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, Wirthmüller 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 $\mathrm{SL}_2 (\mathbb{Z})$ 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 $\mathrm{SL}_2 (\mathbb{Z})$ 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 $E_8$.

中文翻译：

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外尔不变量 $E_8$ 雅可比形式

我们研究了 Weyl 群下根系统 $E_8$ 不变量的 Jacobi 形式。这种 Jacobi 形式在 Frobenius 流形、Gromov-Witten 理论和弦理论中具有重要意义。1992 年，Wirthmüller 证明了任何非 $E_8$ 类型的不可约根系统的 Jacobi 形式的空间是多项式代数。但是对于 $E_8$ 的情况知之甚少。在本文中，我们证明了 Weyl 不变式 $E_8$ Jacobi 形式的双级环不是多项式代数，并证明了每个这样的 Jacobi 形式都可以唯一地表示为 Sakai 引入的具有亚纯系数的九个代数独立的 Jacobi 形式的多项式$\mathrm{SL}_2 (\mathbb{Z})$ 模形式。后一个结果意味着固定索引的外尔不变 $E_8$ Jacobi 形式的空间是 $\mathrm{SL}_2 (\mathbb{Z})$ 模形式环上的一个自由模，并且生成器的数量可以由生成序列计算。我们确定并构造所有小索引的生成器。这些结果将 Chevalley 类型定理适当扩展到 $E_8$ 的情况。