Cluster categories from Fukaya categories

TL;DR

This paper establishes a Calabi-Yau triple structure linking Fukaya categories and cluster categories, and computes morphism spaces using this structure, connecting to Rabinowitz Floer cohomology.

Contribution

It demonstrates that certain Fukaya categories form a Calabi-Yau triple and identifies the resulting quotient as a cluster category, with explicit morphism space calculations.

Findings

The quotient of wrapped and compact Fukaya categories is a cluster category.

The quotient category has a Calabi-Yau structure.

Morphism spaces are computed via Rabinowitz Floer cohomology.

Abstract

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 $L_{Q}$.