Higgs bundles over cell complexes and representations of finitely generated groups

TL;DR

This paper extends the Donaldson-Corlette theorem to cell complexes, establishing a homeomorphism between character varieties of finitely presented groups and moduli spaces of Higgs bundles over these complexes, using harmonic map theory on singular domains.

Contribution

It introduces a framework for Higgs bundles over cell complexes and proves a homeomorphism between group character varieties and Higgs moduli spaces in this setting.

Findings

Homeomorphism between $SL(r, ext{C})$ character variety and Higgs bundle moduli space.

Extension of Donaldson-Corlette theorem to cell complexes.

Application of harmonic map theory on singular domains.

Abstract

The purpose of this paper is to extend the Donaldson-Corlette theorem to the case of vector bundles over cell complexes. We define the notion of a vector bundle and a Higgs bundle over a complex, and describe the associated Betti, de Rham and Higgs moduli spaces. The main theorem is that the $SL(r, \mathbb{C})$ character variety of a finitely presented group $\Gamma$ is homeomorphic to the moduli space of rank $r$ Higgs bundles over an admissible complex $X$ with $\pi_1(X) = \Gamma$. A key role is played by the theory of harmonic maps defined on singular domains.