Type IIA supergravity andM-theory on manifolds withSU(4)structure

Abstract

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 ${\mathbb{R}}^{1,d}\ifmmode\times\else\texttimes\fi{}{\mathcal{M}}_{8}$, $d=1,2$, 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 $\mathrm{K}3\ifmmode\times\else\texttimes\fi{}\mathrm{K}3$, which however do not appear to possess $F$-theory duals with four-dimensional Poincar\'e invariance.