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Non-linear Kaluza-Klein theory for dual fields

We present non-linear uplift ansaetze for all the bosonic degrees of freedom and dual fields in the 7-sphere reduction of D=11 supergravity to maximal SO(8) gauged supergravity and test them for the SO$(7)^{\pm}$ invariant solutions. In particular, we complete the known ansaetze for the internal components of the metric and 4-form flux by constructing a non-linear ansatz for the internal components of the dual 7-form flux. Furthermore, we provide ansaetze for the complete set of 56 vector fields, which are given by more general structures than those available from standard Kaluza-Klein theory. The novel features encountered here have no conventional geometric interpretation and provide a new perspective on Kaluza-Klein theory. We study the recently found set of generalised vielbein postulates and, for the 7-sphere compactification, we show that they reduce to the E7(7) Cartan equation of maximal SO(8) gauged supergravity in four dimensions. The significance of this framework for a higher-dimensional understanding of the embedding tensor and other gauged maximal supergravities is briefly discussed.