Graded rings of integral Jacobi forms

TL;DR

This paper characterizes the algebraic structure of rings of integral weak Jacobi forms, which are important in string theory, modular forms, and Borcherds products, revealing new insights into their composition and applications.

Contribution

It determines the structure of bigraded rings of weak Jacobi forms with integral Fourier coefficients and describes the graded ring of weakly holomorphic Jacobi forms of weight zero and integral index.

Findings

Structure of the bigraded ring of weak Jacobi forms is fully characterized.

The graded ring of weakly holomorphic Jacobi forms of weight zero is described.

Connections to Borcherds products and string theory models are established.

Abstract

We determine the structure of the bigraded ring of weak Jacobi forms with integral Fourier coefficients. This ring is the target ring of a map generalising the Witten and elliptic genera and a partition function of $(0,2)$-model in string theory. We also determine the structure of the graded ring of all weakly holomorphic Jacobi forms of weight zero and integral index with integral Fourier coefficients. These forms are the data for Borcherds products for the Siegel paramodular groups.