# The monotone wrapped Fukaya category and the open-closed string map

Ritter, Alexander F ; Smith, Ivan (2016-08-09)

Article

We build the wrapped Fukaya category $\textit{W}$($\textit{E}$)for any monotone symplectic manifold $\textit{E}$, convex at infinity. We define the open-closed and closed-open string maps, OC : HH$_{*}$($\textit{W}$($\textit{E}$)) → $\textit{SH}^{*}$($\textit{E}$) and CO : $\textit{SH}^{*}$($\textit{E}$) → HH$^{*}$($\textit{W}$($\textit{E}$)). We study their algebraic properties and prove that the string maps are compatible with the $\textit{c}_1$($\textit{TE}$)-eigenvalue splitting of $\textit{W}$($\textit{E}$). We extend Abouzaid’s generation criterion from the exact to the monotone setting. We construct an acceleration functor $\textit{AF}$ : $\textit{F}$($\textit{E}$) → $\textit{W}$($\textit{E}$) from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map $\textit{c}^{*}$ : $\textit{QH}^{*}$($\textit{E}$) → $\textit{SH}^{*}$($\textit{E}$). We define the $\textit{SH}^{*}$($\textit{E}$)-module structure on the Hochschild (co)homology of $\textit{W}$($\textit{E}$) which is compatible with the string maps (this was proved independently for exact convex symplectic manifolds by Ganatra). The module and unital algebra structures, and the generation criterion, also hold for the compact Fukaya category $\textit{F}$($\textit{E}$), and also hold for closed monotone symplectic manifolds. As an application, we show that the wrapped category of $\textit{O}$(−$\textit{k}$) → $\Bbb {CP}^m$ is proper (cohomologically finite) for 1 ≤ $\textit{k}$ ≤ $\textit{m}$. For any monotone negative line bundle $\textit{E}$ over a closed monotone toric manifold $\textit{B}$, we show that $\textit{SH}^{*}$($\textit{E}$) $\neq$ 0, $\textit{W}$($\textit{E}$) is non-trivial and $\textit{E}$ contains a non-displaceable monotone Lagrangian torus $\textit{L}$ on which OC is non-zero.