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 $\mathrm{ AdS}_3$.
Categories: hep-th