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

TL;DR

This paper analyzes the structure of special spinor spaces in eight-manifold compactifications of M-theory, revealing stratifications and stabilizer groups that classify supersymmetric solutions in supergravity.

Contribution

It introduces a detailed geometric framework for understanding constrained spinor spaces and their stratifications in M-theory compactifications on eight-manifolds.

Findings

Describes the chirality and stabilizer stratifications of the manifold.

Identifies stabilizer groups as SU(2), SU(3), G2, or SU(4).

Shows integrability of distributions in AdS3 compactifications.

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 $\mathrm{AdS}_3$.