Computation of quantum cohomology from Fukaya categories

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 $\mathcal{A}\subset \mathrm{Fuk}(X)$ split generates a summand $\mathrm{Fuk}(X{)}_e\subset \mathrm{Fuk}(X)$ corresponding to an idempotent $e\in Q{H}^{\bullet }(X)$ if the Mukai pairing of $\mathcal{A}$ is perfect. Moreover we show $H{H}^{\bullet }(\mathcal{A})\cong Q{H}^{\bullet }(X)e$. 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.

22 pages. To appear at International Mathematical Research Notices