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 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 P. 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 .

M-theory on non-Kähler eight-manifolds

“…Using now that M 8 has a SU(4)-structure and in particular it satisfies equation (2.15), we obtain 1 36) a result that was to be expected since it only depends on M 8 being equipped with a Spin(7)-structure.…”

“…Then, the G 2 -structure φ satisfies τ 2 = 0 and it is therefore a particular case of the general characterization found in references [16,17,36,37] for the most general eleven-dimensional Supergravity supersymmetric compactification background to three-dimensions. It is rewarding to see that although we are considering non-geometric compactification backgrounds, the foliation structure of the most general geometric supersymmetric compactification background is preserved, which also indirectly shows that compactifying in this class of non-geometric compactification background should be possible in principle.…”

“…Then, the solution, even non-geometric, preserves the structure of the most general geometric compactification background of eleven-dimensional Supergravity on an eight-manifold, studied in references [16,17,36,37].…”

Internal circle uplifts, transversality and stratified G-structures

“…On the other hand, it was shown in [5] that the stabilizer stratification induced by ξ 1 and ξ 2 on M has SU(2), SU(3), G 2 and SU(4) strata, whose description is considerably more complex. This stratification of M coincides with a certain coarsening of the preimage of the connected refinement of the canonical Whitney stratification [6,7] of a four-dimensional compact semi-algebraic [8,9] body P ⊂ R 4 through a certain map B : M → R 4 whose image is contained in P. As shown in [5], this complicated stratification generalizes what happens in the much simpler case of N = 1 M-theory flux compactifications on eight-manifolds [10][11][12][13] (which extend the classically fluxless case of [14][15][16]), where the relevant semi-algebraic body is the interval [−1, 1], endowed with its Whitney stratification.…”

“…The complexity of the picture found in [5] may come as a surprise given the relative simplicity of the stabilizer stratification ofM . The purpose of this note is to explain this difference.…”