Koszul duality patterns in Floer theory

Etgü, Tolga; Lekili, Yankı

doi:10.2140/gt.2017.21.3313

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Koszul duality patterns in Floer theory

Tolga Etgü and Yankı Lekili

Geometry & Topology 21 (2017) 3313–3389

DOI: 10.2140/gt.2017.21.3313

Abstract

We study symplectic invariants of the open symplectic manifolds $X_{Γ}$ obtained by plumbing cotangent bundles of 2–spheres according to a plumbing tree $Γ$ . For any tree $Γ$ , we calculate (DG) algebra models of the Fukaya category $ℱ (X_{Γ})$ of closed exact Lagrangians in $X_{Γ}$ and the wrapped Fukaya category $W (X_{Γ})$ . When $Γ$ 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 $ℱ (X_{Γ})$ and $W (X_{Γ})$ are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of $X_{Γ}$ for $Γ = A_{n}, D_{n}$ , based on the Legendrian surgery formula of Bourgeois, Ekholm and Eliashberg.

Keywords

Koszul duality, Floer theory, Legendrian surgery

Mathematical Subject Classification 2010

Primary: 57R58

Secondary: 16E45

References

Publication

Received: 10 April 2015

Revised: 13 September 2016

Accepted: 12 November 2016

Published: 31 August 2017

Proposed: Yasha Eliashberg

Seconded: Ciprian Manolescu, András I. Stipsicz

Authors

Tolga Etgü
Department of Mathematics Koç University 34450 Istanbul Turkey

Yankı Lekili
Department of Mathematics King’s College London London WC2R 2LS United Kingdom

Volume 21, issue 6 (2017)