Local Geometry of the G 2 Moduli Space

Abstract

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.

Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

“…Here we have used F ∈ Λ 2 14 (Y, End(V)), which ensures that 17) where the mapsF , F were introduced in section 5.1.…”

“…1 This space may be endowed by a metric [14][15][16], that shares certain properties with the Kähler metric on a Calabi-Yau moduli space [8,9]. In particular, when used in M theory compactifications, Grigorian and Yau [17] have proposed a local Kähler metric for the combined deformation space of the geometry and M theory flux potential.…”

MODULI SPACES OF G2MANIFOLDS

“…In particular, from [13,20,30], we have Proposition 5 The 3-form ϕ 0 and the corresponding 4-form ψ 0 satisfy the following identities: The above identities can be of course further contracted -the details can be found in [20,30]. These identities and their contractions are crucial whenever any calculations involving ϕ 0 and ψ 0 have to be done.…”

“…Suppose we have χ ∈ Λ 3 , then define π 1 , π 7 and π 27 to be projections of χ onto Λ 3 1 , Λ 3 7 and Λ 3 27 , respectively. Using contraction identities for ϕ and ψ, we get the following relations [20]:…”

“…In the case of deformations in Λ 3 27 the only known way to get results for deformations of the metric and the 4-form ψ is to consider the deformations order by order in ε. This analysis has been carried out in [20], and here we will review those results. So suppose we have a deformatioñ…”