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 $\mathcal{F}(X_{\Gamma })$ 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.

72 pages, 20 figures/tables. Minor corrections and improvements. To appear in Geometry & Topology

References in corpus (5)

• Functors and Computations in Floer homology with Applications Part II
• Symplectic cohomology and duality for the wrapped Fukaya category
• Duality between Lagrangian and Legendrian invariants
• Nontriviality results for the characteristic algebra of a DGA
• Fukaya A_\infty-structures associated to Lefschetz fibrations. II 1/2

Cited by in corpus (20)

• Duality between Lagrangian and Legendrian invariants
• Microlocal Morse theory of wrapped Fukaya categories
• Symplectic Homology of complements of smooth divisors
• Pre-Calabi-Yau structures and moduli of representations
• Multiplicative preprojective algebras are 2-Calabi-Yau
• Generating the Fukaya categories of Hamiltonian G-manifolds
• Symplectic fillings of asymptotically dynamically convex manifolds II--$k$-dilations
• Homological mirror symmetry for Milnor fibers of simple singularities
• Periodic trivial extension algebras and fractionally Calabi-Yau algebras
• Koszul duality via suspending Lefschetz fibrations
• Nonfillable Legendrian knots in the 3-sphere
• Multiplicative preprojective algebras of Dynkin quivers
• Exact Calabi-Yau categories and odd-dimensional Lagrangian spheres
• Homological mirror symmetry for hypertoric varieties II (with an Appendix written jointly with Laurent Côté and Justin Hilburn)
• Algebraic properties of zigzag algebras
• A computation of the ring structure in wrapped Floer homology
• $A_{\infty }$-deformations of zigzag algebras via Ginzburg dg algebras
• Nonexistence of exact Lagrangian tori in affine conic bundles over ${\mathbb{C}}^n$
• Maurer-Cartan deformation of Lagrangians
• Fukaya categories in Koszul duality theory