Koszul duality patterns in Floer theory

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 F (X-Gamma) of closed exact Lagrangians in X-Gamma and the wrapped Fukaya category W (X-Gamma). 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 F (X-Gamma) and W (X-Gamma) 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.