IIA supergravity and M-theory on manifolds with SU(4) structure

We give the general form of supersymmetric backgrounds with two real supercharges of M-theory and type IIA supergravity (with non-zero Romans mass in general) of the form $\mathbb{R}^{1,d} \times \M_8$, 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 four-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 N=2 supergravity. Alternatively, the conditions for supersymmetry can be expressed in terms of a real `superpotential' in accordance with three-dimensional N=1 supergravity. We present explicit examples of M-theory flux vacua on K3 \times K3, which however do not appear to possess F-theory duals with four-dimensional Poincar\'e invariance.