Nonlinear Kaluza-Klein theory for dual fields

Abstract

We present nonlinear uplift Ans\"atze for all the bosonic degrees of freedom and dual fields in the ${S}^{7}$ reduction of $D=11$ supergravity to maximal SO(8) gauged supergravity and test them for the $\mathrm{SO}(7{)}^{\ifmmode\pm\else\textpm\fi{}}$ invariant solutions. In particular, we complete the known Ans\"atze for the internal components of the metric and four-form flux by constructing a nonlinear Ansatz for the internal components of the dual seven-form flux. Furthermore, we provide Ans\"atze 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 generalized vielbein postulates and, for the ${S}^{7}$ compactification, we show that they reduce to the ${\mathrm{E}}_{7(7)}$ 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.