Uniformization in higher-rank Teichmüller spaces and Fock bundles
ABSTRACT
Higher-rank Teichmüller theory pertains to the study of special connected components of character varieties sharing analogous properties to the classical Teichmüller space. By fixing a complex structure on the underlying topological surface, one introduces powerful holomorphic techniques via Higgs bundles, the latter corresponding to fundamental group representations through the non-abelian Hodge correspondence. Yet, a rather adverse aspect of this correspondence is that it fails to transfer the action of the mapping class group on character varieties to the moduli space of Higgs bundles. We will introduce a similar class of augmented bundles over a topological surface that we call Fock bundles which does not require fixing a priori any complex structure on the underlying surface. We will examine the relationship of these objects with representations in the so-called Hitchin components. This is joint work with Charles Reid (Yale) and Alexander Thomas (Lyon).
