The geometry of ${\cal N}=3$ AdS$_4$ in massive IIA

The geometry of the ${\cal N} = 3$, SO(4)--invariant, AdS$_4$ solution of massive type IIA supergravity that uplifts from the ${\cal N} = 3 $ vacuum of $D=4$ ${\cal N} = 8$ dyonic ISO(7) supergravity is investigated. Firstly, a $D=4$, SO(4)--invariant restricted duality hierarchy is constructed and used to uplift the entire, dynamical SO(4)--invariant sector to massive type IIA. The resulting consistent uplift formulae are used to obtain a new local expression for the ${\cal N} = 3 $ AdS$_4$ solution in massive IIA and analyse its geometry. Locally, the internal $S^6$ geometry corresponds to a warped fibration of $S^2$ and a hemisphere of $S^4$. This can be regarded as a warped generalisation of the usual twistor fibration geometry. Finally, the triplet of Killing spinors corresponding to the ${\cal N}=3$ solution are constructed and shown to obey the massive type IIA Killing spinor equations.


Introduction
Massive type IIA supergravity [1] admits a consistent truncation on the six-sphere to maximal supergravity in four dimensions with gauge group ISO(7) = SO(7) ⋉ R 7 [2,3]. The gauging is dyonic, in the sense of [4,5] (see also [6]). By virtue of the consistency of the truncation, all solutions of the four-dimensional theory uplift on S 6 to solutions of massive type IIA supergravity. In particular, the critical points (which can only be AdS) of the four-dimensional scalar potential give rise to supersymmetric and non-supersymmetric ten-dimensional solutions of the form AdS 4 × S 6 . This product is generically warped and the metric on S 6 displays an isometry group G ⊂ SO(7) related to the residual symmetry within ISO(7) supergravity of the critical point it uplifts from. Using this technique, new massive type IIA solutions have been found [2,7,8] and previously known ones [1,9,10] have been recovered. Other supersymmetric AdS 4 solutions of massive type IIA supergravity have been recently found using other methods in [11,12]. Previous constructions of supersymmetric AdS 4 solutions in massive type IIA supergravity include [13,14,15,16,17].
In this paper, we investigate the ten-dimensional uplift of the N = 3 SO(4)-invariant critical point of dyonic ISO (7) supergravity. This D = 4 critical point was found in [18]. A local form of its massive type IIA uplift has already appeared in [8]. Here, we provide an alternate local form of this N = 3 AdS 4 solution of massive IIA supergravity (equation (5.1)) and discuss its geometric features. The internal space of the N = 3 solution is topologically S 6 , endowed with a geometry that can be locally regarded as an S 2 bundle over a half-S 4 . This is a generalisation of the twistor bundle over a quaternionic-Kähler manifold of positive curvature, see e.g. [19,14] for reviews. The twistor fibration allows one to engineer nearly-Kähler or half-flat geometries on six-manifolds M 6 of topology different than S 6 , see e.g. [14]. In turn, a well known class of N = 1 (direct) product solutions AdS 4 × M 6 of massive IIA supergravity entails a nearly-Kähler [9] or a half-flat structure [14,15] on M 6 .
It is suggestive that this N = 3 solution formally corresponds to an elaboration of these N = 1 constructions. This is reminiscent of the situation for a well-known class of D = 11 direct product solutions involving AdS 4 and a tri-Sasaki seven-manifold. Recall that the latter can be regarded as an S 3 bundle over a quaternionic-Kähler base, equipped with an Einstein metric on the total space. This class of solutions is N = 3, see e.g. [20]. On each tri-Sasaki manifold, a second Einstein metric can be obtained by squashing the S 3 fibers by a certain constant amount. The resulting D = 11 AdS 4 solution is N = 1, see [21,20]. The analogy with these N = 3 and N = 1 solutions of D = 11 supergravity should not be taken too far, though. The internal metric of the massive IIA N = 3 solution is certainly not Einstein, unlike the N = 1 nearly-Kähler solutions of [9]. In the IIA N = 3 solution, the S 2 fibers are squashed, not by a constant, but by a warping function of the S 4 hemisphere base. The connection does not have definite duality properties, unlike in the usual twistor fibration. Finally, the N = 3 solution involves a warped, rather than direct, product of AdS 4 and the internal topological S 6 . Like in the D = 11 tri-Sasaki case, though, the SO(3) R-symmetry acts on the fibers of the N = 3 massive IIA solution.
The type IIA N = 3 solution displays a local SO(4) symmetry, inherited from that preserved by the N = 3 critical point of the D = 4 supergravity. More generally, we construct in section 2 the restricted, in the sense of [22], duality hierarchy [23,24] of D = 4 ISO(7) supergravity that is invariant under this SO (4). This result is particularly useful, as it allows us to consistently embed the entire, dynamical SO(4)-invariant sector of the D = 4 N = 8 supergravity into massive type IIA. The explicit consistent uplift formulae are presented in section 3. These formulae give the ten-dimensional uplift of any SO(4)-invariant solution of the ISO(7) supergravity, including solutions with running scalars. The local and global features of this consistent embedding formulae are discussed at length, and generalisations are given. Section 4 discusses further truncations. The truncation to the dynamical G 2 -invariant sector of [3] is recovered, and an example that illustrates the usefulness of the duality hierarchy approach is worked out. In section 5, we turn our attention to the massive IIA uplift of solutions of the D = 4 supergravity, focusing on vacuum solutions. In particular, a new local form of the N = 3 AdS 4 solution in massive IIA is provided. Finally, in section 6, the solution is demonstrated to indeed be N = 3 by explicitly building the triplet of Killing spinors that it preserves.
In a conventional formulation, only the metric and the scalars in (2.4) enter the D = 4 Lagrangian. This reads [22] L = (R − V ) vol 4 + 6 2 dϕ ∧ * dϕ + e 2ϕ dχ ∧ * dχ + 1 2 dφ ∧ * dφ + e 2φ dρ ∧ * dρ , (2.6)   (7)-dyonically-gauged supergravity with invariance equal or larger than the SO(4) subgroup of SO(7) defined in (2.1). For each point we give the residual supersymmetry N and bosonic symmetry G within the full N = 8 theory, its location, the cosmological constant V and the reference where it was first found. We have employed the shorthand c ≡ m/g. All of these data are reproduced from [22].
with the scalar potential given by [22] (2.7) The constants g and m are the electric and magnetic gauge couplings of the parent N = 8 ISO(7) supergravity. When gm = 0, the scalar potential (2.7) contains AdS critical points that spontaneously break the N = 8 supersymmetry and ISO(7) gauge symmetry of the full D = 4 supergravity to some supersymmetry N and residual symmetry G. See table 1 for a summary. The N = 3 SO(4)-invariant point manifests itself as non-supersymmetric within the subtruncation (2.6), (2.7), see [22] for further details. All these critical points are inherent to the dyonic ISO(7) gauging and disappear in the purely electric g = 0, m = 0, or purely magnetic, g = 0, m = 0 limits. Accordingly, these four-dimensional solutions naturally uplift to massive type IIA supergravity on S 6 and do not have direct counterparts in either massless IIA on S 6 or massive IIA on T 6 .
The three-and four-form field strengths of the SO(4)-invariant two-form, B, and threeform potentials, C 1 , C 2 , are These expressions follow from the generic expressions given in (2.8), (2.9) of [22] evaluated on equation (2.5) above. These field strengths are subject to the Bianchi identities These in turn correspond to the SO(4)-invariant truncation of the generic, N = 8 SL (7)covariant Bianchi identities given in in (2.13) of [22].
Not all of the fields in the SO(4)-invariant, restricted tensor hierarchy (2.4) carry independent degrees of freedom: the field strengths of the form potentials are subject to duality relations, see [24,22] for a generic discussion. Particularising the SL(7)-covariant duality equations (2.17), (2.18) of [22] to the present case, we find the following duality relations obeyed by the SO(4)-invariant field strengths: The Bianchi identities (2.9), combined with the duality relations (2.10), reproduce the scalar equations of motion that follow from the Lagrangian (2.6), (2.7). Even though it does not play a critical role in the IIA uplift, it is nevertheless useful to consider the SL(7)-singlet four-form field strength whose duality relation was given in (2.25) of [22]. In the SO(4)-invariant case at hand, this duality relation reads H (4) = e φ+6ϕ g χ 2 3ρ + 4χ − m vol 4 . (2.11) Using (2.10), (2.11), the scalar potential (2.7) can be checked to be related to the four-form field strengths H 1 (4) , H 2 (4) andH (4) through This is the SO(4)-invariant counterpart of the full N = 8 expressions (2.28), (2.29) of [22]. At any of the critical points of the scalar potential (2.7), that were summarised in table 1 above, these four-form field strengths turn out to obey where | 0 denote evaluation at a critical point. We conclude by recovering two interesting sectors of D = 4 N = 8 ISO(7) supergravity from the SO(4)-invariant sector. Firstly, according to the branching rule (2.1), the SO(3) ′ × SO(4) ′ -invariant sector is contained in the SO(4) sector. This is recovered by setting the pseudoscalars to zero, while retaining all other fields in the duality hierarchy (2.4). The SO(3) ′ ×SO(4) ′ -invariant Lagrangian, tensor field strengths, Bianchi identities and duality relations follow by letting χ = ρ = 0 in the expressions above. Secondly, as discussed in [22], the G 2 -invariant sector can be also recovered from the SO(4)-sector. This is apparent from the branching (2.2). The G 2 -invariant sector is recovered from the SO(4)-invariant sector through the identifications ϕ = φ , χ = ρ , B = 0 , C 1 = C 2 ≡ C , (2.15) along with H (3) = 0 and H 1 (4) = H 2 (4) ≡ H (4) . These identifications bring the Lagrangian and duality relations to their G 2 -invariant counterparts, given in section 4 of [22].

Truncation from type IIA supergravity
We are now ready to give the complete, non-linear embedding of the dynamical SO(4)invariant sector of D = 4 N = 8 ISO(7) supergravity into massive type IIA. As discussed in [3], the embedding of the full N = 8 theory is naturally expressed, at the level of the IIA metric, dilaton and form potentials, in terms of the restricted, SL(7)-duality hierarchy introduced in [22]. Accordingly, the complete IIA embedding of the SO(4)-invariant sector is naturally written in terms of the tensor hierarchy discussed in section 2.

Consistent embedding formulae
The SO(4)-invariant consistent embedding formulae can be obtained by particularising the N = 8 formulae given in (3.12), (3.13) of [3] (see also [2]) to the case at hand. It is a matter of simple algebra to find the embedding of the two-and three-form potentials of the D = 4 tensor hierarchy (2.4) into their D = 10 counterparts, using their N = 8 embedding (2.5). In contrast, as is usually the case, the embedding of the D = 4 scalars entails a lengthy computation. Here, we give the final result, referring to appendix A for further details on the relevant geometric structures that arise in the calculation.
In order to express the result, it is convenient to introduce constrained coordinatesμ i , i = 1, 2, 3, on the two-sphere S 2 , and right-invariant one-forms 1 ρ i , i = 1, 2, 3, on the three-sphere S 3 . These are subject to the Maurer-Cartan equations It is also convenient to introduce the following combinations of D = 4 scalars [22] and the following functions of D = 4 scalars and an angle α on the internal S 6 , ∆ 1 = e φ sin 2 α + e ϕ cos 2 α , The right-invariant one-forms ρ i on S 3 shouldn't be confused with the D = 4 pseudoscalar ρ.
Using these definitions, the complete nonlinear embedding of the SO(4)-invariant field content (2.4) of ISO(7) supergravity into type IIA reads, where we use the ten-dimensional Einstein frame conventions of appendix A of [3]. Indices i, j are raised and lowered with δ ij , and ds 2 (S 3 ) is the round metric on the S 3 on which the ρ i are defined. Also, we have introduced the following covariant derivative and one-form A i , These embedding formulae depend on the (non-vanishing) D = 4 electric gauge coupling g, but not on the magnetic coupling m. Thus, they simultaneously describe the embedding of the dynamical SO(4)-invariant sector of the purely electric, m = 0, and dyonic, m = 0, ISO(7) gauging of D = 4 N = 8 supergravity into massless and massive, respectively, type IIA supergravity. The consistent embedding formulae (3.5) are valid in full generality for D = 4 dynamical fields. However, being expressed in terms of the tensor hierarchy (2.4), they contain redundant degrees of freedom. As discussed in general in [3], these redundancies can be eliminated by expressing the consistent embedding in terms of the IIA field strengths and using the D = 4 duality relations. In the case at hand, the only contributions from the D = 4 form field strengths (2.8) happen to occur in the IIA four-formF (4) , where the dots stand for D = 4 scalar and derivative-of-scalar contributions without Hodge dualisations. Equation (3.7) follows from (3.5) after using the D = 4 definitions (2.8). It thus provides a ten-dimensional cross-check on the four-dimensional calculation of section 2. More importantly, equation (3.7) now expresses the consistent embedding in terms of the independent metric and scalar degrees of freedom contained in the D = 4 Lagrangian (2.6), (2.7), when the duality relations (2.10) are employed. A simpler example will be presented in section 4.2.

Local and global structure
For arbitrary values of the D = 4 scalars, the six-dimensional internal local geometry in (3.5) can be regarded as the warped product of an interval I, on which α takes values, and a family of five-dimensional spaces parametrised by α. At fixed α, the five-dimensional space corresponds to an S 2 bundle over S 3 , with connection one-forms A i given in (3.6). All such bundles are trivial. In the present case, this can be seen by the fact that, at fixed α, the curvature of the connection A i is identically zero by the Maurer-Cartan equations (3.2). This local characterisation is useful to discuss the global extension of the geometry, to which we now turn. It is not the only possible local description, though. A different local characterisation will be given below. Globally, the internal geometry extends smoothly into S 6 . This is expected from the fact that the D = 4 theory (2.6), (2.7) arises upon consistent Kaluza-Klein truncation of massive type IIA on S 6 via (3.5), and the Kaluza-Klein deformations are not supposed to change the internal topology. That the topology of the compactification space is indeed S 6 is most easily seen by continuously deforming the geometry into the G 2 -invariant locus (2.15). On this locus, the internal metric in (3.5) reduces to the usual, round Einstein metric (A.2) on S 6 . The local line element (A.2) is adapted to the topological construction of S 6 as the 'join' of S 2 and S 3 , provided the S 6 angle α is restricted to the interval On the G 2 -invariant locus and at α = 0, the S 2 remains finite and the S 3 collapses; at the other endpoint, α = π 2 , the opposite happens. The expression (3.5) makes it straightforward to continuously deform the internal geometry to the round metric on S 6 , since it is given as a function of the D = 4 scalar manifold (2.3). However, once the scalars are fixed to their specific values at some critical point of the potential (2.7), as e.g. in the explicit N = 3 solution (5.1) below, tracking down the deformation into the round S 6 geometry is no longer obvious. In such cases, it is more useful to directly characterise the internal S 6 by verifying that it still corresponds to the join of S 2 and S 3 . Namely, that the shrinking patterns of S 2 and S 3 at each endpoint of the interval I remain valid away from the G 2 -invariant locus. To see this, we use the definitions (3.4) to compute the behaviour of the relevant metric functions at both endpoints of I. At the lower end, Thus, S 2 remains finite and S 3 shrinks to zero size for all values of the D = 4 scalars. At the upper end, 15) and the opposite happens: S 2 shrinks and S 3 remains finite for all D = 4 scalar values.
An alternate local characterisation of the internal geometry in (3.5) may be given as follows. The local internal geometry may also be regarded as an S 2 bundle over the fourdimensional local geometry M 4 ≡ I × S 3 , where I is the interval (3.13) parametrised by α. This local construction is a generalisation of the twistor fibration over a fourdimensional Riemannian space M 4 . In the usual twistor construction, the metric ds 2 (M 4 ) on M 4 is taken to be Einstein with (anti)self-dual Weyl tensor. The local metric on the six-dimensional twistor bundle is see e.g. [29]. Here,μ i parametrise an S 2 as in (3.1), and the covariant derivatives Dμ i are defined as in the left most equation in (3.6), in terms of a M 4 -valued connection A i . Being four-dimensional and Einstein, M 4 is automatically quaternionic-Kähler. The curvature of the connection, is proportional to the quaternionic-Kähler forms J i on M 4 . The self-duality or antiselfduality of the Weyl tensor on M 4 devolves in the antiselfduality or self-duality of F i with respect to the metric ds 2 (M 4 ). For example, the twistor bundle on M 4 = S 4 coincides with the three-dimensional complex projective space, CP 3 . Taking ds 2 (M 4 ) to be the usual round metric on S 4 , (with α here ranging in 0 ≤ α ≤ π) and where the ρ i are the right-invariant Maurer-Carten one-forms on S 3 , subject to (3.2), the twistor bundle metric (3.16) becomes the homogeneous nearly-Kähler metric on CP 3 . The local internal metric in (3.5) is a generalisation of the twistor construction. In our case, M 4 ≡ I × S 3 is the upper S 4 hemisphere, given the range (3.13) of α. The metric ds 2 4 (M 4 ) induced on it is not selfdual Einstein for any values of the D = 4 scalars. On the G 2 -invariant locus (2.15) the S 2 fibration trivialises, A i = 0, and the local geometry becomes locally a warped product of S 2 and I × S 3 . Away from the G 2 -invariant locus, the S 2 is warped (unlike in (3.16)), and non-trivially fibered through (3.6) over I × S 3 . The curvature (3.17) of the connection A i is

20) and its Hodge dual with respect to the metric induced on
The non-trivial connection A i is neither selfdual nor antiself-dual for any values of the D = 4 scalars, as nowhere on the scalar manifold (2.3) do (3.20), (3.21) obey * F i = ±F i . Massive type IIA supergravity admits a class of N = 1 direct product solutions AdS 4 × M 6 where M 6 is nearly-Kähler [9] or half-flat [14,15]. For example, M 6 can be taken to be the round S 6 equipped with its homogeneous nearly-Kähler structure, see appendix A for a review in the present context. On topologies different from S 6 , a natural way to engineer nearly-Kähler geometries or half-flat geometries of the required type is via the usual twistor fibration [14]. For example, M 6 can be taken to be CP 3 with metric (3.16). Our local geometry (3.5) restricted to the G 2 -invariant locus (2.15) reduces to the round, homogeneous nearly-Kähler structure on S 6 . Away from the G 2 locus, as in the N = 3 solution of section 5, the geometry can be locally described by the generalised twistor fibration discussed above.
On the G 2 -invariant locus (2.15), the symmetry of the configuration (3.5) is enhanced to a homogeneously acting G 2 . See section 4.1 for further details. Away from the G 2 locus, the isometry of the internal geometry is the SO(4) subgroup of SO(7) defined in either (2.1) of (2.2). The group SO(4) acts by isometries with cohomogeneity one, and is also preserved by the supergravity forms. The SO(3) d subgroup of SO(4) rotates the S 2 fibers, and SO(3) R acts on the S 3 base. The supersymmetry of the N = 3 solution will be discussed in section 6.
Some generalisations can be envisaged. When the D = 4 scalars are restricted to the G 2 -invariant locus (2.15), the type IIA solution (3.5) depends only on the homogeneous nearly-Kähler structure on S 6 . In this case, the S 6 can be replaced with any other nearly-Kähler manifold. This situation was discussed in [7]. Away from the G 2 locus, the solution can be also generalised. Now, the generalisation entails replacing S 3 with the cyclic lens space S 3 /Z p , with the identification acting on the Hopf fiber. While S 3 /Z p is a smooth manifold, the total six-dimensional geometry corresponding to this generalisation displays orbifold singularities.

Further truncations
It is useful to obtain particular cases of the uplifting formulae obtained above. Here, we will discuss the truncations to the sectors of the D = 4 supergravity with G 2 and SO(3) ′ × SO(4) ′ symmetry.

Truncation to the G 2 sector
The sector of D = 4 ISO(7) supergravity that retains singlets under the G 2 subgroup of SO (7) was analysed in detail in [22], and its explicit ten-dimensional embedding worked out in [3]. Its consistent IIA embedding was recovered from that of the SU(3)-invariant sector in [7]. Here, we will recover the embedding of the G 2 -sector from the SO(4)-invariant consistent truncation formulae of section 3.1.
The D = 4 G 2 -invariant sector is recovered from the SO(4) sector by imposing the identifications (2.15). Bringing these relations to the consistent embedding formulae (3.5), we find that the connection (3.6) trivialises, A i = 0, and that the scalar dependence of the internal metric factorises in front of the round Einstein metric ds 2 (S 6 ) on S 6 foliated as in (A.2). The internal S 6 dependence drops out from the dilaton. Finally, all the dependence of the IIA potentials on the internal S 6 combines into the homogeneous nearly-Kähler structure J , Ω on S 6 , through the expressions (A.14). More concretely, (3.5) reduces to in agreement with the formulae for the consistent truncation to the G 2 -invariant sector given in (4.3) of [3]. Similarly, the constant-scalar field strengths (3.8) reduce to the corresponding contributions in (4.4) of [3].

Dilatons
According to (2.14), the SO(3) ′ × SO(4) ′ -invariant sector of N = 8 ISO(7) supergravity retains only the dilatons φ, ϕ, along with the two-and three-forms in the tensor hierarchy (2.4). From (3.5), (2.14), it is apparent that the field strengths in this subsector will not contain terms in dϕ or dφ , prior to imposing the dualisation (2.10). In other words, equation (3.7) forF (4) is exact (the dots can be disregarded) andĤ (3) andF (2) are zero. Using the dualisation conditions (2.10), the full non-linear embedding of the D = 4 metric plus dilaton sector into massive type IIA reads, at the level of the field strengths, with ∆ 1 , ∆ 3 given by (3.4) with χ = ρ = 0. In this sector, the fibration of S 2 over I × S 3 also trivialises, A i = 0. Accordingly, the symmetry preserved by the configuration (4.2) is the SO(3) ′ ×SO(4) ′ subgroup of SO(7) defined in (2.1), with SO(3) ′ and SO(4) ′ respectively acting on the S 2 and the S 3 . By using the D = 4 duality hierarchy (2.10), the consistent embedding (4.2) becomes expressed in terms of independent four-dimensional degrees of freedom only: the dilatons and their derivatives, and the metric, explicitly and through the Hodge dual.

N = 3 SO(4)-invariant AdS 4 solution of massive type IIA
By the consistency of the embedding, the ten-dimensional metric and dilaton in (3.5), along with the field strengths that follow from the potentials given in that equation, satisfy the equations of motion of massive type IIA supergravity provided the equations of motion that follow from the D = 4 Lagrangian (2.6), (2.7) are imposed. In particular, (3.5), (3.8) evaluated on the critical points of the scalar potential (2.7) summarised in table 1 give rise to AdS 4 solutions of massive type IIA. The N = 1 and N = 0 critical points with G 2 symmetry uplift to the solutions respectively found in [9] and [10]. The nonsupersymmetric SO(7)-invariant critical point gives rise to a solution given in [1]. See [7] for these solutions in our conventions. In all these solutions with at least G 2 symmetry, the fibration trivialises, A i = 0, and the metric becomes the round, SO(7)-symmetric Einstein metric on S 6 . In the G 2 -invariant solutions, the symmetry is reduced by the supergravity forms, which take values along the homogeneous nearly-Kähler structure on S 6 .
Here we are interested in the uplift of the N = 3 critical point of ISO (7) supergravity [18]. Bringing the vacuum expectation values of the D = 4 scalars recorded in table 1 to the formulae (3.5), (3.8), and rescaling the external D = 4 metric and the Freund-Rubin term with the cosmological constant recorded in the table so that AdS 4 is unit radius, as in [7], we find the massive type IIA uplift of the N = 3 solution. In Einstein frame it reads, in the IIA conventions of appendix A of [3]. The covariant derivative ofμ i and the corresponding connection A i are, from (3.6), and we have defined L 2 ≡ 2 − 31 12 3 All the comments made in section 3 for the generic solution away from the G 2 -locus apply to the specific N = 3 solution (5.1). The internal metric and supergravity forms extend smoothly on S 6 . Locally, the solution can be regarded as a (trivial) S 2 bundle over S 3 foliated by α or, alternatively, as the warped generalisation of the twistor fibration discussed in section 3.2. The angle α has range (3.13). The solution displays a cohomogeneity-one (3) R respectively act on the S 2 fibers and the S 3 base. The solution can be generalised by replacing S 3 with the cyclic Lens space S 3 /Z p , a generalisation that introduces orbifold singularities. The N = 3 supersymmetry of the solution is demonstrated in the next section.

Supersymmetry of the N = 3 solution
The gravitini of the D = 4 N = 8 ISO(7) supergravity lie in the spinor representation of SO (7). Under (2.1), this branches as 3 These (super)symmetry groups are preserved by the ten-dimensional uplift, so the above considerations should allow us to identify the G-structures carried by the family of type IIA configurations (3.5). The R 7 that furnishes the fundamental representation of the semisimple, SO (7), part of the D = 4 gauge group is to be identified with the ambient space of the uplifting S 6 . In other words, this SO (7) can be regarded as the generic structure group of the ambient R 7 , with the internal supersymmetry parameters transforming in the 8. On the G 2 -invariant locus (2.15), the type IIA configuration (3.5) is N = 1. The G 2invariant supersymmetry parameter corresponds, via (2.2), to the (1, 1) singlet in (6.1). The structure of R 7 gets reduced to G 2 (holonomy), which in turn descends into S 6 as a nearly-Kähler SU(3)-structure.
Away from the G 2 locus, as in the solution (5.1), the IIA configuration (3.5) is N = 3. The supersymmetry parameter transforms under SO(3) d × SO(3) R as the (3, 1) in (6.1). The ambient R 7 is thus equipped with an SU(2)-structure, with SU(2) ≡ SO(3) R and R-symmetry SO(3) d . Recall that an SU(2)-structure in seven dimensions is characterised by a real one-form and a real two-form, transforming as triplets of the R-symmetry group, see e.g. [30]. Denoting the R 7 coordinates by x I , I = 1, . . . , 7, and splitting I = (i,î), i = 1, 2, 3,î = 0, 1, 2, 3 as in appendix A, the one-and two-forms of our seven-dimensional SO(3) R -structure can be identified as dx i and 1 2 (J i )îĵ dxî∧dxĵ, with (J i )îĵ defined in (A.5). These indeed transform as triplets under the SO(3) d R-symmetry. The SU(2)-structure on R 7 descends on S 6 as an identity structure. The latter is characterised by an SO(3) d triplet of scalars, of one-forms, and of two-forms, that can be constructed as spinor bilinears.
Rather than characterising the identity structure, we will directly contruct the SO(3) d triplet of Killing spinors, focusing on the N = 3 solution (5.1) for definiteness. In principle, one would expect that the consistency of the uplift should determine the relevant Killing spinors from combinations of those of the round S 6 . In practice, however, such formulae have never been worked out (although see e.g. [31] for a discussion). It then turns out to be more efficient, though still a rather demanding exercise, to construct the Killing spinors by direct integration of the type IIA Killing spinor equations on the background (5.1).
Here we give the end result and sketch the main steps to derive it. Further details can be found in appendices D and E.
Letǫ be the ten-dimensional Majorana supersymmetry parameter, let ζ i ± , i = 1, 2, 3, be three of the chiral and antichiral Killing spinors of AdS 4 , and let χ i be an SO(3) d triplet of Dirac spinors on the internal six-dimensional geometry corresponding to the solution (5.1). We takeǫ with the SO(3) d indices contracted, and raised and lowered with δ ij . The superscript c denotes Majorana conjugation. The ten-dimensional spinorǫ given by (6.2) is manifestly Majorana, by the second relation in (D.3). We require that (6.2) annihilates the supersymmetry variations of the type IIA fermions. Using the AdS 4 Killing spinor equations (D.3) obeyed by ζ i ± , this turns out to be equivalent to the following set of equations for χ i and χ ic , defined on the six-dimensional internal geometry: 3b) Here, is the string frame, for convenience, warp factor of the solution (5.1),φ the dilaton therein, F (0) ,F (2) ,Ĥ (3) , the IIA field strengths,Ĝ (4) the internal component ofF (4) The factor of 1 2 e A 2 is chosen for convenience, f 1± , etc., are functions of α, and η i 1 , . . . , η i 4 are independent triplets of spinors on S 2 × S 3 built as tensor products of the Killing spinors of S 2 and S 3 . Specifically, let ψ α , α = 1, 2, be a doublet of spinors of S 2 , constructed from the S 2 Killing spinors, andψ α = (σ 2 ) α β ψ αc for ψ αc the Majorana conjugate of ψ α . The index α here labels the doublet of SO(3) ′ in (2.1) which rotates S 2 . Let ξ α , α = 1, 2, be the two Killing spinors of S 3 that transform as a doublet under the SO(3) L in (2.1) and are singlets under SO(3) R . Then, where σ i are the Pauli matrices andμ i are defined in (3.1). The Pauli matrix σ 2 appears as the SO(3) d charge conjugation matrix. Inserting χ i given by (6.5), (6.6) and its Majorana conjugate χ ic into the Killing spinor equations (6.3a)-(6.3c), an involved calculation produces a(n overdetermined) system of algebraic relations among the functions f 1± , etc., and a differential equation on the interval (3.13) for a combination of them. The details are summarised in appendix D. Significant further massaging allows us to bring the solution of this set of algebraic and differential equations into the form up to an arbitrary overall normalisation. We have defined the following functions of α: The SO(3) d triplet of SO(3) R -invariant spinors χ i given by (6.5) with (6.7), (6.8) solve the Killing spinor equations (6.3a)-(6.3c) on the N = 3 solution (5.1) of massive type IIA supergravity. As an additional check, we have also verified that the three independent N = 1 pure spinors that follow from (6.5) with (6.7), (6.8) solve the pure spinor supersymmetry conditions for AdS 4 solutions of massive IIA supergravity given in [13]. Equipped with the N = 3 Killing spinors, we can proceed to compute the spinor bilinear forms and the torsion classes of the corresponding identity structure. Here we will only give the scalar bilinears. We expect one SO(3) d triplet of scalar bilinears, based on the fact that the six-dimensional identity structure is inherited from an SO(3) R -structure on the ambient R 7 . Let us see how this scalar triplet arises from spinor bilinears. In principle, two such real or purely imaginary scalar bilinears can be constructed out of χ i , namely, χ i † χ j and χ i †γ χ j . Both of these sit in principle in the 3 × 3 → 1 + 3 + 5 of SO(3) d . Direct computation from (6.5), (6.7), (6.8) shows that Thus, for both bilinears, the 5 components vanish identically. The first bilinear is the triplet argued above, and the second one is a singlet which, however, is not independent but is algebraically related to the former. Equation (6.9) provides a further consistency check on our Killing spinors. It was shown in [13] that, for N = 1 supersymmetric warped product solutions of massive IIA supergravity containing AdS 4 , the N = 1 internal Killing spinor χ must satisfy χ †γ χ = 0 and χ † χ ∝ e A , where e 2 A is the string frame warp factor. It is straightforward to see from (6.9) with i = j that each individual χ i , i = 1, 2, 3, satisfies these N = 1 conditions.

Outlook
In this paper we have studied an N = 3 solution of massive IIA supergravity first considered in [8,32]. We have described in detail the sector of ISO(7) supergravity with SO(4) invariance, which includes this solution as a point in its moduli space. This has allowed us to better understand its geometry. The solution consists of a fibration over an interval I of a certain S 2 -bundle M 5 over S 3 , with the S 2 shrinking at one endpoint of the interval and the S 3 at the other, so that the full topology is that of an S 6 (as expected for vacua of the ISO(7) supergravity). Moreover, we have been able to obtain the spinorial parameters χ i , i = 1, 2, 3 under which it is supersymmetric, thus confirming the expectation that it has N = 3 supersymmetry. This expectation was based on the amount of supersymmetry of the vacuum in the four-dimensional ISO(7) supergravity; but while uplift formulas are available for all physical fields, they are not for the supersymmetry parameters, and thus so far a full proof that the solution is N = 3 was lacking.
Our results open the way to several possible developments. First of all, the structure of the spinorial parameters χ i is not completely fixed by the SO(4) invariance. The solution has cohomogeneity one: the SO(4) orbits are copies of the S 2 bundle over S 3 , and thus a priori the isometry group leaves several functions of the coordinate α on I that appear in the χ i undetermined. For the present solution these are fixed by the Killing spinor equations, but it is easy to set up a more general Ansatz where both these functions and those in the physical fields are allowed to vary, without breaking the SO(4) invariance and in particular N = 3 supersymmetry (whose R-symmetry is one of the SO(3) factors in the SO(4)).
Several arguments lead one to suspect the existence of more general N = 3 solutions in massive IIA. On CP 3 , such solutions are predicted to exist by holography [33] and found [34] in first approximation in a regime where the Romans massF (0) is small. VaryingF (0) beyond this regime suggests the existence of a line of solutions. Since CP 3 can be written as a foliation of copies of T 1,1 , it is plausible that such solutions might be related to the ones we are considering here, and that thus there might be a line of deformations in this case, too. A possible analogy is offered by N = 2 solutions: in that case, a line of solutions exists [35] that connects the N = 6 massless solution on CP 3 to an analogue of the solution in [2] obtained by replacing CP 2 with CP 1 × CP 1 .

A Geometric structures on S 6
In this appendix we describe the relevant geometric structures on S 6 that arise in the consistent truncation discussed in the main text. Let µ I , I = 1, . . . , 7, parametrise S 6 as a the locus δ IJ µ I µ J = 1 in R 7 , and let y M , M = 1, . . . , 6, be the S 6 angles. It is convenient to split the µ I according to the SO(3) ′ × SO(4) ′ defined in (2.1) as µ I = (µ i , µî), µ i = cos αμ i , i = 1, 2, 3 , µî = − sin αμî ,î = 0, 1, 2, 3 , where α is one of the y M , and is taken to have range (3.13). In (A.1),μ i ,μî, respectively parametrise an S 2 and an S 3 as δ ijμ iμj = 1 (see equation (3.1)) and similarly forμî. For convenience, the indexî ranges from 0 to 3 as indicated. In terms of these, the round, homogeneous Einstein metric on S 6 , (E.2) of [3], is ds 2 6 = g −2 cos 2 α ds 2 (S 2 ) + dα 2 + sin 2 α ds 2 (S 3 ) , This metric has of course SO (7) isometry, although only the SO(3) ′ × SO(4) ′ that rotates the S 2 and the S 3 is manifest. The local line element (A.2) is adapted to the topological description of S 6 as the join of S 2 and S 3 . For our calculation, we need to write the Killing vectors of S 6 adapted to the splitting (A.1). For this purpose, it is useful to split the local index M on S 6 as M = (α, a,â), where a = 1, 2 andâ = 1, 2, 3 are local indices on S 2 and S 3 , respectively. With the normalisation conventions of appendix E of [3], the non-vanshing components of the Killing vectors of the round metric (A.2) are whereg ab andgâb are the round inverse metrics on S 2 and S 3 andK a ij ,Kâ iĵ their corresponding Killing vectors. The derivatives of the Killing vectors with respect to the y M angles on S 6 are K iî αa = 2g −2 cos 2 αμî ∂ aμ i , K iî αâ = 2g −2 sin 2 αμ i ∂âμî , In these expressions, ∂ M µ I means derivative of µ I = µ I (y M ) with respect to y M , for M = (α, a,â).
With these ingredients, we can calculate the consistent embedding of the D = 4 supergravity of section 2 into type IIA supergravity using the uplift formulae of [2,3]. The ten-dimensional embedding of the tensor hierarchy forms in (2.4) proceeds uneventfully. The embedding of the scalars is much more laborious. This is achieved by bringing the SO(4)-invariant scalar matrix M MN of the D = 4 supergravity, given in (D.23) of [22], to the uplifting formulae (10) of [2]. Manipulating these formulae with the help of (A. , so that (J i ) 0j = δ i j and (J i ) jk = −ǫ i jk . Indices i andî are raised and lowered with the SO(3) ′ and SO(4) ′ invariant metrics δ ij and δîĵ, respectively. The J i are antisymmetric, (J i )îĵ = −(J i )ĵî, anti-selfdual, satisfy the quaternion algebra, and the identity Equations (A.6)-(A.9) need to be used extensively to bring the raw consistent truncation expressions obtained from the formulae in [2,3] to the final form (3.5) presented in the main text. Other useful identities for this purpose include where vol(S 2 ) and vol(S 3 ) are the volume forms corresponding to the metrics ds 2 (S 2 ) and ds 2 (S 3 ) in (A.2), and * 2 and * 3 the corresponding Hodge dual operators. Note that We conclude by retrieving the homogeneous G 2 -invariant nearly-Kähler structure on S 6 from this formalism. The nearly-Kähler forms are given in general by in terms of the constrained µ I that parametrise S 6 and the associative and co-associative forms ψ andψ, on the ambient R 7 , see e.g. appendix E of [3]. It turns out that the non-vanishing components of these forms can be written for the case at hand as Bringing (A.1), (A.13) to (A.12), we find These forms can indeed be checked to satisfy the nearly-Kähler relations

B Flux quantisation and free energies
The fluxes corresponding to the generic configuration (3.5) can be appropriately quantised. The quantisation conditions are with k and N integers and ℓ s the string length. The fields A (3) ,B (2) andφ have been given in (3.5), and from * F (4) , only the contribution corresponding to the Freund-Rubin term (3.10) is relevant. The first relation in (B.1) corresponds to the relation between quantised Romans mass and D = 4 magnetic coupling [2]. The second expression can be written, integrating in the range (3.13) for the angle α, as a relation between the integer N and the electric D = 4 coupling constant g, with v(S 6 ) = 16 15 π 3 the volume of the unit radius round six-sphere. The expression (B.2) coincides with that given in [7,36] from the embedding of the SU(3)-invariant sector. The fact that the present SO(4)-invariant calculation here and the SU(3)-invariant calculation [7,36] agree provides a selfconsistency check, as the relations (B.1), (B.2) must characterise the theory, not merely particular subsectors or solutions.
Equipped with these values of the fluxes, we can proceed to the calculation of the gravitational free energy of the configuration (3.5), along the lines of [36]. For this calculation, we first redefine the external D = 4 metric with the inverse scalar potential V in (2.7) so that the Einstein frame warp factor reads The free energy F is proportional to the inverse of the effective four-dimensional Newton's constant [37]. On the geometry (3.5), (B.3), this evaluates to This again reproduces the expression given in [36]. In order to write the free energy in terms of quantised fluxes, we factorise the scalar potential as V = g 2 (m/g) −1/3Ṽ , where V is the g = m = 1 scalar potential (2.7), and then replace g and m by their values (B.1), (B.2) in terms of N and k. We finally obtain as in [36]. At the critical points of the scalar potential (2.7), recorded in table 1, the free energy (B.5) reduces to values that have been previously given in the literature. For the critical points with at least G 2 symmetry, (B.5) reproduces the corresponding values given in table 1 of [7]. For the N = 3 point, (B.5) gives F = 1 40 3 13/6 π N 5/3 k 1/3 , (B.6) in agreement with [32]. This result also matches the field theory result, equation (8.4) of [38], after correcting a typo there 5 . The formula (B.5) holds for geometries with running D = 4 scalars, not only for scalars frozen at critical points of the D = 4 potential. It was conjectured in [36], based on an SU(3)-invariant calculation, that the free energy (B.5) should further hold at any point of the 70-dimensional coset space E 7(7) /SU(8) of the full D = 4 N = 8 dyonically-gauged ISO(7) supergravity, whereṼ given by the full g = m = 1, N = 8 potential, normalised as in [22]. The present SO(4)-invariant calculation provides further evidence in favour of this conjecture.

C SO(4)-invariant AdS 4 solutions of massive type IIA
The type IIA solutions that we have presented in the main text have the local form where AdS 4 and S 3 are unit radius, µ 0 is a constant, X(α), etc., are functions of the angle α, theμ i , i = 1, 2, 3, are constrained coordinates that define a unit radius S 2 and ρ i are the right-invariant forms on S 3 . The corresponding potentials are of the form In this appendix, we will denote the covariant derivative ofμ i as The function A 0 (α), as well as all other functions of α in (C.1), (C.2) can be read off from the concrete expressions given in the main text.
The configuration (C.1)-(C.3) preserves the SO(4) subgroup of SO(7) defined in (2.1). We will now work out the differential and algebraic equations that the functions X(α), etc., must obey for (C.1) to solve the Bianchi identities and equations of motion of massive type IIA supergravity. Firstly, the functions that specify the potentials (C.2) can be related to those entering the field strengths (C.1) using the corresponding definitions (see (A.3) of [3]). We find (C.4) and We have dropped the explicit α dependence and have denoted with a prime the derivative with respect to it. We used the expressions (C.4)-(C.6) to construct the constant scalar field strengths (3.8) from the potentials (3.5).
Moving to the field equations (see (A.3), (A.1) of [3]), some calculation shows that the type IIA Bianchi identities impose the relations Next, a long calculation shows that theF (4) equation of motion gives and the dilaton equation of motion gives As for the Einstein equation, we have only computed the external components, which produce the following equation for the warp factor: We have employed equations (C.7)-(C.12) to verify that the the generic expressions (3.5), (3.8) evaluated on the critical points of ISO(7) supergravity with at least SO(4) invariance, recorded in table 1 of the main text, are indeed solutions of massive type IIA supergravity. We have verified on a case-by-case basis that all solutions mentioned in section 5, particularly the N = 3 solution (5.1), do satisfy equations (C.7)-(C.12). Up to a check of the internal Einstein equations, this shows that all constant-scalar configurations presented in the main text are indeed solutions of massive type IIA supergravity.

D Derivation of the N = 3 Killing spinors
Following, for convenience, the string frame conventions of [39], the supersymmetry transformations of the type IIA fermions read The IIA dilatino,λ, gravitino,ψ M , and supersymmetry paramater,ǫ, are Majorana. The ten-dimensional gamma matrices areΓ M , with M = 0, 1, . . . , 9, and M = 0, 1, . . . , 9 denoting here ten-dimensional tangent space and local indices, respectively. The slashed forms are defined, as usual, as / H . The ten-dimensional chirality matrix has been denoted bŷ Γ, andΓ M denotes the contraction ofΓ M with the ten-dimensional vielbein. Let us show that the N = 3 solution (5.1) obeys the IIA Killing spinor equations, δλ = 0, δψ M = 0, for the supersymmetry parameters given in section 6. We start by reducing these to the Killing spinor equations (6.3a)-(6.3c) on the internal six-dimensional geometry. In order to do this, we choose a basis for the ten-dimensional gamma matrices such thatΓ with µ = 0, 1, 2, 3 local indices on AdS 4 and M = 1, . . . , 6 now rebranded as a tangent space index on the internal six-dimensional geometry. Here, e 2 A is the string frame warp factor (6.4), γ (4) and γ M four-and six-dimensional gamma matrices (the former contracted with the AdS 4 vielbein),γ (4) = −iγ (4) 0123 andγ = iγ 123456 the respective chirality matrices. The six and ten-dimensional intertwiners B andB (10) , with BB * = I, are such that χ c = Bχ * , where c denotes Majorana conjugation, γ * M = −B −1 γ M B, and similarly forB (10) . Let ζ i ± be (anti)chiral Killing spinors on AdS 4 , and let χ i , i = 1, 2, 3, three arbitrary Dirac spinors on the internal six-dimensional geometry. Writing the ten-dimensional spinor paramaterǫ in terms of ζ i ± and χ i as in (6.2) and making use of the decompositions (D.2) and the AdS 4 Killing spinor equation (D.3), we obtain the six-dimensional Killing spinor equations (6.3a)-(6.3c) from the expressions (D.1) equated to zero.
One method to proceed in general (as pursued in [40,41,42] for N = 2 AdS 4 solutions) is to form spinor bilinears from (6.3a)-(6.3c), use them to show that a geometrically realised SO(3) R-symmetry SO(3) R necessarily emerges and then locally determine the metric, dilaton and fluxes up to PDEs. However given that the solution of section 5 contains S 2 fibred over S 3 , we find it easier to construct all spinors on S 2 ×S 3 that transform as triplets under SO(3) R and couple them to arbitrary spinors on the interval spanned by α, thereby effectively reducing the problem from 6 to 1 dimensions.

D.1 Constructing an SO(3) d triplet in 6d
The first thing that needs addressing is exactly how one constructs a triplet of spinors that transforms under SO(3) R . Clearly the metric on a round S 2 and S 3 preserve SO(3) S 2 and SO(4) S 4 isometries respectively but, as explored in the Minkowski classifications of [43,44,45], each independent Killing spinor on these spheres can only individually be used to form SU(2) doublets. We also have to consider the fact that S 2 is fibred over S 3 in terms of the SU(2) right-invariant 1-forms as (C.3). This means that if one decomposes SO(4) S 4 = SO(3) L × SO(3) R (with L/R standing for left/right) it is only SO(3) R and the diagonal SO(3) d formed from SO(3) ′ ≡ SO(3) S 2 and SO(3) L that are preserved by the full space: the anti-diagonal is broken. Since SO(3) R only involves S 3 , the preceding discussion suggests that we should identify the R-symmetry as SO(3) R = SO(3) d , but we will need to be more explicit to construct its corresponding triplets. As we shall show, the fundamental building blocks of the triplets are actually the SU(2) doublets on S 2 and S 3 , for which we give further details in appendix E. 1. In what follows, we shall parametrise the two SU(2) doublets on S 2 as ψ α andψ α , while the single SU(2) L doublet on S 3 , that is a singlet with respect to SU(2) R , shall be ξ α for α = 1, 2. The Killing vectors of SO(3) ′ we denote by K i and of SO(3) L/R by L i /R i . The key property of the doublets that we shall need is how they transform under the action of the spinorial Lie derivative along these Killing vectors, namely with a corresponding expression for the action of K i onψ α . Here σ i are the Pauli matrices, so that a doublet of a given SU(2) realises the corresponding Lie algebra under its action.
To realise the R-symmetry SO(3) d , then we must construct products of these doublets that transform as The obvious way one might try to construct a triplet is to contract ψ α and ξ α with the Pauli matrices. This is almost correct, but it is actually the matrices σ 2 σ i , which are symmetric, that give the correct transformation properties. It is then not hard to show that obey (D.5) using (D.4) and standard Pauli matrix identities, which gives us two triplets. But this is not the whole story. It is also possible to construct two SO(3) d singlets by contracting the doublets with σ 2 ; so, given that the embedding coordinatesμ i of S 2 transform as a triplet under SO(3) d , also obey (D.5), by Leibniz rule. We can derive (D.6), (D.7) by using ambient space coordinates and group theory. On In order to produce such S 2 spinors, we can work in the ambient R 3 . Here it is clear that our ingredients are constant spinors and the three coordinates; going back to S 2 , these become the ψ α and theμ i . From the ψ α and m copies of the µ i one obtains a representation (m) ⊗ (1/2) = (m − 1/2) ⊕ (m + 1/2). This contains (1/2) or (3/2) for m = 0, 1, 2. Defining µ αβ ≡μ i (σ 2 σ i ) αβ , the doublets one obtains this way can be written as ψ α ,μ αβ ψ β ; it turns out that the latter is simplyψ α . We can now tensor these two doublets with ξ α on S 3 and extract the triplet using (σ 2 σ i ) αβ ; this gives (D.6). On the other hand, the quadruplets one can write in this way can be written asμ (αβ ψ γ) ,μ (αβμγ)δ ψ δ =μ (αβψ γ) . Tensoring these with the ξ γ and contracting one index to extract the triplet, after some Pauli matrix algebra one obtains a linear combination of (D.6) and (D.7). Thus we have obtained a set of triplets . From the embedding coordinates argument we have just given, this set is also exhaustive. One can also see this in the following way. Any 5d spinor can be decomposed in a basis of four linearly independent spinors with complex functional coefficients defined in 5d. This means any additional triplet can be decomposed in a basis of our existing triplets asη i = 4 n=1 (a n ) i j η j n . Since this new triplet needs to transform as (D.5), and η i n already do transform in this fashion, there are only two options. i) We take (a n ) i j constant, in which caseη i is not linearly independent of η i n by definition. ii) We take (a n ) i j to be proportional to the S 2 embedding coordinatesμ j , which leads tõ η i = 4 n=1 c n ǫ ijkμk η j n . At first sight this does seem to give two additional triplets (not four, because those involving η i 3 , η i 4 are proportional to ǫ ijkμjμk = 0). However, one can show that ǫ ijkμk η j 1 = −iη i 2 + η i 3 and ǫ ijkμk η j 2 = −iη i 1 + η i 4 ; so these triplets depend linearly on η i n . The most general SO(3) d triplet of spinors on a fibration of S 2 over S 3 times an interval is then of the form where f n± are functions of α, the coordinate on the interval, to be determined by (6.3a)-(6.3c). Up to this point our supersymmetry discussion has been quite general and will apply to any AdS 4 solution with metric, dilaton and fluxes preserving SO(3) d × SO(3) R with a S 2 × S 3 fibration. We shall now proceed to solve the arbitrary functions of the interval for the solution of section 5. We shall return to this system in full generality in a follow up.
In the remainder of this appendix and in appendix E.1 we will sketch the computation of the functions f n± appearing in (D.8).
The first supersymmetry condition we shall solve is the dilatino variation (6.3b). The reason to start here is that this condition contains no derivatives of the spinors, no Majorana conjugation and the S 2 × S 3 data only appears packaged in the invariant forms. Thus the computation just consists of writing (6.3b) in the form using the the action of the invariant forms listed in (E.27)-(E.28). Once this is done, one knows that all eight X n± must individually vanish, because the 4 triplets η i n are independent, and likewise the positive and negative chirality components of the expression. This leads to eight complex algebraic constraints that the f n± must satisfy. We found it useful to introduce an auxiliary set of complex functions of the interval: and then solve instead for t n± . This is because the dilaton conditions fix several of the t n± in terms of just one other and known functions of the interval. The simplest four of the conditions that follow take the form √ 6s cos 2 α + √ 2∆ 1 (2 + cos 2 α) ± ∆ 1 cos α(2 + √ 3si + cos 2 α) + z 2 sin α √ 6(3 + cos 2 α) cos 2 α ± 4si ∆ 1 (1 + 2 cos 2 α) .
(D. 19) ∆ i are defined in (D.21); we have factored out a common non-vanishing α-dependent factor in Z ± . It is easy to check that (D. 19) contains only 2 independent expressions, namely the middle two of (D.20). The remaining 4 X n± are more complicated and we only quote their solution. Of the eight complex conditions that follow from (6.3b) only 5 are independent. All in all, the solution can be written as We next turn our attention to the AdS 4 gravitino (6.3a), which can be dealt with in much the same fashion as the dilatino. The only additional ingredient one needs is that that χ ic can be expressed in a basis of η i n using the relations between triplets under Majorana conjugation in (D.6)-(D.7). Given (D.20), (6.3a) provides one additional complex and one real constraints on t n± : As these conditions are the only ones that involve complex conjugation, we shall delay solving them until we have fixed more of t n± with the internal gravitino conditions (6.3c). The gravitino conditions on S 2 ×S 3 do require us to take covariant derivatives of spinor (D.24), and H a = 1 2 H abc γ bc now appears; both depend on more than simply the invariant forms. Therefore the action of the forms is insufficient to solve (6.3c) in these directions.
The first thing we need to know is the form of the covariant derivative entering in (6.3c), which requires that we compute the spin connection on M 6 . The vielbein on a generic M 6 consisting of S 2 fibred over S 3 times an interval preserving SO(3) d × SO(3) R may be expressed as e α = e k dα , eâ = 1 2 e C 2 ρâ, e a = e C 1 dy a − λK y a i ρ i , Dμ i = dμ i + λǫ ijk dμ jμk (D. 23) for y a coordinates on S 2 and e K , e C 1 , e C 2 , λ functions of the interval. The spin connection on M 6 defined through de M + Ω M N ∧ e N enters into the definition of (6.3c) as ∇ M = ∂ M + 1 4 Ω M,P Q γ P Q . After computing this it is possible to show that the covariant derivatives along the internal space defined by (D.23) decompose as where D S i y a/â is the covariant derivative on S 2 /S 3 ; we contract the S 2 × S 3 directions with the two Killing vectors that make up SO(3) d . Form expressions should be understood through the Clifford map dx m 1 ∧ . . . ∧ dx m k → γ m 1 ...m k . Elsewhere in the text we have set The other additional object appearing in (6.3c) is H p , but this is not hard to compute if we contract the S 2 × S 3 directions of this along the Killing vectors as in (D.24) and make use of the identities in the last column of (E.4) and (E.4). We proceed by solving these conditions spinor component by spinor component. By making use of the rotation 6 outlined in appendix E.2 it is possible to factorise the S 2 × S 3 data out of each of these components leaving many expression involving only functions of the interval that must vanish. After a lengthy calculation we find that the S 2 × S 3 gravitino imposes just two additional complex constraints that may be expressed as after using (D.20) to tame many expressions. This leaves only one complex function, t 1 − say, to be determined. The last condition we need to deal with is the internal gravitino condition along the interval. For this expression the covariant derivative (D. 24) and H α are expressed in terms on the invariant forms. We can thus once more use the action of the forms of the triplets to massage the interval component of (6.3c) into the form (D.17). Once (D.20) and (D.28) are used to eliminate the other t n± we are left with a single ODE for t 1 − : Although this may appear a little intimidating a closed form solution does in fact exist, and after some effort one finds that However, the fact that c should either be real or purely imaginary is just a consequence of our choice of intertwiner in (D.9). If we include a constant phase in its definition, c can be completely arbitrary and it is this phase that is fixed by the choice of s = ±1. We use this fact to allow for an arbitrary constant in the main text.
We have now completely fixed all 8 auxiliary functions t n± ; inverting (D.18), after significant massaging one is led to the f n± in (6.8). This confirms that a Killing spinor preserving N = 3 super-conformal symmetry does indeed exist.

E Further details on doublets, invariant forms and triplets
In this appendix we shall first give further explicit details about the SU(2) doublets from which the triplets are constucted in appendix E.1. Later we discuss the SO(3) d invariant forms, triplets and how the former acts on the latter in appendix E.2. We will mostly work with a concrete choice of coordinates, y a = (θ 1 , φ 1 ) a on S 2 and yâ = (θ 2 , φ 2 , ψ 2 )â on S 3 .

E.1 SU(2) doublets
In this appendix we construct SU(2) doublets form the Killing spinors on S 2 and S 3 . In principle these calculations where already performed in [43,44,45]; however, we are using slightly different conversions here (notably with the choice of frame on S 3 and sign of the Killing spinor equation on S 2 ), so we provided some additional details here.

E.2 Invariant forms and triplets
In this subsection we will focus on the five-dimensional manifold spanned by the unwarped fibration of S 2 over S 3 ; we give some additional details about the SO(3) d invariant forms, triplets and how the former acts on the latter. Our flat space 5d gamma matrices are γ (5) a = σ a ⊗ I, γ Together, (E.20) and (E.21) can be used to greatly simplify the 8 dimensional spinorial components one needs to solve when plugging (6.5) into (6.3a)-(6.3c). This was extremely useful for deriving the system of sufficient supersymmetry conditions in (D.20), (D.22), (D.28). Using these conditions, it is also not hard to establish that the triplets satisfy the following relations under Majoranna conjugation: and under inner product