M-theory potential from the G ₂ Hitchin functional in superspace

We embed the component fields of eleven-dimensional supergravity into a superspace of the form X × Y where X is the standard 4D, N = 1 superspace and Y is a smooth 7-manifold. The eleven-dimensional 3-form gives rise to a tensor hierarchy of superfields gauged by the diffeomorphisms of Y. It contains a natural candidate for a G₂ structure on Y , and being a complex of superforms, defines a superspace Chern-Simons invariant. Adding to this a natural generalization of the Riemannian volume on X × Y and freezing the (superspin- 3/2 and 1) supergravity fields on X, we obtain an approximation to the eleven-dimensional supergravity action that suffices to compute the scalar potential. In this approximation the action is the sum of the superspace Chern-Simons term and a super-space generalization of the Hitchin functional for Y as a G₂-structure manifold. Integrating out auxiliary fields, we obtain the conditions for unbroken supersymmetry and the scalar potential. The latter reproduces the Einstein-Hilbert term on Y in a form due to Bryant.