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 G₂, SO(3)×SO(3) or SU(3)×U(1)×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 internal three-form flux, as well as for the internal four-form field-strength. The new formula for the four-form only depends on the metric, the flux as well as the four-dimensional scalars and background Killing forms — it does not require to calculate the derivative of the flux. All the Ansätze presented in this work pass a very non-trivial test for a G₂ invariant solution of 11-dimensional supergravity. My results may be generalized to other compactifications, e.g. the reduction from type IIB supergravity to five dimensions.