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

Abstract

We build the wrapped Fukaya category W(E) for any monotone symplectic manifold, convex at infinity. We define the open-closed and closed open-string maps. We study their algebraic properties and prove that the string maps are compatible with the eigenvalue splitting of W(E). We extend Abouzaid's generation criterion from the exact to the monotone setting. We construct an acceleration functor from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map from quantum cohomology QH(E) to symplectic cohomology SH(E). We define the QH(E)- and SH(E)-module structure on the Hochschild (co)homology of W(E) which is compatible with the string maps. 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 any monotone negative line bundle over any projective space is proper (cohomologically finite). For any monotone negative line bundle E over a toric Fano variety, we show that SH(E) is non-trivial and that W(E) contains an essential non-displaceable monotone Lagrangian torus.

78 pages, 31 figures. Version 7: Corrected the claim in 12.3. Final publication in Selecta Mathematica available at Springer via http://dx.doi.org/10.1007/s00029-016-0255-9

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• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Symplectic cohomology and duality for the wrapped Fukaya category
• Fukaya categories and deformations
• Topological quantum field theory structure on symplectic cohomology
• Quilted Floer Cohomology
• Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle
• Floer theory for negative line bundles via Gromov-Witten invariants
• Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry
• Altering symplectic manifolds by homologous recombination
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Cited by in corpus (30)

• On the Fukaya category of a Fano hypersurface in projective space
• Floer theory for negative line bundles via Gromov-Witten invariants
• Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors
• Floer theory and reduced cohomology on open manifolds
• Rabinowitz Floer homology and mirror symmetry
• Automatically generating Fukaya categories and computing quantum cohomology
• Lagrangian sections on mirrors of toric Calabi-Yau 3-folds
• Low-area Floer theory and non-displaceability
• Categorical primitive forms of Calabi-Yau $A_{\infty }$-categories with semi-simple cohomology
• The closed-open string map for $S^1$-invariant Lagrangians
• Invariance of symplectic cohomology and twisted cotangent bundles over surfaces
• Equivariant split generation and mirror symmetry of special isogenous tori
• Disjoinable Lagrangian tori and semisimple symplectic cohomology
• Superfiltered $A_{\infty }$-deformations of the exterior algebra, and local mirror symmetry
• The cyclic open-closed map, u-connections and R-matrices
• The quantum Johnson homomorphism and symplectomorphism of 3-folds
• The wrapped Fukaya category for semi-toric SYZ fibrations
• Peterson conjecture via Lagrangian correspondences and wonderful compactifications
• Packing Lagrangian tori
• Family Floer superpotential's critical values are eigenvalues of quantum product by $c_1$
• Twin Lagrangian fibrations in mirror symmetry
• Floer cohomology of the Chiang Lagrangian
• Quantum characteristic classes, moment correspondences and the Hamiltonian groups of coadjoint orbits
• Orbifold Kodaira-Spencer maps and closed-string mirror symmetry for punctured Riemann surfaces
• A symplectic prolegomenon
• Categorical non-properness in wrapped Floer theory
• On the Filtered Symplectic Homology of Prequantization Bundles
• Periodic leaf-wise intersection points from Lagrangians
• Fukaya categories of blowups
• Maurer-Cartan deformation of Lagrangians