Bounded cohomology, Higgs bundles, and Milnor-Wood inequalities

TL;DR

This paper explores how the generalized Milnor-Wood inequality for certain group representations relates to Higgs bundles via the non-abelian Hodge correspondence, clarifying their interconnections.

Contribution

It establishes a clear translation between the representation theoretic Milnor-Wood inequality and its Higgs bundle counterparts, unifying different versions in the literature.

Findings

Demonstrates the translation of inequalities through non-abelian Hodge correspondence

Clarifies the relation between representation theory and Higgs bundle inequalities

Unifies various Milnor-Wood inequalities in the literature

Abstract

We explain how the generalized Milnor-Wood inequality for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates, under the non-abelian Hodge correspondence, into various kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the relation between the representation theoretic generalized Milnor-Wood inequality and the various different versions of Milnor-Wood inequalities for Higgs bundles that are known in the literature.