More on Gopakumar-Vafa formula: coefficients ${\mathcal{F}}_0$ and ${\mathcal{F}}_1$

In Type IIA compactified on a Calabi-Yau threefold, the genus zero and one terms of the Gopakumar-Vafa (GV) formula describe F-terms that are related to genus zero and one topological amplitudes. While for higher-genus terms ${\mathcal{F}}_g,g\ge 2$, the contribution of a light hypermultiplet can be computed via a sum over Kaluza-Klein harmonics, as has been shown in a recent paper, for $g\le 1$, the sum diverges and it is better to compute ${\mathcal{F}}_0$ and ${\mathcal{F}}_1$ directly in five-dimensional field theory. Such a computation is presented here.