Cyclotomic Gaudin models: construction and Bethe ansatz

Abstract

To any simple Lie algebra $\mathfrak{g}$ and automorphism $\sigma :\mathfrak{g}\to \mathfrak{g}$ we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of $U(\mathfrak{g}{)}^{\otimes N}$ generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case $\sigma =\text{id}$. We go on to construct joint eigenvectors and their eigenvalues for this hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain consisting of a tensor product of Verma modules. To do so we generalize an approach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involving vertex algebras and the Wakimoto construction. As part of this construction, we make use of a theorem concerning cyclotomic coinvariants, which we prove in a companion paper. As a byproduct, we obtain a cyclotomic generalization of the Schechtman-Varchenko formula for the weight function.

51 pages, latex

References in corpus (6)

• Bethe eigenvectors of higher transfer matrices
• On higher order Sugawara operators
• Feigin-Frenkel center in types B, C and D
• Gaudin models for gl(m|n)
• Vertex Lie algebras and cyclotomic coinvariants
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Cited by in corpus (6)

• Local charges in involution and hierarchies in integrable sigma-models
• Cyclotomic Gaudin models with irregular singularities
• Classical $N$-Reflection Equation and Gaudin Models
• 3-dimensional mixed BF theory and Hitchin's integrable system
• Cyclotomic Gaudin models, Miura opers and flag varieties
• Affine opers and conformal affine Toda