One way to realize Teichmüller space of a closed oriented hyperbolic surface Σ is as the space of discrete and faithful representations π_1(Σ) -> PSL(2,R). These constitute a connected component of the character variety of these representations. Higher rank Teichmüller space is a generalization for higher rank Lie groups. More precisely a higher rank Teichmüller space for a group G is a connected component of the G-character variety of representations π_1(Σ) -> G entirely consisting of discrete and faithful representations.

There are however two important differences comparing to the PSL(2,R)-case. Firstly, such spaces do not exist for any group G, even though their existence has long been established whenever G is a split form (the Hitchin components) or Hermitian (maximal components). Secondly, even if they exist for G, not every discrete and faithful representation in G lies in a higher rank Teichmüller space. It was conjectured by Guichard—Labourie—Wienhard that the representations comprising such spaces are positive representations — a special class of discrete and faithful representations — which only exist when G admits a certain structure called positivity.

Non-abelian Hodge theory yields corresponding connected components of the moduli space M(G) of G-Higgs bundles over an associated compact Riemann surface. We will introduce the notion of magical sl_2-triple and show that a group admits a positive structure if and only if it arises from such a magical triple. Then we we will use these magical triples and Higgs bundles to detect and parameterize components of M(G) which should be higher rank Teichmüller spaces (this was subsequently confirmed by Guichard—Labourie—Wienhard using our work). This is joint work with S. Bradlow, B. Collier, O. García-Prada and P. Gothen.