Genus Two Quasi-Siegel Modular Forms and Gromov-Witten Theory of Toric Calabi-Yau Threefolds

TL;DR

This paper develops theories of genus two quasi-Siegel modular forms and applies them to prove that Gromov-Witten potentials of certain Calabi-Yau threefolds are essentially these modular forms, linking geometry and modularity.

Contribution

It introduces new differential rings of quasi-Siegel modular and Jacobi forms for genus two and proves their role in Gromov-Witten theory of toric Calabi-Yau threefolds.

Findings

Gromov-Witten potentials are quasi-Siegel Jacobi and modular forms.

Established the connection between mirror symmetry and genus two modular forms.

Proved the Remodeling Conjecture for specific Calabi-Yau threefolds.

Abstract

We first develop theories of differential rings of quasi-Siegel modular and quasi-Siegel Jacobi forms for genus two. Then we apply them to the Eynard-Orantin topological recursion of certain local Calabi-Yau threefolds equipped with branes, whose mirror curves are genus two hyperelliptic curves. By the proof of the Remodeling Conjecture, we prove that the corresponding open- and closed- Gromov-Witten potentials are essentially quasi-Siegel Jacobi and quasi-Siegel modular forms for genus two, respectively.