Open Gromov-Witten Theory of K_(mathbb P²), K_{{mathbb P¹}times {mathbb P¹}}, K_(Wmathbb P[1,1,2]), K_(mathbb F₁) and Jacobi Forms

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.