Abstract
We build the wrapped Fukaya category \({\mathcal {W}}(E)\) for any monotone symplectic manifold E, convex at infinity. We define the open-closed and closed-open string maps, \({\mathrm {OC}}:{\mathrm {HH}}_*({\mathcal {W}}(E))\rightarrow { SH }^*(E)\) and \({\mathrm {CO}}: { SH }^*(E)\rightarrow {\mathrm {HH}}^*({\mathcal {W}}(E))\). We study their algebraic properties and prove that the string maps are compatible with the \(c_1({ TE })\)-eigenvalue splitting of \({\mathcal {W}}(E)\). We extend Abouzaid’s generation criterion from the exact to the monotone setting. We construct an acceleration functor \({\mathcal {AF}}: {\mathcal {F}}(E)\rightarrow {\mathcal {W}}(E)\) from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map \(c^*:{ QH }^*(E)\rightarrow { SH }^*(E)\). We define the \({ SH }^*(E)\)-module structure on the Hochschild (co)homology of \({\mathcal {W}}(E)\) 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 \({\mathcal {F}}(E)\), and also hold for closed monotone symplectic manifolds. As an application, we show that the wrapped category of \({\mathcal {O}}(-k) \rightarrow \mathbb {C}\mathbb {P}^m\) is proper (cohomologically finite) for \(1\le k \le m\). For any monotone negative line bundle E over a closed monotone toric manifold B, we show that \({ SH }^*(E)\ne 0\), \({\mathcal {W}}(E)\) is non-trivial and E contains a non-displaceable monotone Lagrangian torus \({\mathcal {L}}\) on which \({\mathrm {OC}}\) is non-zero.
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Notes
We are implicitly using canonical identifications of type \({ CF }^*(\varphi _{H_j}(L),L)\equiv { CF }^*(\varphi _{H+H_j}(L),\varphi _{H}(L))\).
Meaning the underlying cohomology level functor is an equivalence. For example, a quasi-isomorphism.
The Maslov index \( \mu _L: H_2(E,L)\rightarrow \mathbb {Z}\) defines a class \(\mu _L\in H^2(E,L)\), and \(\mu _L\mapsto 2c_1({ TE })\) via \(H^2(E,L)\rightarrow H^2(E)\). The Maslov index of a disc bounding L equals the homological intersection number with \({\mathrm {PD}}[\mu _L]\).
By \(\partial _{{\mathbf {q}}_R}\)-equivariance, the \(({\mathbf {q}}^0,{\mathbf {q}}^0)\)-output of \({\mathrm {OC}}({\mathbf {q}}_L{\mathbf {q}}_R\underline{x}',{\mathbf {q}}x_5,x_4,x_3,{\mathbf {q}}x_2,x_1)\) contributes to the \({\mathbf {q}}^1\)-coefficient of \({\mathrm {OC}}({\mathbf {q}}_L{\mathbf {q}}_R\underline{x}',{\mathbf {q}}x_5,{\mathbf {q}}x_4,x_3,{\mathbf {q}}x_2,x_1)\). Compare the Remarks in 3.11 and 3.15.
By \(\partial _{{\mathbf {q}}_L}\)-equivariance, the \(({\mathbf {q}}^0,{\mathbf {q}}^0)\)-output of \(\Delta ^{2|1}(x_2,x_1,{\mathbf {q}}_L{\mathbf {q}}_R\underline{x}',{\mathbf {q}}x_5)\) contributes to the \(({\mathbf {q}}^1,{\mathbf {q}}^0)\)-coefficient of \(\Delta ^{2|1}({\mathbf {q}}x_2,x_1,{\mathbf {q}}_L{\mathbf {q}}_R\underline{x}',{\mathbf {q}}x_5)\). Compare the Remarks in 3.11 and 3.15.
This is the full \({ SH }^*(E)\) for toric negative line bundles, since toric varieties are simply connected.
Remark We already know a priori that (z(y), w(y)) lies at the barycentre of \(\Delta _E\) by (12.5) (for \(X=E\)), but one can also check by hand that \({\mathrm {val}}_t(z(y)) = \tfrac{\lambda _B}{\lambda _E}\, a\) and \({\mathrm {val}}_t(w(y)) = \tfrac{1}{\lambda _E}\) (confirming (12.8) and (12.5)) and in particular \({\mathrm {val}}_t(z,w)>0\) in each entry, so \((z,w)\in {\mathrm {Int}}(\Delta _E)\).
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Acknowledgments
We thank Mohammed Abouzaid for his continuing interest in the project; Denis Auroux and Sheel Ganatra for helpful conversations; the anonymous referee for very extensive comments; Janko Latschev and the referee for pointing out the method in 3.13 to implicitly construct the auxiliary data (previous arXiv versions make explicit constructions); Nick Sheridan and the referee for pointing out how to adapt the proof of the eigensummand decomposition for \({\mathrm {OC}}\) directly to \({\mathrm {CO}}\) in Theorem 9.6, bypassing duality issues.
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Ritter, A.F., Smith, I. The monotone wrapped Fukaya category and the open-closed string map. Sel. Math. New Ser. 23, 533–642 (2017). https://doi.org/10.1007/s00029-016-0255-9
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DOI: https://doi.org/10.1007/s00029-016-0255-9