Non-linear uplift Ans\"atze for the internal metric and the four-form field-strength of maximal supergravity

The uplift of SO(8) gauged N=8 supergravity to 11-dimensional supergravity is well studied in the literature. It is given by consistent relations between the respective vector and scalar fields of both theories. For example, recent work provided non-linear uplift Ans\"atze for the scalar degrees of freedom on the internal manifold: the inverse metric and the three-form flux with mixed index structure. However, one always found the metric of the compactified manifold by inverting the inverse metric --- a task that was only possible in particular cases, e.g. for the G$_2$, SO(3)$\times$SO(3) or SU(3)$\times$U(1)$\times$U(1) invariant solutions of 11-dimensional supergravity. In this paper, I present a direct non-linear uplift Ansatz for the internal metric in terms of the four-dimensional scalars and the Killing forms on the compactified background manifold. Based on this formula, I also find new uplift Ans\"atze for the warp factor and the full three-form flux. Finally, I provide a direct non-linear uplift Ansatz for the internal four-form field-strength in terms of the metric, the flux as well as the four-dimensional scalars and background Killing forms. This new formula does not require to calculate the derivative of the flux. My results may be generalized to other compactifications, e.g. the reduction from type IIB supergravity to five dimensions.

This splitting is called compactification of the (D − 4) extra dimensions. An action including the D-dimensional Einstein-Hilbert term is given by where R D denotes the Ricci scalar in D dimensions. For a consistent compactification, Eq. (2) contains the fourdimensional Einstein-Hilbert action. All other terms correspond to matter. For example, T. Kaluza and O. Klein presented one of the first attempts to unify gravity and electromagnetism [1,2]. They constructed a five-dimensional theory of gravity, such that the extra components of the metric were given by a photon and a scalar field. In that case, the fifth dimension was compactified on a circle, A physicist naturally is in another situation. He 'observes' a four-dimensional theory of gravity coupled to matter and may ask the following question: Is there a higher-dimensional theory, which consistently reduces to the observed theory via compactification of the extra dimensions? This is called an uplift : One constructs the D-dimensional fields (e.g. the metric) out of a given four-dimensional theory of gravity. The main task in establishing such a program is to find Ansätze for the D-dimensional fields in terms of the four-dimensional ones, such that they satisfy the higher-dimensional equations of motion. The uplift is consistent only when the latter is satisfied.
One of the few known examples is the uplift of N = 8 supergravity to 11-dimensional supergravity. N = 8 supergravity represents the low-energy limit of string theory. It is the maximally supersymmetric theory of gravity and contains a local SU(8) gauge symmetry. It was first investigated in the beginning of the 80s [3,4]. At the same time, 11-dimensional supergravity was developed [5], which is the highest dimensional supergravity theory [6]. The respective Lagrangian is also locally SU(8) gauge invariant.
11-dimensional supergravity may spontaneously compactify to SO (8) gauged N = 8 supergravity [7][8][9][10]. The seven extra dimensions therefore compactify on a seven-sphere 1 , This work is based on the uplift of SO (8) gauged N = 8 supergravity to 11-dimensional supergravity [9,[11][12][13][14]. It is given by non-linear Ansätze for the 11-dimensional scalar and vector fields in terms of the four-dimensional ones. These include the correct relations between the 28 vector fields of 11-dimensional supergravity and the 28 vectors of N = 8 supergravity. On the other hand, the 70 scalar degrees of freedom of 11-dimensional supergravity are contained in certain fields that are defined on the internal space (a deformed seven-sphere): the metric g mn , the three-form potential A mnp and the six-form potential A m1···m6 . For the complete uplift, these fields must be related to the 35 scalars u ij IJ and pseudo-scalars v ij IJ of N = 8 supergravity. There is an old explicit formula for the inverse metric ∆ −1 g mn [15], as well as non-linear Ansätze for the full internal six-form potential and the three-form flux with mixed index-structure [14]. There are two technical problems arising here: First, one must invert ∆ −1 g mn 'by hand' in order to obtain ∆g mn . Secondly, one must extract the warp factor ∆ from these expressions by computing their determinants. Both, the inversion of the metric and the calculation of the warp factor can only be done in particular cases, e.g. when the theory is G 2 , SO(3)×SO(3) or SU(3)×U(1)×U (1) invariant [16][17][18][19]. Only in such cases, it is then possible to compute the full internal three-form potential A mnp .
In this paper, I present a new simple non-linear Ansatz for the full internal metric g mn , i.e.
The tensors A m ijkl and B m ijkl are given in terms of the Killing forms on the seven-sphere and the four-dimensional scalar fields (Eqs. (67-70)). In combination with the previous uplift formulas for the inverse metric and the three-form with mixed index structure, I also find new non-linear Ansätze for the warp factor and the full internal three-form potential A mnp . They are given by where the tensor C pq ijkl is defined similarly to A m ijkl and B m ijkl in Eq. (73). The two-forms K mn IJ denote the derivative of the Killing vectors K m IJ on the round seven-sphere. During completion of this paper, a work by Oscar Varela derived similar coordinate-free Ansätze for the metric, the warp factor and the flux [20]. These expressions however, are given in a different form that is based on the tensor hierarchy formalism of gauged supergravity (see Eqs. (24)(25)(26) of [20]). This makes it complicated to actually compare my formulas to those of Varela's work. In order to illustrate the simplicity of the Ansätze above, I test them for a G 2 invariant solution of 11-dimensional supergravity. This essential part of the present work is done in Section VI. It turns out that the new formulas in Eqs. (6)(7)(8) appear to be very suitable for this test.
In the second part of this paper, I derive a new uplift Ansatz for the internal four-form field-strength Here,D m denotes the covariant derivative with respect to the internal background metricg mn . So far, Eq. (9) could only be used in particular cases -when an explicit expression for the internal three-form potential was already given. However, it was rather complicated to compute the derivative of A mnp in such cases, for example to find the G 2 or SO(3)×SO(3) invariant solutions of 11-dimensional supergravity [16,17]. With the new general Ansatz for A mnp above, I derive a simple direct formula for the four-form field-strength, i.e.
A formula for the complete four-form field-strength occurs in Eq. (28) of Varela's work [20]. Again, it is hard to compare both formulas because the expression in [20] is given in a form based on the tensor hierarchy formalism of gauged supergravity. In Section VI, I will demonstrate once more that the present Ansatz above is given in a very convenient form -it can be directly used for a test against the G 2 invariant solution of 11-dimensional supergravity.
The new non-linear Ansatz in Eq. (10) provides another remarkable result: The above expression is 'almost' covariant 2 , which means that raising the indices is simple, Up to now, it was far more complicated to derive F mnpq -by raising each single index of F mnpq with the explicit expression for the inverse metric g mn . For example, this was one of the hardest tasks in verifying the SO(3)×SO (3) invariant solution of 11-dimensional supergravity [17]. In the case of maximally symmetric spacetimes, these results can be used to compute the components of the Ricci tensor via the equations of motion.
In the next section, I collect the main steps to find the consistent uplift of N = 8 supergravity to 11-dimensional supergravity. In Section III, I re-derive the known non-linear Ansätze for the inverse metric ∆ −1 g mn , the three-form with mixed index structure A mn p and the six-form potential A m1···m6 . In Section IV, I present the new uplift Ansätze for the metric g mn , the warp factor ∆ and the full internal three-form potential A mnp . Furthermore, I find the new non-linear Ansatz for the four-form field-strength (F mnpq and F mnpq ) in Section V. In Section VI, I test the new uplift Ansätze for the G 2 invariant solution of 11-dimensional supergravity: I compute the metric and the four-form field-strength using the new formulas in Eqs. (6,10) 3 and compare with the results of [16]. Finally, I conclude in Section VII.

II. THE UPLIFT OF N = 8 SUPERGRAVITY TO 11-DIMENSIONAL SUPERGRAVITY
The bosonic field content of 11-dimensional supergravity is an elfbein E M A (x, y) and a three-form potential A MN P (x, y). The set of coordinates splits into four spacetime (external) coordinates x and seven internal coordinates y. Capital Roman letters denote 11-dimensional indices. These split into external (Greek letters) and internal indices (lower case Roman letters). As a rule of thumb: Letters from the middle of an alphabet always denote curved spacetime indices and letters from the beginning of an alphabet are the corresponding tangent space indices.
The bosonic Lagrangian of 11-dimensional supergravity is written in terms of the elfbein, the three-form potential and the four-form field-strength [7]. The latter is defined by The Lagrangian can also be written in terms of dual fields [21]: for example, one could replace F (4) by its dual seven-form and the three-form potential by its dual six-form A M1···M6 . The latter is the potential for the dual seven-form field-strength, Later, one needs the six-form potential to describe certain vector and scalar degrees of freedom. Let us count the scalar and vector fields in 11-dimensional supergravity. The elfbein is given by It contains the vierbein e µ α (x, y), seven vectors B µ m (x, y) and 28 scalar fields e m a (x, y). On the other hand, the three-form potential splits into the components There are 21 vector fields in A µmn (x, y). Furthermore, A µνm (x, y) contains seven and A mnp (x, y) 35 scalar degrees of freedom. The remaining components A µνρ (x, y) represent the potential for the external field-strength and hence, contain no more scalar or vector degrees of freedom. This is because for all dimensional reductions, The Freund-Rubin parameter f FR is constant for Freund-Rubin compactifications [22] andη µνρσ represents the volume form in four dimensions. All in all, there are 7 + 21 = 28 vectors and 28 + 7 + 35 = 70 scalar degrees of freedom in 11-dimensional supergravity.
The bosonic field content of N = 8 supergravity is a vierbeine µ α (x), 28 'electric' vector fields A µ IJ (x) as well as 35 scalar and 35 pseudo-scalar fields u ij IJ (x), v ij IJ (x). All these fields only depend on the four spacetime coordinates x. The (antisymmetric) bi-vector indices IJ belong to the 28-dimensional representation of SL(8,R) and the (antisymmetric) bi-vector indices ij belong to the 28-dimensional representation of the local SU (8). The bosonic degrees of freedom of both, N = 8 supergravity and 11-dimensional supergravity coincide. This is at least, necessary for a consistent uplift.
In order to uplift N = 8 supergravity to 11-dimensional supergravity, one must explicitly relate the vierbeine, as well as the scalar and vector fields of both theories to each other. In the following, I will restrict to the S 7 compactification [10]. The matching was found by comparing the supersymmetry transformations of the four-and 11-dimensional fields [14,23]. It is based on a global E 7(7) symmetry in N = 8 supergravity [3]. E 7 (7) is not a symmetry of 11-dimensional supergravity. However, one may emphasize the respective E 7(7) structures as much as possible in order to compare the fields with those of N = 8 supergravity.
The correct relation between the vierbeine of N = 8 supergravity and 11-dimensional supergravity is The proportionality factor ∆(x, y) is called the warp factor. Lete m a be the siebenbein for the round seven-sphere andg mn denote the respective background metric and let g mn be the full internal metric of the deformed S 7 [12], g mn =e m ae n a , g mn = e m a e n a .
Then, the warp factor is defined by ∆ = det (e m a ) det (e m a ) = det(g mn ) det (g mn ) .
In order to match the scalar degrees of freedom, one first observes that the 35 scalars and 35 pseudo-scalars of N = 8 supergravity parametrize an element of E 7 /SU (8). This co-set space is indeed, 70-dimensional. Both, scalars and pseudo-scalars together form an elementV M ij (x) in the fundamental representation 56 of E 7 (7) . Its SL(8,R) decomposition is given byV The 56 representation is labeled by indices M, N , . . ., which are raised and lowered with the symplectic form Ω MN (see [3]). The SU(8) indices ij are raised and lowered via complex conjugation, One also writes the scalar fields of 11-dimensional supergravity in an E 7(7) covariant way. Therefore, it is convenient to describe all scalars by the fields e m a , A m1···m6 and A mnp (rather than using A µνm ). Indeed, the internal dual sixform potential A m1···m6 contains the same scalar degrees of freedom as A µνm . In a second step, one converts this scalar field content (e m a , A m1···m6 and A mnp ) into components of a '56-bein' of E 7(7) , i.e. [13,24] These components constitute the GL(7,R) decomposition of the 56-bein The SU(8) indices A, B, . . . are raised and lowered by complex conjugation 4 and the 8×8 Γ-matrices are defined in Appendix A. The correct relation between the 56-bein in 11 dimensions and the four-dimensional scalarsV of N = 8 supergravity was found by considering the respective supersymmetry transformations [14] 5 . It is given by Here, η i A are the eight Killing spinors defined on the internal geometry. The upper index M of the transformation matrix R M N is decomposed under GL(7,R) (Eq. (30)) whereas the lower index N is decomposed under SL(8,R) (Eq. (23)), The non-zero components are [14] R m IJ (y) = They depend on the Killing vectors K m IJ (y) and -forms K mn IJ (y) as well as on the dual volume potentialζ m (y) of the seven-sphere. The Killing vectors and -forms are defined in Appendix A. The (seven dimensional) dual ofζ m (y) is the six-form potential for the internal background volume formη m1···m7 , Note the non-standard normalization ofζ m , which is more convenient for my purposes. m 7 denotes the inverse radius of the round S 7 . Using Eqs. (22,(33)(34)(35)(36), one finally finds the components of In order to match the vector degrees of freedom, one first dualizes the 28 'electric' vector fields A µ IJ (x) in N = 8 supergravity to form 28 'magnetic' vector fields A µIJ (x). Only electric and magnetic vector fields together fit into the 56 representation of E 7(7) : they represent the SL(8,R) decomposition of along the lines of Eq. (23). One also extends the 28 vector fields B µ m and A µmn in 11-dimensional supergravity such that they fit into the 56 representation of E 7 (7) . There are 21 dual vectors A µm1···m5 coming from the six-form potential and seven 'dual graviphotons' that have no physical interpretation [13]. Similar to the case of scalar fields, one defines a 56-bein B µ M of E 7 (7) , which decomposes under GL(7,R) into the various vector degrees of freedom above. Since this work concentrates on the uplift of the scalar fields, I do not give the explicit GL(7,R) decomposition for B µ M here. The interested reader may have a look at [13,14,24]. The consistent relation between the vector fields A µ M (x) of N = 8 supergravity and the 11-dimensional vectors It has also been found by a careful analysis of the supersymmetry transformations in four and 11 dimensions. Here is a simple example for the readers convenience: The task of uplifting N = 8 supergravity to 11-dimensional supergravity is now the following: Starting from Eqs. (31,45), one must seek explicit expressions for the 11-dimensional vector and scalar fields in terms of the fourdimensional ones, In principle, these relations have been found in [14,15]. However, instead of a relation for the metric g mn (x, y), the authors only found an expression for the inverse metric ∆ −1 g mn (x, y), scaled with the warp factor. Furthermore, the Ansätze for the three-form and six-form potentials require the full metric g mn . Until now, the inversion of ∆ −1 g mn is only possible in particular cases, e.g. for G 2 , SO(3)×SO(3) or SU(3)×U(1)×U(1) invariant solutions [16][17][18][19]. Also the warp factor can only be computed from an explicit expression for the metric g mn (by taking the determinant). The reader familiar with the uplift Ansätze presented in [14] may skip the next section, which repeats the derivation of the known scalar uplifts. Section IV then presents new non-linear Ansätze for the full internal metric g mn , the warp factor ∆ and the internal three-form potential A mnp . These hold for the uplift of N = 8 supergravity to 11-dimensional maximally gauged supergravity, even without further restrictions (such as For the readers convenience, I repeat the steps to derive the known uplift relations for the inverse metric ∆ −1 g mn , the three-form with mixed index structure A mn p and the six-form potential A m1···m6 . This was done in [14] and is the basis to understand the new Ansätze for the metric g mn , the warp factor ∆ and the full internal three-form potential A mnp in Section IV. The main problem of comparing the vielbein components in Eqs. (25)(26)(27)(28) and Eqs. (40-43) is the occurrence of the Killing spinors in Eq. (31). However, these are orthonormal and would drop out in non-linear SU(8)-invariant combinations of the vielbeine. For example, let us consider the expression Indeed, the Killing spinors η i A (y) drop out. One now uses Eq. (25) on the lhs and Eq. (40) on the rhs, which results in a non-linear uplift Ansatz for the inverse metric, i.e.
Here, I used the Clifford algebra of the Γ-matrices, given in Appendix A.
In a similar way, one relates which yields a non-linear uplift Ansatz for the three-form. Indeed, using Eqs. (25,26) on the lhs as well as Eqs. (40,41) on the rhs, one finds In order to derive an uplift Ansatz for the internal six-form potential A m1···m6 , I introduce the (seven dimensional) dual one-form Similar to the dual volume potential on the round seven-sphere,ζ m , I use a non-standard normalization for later convenience. The six-form potential A (6) is a tensor in the internal space and its (seven dimensional) dual A (1) is constructed with the full ǫ-tensor. However, one can convert this ǫ-tensor to the tensor densityη (= ±1, 0) using the internal seven-bein e m a ǫ m1···m7 = e m1 a1 . . . e m7 a7η a1···a7 = ∆η m1···m7 .
Here, I used the definition of the warp factor in Eq. (21). Eq. (53) then reads Note that the indices of the six-form potential and its dual are raised and lowered with the full internal metric. Now, let us consider the relation and insert the various vielbein components in Eqs. (25,27) and Eqs. (40,42). This gives an equation for A n , i.e.
When contracting this relation with g np , the first term on the rhs drops out because One finds and dualizes this expression using Eq. (55,37), Here, I suppressed the explicit dependence on the coordinates. The rhs of Eqs. (59,60) further simplifies using the uplift Ansatz for the inverse metric in Eq. (50) and the definition of the Killing two-form in Eq. (A15). It is proportional to which finally gives a simpler non-linear Ansatz for the six-form potential, i.e.
This result has already been derived in [26]. In comparison to Eqs. (59,60), the Ansätze in Eqs. (62,63) do not contain the metric g mn . However, they require an explicit expression for the warp factor, which also can only be given in particular cases. In this section, I derive a new non-linear metric Ansatz for the uplift of SO(8) gauged N = 8 supergravity to 11dimensional supergravity. In combination with the expressions for the inverse metric and the three-form with mixed index structure in Eqs. (50,52), I find further uplift Ansätze for the warp factor and the internal three-form potential A mnp . Note that recent work derived similar coordinate-free formulas (Eqs. (24)(25)(26) of [20]) in a different form.
Following the strategy of the previous section, I consider the relation Let us use Eqs. (25,26) on the lhs: All terms including a factor of A mnp are of the form but such expressions vanish because an antisymmetric index pair [np] is contracted with a symmetric index pair (np). One finally computes the traces of the Γ-matrices using Eq. (A32) and finds that the lhs of Eq. (64) is proportional to the metric g mn , For the rhs, I use Eqs. (40,41) and find that For some readers, Eqs. (66,67) together already represent a useful metric Ansatz in terms of the Killing forms and the four dimensional scalar fields. However, one may simplify the resulting expression further: Using Eqs. (A26,A30) in Appendix A yields where I defined the convenient tensors By definition, these are totally antisymmetric in the SU(8) indices [ijkl] and depend on all 11 coordinates (x, y). Note that a certain linear combination of both tensors is equal to the 'non-metricity' P m ijkl in the SO(8) invariant vacuum [9,26] 7 . One finally finds the metric Ansatz in terms of these tensors, i.e.
This Ansatz is quartic in the four-dimensional scalar fields u ij IJ and v ij IJ , whereas the Ansätze for the inverse metric and the mixed three-form potential were only quadratic.
Let us combine the Ansätze for the metric and the inverse metric in Eqs. (71,50) to get a new Ansatz for the warp factor. This can be done because the new metric Ansatz contains a proportionality factor of ∆ −2 . One finds where the tensor C pq ijkl is defined as Similarly, one combines the Ansatz for the three-form with mixed index structure in Eq. (52) with the metric Ansatz in Eq. (71) to obtain a new Ansatz for the full internal three-form potential, i.e.
The new Ansätze for the warp factor and the three-form potential are sextic in the scalar fields u ij IJ and v ij IJ . It may still be possible to simplify the new Ansätze using some E 7 (7) properties of the u ij IJ and v ij IJ tensors [4,9]. One such simplification concerns the C pq ijkl tensor that occurs in both, the warp factor and the three-form potential. For the rest of this section, I show that it factorizes into 8 where The selfdual tensor K IJKL is defined as a certain combination of Killing vectors in Eq. (A38). It satisfies some useful relations given in Appendix A. The third term in Eq. (75) represents the T -tensor, which is defined in [4], It only depends on spacetime coordinates x and satisfies the property For further relations concerning the T -tensor, see [4,9]. Note that the only difference between C 1 p ijk and the T -tensor is the K IJKL -factor in Eq. (76) instead of a δ IJ KL -factor in Eq. (78). This gives rise to interpret C 1 and C 2 as the y-dependent twins of the T -tensor.
where | [IJKL] + represents the projection onto the selfdual part. This completes the proof of Eq. (75). In order to keep the formulas short, I do not insert the factorization of the C pq ijkl tensor into the uplift Ansätze for the warp factor and the three-form. However, one should always keep in mind that these expressions can still be simplified by Eq. (75).
I must emphasize that the antisymmetry of the three-form potential A mnp is not apparent from the new Ansatz in Eq. (74). This may be a hint that it still can be simplified using the E 7 (7) properties of the u ij IJ and v ij IJ tensors. One should check such a simplification in future work. Note that the recent three-form Ansatz in [20] is given in a coordinate-free form, hence its components are fully antisymmetric by definition.
In Section VI, I will test the new metric Ansatz for the G 2 invariant solution of 11-dimensional supergravity. Note that the Ansätze for the warp factor and the flux originate from the old formulas for ∆ −1 g mn and A mn p using the new metric Ansatz. Since these old expressions were already tested for a G 2 invariant solution [16], I do not re-check Eqs. (72,74) explicitly. For a consistent test, it will be sufficient to compute the metric by Eq. (71) and compare it with the existing expression in [16] 9 .

V. A NEW NON-LINEAR ANSATZ FOR THE FOUR-FORM FIELD-STRENGTH
In this section, I present a new non-linear Ansatz for the four-form field-strength So far, the internal three-form potential was only known in particular cases and it was yet very complicated to compute the derivative of an explicit expression for A mnp . However, I found a new general uplift Ansatz for A mnp in the previous section. In particular, at the level of 11-dimensional vielbein components (Eqs. (51,64)), one finds With a look at Eqs. (25,26) and using Eq. (A22), one has Furthermore, since all SU(8) indices in Eq. (86) are fully contracted, I can replace the 11-dimensional vielbeine by the four dimensional expressions in Eqs. (40-43). This finally yields a general expression for the four-form field-strength, i.e.
One can now evaluate the derivative in general. First, one has hence, one term in F mnpq will be proportional to A [mnpDq] log ∆. Secondly, the covariant background derivativeD m only acts on the y-dependent fields in the vielbein components: the Killing forms and the dual volume potentialζ m . It does not act on the scalars u ij IJ and v ij IJ . In general, Putting all this together, the resulting intermediate expression for F mnpq becomes rather long and I do not display it here. However, it should be clear that it contains the tensorsg mn ,ζ m as well as all four-dimensional vielbeine V M ij . The SU(8) indices ij . . . are fully contracted in pairs. I can therefore replace the V M ij 's by the 11-dimensional vielbein components V M AB . The final step is to use Eqs. (25)(26)(27)(28), which introduces the 11-dimensional fields (e.g. A mnp and A m1···m6 ) as well as Γ-matrices. Using Eqs. (A11) for the traces of products of Γ-matrices, I finally obtain where | [mnpq] denotes antisymmetrized indices mnpq. One eliminates the second term by Eq. (62), For some readers, this expression is already in a desired form. However, one can further simplify this expression. First, the term proportional toη r1···r7 A qr3r4 A r5r6r7 can be replaced using Eq. (57). Together with Eq. (62), this cancels the term proportional toD m log ∆. Finally, one turns the tensor densityη r1···r7 into the tensor ǫ r1···r7 (Eq. (54)) and obtains This formula appears to be more feasible for practical tests than previous expressions [20,26]. It is not difficult to raise all indices with the inverse metric g mn . Therefore, one must keep in mind that the indices of the Killing forms andg mn are raised with the background metric. All other tensors in Eq. (96) are covariant, hence Note the power of the last step: Until now, the field-strength with upper indices has always been found by raising each lower index of F mnpq with the explicit expression for the inverse metric g mn . This was one of the hardest tasks in verifying the SO(3)×SO(3) invariant solution of 11-dimensional supergravity. With the new Ansatz above, it is much simpler to find F mnpq . For maximally symmetric spacetimes, these results may also be used to calculate the Ricci tensor using the Einstein equations.
In the next section, I will test the new Ansatz for the four-form field-strength for the G 2 invariant solution of 11-dimensional supergravity.

VI. TESTING THE NEW UPLIFT ANSÄTZE
This section presents an essential part of this work: I test the new non-linear Ansätze for the metric g mn and the four-form field-strength F mnpq within a G 2 invariant solution of 11-dimensional supergravity. In such a setup, the Ansätze for the inverse metric ∆ −1 g mn (Eq. (50)) and the three-form with mixed index structure (Eq. (52)) were already checked successfully [16]. The same reference computes the warp factor by taking the determinant of the expression for ∆ −1 g mn and the metric g mn by inverting g mn . Finally, it calculates the full internal three-form potential A mnp by lowering the third index with the explicit expression for g mn . It should be clear that a successful test for the metric Ansatz in Eq. (71) includes the tests of the Ansätze for the warp factor and the three-form potential in Eqs. (72,74), since these result from combining the old known Ansätze with the new metric Ansatz.
Here, I compute the metric ∆ −2 g mn by Eqs. (66,67), which is equivalent to use Eq. (71). I follow the strategy of [16]: One first brings the E 7(7) -matrix that encodes the four-dimensional scalars into unitary gauge, where φ IJKL denotes the scalar vacuum expectation value. In this gauge, there is no distinction between SU(8) indices ij . . . and SL(8,R) indices IJ . . . This allows us to write the scalar fields u ij IJ and v ij IJ in terms of the vacuum expectation value φ IJKL . For a G 2 invariant configuration, the latter takes the general form, where C IJKL + is selfdual and C IJKL − is anti-selfdual. The above expression also defines a scalar field λ(x) and a rotation angle α(x). Using the explicit form of the vacuum expectation value in Eq. (100), one finds the four-dimensional scalars u ij IJ and v ij IJ in terms of the G 2 invariants C IJKL ± , i.e.
where p = cosh λ and q = sinh λ. The tensors D IJKL ± are defined as One now expands the C IJKL ± tensors into the (anti-)selfdual bases provided by the Killing forms defined in Eq. (A19,A20), The occurring components ξ, ξ m , ξ mn and S mnp are SO(7) tensors 10 on the round S 7 , hence, its indices are raised and lowered with the background metricg mn . Note that S mnp is totally antisymmetric by construction. Furthermore, one finds the useful relations [16] ξ mng mn = ξ, ξ m ξ n = (9 − ξ 2 )g mn − 6(3 − ξ)ξ mn , ξ m ξ m = (21 + ξ)(3 − ξ), From the decomposition of the C IJKL ± tensors in Eqs. (104,105), one finds the useful contractions as well as for the D IJKL ± tensors, Now, I write the metric g mn in terms of the components ξ, ξ m , ξ mn and S mnp defined above. Therefore, one first using Eq. (67) and expands the scalar fields u IJ KL and v IJ KL in terms of the C IJKL ± and D IJKL ± tensors (Eqs. (101,102)). Secondly, one uses the contractions above together with Eqs. (A27,A28) in Appendix A to bring V mp [IJ V p KL] into the basis provided by Eqs. (A19,A20), The respective coefficients a m , b m n , c m np and d m npq are rather long expressions and I do not display them here. However, it should be clear that they only depend on the SO(7) tensors ξ, ξ m , ξ mn and S mnp . Finally, one computes the metric via Eq. (66). For the contractions of the indices IJKL, one uses Eqs. (A34-A37) and for the contractions of the SO(7) indices, one uses the identities in Eqs. (106,107). This finally results in where I made the following definitions: The test for the inverse metric Ansatz (Eq. (50)) was already performed in [16]. The corresponding final expression is Combining the explicit expressions for the metric and its inverse in Eqs. (117,119) and using the identities in Eqs. (106,107), one finds that This is exactly the combination of the metric and its inverse that defines the warp factor in Eq. (72), hence The explicit expressions for the metric and the warp factor in Eqs. (117,121) reproduce the results of [16]. The reader may also check that the determinant of the metric in Eq. (117) indeed, reproduces Eq. (121). The test is hence, successful.
For the remaining test of the field-strength Ansatz in Eq. (96), I use the explicit expression for A mnp that was found in [16], Note that this expression is slightly simplified using the identities in Eqs. (106,107). Furthermore, the formula for A mnp above differs from the expression given in [16] by a factor of 1/6, which is due to my conventions. However, the definition of the field-strength in Eq. (85) differs from the corresponding definition in [16] by a factor of 6. Hence, the new Ansatz for F mnpq in Eq. (96) should give the same expression as already computed in [16] by calculating the derivative of Eq. (122) directly. A convenient way to use the new Ansatz is such that one may use Eqs. (117,121,122) directly. For the term involving the Killing forms and the four-dimensional scalars, I follow the same strategy as described earlier in this section. I find Putting all together and using Eqs. (106,107) finally results in which matches exactly the expression found in [16]. The test is hence, successful.

VII. CONCLUSION
In this paper, I derive a new non-linear metric Ansatz for the uplift of N = 8 supergravity to 11-dimensional supergravity. An uplift Ansatz for the inverse metric, scaled with the warp factor, ∆ −1 g mn , has already been known for a long time [15]. However, inverting this expression in order to find ∆g mn was only possible in certain cases, for example when the theory is G 2 , SO(3)×SO(3) or SU(3)×U(1)×U(1) invariant [16][17][18][19]. Also the warp factor ∆ could only be extracted by taking the determinant in such particular cases. Following the strategy of [14], I present a new general uplift Ansatz for ∆ −2 g mn in terms of the four-dimensional scalar fields and the Killing forms on the background (Eqs. (66-71)). Note that this Ansatz is similar to a recent coordinate-free expression [20]. However, the formula presented here seems to be more feasible for practical tests: I tested the new metric Ansatz within a G 2 invariant solution of 11-dimensional supergravity in Section VI.
Similarly to [20], the new formula can further be used in order to find non-linear uplift Ansätze for the warp factor and the full internal three-form potential in general. For the warp factor, I combine the old Ansatz for ∆ −1 g mn with the new one for ∆ −2 g mn , which gives a new Ansatz for ∆ −3 (Eq. (72)). Furthermore, I derive a general Ansatz for the full internal three-form potential A mnp (Eq. (74)) by combining the old flux Ansatz for A mn p [14] with the new metric Ansatz. However, this new formula does not reveal the total antisymmetry of the three-form. This may be a hint that one can further simplify the expression for A mnp using some E 7 (7) identities for the four-dimensional scalar fields. I hope that I can provide such a simplification in future work.
In a second part of this paper, I derive a new general non-linear uplift Ansatz for the four-form field-strength F mnpq within the considered uplift of N = 8 supergravity to 11-dimensional supergravity. So far, the simplest way to derive F mnpq was to compute the derivative of the three-form potential. However, this required an explicit expression for the flux, which is only given in particular cases, e.g. the G 2 , SO(3)×SO(3) or SU(3)×U(1)×U(1) invariant solutions of 11-dimensional supergravity. With the new Ansatz for the field-strength (Eq. (96)), there is no need to compute derivatives anymore. It is given in terms of the metric, the flux as well as the four-dimensional scalars and background Killing forms. The formula holds in general and also passes a very non-trivial test for a G 2 invariant solution of 11dimensional supergravity.
The new Ansatz for the field-strength also provides a simple expression for F mnpq (Eq. (97)) in terms of the inverse metric, the flux as well as the four-dimensional scalars and background Killing forms. This new formula makes it redundant to raise each index of F mnpq with the explicit expression for the inverse metric, g mn , which was so far, the only way to derive F mnpq . The new direct Ansatz for F mnpq is also much more effective than this old methodin order to verify the SO(3)×SO(3) invariant solution of 11-dimensional supergravity, the index-raising of F mnpq was one of the hardest tasks [26].
In future, one may also find new Ansätze for the Christoffel connections in 11-dimensional supergravity in terms of the four-dimensional scalars and background Killing forms. Since they are given by the first derivative of the metric, one could find new simple expressions in full analogy to the derivation of the field-strength Ansatz. Similarly, one could derive a non-linear Ansatz for the Riemann tensor.
In this paper, all Ansätze are derived within the S 7 reduction of 11-dimensional supergravity. This leads to the compact gauging SO (8). However, the methods provided here should also apply in general for other truncations. As a first example, one may extend the theory to the non-compact CSO(p, q, r) gaugings [27,28]. In this case, the IJ indices of the Killing forms are raised and lowered with the CSO(p, q, r)-metric η IJ instead of the SO(8) metric δ IJ . This effects the definition of the matrix R M N in Eqs. (33-36) and hence, the A m ijkl and B m ijkl tensors in Eqs. (69,70). Thus, the new Ansätze for the metric, the three-form and warp-factor will be slightly modified. However, the new Ansatz for the four-form field-strength will change more dramatically: Eqs. (90-93) do not hold if the IJ indices of the Killing forms are raised and lowered with the full CSO(p, q, r) metric. Since the new Ansatz for F mnpq depends on those identities, it will take much more effort to derive an adapted Ansatz for the four-form within the non-compact gaugings. Finally, the presented methods may also be used for the reduction from type IIB supergravity to five dimensions [29][30][31].
Here,D m is the covariant derivative with respect to the internal background metricg mn and m 7 is the inverse S 7 radius.
The Killing spinors define a set of Killing vectors and their derivatives, K mn IJ =η IΓ mn η J , (A13) Using Eq. (A3), one verifies that K mn IJ is indeed, proportional to the derivative of K m IJ , D n K m IJ = m 7 K mn IJ ,D p K mn IJ = 2m 7gp[m K n] IJ .
Using Eq. (A8), one also finds that K m1···m5 IJ is the (seven dimensional) dual to K mn IJ , Note that curved seven dimensional indices of the Killing vectors and their derivatives are always raised and lowered with the background metricg mn .
The following bi-linears in the Γ-matrices represent a basis for (anti-)selfdual SU (8)