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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 Painlevé 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.
Buryak A., Rossi P., International Mathematics Research Notices 2025 Vol. 2025 No. 20 P. 1–21
The Riemann hierarchy is the simplest example of rank one, (+)-dimensional integrable system of nonlinear evolutionary PDEs. It corresponds to the dispersionless limit of the Korteweg–de Vries hierarchy. In the language of formal variational calculus, we address the classification problem for deformations of the Riemann hierarchy satisfiying different extra requirements (general deformations, deformations as systems ...
Added: November 27, 2025
E. Ievlev, Pichugina P., A. Yung, International Journal of Modern Physics A 2025 Vol. 40 No. 24 Article 2550091
It has been demonstrated that the non-Abelian solitonic vortex string in four-dimensional (4D) N=2 supersymmetric QCD (SQCD) with gauge group U(2) and Nf=4 quark hypermultiplets behaves as a critical superstring. This string propagates in a ten-dimensional space comprising the flat 4D space and an internal Calabi–Yau noncompact threefold, specifically, the conifold. The lowest state of this string is a ...
Added: October 2, 2025
Huang G., Kuksin S., Piatnitski A., Journal of Dynamics and Differential Equations 2024
We are concerned with averaging theorems for ε-small stochastic perturbations of integrable equations in Rd×Tn={(I,φ)} (Formula presented.) and in R2n={v=(v1,⋯,vn),vj∈R2}, (Formula presented.) where I=(I1,⋯,In) is the vector of actions, Ij=12‖vj‖2. The vector-functions θ and W are locally Lipschitz and non-degenerate. Perturbations of these equations are assumed to be locally Lipschitz and such that some few ...
Added: March 20, 2025
Basalaev A., Letters in Mathematical Physics 2024 Vol. 114 Article 120
According to Zuo and an unpublished work of Bertola, there is a two-index series of Dubrovin–Frobenius manifold structures associated to a B-type Coxeter group. We study the relations between these structures for the different values of these indices. We show that part of the data of such Dubrovin–Frobenius manifold indexed by (k, l) can be recovered ...
Added: December 4, 2024
Zabrodin A., Успехи математических наук 2023 Т. 78 № 2(470) С. 149–188
We find integrals of motion for the recently introduced
deformed Ruijsenaars–Schneider many-body system, which is the dynamical system for poles of elliptic solutions to the Toda lattice with constraint
of type B. Our method is based on the fact that the equations of motion
for this system coincide with those for pairs of Ruijsenaars–Schneider particles which stick together ...
Added: December 1, 2023
Krichever I., A. Zabrodin, Physica D: Nonlinear Phenomena 2023 Vol. 453 Article 133827
We introduce a new integrable hierarchy of nonlinear differential-difference equations which is a subhierarchy of the 2D Toda lattice defined by imposing a constraint to the Lax operators of the latter. The 2D Toda lattice with the constraint can be regarded as a discretization of the BKP hierarchy. We construct its algebraic-geometrical solutions in terms ...
Added: November 30, 2023
Springer, 2016.
This volume presents modern trends in the area of symmetries and their applications based on contributions from the workshop "Lie Theory and Its Applications in Physics", held near Varna, Bulgaria, in June 2015. Traditionally, Lie theory is a tool to build mathematical models for physical systems.
Recently, the trend has been towards geometrization of the mathematical ...
Added: October 18, 2023
Basalaev A., Journal of Physics A: Mathematical and Theoretical 2022 Vol. 55 No. 29 Article 295202
We investigate the solutions to open WDVV equation, associated to type A and D Dubrovin–Frobenius manifolds. We show that these solutions satisfy some stabilization condition and associate to both of them the systems of commuting PDEs. In the type A we show that the system of PDEs constructed coincides with the dispersionless modified KP hierarchy ...
Added: December 4, 2022
Leonid O. Chekhov, Michael Shapiro, International Mathematics Research Notices 2022 Vol. 2009 P. 53–54
Using Fock–Goncharov higher Teichmüller space variables we derive log-canonical coordinate representation for entries of general symplectic leaves of the AnAn groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric RR-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson ...
Added: November 25, 2022
Sarkissian G., Spiridonov V., Journal of Physics A: Mathematical and Theoretical 2022 Vol. 55 No. 38 Article 385203
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. ...
Added: August 30, 2022
Spiridonov V., Sarkissian G. A., / Series arXiv "math". 2021. No. arXiv:2105.15031.
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. ...
Added: November 9, 2021
Cham: Birkhäuser, 2020.
The book consists of articles based on the XXXVIII Białowieża Workshop on Geometric Methods in Physics, 2019. The series of Białowieża workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. The works in this book, based on presentations given at the workshop, ...
Added: November 3, 2021
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
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
Runov B. A., Classical and Quantum Gravity 2021 Vol. 38 No. 6 Article 065021
Einstein-Rosen waves with two polarizations are cylindrically symmetric solutions to vacuum Einstein equations. Einstein equations in this case reduce to an integrable system. In 1971, Geroch has shown that this system admits an infinitedimensional group of symmetry transformations known as the Geroch group. The phase space of this system can be parametrized by a matrix-valued ...
Added: October 21, 2021
Beketov M., Lyashik A., Zabrodin A. et al., Nuclear Physics B 2016 Vol. 903 P. 150–163
We extend the quantum–classical duality to the trigonometric(hyperbolic) case. The duality establishes an explicit relationship between the classical N-body trigonometric Ruijsenaars–Schneider model and the inhomogeneous twisted XXZ spin chain on N sites. Similarly to the rational version, the spin chain data fixes a certain Lagrangian submanifold in the phase space of the classical integrable system. The inhomogeneity parameters are equal to the coordinates of particles ...
Added: October 5, 2021
Prokofev V. V., Zabrodin A., Theoretical and Mathematical Physics 2021 Vol. 208 No. 2 P. 1093–1115
We consider solutions of the 2D Toda lattice hierarchy that are elliptic functions of the "zeroth" time t(0) = x. It is known that their poles as functions of t1 move as particles of the elliptic RuijsenaarsSchneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians that govern the dynamics of poles with respect to the mth hierarchical times t(m) and (t) over bar (m) of the 2D Toda lattice hierarchy are obtained from the expansion of the spectral curve for the Lax matrix of the Ruijsenaars-Schneider model at the marked points. ...
Added: September 7, 2021
Atalikov K., Zotov A., Journal of Geometry and Physics 2021 Vol. 164 Article 104161
We give detailed description for continuous version of the classical IRF-Vertex relation, where on the IRF side we deal with the Calogero-Moser-Sutherland models. Our study is based on constructing modifications of the Higgs bundles of infinite rank over elliptic curve and its degenerations. In this way the previously predicted gauge equivalence between L-A pairs of the Landau-Lifshitz type equations and 1 + 1 field theory generalization of the Calogero-Moser-Sutherland models is described. In this paper the sl(2) case is studied. Explicit changes of variables are obtained between the rational, trigonometric and elliptic models. ...
Added: September 7, 2021
Galkin S., Belmans P., Mukhopadhyay S., / 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
Ovsienko V., Shapiro M., Electronic Research Announcements in Mathematical Sciences 2019 Vol. 26 P. 1–15
We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of "extended quivers," which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared ...
Added: February 25, 2021
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
Finkelberg Michael, Fujita R., Representation Theory 2021 Vol. 25 P. 67–89
The convolution ring of loop rotation equivariant K-homology of the affine Grassmannian of GL(n) was identified with
a quantum unipotent cell of the loop group of SL(2) by Cautis and Williams. We identify the basis formed by
the classes of irreducible equivariant perverse coherent sheaves with the dual
canonical basis of the quantum unipotent cell. ...
Added: January 29, 2021
Talalaev D., Шарыгин Г. И., Journal of Noncommutative Geometry 2017 Vol. 11 No. 2 P. 741–756
In this paper we address the following question: is it always possible to choose a deformation quantization of a Poisson algebra A so that certain Poisson-commutative subalgebra Cin it remains commutative? We define a series of cohomological obstructions to this, that take values in the Hochschild cohomology of C with coefficients in A. In some ...
Added: October 28, 2020