Quantitative embedded contact homology

TL;DR

This paper introduces ECH capacities, a sequence of real numbers derived from embedded contact homology, to provide sharp symplectic embedding obstructions and explore their asymptotic relation to symplectic volume.

Contribution

It defines ECH capacities for Liouville domains, computes them for key examples, and demonstrates their effectiveness in symplectic embedding problems, linking to volume asymptotics.

Findings

ECH capacities are monotone under symplectic embeddings.

Computed ECH capacities for ellipsoids, polydisks, and certain cotangent bundle subsets.

ECH capacities provide sharp obstructions for embedding problems like ellipsoid into ball.

Abstract

Define a "Liouville domain" to be a compact exact symplectic manifold with contact-type boundary. We use embedded contact homology to assign to each four-dimensional Liouville domain (or subset thereof) a sequence of real numbers, which we call "ECH capacities". The ECH capacities of a Liouville domain are defined in terms of the "ECH spectrum" of its boundary, which measures the amount of symplectic action needed to represent certain classes in embedded contact homology. Using cobordism maps on embedded contact homology (defined in joint work with Taubes), we show that the ECH capacities are monotone with respect to symplectic embeddings. We compute the ECH capacities of ellipsoids, polydisks, certain subsets of the cotangent bundle of T2, and disjoint unions of examples for which the ECH capacities are known. The resulting symplectic embedding obstructions are sharp in some interesting cases, for example for the problem of embedding an ellipsoid into a ball (as shown by work of McDuff-Schlenk) or embedding a disjoint union of balls into a ball. We also state and present evidence for a conjecture under which the asymptotics of the ECH capacities of a Liouville domain recover its symplectic volume.