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

TL;DR

This paper computes the genus zero and one terms of the Gopakumar-Vafa formula in Type IIA string theory compactified on Calabi-Yau threefolds directly in five-dimensional field theory, addressing divergence issues present in higher-genus calculations.

Contribution

It provides a direct five-dimensional field theory computation of  and  terms of the GV formula, clarifying their structure and divergence issues.

Findings

Explicit computation of  and  terms in 5D field theory

Resolution of divergence issues for g  in hypermultiplet sums

Enhanced understanding of topological amplitudes in string compactifications

Abstract

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 $g \leq 1$, 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.