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## Complex hypergeometric functions and integrable many body problems

Cornell University
,
2021.
No. arXiv:2105.15031.

Spiridonov V., Sarkissian G. A.

General reduction of the elliptic hypergeometric equation to the level of complex hypergeometric functions is described. Derived equation is generalized to the Hamiltonian eigenvalue problem for new rational integrable N-body systems emerging from particular degenerations of the elliptic Ruijsenaars and van Diejen models.

Keywords: integrable systemsdifference equationsразностные уравнениядискретные интегрируемые моделиГипергеометрические функцииHypergeometric functions

Publication based on the results of:

Esterov A. I., Takeuchi K., Nagoya Mathematical Journal 2018 Vol. 231 P. 1-22

We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials. In particular, we extend one of the results of Gelfand et al. [Generalized Euler integrals and A-hypergeometric functions, Adv. Math. 84 (1990), 255–271] to various directions. In the course of the proof, some properties of vanishing cycles of perverse sheaves ...

Added: October 31, 2018

Frolenkov D., Journal of Number Theory 2020 Vol. 207 No. 2 P. 247-281

We prove an explicit formula for the cubic moment of central values of automorphic
$L$-functions associated to primitive cusp forms of level one and large weight.
The resulting explicit formula contains the main term predicted by the random matrix theory conjectures, while the error term is expressed as the fourth moment of the Riemann zeta function weighted ...

Added: September 25, 2020

Balkanova O., Bhowmik G., Frolenkov D. et al., Proceedings of the London Mathematical Society 2020 Vol. 121 No. 2 P. 177-219

Let $ \mathfrak{f} $ run over the set $ H_{4k} $ of primitive cusp forms of level one and weight $ 4k $, $ k \in \N $.
We prove an explicit formula for the mixed moment of the Hecke $L$-function $ L(\mathfrak{f}, 1/2) $ and the symmetric square $L$-function $L(\sym^2\mathfrak{f}, 1/2)$, relating it to the ...

Added: September 25, 2020

Kharchev S., Levin A., Olshanetsky M. et al., Journal of Mathematical Physics 2018 Vol. 59 No. 103509 P. 1-36

We define the quasi-compact Higgs G -bundles over singular curves introduced in our previous paper for the Lie group SL(N). The quasi-compact structure means that the automorphism groups of the bundles are reduced to the maximal compact subgroups of G at marked points of the curves. We demonstrate that in particular cases, this construction leads ...

Added: October 20, 2018

Derbyshev A. E., Povolotsky A. M., Priezzhev V. B., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2015 Vol. 91 P. 022125

The generalized totally asymmetric exclusion process (TASEP) [J. Stat. Mech. (2012) P05014] is an integrable generalization of the TASEP equipped with an interaction, which enhances the clustering of particles. The process interpolates between two extremal cases: the TASEP with parallel update and the process with all particles irreversibly merging into a single cluster moving as ...

Added: February 19, 2015

Krichever I. M., Функциональный анализ и его приложения 2012 Т. 46 № 2 С. 37-51

Using meromorphic differentials with real periods, we prove Arbarello's conjecture that any compact complex cycle of dimension g−n in the moduli space M_g of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n. ...

Added: April 17, 2014

Sechin I., Zotov A., Physics Letters B 2018 Vol. 781 P. 1-7

In this paper we discuss R -matrix-valued Lax pairs for sl N Calogero-Moser model and their relation to integrable quantum long-range spin chains of the Haldane-Shastry-Inozemtsev type. First, we construct the R -matrix-valued Lax pairs for the third flow of the classical Calogero-Moser model. Then we notice that the scalar parts (in the auxiliary space) of the M -matrices ...

Added: September 18, 2018

Булеков А. А., Journal of Physics: Conference Series 2021 Vol. 1740 Article 012069

The paper is devoted to the construction of spectral series and the estimation of the approximation accuracy for the operator of the Curie – Weiss model. In the course of work, the operator is reduced to a tridiagonal form in the subspace of the original space, then to a secondorder difference equation. The admissibility of ...

Added: June 9, 2021

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

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

A. Zabrodin, A. Zotov, 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

Khoroshkin S. M., Tsuboi Z., Journal of Physics A: Mathematical and Theoretical 2014 Vol. 47 P. 1-11

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the Uq(\hat{sl}(2)) case. ...

Added: December 8, 2014

Nirov Khazret S., Razumov A. V., Journal of Geometry and Physics 2017 Vol. 112 P. 1-28

A detailed construction of the universal integrability objects related to the integrable
systems associated with the quantum loop algebra Uq(L(sl2)) is given. The full proof of the
functional relations in the form independent of the representation of the quantum loop
algebra on the quantum space is presented. The case of the general gradation and general
twisting is treated. The ...

Added: January 29, 2018

Marshall I., Letters in Mathematical Physics 2017 Vol. 107 No. 4 P. 619-642

Presentation of a method for generating Lax pairs for systems obtaibed by means of Hamilton reduction ...

Added: December 8, 2016

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

Duval C., Shevchishin V., Valent G., Journal of Geometry and Physics 2015 Vol. 87 P. 461-481

We obtain, in local coordinates, the explicit form of the two-dimensional, superintegrable
systems of Matveev and Shevchishin involving linear and cubic integrals. This enables us
to determine for which values of the parameters these systems are indeed globally defined
on S^2. ...

Added: March 23, 2015

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

Povolotsky A. M., Journal of Physics A: Mathematical and Theoretical 2013 Vol. 46 No. 46 P. 465205

The conditions of the integrability of general zero range chipping models with factorized steady states, which were proposed in Evans et al (2004 J. Phys. A: Math. Gen. 37 L275), are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle ...

Added: November 14, 2013

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

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

Синцова К. А., Лялинов М. А., Wave Motion, 2018

In this work we construct and discuss special solutions of a homogeneous problem for the Laplace
equation in a domain with the cone-shaped boundaries. The problem at hand is interpreted as that
describing oscillatory linear wave movement of a uid under gravity in such a domain. These solutions are found in terms of the Mellin transform and ...

Added: November 27, 2019

Marshakov A., Семенякин Н. С., Journal of High Energy Physics 2019 Vol. 100 No. 10 P. 1-52

We discuss relation between the cluster integrable systems and spin chains in the context of their correspondence with 5d supersymmetric gauge theories. It is shown that glN XXZ-type spin chain on M sites is isomorphic to a cluster integrable system with N × M rectangular Newton polygon and N × M fundamental domain of a ...

Added: October 21, 2019

Povolotsky A. M., Journal of Statistical Mechanics: Theory and Experiment 2019 No. 074003 P. 1-22

We establish the exact laws of large numbers for two time additive quantities in the raise and peel model, the number of tiles removed by avalanches and the number of global avalanches happened by given time. The validity of conjectures for the related stationary state correlation functions then follow. The proof is based on the ...

Added: October 8, 2019