Decomposable (4, 7) solutions in eleven-dimensional supergravity

@article{Alekseevsky2019Decomposable7,
  title={Decomposable (4, 7) solutions in eleven-dimensional supergravity},
  author={Dmitri Alekseevsky and Ioannis Chrysikos and Arman Taghavi-Chabert},
  journal={Classical and Quantum Gravity},
  year={2019}
}
Consider an oriented four-dimensional Lorentzian manifold $(\widetilde{M}^{3, 1}, \widetilde{g})$ and an oriented seven-dimensional Riemannian manifold $(M^{7}, g)$. We describe a class of decomposable eleven-dimensional supergravity backgrounds on the product manifold $({\mathcal{M}}^{10, 1}=\widetilde{M}^{3,1} \times M^7, g_{{\mathcal{M}}}=\widetilde{g}+g)$, endowed with a flux form given in terms of the volume form on $\widetilde{M}^{3, 1}$ and a closed $4$-form $F^{4}$ on $M^{7}$. We show… 
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