Higgs bundles over cell complexes and representations of finitely generated groups

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.

20 pages

References in corpus (1)

• On the fundamental groups of normal varieties

Cited by in corpus (1)

• Principal bundles on 2-dimensional CW-complexes with disconnected structure group