Apparaît dans la collection : 2025 - T2 - Higher rank geometric structures

This minicourse will focus on Slodowy slices in the moduli space of $G$-Higgs bundles on a closed Riemann surface, where $G$ is a complex semisimple Lie group. Under the nonabelian Hodge correspondence, these spaces describe specific ways of deforming Fuchsian representations of a closed surface group in the G-character variety via a group homomorphism from $SL(2,C)$ into $G$. For the principal $SL(2,C)$ in $G$, the Slodowy slice is the well known and extensively studied Hitchin section, and the associated representations (known as Hitchin representations) define "higher rank Teichmuller components" of the character variety for the split real form of $G$. That is, components consisting entirely of Anosov representations. For other special groups homomorphisms of $SL(2,C)$ into $G$, known as magical sl2s, the Slodowy slice parameterizes the higher rank Teichmuller spaces associated to positive representations, in the sense of Guichard-Labourie-Wienhard. Conjecturally, this describes all higher rank Teichmuller spaces for closed surfaces. I will spend the first 2 talks setting up these objects, discussing their relations to the component count of the character variety of representations of a closed surface group into real semisimple Lie groups.

In general, the Slodowy slice is closed in the moduli space but not open. In the final talk, I will discuss a conjectural picture where the Slodowy slice is a generalization of the Fuchsian locus in Quasifuchsian space. One aim of this conjectural picture is to prove that, under the nonabelian Hodge correspondence, the representations associated to Higgs bundles in general Slodowy slices are all Anosov. The last lecture will be given by Samuel Bronstein and Junming Zhang. Each will explain their recent works using Higgs bundle techniques to prove certain Higgs bundles in Slodowy slices have Anosov holonomy.