An elementary alternative to ECH capacities

TL;DR

This paper introduces a new sequence of symplectic capacities in four dimensions that are easier to define and compute, using only basic holomorphic curve notions, and are effective in obstructing symplectic embeddings.

Contribution

It provides an elementary alternative to ECH capacities that avoids Seiberg-Witten theory, matching ECH capacities on key examples and aiding in embedding obstructions.

Findings

New capacities agree with ECH capacities on convex and concave toric domains.

Capacities satisfy key properties similar to ECH capacities.

Useful for obstructing embeddings into closed symplectic four-manifolds.

Abstract

The ECH capacities are a sequence of numerical invariants of symplectic four-manifolds which give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their basic properties currently requires Seiberg-Witten theory. In this note we define a new sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves. The new capacities satisfy the same basic properties as ECH capacities and agree with the ECH capacities for the main examples for which the latter have been computed, namely convex and concave toric domains. The new capacities are also useful for obstructing symplectic embeddings into closed symplectic four-manifolds. This work is inspired by a recent preprint of McDuff-Siegel giving a similar elementary alternative to symplectic capacities from rational SFT.