Eigenvalues of Bethe vectors in the Gaudin model

TL;DR

This paper explicitly calculates the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors using Harish-Chandra images, connecting affine vertex algebra centers, classical $W$-algebras, and $q$-characters.

Contribution

It combines recent calculations of Harish-Chandra images with known theorems to explicitly determine eigenvalues in the Gaudin model, linking $W$-algebras and $q$-characters.

Findings

Explicit formulas for eigenvalues of Gaudin Hamiltonians.

Connection established between $q$-characters and classical $W$-algebras.

Harish-Chandra images interpreted as elements of classical $W$-algebras.

Abstract

A theorem of Feigin, Frenkel and Reshetikhin provides expressions for the eigenvalues of the higher Gaudin Hamiltonians on the Bethe vectors in terms of elements of the center of the affine vertex algebra at the critical level. In our recent work, explicit Harish-Chandra images of generators of the center were calculated in all classical types. We combine these results to calculate the eigenvalues of the higher Gaudin Hamiltonians on the Bethe vectors in an explicit form. The Harish-Chandra images can be interpreted as elements of classical $W$-algebras. We provide a direct connection between the rings of $q$-characters and classical $W$-algebras by calculating classical limits of the corresponding screening operators.