Gaudin model and Deligne’s category

We show that the construction of the higher Gaudin Hamiltonians associated with the Lie algebra $$\mathfrak {gl}_{n}$$ gl n...

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Summary:	We show that the construction of the higher Gaudin Hamiltonians associated with the Lie algebra $$\mathfrak {gl}_{n}$$ gl n admits an interpolation to any complex number n. We do this using the Deligne’s category $$\mathcal {D}_{t}$$ D t , which is a formal way to define the category of finite-dimensional representations of the group $$GL_{n}$$ G L 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 $$\mathfrak {gl}_{n\vert n'}$$ gl n \| n ′ , we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy.

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