G2 geometry and integrable systems

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.