Integrals of motion from quantum toroidal algebras

Abstract

We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors. That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model. We also discuss the (gl(m),gl(n)) duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ansatz equations associated to affine sl(2).

arXiv commentLatex, 28 pages, minor revision, some misprints corrected

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• Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy
• The $({\mathfrak{gl}}_m,{\mathfrak{gl}}_n$) duality in the quantum toroidal setting
• The $R$-matrix of the quantum toroidal algebra $U_{q,t}({\stackrel{..}{\mathrm{gl}}}_1)$ in the Fock module
• Towards trigonometric deformation of ${\widehat{\mathfrak{sl}}}_2$ coset VOA
• Integrable ${\mathbb{Z}}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories
• Pentad and triangular structures behind the Racah matrices
• Kondo line defects and affine Gaudin models
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
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• Generalized Gibbs Ensemble of 2D CFTs with U(1) Charge from the AGT Correspondence
• Affine opers and conformal affine Toda
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• Combinatorics of vertex operators and deformed $W$-algebra of type D$(2,1;\alpha )$
• Wreath Macdonald operators
• On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta
• On the Integrable Structure of the SU(2) Wess-Zumino-Novikov-Witten Model
• Quantum Integrability and Quantum Groups: a special issue in memory of Petr P. Kulish