# Cyclotomic Gaudin models: construction and Bethe ansatz

Benoit Vicedo, Charles A. S. Young

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

## 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.
Original language English 971-1024 53 Communications in Mathematical Physics 343 3 24 Mar 2016 https://doi.org/10.1007/s00220-016-2601-3 Published - 22 Apr 2016

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