Topological String on elliptic CY 3-folds and the ring of Jacobi forms

TL;DR

This paper demonstrates that topological string amplitudes on elliptic Calabi-Yau threefolds can be expressed using meromorphic Jacobi forms, revealing deep modular structures and potential connections to known automorphic forms.

Contribution

It provides evidence that all genus amplitudes are governed by Jacobi forms with specific growth properties, suggesting a new automorphic framework for topological string theory on these geometries.

Findings

Amplitudes expressed in terms of meromorphic Jacobi forms with universal denominators.

Poles of the forms are only at torsion points, indicating a special modular structure.

Results agree with curve counting data, confirming the theoretical predictions.

Abstract

We give evidence that the all genus amplitudes of topological string theory on compact elliptically fibered Calabi-Yau manifolds can be written in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base degree. The denominators of these forms have a simple universal form with the property that the poles of the meromorphic form lie only at torsion points. The modular parameter corresponds to the fibre class while the role of the string coupling is played by the elliptic parameter. This leads to very strong all genus results on these geometries, which are checked against results from curve counting. The structure can be viewed as an indication that an N=2 analog of the reciprocal of the Igusa cusp form exists that might govern the topological string theory on these Calabi-Yau manifolds completely.