Computing higher symplectic capacities I

TL;DR

This paper introduces recursive formulas for higher symplectic capacities of convex toric domains, applies homological perturbation theory to compute structure coefficients, and derives new obstructions and curve counts in symplectic embedding problems.

Contribution

It provides the first recursive formulas for higher symplectic capacities and links algebraic structures to pseudoholomorphic curve counts in symplectic geometry.

Findings

New obstructions for stabilized embeddings of ellipsoids and polydisks.

Counts of pseudoholomorphic curves with tangency constraints.

Recursive formulas for higher symplectic capacities.

Abstract

We present recursive formulas which compute the recently defined "higher symplectic capacities" for all convex toric domains. In the special case of four-dimensional ellipsoids, we apply homological perturbation theory to the associated filtered L-infinity algebras and prove that the resulting structure coefficients count punctured pseudoholomorphic curves in cobordisms between ellipsoids. As sample applications, we produce new previously inaccessible obstructions for stabilized embeddings of ellipsoids and polydisks, and we give new counts of curves with tangency constraints.