Local geometry of the G2 moduli space

We consider deformations of torsion-free G2 structures, defined by the G2-invariant 3-form $phi$ and compute the expansion of the Hodge star of $phi$ to fourth order in the deformations of $phi$. By considering M-theory compactified on a G2 manifold, the G2 moduli space is naturally complexified, and we get a Kahler metric on it. Using the expansion of the Hodge star of $phi$ we work out the full curvature of this metric and relate it to the Yukawa coupling.