# A geometric realization of the $m$-cluster categories of type $\tilde{D_n}$

## TL;DR

This paper provides a geometric model for the $m$-cluster categories of type $ ilde{D_n}$ using polygons with inner polygons, linking algebraic mutations to geometric flips.

## Contribution

It introduces a geometric realization of $m$-cluster categories of type $ ilde{D_n}$ via polygonal arcs and demonstrates the compatibility of mutations with geometric flips.

## Key findings

- Geometric realization of $m$-cluster categories using polygons with inner polygons.
- Mutation of quivers corresponds to flips in the polygonal model.
- Explicit example for type $ ilde{D_7}$ illustrates the theory.

## Abstract

We show that a subcategory of the $m$-cluster category of type $\tilde{D_n}$ is isomorphic to a category consisting of arcs in an $(n-2)m$-gon with two central $(m-1)$-gons inside of it. We show that the mutation of colored quivers and $m$-cluster-tilting objects is compatible with the flip of an $(m+2)$-angulation. In the final part of this paper, we detail an example of a quiver of type $\tilde{D_7}$.