Symplectic capacities from positive $S^1$–equivariant symplectic homology

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.