We show that the construction of the higher Gaudin Hamiltonians associated with the Lie algebra \(\mathfrak {gl}_{n}\) admits an interpolation to any complex number n. We do this using the Deligne’s category \(\mathcal {D}_{t}\), which is a formal way to define the category of finite-dimensional representations of the group \(GL_{n}\), when n is not necessarily a natural number. We also obtain interpolations to any complex number n of the no-monodromy conditions on a space of differential operators of order n, which are considered to be a modern form of the Bethe ansatz equations. We prove that the relations in the algebra of higher Gaudin Hamiltonians for complex n are generated by our interpolations of the no-monodromy conditions. Our constructions allow us to define what it means for a pseudo-differential operator to have no monodromy. Motivated by the Bethe ansatz conjecture for the Gaudin model associated with the Lie superalgebra \(\mathfrak {gl}_{n\vert n'}\), we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy.

我们证明了与李代数相关的高丁哈密顿量的构造\(\mathfrak {gl}_{n}\) 允许对任何复数进行插值n。我们使用德利涅范畴 \(\mathcal {D}_{t}\) 来做到这一点，这是定义有限范畴的正式方法群的维度表示 \(GL_{n}\)，当 n不一定是自然数。我们还获得了阶微分算子空间上的非单性条件的任意复数 n 的插值 n，被认为是 Bethe ansatz 方程的现代形式。我们证明了复数 n 的高阶高丁哈密顿量的代数关系是通过我们对非单性条件的插值生成的。我们的构造允许我们定义伪微分算子不具有单性的含义。受到与李超代数相关的高丁模型的 Bethe ansatz 猜想的启发\(\mathfrak {gl}_{n\vert n'}\)，我们证明无单性微分算子的比率是无单性微分算子的伪微分算子。