Exponential Networks, WKB and Topological String

We propose a connection between $3\mathrm{d}$-$5\mathrm{d}$ exponential networks and exact WKB for difference equations associated to five dimensional Seiberg–Witten curves, or equivalently, to quantum mirror curves to toric Calabi–Yau threefolds $X$: the singularities in the Borel planes of local solutions to such difference equations correspond to central charges of $3\mathrm{d}$-$5\mathrm{d}$ BPS KK-modes. It follows that there should be distinguished local solutions of the difference equation in each domain of the complement of the exponential network, and these solutions jump at the walls of the network. We verify and explore this picture in two simple examples of $3\mathrm{d}$-$5\mathrm{d}$ systems, corresponding to taking the toric Calabi–Yau $X$ to be either $\mathbb{C}^3$ or the resolved conifold. We provide the full list of local solutions in each sector of the Borel plane and in each domain of the complement of the exponential network, and find that local solutions in disconnected domains correspond to non-perturbative open topological string amplitudes on $X$ with insertions of branes at different positions of the toric diagram. We also study the Borel summation of the closed refined topological string free energy on $X$ and the corresponding non-perturbative effects, finding that central charges of $\mathrm{5d}$ BPS KK-modes are related to the singularities in the Borel plane.