Symplectic capacities from positive S^1-equivariant symplectic homology

TL;DR

This paper introduces a new sequence of symplectic capacities derived from positive S^1-equivariant symplectic homology, enabling broader computations and applications in symplectic embedding problems.

Contribution

It defines new symplectic capacities c_k that are conjecturally equal to Ekeland-Hofer capacities but are more computationally accessible, especially for toric domains.

Findings

Capacities c_k are explicitly computed for convex and concave toric domains.

Optimal symplectic embeddings of cubes into toric domains are determined.

Capacities c_k are extended to functions of Liouville domains, broadening their applicability.

Abstract

We use positive S^1-equivariant symplectic homology to define a sequence of symplectic capacities c_k for star-shaped domains in R^{2n}. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but they satisfy axioms which allow them to be computed in many more examples. In particular, we give combinatorial formulas for the capacities c_k of any "convex toric domain" or "concave toric domain". As an application, we determine optimal symplectic embeddings of a cube into any convex or concave toric domain. We also extend the capacities c_k to functions of Liouville domains which are almost but not quite symplectic capacities.