Consistency of the AdS7×S4 reduction and the origin of self-duality in odd dimensions

Abstract

We discuss the full nonlinear Kaluza–Klein (KK) reduction of the original formulation of d=11 supergravity on AdS7×S4 to gauged maximal (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 the general solution. Requiring that the expressionδF=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 the 3-form A modifies the geometrical proposal by Freed et al. by including d=7 scalar fields. A first order formulation for A in d=11 is needed to obtain the d=7 supersymmetry laws and the action for the nonabelian selfdual antisymmetric tensor field Sαβγ,A. Therefore selfduality in odd dimensions originates from a first order formalism in higher dimensions.