Refined Elementary Capacities from Symplectic Field Theory

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.