Open Gromov-Witten Theory of $K_{{\mathbb{P}}^2},K_{{\mathbb{P}}^1\times {\mathbb{P}}^1},K_{W\mathbb{P}[1,1,2]},K_{{\mathbb{F}}_1}$ and Jacobi Forms

Abstract

It was known through the efforts of many works that the generating functions in the closed Gromov-Witten theory of $K_{{\mathbb{P}}^2}$ are meromorphic quasi-modular forms basing on the B-model predictions. In this article, we extend the modularity phenomenon to $K_{{\mathbb{P}}^1\times {\mathbb{P}}^1},K_{W\mathbb{P}[1,1,2]},K_{{\mathbb{F}}_1}$. More importantly, we generalize it to the generating functions in the open Gromov-Witten theory using the theory of Jacobi forms where the open Gromov-Witten parameters are transformed into elliptic variables.

2 figures, revised version