Symplectic embeddings of four-dimensional polydisks into balls

Abstract

We obtain new obstructions to symplectic embeddings of the four-dimensional polydisk $P\left(a,1\right)$ into the ball $B\left(c\right)$ for $2\le a\le \left(\sqrt{7}-1\right)∕\left(\sqrt{7}-2\right)\approx 2.549$, extending work done by Hind and Lisi and by 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 (2016). We also observe that the computational complexity of this criterion can be reduced from $O\left(2^n\right)$ to $O\left(n^2\right)$.