Bethe algebra of the gl_{N+1} Gaudin model and algebra of functions on the critical set of the master function

TL;DR

This paper explores the relationship between the Bethe algebra of the gl_{N+1} Gaudin model and the algebra of functions on the critical set of the master function, establishing an isomorphism and proving linear independence of Bethe vectors.

Contribution

It demonstrates an isomorphism between the algebra of functions on the critical set and the Bethe algebra, and proves linear independence of Bethe vectors.

Findings

Isomorphism between function algebra and Bethe algebra established

Linear independence of Bethe vectors proved

Nonzero Bethe vectors for different critical points confirmed

Abstract

Consider a tensor product of finite-dimensional irreducible gl_{N+1}-modules and its decomposition into irreducible modules. The gl_{N+1} Gaudin model assigns to each multiplicity space of that decomposition a commutative (Bethe) algebra of linear operators acting on the multiplicity space. The Bethe ansatz method is a method to find eigenvectors and eigenvalues of the Bethe algebra. One starts with a critical point of a suitable (master) function and constructs an eigenvector of the Bethe algebra. In this paper we consider the algebra of functions on the critical set of the associated master function and show that the action of this algebra on itself is isomorphic to the action of the Bethe algebra on a suitable subspace of the multiplicity space. As a byproduct we prove that the Bethe vectors corresponding to different critical points of the master function are linearly independent and, in particular, nonzero.