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

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