Boundary Depth and Deformations of Symplectic Cohomology

TL;DR

This paper explores the relationship between two versions of symplectic cohomology for Liouville domains, showing how the ambient version deforms the intrinsic one under certain boundary conditions, with applications to mirror symmetry.

Contribution

It establishes a deformation framework connecting ambient and intrinsic symplectic cohomology, introducing a filtration and quantitative tools for analysis.

Findings

Ambient symplectic cohomology deforms the intrinsic version under small boundary depth.

Constructed a filtration whose associated graded matches the intrinsic theory.

Applied results to construct local SYZ mirror pieces.

Abstract

We study the relation between two versions of symplectic cohomology associated to a Liouville domain $D$ embedded in a symplectic manifold $M$: the ambient version $SC^*_M(D)$ defined over the Novikov field and depending on the embedding, and the intrinsic version $SC^*_{\theta}(D)$ depending on the choice of a local Liouville form and defined over the ground field. We show that when $D$ has sufficiently small boundary depth, the ambient version can be viewed as a deformation of the intrinsic one. This is achieved by constructing a filtration whose associated graded reproduces the intrinsic theory, and developing quantitative tools to control the deformation. We apply our results to constructing local pieces of the SYZ mirror.