#### Volume 21, issue 6 (2017)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Other MSP Journals
Koszul duality patterns in Floer theory

### Tolga Etgü and Yankı Lekili

Geometry & Topology 21 (2017) 3313–3389
##### Abstract

We study symplectic invariants of the open symplectic manifolds ${X}_{\Gamma }$ obtained by plumbing cotangent bundles of 2–spheres according to a plumbing tree $\Gamma$. For any tree $\Gamma$, we calculate (DG) algebra models of the Fukaya category $\mathsc{ℱ}\left({X}_{\Gamma }\right)$ of closed exact Lagrangians in ${X}_{\Gamma }$ and the wrapped Fukaya category $\mathsc{W}\left({X}_{\Gamma }\right)$. When  $\Gamma$ is a Dynkin tree of type ${A}_{n}$ or ${D}_{n}$ (and conjecturally also for ${E}_{6},{E}_{7},{E}_{8}$), we prove that these models for the Fukaya category $\mathsc{ℱ}\left({X}_{\Gamma }\right)$ and $\mathsc{W}\left({X}_{\Gamma }\right)$ are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of ${X}_{\Gamma }$ for $\Gamma ={A}_{n},{D}_{n}$, based on the Legendrian surgery formula of Bourgeois, Ekholm and Eliashberg.

##### Keywords
Koszul duality, Floer theory, Legendrian surgery
Primary: 57R58
Secondary: 16E45