Koszul duality patterns in Floer theory

Etgü, Tolga¹ ; Lekili, Yankı²

¹ Department of Mathematics, Koç University, 34450 Istanbul, Turkey
² Department of Mathematics, King’s College London, London, WC2R 2LS, United Kingdom

Geometry & topology, Tome 21 (2017) no. 6, pp. 3313-3389.

Résumé

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

Détail

DOI : 10.2140/gt.2017.21.3313

Classification : 57R58, 16E45
Keywords: Koszul duality, Floer theory, Legendrian surgery

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