A proof of the Gaudin Bethe Ansatz conjecture

2016/08/16 by Rybnikov, Leonid · 1 citation
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1608.04625

Abstract

Gaudin algebra is the commutative subalgebra in U(\mathfrakg)^{⊗ N} generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra \mathfrakg. This algebra depends on a collection of pairwise distinct complex numbers z₁,…,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 \mathfrakg-modules, for all values of the parameters z₁,…,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 \mathfrakg-modules are in 1-1 correspondence with monodromy-free ^LG-opers on the projective line with regular singularities at the points z₁,…,z_N,∞ and the prescribed residues at the singular points.

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