Geometry and periods of G2-moduli spaces

This paper is concerned with the geometry of the moduli space $M$ of torsion-free $G_2$-structures on a compact $G_2$-manifold M, equipped with the volume-normalised $L^2$-metric $G$. When $b^1\left(M\right)=0$, this metric is known to be of Hessian type and to admit a global potential. Here we give a new description of the geometry of $M$, based on the observation that there is a natural way to immerse the moduli space into a homogeneous space $D$ diffeomorphic to $GL(n+1)/\left(\right\{\pm 1\}\times O(n\left)\right)$, where $n=b^3\left(M\right)-1$. We point out that the formal properties of this immersion $\Phi :M\to D$ are very similar to those of the period map defined on the moduli spaces of Calabi–Yau threefolds. With a view to understand the curvatures of $G$, we also derive a new formula for the fourth derivative of the potential and relate it to the second fundamental form of $\Phi \left(M\right)\subset D$.