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

Understand

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.