The monotone wrapped Fukaya category and the open-closed string map

Abstract

Abstract. We build the wrapped Fukaya category W(E) for any monotone symplectic manifold E, convex at infinity. We define the open-closed and closed open-string maps, OC : HH * (W(E)) → SH * (E) and CO : SH * (E) → HH * (W(E)). We study their algebraic properties and prove that the string maps are compatible with the c 1 (T E)-eigenvalue splitting of W(E). We extend Abouzaid's generation criterion from the exact to the monotone setting. We construct an acceleration functor AF : F (E) → W(E) from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map c * : QH * (E) → SH * (E). We define the SH * (E)-module structure on the Hochschild (co)homology of W(E) which is compatible with the string maps (this was proved independently for exact convex symplectic manifolds by Ganatra). The module and unital algebra structures, and the generation criterion, also hold for the compact Fukaya category F (E), and also hold for closed monotone symplectic manifolds.As an application, we show that the wrapped category of O(−k) → CP m is proper (cohomologically finite) for 1 ≤ k ≤ m. For any monotone negative line bundle E over a closed monotone toric manifold B, we show that SH * (E) = 0, W(E) is non-trivial, and E contains a non-displaceable monotone Lagrangian torus L on which OC is non-zero.

Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

“…There is also a reduced symplectic cohomology group SH * (E) red (see e.g. [3]), which fits into a long exact sequence As should already be obvious from the discussion so far, we do not require that [ω E ] ∈ H 2 (E; R) is trivial, or even that it vanishes on spherical homology classes (such assumptions occur in most, but not all, of the literature concerning symplectic cohomology; among the exceptions are [33,18,17]). Eventually, we will impose a vanishing condition, but one which is weaker and not expressed in terms of classical topology:…”

“…The previous discussion of connections on A ∞ -algebras carries over without any difficulties to the categorical case. The Hochschild (co)homology of W(E) is related to symplectic cohomology by open-closed string maps [1,10,33] SH * (E) −→ HH * (W(E), W(E)),…”

Twin Lagrangian fibrations in mirror symmetry

“…Denote by C the full subcategory formed by T 1 , · · ·, T p , from the above we see that the open-closed map OC restricted on C hits an invertible element of QH * Bl K (C 2 ) λ . By the generation criterion obtained in [38], the claim follows.…”

Quantum Cohomology and Closed-String Mirror Symmetry for Toric Varieties

“…If X is non-compact then, as in Remark 1.7, we should instead work with the wrapped Fukaya category W(X). We should also replace quantum cohomology with symplectic cohomology (the 'wrapped' equivalent) in the closed-open map, which yields the informal conjectural isomorphism There is expected to be an acceleration map QH * (X) → SH * (X) (and a corresponding functor F(X) → W(X); see [36]) and Corollary 3 identifies its domain as the right-hand side of (4) but with the 'numerator' replaced by its subring Λ H ∆ .…”