Cyclic Higgs bundles and the affine Toda equations

TL;DR

This paper introduces cyclic Higgs bundles within the Hitchin component and establishes their equivalence to solutions of affine Toda equations, linking geometric structures with integrable systems and harmonic maps.

Contribution

It demonstrates that cyclic Higgs bundles correspond to affine Toda solutions, providing new insights into the geometry of Hitchin components and integrable systems.

Findings

Cyclic Higgs bundles are characterized within the Hitchin component.

A correspondence between cyclic Higgs bundles and affine Toda solutions is established.

The relationship is explained via lifts of harmonic maps.

Abstract

We introduce a class of Higgs bundles called cyclic which lie in the Hitchin component of representations of a compact Riemann surface into the split real form of a simple Lie group. We then prove that such a Higgs bundle is equivalent to a certain class of solutions to the affine Toda equations. We further explain this relationships in terms of lifts of harmonic maps.