Koszul duality patterns in Floer theory

We study symplectic invariants of the open symplectic manifolds [math] obtained by plumbing cotangent bundles of 2–spheres according to a plumbing tree [math] . For any tree [math] , we calculate (DG) algebra models of the Fukaya category [math] of closed exact Lagrangians in [math] and the wrapped Fukaya category [math] . When [math] is a Dynkin tree of type [math] or [math] (and conjecturally also for [math] ), we prove that these models for the Fukaya category [math] and [math] are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of [math] for [math] , based on the Legendrian surgery formula of Bourgeois, Ekholm and Eliashberg.