Weyl invariant E8 Jacobi forms

Wang, Haowu

doi:10.4310/CNTP.2021.v15.n3.a3

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Weyl invariant E₈ Jacobi forms

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Wang, Haowu
Max Planck Institute for Mathematics, Max Planck Society;

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https://dx.doi.org/10.4310/CNTP.2021.v15.n3.a3
(Publisher version)

https://doi.org/10.48550/arXiv.1801.08462
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Citation

Wang, H. (2021). Weyl invariant E₈ Jacobi forms. Communications in Number Theory and Physics, 15(3), 517 -573. doi:10.4310/CNTP.2021.v15.n3.a3.

Cite as: https://hdl.handle.net/21.11116/0000-0009-2EFD-8

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 $E_8$.