The landscape of G-structures in eight-manifold compactifications of M-theory

We consider spaces of "virtual" constrained generalized Killing spinors, i.e. spaces of Majorana spinors which correspond to "off-shell" s-extended supersymmetry in compactifications of eleven-dimensional supergravity based on eight-manifolds M. Such spaces naturally induce two stratifications of M, called the chirality and stabilizer stratification. For the case s = 2, we describe the former using the canonical Whitney stratification of a three-dimensional semi-algebraic set R. We also show that the stabilizer stratification coincides with the rank stratification of a cosmooth generalized distribution D-0 and describe it explicitly using the Whitney stratification of a four-dimensional semi-algebraic set B. The stabilizer groups along the strata are isomorphic with SU(2), SU(3), G(2) or SU(4), where SU(2) corresponds to the open stratum, which is generically non-empty. We also determine the rank stratification of a larger generalized distribution D which turns out to be integrable in the case of compactifications down to AdS(3).