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.

v1: 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 (3)

• Fukaya categories and deformations
• Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair
• String topology and the based loop space

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• Equivariant split generation and mirror symmetry of special isogenous tori
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• The Bagger-Lambert model and Type IIA string theory
• Algebraic sheaves of Floer homology groups via algebraic torus actions on the Fukaya category