In 1992 Wirthmüller showed that for any irreducible root system not of type $E_8$ the ring of weak Jacobi forms invariant under Weyl group is a polynomial algebra. However, it has recently been proved that for $E_8$ the ring is not a polynomial algebra. Weyl invariant $E_8$ Jacobi forms have many applications in string theory and it is an open problem to describe such forms. The scaled refined free energies of $E$-strings with certain $\eta$-function factors are conjectured to be Weyl invariant $E_8$ quasi-holomorphic Jacobi forms. It is further observed that the scaled refined free energies up to some powers of $E_4$ can be written as polynomials in nine Sakai’s $E_8$ Jacobi forms and Eisenstein series $E_2, E_4, E_6$. Motivated by the physical conjectures, we prove that for any Weyl invariant $E_8$ Jacobi form $\phi_t$ of index $t$ the function $E^{[t/5]}_4 \Delta^{[5t/6]} \phi_t$ can be expressed uniquely as a polynomial in $E_4$, $E_6$ and Sakai’s forms, where $[x]$ is the integer part of $x$. This means that a Weyl invariant $E_8$ Jacobi form is completely determined by a solution of some linear equations. By solving the linear systems, we determine the generators of the free module of Weyl invariant $E_8$ weak (resp. holomorphic) Jacobi forms of given index $t$ when $t \leq 13$ (resp. $t \leq 11$).

中文翻译：

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Weyl 不变量 $E_8$ 雅可比形式和 $E$ 字符串

1992 年，Wirthmüller 证明，对于任何非 $E_8$ 类型的不可约根系，Weyl 群下的弱雅可比环形式不变是多项式代数。然而，最近已经证明，对于$E_8$，环不是多项式代数。韦尔不变量 $E_8$ 雅可比形式在弦理论中有许多应用，描述此类形式是一个悬而未决的问题。具有某些 $\eta$ 函数因子的 $E$ 弦的缩放精制自由能被推测为 Weyl 不变量 $E_8$ 准全纯雅可比形式。进一步观察到，达到 $E_4$ 某些幂的缩放精炼自由能可以写成九个 Sakai 的 $E_8$ 雅可比形式和爱森斯坦级数 $E_2、E_4、E_6$ 的多项式。受物理猜想的启发，我们证明对于索引 $t$ 的任意 Weyl 不变量 $E_8$ 雅可比形式 $\phi_t$，函数 $E^{[t/5]}_4 \Delta^{[5t/6]} \phi_t$ 可以唯一地表示为 $E_4$、$E_6$ 和 Sakai 形式的多项式，其中 $[x]$ 是 $x$ 的整数部分。这意味着 Weyl 不变量 $E_8$ 雅可比形式完全由一些线性方程的解确定。通过求解线性系统，我们确定当 $t \leq 13$（或 $t \leq 11$ 时）给定索引 $t$ 的 Weyl 不变量 $E_8$ 弱（或全纯）雅可比形式的自由模的生成元）。