Computing Higher Symplectic Capacities I

Abstract

Abstract We present recursive formulas that 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 $\mathcal{L}_\infty $ 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.