Cyclic Higgs bundles and the affine Toda equations

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 Higgs bundles correspond to a class of solutions to the affine Toda equations. This relationship is further explained in terms of lifts of harmonic maps.