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## Complex hypergeometric functions and integrable many-body problems

Journal of Physics A: Mathematical and Theoretical. 2022. Vol. 55. No. 38. Article 385203.

Sarkissian G., Spiridonov V.

General reduction of the elliptic hypergeometric equation to the level of complex hypergeometric functions is described. The 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: интегрируемые системыintegrable systems elliptic hypergeometric equationcomplex hypergeometric functionэллиптическое гипергеометрическое уравнениекомплексная гипергеометрическая функция

Publication based on the results of:

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

Arsie A., Buryak A., Lorenzoni P. et al., Communications in Mathematical Physics 2021 Vol. 388 P. 291-328

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of an F-CohFT starting from a semisimple ...

Added: October 29, 2021

Marshakov A., Fock V., / Cornell University. Series math "arxiv.org". 2014.

We describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups PGLˆ(N), which can be enumerated by cyclically irreducible elements the co-extended affine Weyl groups (Wˆ×Wˆ)♯. Their phase spaces admit cluster coordinates, whereas the integrals of motion are cluster functions. We show, that this class of integrable systems coincides with the ...

Added: October 29, 2014

Galkin S., Belmans P., Mukhopadhyay S., / Cornell University. Series math "arxiv.org". 2020. No. 2009.05568.

We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. These graphs encode degenerations of curves to rational curves, and graph potentials encode degenerations of the moduli space of rank 2 bundles with fixed determinant. We show that the birational type of the graph potential only depends on the homotopy type of ...

Added: April 15, 2021

Kazaryan M., Lando S., Успехи математических наук 2015 Т. 70 № 3 С. 70-106

This paper reviews modern approaches to the construction of formal solutions to integrable hierarchies of mathematical physics whose coefficients are answers to various enumerative problems. The relationship between these approaches and the combinatorics of symmetric groups and their representations is explained. Applications of the results to the construction of efficient computations in problems related to ...

Added: September 21, 2015

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

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

Маршал Я., International Mathematics Research Notices 2014 Vol. 252

A Poisson structure is defined on the space W of twisted polygons in R^\nu. Poisson reductions with respect to two Poisson group actions on 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 Virasoro structure, the second Toda lattice ...

Added: November 28, 2014

Kamnitzer J., Halacheva I., Weekes A. et al., / Cornell University. Series math "arxiv.org". 2017.

Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for g-representations. These algebras depend on a parameter which is a point in the Deligne-Mumford moduli space of marked stable genus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the ...

Added: October 17, 2017

Buryak A., Rossi P., Advances in Mathematics 2021 Vol. 386 No. 6 Article 107794

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's r-spin classes. They are parameterized by a phase space which has one extra dimension and in genus 0 they correspond to the extended r-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while ...

Added: October 29, 2021

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

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

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

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

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

Buryak A., Dubrovin B., Guere J. et al., Communications in Mathematical Physics 2018 Vol. 363 No. 1 P. 191-260

In this paper we continue the study of the double ramification hierarchy introduced by the first author. After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton and divisor equations plus some important degree ...

Added: September 27, 2020

Grekov A., Zotov A., Journal of Physics A: Mathematical and Theoretical 2018 Vol. 51 No. 315202 P. 1-26

The article is devoted to the study of R-matrix-valued Lax pairs for N-body (elliptic) Calogero-Moser models. Their matrix elements are given by quantum GLN~ R-matrices of Baxter-Belavin type. For N~=1 the widely known Krichever's Lax pair with spectral parameter is reproduced. First, we construct the R-matrix-valued Lax pairs for Calogero-Moser models associated with classical root systems. For this purpose we study generalizations ...

Added: September 18, 2018

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

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

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

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

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

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

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