Global structure of G2-moduli spaces

Characterize the global structure of the moduli space M of torsion-free G2-structures on a compact oriented 7-manifold M, beyond the established local descriptions.

Background

Joyce proved that the moduli space of torsion-free G2-structures on a compact 7-manifold is a smooth affine manifold of dimension b3(M), with local models in cohomology. However, these results are local in nature and do not determine global geometric properties of the moduli space.

The paper emphasizes that the absence of an analogue of Yau's theorem in G2-geometry hinders global understanding. The authors introduce a new period-type immersion into a homogeneous space as a perspective, but the comprehensive global description of the moduli space remains unresolved.

References

However, all of these results are local, and nothing is known about the global structure of the moduli space, partly because of the lack of an analog of Yau's theorem in $G_2$-geometry.

— Geometry and periods of $G_2$-moduli spaces  (2410.09987 - Langlais, 2024) in Introduction and motivation (Section 1)

For the geometries studied here, the situation is more subtle: the background is described in terms of a local $\Gx 2$ structure which may enhance to $\Spin 7$ on subsets of $X$, and it is not clear that the corresponding space of structures admits a comparably useful description.

— A real superpotential for M-theory on eight-manifolds  (2610.12439 - Ashmore et al., 8 Oct 2026) in Section 6, Discussion