ECH (embedded contact homology) capacities give obstructions to symplectically
embedding one four-dimensional symplectic manifold with boundary into another.
These obstructions are known to be sharp when the domain and target are
ellipsoids (proved by McDuff), and more generally when the domain is a
“concave toric domain” and the target is a “convex toric domain” (proved by
Cristofaro-Gardiner). However ECH capacities often do not give sharp obstructions,
for example in many cases when the domain is a polydisk. This paper uses
more refined information from ECH to give stronger symplectic embedding
obstructions when the domain is a polydisk, or more generally a convex toric
domain. We use these new obstructions to reprove a result of Hind and Lisi
on symplectic embeddings of a polydisk into a ball, and generalize this to
obstruct some symplectic embeddings of a polydisk into an ellipsoid. We also
obtain a new obstruction to symplectically embedding one polydisk into
another, in particular proving the four-dimensional case of a conjecture of
Schlenk.
