Noncontractible loops of symplectic embeddings between convex toric domains

TL;DR

This paper demonstrates the existence of noncontractible loops of symplectic embeddings between convex toric domains, extending known results from ellipsoids and employing holomorphic cylinder moduli spaces.

Contribution

It generalizes noncontractibility results from ellipsoids to convex toric domains using ECH capacities and holomorphic curve techniques.

Findings

Loops of symplectic embeddings are noncontractible under certain capacity inequalities.

Constructed loops become contractible when the target domain is sufficiently large.

Holomorphic cylinders are used to analyze the topology of embedding spaces.

Abstract

Given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result. Given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible. We show how the constructed loops become contractible if the target domain becomes large enough. The proof involves studying certain moduli spaces of holomorphic cylinders in families of symplectic cobordisms arising from families of symplectic embeddings.