Consistency of the $Ad{S}_7\times S_4$ reduction and the origin of self-duality in odd dimensions

We discuss the full nonlinear Kaluza-Klein (KK) reduction of the original formulation of d=11 supergravity on $Ad{S}_7\times S_4$ 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 $\delta F=d\delta A$ind=11canbewrittenas$\delta d(fields in d=7)$, 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 $A_{\mathrm{\Lambda }\mathrm{\Pi }\mathrm{\Sigma }}$ in d=11 is needed to obtain the d=7 supersymmetry laws and the action for the nonabelian selfdual antisymmetric tensor field $S_{\alpha \beta \gamma ,A}$. Therefore selfduality in odd dimensions originates from a first order formalism in higher dimensions.