Moduli of Coassociative Submanifolds and Semi-Flat Coassociative Fibrations

Abstract

We study the natural structure on the moduli space of deformations of compact coassociative submanifolds. We show that a G2-manifold with a T^4-action of isomorphisms such that the orbits are coassociative tori is locally equivalent to a minimal 3-manifold in R^{3,3} = H^2(T^4,R) with positive induced metric. By studying minimal surfaces in quadrics we show how to construct minimal 3-manifold cones in R^{3,3} and hence G2-metrics from equations similar to a set of affine Toda equations. The relation to semi-flat special Lagrangian fibrations and the Monge-Amp\`ere equation are explained.