Consistent truncations of M-theory for general SU(2) structures

In seven dimensions any spin manifold admits an SU(2) structure and therefore very general M-theory compactifications have the potential to allow for a reduction to N=4 gauged supergravity. We perform this general SU(2) reduction and give the relation of SU(2) torsion classes and fluxes to gaugings in the N=4 theory. We furthermore show explicitly that this reduction is a consistent truncation of the eleven-dimensional theory, in other words classical solutions of the reduced theory also solve the eleven-dimensional equations of motion. This reduction generalizes previous M-theory reductions on Tri-Sasakian manifolds and type IIA reductions on Calabi-Yau manifolds of vanishing Euler number. Moreover, it can also be applied to compactifications on certain G2 holonomy manifolds and to more general flux backgrounds.


Introduction
It has been argued that the vanishing of certain topological indices of the compactification manifold restricts the appearance of string corrections due to the appearance of additional (spontaneously broken) supercurrent [1]. This was exemplified in [1] for Calabi-Yau compactifications of type II for the case of vanishing Euler number. The vanishing Euler number ensures the existence of an SU(2) structure on the Calabi-Yau manifold. Reduction on such an SU(2) structure leads to N = 4 gauged supergravities [1][2][3][4][5][6], which can be conveniently described by the embedding tensor formalism [7,8].
Non-renormalization theorems of N = 4 supergravity then explain the vanishing of perturbative string corrections for these compactifications and lead to the conjecture that also certain non-perturbative string corrections must vanish for these backgrounds. In the discussed case of Calabi-Yau manifolds with vanishing Euler number it could be shown by applying mirror symmetry that this conjecture indeed is true.
These findings suggest that spontaneously broken supercurrents play a far more important role in string compactifications than considerations of effective actions would suggest. In particular this suggests that there is a general scheme to understand string corrections for general G-structure backgrounds. The most pressing question is whether spontaneously broken supercurrents can also restrict string corrections when only N = 1 remains unbroken.
A particularly interesting case to address this question are M-theory compactifications to four dimensions. It has been known for a long time that any seven-dimensional spin manifold admits an SU(2) structure [9][10][11]. This suggests that we should be able to find for many such M-theory compactifications a reduction to N = 4 gauged supergravity, which might give strong constraints on membrane instanton corrections in these backgrounds. In this paper we will perform such an SU(2) structure reduction to four dimensions and determine the corresponding N = 4 gauged supergravity by identifying the corresponding gaugings.
We show that the reduction performed in this paper is in fact a consistent truncation. Consistent truncations to gauged supergravities have been performed for many particular AdS backgrounds, see for instance [12][13][14][15][16][17][18]. Our reduction generalizes the known M-theory reductions to N = 4 gauged supergravity. Therefore it might help to understand more general four-dimensional AdS backgrounds.
The given reduction is applicable both to compactifications to Minkowski and AdS spacetimes. A particularly interesting application would be to understand the corrections to M-theory compactifications on G 2 manifolds: Some of the known Joyce manifolds of [19] are dual heterotic Calabi-Yau backgrounds with vanishing Euler number [20], where the techniques of [1] could be used.
The paper is organized as follows. In Section 2 we discuss general SU(2) structure manifolds in seven dimensions and thereby set the stage for performing the reduction. In Section 3 we will make the reduction ansatz and then perform the SU(2) reduction to four dimensions. The embedding tensor components of the corresponding N = 4 gauged supergravity are identified in Section 4. In Section 5 we discuss the consistency of the truncation, and in Section 6 we make contact with some classes of AdS vacua in the literature. Some of the technical details as well as our conventions regarding N = 4 gauged supergravity are presented in three appendices.
We denoted in (2.1) the index of the broken SU(2) 3 that is related to the SO(3) by i and the index of the broken SU(2) 4 inside SO(4) byî. The third SU (2) subgroup is the (unbroken) structure group.

(2.4)
Taking the products of these bilinears and using (2.1) yields the relations The Fierz identities guarantee also the existence of an almost product structure P : T Y → T Y , P 2 = id, on the manifold, defined locally via where the vectorsK a are defined by The eigenspaces T 3 Y and T 4 Y of P to the eigenvalues +1 and −1 respectively yield a global decomposition of the tangent space, The subbundle T 3 Y is trivial, spanned byK a . By definition of P , the K a (Jâ) are trivial on T 4 Y (T 3 Y ). Note that the splitting (2.9) and the definition of such an SU(2) 3 triple of one-form K a and an SU(2) 4 triple of two-forms Jâ,â = 1, 2, 3, which satisfy the conditions (2.5) and (2.6), allow for a definition of the SU(2)-structure with no reference to spinors.
Let us now discuss the frame bundle over spacetime times Y . We choose a section (vielbein) 1 e A = (e µ , K a , e α ) , (2.10) where the e µ live in spacetime and depend only on spacetime coordinates. In contrast, the K a = k a b (v b + G a ) consist of one-forms v a in T * 3 and spacetime gauge fields G a (the Kaluza-Klein vectors) that parameterize the fibration of T * 3 over spacetime, as well as the coefficient k a b , which is a spacetime scalar. 2 Furthermore, the e α are one-forms on T * 4 such that Jâ = 1 2 (Iâ) α β e α ∧ e β , (2.11) with constant coefficients (Iâ) α β that are the generators of the SU(2) 4 algebra of complex structures on the frame bundle, i.e.
Similarly, theĨ a are generators of SO(3) given by (Ĩ a ) b c = ǫ abc . The dual vielbein to (2.10) isê wherev a are the vector fields dual in T * 3 to the vielbein component v a . Next, we consider the Levi-Civita connection one-form Ω, which is the unique torsionfree connection satisfying the Maurer-Cartan equation (2.14) The corresponding curvature two-form is defined by The Ricci tensor (in flat indices) is defined by contraction with the dual vielbein, , where we have called the full SO(7) connection Θ. Using SO(3) ≡ (SU(2) 3 )/Z 2 and SO(4) ≡ (SU(2) 4 × SU(2))/Z 2 , we can further decompose the adjoint representation of SO(7) and thus the internal connection Θ as where su(2) is the adjoint of the SU(2) structure group, su(2) 4 is spanned by the Iâ and su(2) 3 by (I a ) b c = ǫ abc . Using this decomposition and the vielbein (2.10), the Maurer-Cartan equations (2.14) read in components (2.20) Note that the connection component θ is the torsionful SU(2) connection. Its internal torsion two-form T can be expressed in terms of the other components of Θ. On T 3 Y the internal torsion is given by T a = dK a and the component on T 4 is Similar to the connection one-form we can also decompose the Ricci tensor grouptheoretically. In particular, we are interested in the 'symmetric' representation S 2 T * Y , which decomposes as Here, the (1, 3, 3) representation S 2 0 T * 4 is spanned by the products of generators of su(2) 4 and su (2). In other words, since the elements of su(2) and su(2) 4 commute, the representation can be written as (2.23)

Dimensional reduction from M-theory
In this section, we will reduce the eleven-dimensional supergravity action to four dimensions. Here, G 4 = dC 3 is the form field strength of the three-form gauge field.

The reduction ansatz
The almost product structure (2.7) on Y will play a central role in the choice of our reduction ansatz. T 3 has trivial structure group and is therefore parallelizable. We hence introduce a basis of three global one-forms v a , a = 1, 2, 3, on this subbundle, yielding three one-forms, three two-and a three-form (their wedge products) as expansion forms.
On T 4 our ansatz similarly to [1] contains SU(2) singlets and triplets. It is easily checked that SU(2) doublets exactly correspond to odd forms on T 4 . Therefore, the ansatz will consist of two-forms ω I , I = 1, . . . , n, that all square to the same volume form vol where η is a metric with signature (3, n − 3), reflecting the number of singlet and triplet representations as discussed above. Furthermore, we include all wedge products of ω I and v a in the reduction ansatz. For instance, we expand the forms Jâ and K a of (2.3) that specify the SU(2) structure in the set of modes ω I , I = 1, . . . , n, and v a , a = 1, 2, 3, i.e. Jâ = e ρ 4 /2 ζâ I ω I , 3 . Note that a consequence of the second equation in (3.3) is that which in particular means that det(k) = 6.
Note that the presence of internal one-forms in our ansatz gives rise to Kaluza-Klein vectors G i , i.e. mixed spacetime and internal components of the ten-dimensional metric. The expansion coefficients ζâ I , ρ 4 , ρ 3 and k a b depend on the spacetime coordinates and give rise to scalar fields in four dimensions. Furthermore, (2.5) yields the relations The four-dimensional fields ρ 3/4 describe the volume moduli of T 3/4 while the ζâ I describe the SU(2)-structure geometry and k a b describes the three-dimensional geometry. We can also expand the three-form gauge field in terms of this basis. This gives We will also describe fluxes in this setup, therefore our ansatz for the four-form field strength will be where we define with f 0 , g 0 and g a I being constants, and we demand the Bianchi identity dG 4 = 0 . is only defined up to an exact piece, so that only a subset of the numbers (g 0 , g aI ) are actual flux numbers. Also, note that the flux piece in (3.9) proportional to vol 4 (Mink) has a dependence on the volume factors because it originates from dual seven-form flux Note that the flux piece in (3.9) proportional to vol 4 (Mink) can be absorbed in dĈ but will reoccur later when we introduce dual fields. We discuss this seven-form flux again in Appendix B.1.
The Jâ and the K a in general define the Hodge star, which splits into a spacetime component and two components * 3 and * 4 acting on forms on T 3 and T 4 , respectively. The precise form of * 3 and * 4 is fixed by The latter can be translated into In the following reduction, we will assume that the internal volume is normalized, (3.14) To perform the reduction, we must next specify the differentials of the expansion forms {v i , ω I }. As remarked above, we will require that the differential algebra of modes they span closes, i.e. dv a = 1 Here, the coefficients t ab , t a I andT I aJ are constants that parameterize the SU(2) structure reduction ansatz for a particular manifold. In particular we exclude any terms on the right-hand side of the above equations involving SU(2) doublets. Note also that in the second equation of (3.15) a possible term proportional to v 1 ∧ v 2 ∧ v 3 is immediately set to zero by the constraint that d(ω I ∧ ω J ∧ ω K ) = 0.
The t ab , t a I andT I aJ specify the torsion classes of Y . We choose them and hence the torsion classes of Y to be constant. These constants are constrained by the fact that the exterior derivative squares to zero and the integral of d(v a ∧ v b ∧ ω I ∧ ω J ) over Y should vanish. The constraints are encapsulated by algebraic relations, given by The last equation determines the symmetric part ofT I jK η KJ , j = 1, 2, so that where T I aK is a triple of so(3, n − 3) matrices, i.e.
The fourth condition just states that the T I aJ form an so(3, n − 3) subalgebra S defined by In particular, S is Abelian if t ab = 0 and simple if t ab has rank three. The remaining condition is If t ab is zero, the t a I are invariant under S. If t ab is non-zero, the t a I form a non-trivial representation under S. The Bianchi identity (3.10) also leads to constraints on the flux numbers appearing in (3.9), given by (3.21)

Reduction of gravity
In this section we dimensionally reduce the gravitational term in the eleven-dimensional supergravity action (3.1), For this we have to compute the eleven-dimensional Ricci scalar in terms of the ansatz (3.3). We start by computing the connection Ω from the Maurer-Cartan equations (2.20) and (3.15) in Appendix A.2. There we find for the components of the connection 23) with the projector P I J = δ I J − ζâ I ζâ J and the covariant derivatives given by (3.24) In Appendix A.3 we compute from this connection the components of the ten-dimensional Ricci curvature. For the reduction we only need the Ricci scalar r 11 , given by where r 4 is the four-dimensional Ricci scalar. Note that we can rewrite with the definition of the covariant derivative as The eleven-dimensional volume form includes a prefactor e ρ 4 +ρ 3 that describes the scaling of the internal volume. Thus the reduction of the eleven-dimensional Einstein-Hilbert action to four dimensions in (3.22) is (3.28) We perform a Weyl rescaling e µ → e −(ρ 3 +ρ 4 )/2 e µ , (3.29) to bring the four-dimensional Einstein-Hilbert term into its canonical form and get where we defined φ = ρ 4 + 2 3 ρ 3 with

Reduction of the four-form field strength
Next we want to reduce the four-form field strength action For this we compute the four-form field strength G 4 , defined in (3.8), using (3.15) and (3.9). We find where we defined DG a in (3.23) and the other covariant derivatives are Now we can insert this into (3.32) and perform the Weyl rescaling (3.29). We find for the kinetic term To evaluate the topological term correctly in the presence of four-form flux, we assume eleven-dimensional spacetime to be the boundary of a fictional twelve-dimensional space and write the topological term of (3.32) as [21,22] where we used that the flux G flux 4 squares to zero, cf. (3.9), and defined C T to be the part of C 3 with two or more external legs, while C V is the component of C 3 with one or less external leg. In other words, Now we integrate out the three-formĈ, or, more easily, integrate out its field strengtĥ F = dĈ +Ĉ a ∧ DG a , sinceĈ does not appear by itself in the action. The equation of motion forF iŝ and inserting this forF in the action leads to an additional term for the potential Note that the topological term then reduces to where S top,vec only depends on the vector fields C I and C a .
Finally, we want to introduce scalar fields γ a so that the kinetic term of theĈ a can be replaced. To be consistent, we also have to introduce magnetic vector fieldsC I and C a that are dual to C I and C a , as well as a number of auxiliary two-form fields. Also, we want to perform an electric-magnetic duality between C a andC a to end up in the standard frame of N = 4 gauged supergravity. Since the scalars c aI and c 0 are charged under C a ,C I andC a will be charged under their dual two-formsĈ a I andĈ 0 , which must be introduced as well. Note that this very much complicates the situation compared to [1,18].
When dualizing fields, Bianchi identities and field equations are swapped. This means that from the Bianchi identities ofF a = DĈ a − ǫ abc C b ∧ DG c and F a = DC a + c 0 DG a we can deduce the couplings of their dual fields γ a andC a in the Lagrangian. On the other hand, the field equations ofF a and F a tell us what should be the covariant derivatives of γ a andC a . In particular, we can see that if C a appears in the covariant derivative of scalar fields, their dual tensors have to appear in the covariant derivative ofC a , and there should be an additional topological coupling of this tensor to C a in the final Lagrangian. Furthermore, ifĈ a appears in the covariant derivative of an electric gauge field, the scalar γ a must be gauged under the magnetic dual of the gauge field, andĈ a should be topologically coupled to this magnetic vector.
In Appendix B.1 we perform the duality transformation fromĈ a and C a to γ a and C a . BothĈ a and C a become auxiliary fields without kinetic terms. Moreover, also a new auxiliary vector fieldC I and the auxiliary tensorsĈ 0 andĈ a I appear in the dual Lagrangian. These dual fields will mostly appear through their covariant derivatives (3.42) The kinetic terms in the dual Lagrangian can be computed tõ the potential is given bỹ (3.44) and the topological terms arẽ Variation with respect to the auxiliary tensor fields leads to the duality relations between electric and magnetic vectors 46) while variation with respect to the magnetic vectors gives the duality relations between tensors and scalars (3.47) Note that each of the relations in (3.47) and (3.46) gets multiplied by certain charge components. Thus, if certain charges are vanishing, the corresponding duality equation is eliminated. At the same time, the corresponding couplings in the covariant derivative vanish and the corresponding auxiliary field is removed from the Lagrangian altogether. In the generic case of non-vanishing couplings, we can use the duality relations (3.47) and (3.46) in order to eliminate the fields we have introduced above and come back to the Lagrangian of (3.35) and (3.41) that we obtained from the reduction.
The action obtained in (3.43), (3.44) and (3.45) together with (3.30) fits perfectly into the framework of N = 4 gauged supergravity. We make the identification with the standard notation in the next section.

Matching with N = supergravity
In this section we want to match the results of the dimensional reduction on a sevendimensional SU(2)-structure manifold with the standard formulation of N = 4 gauged supergravity, which is reviewed in Appendix C. We organize the vector fields as so that the metric η M N is in the standard form Note that this involves an electric-magnetic duality transformation betweenC a and C a , which can be performed in the standard way following [8].
Next, the scalars c 0 and ρ 3 combine into the N = 4 axiodilaton τ as τ = (−c 0 + i e ρ 3 ) so that the covariant derivative reads Therefore we find for the embedding tensor components ξ αM that the only non-vanishing component is ξ +a = −ǫ abc t bc . is given in terms of the scalars (φ, g 3 ab , ζâ I , γ a , c aI ) by (4.6) The corresponding vielbein V is given by Vb a =ζb I c aI , Vb a =0 , so that From the covariant derivatives we can read off the remaining embedding tensor components f +abc =f 0 ǫ abc , Moreover, we identify

Consistent truncation
In this section we want to show that the SU(2) reduction to four dimensions is indeed a consistent truncation of the eleven-dimensional supergravity action. In other words, we want to show that the four-dimensional equations of motion imply the ten-dimensional ones. In [1] a heuristic argument was already given why SU(2) structure reductions for modes ω I and v a that obey the constraints (3.2) and (3.15) are consistent truncations.
In this section we will prove this claim by an explicit check of the eleven-dimensional equations of motion.
The four-dimensional relevant equations are the Einstein equation and for the scalars originating from the Lagrangian of [8] discussed in Appendix C. The eleven-dimensional equations of motion originate from the action (3.1) and are given by The major work consists of showing that the eleven-dimensional Einstein equations are satisfied if the four-dimensional equations of motion (and Bianchi identities) hold. The technical details for determining the Ricci curvature and the energy-momentum tensor are delegated to the first two appendices. In Appendix A.4 we give the Ricci curvature in the Einstein frame. In Appendix B.2, we also compute the energy-momentum tensor generated by G 4 . When we insert these results into the eleven-dimensional Einstein equation, we see that the equations reduce to the four-dimensional equations of motion in the following way: • The trace of the Einstein equations is satisfied by the equations of motion for φ, ρ 3 and by the trace of the four-dimensional Einstein equation.
• The Einstein equations with indices (µν) give the four-dimensional Einstein equations.
• For the indices (µa) we recover the equations of motion for the Kaluza-Klein vector G a µ .
• For the higher form field components we then use the four-dimensional equation of motion forF (3.39) to eliminateF , the scalar-tensor duality relation (3.47) to replace the tensor fields by their dual scalars and the electro-magnetic duality relation (3.46) to replace all magnetic vector fields by their electric counterparts. In this way we can rewrite the eleven-dimensional Einstein equations in terms of four-dimensional scalars and electric vector fields (up to appearances of the magnetic vectors and tensors in the gaugings). As expected, this completely reproduces the four-dimensional equations of motions for these scalars and vector fields.

Simple supersymmetric backgrounds
Let us now briefly discuss some classes of supersymmetric AdS vacua. We will only discuss the simple examples of N = 4 AdS vacua discussed in [23] and of N = 3 AdS vacua from Tri-Sasakian manifolds whose consistent truncation has been worked out in [17]. The discussion of cases with N ≤ 2 goes beyond the scope of this paper.

N = 4 AdS vacua
In [23] four-dimensional N = 4 AdS vacua had been classified. A necessary requirement for such backgrounds is that there is one electrically and one magnetically gauged SU (2) in the theory whose gauge bosons are graviphotons. Let us apply the findings of [23] to the gaugings of SU(2) structures given in (4.4) and (4.9). The embedding tensor component ξ in (4.4) must be zero in these vacua, which means that t ab is symmetric. Moreover, the electric and magnetic gaugings obey the relationship where * 6 is the Hodge star in the six-dimensional space of graviphotons and τ is the axiodilaton. For the possible gaugings given in (4.9), this means that where the supergravity vielbein has been given in (4.7). But N = 4 supersymmetry also requires thatṼã d Vb I Vĉ J f +dIJ = 0 , where the dual vielbeinṼã is given bỹ Comparing the two formulas (6.2) and (6.3), using (4.7) and the above formula, shows the contradiction. Thus, in the class of supergravities obtained from SU(2) structure truncations no N = 4 AdS vacua can be found.

Tri-Sasakian manifolds
The consistent truncations worked out in [17] for Tri-Sasakian manifolds admit N = 3 AdS vacua. These truncations are minimal in that I runs I = 1, 2, 3. There the ansatz is A classification of N = 3 vacua is beyond the scope of this paper, but note that one could easily for instance add a non-trivial four-form flux by switching on g a I ∼ δ a I . These four-form fluxes effectively just change the value of the axiodilaton τ but do not modify the solution in any other way.
An interesting special case is the reduction on S 7 . We have already shown that the discussed SU(2) structure reductions do not allow for N = 4 AdS vacua. And indeed, the consistent truncation presented here does not distinguish between S 7 and any other Tri-Sasakian manifold. Thus while the full vacuum of AdS 4 × S 7 preserves N = 8 supersymmetry, the truncation to this N = 4 gauged supergravity (i.e. the truncation to SU(2) singlet and triplet modes) preserves only N = 3.

Acknowledgments
We would like to thank A. Kashani-Poor and R. Minasian for their motivational support as well as for useful discussions and for comments on the final draft.
It also implies From the algebra (3.15) we find and dvol Another useful formula will be where we defined the covariant derivative and From (3.15) and (3.2) we also deduce that Note also that we find from (3.17) that ǫâbĉT I aJ ζb I ζĉ J = ǫâbĉT I aJ ζb I ζĉ J , (A.9) Eq. (3.5) implies (∂ µ ζâ I )η IJ ζb J = 0 , (A. 10) which means that over four-dimensional spacetime, the Jâ do not rotate into each other and therefore really move in SO(3,n) SO(3)×SO(n) . Note that from (2.11) and (3.3) we also find (Iâ) α β = e ρ 4 /2 ζâ I ω I αβ . (A.11) The decomposition of ω I into representations of SU (2) reads where P I J = δ I J − ζâ I ζâ J . The latter term in (A.12) is invariant under the Iâ. We hence find

A.2 Connection
In this section we want to compute the connection from the three Maurer-Cartan equations (2.20), using (3.3) and (3.15). From the first equation in (2.20) we see that both λ µ α (ê β ) and γ µ a (K b ) are symmetric in their lower indices. Now let us first solve the second equation. From the explicit form of K a in (3.3) we find with help of (3.15) where the explicit form of DG a can be found in (A.6) and we defined (A.20) Comparison with (2.20) gives for the connection components where γ a µν is symmetric in its lower indices. Note that we used (2.23) for the decomposition of τ a α . If we use the explicit expressions for the Jâ given in This can be solved for the connection components as (A.23) where we defined the covariant derivatives (A.24) and the projector P I J = (δ I J − ζb I ζb J ). Finally, we solve for the first Maurer-Cartan equation in (2.20), by using the explicit form of λ α µ and γ a µ , given in (A.21) and (A.23). The result is whereω is the four-dimensional connection.
In total, we find for the components of the connection Note that the scalar ρ 3 + ρ 4 is ungauged.
In the next section, we compute the Ricci curvature from the Levi-Civita connection.
For that we will also need the differential identities (A.28) that can be computed with help of (2.20), (3.15) and (A.26). Note that ∇ θ denotes the SU(2) connection.

A.3 Ricci curvature
The components of the Ricci curvature are given by Let us now compute these components one by one. We start with Ric µν . We compute This gives the external Ricci curvature (A.31) For Ric µa we compute , (A.32) and therefore find where we used in the last equation that dω I = ∇( 1 2 ω I βα e β ) ∧ e α . Therefore we find Let us now compute the internal component Ric ab . We find where we defined g 3 ab = k c a k c b , |t| = g 3 ab t ab and t A = 1 2 (t − t T ). Then we get Let us now compute the internal component Ric aα . We find Let us now compute the internal component Ric αβ . We find where Ric θ αβ is the Ricci curvature of the SU(2) connection θ. We can compute Ric θ αβ by taking the SU(2)-covariant derivative of the torsion of the connection θ [1] Ric θ αβ = 1 6 (dT α + θ α λ ∧ T λ )(ê γ ,ê δ ,ê ρ )ǫ γδρβ . (A.41) From (2.20) we find that the component T α of the torsion torsion tensor of θ is (A.42) From this we find for the SU(2) Ricci curvature (A.43) Now we are able to determine Ric αβ . We find From (A.31), (A.37) and (A.44) we can compute the Ricci scalar. It reads (A.45)

A.4 Ricci curvature in the Einstein frame
In order to define the four-dimensional theory in the Einstein frame, we have to perform the Weyl rescaling (3.29). As the scalar fields only depend on four-dimensional spacetime, this mostly affects the Ricci curvature component Ric µν , given in (A.31). It reads in the four-dimensional Einstein frame (A.46) for scalars γ a and dualizing into a standard electric-magnetic duality frame of N = 4 gauged supergravity, as described in [8]. As we will see, this requires to exchange the gauge fields C a for their magnetic duals, which we will denote byC a . However, to perform the dualization in a consistent way, we also have to introduce further dual auxiliary fields: A magnetic vector fieldC I and the two-formsĈ 0 andĈ a I . The magnetic vector fieldC I will appear in the covariant derivative of γ a , becauseĈ a appears in the covariant derivative of C I . The new two-forms will appear in the covariant derivative ofC a , since C a appears in the covariant derivatives of the scalars c aI and c 0 .
We perform the field dualizations by showing that the set of Bianchi identities and equations of motions are the same, with Bianchi identities swapped for the equations of motion and vice versa. Therefore, let us start by deriving the Bianchi identities from dG 4 = 0. This gives for the field strengthsF a = DĈ a −ǫ abc C b ∧DG c , F I = DC I +c aI DG a and F a = DC a + c 0 DG a the identities dF a + ǫ abc t dc G b ∧F d + ǫ abc F b ∧ DG c = 0 , Furthermore, we vary the Lagrangian with respect to the fieldsĈ a , C a and C I to determine their equations of motion to be 0 =d(e 2φ g ab 3 * F b ) + t ab ǫ bcd G c ∧ (e 2φ g de * F e ) + e φ−ρ 3 t ab g 3 bc * F c + e ρ 3 t a I H IJ * F J + 1 2 ǫ abc η IJ Dc bI ∧ Dc cJ + f 0 DG a + (g 0 + c aI η IJ t a J )F a + (g a I + c 0 t a I + ǫ abc T J bI c cJ + t (ab) c bI )η IJ F J , 0 =(d + ǫ bcd t bc G d ∧)(e φ−ρ 3 g 3 ae * F e ) − ǫ abc t db G c ∧ (e φ−ρ 3 g 3 de * F e ) − ǫ abc DG b ∧ (e 2φ g cd 3 * F d ) + ǫ abc t bc (e −2ρ 3 * Dc 0 ) − f 0 ǫ abc G b ∧ DG c − ǫ abc t c I H IJ e −φ g bd * Dc dJ + (g 0 + c bI η IJ t b J )F a + η IJ Dc aI ∧ F J , 0 =d(e ρ 3 H J I * F J ) − T J aI G a ∧ (e ρ 3 H K J * F K ) − 1 2 ǫ abc t bc G a ∧ (e ρ 3 H J I * F J ) − T J aI H K J g ab 3 e −φ * Dc bK − 1 2 ǫ abc t bc H J I g ad 3 e −φ * Dc dJ + (g a I + c 0 t a I + ǫ abc T J bI c cJ + t (ab) c bI )F a + Dc aI ∧ F a + Dc 0 ∧ F I .

(B.2)
We now want to introduce dual fields with field strengths Γ a ,F a andF I in such a way that on-shell the duality relations Γ a =e 2φ g ab 3 * F b , F a =e φ−ρ 3 g 3 ab * F b , should hold. Similarly, we need to introduce two-formsĈ 0 andĈ a I that are related to the scalars c 0 and c aI by similar duality relations. We also need to include seven-form flux Now we are in the position to give the dual Lagrangian.
S dual = − 1 4κ 2 4 e ρ 3 −φ g ab 3 (DC a − c aI η IK DC K − ǫ acd γ c DG d − 1 2 c aI η IK c cK DG c ) ∧ * (DC b − c bJ η JL DC L − ǫ bef γ e DG f − 1 2 c bJ η JL c eL DG e ) − e −2φ g 3 ab (Dγ a + 1 2 ǫ acd c cI η IJ Dc dJ ) ∧ * 4 (Dγ b + 1 2 ǫ bef c eI η IJ Dc f J ) − 2c 0 DC a ∧ DG a + g 0 (2DC a − t baĈ b ) ∧Ĉ a + 2DC a ∧ ǫ abc (t bcĈ (B.8) The first two terms are the new kinetic terms that replace the ones ofĈ a and C a . The next term is a topological term to complete the equations of motion of (B.7). The remaining topological terms ensure that variation with respect to the auxiliary fields gives the duality relations (B.3) as well as 4 F a =e −2φ g 3 ab * (Dγ b + 1 2 ǫ bcd c cJ η JK Dc dK ) , F 0 =e −2ρ 3 * Dc 0 , F a I =e −φ H J I g ab 3 * Dc bJ . (B.9) Note that γ a cannot have a potential term since it inherits the shift symmetry from its dual tensorĈ a .

B.2 Energy-Momentum tensor
In this appendix we want to compute the eleven-dimensional energy-momentum tensor that appears in the Einstein field equations (5.5). For this, we use the form (3.33) and the field dualizations that have been discussed in more detail in Section 3.3 and in Appendix B.1.
First of all note that the components T µα and T aα both are identical zero, due to the absence of SU(2) doublet degrees of freedom in our ansatz. This fits nicely together with (A.35) and (A.39). For the remaining components, let us first computeT AB = 1 where we defined the totally antisymmetric tensor M M N P QRS = ǫ mnpqrs ν M m ν N n ν P p ν Q q ν R r ν S s , (C. 8) from the SO(6, n) vielbein ν M m . Finally, the embedding tensor components obey a number of quadratic constraints that are necessary in order to ensure locality of the supergravity. These constraints are given by