Symplectic embeddings from concave toric domains into convex ones

TL;DR

This paper proves that ECH capacities provide sharp obstructions for symplectic embeddings of all concave toric domains into convex ones, extending previous results and offering new formulas for convex domains.

Contribution

It establishes the sharpness of ECH capacity obstructions for all concave-to-convex symplectic embeddings and introduces a new formula relating convex toric domains to collections of balls.

Findings

ECH capacities are sharp obstructions for all concave-to-convex embeddings

New formula for ECH capacities of convex toric domains

ECH capacities of convex domains determined by associated ball collections

Abstract

Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. In "Symplectic embeddings into four-dimensional concave toric domains", the author, Choi, Frenkel, Hutchings and Ramos computed the ECH capacities of all "concave toric domains", and showed that these give sharp obstructions in several interesting cases. We show that these obstructions are sharp for all symplectic embeddings of concave toric domains into "convex" ones. In an appendix with Choi, we prove a new formula for the ECH capacities of convex toric domains, which shows that they are determined by the ECH capacities of a corresponding collection of balls.