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Journal Article

Weyl invariant E8 Jacobi forms


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

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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: http://hdl.handle.net/21.11116/0000-0009-2EFD-8
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 $E_8$.