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We study the refined topological string partition function of a class oftoric elliptically fibered Calabi-Yau threefolds. These Calabi-Yau threefoldsgive rise to five dimensional quiver gauge theories and are dual toconfigurations of M5-M2-branes. We determine the Gopakumar-Vafa invariants forthese threefolds and show that the genus $g$ free energy is given by the weight$2g$ Eisenstein series. We also show that although the free energy at allgenera are modular invariant the full partition function satisfies thenon-perturbative modular transformation property discussed by Lockhart and Vafain arXiv:1210.5909 and therefore the modularity of free energy is up tonon-perturbative corrections.

Elliptic CY3folds and Non-Perturbative Modular Transformation