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Koszul duality patterns in Floer theory

Tolga Etgü and Yankı Lekili

Geometry & Topology 21 (2017) 3313–3389

We study symplectic invariants of the open symplectic manifolds XΓ 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 (XΓ) of closed exact Lagrangians in XΓ and the wrapped Fukaya category W(XΓ). When  Γ is a Dynkin tree of type An or Dn (and conjecturally also for E6,E7,E8), we prove that these models for the Fukaya category (XΓ) and W(XΓ) are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of XΓ for Γ = An,Dn, based on the Legendrian surgery formula of Bourgeois, Ekholm and Eliashberg.

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Koszul duality, Floer theory, Legendrian surgery
Mathematical Subject Classification 2010
Primary: 57R58
Secondary: 16E45
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
Tolga Etgü
Department of Mathematics
Koç University
34450 Istanbul
Yankı Lekili
Department of Mathematics
King’s College London
United Kingdom