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Non-linear uplift Ansätze 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 Ansätze 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 Ansätze 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.