Symplectic embeddings of four-dimensional polydisks into balls

TL;DR

This paper establishes new optimal obstructions for symplectic embeddings of four-dimensional polydisks into balls, extending previous work and simplifying the computational criteria for such embeddings.

Contribution

It introduces improved bounds for embedding polydisks into balls and reduces the complexity of the embedding criterion from exponential to quadratic time.

Findings

New bounds for embedding polydisks into balls are proven.

The optimality of the bounds is confirmed using Schlenk's folding construction.

A simplified criterion reduces computational complexity from exponential to quadratic.

Abstract

In this paper we obtain new obstructions to symplectic embeddings of the four-dimensional polydisk $P(a,1)$ into the ball $B(c)$ for $2\leq a<\frac{\sqrt{7}-1} {\sqrt{7}-2} \approx 2.549$, extending work done by Hind-Lisi and Hutchings. Schlenk's folding construction permits us to conclude our bound on $c$ is optimal. Our proof makes use of the combinatorial criterion necessary for one "convex toric domain" to symplectically embed into another introduced by Hutchings in \cite{Beyond}. Additionally, we prove that if certain symplectic embeddings of four dimensional convex toric domains exist then a modified version of this criterion from \cite{Beyond} must hold, thereby reducing the computational complexity of the original criterion from $O(2^n)$ to $O(n^2)$.