Minimal $D=4$ truncations of type IIA

Consistent embeddings are found of the minimal $\mathcal{N} = 2$ and $\mathcal{N} = 3$ gauged supergravities in four dimensions into its maximally supersymmetric, $\mathcal{N} = 8$, counterpart with a dyonic ISO(7) gauging. These minimal truncations retain the metric along with relevant U(1) and SO(3) R-symmetry gauge fields selected from the ISO(7) ones. The remaining ISO(7) gauge fields are turned off, with subtleties introduced by the dyonic gauging, and the scalars are fixed to their expectation values at the $\mathcal{N} = 2$ and $\mathcal{N} = 3$ vacua of the $\mathcal{N} = 8$ theory. Using the truncation formulae for massive type IIA supergravity on the six-sphere to $D=4$ $\mathcal{N} = 8$ ISO(7) supergravity, the minimal $D=4$ $\mathcal{N} = 2$ and $\mathcal{N} = 3$ gauged supergravities are then uplifted consistently to ten dimensions.


Introduction
Despite their limited field contents, minimal, or pure, gauged supergravitites prove very useful for holography when embedded in string or M-theory. These theories involve the gravity multiplet only, stripped of additional matter multiplets. Thus, they capture holographically universal aspects of large classes of superconformal field theories that are governed exclusively by the R-symmetry, and that are independent of details involving particular matter couplings or flavour symmetries, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In this paper, I will present dimensional reductions of ten-dimensional massive type IIA supergravity [18] to the minimal N = 2 [19,20] and N = 3 [20] gauged supergravities in four dimensions. These will be obtained by consistent truncation on the internal six-dimensional geometries corresponding to the N = 2 and N = 3 AdS 4 solutions of type IIA recently constructed in [21] and [22,23]. Such consistent truncations should exist, based on the general arguments of [24,25] (see also [26,27]). I will show by direct construction that this is indeed the case.
A natural strategy to build consistent truncations of string or M-theory down to pure gauged supergravities relies on the existence of a G-structure description [28] of the background geometry. A Kaluza-Klein truncation ansatz can be constructed involving the lower-dimensional fields together with forms on the internal geometry suitably selected among those defining the G-structure [24]. Consistency is then shown by enforcing the higher-dimensional equations of motion on the lower-dimensional ones while making use of the torsion classes. This technique or variations thereof has been fruitfully applied in various contexts [24,25,[29][30][31][32][33][34][35][36][37][38][39][40]. G-structure analyses exist [41][42][43] for the N = 2 AdS 4 type IIA solution of [21] and generalisations thereof. The construction of the associated consistent truncation would thus be amenable in this case to the use of this technology. Unfortunately, for the N = 3 AdS 4 solution, a workable G-structure description is not readily available -although it should follow from the Killing spinor analysis of [23].
A different strategy can nevertheless be employed for the cases at hand, which relies on the existence of a truncation to a larger (in fact, maximal) supergravity on the relevant geometries. More concretely, the internal space for both the N = 2 [21] and N = 3 [22,23] AdS 4 solutions corresponds to a topological six-sphere, S 6 (equipped with metrics that display isometry groups smaller than the largest possible one, SO (7)). Not unrelated to this fact is the existence of a truncation of massive IIA supergravity on S 6 [21,44] (see also [45][46][47]) down to D = 4 N = 8 supergravity with a dyonically gauged [48][49][50] ISO(7) = SO(7) ⋉ R 7 gauge group [51]. In fact, the N = 2 [21] and N = 3 [22,23] AdS 4 solutions of massive IIA arise as the S 6 uplifts of critical points of ISO (7) supergravity that break the N = 8 supersymmetry of the theory to N = 2 [21] and N = 3 [52]. In this context, the argument of [24] indirectly implies that truncations should exist, within four dimensions, of the N = 8 ISO(7) theory to the minimal N = 2 and N = 3 supergravities around the corresponding critical points. I will show that this is indeed the case.
The desired type IIA truncations can thus be constructed by a two-step process. First of all, truncate D = 4 N = 8 ISO(7) supergravity to the minimal N = 2 and N = 3 gauged supergravities around the corresponding critical points. This is achieved by leaving the D = 4 metric and suitable combinations of the massless vectors dynamical, while placing restrictions on the remaining fields. These restrictions amount to turning off the remaining massless vectors and the massive ones, as well as freezing the scalars to their vacuum expectation values (vevs). Some of the gauge fields that need to be truncated out are the dyonically gauged non-compact ones of ISO (7). Interestingly, this is done by writing them in terms of the surviving compact R-symmetry gauge fields and their Hodge duals, rather than by setting them to zero. Secondly, bring these field restrictions to the D = 10 to D = 4 consistent truncation formulae of [21,44], in order to find the embedding into massive type IIA. The consistency of the IIA truncation to the full N = 8 supergravity [21,44], along with the consistency of the further truncations within D = 4, translates into the consistency of the IIA truncation to the minimal D = 4 N = 2 and N = 3 theories.
The minimal N = 2 and N = 3 theories arise as different, non-overlapping subsectors of the parent N = 8 supergravity. Indeed, the relevant U(1) and SO(3) R-symmetry gauge groups are embedded differently into the SO(7) compact subgroup of the parent ISO(7) gauge group of the N = 8 theory. More precisely, this U(1) is not the Cartan subgroup of this SO (3). The N = 3 gauged supergravity can nevertheless be truncated to N = 2 by retaining the U(1) Cartan subgroup of SO (3). This yields an alternative embedding of N = 2 supergravity into N = 8 and type IIA. In any case, in order to construct the embeddings into N = 8, it is thus convenient to start from a smaller sector of the N = 8 supergravity that is more manageable than the full theory, yet is large enough to contain both minimal truncations. A suitable such sector is the N = 4 one built in [53], which is reviewed in section 2 for convenience. The minimal theories are then embedded into this N = 4 sector and the full N = 8 theory in section 3, and then uplifted to type IIA in section 4. Section 5 concludes, and technical details are left for the appendices.

An intermediate N = 4 sector
With the aim of finding consistent truncations of the D = 4 N = 8 theory to minimal D = 4 N = 2 and N = 3 supergravites, it is useful to start from a suitable subsector of the N = 8 theory. The relevant subsector should be small enough so that an explicit parametrisation for its fields can be introduced, and large enough to contain both disjoint pure N = 2 and N = 3 subsectors. The N = 4 subsector constructed in [53] suits that purpose. In this section, I will review the aspects of that N = 4 model that are relevant for the present discussion, referring to that reference for further details 1 . This N = 4 model arises as the subsector of the N = 8 theory that retains all singlets under a certain SU(2) subgroup of the ISO(7) gauge group of the N = 8 theory. This SU(2) is defined via the following embedding into SO(7) ⊂ ISO (7), with the triplet of SU(3) branching as 3 → 2 + 1 under SU (2). Equivalently, this SU(2) ≡ SO(3) R is also embedded in SO (7) through The sector of N = 8 ISO (7)  It also contains both minimal N = 2 and N = 3 subsectors. The SO(3) R -invariant sector of N = 8 ISO(7) supergravity corresponds to an N = 4 supergravity coupled to three vector multiplets. The scalar manifold is therefore .
submanifold of the second factor in (2.3) with coset representative ν given in (2.9) of [53] and scalar matrix m ≡ ν T ν. The components of m will be denoted m ij .
The nine vectors of the model gauge (dyonically, in the frame inherited from the N = 8 theory), an gauge group, where SO(3) ′ and SO(3) L were defined in (2.2), and R 3 are the three translations of ISO(7) ≡ SO(7) ⋉ R 7 that commute with SO(4) ′ . The electric gauge fields associated to each factor on the r.h.s. of (2.4) are respectively denoted A ′i , A (t)i , A (L)î , their field strengths H ′i (2) , H (t)i (2) , H (L)î (2) , and the corresponding magnetic (2)î . Collectively, these are denoted as 5) and similarly for the field strengths, H Λ (2) , and the magnetic duals,Ã Λ ,H (2)Λ . In the gauged theory, the electric field strengths explicitly read and their magnetic counterparts, In (2.6), (2.7), g and m are the electric and magnetic gauge couplings of the parent N = 8 ISO(7) theory, and B i , B i j , Bî are SO(3) R -invariant [53] two-form potentials in the restricted [51] tensor hierarchy [54]. Their explicit three-form field strengths will not be needed. The three-form potentials in the hierarchy will not be needed either, short of generically keeping track of an auxiliary three-form whose four-form field strength becomes the Freund-Rubin term upon uplift to type IIA, as in [23,44,55]. The bosonic Lagrangian of this N = 4 subsector reads [53] In this paper, no significant role will be played by either the non-linear scalar selfcouplings, or the higher-rank tensor hierarchy fields as already indicated. For that reason, I simply refer to (2.15)-(2.17) of [53] for the scalar kinetic terms h uv Dq u ∧ * Dq v , to (2.25) of that reference for the scalar potential V , and to appendix B therein for the dualisation conditions for the two-and three-form potentials in the tensor hierarchy. The gauge field self-couplings via their field strengths (2.6), (2.7) will be important, as will their couplings to the scalars. These couplings occur minimally, through the covariant derivatives Dq u , explicitly given by and non-minimally through the gauge kinetic matrix N ΛΣ = R ΛΣ + iI ΛΣ . In the basis (2.5), this is Defining the scalar-dependent 3 × 3 matrix, the blocks that compose (2.10) read Of the tensor dualisation conditions, the only ones that will play a role here are the vectorvector duality relations,H with R ΛΣ and I ΛΣ the scalar matrices just defined. For future reference, the equations of motion that derive from the Lagrangian (2.8), upon variation of the scalars q u , the electric vectors A Λ , and the metric g µν are with k u Λ the Killing vectors on the second factor of the scalar manifold (2.3) that can be read off from the covariant derivatives (2.9). In addition, the variation w.r.t. the magnetic gauge fieldÃ (t) i yields the last three components, in the basis (2.5), of the vector duality relations (2.13). Finally, the variation w.r.t. the two-form potential B i gives a dualisation condition for its field strength H i (3) , with k ui three of the Killing vectors k u Λ . The vector equations of motion in (2.14) are equivalent to the Bianchi identities for the magnetic field strengths (2.7), once the duality hierarchy [56] is employed. The Bianchi identities for their electric counterparts (2.6) read with the covariant derivatives defined as . (2.17) Finally, the covariant derivatives of the magnetic field strengths (2.7), which feature in the vector equations of motion in (2.14), are As argued briefly above and in more detail in [53], the present N = 4 model contains the SU(3)-and SO(4)-invariant sectors of N = 8 ISO(7) supergravity constructed in [51]. In that reference, the SU(3)-invariant scalars were denoted by ϕ, χ, φ, a, ζ,ζ, the electric and magnetic gauge fields by A 0 , A 1 ,Ã 0 ,Ã 1 and their field strengths by H 0 (2) , H 1 (2) ,H (2)0 , H (2)1 . This SU(3)-invariant field content is recovered from the N = 4 model by identifying ϕ, χ here and there, and further identifying the scalars as the electric vectors as

20) and their magnetic duals as
With these identifications, the present N = 4 model reduces to the SU(3)-invariant sector as given in section 3 of [51]. The SO(4)-invariant sector, in turn, contains four scalars, ϕ ′ , χ ′ , φ ′ , ρ ′ (denoted in [51] with no primes), and no vectors. Setting to zero the vectors of the N = 4 model and identifying the scalars as the SO(4)-invariant sector is recovered, as given in section 5 of [51].
The N = 4 model contains all previously known AdS vacua of N = 8 dyonic ISO(7) supergravity (see table 1 of [51] for a summary) and, as shown in [53], some new nonsupersymmetric vacua. In particular, the N = 2, SU(3) × U(1)-invariant AdS vacuum is located within the N = 4 theory at while the N = 3, SO(4)-invariant critical point occurs at Unlike in the SU(3) and SO(4)-invariant sectors which respectively contain them, these vacua preserve their full N = 2 and N = 3 supersymmetries within the N = 4, SO(3) Rinvariant sector [53]. See that reference for the allocation of the spectra within the N = 4 theory in OSp(4|2) and OSp(4|3) multiplets, and [52,57] for the supersymmetric spectra within the full N = 8 ISO (7) supergravity.

Minimal truncations of ISO(7) supergravity
I will now move on to show that the N = 4 subsector of N = 8 dyonic ISO(7) supergravity discussed in [53] and reviewed in the previous section can be further truncated around the vacua (2.23), (2.24) to the minimal N = 2 and N = 3 gauged supergravities (A.2), (A.6). The conventions for these theories are specified in appendix A.1.

General strategy
The goal is to show that the Bianchi identities (2.16) and the equations of motion (2.14) of the N = 4 model are satisfied on the field equations (A.1), (A.4) of the minimal theories.
In order to do this, the metric must be left dynamical; the scalars must be frozen to the respective vevs (2.23), (2.24); the two-form potentials set to zero, B i = B i j = Bî = 0; the three-form potentials similarly turned off (except for the generic three-form argued above); and the N = 2 and N = 3 dynamical vectors must be conveniently selected among those of the N = 4 theory that gauge the residual R-symmetry. Thus, for the N = 2 and the N = 3 minimal truncations, the retained vectors must respectively gauge a U(1) and the diagonal SO(3) d of the compact subgroup SO(3) ′ × SO(3) L of the N = 4 gauge group (2.4). The vectors that gauge the remaining compact generators must be truncated out, even if they remain massless at the relevant vacuum. The vectors A (t)i that gauge the non-compact R 3 factor of the gauge group (2.4) become massive at both, and any other, vacua. The gauge fields A (t)i must therefore be truncated out too, albeit not by naively setting them to zero but, as will be argued below, by relating their field strengths to the R-symmetry vector field strengths and their Hodge duals. To see this, note that the requirement that the scalars q u be frozen to their vevs can be phrased in a covariant way by requiring that they are covariantly constant, Dq u = 0. This must happen even if some of the vectors A Λ ,Ã Λ are left dynamical. Bringing the vevs (2.23) and (2.24) to the definition (2.9) of the covariant derivatives, this requirement translates into the algebraic relation for the non-compact gauge fields at both the N = 2 and the N = 3 vacua. Of course, i = 0 is a valid solution to the constraint (3.1). However, for the cases of interest, it can be checked that the equations of motion then imply that all vectors must vanish A Λ = 0,Ã Λ = 0. This choice thus leads to the N = 2 and N = 3 AdS vacuum solutions. One is thus led to enquire whether there are more general solutions with i non-vanishing, along with some of the compact vectors. It turns out that there are. Using the definitions (2.6), (2.7) of the vector field strengths, (3.1) can be seen to imply the constraint gH at the level of the field strengths. Equation (3.2) can be rewritten with the help of the duality relations (2.13), by trading the magnetic field strengthsH (t) (2)i for q u -dependent combinations of the electric field strengths H Λ (2) and their Hodge duals. Taking the Hodge dual of the resulting equation, further algebraically independent constraints are generated that allow one to solve for H (2)i in terms of the dynamical R-symmetry gauge fields.
Under these assumptions, the N = 4 field equations must then be shown to reduce to those, (A.1), (A.4), of the minimal N = 2 and N = 3 theories. With q u set to their vevs and Dq u = 0, the N = 4 equations of motion (2.14) and Bianchi identities (2.16) give rise to the following algebraic constraints on the gauge field strengths and to the equations The gauge covariant derivatives here are given in (2.17) and (2.18). In (3.3), the derivatives ∂ u I ΛΣ and ∂ u R ΛΣ w.r.t. the scalar fields q u are computed from the explicit expression (2.10)-(2.12) of the gauge kinetic matrix, and then evaluated at the scalar vevs. In (3.4), the potential V becomes the cosmological constant at each critical point (see e.g. tables 3 and 4 of [51]) and the gauge kinetic matrix I ΛΣ is also evaluated at the constant scalar vevs. At this stage, the constraint (3.3) on the vector field strengths must be identically satisfied, and (3.4) must reduce to the field equations (A.1), (A.4) of the N = 2 and N = 3 theories upon suitable rescalings of the non-vanishing gauge fields and the metric. In particular, the N = 4 (and N = 8) metric g µν and the metricsḡ µν of the N = 2 and N = 3 theories are related by the constant rescalinḡ in terms of the relevant cosmological constants V . These rescalings involve the N = 4 (and N = 8) non-vanishing gauge couplings g and m through V . The magnetic gauge coupling m turns out to be eventually rescaled away, while the electric gauge coupling survives as the coupling g of the minimal N = 2 and N = 3 theories. The restricted tensor hierarchy fields can be safely disregarded in all this process. The three-form field strengths are set to zero. Their dualisation conditions are equivalent to projections of the scalar equations of motion, and these are identically satisfied for Dq u = 0 and vanishing three-form field strengths (see (2.15) and, more generally, appendix B of [53]). A non-vanishing three-form potential related to the type IIA Freund-Rubin term plays an auxiliary role.

Truncation to minimal N = 2 supergravity
Let us make the above discussion more explicit for the truncation to minimal N = 2 supergravity around the N = 2 vacuum. Similar operations were performed in [12] starting from the N = 2 SU(3)-invariant sector [51] of N = 8 ISO(7) supergravity, to show that a particular black hole was a solution of the N = 8 theory. The present analysis turns out to be a simple extension of [12] to general dynamical metric and graviphoton.
First of all, the scalars need to be frozen to their vevs (2.23). Since the dynamics must be SU(3)-invariant, the N = 4 electric gauge fields must in turn be truncated as in (2.20), with similar relations for the magnetic duals, and Bringing (2.23), (3.6), (3.7) to the scalar covariant derivatives (2.9), all of them become zero, even for A = 0, except Da ij which produces the constraint (3.1) with 3 = 0 as well, but then the equations of motion enforce A = 0 and the N = 2 AdS vacuum is recovered.
Instead, one can proceed with A (t)3 = 0,Ã (t) 3 = 0. Setting to zero the two-form potentials and using (3.6), (3.7) the electric vector field strengths (2.6) become and give the following non-vanishing Abelian field strength: . (3.9) The compact gauge field A ′3 = − 1 2 A (L)3 gauges the residual U(1) R-symmetry. In (3.7) and (3.9), this gauge field has been identified with the graviphoton A of the N = 2 theory (A.2) and its field strength F , up to a suitably chosen normalisation. On (2.23), (3.8), the vector duality relations (2.13) simplify tõ and 11) and the constraints (3.1), (3.2) reduce to Combining (3.11) with the second relation in (3.12) and further taking Hodge duals, a set of equations is obtained that allows one to solve for H Finally, the metric is rescaled as in (3.5), (3.14) At this point, all of the vectors of the N = 4 model have either been set to zero or written in terms of the N = 2 graviphoton and its Hodge dual. It only remains to verify that, with these definitions, the constraints (3.3) are satisfied and the equations (3.4) give rise to the N = 2 field equations (A.1). Bringing (3.8), (3.9) and H (t)3 (2) given in (3.13) to (3.3), and using the explicit expression (2.10)-(2.12) for the gauge kinetic matrix, some calculation reveals that these constraints are indeed satisfied identically for any F . Next, the Bianchi identities and vector equations of motion in (3.4)   To summarise, the identifications (2.23), (3.6)-(3.10), (3.13), (3.14) have been shown to define a consistent truncation of the D = 4 N = 4 model [53] of section 2, to minimal D = 4 N = 2 gauged supergravity (A.2). In retrospect, it was critical to get rid of the magnetically gauged non-compact vector A (t)3 (2) with field strength H (t)3 (2) , not by naively setting it to zero, but by writing it in terms of the surviving graviphoton F and its Hodge dual through the relations (3.13). Setting H (t)3 (2) = 0 also leads to F = 0.

Truncation to minimal N = 3 supergravity
The truncation to minimal N = 3 supergravity proceeds similarly. First of all, the scalars are fixed to their N = 3 vevs (2.24). On this vacuum, only the vectors 2 Thus, the latter combinations must be truncated out by identifying the SO(3) graviphoton A i of the N = 3 theory (A.6) with Together with (3.17) we also have, from (A.3) and (2.6), the following identifications at the level of the electric field strengths: .
Satisfactorily, the covariant derivatives of H ′i (2) and H (L)î (2) in (2.17) then coincide with themselves and with that of F i defined in (A.5). Bringing the vevs (2.24) and the vector identifications (3.17) to (2.9), all the covariant derivatives are seen to vanish, except the ones of the Stückelberg scalars a ij , which produce the relation (3.1). The latter can be solved by letting A (t)i = 0,Ã (t) i = 0, but then the equations of motion also set F i = 0. This leads to the N = 3 AdS vacuum solution. Alternatively, i can be left non-vanishing but still subject to the constraint (3.1), with their field strengths H Combining the equation forH (t) (2)i in (3.19) with the constraint (3.2), and further taking Hodge duals, a set of equations is obtained that allows one to solve for the electric field strength H (t)i (2) and the magneticH (t) (2) in terms of F i and * F i as Finally, the metric must be rescaled as in (3.5): Now, it only remains to verify that, with these definitions, the constraints (3.3) are satisfied and the equations (3.4) give rise to the N = 3 field equations (A.4). Bringing (3.18) and H (t)i (2) given in (3.20) to (3.3), some calculation shows that these constraints indeed check out identically for arbitrary F i . The Bianchi identities and equations of motion for H ′i (2) and H (L)î (2) immediately reduce to their N = 3 counterparts in (A.4), while those for H (t)i (2) give The last parenthesis is identically zero, as can be easily

Embedding into N = 8 supergravity
The minimal N = 2 and N = 3 supergravities can be finally embedded into the full N = 8 ISO(7) theory by combining their embeddings, given in sections 3.2 and 3.3 above, into the intermediate N = 4 model, with the embedding [53] of the latter into the parent N = 8 theory. With the metric rescaled as in (3.5), the two-form potentials in the restricted tensor hierarchy set to zero, and simply keeping an eye on the auxiliary three-form potential that will eventually give rise to the Freund-Rubin term upon uplift to IIA, the problem reduces to keeping track of the gauge fields. The electric vectors (2.5) and their magnetic duals into the N = 8 magnetic duals as [53] A ij = ǫ ij kÃ′ k ,Ã ia = 0 ,Ã ab = −(Jî − ) abÃ The indices here have been split as I = (i, a), where i = 1, 2, 3 and a = 0, 1, 2, 3 with the index conventions of footnote 1. The antisymmetric, anti-selfdual, quaternionic matrices (Jî − ) ab are defined in appendix A of [53]. The field strengths (2.6), (2.7) are similarly embedded into the N = 8 ones, H IJ (2) , H I (2) , H (2)I J ,H (2)I . Bringing (3.8) and (3.13) to the field strengths version of (3.24), (3.25), the N = 2 graviphoton is finally embedded into the N = 8 vector field strengths. Note that this embedding involves both the minimal N = 2 graviphoton field strength F and its Hodge dual * F . Similarly, bringing (3.18), (3.20) to (3.24), (3.25) written for the field strengths, the N = 3 graviphoton field strength F i and its Hodge dual * F i are embedded into the N = 8 vector field strengths. Equations (3.24), (3.25) and their analogues for the field strengths are useful to work out the ten-dimensional uplift of the minimal theories, to which I now turn.

Truncation to minimal N = 2 supergravity
Particularising the N = 8 consistent truncation formulae of [21,44] to the relevant N = 4 subsector and on to the minimal N = 2 theory as discussed in sections 3.2 and 3.4, a consistent truncation of massive IIA supergravity to D = 4 N = 2 minimal gauged supergravity is obtained. For this purpose, the SU(3)-invariant truncation formulae of [55] also come in handy, along with the generic formulae of [21,44]. This is because the minimal N = 2 theory discussed here is also a subsector of the SU(3)-invariant sector of ISO(7) supergravity. The minimal truncation can be expressed in terms of the same geometric structures on the internal S 6 discussed in [55]. The resulting truncation formulae are formally identical to those found in [12] for the IIA uplift of a particular D = 4 black hole. The D = 4 metric and graviphoton here are generic, though: they are only required to obey the D = 4 N = 2 field equations (A.1).
Here, ds 2 4 and F are the metric and field strength of the D = 4 N = 2 theory (A.2), and * F is the Hodge dual with respect to the former. The S 6 angle α ranges as 0 ≤ α ≤ π, while J is the Kähler form on the complex projective plane; the one-formη is defined aŝ in terms of the S 6 angle ψ, with 0 ≤ ψ ≤ 2π, a one-form σ such that dσ = 2 J, and the D = 4 N = 2 gauge field A affected by the coupling constant g. Finally, the metric ds 2 6 and the formsF 0 (4) ,Ĥ 0 (3) andF 0 (2) correspond to the background values of the N = 2 AdS 4 solution of [21], shifted with the D = 4 gauge potential A throughη in (4.2), namely, ds 2 6 = L 2 3 + cos 2α 1/2 5 + cos 2α 1/8 3 2 dα 2 + 6 sin 2 α 3 + cos 2α ds 2 (CP 2 ) + 9 sin 2 α 5 + cos 2αη 2 , eφ = e φ 0 5 + cos 2α In these expressions, ds 2 (CP 2 ) and vol(CP 2 ) are the Fubini-Study metric, normalised so that its Ricci tensor equals 6 times the metric, and the corresponding volume form. In this minimal N = 2 truncation, the dilaton eφ and the Romans massF (0) take on their exact background values [21]. Although the expressions (4.1) were given in terms of the N = 8 couplings g and m, these should be traded for the classical parameters L and e

Truncation to minimal N = 3 supergravity
The uplift of the minimal N = 3 theory to massive type IIA proceeds similarly. Together with the general uplifting formulae of [21,44], the SO(4)-invariant uplifting formulae of [23] are useful for this purpose, even though the N = 3 minimal theory is not a subsector of the SO(4)-invariant sector of the full N = 8 supergravity. In any case, the general embedding formulae of [21,44] provide the contributions of the SO(3) d Yang-Mills gauged fields on top of the N = 3 background solution as constructed in [23]. I will drop the hat on the SO(3) L indexî in order to label SO(3) d triplets as i = 1, 2, 3 (see footnote 1), and will follow the conventions of [23] for the quantities that pertain to the background geometry.

Discussion
Consistent truncations have been presented to the pure N = 2 and N = 3 gauged supergravities in D = 4, both from a larger theory also in D = 4, and from ten-dimensional massive type IIA supergravity. Similar two-step truncations, from string/M-theory down to a matter-coupled (or even maximal) D-dimensional gauged supergravity, followed by a further truncation to a pure gauged supergravity have been constructed in [59][60][61][62][63][64][65][66][67][68]. The strategy to strip off the matter couplings of a D-dimensional supergravity and truncate it to the gravity multiplet consists in fixing the scalars to their vevs at a suitable vacuum, and truncating out the massive gauge fields at that vacuum (possibly along with some massless gauge fields as well).
In this sense, the truncations constructed in this paper are in the same spirit than the minimal truncation constructed recently in section 3.3 of [68]. In that reference, minimal D = 4 N = 2 gauged supergravity was embedded into D = 4 N = 8 SO(8)gauged supergravity [69] around its N = 2 vacuum [70], and then uplifted [71,72] to D = 11 supergravity [73] on the N = 2 solution of [74]. Unlike in [68], in the present case some of the massive vector fields to be truncated out gauge dyonically shift symmetries of Stückelberg scalars. As shown in section 3, the procedure to switch those off is not simply to set them to zero. Instead, the vector duality relations (2.13) must be employed to write these gauge fields in terms of the surviving R-symmetry gauge fields and their Hodge duals. Essentially the same process was performed in e.g. section 6.1 of [65], in a symplectic frame where the Stückelberg scalars there appeared dualised into tensors.
These subtruncations of larger theories to pure supergravities do not follow from any obvious symmetry principles. Thus, their consistency must be checked at the level of the field equations. This has been done in section 3 for the truncations presented in this paper. In contrast, the intermediate N = 4 subsector of the N = 8 ISO(7) theory, that was used for convenience, does arise from the latter as a singlet sector truncation. This N = 4 theory corresponds to the sector of the N = 8 supergravity that is invariant under the SU(2) or SO(3) R in (2.1) or (2.2) [53].
Using the general N = 8 consistent truncation formulae of [21,44], the minimal N = 2 and N = 3 theories have been uplifted to ten dimensions. In this way, consistent truncations have been constructed of massive type IIA supergravity on the internal spaces of the N = 2 and N = 3 AdS 4 solutions of [21] and [22,23], to the minimal D = 4 N = 2 and N = 3 gauged supergravities. The N = 2 truncation formulae of section 4.1 are straightforward extensions of the formulae given in [12] for the IIA embedding of a particular D = 4 black hole. In section 4.1, the only restriction on the D = 4 N = 2 fields is that they obey the minimal supergravity field equations (A.1). Thus, not only the black hole considered in [12] uplifts to D = 10, but in fact any other solution of minimal N = 2 supergravity does through those consistent embedding formulae. For example, the SU(3)-invariant, supersymmetric Reissner-Nordström black hole with constant scalars discussed in [75,76] can be more economically regarded as a solution of minimal N = 2 gauged supergravity. Also this black hole uplifts to IIA using the same formulae in [12] and section 4.1 here, as any other solution, supersymmetric or otherwise, of minimal D = 4 N = 2 supergravity does. Explicit calculations in related contexts and in [12] for their specific black hole, make supersymmetry expected to be preserved in general by the uplifting process. Similar statements apply to the N = 3 uplift. The supersymmetric solutions to minimal D = 4 N = 2 gauged supergravity have been classified in [77].
In addition to the IIA uplift of N = 2 supergravity given in section 4.1, an alternative uplift of this theory can be given. This follows from the results of section 4.2 simply by bringing to that section the restrictions (A.7) for the further truncation of the D = 4 N = 3 theory to N = 2. While various consistent truncations of string or M-theory are known to D = 4 N = 2 pure gauged supergravity [24,29,40,68], truncations to the pure N = 3 gauged theory are less common. One such truncation has been constructed, from D = 11, in [30] (see also [63]).
The internal spaces corresponding to the IIA truncations of section 4 correspond to smooth geometries on topological six-spheres. Some (singular) generalisations for the background geometries can be easily engineered [21,23] (see also [78]), that are carried over to the corresponding minimal truncations. For example, the N = 2 truncation of section 4.1 is still valid if CP 2 is replaced, along with its related quantities J and σ, with any local Kähler-Einstein four-dimensional space of positive curvature. Similarly, the N = 3 truncation of section 4.2 still holds if the S 3 on which the ρ i take values is replaced with the lens space S 3 /Z p , with the discrete identification acting on the Hopf fiber.
In any case, the results of this paper deliver the consistent truncations to minimal gauged supergravities envisaged in general in [24,25], corresponding to the supersymmetric AdS 4 solutions of massive type IIA supergravity constructed in [21] and [22,23].
A Minimal N = 2 and N = 3 gauged supergravities

A.1 Conventions
The bosonic sector of minimal D = 4 N = 2 gauged supergravity [19,20] includes the metric 3ḡ µν , with line element ds 2 4 , and a gauge field A with field strength F = dA, subject to the field equations The latter two derive from the Einstein-Maxwell Lagrangian with a negative cosmological constant −6g 2 , The AdS vacuum is attained for ds 2 4 = g −2 ds 2 (AdS 4 ), with ds 2 (AdS 4 ) the unit-radius anti-de Sitter metric, and A = 0.
The bosonic field content of minimal D = 4 N = 3 gauged supergravity [20] contains the metricḡ µν , with line element ds 2 4 , and SO(3) gauge fields A i with field strengths These fields obey the field equations where the SO(3) indices i = 1, 2, 3 are raised and lowered with δ ij , and the covariant derivatives are The equations of motion (the latter two equations in (A.4)) derive from the Einstein-Yang-Mills Lagrangian with a cosmological constant −6g 2 , The AdS vacuum is attained for ds 2 4 = g −2 ds 2 (AdS 4 ), where ds 2 (AdS 4 ) is again the unitradius anti-de Sitter metric, and A i = 0. The N = 3 theory (A.6) can be consistently truncated to the N = 2 theory (A.2) through the identifications A.2 SO(7) embedding of the N = 2 and N = 3 gauge groups It is interesting to determine the generators of SO (7), and ultimately of SL(8) ⊂ E 7 (7) , corresponding to the U(1) and SO(3) d subgroups that are gauged in the N = 2 and N = 3 minimal models. This U(1) is the commutant of the SU(3) of (2.1) inside SO (7). The group SO(3) d is the one that appears in (2.2), and its generators have already been given in (A.5) of [53]. In any case, the relevant generators can be found by bringing (3.24), to the general expression for the ISO(7) gauge fields coupled to SL(8) generators [51]. If t A B , A = 1, . . . , 8, are the generators of the SL(8) subalgebra of E 7 (7) , then SO (7) is generated by T IJ ≡ 2 t [I K δ J]K , with A = (I, 8), I = 1, . . . , 7. The U(1) and SO(3) d R-symmetry groups of the minimal N = 2 and N = 3 supergravities turn out to be generated by the generators T and T k , k = 1, 2, 3, of SO(7) given by: with the index I = (i, a) split as below (3.25) (see also footnone 1), and the antisymmetric, anti-selfdual matrices (Jî − ) ab as defined in appendix A of [53]. In particular this U (1)

B.1 Truncation to minimal N = 2 supergravity
The massive type IIA configuration (4.1), (4.3) has the local form In these expressions, ds 2 4 and F = dA are the fields of minimal N = 2 gauged supergravity (A.2), µ 0 is a constant, X(α), etc., are functions of the angle α that can be read off from (4.1), (4.3), and J = 1 2 dσ is the Kähler form on CP 2 . This configuration, with the explicit functions X(α), etc. that can be read off from (4.1), (4.3) of the main text, was obtained in section 4.1 by consistent uplift using the formulae of [21,44,55]. Here, I will explicitly verify at the level of the field equations of the tendimensional form fields that these expressions do indeed define a consistent truncation of massive IIA supergravity down to minimal N = 2 gauged supergravity in four dimensions.
Imposing the Bianchi identities of massive type IIA, (A.4) of [44] in our conventions, on the configuration (B.1), and using the Bianchi identity, dF = 0, and Maxwell equation, d * F = 0, of the D = 4 graviphoton, the relations (B.4) of [55] Here and subsequently, a prime denotes derivative with respect to α, and the explicit α dependence is suppressed from the functions. These and the following relations are obtained by assuming that F , * F , F ∧ F and F ∧ * F are non-vanishing and independent. Next, I turn to check the equations of motion of the type IIA form fields, collected in (A.5) of [44]. TheF (4)

B.2 Truncation to minimal N = 3 supergravity
The massive type IIA configuration (4.8), (4.9) is of the (also massive) local form where ds 2 4 and F i are the fields of D = 4 N = 3 gauged supergravity (A.6), µ 0 is a constant, X(α), etc., are all real functions of the angle α, theμ i , i = 1, 2, 3, are constrained coordinates that define a unit radius S 2 through (4.4), andρ i correspond to the rightinvariant Maurer-Cartan forms on S 3 , shifted by the D = 4 N = 3 SO(3) gauge field A i , as in (4.6). In this appendix, I will denote the covariant derivative ofμ i for convenience aŝ The function A 0 (α), as well as all other functions of α in (B.7), can be read off from the concrete expressions given section 4.2 of the main text. It is worth emphasising that all of the above functions are real. Bars have been used as a mere notational device, in order to neutralise the real threat of running out of symbols. For example, M 2 (α) and M 2 (α), etc., are real and unrelated.
Let us now find the equations that these functions must obey for the configuration (B.7), (B.8) to solve the massive type IIA field equations, assuming that the D = 4 Yang-Mills field strength F i is covariantly closed and co-closed as dictated by its Bianchi identity and equation of motion, (A.4). For that purpose, one needs to use the identities recorded in equation (A.10) of [23] as well as the following ones involving the D = 4 gauge field strength: Equipped with these identities, and requiring that F i , * F i , F i ∧ F i ,μ iμj F i ∧ F j , F i ∧ * F i andμ iμj F i ∧ * F j be independent quantities, a very lengthy calculation allows one to obtain the set of algebraic and differential relations that the functions of α in (B.7) must obey for that configuration to solve the type IIA field equations. TheF (4) Bianchi identity gives the first equation in (C.7) of [23] and (B.10) theĤ (3) Bianchi identity gives the second equation in (C.7) of [23], together with and theF (2) Bianchi identity gives the third through sixth equations in (C.7) of [23] and