Elliptic CY3folds and Non-Perturbative Modular Transformation

TL;DR

This paper investigates the refined topological string partition function of elliptically fibered Calabi-Yau threefolds, revealing their modular properties and non-perturbative corrections, with implications for five-dimensional gauge theories and M-brane configurations.

Contribution

It determines Gopakumar-Vafa invariants for these threefolds and demonstrates the non-perturbative modular transformation behavior of the full partition function.

Findings

Genus g free energy expressed as Eisenstein series

Partition function exhibits non-perturbative modular invariance

Gopakumar-Vafa invariants explicitly computed

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 $2g$ 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.