We modify the Laplacian coflow of co-closed G 2 -structures -d dt ψ = ∆ψ where ψ is the closed dual 4-form of a G 2 -structure ϕ. The modified flow is now weakly parabolic in the direction of closed forms up to diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified flow.

We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonionic covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the Riemann curvature that lies in the 7-dimensional representation of G2. We also interpret the choice of a particular G2-structure within the same metric class as a choice of gauge and show that under a change of this gauge, the torsion does transform as an octonion-valued connection 1-form. Finally, we also show an explicit relationship between the octonion bundle and the spinor bundle, define an octonionic Dirac operator and explore an energy functional for octonion sections. We then prove that critical points correspond to divergence-free torsion, which is shown to be an octonionic analog of the Coulomb gauge.

We consider deformations of torsion-free G 2 structures, defined by the G 2 -invariant 3-form ϕ and compute the expansion of * ϕ to fourth order in the deformations of ϕ. By considering M -theory compactified on a G 2 manifold, the G 2 moduli space is naturally complexified, and we get a Kähler metric on it. Using the expansion of * ϕ we work out the full curvature of this metric and relate it to the Yukawa coupling.

This paper is a review of current developments in the study of moduli spaces of G 2 manifolds. G 2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G 2 . Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi-Yau manifolds in 7 dimensions. They play an important role in physics as natural candidates for supersymmetric vacuum solutions of M -theory compactifications. Despite the physical motivation, many of the results are of purely mathematical interest. Here we cover the basics of G 2 manifolds, local deformation theory of G 2 structures and the local geometry of the moduli spaces of G 2 structures.