The generating function of the embedding capacity for 4-dimensional symplectic ellipsoids

TL;DR

This paper introduces a generating function that encodes the ECH capacities sequences of 4-dimensional symplectic ellipsoids, providing new formulations of McDuff's theorem on symplectic embeddings.

Contribution

It presents a novel generating function approach to encode ECH capacities, offering new perspectives on symplectic embedding criteria for 4D ellipsoids.

Findings

Generating function encodes ECH capacities sequences.

Provides new equivalent formulations of McDuff's theorem.

Enhances understanding of symplectic embedding conditions.

Abstract

Quite recently, McDuff showed that the existence of a symplectic embedding of one four-dimensional ellipsoid into another can be established by comparing their corresponding sequences of ECH capacities. In this note we show that these sequences can be encoded in a generating function, which gives several new equivalent formulations of McDuff's theorem.