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Article

A Proof of the Gaudin Bethe Ansatz Conjecture

International Mathematics Research Notices. 2020. Vol. 2020. No. 22. P. 8766-8785.

The Gaudin algebra is the commutative subalgebra in U(g)^⊗N generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra g⁠. This algebra depends on a collection of pairwise distinct complex numbers z1,…,zN. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible g-modules, for all values of the parameters z1,…,zN⁠. We deduce from this result the Bethe Ansatz conjecture in the Feigin–Frenkel form that states that the joint eigenvalues of the higher Gaudin Hamiltonians on the tensor product of irreducible finite-dimensional g-modules are in 1-1 correspondence with monodromy-free LG-opers on the projective line with regular singularities at the points z1,…,zN,∞⁠, and the prescribed residues at the singular points.