We construct cubic Hamiltonians for quantum Gaudin models of affine types $\hat{\mathfrak{sl}}_M$. They are given by hypergeometric integrals of a form we recently conjectured in arXiv:1804.01480. We prove that they commute amongst themselves and with the quadratic Hamiltonians. We prove that their vacuum eigenvalues, and their eigenvalues for one Bethe root, are given by certain hypergeometric functions on a space of affine opers.

中文翻译：

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仿射 Gaudin 模型中运动的三次超几何积分

我们为仿射类型 $\hat{\mathfrak{sl}}_M$ 的量子 Gaudin 模型构造三次哈密顿量。它们由我们最近在 arXiv:1804.01480 中推测的形式的超几何积分给出。我们证明了它们之间以及与二次哈密顿量之间的通勤。我们证明了它们的真空特征值和一个 Bethe 根的特征值是由仿射运算空间上的某些超几何函数给出的。