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A Proof of the Gaudin Bethe Ansatz Conjecture
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.