Computation of quantum cohomology from Fukaya categories

TL;DR

This paper links the structure of Fukaya categories to quantum cohomology, providing a method to compute quantum cohomology from Fukaya categories and applying it to a specific symplectic manifold.

Contribution

It establishes a connection between split-generating subcategories of Fukaya categories and quantum cohomology, enabling explicit computations for certain blow-ups.

Findings

Fukaya subcategory split generates a summand of Fukaya category under perfect pairing.

Isomorphism between Hochschild cohomology of subcategory and quantum cohomology element.

Explicit computation of quantum cohomology for a blow-up of  P^2 at four points.

Abstract

Assume the existence of a Fukaya category $\mathrm{Fuk}(X)$ of a compact symplectic manifold $X$ with some expected properties. In this paper, we show $\mathscr{A} \subset \mathrm{Fuk}(X)$ split generates a summand $\mathrm{Fuk}(X)_e \subset \mathrm{Fuk}(X)$ corresponding to an idempotent $e \in QH^\bullet(X)$ if the Mukai pairing of $\mathscr{A}$ is perfect. Moreover we show $HH^\bullet(\mathscr{A}) \cong QH^\bullet(X) e$. As an application we compute the quantum cohomology and the Fukaya category of a blow-up of $\mathbb{C} P^2$ at four points with a monotone symplectic structure.