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Title of Journal: J High Energ Phys

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Abbravation: Journal of High Energy Physics

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Springer Berlin Heidelberg

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DOI

10.1007/978-3-319-18257-5_28

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1029-8479

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Planck constant as spectral parameter in integrabl

Authors: A Levin M Olshanetsky A Zotov
Publish Date: 2014/10/20
Volume: 2014, Issue: 10, Pages: 109-
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Abstract

We construct special rational gl N KnizhnikZamolodchikovBernard KZB equations with Ñ punctures by deformation of the corresponding quantum gl N rational Rmatrix They have two parameters The limit of the first one brings the model to the ordinary rational KZ equation Another one is τ At the level of classical mechanics the deformation parameter τ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems Next we notice that the identities underlying generic elliptic KZB equations follow from some additional relations for the properly normalized Rmatrices The relations are noncommutative analogues of identities for scalar elliptic functions The simplest one is the unitarity condition The quadratic in R matrices relations are generated by noncommutative Fay identities In particular one can derive the quantum YangBaxter equations from the Fay identities The cubic relations provide identities for the KZB equations as well as quadratic relations for the classical rmatrices which can be treated as halves of the classical YangBaxter equation At last we discuss the Rmatrix valued linear problems which provide gl Ñ CM models and Painlevé equations via the above mentioned identities The role of the spectral parameter plays the Planck constant of the quantum Rmatrix When the quantum gl N Rmatrix is scalar N = 1 the linear problem reproduces the Krichever’s ansatz for the Lax matrices with spectral parameter for the gl Ñ CM models The linear problems for the quantum CM models generalize the KZ equations in the same way as the Lax pairs with spectral parameter generalize those without itThis article is published under an open access license Please check the Copyright Information section for details of this license and what reuse is permitted If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and reuse information please contact the Rights and Permissions team

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