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

SIGMA 15 (2019), 010, 32 pages      arXiv:1809.06786
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

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.

pdf (564 kb)   tex (36 kb)


  1. Aparicio-Arroyo M., The geometry of ${\rm SO}(p,q)$ Higgs bundles, Ph.D. Thesis, Facultad de Ciencias de la Universidad de Salamanca, 2009.
  2. Aparicio-Arroyo M., Bradlow S., Collier B., García-Prada O., Gothen P.B., Oliveira A., ${\rm SO}(p,q)$-Higgs bundles and higher Teichmüller components, arXiv:1802.08093.
  3. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
  4. Baraglia D., Schaposnik L.P., Cayley and Langlands type correspondences for orthogonal Higgs bundles, arXiv:1708.08828.
  5. Biswas I., Ramanan S., An infinitesimal study of the moduli of Hitchin pairs, J. London Math. Soc. 49 (1994), 219-231.
  6. Bradlow S.B., García-Prada O., Gothen P.B., Homotopy groups of moduli spaces of representations, Topology 47 (2008), 203-224, arXiv:math.AG/0506444.
  7. Burger M., Iozzi A., Wienhard A., Surface group representations with maximal Toledo invariant, Ann. of Math. 172 (2010), 517-566, arXiv:math.DG/0605656.
  8. Burger M., Iozzi A., Wienhard A., Higher Teichmüller spaces: from ${\rm SL}(2,{\mathbb R})$ to other Lie groups, in Handbook of Teichmüller Theory, Vol. IV, IRMA Lect. Math. Theor. Phys., Vol. 19, Eur. Math. Soc., Z\"urich, 2014, 539-618, arXiv:1004.2894.
  9. Collier B., ${\rm SO}(n,n+1)$-surface group representations and their Higgs bundles, arXiv:1710.01287.
  10. Collier B., Tholozan N., Toulisse J., The geometry of maximal representations of surface groups into $\mathrm{SO}(2,n)$, arXiv:1702.08799.
  11. García-Prada O., Gothen P., Mundet i Riera I., The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations, arXiv:0909.4487.
  12. García-Prada O., Oliveira A., Connectedness of Higgs bundle moduli for complex reductive Lie groups, Asian J. Math. 21 (2017), 791-810, arXiv:1408.4778.
  13. Goldman W.M., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200-225.
  14. Goldman W.M., Topological components of spaces of representations, Invent. Math. 93 (1988), 557-607.
  15. Goldman W.M., Millson J.J., Local rigidity of discrete groups acting on complex hyperbolic space, Invent. Math. 88 (1987), 495-520.
  16. Guichard O., Wienhard A., Positivity and higher Teichmüller theory, arXiv:1802.02833.
  17. Guichard O., Wienhard A., Anosov representations: domains of discontinuity and applications, Invent. Math. 190 (2012), 357-438, arXiv:1108.0733.
  18. Hausel T., Thaddeus M., Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003), 197-229, arXiv:math.AG/0205236.
  19. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  20. Hitchin N.J., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  21. Hitchin N.J., Lie groups and Teichmüller space, Topology 31 (1992), 449-473.
  22. Kostant B., The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973-1032.
  23. Labourie F., Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006), 51-114, arXiv:math.DG/0401230.
  24. Li J., The space of surface group representations, Manuscripta Math. 78 (1993), 223-243.
  25. Li Q., An introduction to Higgs bundles via harmonic maps, arXiv:1809.05747.
  26. Macdonald I.G., Symmetric products of an algebraic curve, Topology 1 (1962), 319-343.
  27. Milnor J., On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958), 215-223.
  28. Narasimhan M.S., Seshadri C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-567.
  29. Nitsure N., Moduli space of semistable pairs on a curve, Proc. London Math. Soc. 62 (1991), 275-300.
  30. Ramanathan A., Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129-152.
  31. Simpson C.T., Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867-918.
  32. Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5-95.
  33. Toledo D., Representations of surface groups in complex hyperbolic space, J. Differential Geom. 29 (1989), 125-133.
  34. Wienhard A., An invitation to higher Teichmüller theory, arXiv:1803.06870.

Previous article  Next article   Contents of Volume 15 (2019)