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## Cluster Toda Chains and Nekrasov Functions

Theoretical and Mathematical Physics. 2019. Vol. 198. No. 2. P. 157-188.

We extend the relation between cluster integrable systems and q-difference equations beyond the Painlev*é* case. We consider the class of hyperelliptic curves where the Newton polygons contain only four boundary points. We present the corresponding cluster integrable Toda systems and identify their discrete automorphisms with certain reductions of the Hirota difference equation. We also construct nonautonomous versions of these equations and find that their solutions are expressed in terms of five-dimensional Nekrasov functions with Chern–Simons contributions, while these equations in the autonomous case are solved in terms of Riemann theta functions.

Galkin S., Belmans P., Mukhopadhyay S., Graph potentials and moduli spaces of rank two bundles on a curve / 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

Bershtein M., Gavrylenko P., Marshakov A., Journal of High Energy Physics 2018 Vol. 2018 No. 2 P. 1-33

We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of ...

Added: October 14, 2018

Bucher E., Machacek J., Shapiro M., Science China Mathematics 2019 Vol. 62 No. 7 P. 1257-1266

We initiate a study of the dependence of the choice of ground ring on the problem on whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and upper cluster algebra is given by using a variation of Muller’s theory of cluster localization. ...

Added: February 25, 2021

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

Тонконог Д. И., Singularities of integrable Hamiltonian systems: a criterion for non-degeneracy, with an application to the Manakov top / Cornell University. Series math "arxiv.org". 2012.

Let (M,\omega) be a symplectic 2n-manifold and h_1,...,h_n be functionally independent commuting functions on M. We present a geometric criterion for a singular point P\in M (i.e. such that {dh_i(P)}_{i=1}^n are linearly dependent) to be non-degenerate in the sence of Vey-Eliasson.
Then we apply Fomenko's theory to study the neighborhood U of the singular Liouville fiber ...

Added: February 20, 2013

Levin A., Olshanetsky M., Smirnov A. V. et al., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2012 Vol. 8 No. 095

We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint G-bundles of different topological types over complex curves Sigma(g,n) of genus g with n marked points. The bundles are defined by their characteristic classes - elements of H-2 (Sigma(g,n), Z(G)), where Z (G) is a center of the ...

Added: February 7, 2013

Zotov A., Levin A., Olshanetsky M., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2009 Vol. 5 No. 065 P. 1-22

Modifications of bundles over complex curves is an operation that allows one to
construct a new bundle from a given one. Modifications can change a topological type of
bundle. We describe the topological type in terms of the characteristic classes of the bundle.Being applied to the Higgs bundles modifications establish an equivalence between different classical integrable systems. ...

Added: October 15, 2012

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

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

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

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

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

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

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

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

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

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

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

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

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

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., 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

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