Studying Deformations of Fuchsian Representations with Higgs Bundles

Symmetry Integrability and Geometry-Methods and Applications

article

Brian Collier¹ 

[1] Department of Mathematics, University of Maryland, College Park

关键词: hyperelliptic curve;    soliton solution;    KP hierarchy;    Sato Grassmannian;   

DOI  :  10.3842/SIGMA.2019.010

来源: National Academy of Science of Ukraine

【 摘 要 】

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].

【 授权许可】

Unknown