G2 geometry and integrable systems

TL;DR

This paper explores the geometry of G2 structures, Higgs bundles, and integrable systems, establishing links between surface representations, affine Toda equations, and special geometric distributions in 5 dimensions.

Contribution

It introduces a novel connection between cyclic Higgs bundles, affine Toda equations, and G2 geometry, expanding understanding of moduli spaces and geometric structures on surfaces.

Findings

Cyclic Higgs bundles correspond to affine Toda equations.

The geometry of 2-plane distributions in 5D relates to split G2 structures.

Local models of coassociative fibrations relate to minimal 3-manifolds in R^{3,3}.

Abstract

We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real simple Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspond to a form of the affine Toda equations. In each case we relate cyclic Higgs bundles to geometric structures on the surface. We elucidate the geometry of generic 2-plane distributions in 5 dimensions, relating it to a parabolic geometry associated to the split real form of $G_2$ and a conformal geometry with holonomy in $G_2$. We prove the distribution is the bundle of maximal isotropics corresponding to the annihilator of a spinor satisfying the twistor-spinor equation. We study the moduli space of coassociative submanifolds of a $G_2$-manifold with an aim towards understanding coassociative fibrations. We consider coassociative fibrations where the fibres are orbits of a $T^4$-action of isomorphisms and prove a local equivalence to minimal 3-manifolds in $R^{3,3}\cong H^2(T^4,\mathbb{R})$ with positive induced metric.