# Superpotential of Three Dimensional ${\cal N}=1$ Heterotic Supergravity

## TL;DR

This paper derives the superpotential for three-dimensional ${
m N}=1$ heterotic supergravity compactified on $G_2$ structure manifolds, linking extrema of the superpotential to supersymmetric solutions.

## Contribution

It explicitly constructs the superpotential functional for the reduced theory and confirms its critical points correspond to supersymmetric heterotic compactifications on $G_2$ manifolds.

## Key findings

- Superpotential functional derived for 3D ${
m N}=1$ heterotic theory.
- Extrema of the superpotential match supersymmetric $G_2$ compactifications.
- Comments on moduli stabilization and instanton bundle roles.

## Abstract

We dimensionally reduce the ten dimensional heterotic action on spacetimes of the form ${\cal M}_{(2,1)}\times Y$, where ${\cal M}_{(2,1)}$ is three dimensional maximally symmetric Anti de Sitter or Minkowski space, and $Y$ is a compact seven dimensional manifold with $G_2$ structure. In doing so, we derive the real superpotential functional of the corresponding three dimensional ${\cal N}=1$ theory. We confirm that extrema of this functional precisely correspond to supersymmetric heterotic compactifications on manifolds of $G_2$ structure. We make some comments on the role of the superpotential functional with respect to the coupled moduli problem of instanton bundles over $G_2$ manifolds.