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Computation of quantum cohomology from Fukaya categories (1712.03924v3)
Published 11 Dec 2017 in math.SG
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 $\mathbb{C} P2$ at four points with a monotone symplectic structure.