Consistent truncation of d = 11 supergravity on AdS_4 x S^7

TL;DR

This work resolves subtle inconsistencies in the non-linear embedding of ${oldsymbol d}=11$ supergravity on ${ m AdS}_4 imes S^7$ into gauged ${ m N}=8$, d=4 supergravity by exploiting ${ m E}_{7(7)}$ symmetry to classify all solutions of the generalized vielbein postulate (GVP) and the ${rak A}$-equations. It shows that a previously overlooked homogeneous kernel correction is unique and necessary to enforce the correct 11D flux tensor structure at every point on the ${ m E}_{7(7)}/{ m SU}(8)$ coset, and that the Freund-Rubin and internal fluxes are invariants of the linearized system, computable via two covariant projection methods. The authors provide analytic checks in the ${ m SO}(8)$, ${ m SO}(7)^-$, and ${ m SO}(7)^+$ sectors and extensive numerical tests across additional vacua, confirming the lift formulae and giving new, non-trivial tests of the flux-explicit embeddings. These results complete the consistency proof of the ${ m S^7}$ truncation and offer practical flux-lift formulae applicable to a wider class of ${ m AdS}_4$ vacua, with potential relevance to ${ m AdS}_5 imes S^5$.

Abstract

We study the system of equations derived twenty five years ago by B. de Wit and the first author [Nucl. Phys. B281 (1987) 211] as conditions for the consistent truncation of eleven-dimensional supergravity on AdS_4 x S^7 to gauged N = 8 supergravity in four dimensions. By exploiting the E_7(7) symmetry, we determine the most general solution to this system at each point on the coset space E_7(7)/SU(8). We show that invariants of the general solution are given by the fluxes in eleven-dimensional supergravity. This allows us to both clarify the explicit non-linear ansatze for the fluxes given previously and to fill a gap in the original proof of the consistent truncation. These results are illustrated with several examples.