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## Quantum Gaudin model and classical KP hierarchy

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 Gaudin model with twisted boundary conditions (the master T-operator) and show that it satisfies the bilinear identity and Hirota equations for the classical KP hierarchy. This implies that zeros of eigenvalues of the master T-operator in the spectral parameter have the same dynamics as the Calogero-Moser system of particles.

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., Alexandrov A., Leurent S. et al., 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 ...

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

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

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

Gavrylenko P., Santachiara R., Journal of High Energy Physics 2019 Vol. 2019 No. 11 P. 1-36

We present an approach that gives rigorous construction of a class of crossing invariant functions in c = 1 CFTs from the weakly invariant distributions on the moduli space \( {\mathcal{M}}_{0,4}^{\mathrm{SL}\left(s,\mathbb{C}\right)} \) of SL(2, ℂ) flat connections on the sphere with four punctures. By using this approach we show how to obtain correlation functions in the Ashkin-Teller and the Runkel- ...

Added: May 14, 2020

Tsuboi Z., Zabrodin A., Zotov A., Journal of High Energy Physics 2015 Vol. 2015 No. 5, Article number 86

For integrable inhomogeneous supersymmetric spin chains (generalized graded magnets) constructed employing Y(gl(N|M))-invariant R-matrices in finite-dimensional representations we introduce the master T-operator which is a sort of generating function for the family of commuting quantum transfer matrices. Any eigenvalue of the master T-operator is the tau-function of the classical mKP hierarchy. It is a polynomial in ...

Added: September 7, 2015

Zabrodin A., Journal of Physics A: Mathematical and Theoretical 2013 Vol. 46 No. 18 P. 185203

We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in a transverse direction. Similarly to the Laplacian growth in radial geometry, this problem can be embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. However, the relevant solution ...

Added: April 29, 2013

Zabrodin A., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2014 Vol. 10 No. 006 P. 18

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], we show how to construct the master T-operator for the quantum inhomogeneous GL(N) XXX spin chain with twisted boundary conditions. It satisfies the bilinear identity and Hirota equations for the ...

Added: July 15, 2014

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

Alexandrov A., Mironov A., Morozov A. et al., Journal of Physics A: Mathematical and Theoretical 2012 No. 45 P. 1-10

We construct partition functions that are tau-functions of integrable hierarchies. ...

Added: September 19, 2012

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

Smirnov A., Matveenko S., Semenova E., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2015 Vol. 11

In the article, we describe three-phase finite-gap solutions of the focusing nonlinear Schrödinger equation and Kadomtsev-Petviashvili and Hirota equations that exhibit the behavior of almost-periodic ''freak waves''. We also study the dependency of the solution parameters on the spectral curves. ...

Added: October 15, 2015

Gavrylenko P., Marshakov A., / Cornell University. Series "Working papers by Cornell University". 2015. No. 1507.08794.

We consider the conformal blocks in the theories with extended conformal W-symmetry for the integer Virasoro central charges. We show that these blocks for the generalized twist fields on sphere can be computed exactly in terms of the free field theory on the covering Riemann surface, even for a non-abelian monodromy group. The generalized twist ...

Added: October 14, 2015

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

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

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

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

Gavrylenko P., Marshakov A., Journal of High Energy Physics 2014 No. 5 P. 97

We study the extended prepotentials for the S-duality class of quiver gauge theories, considering them as quasiclassical tau-functions, depending on gauge theory condensates and bare couplings. The residue formulas for the third derivatives of extended prepotentials are proven, which lead to effective way of their computation, as expansion in the weak-coupling regime. We discuss also ...

Added: October 20, 2014

Pyatov P. N., Ogievetsky O., / 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

Zabrodin A., Journal of Mathematical Physics 2019 Vol. 60 P. 033502-1-033502-14

We introduce the discrete time version of the spin Calogero-Moser system. The equations of motion follow from the dynamics of poles ofrational solutions to the matrix Kadomtsev-Petviashvili hierarchy with discrete time. The dynamics of poles is derived using the auxiliarylinear problem for the discrete flow ...

Added: May 30, 2019

Zabrodin A., / 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

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021