Gaudin Model, Bethe Ansatz and Critical Level

TL;DR

This paper introduces a novel diagonalization method for Gaudin model Hamiltonians using Wakimoto modules at the critical level, linking integrable models with conformal field theory through bosonic correlation functions.

Contribution

It develops a new approach to diagonalize Gaudin Hamiltonians via Wakimoto modules, connecting algebraic Bethe ansatz with conformal field theory techniques.

Findings

Eigenvectors constructed from invariant functionals on Wakimoto modules.

Bethe ansatz equations appear as Kac-Kazhdan equations.

Connection established between Gaudin models and WZNW correlation functions.

Abstract

We propose a new method of diagonalization of hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on Wakimoto modules over affine algebras at the critical level. We construct eigenvectors of these hamiltonians by restricting certain invariant functionals on tensor products of Wakimoto modules. In conformal field theory language, the eigenvectors are given by certain bosonic correlation functions. Analogues of Bethe ansatz equations naturally appear as Kac-Kazhdan type equations on the existence of certain singular vectors in Wakimoto modules. We use this construction to expalain a connection between Gaudin's model and correlation functions of WZNW models.