Higgs bundles over cell complexes and representations of finitely presented groups

Georgios Daskalopoulos, Chikako Mese and Graeme Wilkin

Vol. 296 (2018), No. 1, 31–55

DOI: 10.2140/pjm.2018.296.31

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 notions 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 $SL\left(r,ℂ\right)$ 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 ${\pi }_1\left(X\right)=\Gamma$. A key role is played by the theory of harmonic maps defined on singular domains.

Keywords

Higgs bundles, harmonic maps, simplicial complexes

Mathematical Subject Classification 2010

Primary: 58E20

Secondary: 53C07, 58D27

Milestones

Received: 3 December 2016

Revised: 8 February 2018

Accepted: 23 February 2018

Published: 1 May 2018