Vortices on Cylinders and Warped Exponential Networks

We study 3d $\mathcal{N}=2$ $U(1)$ Chern-Simons-matter QFT on a cylinder $C\times\mathbb{R}$. The topology of $C$ gives rise to BPS sectors of low-energy solitons known as kinky vortices, which interpolate between (possibly) different vacua at the ends of the cylinder and at the same time carry magnetic flux. We compute the spectrum of BPS vortices on the cylinder in an isolated Higgs vacuum, through the framework of \emph{warped} exponential networks, which we introduce. We then conjecture a relation between these and standard vortices on $\mathbb{R}^2$, which are related to genus-zero open Gromov-Witten invariants of toric branes. More specifically, we show that in the limit of large Fayet-Iliopoulos coupling, the spectrum of kinky vortices on $C$ undergoes an infinite sequence of wall-crossing transitions, and eventually stabilizes. We then propose an exact relation between a generating series of stabilized CFIV indices and the Gromov-Witten disk potential, and discuss its consequences for the structure of moduli spaces of vortices.