss of generality, which leaves (3.62a), (3.62b) and (3.62f) to solve -this turns out to be impossible. To see this, one needs to consider the combination 4(3.62f)+ f 1 (3.62a)∧dρ + f 2 (3.62a) ∧ Vol(S 2 ). When one tunes which cannot be solved without violating the the initial assumptions that lead to this case. We conclude that there exist no solutions.

Sub case: Special value of β, H
The final case in IIB requires us to tune tan β to a specific value. After redefining δ → δ + α, this value is In addition, we rotate the vielbein (with respect to (3.44)) as In what follows we assume that the undefined functions of the spinor ansatz are bounded as 0 < 2α < π and 0 < δ + α < π 2 , as the upper and lower limits have been dealt with in the proceeding sections. It is then possible to show that the necessary and sufficient conditions for supersymmetry for this case are are where Vol(S 2 ) is the volume form on the S 2 spanned by (θ, φ). We solve these conditions by first using (3.68b)-(3.68c) to define the vielbein locally without loss of generality as where we have taken θ, φ as local coordinates coordinates and introduced the additional coordinates ψ and ρ = e A+C cos α sin δ sin(α + δ) . (3.71) We can invert this conditions then use (3.68a) to define A, C, Φ, δ in terms of α, ρ and some integration constants c i as The second equality in (3.68d) implies that α is itself a function of ρ only, so we realise that ∂ ψ is an isometry of the solution and that M 4 is foliation of a (SU (2) × U (1) preserving) squashed 3-sphere over an interval. We now turn our attention to the fluxes. We have that (3.68d) simply defines the NSNS flux in such a way that it is automatically closed, while the RR fluxes are defined through the 4d fluxes that follow from (3.68e). We could use the definitions of the functions in (3.72) and vielbein in (3.69) to calculate the 10d fluxes immediately, however we already have enough information to first fix α. The 3-form component of 4 λ(G − ) is necessarily parallel to v 1 ∧ w 1 ∧ v 2 from which it follows that the 10d flux F 1 is parallel to w 2 . As this vielbein is fibred over the S 2 we have that dF 1 = 0 iff ( 4 λ(G − )) 3 = 0. For cos(α + δ) = 0 this imposes d(c 2 1 c 2 2 sin 2 α + c 3 ρ 2 ) = 0, which implies that and as a result, every function has be solved in terms of ρ and the four integration constants c i . We are now ready to calculate the fluxes. The non-trivial ones take the form where dB = H. Clearly the Bianchi identites of the fluxes are implied automatically and one can show that this is true of (3.68f) also. So this case contains a single example, expressed in terms of 4 integration constants. The 10d metric, warp factor and dilation then take the form This solution preserves an SO(4) R-symmetry realised by one SU (2) factor of the round S 3 and the SU (2) of the squashed sphere -the residual symmetries of the spheres make up an SU (2)×U (1) flavour symmetry. Despite our assumption that α + δ = π 2 (which is when c 3 = 0) when deriving (3.68a)-(3.68f) there is in fact no issue with taking this limit, which merely collapses this solution to that of section 3.2.1. There is good reason for this, as one can actually generate this solution from section 3.2.1 by first T-dualising on ∂ ψ then performing a formal U-duality 6 on the Mink 3 followed by another T-duality on ∂ ψ . Additionally, this solution is also contained in section 4.2.2: it can be obtained by imposing that the coordinate x there (which should be identified with ψ in this section) is an isometry and then T-dualising it.
This concludes our IIB classification, we shall now turn our attention towards IIA.

Mink 3 with an S 3 factor in IIA
The type IIA supersymmetry conditions obtained from plugging (2.14) into the seven-dimensional supersymmetry constraints (2.4) lead to the following constraints on the four-dimensional bispinors while the fluxes are defined through and must additionally satisfy As before, we will first examine the 0-form constraints. These are given by two of the three constraints that were found for type IIB: Again, the solutions branch off similar to type IIB, with an α = 0 and an α = 0 branch. We parameterise Branch I as in (3.9) and Branch II as in (3.11).

Branch I: Solutions with α = 0
As was the case in IIB we first study the lower form conditions that follow from (3.1). After once more rotating the canonical frame of (B.3) by (3.13) we extract the necessary, but not sufficient, supersymmetry conditions where cos β(...) represents further terms which vanish when cos β = 0. These are sufficient to truncate the ansatz considerably. We note from (4.5a) that if sin δ = 0 then either H 0 = 0 or cos β = 0, however the latter also leads to H 0 = 0 because of (4.5b) -so sin δ = 0 implies H 0 = 0. We also observe that if sin δ = 0 then (4.5c) requires dθ = dφ = 0, or naively cos θ = 0 but this is a subcase of the former (see (3.9)), so sin δ = 0 also implies dθ = dφ = 0. Our task now is to show that, as in IIB, sin δ = 0 is a necessary condition: first we note that if we set cos δ = 0 then there is no solution as (4.5c) sets the vielbein to zero, thus we can restrict our considerations to 0 < sin δ < π 2 where (w 1 , v 2 ) must span an S 2 . However, as was the case in IIB, (4.5e) can be rewritten as d(e 2A+2C−Φ sin β(cos δw 1 − sin δv 2 )) + 2νe 2A+C−Φ (sin βw 1 ∧ v 2 + cos β sin δv 1 ∧ w 2 ) = 0, (4.6) using (4.5d) which excludes this because w 1 ∧ v 2 gives rise to a Vol(S 2 ) term that can not be cancelled by the parts involving (v 1 , w 2 ). Thus we can once more conclude that Given this, we can write the necessary and sufficient solutions for supersymmety in the α = 0 limit in a relatively simple way. After rotating the canonical frame, this time as in (3.16), we find This is as far as we can go without making assumptions about β, which we now proceed to do.

Sub Case: β = 0
Setting β = 0 in (4.8) immediately leads to H 3 = 0, the rest of the conditions are implied by The first thing we note is that given (4.9c)-(4.9d), G + must be a 0-form and G − a 1-form which means (4.9e) is solved automatically. Next, we solve (4.9a) by using it to define the vielbein where g is a function parametrising a potential coordinate transformation in x. From eq. (4.9c), we see that the combination e A−Φ only depends on x and that e A , e Φ and e C are functions of x, ρ only, so that ∂ ψ 1 and ∂ ψ 2 are isometries. We thus choose to parametrise the latter of which is a convenient choice we make without loss of generality. For the fluxes, we use the v 1 , v 2 , w 1 , w 2 vielbein on M 4 to compute the Hodge duals from eq. (4.9c) and (4.9d), arriving at the ten-dimensional fluxes The Bianchi identities reduce to dF 0 = dF 4 = 0 which leads to the former of which can be immediately integrated as f = (c + F 0 x), dc = 0. The metric takes the form (4.14) This solution corresponds to an intersecting D4-D8 brane system, where the localised D4-branes are embedded in the D8-branes [40].

Sub Case
The β = π 2 limit of (4.8) leads to H 3 ∝ v 1 ∧ w 1 ∧ w 2 with the remaining conditions implied by where we introduce u = (v 1 , w 1 , w 2 ). (4.16) to ease notation, and to make clear the cyclic property of these vielbein. The first thing we note is that, given (4.15c), the second of (4.15e) reads 4 H 3 ∧v 1 ∧w 1 ∧w 2 = 0, but since H 3 ∝ v 1 ∧w 1 ∧w 2 we must set H 3 = 0 . Next one can show that both (4.15a) and (4.15b) together imply the useful identities so we must have Consistency of the first of these with (4.15a) implies that c i = 0, and with this fixed (4.15b) is also follow from (4.19). We can use the standard trick of taking (4.19) to define a vielbein in terms of local coordinates, namely Having defined the vielbein, it is then a simple matter to solve the first of (4.15e) by introducing a free function All that remains is to calculate the fluxes, and impose their Bianchi identities. Using (4.20) to take the Hodge duals of 4.15c and 4.15d we find the 10d fluxes which clearly means the Bianchi identities, away from localised sources, follow from The metric takes the form , (4.24) The solution corresponds to an intersecting D2-D6 brane system [44].

Sub Case: Generic β
For 0 < β < π 2 one is able to divide by sin β, cos β freely when simplifying (4.8). Assuming that cos β = 0 the result is where H 3 is closed given (4.25a) and (4.25c). As usual we solve (4.25a) and (4.25b) by using them to define a vielbein in terms of local coordinates where g(x 1 ) is a function parametrising a potential diffeomorphism in x 1 . With local coordinates introduced we can solve (4.25c) in terms of them as so that ∂ ψ i are necessarily isometry directions. We can then calculate the ten-dimensional fluxes as before -first we note that (4.28) should be constant. We shall restrict ourselves to the case ∂ x c = 0. For generic β then the fluxes may be expressed as We note that the Bianchi identities follow when anything not coupled to B 2 is closed, and since these terms reproduce (4.12) the Bianchi identities imply the PDEs of (4.13) once more. This is because the class of solutions in this section can be generated via U-duality, from the intersecting D4-D8 system in section 4.1.1. For completeness the metric takes the form 30) where T 2 is spanned by (ψ 1 , ψ 2 ).

Branch II: α non-Zero
For the second branch with 0 < α < π, we begin by studying the lower form conditions that follow from (4.1). As in IIB we find it useful to rotate the canonical frame of (B First from (4.31a) we observe that when cos(α − δ) = 0 (cos β = 0 is a subcase of this) we necessary have H 0 = 0. Next we observe that generically (4.31b)-(4.31e) can be used to locally define the vielbein on M 4 , the only exception is when sin β = 0 (cos α = 0 is a subcase of this). Setting sin β = 0 means that in order to solve (4.31d)-(4.31e) we must take δ = π 2 , additionally (θ, φ) drop out of the definition of the spinors so we can fix without loss of generality. Interestingly one doesn't need to set H 0 = 0, however as we shall see in section 4.2.1 this case actually contains no solutions. If one assumes sin β = 0 we see that v 1 , w 1 must span S 2 while v 2 , w 2 can be expressed in terms of local coordinates in such a way that they have no legs in S 2 . This is a problem for (4.31f) which generically has an Vol(S 2 ) factor, due to the v 1 ∧ w 1 term which sits orthogonal to everything else. Thus the only resolution is to fix cos(α − δ) = 0 which leads to H 0 = 0 also. This actually leads to a novel class of solutions that we shall derive in section 4.2.2.

Sub Case: β = 0
Upon setting β = 0 we are lead to the following conditions for supersymmetry where here as elsewhere u = (v 1 , w 1 , w 2 ). Using the same techniques as are spelled out in section 3.1.2, it is possible to establish that however plugging this back into (4.33a) leads to which cannot be solved.

Sub Case
The final case we consider is when δ = α + π 2 and contains β = π 2 as a sub case. In addition to the rotating the canonical frame (B.3) by (3.44) we find it useful to send v 1 + iw 1 → e −iβ (v 1 + iw 1 ), then the necessary and sufficient conditions for supersymmetry are e C cos α sin βdθ + 2ν sin αv 1 = e C cos α sin β sin θdφ + 2ν sin αw 1 = 0 , (4.38c) where we introduce k 1 = (sin α cos βw 2 + sin βv 2 ), k 2 = (sin α cos βv 2 − sin βw 2 ), k 3 = (cos αv 2 − sin α sin βw 2 ), k 4 = (cos αw 2 + sin α sin βw 1 ), ∆ = sin 2 β + cos 2 β sin 2 α, Vol(S 2 ) = sin θdθ ∧ dφ, (4.39) to ease presentation. We can use (4.38c)-(4.38d) to locally define the vielbein through where (θ, φ, x) and ρ = e We can now turn our attention to (4.38a) and (4.38e) which lead to the PDEs and tell us that e 2A , e 2C , e Φ , α, β are functions of (ρ, x) only, which means these solutions support an additional SU (2) isometry due to round S 2 spanned by (θ, φ). Actually this SU (2) is an additional part of an enhanced R-symmetry which together with the SU (2) R of S 3 gives SO(4) R -there is also an SU (2) flavour symmetry. It is now a simple matter to calculate the Hodge dual of the fluxes from and then the fluxes themselves given (4.38d) and our local vielbein (4.40). At first G ± take a complicated form that we will not quote here, however, we are yet to impose also (4.38g): doing so and making extensive use of (4.43a)-(4.43c) one can express the ten-dimensional fluxes as: where (4.38g) can be expressed in terms of F 0 as Imposing that F 0 is constant together with (4.43a)-(4.43c) and (4.45) then implies dH = 0 and the Biachi identities of the remaining fluxes. This system is quite complicated, however taking inspiration from section 4.1 of [36] (which re-derives [49]) we anticipate that the system can be further simplified if we treat the cases F 0 = 0 and F 0 = 0 separately.
If we set F 0 = 0 then (4.45) and (4.44) impose that ρ e 2A sin α = L 2 , dL = 0 (4.46) This leaves (4.43a)-(4.43c) to solve. We first integrate (4.43a) as e A−C sin α = c, dc = 0, (4.47) then use it to express (4.43b) as We note that this defines an integrability condition that we can solve by introducing an auxiliary function h(ρ, x) such that Plugging these definitions into (4.43c) and making use of (4.46)-(4.47) we arrive at This is a 3d Laplacian expressed in axially symmetric cylindrical polar coordinates (up to rescaling x) . Solution in this class are in one to one correspondence with solution to this Laplace equation. The physical data can be expressed in terms of h and the 2 constants (c, L), as It is interesting that this class depends on axially symmetric Laplacian, indeed the same is true of the class of AdS 5 × S 2 solutions in IIA [50] one obtains by dimensionally reducing the M-theory class of Lin-Lunin-Maldecena [51]. The M-theory class actually depend on a 3d Toda equation, which is equivalent to the Laplacian only when one imposes an additional U (1) isometry, which one then uses to reduce to IIA. As the class of this section is in massless IIA it can be lifted to M-theory, so an obvious question poses itself: Is there a class in M-theory governed by a 3d Toda from which the backgrounds in this section descend? It would be interesting to look into this and what connection, if any, this class has to AdS 5 × S 2 or indeed any AdS class.
We expect to be able to perform a similar simplification of the sytem of PDE's for F 0 = 0 case, however up to this point we have failed to do so in general. However there is a special case which is far more simple, namely β = π 2 . Here (4.43a)-(4.43c) and (4.45) force all that remains is to ensure that dF 0 = 0 which is ensured as long as This solution has all the generic fluxes except the internal part of F 6 turned on, it bears some resemblance to D8-branes on some sort of cone, but the precise picture depends on what values the free constant α takes.
We shall come back to study the solutions that follow from these massless and massive systems in [71].

The unique type II AdS 4 × S 3 background
We have classified all Mink 3 × S 3 with internal Killing spinors of equal norm, up to certain PDE determining various warp factors. As equal norms is a requirement 7 for the existence of AdS 4 7 It is established in Appendix D that the 7d spinors χ 1 , χ 2 must obey the relation |χ 1 | 2 ± |χ 2 | 2 = c ± e ±A where c ± are constants. We can, without loss of generality solve these conditions in terms of unit norm spinors χ 0 i and an angle ζ as To make a Mink 3 solution AdS 4 requires us to fix the dependence of e 2A on the AdS radius, but since e 2A sin ζ is constant, we must either set ζ = c − = 0 or fix ζ such that it also depends on the AdS radius. The latter contradicts the assumption of an SO(2, 3) isometry, so we conclude that AdS 4 requires c − = 0; ie equal 7d spinor norms.
it is reasonable to ask whether such solutions are contained within our classification. Any AdS 4 solution can be expressed as a Mink 3 solution, one need only parametrise AdS as the Poincaré patch. This comes about quite naturally in terms of the Mink 3 × M 7 set up by imposing that We will now show that there is a unique compact 8 AdS 4 solution, at least locally, for the class of solutions of section 4.1.2. This background corresponds to a foliation of AdS 4 × S 3 × S 2 over an interval and is the near-horizon of a D2-D6 brane intersection, and can also be obtained by dimensionally reducing a certain Z k orbifold of AdS 4 × S 7 . Starting from M-theory one first parameterises S 7 as a foilation of S 3 ×S 3 over a closed interval, then performs both the orbifolding and reduction to IIA on the Hopf fibre of one of the S 3 's (see e.g. [52]) -there by preserving 16 supercharges. For this purpose, we take the metric (4.24), expressed in the form and assume an AdS 4 factor, which requires e 2A = r 2 e 2Ã , with the rescaled warp factorÃ as well as the dilaton Φ undetermined functions independent of the AdS 4 radial coordinate r. Furthermore, the background is only SO(2, 3) invariant if the internal metric is independent of r, which fixes ρ and the x i to scale as ρ ∼ r 1/2 and x i ∼ r. Keeping this in mind, we parametrise x 2 = r q(µ) sin θ sin φ , where q(µ) and h(µ) are undetermined functions of some coordinate µ, and the (θ, φ) directions parametrise a 2-sphere such that the R 3 spanned by x i is written in polar coordinates with radius r q(µ). Now we have to ensure that the metric is diagonal with respect to the r-direction, i.e. set g r µ = 0, and that it shows the 1/r 2 behaviour for g rr , which amounts to imposing g rr = e 2Ã /r 2 . These two conditions lead to the following expressions forÃ and Φ in terms of q(µ), h(µ) and independent of (θ, φ): These expressions imply, once inserted in the first eq. of (4.15e), the following ODE for the q and h functions: which can be solved in closed form as h = h q(µ) and also implies the Bianchi identities of the fluxes. As h is a function of q, rather than µ, we can use diffeomophism invariance to fix q such that h is simple, without loss of generality we choose where L and k are constants. This leads to The resulting metric is of the form with fluxes This is the IIA reduction of AdS 4 × S 7 /Z k with length scale L and k D6-branes, as in eq. (2.8) of [52]. The fact that (θ, φ) are isometry directions of this solution means that there is an additional SU (2) S 2 symmetry due to the round S 2 factor in the metric and fluxes. The spinors of this solution are then charged under SU (2) S 2 and just one of the SU (2)'s of S 3 (see the ν dependence of (4.22)), SU (2) + say. Since S 2 and S 3 appear as a product the spinors are actually charged under SU (2) + ×SU (2) S 2 which realises an enhanced SO(4) R-symmetry as required by the N = 4 super-conformal algebra in 3d -SU (2) − , under which the spinors are not charged, is a flavour symmetry.

Type II with a single Killing spinor
In the previous two sections, we have worked out the supersymmetry conditions making use of the pure spinor equations (2.4), which are valid only in case |χ 1 | 2 = |χ 2 | 2 . Note that this is a necessary condition for the existence of D-branes which do not break background supersymmetry.The supersymmetry condition for a D p -brane is given by Γ (p) 1 = 2 . Since Γ (p) is unitary, squaring this equation leads to the conclusion that left-and right-handside must have equal norm. 9 We will examine the simplest non-equal norm case, namely the one where We could either make use of the generalised geometrical reformulation of supersymmetry which incorporates |χ 1 | 2 −|χ 2 | 2 = 0 as deduced in appendix D, or use the actual Killing spinor equations.
Considering the simplicity of this case, we will use the latter. Much of the work has however already been done: the conditions for seven-dimensional pure NSNS solutions have been deduced in [29,56,57] up to some ansätze. We will merely show that the ansätze made in [29,56,57] (no warp factor, no external NSNS flux) are in fact enforced by supersymetry, and then proceed to plug in the decomposition resulting from M 7 = S 3 × M 4 . This leads to a pair of explicit pure NS backgrounds: the NS5-brane and the U-dual to the IIB conical backgrounds of section 3.2.1. We will also analyse the seven-dimensional RR-sector, which is new, but the conclusion is that all RR-fluxes vanish.

Seven-dimensional decomposition
Our starting point are the democratic supersymmetry equations, which read as follows for 2 = 0: As can be seen, the NSNS and RR sectors decouple. We impose a similar 3 + 7 decomposition as before: the metric and RR flux is given by (2.1), while the Killing spinor 1 is given by (2.10) and 2 = 0. We generalize the NSNS 3-form flux by allowing a term h e 3A Vol 3 . Using the convention γ µνρ = µνρ , plugging the above decompositions into (6.2) leads to the 9 The argument is slightly more complicated due to the fact that Spin(1, 9) spinors do not admit a non-trivial norm, and hence one should decompose to Spin(9) first. See [53] for details. Also note that strictly speaking, the norms need only be equivalent on the brane.
following 7d equations: As can be seen, the NSNS and RR sector split and can thus be analysed independently. The existence of a globally defined nowhere-vanishing Spin (7) Majorana spinor χ reduces the structure group of M 7 to G 2 . More concretely, the following bilinears can be defined: where we have normalized χ. The other bilinears, i.e., the 1-, 2-, 5-and 6-form vanish. As has been deduced in [29,56,57], (6.3a) can be rewritten in terms of the G 2 -structure as We will analyse the solution to the NSNS sector by requiring a further splitting of M 7 = S 3 × M 4 . On the other hand, we will show that the RR-fluxes vanish for any M 7 .

NSNS sector
Considering the case M 7 = S 3 × M 4 , we further decompose the spinor as χ = ξ ⊗ (sin(α/2)η + cos(α/2)γη) + m.c., η † η = 1, η †γ η = 0 , (6.7) with m.c. the Majorana conjugate. This leads to a further reduction of the structure group. Since S 3 is parallelisable, it has trivial structure group, leading to a Spin(4) structure group on M 4 . Generically, the structure group need not reduce on M 4 . 10 In the case where either η + or η − is nowhere vanishing, the structure group reduces to SU (2), in case both are nowhere-vanishing, the structure group is trivial. As everywhere else, our analysis is purely local and we will work with a local trivial structure, parametrising possible vanishing of either chiral spinor by the angle α.
First, as in [58], we make use of an auxiliary SU (3)-structure (J, Ω) to express the G 2 -structure as . (6.8) Next, we decompose the SU (3)-structure in terms of the vielbeine as Inserting this into (6.6), one finds When α = π 2 , by taking linear combinations, exterior derivatives and wedge products with the vielbein of the equations in (6.6), one can derive where, since u i form a basis of independent 1-forms, the terms in square parentheses must vanish. This is sufficient to conclude that It is then not hard to establish that in a similar fashion. This means we can locally parametrise Plugging this back into (6.6) we find that c i = e −C ν tan α, (6.14) so either dC = 0 or c = α = 0.
Case 1: When α = 0, u i are the vielbeine of T 3 so we can simply take All that is left to do is calculate H and impose its Bianchi identity. We find Closure of the flux then implies that e 2Φ is harmonic, leading to 17) which is consistent with the definition of ρ. Finally, we note that the metric is given by This is an NS5-brane, dual to the D5-brane solution in section 3.1.2 [41].

B The bispinors of M 4
We consider gamma matrices satisfying γ abcd = abcd , γ (4−n) = (−1) which furnish an SU (2)-structure. Given two globally well-defined nowhere vanishing chiral spinors of opposite chirality, the structure group reduces to a trivial structure [60]. Generically, supersymmetry requires a nowhere vanishing spinor η, which can admit a chiral locus. This ensures that, although the structure group of M 4 cannot be globally reduced, it is possible to reduce the structure group of the generalised cotangent bundle T M 4 ⊕ T * M 4 to SU (2) × SU (2), completely analogously to the well-known situation of SU (3)-structures [30]. Since the supersymmetry constraints are local, we will always work with the vielbeine determining the local trivial structure. Using the conventions of [61] with η = (η + , η − ), we set Although some care must be taken on the chiral locus, where the above 1-forms all vanish, it turns out that no solutions exist on the chiral locus, as discussed in sections 3 and 4. We can expand the locally defined 4d components of the Killing spinors η 1,2 in terms of η as where a, b, c and d are subject to We can then calculate the 4d bispinors appearing in (3.1a)-(3.1f) and (4.1a)-(4.1f), where to do so we find it useful to parametrise for a i , b i , c i and d i real. However we first note that in both IIA and IIB we must solve the 0-form constraints We can solve these in general by fixing which turns (B.6) into In terms of this parametrisation the 4d bispinors are given by C The SU (2) doublets of S 3 There exist two independent spinors on S 3 that obey the Killing spinor relations each of which preserves two supercharges. Additionally the global isometry group of S 3 can be decomposed as SO(4) = SU (2) + × SU (2) − , so S 3 supports two sets of SU (2) Killing vectors K i ± , i = 1, 2, 3, that are dual to one forms that obey i.e. the right-/left-invariant forms of SU (2). It is possible to use the spinors on S 3 to construct SU (2) ± doublets. Consider the following vector with spinor entries These transform under the action of the spinoral Lie derivative as 11 for σ i the Pauli matrices, which means that ξ a ± transforms as a doublet under local SU (2) ± transformations and a singlet under SU (2) ∓ .

D Supersymmetry conditions for three-dimensional external spacetimes
In [31], supersymmetry conditions for 3+7 dimensional compactifications are given in terms of bispinors. The repackaging of the supersymmetry conditions was done under the following conditions: • The external space is Minkowski.
• The spinors have equivalent length.
• The NSNS flux H does not have an external component.
In this section, we will look at relaxing the latter two conditions to obtain more general solutions. Our starting point will be the ten-dimensional bispinor description of the supersymmetry 11 The spinoral lie derivative along a Killing vector K is defined as The easiest way to see that this leads to the claimed transformation property, is to parametrise the vielbein on S 3 as e 1 = 1 2 dθ, e 2 = 1 2 sin θdφ, e 3 = 1 2 (dψ + cos θdφ) and take the flat space gamma-matrices to be the Pauli matrices σ i . Then (C.1) is solved by ξ + = e i 2 θσ1 e i 2 φσ3 ξ 0 + , ξ − = e − i 2 ψσ3 ξ 0 − for ξ 0 ± constant 2d spinors. The SU (2) ± forms are then precisely constraints, as described in [37]: The final two equations are known as the pairing equation; we refer to [37] for more details, and will follow along the lines of section (4.2) in the following.
We consider the case where the Killing spinors are given by (2.10). Due to the properties of Spin(1, 2), we can define with the other bilinears vanishing. Since we are considering flat space, ζ are covariantly constant, hence dv = 0. Making use of the spinor decomposition, it follows that where we have defined Φ + + iΦ − = 8e −A χ 1 ⊗ χ † 2 and K,K should be read as 1-forms in ten dimensions. Using the flux decomposition (D.5) We will first solve (D.1c): since by construction, v, K are Killing vectors, we must have Next lets consider (D.1b), which leads to Next, let us consider (D.1a). We find that In addition, the fact that c − = 0 does not change the argument of [37], so the pairing equations remain unchanged, leading to

E M-theory
The focus of this paper are backgrounds in type IIA and type IIB. In this appendix, we will discuss M-theory backgrounds on R 1,2 × S 3 × M 5 . Given equivalent internal spinor norms, our (massive) IIA classification is complete (up to finding solutions to PDE). Therefore, a significant number of backgrounds one would obtain from a similar analysis of M-theory are those which one can obtain from uplifting our massless IIA backgrounds. Novel solutions from a complete M-theory analysis would be backgrounds satisfying one of the two conditions: either M 5 does not admit an S 1 factor to be integrated out to perform the dimensional reduction to IIA, or the internal component of the Killing spinor on M 8 = S 3 × M 5 is such that after the reduction, the resulting seven-dimensional internal components of the IIA spinors are not of equal norm.
Such a full M-theory classification is beyond the scope of this paper. Instead, we aim to make contact with the literature of M-theory on R 1,2 × M 8 , which is much studied (see for example [62][63][64][65][66][67]). We will derive the decomposed supersymmetry conditions, and give several simple classes of solutions.
For N = 1 solutions to the supersymmetry constraints on R 1,2 × M 8 , the Killing spinor decomposes as where ξ is a Majorana spinor of Spin(1, 2) and χ ± are chiral Majorana spinors of Spin (8).
Generically, χ ± can have zeroes, and the structure group of M 8 is SO(8), although a Spin(7)structure can be defined on the auxiliary space M 8 × S 1 [66]. In the case where one of the two does not vanish, the structure group reduces to Spin(7) [65]. If both chiral spinors have no zeroes, both of them define a Spin(7)-structure: the intersection of the two leads to a reduction of the structure group to G 2 . 12 The reduction of the structure group leads to the existence of globally defined invariant tensors. Instead, we will work locally, and consider patches where either one or both are non-zero.

E.1 Spin(7) holonomy
Let us first examine the case with Following the conventions of [65], the general solution to the M-theory supersymmetry constraints with these ansätze is that where ds 2 (M 8 ) a metric of Spin (7) holonomy. The four-form F lies in the 27 of Spin (7), i.e., it satisfies with Ψ mnpq =χγ mnpq χ the invariant four-form defining the Spin(7)-structure. In addition, the Bianchi identity and equation of motion for F require that F is harmonic and satisfies away from M2-brane sources.
(E.8) 12 Note that this is only the case for two Spin(7)-structures defined in terms of opposite chirality spinors.
Given two same chirality globally well-defined nowhere vanishing spinors, the structure group instead reduces to Spin(6) SU (4).
By making use of this decomposition, the Spin (7) four-form decomposes in terms of the I × SU (2)-structure as . (E.10) Using the decomposed Ψ, we examine the supersymmetry conditions. First, we consider the flux component F , which we decompose as with F 1 ,F 1 one-forms on M 5 . Inserting this and (E.9) into (E.4), it follows from (m, n) = (a, b) that F 1 ∼F 1 , F a 1 V a = 0 and from (m, n) = (α, a) that F a 1 ω ab = F a 1 ω * ab = F a 1 J ab = 0. Hence In general, this means that locally V = e −3C dτ , and we will write ds 2 (M 8 ) = e 2C ds 2 (S 3 ) + e −6C dτ 2 + ds 2 4 (E.14) Let us give some simple classes of examples for which the above conditions are solved.
• Next, let us examine Sasaki-Einstein structures, as well as a class of generalizations. It can be shown that any five-dimensional Sasaki-Einstein can be defined by means of a set of real forms (Ṽ , ω j ), j = 1, 2, 3 withṼ a one-form and ω j two-forms. These satisfy [15] dω 1 = d(ω 2 + iω 3 ) + 3iṼ ∧ (ω 2 + iω 3 ) = 0 . (E.17) A more general class of spaces are the so-called hypo manifolds [69], satisfying dω 1 = d(Ṽ ∧ ω 2 ) = d(Ṽ ∧ ω 3 ) = 0, which themselves are a subclass of balanced manifolds [70], satisfying By setting J = ω 1 , e c V =Ṽ , Reω = ω 2 , Imω = ω 3 , it follows that any solution to the supersymmetry constraints is a balanced metric. On the other hand, any solution to the supersymmetry constraints which is hypo automatically is such that ds 2 4 is Calabi-Yau. This leads to the conclusion that the spinors do not define a Sasaki-Einstein on M 5 , as the base space of a Sasaki-Einstein manifold is not Ricci-flat. Another way to see this is to note that Sasaki-Einstein metrics can be written as a fibration over a Kähler-Einstein base, but it is clear that since the supersymmetry constraints are invariant under permutations of (J, Reω, Imω), ds 2 4 cannot be non-Calabi-Yau Kähler.
We now express the G 2 -structure (E.22) in terms of the trivial structure of S 3 × M 5 . The first bilinear we calculate is K = e C 2 (Ima 0 K 1 + Rea 0 K 2 − ImaK 3 ) − bu 2 − ReaV, (E. 28) and the only way to make this compatible with (E.23a) is to set a 0 = Ima = 0 (E.29) so that the spinors η 1,2 are nowhere parallel. We are now free to parametrise b = cos α, Rea = sin α (E. 30) and rotate to a frame where K = V. where Vol 7 is the volume form of the manifold spanned by the warped left-invariant forms of S 3 and the vielbein, with orientation e C 2 K 1 , e C 2 K 2 , e C 2 K 3 , u 1 , e 1 , e 2 , e 3 .