Refined Elementary Capacities from Symplectic Field Theory

TL;DR

This paper generalizes symplectic capacities by incorporating a constraint on the number of positive ends, providing new sharp embedding obstructions for convex toric domains through a novel proof method involving neck-stretching.

Contribution

It introduces a generalized capacity formula for convex toric domains that unifies and extends previous capacities, with a new proof technique for curve existence.

Findings

New capacity formula for convex toric domains

Provides sharp embedding obstructions in various cases

Introduces a novel neck-stretching proof method

Abstract

We extend the family of capacities given by McDuff and Siegel by including a constraint $\ell$ on the number of positive asymptotically cylindrical ends of curves showing up in the definition. We prove a generalized computation formula for four-dimensional convex toric domains that offers new, sometimes sharp, embedding obstructions in stabilized and unstabilized cases. The formula restricts to the McDuff-Siegel capacities for $\ell = \infty$ and to the Gutt-Hutchings capacities for $\ell = 1$. To verify the formula, we must prove the existence of certain curves in the convex toric domain $X_\Omega$, and this requires a new method of proof compared to McDuff-Siegel. We neck-stretch along $\partial X_\Omega$ with curves known to exist in a well-chosen ellipsoid containing $X_\Omega$, and we obtain the desired curves in the bottom level of the resulting psuedoholomorphic building.