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# Advances in Theoretical and Mathematical Physics

## Volume 21 (2017)

### Number 7

### Special Issue: Proceedings of the Strings 2016 Conference in Beijing

Guest Editors: J. Maldacena (Institute for Advanced Study), H. Ooguri (California Institute of Technology), H. Babak (Harvard University), S. Li (Tsinghua University), W. Song (Tsinghua University), and H. Lin (Tsinghua University)

### $M$-theory potential from the $G_2$ Hitchin functional in superspace

Pages: 1613 – 1634

DOI: https://dx.doi.org/10.4310/ATMP.2017.v21.n7.a1

#### Authors

#### Abstract

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 $G_2$ 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 ($\textrm{superspin-} \frac{3}{2}$ 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 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.

Published 19 March 2018