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## The master T-operator for the Gaudin model and KP hierarchy.

Nuclear Physics B. 2014. Vol. 883 . No. . P. 173-223.

Following the approach of [Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages, arXiv:1112.3310], we construct the master T-operator for the quantum Gaudin model with twisted boundary conditions and show that it satisfies the bilinear identity and Hirota equations for the classical KP hierarchy. We also characterize the class of solutions to the KP hierarchy that correspond to eigenvalues of the master T-operator and study dynamics of their zeros as functions of the spectral parameter. This implies a remarkable connection between the quantum Gaudin model and the classical Calogero-Moser system of particles.

Zabrodin A., Proceedings of Physics and Mathematics of Nonlinear Phenomena 2014 Vol. 482 No. 012047 P. 10

This short note is a review of the intriguing connection between the quantum Gaudin model and the classical KP hierarchy recently established in [A.Alexandrov, S.Leurent, Z.Tsuboi, A.Zabrodin, The master T-operator for the Gaudin model and KP hierarchy, Nuclear Physics B 883 (2014) 173-223]. We construct the generating function of integrals of motion for the quantum ...

Added: July 15, 2014

Gorsky A., Zabrodin A., Zotov A., Journal of High Energy Physics 2014 No. 01 P. 070,28

In this paper we clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous glninvariant XXX spin chain on N sites with twisted ...

Added: July 15, 2014

Zabrodin A., (Mathematical Sciences 2013 No. 596 P. 7-12

We review the role of the Hirota equation and the tau-function in the theory of classical and quantum integrable systems. ...

Added: February 16, 2013

Васильев М., Zabrodin A., Zotov A., Nuclear Physics B - Proceedings Supplements 2020 Vol. 952 No. 114931 P. 1-20

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all ...

Added: August 20, 2020

Zabrodin A., Alexandrov A., Kazakov V. et al., Journal of High Energy Physics 2013 Vol. 09 P. 064

For an arbitrary generalized quantum integrable spin chain we introduce a “master T-operator” which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of ...

Added: November 14, 2013

Zabrodin A., Alexandrov A., Journal of Geometry and Physics 2013 Vol. 67 P. 37-80

We review the formalism of free fermions used for construction of tau-functions of classical integrable hierarchies and give a detailed derivation of group-like properties of the normally ordered exponents, transformations between different normal orderings, the bilinear relations, the generalized Wick theorem and the bosonization rules. We also consider various examples of tau-functions and give their ...

Added: February 16, 2013

Zabrodin A., The master T-operator for vertex models with trigonometric R-matrices as classical tau-function / ИТЭФ. Series "ITEP-TH-17/12". 2012. No. 17.

The construction of the master T-operator recently suggested is applied to integrable vertex models and associated quantum spin chains with trigonometric R-matrices. The master T-operator is a generating function for commuting transfer matrices of integrable vertex models depending on infinitely many parameters. At the same time it turns out to be the tau-function of an ...

Added: May 24, 2012

Zabrodin A., Zotov A., Nuclear Physics B 2018 Vol. 927 P. 550-565

We discuss a self-dual form or the Backlund transformations for the continuous (in time variable) glN Ruijsenaars-Schneider model. It is based on the first order equations in N+M complex variables which include N positions of particles and M dual variables. The latter satisfy equations of motion of the glM Ruijsenaars-Schneider model. In the elliptic case ...

Added: February 15, 2018

Pyatov P. N., de Gier J., Zinn-Justin P., Journal of Combinatorial Theory, Series A 2009 Vol. 116 P. 772-794

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik–Zamolodchikov equation with reflecting boundaries in the Dyck path representation of the Temperley–Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ 2-weighted ...

Added: October 16, 2012

Pyatov P. N., Ogievetsky O., Cayley-Hamilton Theorem for Symplectic Quantum Matrix Algebras / Cornell University. Series math "arxiv.org". 2020.

We establish the analogue of the Cayley--Hamilton theorem for the quantum matrix algebras of the symplectic type. ...

Added: January 26, 2021

Feigin B. L., Jimbo M., Mukhin E., Communications in Mathematical Physics 2019 No. 367 P. 455-481

On a Fock space constructed from mn free bosons and lattice Z mn , we give a level n action of the quantum toroidal algebra E m associated to gl m , together with a level m action of the quantum toroidal algebra E n associated to gl n . We prove that the E ...

Added: December 10, 2019

Buryak A., Rossi P., Communications in Mathematical Physics 2016 Vol. 342 No. 2 P. 533-568

In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced by the first author using intersection theory of the double ramification cycle in the moduli space of stable curves. In particular, we prove a recursion formula that recovers the full hierarchy starting from just one of the ...

Added: September 28, 2020

Marshakov A., International Journal of Modern Physics A 2013 Vol. 28 No. 3-4 P. 1340007

We propose an explicit construction for the integrable models on Poisson submanifolds of the Lie groups. The integrals of motion are computed in cluster variables via the Lax map. This generalized construction for the co-extended loop groups allows to formulate, in general terms, some new classes of integrable models. ...

Added: March 28, 2013

Ogievetsky O., Pyatov P. N., Journal of Geometry and Physics 2021 Vol. 165 Article 104211

We establish the analogue of the Cayley–Hamilton theorem for the quantum matrix algebras of the symplectic type. We construct the algebra in which the quantum characteristic polynomial acquires a factorized form. The low-dimensional examples and the classical limit are discussed. ...

Added: March 18, 2021

Gurevich D., Rubtsov V., Saponov P. A. et al., Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics 2015 Vol. 16 No. 7 P. 1689-1707

The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra. Motivated by this analogy, we realize a braiding of the mentioned Poisson structure, i.e. we introduce a “braided Poisson” algebra associated with an involutive solution to the quantum Yang–Baxter equation. Also, we ...

Added: September 8, 2014

Gavrylenko P., Journal of High Energy Physics 2015 No. 09 P. 167

We study the solution of the Schlesinger system for the 4-point $\mathfrak{sl}_N$ isomonodromy problem and conjecture an expression for the isomonodromic τ-function in terms of 2d conformal field theory beyond the known N = 2 Painlevé VI case. We show that this relation can be used as an alternative definition of conformal blocks for the ...

Added: October 9, 2015

Zabrodin A., Zotov A., Journal of Physics A: Mathematical and Theoretical 2017 Vol. 50 No. 20 P. 1-12

We discuss the correspondence between the Knizhnik–Zamolodchikov equations associated with GL(N) and the n-particle quantum Calogero model in the case when n is not necessarily equal to N. This can be viewed as a natural 'quantization' of the quantum-classical correspondence between quantum Gaudin and classical Calogero models. ...

Added: June 7, 2017

Alexandrov A., Mironov A., Morozov A. et al., Journal of High Energy Physics 2014 Vol. 11 No. 80 P. 1-31

There is now a renewed interest to a Hurwitz tau-function, counting the
isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and
Grothiendicks’s dessins d’enfant. It is distinguished by belonging to a particular family
of Hurwitz tau-functions, possessing conventional Toda/KP integrability properties. We
explain how the variety of recent observations about this function fits into the ...

Added: December 2, 2014

Marshall I., International Mathematics Research Notices 2015 Vol. 18 P. 8925-8958

A Poisson structure is defined on the space {\mathcal {W}} of twisted polygons in {\mathbb {R}}^{\nu }. Poisson reductions with respect to two Poisson group actions on {\mathcal {W}} are described. The \nu =2 and \nu =3 cases are discussed in detail. Amongst the Poisson structures arising in examples are to be found the lattice ...

Added: November 28, 2014

Mironov A., Morozov A., Natanzon S. M., Journal of High Energy Physics 2011 No. 11(097) P. 1-31

Correlators in topological theories are given by the values of a linear form on the products of operators from a commutative associative algebra (CAA). As a corollary, partition functions of topological theory always satisfy the generalized WDVV equations of. We consider the Hurwitz partition functions, associated in this way with the CAA of cut-and-join operators. ...

Added: October 12, 2012

Marshakov A., Journal of Geometry and Physics 2012 Vol. 003 P. 16-36

We discuss the Poisson structures on Lie groups and propose an explicit construction of the integrable models on their appropriate Poisson submanifolds. The integrals of motion for the SL(N)-series are computed in cluster variables via the Lax map. This construction, when generalised to the co-extended loop groups, gives rise not only to alternative descriptions of relativistic Toda systems, but allows ...

Added: February 11, 2013

Takebe T., Tokyo: Research Center for Mathematical Physics, Rikkyo Universty, 2014

This is a lecture note based on the series of lectures on the dispersionless integrable hierarchies delivered by the authore in June, 2013, at the Rikkyo University, Tokyo, Japan. The contents are survey on dispersionless integrable hierarchies, including introduction to integrable systems in general, and on their connections with complex analysis. ...

Added: June 21, 2014

Buryak A., Dubrovin B., Guere J. et al., International Mathematics Research Notices 2020 Vol. 2020 No. 24 P. 10381-10446

In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the 1st author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus 1 quantum correction and, as an application, compute completely the quantization ...

Added: April 21, 2020

Marshakov A., Миронов А. Д., Морозов А. Ю., Journal of Geometry and Physics 2011 Vol. 61 P. 1203-1222

We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block ...

Added: February 28, 2013