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

Abstract

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 ${\cal R}$. We also show that the stabilizer stratification coincides with the rank stratification of a cosmooth generalized distribution ${\cal D}_0$ and describe it explicitly using the Whitney stratification of a four-dimensional semi-algebraic set $\mathfrak{P}$. The stabilizer groups along the strata are isomorphic with $\mathrm{SU}(2)$, $\mathrm{SU}(3)$, $\mathrm{G}_2$ or $\mathrm{SU}(4)$, where $\mathrm{SU(2)}$ corresponds to the open stratum, which is generically non-empty. We also determine the rank stratification of a larger generalized distribution ${\cal D}$ which turns out to be integrable in the case of compactifications down to $\mathrm{AdS}_3$.