The eleven dimensional supergravity equations on edge manifolds

We study the eleven dimensional supergravity equations which describe a low energy approximation to string theories and are related to M-theory under the AdS/CFT correspondence. These equations take the form of a non-linear differential system, on ${\mathbb{B}}^7\times {\mathbb{S}}^4$ with the characteristic degeneracy at the boundary of an edge system, associated to the fibration with fiber ${\mathbb{S}}^4\mathrm{.}$ We compute the indicial roots of the linearized system from the Hodge decomposition of the 4-sphere following the work of Kantor, then using the edge calculus and scattering theory we prove that the moduli space of solutions, near the Freund–Rubin states, is parametrized by three pairs of data on the bounding 6-sphere.