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3 - Representations of surface groups and Higgs bundles

Published online by Cambridge University Press:  05 April 2014

By
Peter B. Gothen
Edited by
Leticia Brambila-Paz ,
Peter Newstead ,
Richard P. Thomas  and
Oscar García-Prada
Peter B. Gothen
Affiliation:
Universidade de Porto
Leticia Brambila-Paz
Affiliation:
Centro de Investigación en Matemáticas A.C. (CIMAT), Mexico
Peter Newstead
Affiliation:
University of Liverpool
Richard P. Thomas
Affiliation:
Imperial College of Science, Technology and Medicine, London
Oscar García-Prada
Affiliation:
Consejo Superior de Investigaciones Cientificas, Madrid
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Summary

Introduction

In this chapter we give an introduction to Higgs bundles and their application to the study of surface group representations. This is based on two fundamental theorems. The first is the theorem of Corlette and Donaldson on the existence of harmonic metrics in flat bundles, which we treat in Lecture 1 (Section 3.2), after explaining some preliminaries on surface group representations, character varieties and flat bundles. The second is the Hitchin—Kobayashi correspondence for Higgs bundles, which goes back to the work of Hitchin and Simpson; this is the main topic of Lecture 2 (Section 3.3). Together, these two results allow the character variety for representations of the fundamental group of a Riemann surface in a Lie group G to be identified with a moduli space of holomorphic objects, known as G-Higgs bundles. Finally, in Lecture 3 (Section 3.4), we show how the ℂ*-action on the moduli space G-Higgs bundles can be used to study its topological properties, thus giving information about the corresponding character variety.

Owing to lack of time and expertise, we do not treat many other important aspects of the theory of surface group representations, such as the approach using bounded cohomology (e.g. [9, 10]), higher Teichmüller theory (e.g. [17]), or ideas related to geometric structures on surfaces (e.g. [28]). We also do not touch on the relation of Higgs bundle moduli with mirror symmetry and the Geometric Langlands Programme (e.g. [33], [39]).

Type
Chapter
Information
Moduli Spaces , pp. 151 - 178
Publisher: Cambridge University Press
Print publication year: 2014

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