A note on fluxes and superpotentials in M-theory compactifications on manifolds of G₂ holonomy

We consider the breaking of N = 1 supersymmetry by non-zero G-flux when M-theory is compactified on a smooth manifold X of G₂ holonomy. Gukov has proposed a superpotential W to describe this breaking in the low-energy effective theory. We check this proposal by comparing the bosonic potential implied by W with the corresponding potential deduced from the eleven-dimensional supergravity action. One interesting aspect of this check is that, though W depends explicitly only on G-flux supported on X, W also describes the breaking of supersymmetry by G-flux transverse to X.