Geometry and periods of $G_2$-moduli spaces

TL;DR

This paper explores the geometry of the moduli space of torsion-free G_2-structures on compact G_2-manifolds, introducing a new immersion into a homogeneous space and relating it to period maps, with implications for understanding the metric's curvature.

Contribution

It provides a novel description of the G_2-moduli space geometry via an immersion into a homogeneous space, drawing parallels with Calabi-Yau period maps and deriving new curvature formulas.

Findings

The moduli space can be immersed into a homogeneous space similar to period maps.

A new formula for the fourth derivative of the potential is derived.

The second fundamental form relates to the curvature of the moduli space.

Abstract

This paper is concerned with the geometry of the moduli space $\mathscr{M}$ of torsion-free $G_2$-structures on a compact $G_2$-manifold $M$, equipped with the volume-normalised $L^2$-metric $\mathscr{G}$. 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 $\mathscr{M}$, 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 $\Phi : \mathscr{M} \rightarrow \mathfrak{D}$ 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 $\mathscr{G}$, we also derive a new formula for the fourth derivative of the potential and relate it to the second fundamental form of $\Phi(\mathscr{M}) \subset \mathfrak{D}$.