Moduli of Coassociative Submanifolds and Semi-Flat Coassociative Fibrations

TL;DR

This paper investigates the deformation space of compact coassociative submanifolds in G2-manifolds, linking their moduli to minimal 3-manifolds in a pseudo-Euclidean space and exploring their relation to semi-flat fibrations and integrable systems.

Contribution

It establishes a correspondence between coassociative deformation moduli and minimal 3-manifolds in R^{3,3}, and constructs G2-metrics via affine Toda-like equations.

Findings

Moduli space of coassociative submanifolds relates to minimal 3-manifolds in R^{3,3}.

Constructs G2-metrics from minimal surface equations similar to affine Toda equations.

Explains connection to semi-flat special Lagrangian fibrations and Monge-Ampère equations.

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.