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

By Alexander F. Ritter, I. Smith. 2012

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We build the wrapped Fukaya category $${\mathcal {W}}(E)$$W(E) for any monotone symplectic manifold E, convex at infinity. We define the open-closed and closed-open string maps, $${\mathrm {OC}}:{\mathrm {HH}}_*({\mathcal {W}}(E))\rightarrow { SH }^*(E)$$OC:HH∗(W(E))→SH∗(E) and $${\mathrm {CO}}: { SH }^*(E)\rightarrow {\mathrm {HH}}^*({\mathcal {W}}(E))$$CO:SH∗(E)→HH∗(W(E)). We study their algebraic properties and prove that the string maps are compatible with the $$c_1({ TE })$$c1(TE)-eigenvalue splitting of $${\mathcal {W}}(E)$$W(E). We extend Abouzaid’s generation criterion from the exact to the monotone setting. We construct an acceleration functor $${\mathcal {AF}}: {\mathcal {F}}(E)\rightarrow {\mathcal {W}}(E)$$AF:F(E)→W(E) from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map $$c^*:{ QH }^*(E)\rightarrow { SH }^*(E)$$c∗:QH∗(E)→SH∗(E). We define the $${ SH }^*(E)$$SH∗(E)-module structure on the Hochschild (co)homology of $${\mathcal {W}}(E)$$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 $${\mathcal {F}}(E)$$F(E), and also hold for closed monotone symplectic manifolds. As an application, we show that the wrapped category of $${\mathcal {O}}(-k) \rightarrow \mathbb {C}\mathbb {P}^m$$O(-k)→CPm is proper (cohomologically finite) for $$1\le k \le m$$1≤k≤m. For any monotone negative line bundle E over a closed monotone toric manifold B, we show that $${ SH }^*(E)\ne 0$$SH∗(E)≠0, $${\mathcal {W}}(E)$$W(E) is non-trivial and E contains a non-displaceable monotone Lagrangian torus $${\mathcal {L}}$$L on which $${\mathrm {OC}}$$OC is non-zero.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1e567328397b047deb28ebae78f7f15245953dd3

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Citations in the corpus

Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “As should already be obvious from the discussion so far, we do not require that [ωE ] ∈ H(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]).”
Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors (2017, methodology): “ersion of the wrapped Fukaya category for monotone Lagrangians, as we proceed to explain. Thewrapped Fukaya category as well as symplectic cohomology were deﬁned in the monotone symplectic setting in [34] using coeﬃcients in the Novikov ﬁeld. When this construction is applied to exact Lagrangians in”