Maximal surface group representations in isometry groups of classical   Hermitian symmetric spaces
  Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant. 
Steven B. Bradlow (University of Illinois), Oscar Garcia-Prada (CSIC,   Madrid), Peter B. Gothen (Universidade do Porto)