Séminaires : Séminaire d'Algèbre

The (decorated) higher Teichmüller space of a marked surface is a space of local systems valued in a split simple Lie group of a Dynkin type I. There is a corresponding cluster algebra, which gives rise to coordinates on the higher Teichmüller space. We will discuss additive categorifications of these cluster algebras. We first associate a (relative) 3-Calabi--Yau dg category with the surface and Dynkin type I. This dg category arises by gluing, along a perverse (co)sheaf of categories. The fundamental building block is associated with the 3-gon surface (=the basic triangle). The 3-CY category of the basic triangle was introduced and studied in recent work of B. Keller and M. Liu. We then discuss an equivalence between the following three Frobenius exact dg/infinity-categories, categorifying the cluster algebra:

1) The Higgs category associated with the 3-CY category.

2) The cosingularity category of the 3-CY category.

3) The topological Fukaya category of the surface valued in the 2-periodic 1-CY cluster category of type I.

This talk will take place in hybrid mode at the Institut Henri Poincaré.