A geometric model for cluster categories of type Dn

We give a geometric realization of cluster categories of type Dn using a polygon with n vertices and one puncture is its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices. 0 Introduction Cluster categories were introduced in [BMRRT] and, independently, in [CCS1] for type An, as a means for better understanding of the cluster algebras of Fomin and Zelevinsky [FZ1, FZ2]. Since then cluster categories have been the subject of many investigations, see, for instance, [ABST1, ABST2, BMR1, BMR2, BMRT, CC, CCS2, CK1, CK2, K, KZ, Z1]. In the approach of [BMRRT], the cluster category CA is defined as the quotient DA/F of the derived category DA of a hereditary algebra A by the endofunctor F = τ DbA [1], where τDbA is the Auslander-Reiten translation and [1] is the shift. On the other hand, in the approach of [CCS1], which is only valid in type An, the cluster category is realized by an ad-hoc method as a category of diagonals of a regular polygon with n+ 3 vertices. The morphisms between diagonals are constructed geometrically using so called elementary moves and mesh relations. In that realization, clusters are in one-to-one correspondence with triangulations of the polygon and mutations are given by flips of diagonals in the triangulation. Recently, Baur and Marsh [BM] have generalized this model to m-cluster categories of type An. In this paper, we give a geometric realization of the cluster categories of type Dn in the spirit of [CCS1]. The polygon with (n + 3) vertices has to be replaced by a polygon with n vertices and one puncture in the center, and instead of looking at diagonals, which are straight lines between two vertices, one has to consider homotopy classes of paths between two vertices, which we will call edges. This punctured polygon model has appeared recently in the work of Fomin, Shapiro and Thurston [FST] on the relation between cluster algebras and triangulated surfaces. Let us point out that they work in a vastly more general context and the punctured polygon is only one example of their theory. We define the cluster category by an ad-hoc method as the category