Symplectic cohomology and duality for the wrapped Fukaya category

TL;DR

This paper establishes a duality in the wrapped Fukaya category relating Hochschild homology, symplectic cohomology, and Hochschild cohomology, using advanced geometric and algebraic techniques, under certain non-degeneracy conditions.

Contribution

It introduces a duality framework for the wrapped Fukaya category, generalizing Calabi-Yau structures with new geometric and algebraic tools like Fourier-Mukai theory and holomorphic quilts.

Findings

Isomorphisms between Hochschild homology and symplectic cohomology

Isomorphisms between symplectic cohomology and Hochschild cohomology

Development of new geometric operations and duality principles

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.