Cluster theory of topological Fukaya categories. Part II: Higher Teichmüller theory

Abstract: We construct relative $3$-Calabi--Yau categories related with higher Teichm\"uller theory. We further study their corresponding cosingularity categories and the additive categorification of the corresponding cluster algebras. The input for our constructions is a marked surface with boundary and a Dynkin quiver $I$. In the case of the triangle, these categories have been described in recent work of Keller--Liu. For general surfaces, the categories are constructed via gluing along a perverse schober, categorifying the amalgamation of cluster varieties. The case $I=A_1$ was subject of the prequel paper. We show that the cosingularity category is equivalent to the corresponding Higgs category and to the topological Fukaya category of the marked surface valued in the $1$-Calabi--Yau cluster category of type $I$.