Geometry and periods of $G_2$-moduli spaces

This paper is concerned with the geometry of the moduli space $\mathcal{M}$ of torsion-free $G_2$-structures on a compact $G_2$-manifold $M$, equipped with the volume-normalised $L^2$-metric $\mathcal{G}$. When $b^1(M)=0$, 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 $\mathcal{M}$, based on the observation that there is a natural way to immerse the moduli space into a homogeneous space $\mathfrak{D}$ diffeomorphic to $GL(n+1)\mathrm{/}(\{\pm 1\}\times O(n))$, where $n=b^3(M)-1$. We point out that the formal properties of this immersion $\mathrm{\Phi }:\mathcal{M}\to \mathfrak{D}$ 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 $\mathcal{G}$, we also derive a new formula for the fourth derivative of the potential and relate it to the second fundamental form of $\mathrm{\Phi }(\mathcal{M})\subset \mathfrak{D}$.