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

TL;DR

This paper presents a detailed nonlinear Kaluza-Klein reduction of 11-dimensional supergravity on AdS7×S4 to 7-dimensional gauged supergravity, revealing how self-duality in odd dimensions arises from a higher-dimensional first order formalism.

Contribution

It provides the full nonlinear embedding ansatz for the reduction, checks its consistency, and clarifies the origin of self-duality in odd dimensions from a higher-dimensional perspective.

Findings

Derived the full nonlinear embedding of 7D fields in 11D supergravity.

Obtained the correct 7D scalar potential and supersymmetry laws.

Showed that self-duality in odd dimensions stems from a first order formalism in higher dimensions.

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 $\delta F=d\delta A$ 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 A modifies the geometrical proposal by Freed et al. by including d=7 scalar fields. A first order formulation for the three index photon $A_{\Lambda\Pi\Sigma}$ 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.