Nonabelianization of Higgs bundles

TL;DR

This paper explores the structure of Higgs bundle moduli spaces for certain non-abelian Lie groups, revealing that their fibers are moduli spaces of rank 2 bundles on spectral curves, not abelian varieties.

Contribution

It demonstrates that fibers of the integrable systems for these Higgs bundles are moduli spaces of rank 2 bundles, providing a nonabelian generalization of classical abelian cases.

Findings

Fibers are moduli spaces of rank 2 bundles on spectral curves.

These fibers satisfy natural stability conditions.

The structure differs from classical abelian varieties in integrable systems.

Abstract

The character varieties of representations of a surface group into the Lie groups SL(m,H), SO(2m,H) and Sp(m,m) have a holomorphic description in terms of the moduli space of Higgs bundles. We show that the fibres of the integrable system in these cases are not abelian varieties, but are instead moduli spaces of rank 2 bundles on a spectral curve, satisfying natural stability conditions.