A geometric realization of the $m$-cluster category of type $\tilde{A}$

TL;DR

This paper provides a geometric model for a subcategory of the $m$-cluster category of type $ ilde{A}$ using $(m+2)$-angulations of an annulus, linking combinatorics with algebraic structures.

Contribution

It introduces a geometric realization of the $m$-cluster category of type $ ilde{A}$ and establishes a bijection with mutation classes of coloured quivers.

Findings

Geometric realization via $(m+2)$-angulations of an annulus.

Bijection between angulations and mutation classes of coloured quivers.

Connection between combinatorial models and algebraic categories.

Abstract

We give a geometric realization of a subcategory of the $m$-cluster category $\mathcal{C}^m$ of type $\widetilde{A}_{p,q}$, by using $(m+2)$-angulations of an annulus with $p+q$ marked points. We also give a bijection between an equivalence class of $(m+2)$-angulations and the mutation class of coloured quivers of type $\widetilde{A}_{p,q}$.