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Cyclotomic Gaudin models: construction and Bethe ansatz

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JournalCommunications in Mathematical Physics
DateAccepted/In press - 13 Feb 2016
DateE-pub ahead of print - 24 Mar 2016
DatePublished (current) - Mar 2016
Issue number3
Volume343
Number of pages53
Pages (from-to)971-1024
Early online date24/03/16
Original languageEnglish

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.

Bibliographical note

This is a pre-copyedited author produced PDF of an article accepted for publication in Communications in Mathematical Physics, Benoit, V and Young, C, 'Cyclotomic Gaudin models: construction and Bethe ansatz', Commun. Math. Phys. (2016) 343:971, first published on line March 24, 2016.

The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-016-2601-3

© Springer-Verlag Berlin Heidelberg 2016

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