Higgs bundles, the Toledo invariant and the Cayley correspondence

TL;DR

This paper introduces a new approach to defining and bounding the Toledo invariant for G-Higgs bundles on Riemann surfaces, establishing rigidity and Cayley correspondence results for maximal invariants, and providing a unified proof of the Milnor-Wood inequality.

Contribution

It defines the Toledo invariant for G-Higgs bundles, proves a Milnor-Wood type bound, and establishes rigidity and Cayley correspondence results in a unified framework.

Findings

Proved a Milnor-Wood type inequality for the Toledo invariant.

Established rigidity results for maximal Toledo invariant.

Provided a new proof of the Milnor-Wood inequality using Higgs bundle theory.

Abstract

We define the Toledo invariant of a G-Higgs bundle on a Riemann surface, where G is a real semisimple group of Hermitian type, and we prove a Milnor-Wood type bound for this invariant when the bundle is semistable. We prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley correspondence when the symmetric space defined by G is of tube type. This gives a new proof of the Milnor-Wood inequality of Burger-Iozzi-Wienhard for representations of the fundamental group of a Riemann surface into G. Compared to previous results using Higgs bundles, it uses general theory and avoids any case by case study.