We give the general form of supersymmetric backgrounds with two real supercharges of M theory and type IIA supergravity (with nonzero Romans mass in general) of the form ${\mathbb{R}}^{1,d}\times {\mathcal{M}}_8$, $d=1,2$, on eight-dimensional manifolds with $SU(4)$ structure. We point out a subtlety in the integrability theorems for low-dimensional supersymmetric compactifications. As a special case we examine Calabi-Yau flux vacua, and we show that unbroken supersymmetry does not in general require the 4-form flux to be (2,2) or primitive. Our results could be used to construct novel higher dimensional analogues of the Klebanov-Strassler geometry. In the case of M-theory large-volume Calabi-Yau flux vacua, our results are in agreement with partial supersymmetry breaking in three-dimensional $\mathcal{N}=2$ supergravity. Alternatively, the conditions for supersymmetry can be expressed in terms of a real “superpotential” in accordance with three-dimensional $\mathcal{N}=1$ supergravity. We present explicit examples of M-theory flux vacua on $\mathrm{K}3\times \mathrm{K}3$, which however do not appear to possess $F$-theory duals with four-dimensional Poincaré invariance.

• Received 16 December 2013

DOI:https://doi.org/10.1103/PhysRevD.89.064030