Weak $G_2$-manifolds and scale separation in M-theory from type IIA backgrounds

This work provides evidence for the existence of supersymmetric and scale-separated AdS$_4$ vacua in M-theory of the Freund-Rubin type. The internal space has weak $G_2$-holonomy, which is obtained from the lift of AdS vacua in massless type IIA on a specific SU(3)-structure with O6-planes. Such lifts require a local treatment of the O6-planes, therefore going beyond the usual smeared approximation. The setup is analysed by solving the pure spinor equations and the Bianchi identities perturbatively in a small backreaction parameter, preserving supersymmetry manifestly and therefore extending on previous work. This approach is applicable to lifts of other type IIA vacua on half-flat SU(3)-structures, including those with D6-brane sources. The resulting 7d manifold presented here exhibits singularities originating from the O6-planes loci in type IIA theory. Additionally, scale separation in M-theory arises from a decoupling between the Ricci curvature and the first eigenvalue of the Laplacian of the proposed 7d manifold, thereby challenging certain conjectures in the swampland program.