Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

TL;DR

This paper proves that the wrapped Fukaya category of Weinstein manifolds and sectors is generated by unstable manifolds of critical points, establishing an isomorphism between Hochschild homology and symplectic cohomology, confirming a conjecture of Seidel.

Contribution

It provides a geometric proof that the wrapped Fukaya category is generated by certain unstable manifolds and confirms the open-closed map as an isomorphism, extending to sectors and various setups.

Findings

Wrapped Fukaya category generated by unstable manifolds of index n critical points.

Open-closed map from Hochschild homology to symplectic cohomology is an isomorphism.

Results apply to Weinstein sectors and multiple categorical setups.

Abstract

We prove that the wrapped Fukaya category of any $2n$-dimensional Weinstein manifold (or, more generally, Weinstein sector) $W$ is generated by the unstable manifolds of the index $n$ critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer cohomology and the fairly simple observation that Floer cohomology vanishes for Lagrangian submanifolds that can be disjoined from the isotropic skeleton of the Weinstein manifold. Note that we do not need any additional assumptions on this skeleton. By applying our generation result to the diagonal in the product $W \times W$, we obtain as a corollary that the open-closed map from the Hochschild homology of the wrapped Fukaya category of $W$ to its symplectic cohomology is an isomorphism, proving a conjecture of Seidel. We work mainly in the "linear setup" for the wrapped Fukaya category, but we also extend the proofs to the "quadratic" and "localisation" setup. This is necessary for dealing with Weinstein sectors and for the applications.