A proof of the Gaudin Bethe Ansatz conjecture

TL;DR

This paper proves the Gaudin Bethe Ansatz conjecture by demonstrating the existence of a cyclic vector in the Gaudin algebra's action on tensor products of irreducible modules, establishing a correspondence with monodromy-free opers.

Contribution

It provides a proof of the Gaudin Bethe Ansatz conjecture, linking eigenvalues of Gaudin Hamiltonians to monodromy-free opers, confirming a key conjecture in integrable systems and representation theory.

Findings

Existence of a cyclic vector in the Gaudin algebra

Bijective correspondence between eigenvalues and monodromy-free opers

Validation of the Bethe Ansatz conjecture in the Feigin-Frenkel form

Abstract

Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of pairwise distinct complex numbers $z_1,\ldots,z_N$. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible $\mathfrak{g}$-modules, for all values of the parameters $z_1,\ldots,z_N$. We deduce from this result the Bethe Ansatz conjecture in the Feigin-Frenkel form which states that the joint eigenvalues of the higher Gaudin Hamiltonians on the tensor product of irreducible finite-dimensional $\mathfrak{g}$-modules are in 1-1 correspondence with monodromy-free ${}^LG$-opers on the projective line with regular singularities at the points $z_1,\ldots,z_N,\infty$ and the prescribed residues at the singular points.