A geometric description of $m$-cluster categories

Abstract

We show that the $m$-cluster category of type $A_{n-1}$ is equivalent to a certain geometrically-defined category of diagonals of a regular $nm+2$-gon. This generalises a result of Caldero, Chapoton and Schiffler for $m=1$. The approach uses the theory of translation quivers and their corresponding mesh categories. We also introduce the notion of the $m$th power of a translation quiver and show how it can be used to realise the $m$-cluster category in terms of the cluster category.