Weyl invariant $E_8$ Jacobi forms

TL;DR

This paper studies Weyl invariant $E_8$ Jacobi forms, showing their ring structure is not polynomial and expressing them via algebraically independent forms, extending Chevalley's theorem.

Contribution

It proves that the ring of Weyl invariant $E_8$ Jacobi forms is not polynomial and provides a unique polynomial expression for each form in terms of nine generators.

Findings

The ring of $E_8$ Jacobi forms is not polynomial.

Every $E_8$ Jacobi form can be expressed as a polynomial in nine generators.

The space of forms of fixed index is a free module over SL(2,Z) modular forms.

Abstract

We investigate the Jacobi forms for the root system $E_8$ invariant under the Weyl group. This type of Jacobi forms has significance in Frobenius manifolds, Gromov--Witten theory and string theory. In 1992, Wirthm\"{u}ller proved that the space of Jacobi forms for any irreducible root system not of type $E_8$ is a polynomial algebra. But very little has been known about the case of $E_8$. In this paper we show that the bigraded ring of Weyl invariant $E_8$ Jacobi forms is not a polynomial algebra and prove that every such Jacobi form can be expressed uniquely as a polynomial in nine algebraically independent Jacobi forms introduced by Sakai with coefficients which are meromorphic SL(2,Z) modular forms. The latter result implies that the space of Weyl invariant $E_8$ Jacobi forms of fixed index is a free module over the ring of SL(2,Z) modular forms and that the number of generators can be calculated by a generating series. We determine and construct all generators of small index. These results give a proper extension of the Chevalley type theorem to the case of $E_8$.