Symplectic cohomology and duality for the wrapped Fukaya category

Abstract

Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild homology of W to symplectic cohomology and from symplectic cohomology to the Hochschild cohomology of W are isomorphisms, in a manner compatible with ring and module structures. This is a consequence of a more general duality for the wrapped Fukaya category, which should be thought of as a non-compact version of a Calabi-Yau structure. The new ingredients are: (1) Fourier-Mukai theory for W via a wrapped version of holomorphic quilts, (2) new geometric operations, coming from discs with two negative punctures and arbitrary many positive punctures, (3) a generalization of the Cardy condition, and (4) the use of homotopy units and A-infinity shuffle products to relate non-degeneracy to a resolution of the diagonal.

arXiv commentv1: 166 pages, 26 figures. Feedback and comments are welcome! This paper will (eventually) be split into two papers. arXiv admin note: text overlap with arXiv:1001.4593 by other authors

References in corpus (5)

• Fukaya categories and deformations
• Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair
• Symplectic cohomology and q-intersection numbers
• Symplectic homology product via Legendrian surgery
• String topology and the based loop space

Cited by in corpus (46)

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• Mirror symmetry: from categories to curve counts
• On the symplectic cohomology of log Calabi-Yau surfaces
• Floer theory and reduced cohomology on open manifolds
• The symplectic arc algebra is formal
• Arithmetic mirror symmetry for the 2-torus
• Symplectic Homology of complements of smooth divisors
• Floer theory and flips
• Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures
• Automatically generating Fukaya categories and computing quantum cohomology
• Orlov and Viterbo functors in partially wrapped Fukaya categories
• Simplifying Weinstein Morse functions
• The Fukaya category pairs with Lagrangian cobordisms
• Multiplicative preprojective algebras are 2-Calabi-Yau
• Formal groups and quantum cohomology
• Wrapped Floer cohomology and Lagrangian correspondences
• Prime-localized Weinstein subdomains
• Homological mirror symmetry for punctured spheres
• Geometric and algebraic presentations of Weinstein domains
• Intrinsic mirror symmetry and categorical crepant resolutions
• A spectral sequence of the Floer cohomology of symplectomorphisms of trivial polarization class
• Koszul duality via suspending Lefschetz fibrations
• Speculations on homological mirror symmetry for hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$
• Donaldson-Thomas invariants of length 2 flops
• Equivariant split generation and mirror symmetry of special isogenous tori
• Invariance of Immersed Floer cohomology under Maslov flows
• Pairings in mirror symmetry between a symplectic manifold and a Landau-Ginzburg $B$-model
• Multiplicative preprojective algebras of Dynkin quivers
• Secondary products in supersymmetric field theory
• Spherical twists and Lagrangian spherical manifolds
• Versality in mirror symmetry
• Fukaya categories of Lagrangian cobordisms and duality
• Open Gromov-Witten invariants from the Fukaya category
• Applications of the theory of Floer to symmetric spaces
• Iterations of symplectomorphisms and p-adic analytic actions on the Fukaya category
• Crepant resolutions of stratified varieties via gluing
• Categorical non-properness in wrapped Floer theory
• An exposition of the equivalence of Heegaard Floer homology and embedded contact homology
• Nonexistence of exact Lagrangian tori in affine conic bundles over ${\mathbb{C}}^n$
• The Deligne-Mumford operad as a trivialization of the circle action
• Algebraic sheaves of Floer homology groups via algebraic torus actions on the Fukaya category
• The Bagger-Lambert model and Type IIA string theory