Consistency of the AdS7 x S4 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 AdS(7) x S-4 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 cl = 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 expression delta F = d delta A in d = 11 can be written as delta d(fields in d = 7), we find the ansatz for the 4-form F. It satisfies the Bianchi identities. The corresponding ansatz for the S-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 S-alpha beta gamma,S-A. Therefore selfduality in odd dimensions originates from a first order formalism in higher dimensions. (C) 2000 Elsevier Science B.V. All rights reserved.