Elliptic CY3folds and non-perturbative modular transformation

Abstract

We study the refined topological string partition function of a class of toric elliptically fibered Calabi–Yau threefolds. These Calabi–Yau threefolds give rise to five dimensional quiver gauge theories and are dual to configurations of M5-M2-branes. We determine the Gopakumar–Vafa invariants for these threefolds and show that the genus g free energy is given by the weight 2 g Eisenstein series. We also show that although the free energy at all genera are modular invariant, the full partition function satisfies the non-perturbative modular transformation property discussed by Lockhart and Vafa in arXiv:1210.5909 and therefore the modularity of free energy is up to non-perturbative corrections.