Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 010, 32 pages      arXiv:1809.06786      https://doi.org/10.3842/SIGMA.2019.010
Contribution to the Special Issue on Geometry and Physics of Hitchin Systems

Studying Deformations of Fuchsian Representations with Higgs Bundles

Brian Collier
Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Received October 16, 2018, in final form February 02, 2019; Published online February 12, 2019

Abstract
This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular interest and is related to the field of higher Teichmüller theory. Our main tool is the theory of Higgs bundles. We try to develop the general theory of Higgs bundles for real groups and indicate where subtleties arise. However, the main emphasis is placed on concrete examples which are our motivating objects. In particular, we do not prove any of the foundational theorems, rather we state them and show how they can be used to prove interesting statements about components of the character variety. We have also not spent any time developing the tools (harmonic maps) which define the bridge between Higgs bundles and the character variety. For this side of the story we refer the reader to the survey article of Q. Li [arXiv:1809.05747].

Key words: Higgs bundles; character varieties; higher Teichmüller theory.

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