Gaudin model and Deligne’s category

Abstract

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.