Koszul duality patterns in Floer theory

TL;DR

This paper investigates symplectic invariants of certain open symplectic manifolds constructed via plumbing, providing algebraic models of their Fukaya categories and revealing Koszul duality relationships, with explicit computations for specific cases.

Contribution

It computes algebraic models of Fukaya categories for plumbing manifolds and establishes Koszul duality relations for Dynkin trees, extending understanding of symplectic invariants.

Findings

Models of Fukaya categories are explicitly calculated for plumbing manifolds.

Koszul duality relates these models for Dynkin type A and D trees.

Explicit symplectic cohomology computations are provided for certain cases.

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 $\mathcal{F}(X_\Gamma)$ of closed exact Lagrangians in $X_\Gamma$ and the wrapped Fukaya category $\mathcal{W}(X_\Gamma)$. 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 $\mathcal{F}(X_\Gamma)$ and $\mathcal{W}(X_\Gamma)$ 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.