Research output: Contribution to journal › Article › peer-review

**Cyclotomic Gaudin models: construction and Bethe ansatz.** / Vicedo, Benoit; Young, Charles A. S.

Research output: Contribution to journal › Article › peer-review

Vicedo, B & Young, CAS 2016, 'Cyclotomic Gaudin models: construction and Bethe ansatz', *Communications in Mathematical Physics*, vol. 343, no. 3, pp. 971-1024. https://doi.org/10.1007/s00220-016-2601-3

Vicedo, B., & Young, C. A. S. (2016). Cyclotomic Gaudin models: construction and Bethe ansatz. *Communications in Mathematical Physics*, *343*(3), 971-1024. https://doi.org/10.1007/s00220-016-2601-3

Vicedo B, Young CAS. Cyclotomic Gaudin models: construction and Bethe ansatz. Communications in Mathematical Physics. 2016 Apr 22;343(3):971-1024. https://doi.org/10.1007/s00220-016-2601-3

@article{f98778c92ec947c4846f17e7c231fcdb,

title = "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.",

keywords = "math.QA",

author = "Benoit Vicedo and Young, {Charles A. S.}",

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 {\textcopyright} Springer-Verlag Berlin Heidelberg 2016",

year = "2016",

month = apr,

day = "22",

doi = "10.1007/s00220-016-2601-3",

language = "English",

volume = "343",

pages = "971--1024",

journal = "Communications in Mathematical Physics",

issn = "0010-3616",

publisher = "Springer New York",

number = "3",

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T1 - Cyclotomic Gaudin models: construction and Bethe ansatz

AU - Vicedo, Benoit

AU - Young, Charles A. S.

N1 - 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

PY - 2016/4/22

Y1 - 2016/4/22

N2 - 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.

AB - 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.

KW - math.QA

U2 - 10.1007/s00220-016-2601-3

DO - 10.1007/s00220-016-2601-3

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JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

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