Cluster categories from Fukaya categories

We show that the derived wrapped Fukaya category $D^{\pi }\mathcal{W}(X_Q^{d+1})$, the derived compact Fukaya category $D^{\pi }\mathcal{F}(X_Q^{d+1})$ and the cocore disks $L_Q$ of the plumbing space $X_Q^{d+1}$ form a Calabi–Yau triple. As a consequence, the quotient category $D^{\pi }\mathcal{W}(X_Q^{d+1})\mathrm{/}{D}^{\pi }\mathcal{F}(X_Q^{d+1})$ becomes the cluster category associated to $Q$. One of its properties is a Calabi–Yau structure. Also it is known that this quotient category is quasi-equivalent to the Rabinowitz Fukaya category due to the work of Ganatra–Gao–Venkatesh. We compute the morphism space of $L_Q$ in $D^{\pi }\mathcal{W}(X_Q^{d+1})\mathrm{/}{D}^{\pi }\mathcal{F}(X_Q^{d+1})$ using the Calabi–Yau structure, which is isomorphic to the Rabinowitz Floer cohomology of $L_Q$.