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 $D^\pi\mathcal{W}(X_{Q}^{d+1})/D^\pi\mathcal{F}(X_{Q}^{d+1})$ 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 $D^\pi\mathcal{W}(X_{Q}^{d+1})/D^\pi\mathcal{F}(X_{Q}^{d+1})$ using the Calabi--Yau structure, which is isomorphic to the Rabinowitz Floer cohomology of $L_{Q}$.Comment: 19 pages, Comments are welcom

Categorical entropies on symplectic manifolds

In this paper, being motivated by symplectic topology, we study categorical entropy. Specifically, we prove inequalities between categorical entropies of functors on a category and its localization. We apply the inequalities to symplectic topology to prove equalities between categorical entropies on wrapped, partially wrapped, and compact Fukaya categories if the functors are induced by the same compactly supported symplectic automorphisms. We also provide a practical way to compute the categorical entropy of symplectic automorphisms by using Lagrangian Floer theory if their domains satisfy a type of Floer-theoretical duality. Our main examples of symplectic manifolds satisfying the duality conditions are the plumbings of cotangent bundles of sphere along a tree. Moreover, for symplectic automorphisms of Penner type, we prove that our computation of categorical entropy becomes a computation by simple linear algebra.Comment: 38 page