Refined Elementary Capacities from Symplectic Field Theory

We extend the family of capacities given by McDuff and Siegel by including a constraint $\mathrm{\ell }$ 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 $\mathrm{\ell }=\mathrm{\infty }$ and to the Gutt-Hutchings capacities for $\mathrm{\ell }=1$. To verify the formula, we must prove the existence of certain curves in the convex toric domain $X_{\mathrm{\Omega }}$, and this requires a new method of proof compared to McDuff-Siegel. We neck-stretch along $\mathrm{\partial }{X}_{\mathrm{\Omega }}$ with curves known to exist in a well-chosen ellipsoid containing $X_{\mathrm{\Omega }}$, and we obtain the desired curves in the bottom level of the resulting psuedoholomorphic building.