Book contents
- Frontmatter
- Contents
- Preface
- List of contributors
- 1 Introduction to algebraic stacks
- 2 BPS states and the P = W conjecture
- 3 Representations of surface groups and Higgs bundles
- 4 Introduction to stability conditions
- 5 An introduction to d-manifolds and derived differential geometry
- 6 13/2 ways of counting curves
- References
3 - Representations of surface groups and Higgs bundles
Published online by Cambridge University Press: 05 April 2014
Book contents
- Frontmatter
- Contents
- Preface
- List of contributors
- 1 Introduction to algebraic stacks
- 2 BPS states and the P = W conjecture
- 3 Representations of surface groups and Higgs bundles
- 4 Introduction to stability conditions
- 5 An introduction to d-manifolds and derived differential geometry
- 6 13/2 ways of counting curves
- References
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]).
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- Information
- Moduli Spaces , pp. 151 - 178Publisher: Cambridge University PressPrint publication year: 2014
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