Cluster categories from Fukaya categories

Let [Formula: see text] be a quiver whose underlying graph is a tree and let [Formula: see text] be the result of plumbing copies of [Formula: see text] along the quiver [Formula: see text]. We show that the triple [Formula: see text] is a Calabi–Yau triple, where [Formula: see text] and [Formula: see text] denote the derived wrapped and compact Fukaya categories of [Formula: see text] and [Formula: see text] denotes the direct sum of cocore disks. As a consequence, the Verdier quotient [Formula: see text] (which is equivalent to the derived Rabinowitz Fukaya category) becomes the cluster category associated to [Formula: see text] and in particular, inherits a Calabi–Yau structure. We compute the endomorphism space of [Formula: see text] in this quotient using the Calabi–Yau structure, giving the Rabinowitz Floer cohomology of [Formula: see text].