Gaudin model and Deligne’s category

Letters in Mathematical Physics. 2024. Vol. 114. No. 1. Article 3.

B. Feigin, L. Rybnikov, F. Uvarov

We show that the construction of the higher Gaudin Hamiltonians associated with the Lie algebra gl(n) admits an interpolation to any complex number n. We do this using the Deligne’s category 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 gl(n|n'), we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy.

Keywords: Bethe ansatz, Gaudin model, Deligne's category, Pseudo-differential operators

Publication based on the results of: Representation theory of vertex algebras and quantum groups with applications to knot theory, integrable systems, geometry of moduli spaces, and enumerative invariants (2024)