The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz

2024/12/31 by Cheong, Wan Keng, Lam, Ngau · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2501.00401

Abstract

Let \mathfrakB_m|n(\underline\boldsymbolz) be the Gaudin algebra of the general linear Lie superalgebra \mathfrakgl_m|n with respect to a sequence \underline\boldsymbolz ∈ ℂ^ℓ of pairwise distinct complex numbers, and let M be any ℓ-fold tensor product of irreducible polynomial modules over \mathfrakgl_m|n. We show that the singular space M^{\rm sing} of M is a cyclic \mathfrakB_m|n(\underline\boldsymbolz)-module and the Gaudin algebra \mathfrakB_m|n(\underline\boldsymbolz)_{M^{\rm sing}} of M^{\rm sing} is a Frobenius algebra. We also show that \mathfrakB_m|n(\underline\boldsymbolz)_{M^{\rm sing}} is diagonalizable with a simple spectrum for a generic \underline\boldsymbolz and give a description of an eigenbasis and its corresponding eigenvalues in terms of the Fuchsian differential operators with polynomial kernels. This may be interpreted as the completeness of a reformulation of the Bethe ansatz for \mathfrakB_m|n(\underline\boldsymbolz)_{M^{\rm sing}}.

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