M-theory Potential from the $G_2$ Hitchin Functional in Superspace

TL;DR

This paper develops a superspace formulation of eleven-dimensional supergravity compactified on a 7-manifold with a $G_2$ structure, deriving a scalar potential from a superspace Hitchin functional that captures key geometric and physical features.

Contribution

It introduces a superspace approach to embed 11D supergravity fields, constructs a superspace $G_2$ Hitchin functional, and derives the scalar potential relevant for supersymmetry breaking analysis.

Findings

Derived conditions for unbroken supersymmetry.

Reproduced the Einstein-Hilbert term on the 7-manifold.

Formulated a superspace scalar potential from the Hitchin functional.

Abstract

We embed the component fields of eleven-dimensional supergravity into a superspace of the form $X\times Y$ 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 $G_2$ 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 $X\times Y$ and freezing the (superspin-$\frac32$ 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 superspace generalization of the Hitchin functional for $Y$ as a $G_2$-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.