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.

Segal–Sugawara vectors for the Lie algebra of type G2

“…. ⊗ M λ ℓ for the commutative subalgebra A(g) µ in this tensor product can be found in [9] (also reproduced in [19]). They are parameterized by a set of distinct complex numbers w 1 , .…”

“…Furthermore, using the connections with the Gaudin model as discovered in [9], we get formulas for higher Gaudin Hamiltonians associated with g and calculate their eigenvalues on the Bethe vectors; see also [10]. In the classical types such formulas were given in a recent work [19].…”

The Yangian relations of Heisenberg spin chain model

“…Yangian represents a central concept within the framework of the physical integrable model and its mathematic foundation. Yangian algebra has many implementations in some areas and physical models, such as Lie algebras 10 , 13 – 17 , the scattering matrices and reflection matrices of the Yangian 18 , 19 , the RTT realization of Yangian with D-type Lie algebra 20 , the theory of classical W-algebras and affine vertex algebras 21 – 23 , two-dimensional example (Gross-Neveu model) 24 , Hubbard model 25 – 27 and integrable spin chains 28 .…”

Center of the quantum affine vertex algebra in type A

“…Explicit formulas for generators of the Feigin-Frenkel center z( g) were given in [2] and [3] for type A (see also [25]), in [23] for types B, C and D; and in [26] for type G 2 . Due to general results of [8], [9] and [29], these formulas lead to explicit constructions of commutative subalgebras of the universal enveloping algebra U(g) and to explicit higher order Hamiltonians and their eigenvalues on the Bethe vectors in the Gaudin model associated with g; see also [14], [24].…”