Embedding the SU(3) sector of SO(8) supergravity in D = 11

The SU(3)-invariant sector of maximal supergravity in four dimensions with an SO(8) gauging is uplifted to D ¼ 11 supergravity. In order to do this, the SU(3)-neutral sector of the tensor and duality hierarchies of the D ¼ 4 N ¼ 8 supergravity is first worked out. The consistent D ¼ 11 embedding of the full, dynamical SU(3) sector is then expressed at the level of the D ¼ 11 metric and three-form gauge field in terms of these D ¼ 4 tensors. The redundancies introduced by this approach are eliminated at the level of the D ¼ 11 four-form field strength by making use of the D ¼ 4 duality hierarchy. Our results encompass previously known truncations of D ¼ 11 supergravity down to sectors of SO(8) supergravity with symmetry larger than SU(3), and include new ones. In particular, we obtain a new consistent truncation of D ¼ 11 supergravity to minimal D ¼ 4 N ¼ 2 gauged supergravity. The three-form field strengths are dual to scalar-dependent combinations of derivatives of scalars:


I. INTRODUCTION
Being complicated theories with large field contents, it proves useful for applications to truncate maximal gauged supergravities to smaller subsectors that are invariant under some symmetry group. In this paper, we will be interested in D ¼ 4 N ¼ 8 supergravity with an electric SO(8) gauging [1] and one of its most fruitful sectors: the one invariant under the SU(3) subgroup of SO (8). This sector preserves N ¼ 2 supersymmetry and retains, along with the N ¼ 2 gravity multiplet, a vector multiplet and a hypermultiplet with an Abelian gauging. The (AdS) vacuum structure in this sector has been completely charted [2] and the corresponding mass spectra within the full N ¼ 8 theory determined [3,4]. Holographic duals have been established for some of these vacua as distinct superconformal phases [5,6] of the M2-brane field theory. Other interesting solutions of, for example, domain wall [7,8], defect [9], black hole [10] or Euclidean [11] type have been constructed in this sector that enjoy precise holographic interpretations [6,12].
The relevance for holography of D ¼ 4 N ¼ 8 SO(8)gauged supergravity [1] is intimately linked to the fact that it can be obtained as a consistent truncation of D ¼ 11 supergravity [13] on the seven-sphere, S 7 [14,15].
Further results on the consistency of the truncation have been given more recently in [16][17][18][19][20][21][22][23][24][25][26]. The goal of this paper is to provide the consistent uplift of the SU(3) sector of SO(8) gauged supergravity into D ¼ 11 by using the uplifting formulas of [25], thus putting them to the test. We extend previous results on the consistent D ¼ 11 embedding of further subsectors contained in the SU(3) sector [4,27,28], and provide a unified treatment. We make contact with those previously known consistent truncations and establish new ones. In particular, we construct a new consistent embedding of D ¼ 4 N ¼ 2 pure gauged supergravity into D ¼ 11, where the internal geometry on S 7 corresponds to the N ¼ 2 SUð3Þ × Uð1Þinvariant solution obtained by Corrado-Pilch-Warner (CPW) [27].
A systematic approach to the consistent uplift of D ¼ 4 N ¼ 8 SO(8) supergravity to D ¼ 11 was proposed in [25], similar to the method employed in [29,30] to uplift D ¼ 4 N ¼ 8 ISO (7) supergravity [31] into type IIA. This approach relies on the tensor hierachy [32,33] of maximal four-dimensional supergravity-the extension of its field content to include the magnetic gauge fields along with higher rank potentials in representations of E 7ð7Þ . The full D ¼ 11 embedding of the bosonic sector of SO(8) supergravity can be expressed at the level of the D ¼ 11 metric and three-form potential in terms of a subset, dubbed restricted in [25], of the D ¼ 4 tensor hierarchy that is still N ¼ 8 but only covariant under SLð8Þ ⊂ E 7ð7Þ . The D ¼ 4 tensor hierarchy carries redundant degrees of freedom (d.o.f.) beyond those contained in the conventional N ¼ 8 Lagrangian, and these are carried over to the D ¼ 11 embedding. These redundancies can be eliminated in D ¼ 4 by imposing suitable duality relations among the field strengths of the tensor hierarchy [34]. Expressing the D ¼ 11 embedding at the level of the four-form field strength and employing these D ¼ 4 dualizations, redundancy-free uplifting formulas are obtained that contain only the dynamically independent fields (that is, the metric, the scalars and the electric vectors) that feature in the conventional D ¼ 4 N ¼ 8 Lagrangian.
Some aspects of the SU(3)-invariant sector of SO(8)gauged supergravity are summarized in Sec. II, and the SU(3)-invariant restricted tensor and duality hierarchies are constructed. Section III discusses the consistent uplift of the SU(3)-invariant sector into D ¼ 11 supergravity following the tensor and duality hierarchy approach. Contact with the consistent uplift of previously known subsectors is made and a new D ¼ 11 embedding of D ¼ 4 N ¼ 2 pure gauged supergravity is established. Section IV further tests our formalism by recovering known AdS 4 solutions in D ¼ 11 from uplift of critical points, and Sec. V concludes. Some technical details are contained in the Appendixes. Our conventions for D ¼ 11 and D ¼ 4 N ¼ 8 supergravity are those of [25].

II. THE SU(3)-INVARIANT SECTOR OF SO(8) SUPERGRAVITY
Let us start by reviewing some aspects of the SU(3) sector of SO(8)-gauged supergravity. We choose a triangular, or Iwasawa, parametrization for the [SU(3)-invariant truncation of the] E 7ð7Þ =SUð8Þ coset representative. Since previous literature often chooses the unitary gauge for the coset, we believe that our presentation has some intrinsic value even if the material that is covered (the Lagrangian in Sec. II A, the further subsectors in II C, and the vacuum structure in II D) is mostly review. The SU(3)-invariant, restricted tensor and duality hierarchies worked out in Sec. II B are new.

A. Field content and Lagrangian
The SU(3)-invariant sector of SO(8)-gauged maximal four-dimensional supergravity [1] corresponds to an N ¼ 2 supergravity coupled to a vector and a hypermultiplet. In addition to the fields entering these N ¼ 2 multiplets, we wish to consider the SU(3)-singlets in the (restricted, in the sense of [25]) N ¼ 8 tensor hierarchy [32,33]. The relevant bosonic matter content thus includes the metric∶ ds 2 4 ; 6 scalars∶ φ; χ; ϕ; a; ζ;ζ; 2 electric vectors and their magnetic duals∶ A 0 ; A 1 ;Ã 0 ;Ã 1 ; 5 two-form potentials∶ B 0 ; B 2 ; B ab ¼ B ðabÞ ; 4 three-form potentials∶ C 1 ; C ab ¼ C ðabÞ ; ð2:1Þ all of them real. The superscripts on B 0 , B 2 and C 1 are just labels without further meaning. The electric and magnetic vectors can be collectively denoted A Λ andÃ Λ , with the index Λ ¼ 0, 1 formally labeling "half" the fundamental representation of Spð4; RÞ. The indices on B ab and C ab take on two values which, for convenience, are labeled a ¼ 7, 8. The index a formally labels a doublet of SL(2), but we do not attach any significance to its position as it can be raised and lowered with δ ab . See Appendix A for the embedding of the SU(3)-invariant fields (2.1) into their parent N ¼ 8 counterparts.
where g is the gauge coupling constant. Following [31], here and throughout we have employed the shorthand definitions ð2:5Þ The covariant derivatives (2.4) correspond to an electric gauging of the Uð1Þ 2 Cartan subgroup of SUð2Þ × Uð1Þ ⊂ SUð2; 1Þ generated by and derives from the following real superpotential (squared) through the usual formula Here, G mn , m ¼ 1; …; 6, denotes the nonlinear sigma model metric on (2.2), and G mn its inverse, which can be read off from the scalar kinetic terms in the Lagrangian (2.3). Finally, the gauge kinetic matrix is ð2:10Þ and the (electric) gauge two-form field strengths that appear in (2.3) are simply We have computed the SU(3)-invariant Lagrangian (2.3) and the quantities that define it using the D ¼ 4 N ¼ 8 embedding tensor formalism [35] (see [36] for a recent review) with the conventions of [25] for the SO(8) gauging [1]. The superpotential (2.8) corresponds to one of the eigenvalues of the N ¼ 8 gravitino mass matrix restricted to the SU(3)-singlet space. See [4] for the N ¼ 2 special geometry of the model, in unitary gauge for the scalar coset. Superpotentials have previously appeared, also in unitary gauge, in [8,37].

B. Restricted tensor and duality hierarchies
Besides the electric gauge fields that enter the conventional supergravity Lagrangian, one may consider a set of other gauge potentials in the so-called tensor hierarchy. The full N ¼ 8 tensor hierarchy includes all vectors, both electric and magnetic, along with higher-rank (two-, three-, and four-form) gauge potentials, in representations of the duality group of the ungauged theory, E 7ð7Þ [32,33]. The full tensor hierarchy corresponding to the N ¼ 2 subsector at hand is obtained by retaining the singlets under the decomposition of those E 7ð7Þ representations under SU(3). Here, we are only interested in a subset of the N ¼ 8 tensor hierarchy. The reason is that not all E 7ð7Þ -covariant fields in the hierarchy are necessary to describe the full D ¼ 11 embedding of N ¼ 8 SO(8)-gauged supergravity, as argued in [25]. Only the vectors and some two-and three-form potentials in representations of the maximal SLð8; RÞ subgroup of E 7ð7Þ are relevant for this purpose. This subset was dubbed the restricted tensor hierarchy in [25]. Thus, the tensor fields that we want to consider are the singlets under SUð3Þ ⊂ SLð8; RÞ of the N ¼ 8 restricted tensor hierarchy. The complete list is given in (2.1). See Appendix A for further details.
The field strengths of the SU(3)-invariant, restricted tensor hierarchy fields can be obtained by particularizing the N ¼ 8 expressions given in [25], with the help of the expressions contained in Appendix A for their embedding into their N ¼ 8 counterparts. The electric vector field strengths have already been given in (2.11), while the magnetic field strengths arẽ The three-form field strengths read, in turn, Finally, the four-form field strengths are with DC ab ¼ dC ab þ 2gϵ cða A 0 ∧ C bÞ c . The field strengths (2.11)-(2.14) are subject to the Bianchi identities  (14) of [25] to the present case.
All of the fields in the restricted tensor hierarchy carry d.o.f., although not independent ones. They are instead subject to a duality hierarchy [34]. The magnetic two-form field strengths can be written as scalar-dependent combinations of the electric gauge field strengths and their Hodge duals: ; þ3χe 2φ ð2X − 1ÞH 0 ð2Þ þ 6χX 2 H 1 ð2Þ : ð2:16Þ The three-form field strengths are dual to scalar-dependent combinations of derivatives of scalars: ð2:17Þ Finally, the four-form field strengths correspond to the following scalar-dependent top forms on four-dimensional spacetime:    (2.6). These are the only symmetries of the SU(3)-invariant potential (2.7). The symmetry is enhanced in the subsectors that we now turn to discuss.

C. Some further subsectors
It is interesting to consider further subsectors contained in the SU(3)-invariant sector in the notation that we are using. A natural way to obtain those is to impose invariance under a subgroup G of SO(8) that contains SU(3). The relevant tensor hierarchy field strengths and their dualization conditions are obtained by bringing the G-invariant restrictions specified on a case-by-case basis below to (2.11)-(2.14) and (2.16)-(2.18). The field content in each of these subsectors is summarized for convenience in Table I.
An obvious yet still interesting sector is attained by requiring an additional invariance under the Uð1Þ 2 with which SU(3) commutes inside SO (8). The resulting SUð3Þ × Uð1Þ 2 -invariant sector throws out the hypermultiplet and sets identifications on the restricted tensor hierarchy, 2 This sector thus reduces to N ¼ 2 supergravity coupled to a vector multiplet with a Fayet-Iliopoulos gauging, namely, to the Uð1Þ 4 -invariant sector (i.e., the gauged STU model) with all three vector multiplets identified, along with the relevant tensor hierarchy fields. Inserting (2.23) in (2.3), the Lagrangian indeed reduces to e.g., (6.28), (6.29) of [28] with the fields and coupling constants here and there identified as The potential of the SUð3Þ × Uð1Þ 2 -invariant sector, (2.7) with (2.23), acquires a symmetry under the compact generator, k½E 0 − k½F 0 in the notation of (A15), of the vector multiplet scalar manifold. The field redefinition in the first line of (2.24) is a Uð1Þ ⊂ SLð2; RÞ transformation generated by this Killing vector, followed by a change of sign of χ.
One may also consider SUð3Þ × Uð1Þ-invariant sectors, with U(1) chosen to be one of the three trialityinequivalent 3 Uð1Þ v , Uð1Þ s or Uð1Þ c factors with which SU(3) commutes inside SO (8). These invariant sectors are attained by setting while retaining both vectors and their magnetic duals.
Only the SUð3Þ × Uð1Þ s -invariant subtruncation is supersymmetric, and coincides with the SUð3Þ × Uð1Þ 2 sector discussed above-in other words, invariance under Uð1Þ s cannot be enforced on top of SU(3) without also imposing Uð1Þ c invariance, but not the other way around. The other two subtruncations retain the would-be vector multiplet and "half" a hypermultiplet: either the scalars ϕ, a in the SUð3Þ × Uð1Þ v sector, or the pseudoscalars ζ,ζ in the SUð3Þ × Uð1Þ c sector, with ϕ a function of the  [38], with labels ðv; þ; −Þ there denoted ðv; s; cÞ here. We follow the spectrum conventions of e.g., [39] whereby, at the SO(8) vacuum, the (graviton, gravitini, vectors, spinors, scalars, pseudoscalars) of N ¼ 8 supergravity lie in the ð1; 8 s ; 28; 56 s ; 35 v ; 35 c Þ of SO (8).
pseudoscalars in the latter case. The covariant derivatives (2.4) simplify accordingly. In the SUð3Þ × Uð1Þ v sector, ϕ, a remain charged under A 0 and no field is charged under A 1 . In the SUð3Þ × Uð1Þ c sector the covariant derivatives reduce to showing that ζ,ζ become a doublet charged only under the combined gauge field A 0 þ 3A 1 . It is possible to further truncate the SUð3Þ × Uð1Þ c sector to a two-scalar model retaining ðφ; ζÞ along with B 77 ¼ B 88 and C 1 , 3) with these identifications and the superpotential reduces, from (2.8), to where e 2ϕ is shorthand for the expression in terms of ζ ¼ζ that appears in (2.26). This is the model considered in [27].
The identifications [the second equation implies e 2ϕ here ¼ cosh 2 χ there on (2.26)] indeed bring the superpotential (2.29) to (3.9) of [27], up to normalization. The SUð3Þ × Uð1Þ-invariant sectors can be further reduced by imposing a larger SOð6Þ ∼ SUð4Þ symmetry. The corresponding sectors are obtained by letting Again, only the SUð4Þ s -invariant sector is supersymmetric: it truncates out the vector multiplet of the SUð3Þ × Uð1Þ s sector, leading to minimal N ¼ 2 gauged supergravity.
Setting all scalars to zero as in (2.33), further setting consistently B 0 ¼ 2 3 B 2 ¼ 0, and rescaling for convenience the metric and the graviphoton as Eq. (2.3) reduces to the bosonic Lagrangian of pure N ¼ 2 gauged supergravity, withF ≡ dĀ. For later reference, we note that the only tensor hierarchy field strengths that are active in the SUð4Þ s sector are where the bars refer to the rescaled quantities (2.34). The other two truncations (2.31), (2.32) are manifestly nonsupersymmetric. Imposing invariance under SOð6Þ v selects the proper scalars φ, ϕ, a along with the gauge field A 0 , while invariance under SUð4Þ c retains the pseudoscalars χ, ζ,ζ along with A 0 þ A 1 . In the latter case, the scalars become functions of the pseudoscalars as indicated in (2.32). It was noted in [4] that the SUð4Þ c -invariant sector coincides with a subtruncation, considered in [40], of the D ¼ 4 N ¼ 2 gauged supergravity obtained upon consistent truncation of M-theory on any (skew-whiffed) Sasaki-Einstein seven-manifold [41]. Indeed, using (2.32) and further identifying the pseudoscalars and vectors here and in [40] as The SOð7Þ s truncation gives minimal N ¼ 1 gauged supergravity while the SOð7Þ v and the SOð7Þ c sectors are nonsupersymmetric. They respectively retain one dilaton (φ ¼ ϕ) and one axion [χ, together with the identifications (2.39)], along with the relevant tensors in the hierarchy. All three SO (7) sectors are contained within the G 2 -invariant sector. This corresponds to N ¼ 1 supergravity coupled to a chiral multiplet with a scalar manifold SLð2Þ=SOð2Þ which is diagonally embedded in (2.2) via The Lagrangian in this sector is (2.3) with the identifications (2.41). It can be cast in canonical N ¼ 1 form, in the conventions of e.g., Sec. 4.2 of [31], in terms of the following Kähler potential and holomorphic superpotential All of the above further truncations arise from symmetry principles, by retaining the fields that are neutral under the relevant invariance groups. For this reason, the above truncations can be directly implemented at the level of the Lagrangian (2.3). In particular, a consistent truncation to minimal N ¼ 2 supergravity is obtained by retaining singlets under SUð4Þ s , as noted above. We conclude this section by noting an alternate truncation of the SU(3) sector to minimal N ¼ 2 supergravity that is inequivalent to the SUð4Þ s -invariant truncation. In fact, this alternative minimal truncation is not driven by symmetry principles in any obvious way, so we have verified its consistency at the level of the field equations. First, freeze the scalars to their vacuum expectation values (vevs) at the SUð3Þ × Uð1Þ cinvariant vacuum (see Sec. II D), Second, identify the electric and magnetic vectors as turn off the two-form potentials, and retain an auxiliary three-form potential as Finally, rescale the metric for convenience: withF ≡ dĀ. The relations here for the magnetic field strengths are compatible with the vector duality relations (2.16) evaluated on the scalar vevs (2.43), and the last equality for the magnetic graviphoton field strength˜F is fixed by˜F ¼ ∂L=∂F, with L as in (2.35). Moving on to the three-form field strengths, we find that all of them are zero by bringing (2.44), (2.45) to their definitions (2.13) in terms of potentials. This was expected, as the three-form field strengths are dual to combinations (2.17) of (Hodge duals of) derivatives of scalars, and these have been frozen to their vevs (2.43). Finally, for the four-form field strengths we obtain, from (2.14) with (2.45), H 78 The list of vacua of D ¼ 4 N ¼ 8 supergravity with an electric SO(8) gauging [1] that preserve at least a subgroup SU(3) of SO(8) was elucidated in [2]. All of them are AdS. These vacua arise as extrema of the scalar potential (2.7), in our conventions, and for convenience we have summarized them in Table II. The table includes the residual supersymmetry N and bosonic symmetry G 0 for each vacuum, as well as its location in the scalar space (2.2) in the parametrization that we are using. The corresponding cosmological constant, given by (2.7), and the scalar mass spectrum within the SU(3)-invariant sector is also given. See [4] for the bosonic spectra within the full N ¼ 8 supergravity. All three supersymmetric points are also extrema of the superpotential (2.8). On the SO(8) and the G 2 points, the F-terms that derive from the holomorphic superpotential (2.42) also vanish.
It was argued in [25] that some combinations of the fourform field strengths of the duality hierarchy ought to vanish at critical points of the scalar potential, thus yielding necessary conditions for critical points. In our SU(3)invariant case, these conditions read Using the dualization conditions (2.18), it can be checked that the relations (2.49) do indeed hold at the critical points summarized in Table II.

III. D = 11 UPLIFT
We now switch gears and present the D ¼ 11 embedding of the SU(3)-invariant sector considered in the previous section. We will use the consistent S 7 uplifting formulas given in [25]. It is a tedious, but otherwise mechanical, exercise to particularize the general N ¼ 8 uplifting formulas in that reference to the SU(3)-invariant sector at hand. Section III A contains the D ¼ 11 uplift of the entire SU(3)-invariant sector while Sec. III B particularizes to some relevant subsectors and makes contact with previous literature. Section III C contains a new consistent truncation of D ¼ 11 supergravity to minimal D ¼ 4 N ¼ 2 gauged supergravity.

A. Uplift of the SU(3) sector
We first find it useful to present the result in terms of R 8 "embedding coordinates" μ A , A ¼ 1; …; 8, in the 8 v of SO (8), that define the S 7 as the locus In maintaining an explicitly real notation, it is thus convenient to split R 8 ¼ R 6 × R 2 , and the indices as A ¼ ði; aÞ, with i ¼ 1; …; 6 and a ¼ 7, 8 respectively labeling the first and second factors.
ij (real) and Ω ijk (complex) that define the natural Calabi-Yau structure of R 6 . See (A6) for our conventions. Inside R 8 , these tensors are respectively invariant under SOð6Þ v × SOð2Þ, SUð3Þ × Uð1Þ 2 and SUð3Þ × Uð1Þ c , where SO(2) rotates the R 2 factor in R 8 ¼ R 6 × R 2 . Indices on R 6 and R 2 are raised and lowered with δ ij and δ ab , respectively.
Only the D ¼ 4 metric, the scalars, and the electric gauge fields in the SU(3)-invariant restricted duality TABLE II. All critical points of D ¼ 4 N ¼ 8 supergravity with an electric SO(8) gauging with at least SU(3) invariance, reproducing the results of [2] in our parametrization. For each point we give the residual supersymmetry N and bosonic symmetry G 0 within the full N ¼ 8 theory, their location in the parametrization that we are using, the cosmological constant V 0 and the scalar mass spectrum within the SU(3)-invariant sector. The masses are given in units of the AdS radius, L 2 ¼ −6=V 0 . We have abbreviated Uð3Þ c ≡ SUð3Þ × Uð1Þ c . hierarchy (2.1) enter the D ¼ 11 metric dŝ 2 11 . In order to express the result, it is useful to introduce a symmetric matrix h ab of D ¼ 4 scalars and its inverse as 4 ð3:2Þ and the following combination of D ¼ 4 scalars and constrained coordinates μ i , μ a , With these definitions, the embedding into the D ¼ 11 metric reads where ϵ ab is the totally antisymmetric symbol with two indices, and the covariant derivatives are defined as ð3:5Þ For generic values of the D ¼ 4 scalars, the metric (3.4) enjoys an SUð3Þ × Uð1Þ v isometry. Moving on to the D ¼ 11 three-formÂ ð3Þ , all the D ¼ 4 fields in the tensor hierarchy (2.1), except for the metric, enter its expression. A long calculation yieldsÂ where A is a three-form on the internal S 7 that depends on the D ¼ 4 scalars: ijk ÞDμ i ∧ Dμ j ∧ Dμ k : ð3:7Þ Here, we have defined the shorthand functions and one-forms The field strength four-formF ð4Þ ¼ dÂ ð3Þ is computed to bê ij Dμ i ∧ Dμ j þ ijk Þμ i Dμ j ∧ Dμ k ∧ H 1 ð2Þ þ dA scalars : ð3:10Þ In this expression, H 1 ð4Þ , H ab ð4Þ , etc., turn out to reproduce the D ¼ 4 four-, three-and magnetic two-form field strengths (2.12)-(2.13) of the restricted tensor hierarchy (2.1). This provides a D ¼ 11 crosscheck of the D ¼ 4 calculation of Sec. II B. The terms that contain the electric two-form field strengths H 0 ð2Þ , H 1 ð2Þ , come from the vector contributions in the covariant derivatives Dμ i and Dμ a in (3.7). Finally, dA scalars contains two types of terms. The first type includes contributions of covariant derivatives of D ¼ 4 scalars, wedged with three-forms on the internal S 7 . The second type includes internal four-forms with coefficients that depend on the D ¼ 4 scalars algebraically only. The presence inÂ ð3Þ of J In particular, the Freund-Rubin term [the first two contributions on the right-handside of (3.10)], can be simplified by using the identities (2.19), (2.22) that relate the dualized four-form field strengths (2.18) to the scalar potential (2.3) and its derivatives: At a critical point, the terms in derivatives of the potential drop out and the Freund-Rubin term becomes proportional to the AdS 4 cosmological constant, in agreement with the general N ¼ 8 discussion of [25]. See also [24] for a related discussion. All the Freund-Rubin terms that we write for the truncations to specific subsectors in Sec. III B and for the concrete AdS 4 solutions in Sec. IV agree with the generic expression (3.11).

B. Uplift of some further subsectors
The uplifting formulas of Sec. III A simplify by imposing a symmetry enlargement, carried over to D ¼ 11 by restricting the D ¼ 4 fields as in Sec. II C. Introducing intrinsic S 7 angles by solving the constraint (3.1) is also facilitated in further subsectors, as some intrinsic angles are better suited than others to make the relevant symmetry apparent in D ¼ 11. See Appendix B for some relevant geometric structures on S 7 .

SU(4)-invariant sectors
While the deformations inflicted on the internal S 7 by the SU (3) ð3:18Þ This coincides with the consistent truncation of D ¼ 11 supergravity down to minimal N ¼ 2 gauged supergravity obtained in [43], with straightforward identifications. An alternate D ¼ 11 embedding of minimal N ¼ 2 supergravity will be given in Sec. III C.

G 2 -invariant sector
The D ¼ 11 embedding formulas of Sec. III A particularized to the G 2 -invariant sector (2.41) become, in the relevant set of intrinsic coordinates described in Appendix B, ðe 2φ X −3 Δ 1 dβ 2 þ sin 2 βds 2 ðS 6 ÞÞ; A ð3Þ ¼ C 1 sin 2 β þ C 88 cos 2 β þ 4g −1 sin β cos βB 77 ∧ dβ þ g −3 χΔ −1 1 sin 2 β½e 2φ X −1 Δ 1 J ∧ dβ þ X 2 sin β cos βReΩ þ e 2φ Xsin 2 βImΩ; ð3:22Þ where β is an angle on S 7 , ds 2 ðS 6 Þ is the round metric on S 6 normalized so that the Ricci tensor equals five times the metric, J and Ω are the homogeneous nearly Kähler forms on S 6 and the function Δ 1 is, from (3.3) with (B22), The associated four-form field strength readŝ C. Minimal N = 2 gauged supergravity from D = 11 It was noted in Sec. II C that the SUð4Þ s sector coincides with minimal N ¼ 2 gauged supergravity. In Sec. III B 2, the corresponding D ¼ 11 uplift was obtained and shown to coincide with the consistent embedding of [43]. It was also discussed at the end of Sec. II C that the SU(3) sector admits an alternative truncation to minimal N ¼ 2 supergravity, by fixing the scalars to their vevs (2.43) at the N ¼ 2, SUð3Þ × Uð1Þ c -invariant point and selecting the N ¼ 2 graviphoton as in (2.44). Bringing these D ¼ 4 identifications to the general SU(3)-invariant consistent uplifting formulas of Sec. III A, we obtain a new embedding of pure N ¼ 2 gauged supergravity into D ¼ 11.
We find it convenient to present the result in local intrinsic S 7 coordinates ψ 0 , τ 0 , α, and in terms of a local five-dimensional Sasaki-Einstein structure η 0 , J 0 and Ω 0 . The former are locally related to the global coordinates ψ, τ, α, defined in (B1), that are adapted to the topological description of S 7 as the join of S 5 and S 1 , with α here identified with that in (B1) and The local five-dimensional Sasaki-Einstein structure forms η 0 , J 0 and Ω 0 are related to their globally defined counterparts η ð5Þ , J ð5Þ and Ω ð5Þ discussed in Appendix B and the global coordinate ψ via

ð3:26Þ
The real two-form J 0 coincides with the Kähler form on CP 2 , σ is a one-form on the latter such that dσ ¼ 2J 0 [given e.g., by (B11)] and the constant phase e i π 4 in the complex two-form Ω 0 has been chosen for convenience, in order to simplify the resulting expressions. The primed forms defined in (3.26) satisfy the Sasaki-Einstein conditions (B5) and (B6).
Bringing all these definitions, along with the D ¼ 4 restrictions (2.43)-(2.46), to the uplifting formulas (3.4), (3.6), (3.7), we find a new consistent embedding of minimal D ¼ 4 N ¼ 2 gauged supergravity (2.35) into the D ¼ 11 metric and three-form: These expressions depend explicitly on the dynamical D ¼ 4 metric ds 2 4 and graviphotonĀ. The former only features in dŝ 2 11 but not inÂ ð3Þ . The latter appears both in dŝ 2 11 and inÂ ð3Þ , but only through the gauge covariant derivative This singles out ψ 0 as the angle on the local N ¼ 2 "Reeb" direction and thus justifies the primed coordinates (3.25) that we chose to present the result. Two other D ¼ 4 fields enter the consistent embedding through the three-form (3.28): the magnetic dual,˜Ā, of the D ¼ 4 graviphoton, and the auxiliary three-form potential C 1 .
The four-form field strength corresponding toÂ ð3Þ in (3.27) can be computed with the help of (the primed version of) the Sasaki-Einstein conditions (B5), (B6). We find

IV. RECOVERING D = 11 AdS 4 SOLUTIONS
Setting the scalars to the vevs at each critical point with at least SU(3) invariance that were recorded in Table II, and turning off the relevant tensor hierarchy fields, the consistent embedding formulas of Sec. III produce AdS 4 solutions of D ¼ 11 supergravity. All these D ¼ 11 solutions are known, so our presentation must necessarily be brief. Our main motivation to work out these solutions is rather to test the consistency of the uplifting formulas of [25] [and their particularization to an explicit, SU(3)invariant, subsector]. Except for the more involved D ¼ 11 Einstein equation, we have indeed verified that the metrics and four-forms that we write below do indeed solve the eleven-dimensional field equations. Please refer to Appendix D for details.
We present the solutions in the appropriate intrinsic S 7 angles defined in Appendix B. These have already been employed in Sec. III B to write the consistent D ¼ 11 embedding of various further subsectors. Also, AdS 4 is always taken to be unit radius (so that the Ricci tensor equals −3 times the metric). As a consequence, the metric ds 2 ðAdS 4 Þ that appears in the expressions below is related to the metric ds 2 4 that appears in the D ¼ 4 Lagrangian (2.3) and D ¼ 11 embedding (3.4) by a rescaling where V 0 is the cosmological constant at each critical point given in Table II. The Freund-Rubin term is rescaled accordingly with respect to (3.11). Let us first discuss the supersymmetric solutions. The N ¼ 8, SO(8) point uplifts to the Freund-Rubin solution [44] for which the internal four-form vanishes and the internal metric is the round, Einstein metric ds 2 ðS 7 Þ, given in e.g., (B3) or (B17). The N ¼ 2, SUð3Þ × Uð1Þ c critical point uplifts to the D ¼ 11 CPW solution [27]. A local form of this solution can be obtained from the expressions in Sec. III C by turning off the D ¼ 4 graviphoton,Ā ¼ 0, F ¼ 0, and fixing the metric to ds 2 4 ¼ g −2 ds 2 ðAdS 4 Þ. As a check, we have verified that the solution in R 8 embedding coordinates μ A , directly obtained from the formulas in Sec. III A, perfectly agrees with the CPW solution as given in [45]. Finally, the N ¼ 1 G 2 -invariant solution can be written, using the results and the notation of Sec. III B 3, in terms of the homogeneous nearly Kähler structure of the S 6 inside S 7 as A ¼ sin 2 β 3 3=4 ffiffi ffi 2 p g 3 ð2 þ cos 2βÞ ½3 sin β cos βReΩ þ ffiffi ffi 3 p sin 2 βImΩ þ ð2 þ cos 2βÞJ ∧ dβ: ð4:3Þ This solution was first obtained by de Wit, Nicolai and Warner [15]. Turning to the nonsupersymmetric solutions, the SO(7) critical points can again be uplifted using the results and conventions of Sec. III B 3. The SOð7Þ v solution uplifts to a solution first written by de Wit and Nicolai [46]. In our conventions, we get while the SOð7Þ c point uplifts to Englert's solution [47] 10 with internal three-form In the SOð7Þ c solution, ds 2 ðS 7 Þ is, as always, the round, SO(8)-invariant metric. It should be understood in this context as the sine-cone form (B23). Since SOð7Þ c ⊃ SUð4Þ c , this solution can also be reobtained from the SUð4Þ c -invariant truncation of Sec. III B 2 and written in terms of the homogeneous Sasaki-Einstein structure on S 7 . The D ¼ 11 metric is the same appearing in (4.5) with ds 2 ðS 7 Þ now understood as the Hopf fibration (B17), and the four-form is given bŷ with internal three-form The metric in (4.5) and four-form (4.7) for the SOð7Þ c solution coincide with (3.11) of [40] upon using the redefinitions (2.37), and making an appropriate choice for the phase of the complex scalar χ there ≡ − 1 ffiffi 3 p ðζ here þ iζ here Þ, which is unfixed at the critical point. We obtain perfect agreement with [40] upon shifting that phase by π.
Finally, the SUð4Þ c -invariant point gives rise to the Pope-Warner solution [48] in eleven dimensions. Using the results of Sec. III B 2, this solution can also be written in terms of the homogeneous Sasaki-Einstein structure on S 7 as where the internal three-form potential is now We again find agreement with [40]: (4.9) coincides with (3.8) of that reference when the identifications (2.37) are taken into account and the phase of χ there ≡ − 1 ffiffi 3 p ðζ here þ iζ here Þ, which is again unfixed at the critical point, is shifted by π 4 .

V. DISCUSSION
The main goal of this paper was to test the formulas of [25] for the consistent truncation [14] of D ¼ 11 supergravity [13] on S 7 down to D ¼ 4 N ¼ 8 SO(8)-gauged supergravity [1]. We have done so by particularizing these formulas to the SU(3)-invariant sector of the D ¼ 4 supergravity, using an explicit parametrization. When further restricted appropriately, our results correctly reproduce previously known consistent embeddings of sectors that preserve symmetries larger than SU(3). Our formalism thus extends previous literature and provides a unified D ¼ 11 embedding of the full SU(3)-invariant sector of SO (8) supergravity including all dynamical (bosonic) fields. It does so systematically, by using the restricted tensor hierarchy approach of [25].
As another crosscheck on the formulas of [25], we have rederived the known AdS 4 solutions of D ¼ 11 supergravity that arise upon consistent uplift of the critical points of SO(8) supergravity with at least SU(3) symmetry [2]. Again, we have found perfect agreement with the existing literature. As a further test, we have checked that the D ¼ 11 field equations are indeed verified on these AdS 4 solutions. Moreover, we have done this in a unified way for all of them; please refer to Appendix D for the details. This should again be regarded as a stringent test on the consistency of our formalism. Although we have not explicitly verified the D ¼ 11 Einstein equation due to its more involved structure, we have reproduced known solutions, like the ones presented in [40], for which the Einstein equation has been verified.
We have also obtained new embeddings of minimal D ¼ 4 N ¼ 2 gauged supergravity both into its parent D ¼ 4 N ¼ 8 SO(8)-gauged supergravity and into D ¼ 11 supergravity. A previously known embedding is obtained by fixing the scalars to their vevs at the SO(8) point and then selecting the graviphotonĀ as an appropriate combination of the two SU(3)-invariant vectors A Λ , Λ ¼ 0, 1. The resulting D ¼ 11 consistent uplift coincides with a previously known one, constructed in Sec. 2 of [43], that is in fact valid for any Sasaki-Einstein seven-manifold. The consistency of this truncation, at least within D ¼ 4 theories, is guaranteed by symmetry principles. This is because this embedding of minimal N ¼ 2 supergravity into N ¼ 8 coincides with the SUð4Þ s -invariant sector of the latter.
More interestingly, we have shown N ¼ 8 SO(8) supergravity to admit an alternative truncation to minimal N ¼ 2 supergravity by similarly fixing the scalars to their vevs at, now, Warner's N ¼ 2 SUð3Þ × Uð1Þ c point [2] and again selecting the graviphotonĀ appropriately. Although this alternative truncation is not driven by any apparent symmetry principle, it is nevertheless consistent. We have explicitly verified this at the level of the D ¼ 4 equations of motion that follow from the Lagrangian (2.3), including Einstein. Using our formalism, we have then uplifted this minimal N ¼ 2 supergravity to D ¼ 11 in Sec. III C. Again, we have explicitly verified the consistency of the D ¼ 11 embedding; see Appendix C. Thus, we have constructed the consistent truncation of D ¼ 11 supergravity on the N ¼ 2 AdS 4 solution of CPW [27] down to minimal D ¼ 4 N ¼ 2 gauged supergravity, predicted to exist by the general conjecture of [43].  [31]. The SOð8Þ ⊂ SLð8; RÞ ⊂ E 7ð7Þ subgroup is generated by T AB ≡ 2t ½A C δ BC . The generators of SUð3Þ ⊂ SOð8Þ can then be taken to beλ α , α ¼ 1; …; 8, defined as These generators indeed close into the SU(3) commutation relations with f αβγ ¼ f ½αβγ Gell-Mann's structure constants, Inside E 7ð7Þ , the SU(3) generated by (A1) commutes with SLð2; RÞ × SUð2; 1Þ, with the first factor generated by V ¼ e −χE 0 e − 1 2 φH 0 e 1 ffi ffi and Moving on, we need to specify how the SU(3)-invariant tensor fields in (2.1) are embedded into their N ¼ 8 counterparts. Recall that the restricted N ¼ 8 tensor hierarchy contains 28 0 electric vectors A AB , 28 magnetic vectorsÃ AB , 63 two-forms B A B and 36 three-forms C AB , in representations of SLð8; RÞ [25]. In order to determine the embedding of the SU(3)-invariant vectors A Λ ,Ã Λ , Λ ¼ 0, 1, into their N ¼ 8 counterparts, we note that SU(3) commutes inside SOð8Þ ⊂ E 7ð7Þ with the Uð1Þ 2 generated, in the notation of (A5), by ðE 2 − F 2 Þ and H 2 or, equivalently, by K 0 and K 3 defined in (A9). These are the Cartan generators of the maximal compact subgroup SUð2Þ × Uð1Þ of the hypermultiplet scalar manifold. Splitting again the N ¼ 8 index as below (A5), A ¼ ði; aÞ, and fixing the normalizations for convenience we have the following embedding into the N ¼ 8 vectors: Similarly, for the two-form potentials we define and for the three-form potentials, The field strengths and couplings brought to Sec. II can be obtained by inserting these expressions into the N ¼ 8 equations given in [25]. For example, the gauge covariant derivative acting on the scalars reduce to 1. S 7 as the join of S 1 and a Sasaki-Einstein S 5 The first set of coordinates solves the constraint (3.1) by splitting μ A , A ¼ 1; …; 8, as μ i ¼ cos αμ i ; i¼ 1; …; 6; μ 7 ¼ sin α cos ψ; with 0 ≤ α ≤ π=2, 0 ≤ ψ < 2π, andμ i , i ¼ 1; …; 6, defining in turn an S 5 , i.e., subject to the constraint δ ijμ iμj ¼ 1. The intrinsic coordinates (B1) are adapted to the topological description of S 7 as the join of S 5 and S 1 , for which the round, Einstein, SO(8)-invariant metric, ðB2Þ on S 7 displays only a manifest SOð6Þ v × SOð2Þ symmetry, with ds 2 ðS 5 Þ ¼ δ ij dμ i dμ j the round, Einstein metric on S 5 normalized so that the Ricci tensor equals four times the metric. This S 5 comes naturally equipped with the Sasaki-Einstein structure (η ð5Þ , J ð5Þ , Ω ð5Þ ) endowed upon it from the Calabi-Yau forms J ð6Þ , Ω ð6Þ , (A6), on the R 6 factor of ij dμ i ∧ dμ j ; These satisfy J ð5Þ ∧ Ω ð5Þ ¼ 0; ij dμ i ∧ dμ j ¼ cos 2 αJ ð5Þ − sin α cos αdα ∧ η ð5Þ ; The round metric ds 2 ðS 5 Þ in (B3) naturally adapts itself to the Sasaki-Einstein structure (B4) when written as with ds 2 ðCP 2 Þ the Fubini-Study metric on the complex projective plane, normalized so that the Ricci tensor equals six times the metric, 0 ≤ τ < 2π an angle on the S 5 Hopf fiber, and σ a one-form on CP 2 such that dσ ¼ 2J ð4Þ with J ð4Þ the Kähler form on CP 2 , so that η ð5Þ ≡ dτ þ σ and J ð5Þ ≡ J ð4Þ . For completeness, we note that ds 2 ðCP 2 Þ can be written in terms of complex projective coordinates ξ i , i ¼ 1, 2, as by introducing complex coordinates on R 6 ¼ C 3 through In these coordinates, the one-form σ in (B8) reads 2. S 7 with its homogeneous Sasaki-Einstein structure A second set of intrinsic coordinates on S 7 can be chosen that adapt themselves to its two natural, homogeneous seven-dimensional Sasaki-Einstein structures. These descend on S 7 from the Calabi-Yau forms J These are subject to The round metric on S 7 adapted to seven-dimensional Sasaki-Einstein structure reads, similarly to (B8), where ds 2 ðCP 3 AE Þ is the Fubini-Study metric, normalized so that the Ricci tensor equals eight times the metric. The AE refers to two different embeddings of CP 3 into S 7 , with isometry group SUð4Þ c ⊂ SOð8Þ for the þ sign and SUð4Þ s ⊂ SOð8Þ for the − sign. The angles ψ AE have period 2π and the one-forms σ AE in (B17) obey dσ AE ¼ 2J ð7Þ AE so that η ð7Þ AE ≡ dψ AE þ σ AE . It is also useful to make manifest the CP 2 that resides inside CP 3 AE , which is equipped with the complex projective coordinates ξ i , i ¼ 1, 2, that appear in (B10) and the metric (B9). This can be achieved by writing where τ AE are angles of period 2π. The metrics ds 2 ðCP 3 AE Þ and one-forms σ AE inside the round S 7 metric (B17) can be written in terms of the coordinates (B18) as and with ds 2 ðCP 2 Þ and σ respectively given by (B9) and (B11). The round S 7 metrics (B3) with (B8) and (B17) with (B19) are of course diffeomorphic: they are brought into each other by the change of coordinates 3. S 7 as the sine-cone over a nearly Kähler S 6 A third and final set of intrinsic angles on S 7 is better suited to describe the solutions with at least G 2 symmetry. First split the μ A , A ¼ 1; …; 8, as μ A ¼ ðμ I ; μ 8 Þ, with I ¼ 1; …; 7, and then let where 0 ≤ β ≤ π, andν I , I ¼ 1; …; 7, define an S 6 through the constraint δ IJν IνJ ¼ 1. In these coordinates, the round metric (B2) takes on the local sine-cone form ds 2 ðS 7 Þ ¼ dβ 2 þ sin 2 βds 2 ðS 6 Þ; ðB23Þ where ds 2 ðS 6 Þ ¼ δ IJ dν I dν J is the round, Einstein metric on S 6 normalized so that the Ricci tensor equals five times the metric. This S 6 is naturally endowed with the homogeneous nearly Kähler structure 5 ðJ ; ΩÞ inherited from the closed associative and co-associative forms, on the R 7 factor of R 8 ¼ R 7 × R in which S 6 is embedded: The nearly Kähler forms are subject to It is also useful to note the following relations between the associative and co-associative forms ψ,ψ written in constrained R 8 coordinates μ A ¼ ðμ I ; μ 8 Þ, the S 7 coordinate β in (B22), and the nearly Kähler forms (B26): 5 The typography we use for the nearly Kähler forms on S 6 differentiates them from the Calabi-Yau forms (A6) on R 6 . For that reason, we omit labels ð6Þ for the former. Similarly, we omit labels ð7Þ for the associative and co-associative forms on R 7 .