Geometry and periods of $G_2$-moduli spaces

Abstract

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 $GL(n+1)/(\{\pm 1\}\times O(n))$, where $n=b^3(M)-1$. We point out that the formal properties of this immersion $\Phi :\mathcal{M}\to \mathfrak{D}$ 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 $\Phi (\mathcal{M})\subset \mathfrak{D}$.

arXiv commentSecond version, 35 pages. Minor typos fixed, exposition improved, proofs of the main results in Section 4 (Theorem 4.9 and Proposition 4.12) clarified. To appear in Adv. Math

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