We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n Jacobi forms in terms of Jacobi theta functions and modular forms. By using them we obtain Seiberg-Witten curves of type E_6 and E_7 for the E-string theory. The coefficients of each curve are E_n weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthm\"uller.

中文翻译：

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$E_n$ Jacobi 形式和 Seiberg-Witten 曲线

我们讨论在类型 E_n (n=6,7,8) 的 Weyl 群的作用下不变的雅可比形式。对于 n=6,7，我们明确地构造了 E_n 弱雅可比形式的代数的全套生成器。我们首先根据 Jacobi theta 函数和模形式构造 n+1 个独立的 E_n Jacobi 形式。通过使用它们，我们获得了 E 弦理论的 E_6 和 E_7 类型的 Seiberg-Witten 曲线。每条曲线的系数都是根系统指定的特定权重和指数的 E_n 弱雅可比形式，实现了 Wirthm\"uller 前段时间证明存在的生成器。