Eigenvalues of Bethe vectors in the Gaudin model

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.

29 pages

References in corpus (2)

• Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence
• On higher order Sugawara operators

Cited by in corpus (5)

• Equivalences between three presentations of orthogonal and symplectic Yangians
• Center of the quantum affine vertex algebra in type A
• Segal-Sugawara vectors for the Lie algebra of type $G_2$
• On the Gaudin model associated to Lie algebras of classical types
• Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert Calculus