Lattice Formulas For Rational SFT Capacities

TL;DR

This paper develops new bounds and formulas for rational SFT capacities of toric domains using lattice optimization and algebraic geometry, with applications to symplectic geometry and capacity computations.

Contribution

It introduces a simple lattice formula for RSFT capacities of convex toric domains and provides new bounds and asymptotic laws using toric algebraic geometry.

Findings

Sharp bounds for RSFT capacities in dimension 4

A simple lattice formula for many RSFT capacities

Asymptotic behavior of RSFT capacities for convex toric domains

Abstract

We initiate the study of the rational SFT capacities of Siegel using tools in toric algebraic geometry. In particular, we derive new (often sharp) bounds for the RSFT capacities of a strongly convex toric domain in dimension $4$. These bounds admit descriptions in terms of both lattice optimization and (toric) algebraic geometry. Applications include (a) an extremely simple lattice formula for for many RSFT capacities of a large class of convex toric domains, (b) new computations of the Gromov width of a class of product symplectic manifolds and (c) an asymptotics law for the RSFT capacities of all strongly convex toric domains.