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

TL;DR

This paper extends the known modularity properties of closed Gromov-Witten generating functions to open cases and new geometries, using Jacobi forms and elliptic variables.

Contribution

It generalizes the modularity phenomenon to open Gromov-Witten invariants and additional geometries via Jacobi forms.

Findings

Generating functions are meromorphic quasi-modular forms for new geometries.

Extension of modularity to open Gromov-Witten theory using Jacobi forms.

Transformation of open Gromov-Witten parameters into elliptic variables.

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.