Consistency of the $AdS_7\times S_4$ reduction and the origin of self-duality in odd dimensions

TL;DR

This work resolves the nonlinear Kaluza-Klein truncation problem for the original 11D supergravity on $AdS_7\times S_4$ by constructing a complete nonlinear embedding of 7D ${\cal N}=4$ gauged supergravity within the 11D fields. It develops a detailed ansatz for the 11D metric, 4-form flux, and a first-order formulation of the 3-index tensor, linked through a field-dependent matrix $U(y,x)$ that ensures the 7D SUSY rules emerge from the 11D theory. The authors demonstrate consistency by matching 7D transformation laws, equations of motion, and the scalar potential, and they expose the origin of self-duality in odd dimensions as a consequence of a higher-dimensional first-order action. The results unify known truncations and argue for preferring the standard 11D framework over the SU(8) reformulation for compactifications, with implications for lifting lower-dimensional solutions to M-theory.

Abstract

We discuss the full nonlinear Kaluza-Klein (KK) reduction of the original formulation of d=11 supergravity on $AdS_7\times S_4$ to gauged maximal ({\cal N}=4) supergravity in 7 dimensions. We derive the full nonlinear embedding of the d=7 fields in the d=11 fields (``the ansatz'') and check the consistency of the ansatz by deriving the d=7 supersymmetry laws from the d=11 transformation laws in the various sectors. The ansatz itself is nonpolynomial but the final d=7 results are polynomial. The correct d=7 scalar potential is obtained. For most of our results the explicit form of the matrix U connecting the d=7 gravitino to the Killing spinor is not needed, but we derive the equation which U has to satisfy and present a solution. Requiring that the expression $δF=dδA$ in d=11 can be written as $δd(fields in d=7)$, we find the ansatz for the 4-form F. It satisfies the Bianchi identities. The corresponding ansatz for A modifies the geometrical proposal by Freed et al. by including d=7 scalar fields. A first order formulation for the three index photon $A_{ΛΠΣ}$ in d=11 is needed to obtain the d=7 supersymmetry laws and the action for the nonabelian selfdual antisymmetric tensor field $S_{αβγ,A}$. Therefore selfduality in odd dimensions originates from a first order formalism in higher dimensions.