Fluxes in M-theory on 7-manifolds and G structures

TL;DR

Behrndt and Jeschek address whether warp compactifications of M-theory on 7-manifolds with 4-form flux can yield a flat 4-dimensional Minkowski space with four unbroken supercharges. They derive and solve the 11D Killing spinor equations under a warped ansatz, treating a general spinor decomposition beyond a direct-product ansatz. They show that a single Killing spinor forbids nontrivial 4-form flux in a flat external space, while the presence of at least two Killing spinors enables fluxes through a reduction to a 6-manifold with nontrivial $SU(3)$ structure. This clarifies the geometric conditions under which flux can be included in M-theory compactifications and highlights $SU(3)$ structure as the key ingredient for preserving flat Minkowski vacua with flux.

Abstract

We consider warp compactifications of M-theory on 7-manifolds in the presence of 4-form fluxes and investigate the constraints imposed by supersymmetry. As long as the 7-manifold supports only one Killing spinor we infer from the Killing spinor equations that non-trivial 4-form fluxes will necessarily curve the external 4-dimensional space. On the other hand, if the 7-manifold has at least two Killing spinors, there is a non-trivial Killing vector yielding a reduction of the 7-manifold to a 6-manifold and we confirm that 4-form fluxes can be incorporated if one includes non-trivial SU(3) structures.