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

Abstract

We describe a class of decomposable eleven-dimensional supergravity backgrounds which are products of a four-dimensional Lorentzian manifold and a seven-dimensional Riemannian manifold, endowed with a flux form given in terms of the volume form on and a closed 4-form F4 on M7. We show that the Maxwell equation for such a flux form can be read in terms of the co-closed 3-form . Moreover, the supergravity equation reduces to the condition that is an Einstein manifold with negative Einstein constant and (M7,g,F) is a Riemannian manifold which satisfies the Einstein equation with a stress-energy tensor associated to the 3-form . Whenever this 3-form is generic, we show that the Maxwell equation induces a weak -structure on M7 and obtain decomposable supergravity backgrounds given by the product of a weak -manifold with a Lorentzian Einstein manifold . We also construct examples of compact homogeneous Riemannian 7-manifolds endowed with non-generic invariant 3-forms which satisfy the Maxwell equation, but the construction of decomposable homogeneous supergravity backgrounds of this type remains an open problem.