Cyclotomic Gaudin models: construction and Bethe ansatz

TL;DR

This paper constructs a family of cyclotomic Gaudin algebras associated with simple Lie algebras and automorphisms, and develops a Bethe ansatz method to find their eigenvectors, extending known results to a cyclotomic setting.

Contribution

It introduces cyclotomic Gaudin algebras for any simple Lie algebra and automorphism, and generalizes the Bethe ansatz approach to this new context.

Findings

Construction of cyclotomic Gaudin algebras for arbitrary simple Lie algebras.

Development of a Bethe ansatz method for eigenvector construction.

Generalization of the Schechtman-Varchenko formula to the cyclotomic case.

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.