Dimers and M-curves

We explore various aspects of the correspondence between dimer models and integrable systems recently introduced by Goncharov and Kenyon. Dimer models give rise to relativistic integrable systems that match those arising from 5d N=1 gauge theories studied by Nekrasov. We apply the correspondence to dimer models associated to the Y^{p,0} geometries, showing that they give rise to the relativistic generalization of the periodic Toda chain originally studied by Ruijsenaars. The correspondence reduces the calculation of all conserved charges to a straightforward combinatorial problem of enumerating non-intersecting paths in the dimer model. We show how the usual periodic Toda chain emerges in the non-relativistic limit and how the Lax operator corresponds to the Kasteleyn matrix of the dimer model. We discuss how the dimer models for general Y^{p,q} manifolds give rise to other relativistic integrable systems, generalizing the periodic Toda chain and construct the integrable systems for general Y^{p,p} explicitly. The impurities introduced in the construction of Y^{p,q} quivers are identified with impurities in twisted sl(2) XXZ spin chains. Finally we discuss how the physical concept of higgsing a dimer model provides an efficient method for producing new integrable systems starting from known ones. We illustrate this idea by constructing the integrable systems for higgsings of Y^{4,0}.

We give a geometric proof of a formula, due to Segal and Wilson, which describes the order of vanishing of the Riemann theta function in the direction which corresponds to the direction of the tangent space of a Riemann surface at a marked point. While this formula appears in the work of Segal and Wilson as a by-product of some nontrivial constructions from the theory of integrable systems (loop groups, infinite-dimensional Grassmannians, tau functions, Schur polynomials, etc.) our proof only uses the classical theory of linear systems on Riemann surfaces.

This paper focuses on the relevance of a certain class of left-invariant Hamiltonians (affine-quadratic) on a reductive semi-simple Lie algebra g for the theory of integrable systems and the equations of applied mathematics.Any semi-simple Lie group G that contains a closed subgroup K is reductive, in the sense, that the orthogonal complement p in g of the Lie algebra k of K , relative to the Killing form, satisfies [k, p] p.This implies that K acts (by adjoint action) on p and therefore induces the semi-direct product p K .Consequently, g , as a vector space, carries two Lie algebra structures: semi-simple, and semi-direct.Hence, the dual g * carries two Poisson structures as well.Any affine-quadratic function H on g can be simultaneously viewed as a Hamiltonian for either Poisson structure.We will show that certain coadjoint orbits relative to the semi-direct action are the cotangent bundles of SO(n).This implies that the equations of an n -dimensional top can be represented on such coadjoint orbits.In this situation there is a canonical affine-quadratic Hamiltonian whose Hamiltonian equations on these coadjoint orbits coincide with the equations of the top.This implies that the integrable cases of the top correspond to the the integrable cases of the overseeing affine Hamiltonian.More generally, we will identify a subclass of affine-quadratic Hamiltonians, called isospectral, that provides new insights into the theory of integrable systems based on the contributions of S. V. Manakov, A. T. Fomenko, A. S. Mischenko and O. Bogoyavlensky listed in the references.

This is a very interesting collection of introductory and review articles on the theory and applications of classical and quantum integrable systems. The book reviews several integrable systems such as the KdV equation, vertex models, RSOS and IRF models, spin chains, integrable differential equations, discrete systems, Ising, Potts and other lattice models and reaction--diffusion processes, as well as outlining major methods of solving integrable systems. These include Lax pairs, Bäcklund and Miura transformations, the inverse scattering method, various types of the Bethe Ansatz, Painlevé methods, the dbar method and fusion methods to mention just a few.

Pseudo differential operators and integrable systems in AXIOM

In the theory of integrable Hamiltonian systems with two degrees of freedom, widely known are integrable systems having integrals of high degrees, namely, 3 and 4. Examples are the Kovalevskaya system and its generalizations—the Kovalevskaya–Yehia system and the Kovalevskaya system on the Lie algebra so(4) the Goryachev–Chaplygin–Sretensky, Sokolov, and Dullin–Matveev systems. It is shown that, at a number of isoenergy 3-surfaces, the third and fourth degrees of integrals of these systems can be reduced by using integrable billiards bounded by arcs of confocal quadrics. Moreover, the integrals of degree 3 and 4 are reduced to the same canonical quadratic integral on a billiard.

This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock’s elliptic weights (Fock, Inverse spectral problem for GK integrable system. arXiv e-prints arXiv:1503.00289 , 2015). Specific instances of such models were studied in Boutillier et al. (Invent Math 208(1):109–189, 2017), Boutillier et al. (Probability theory and related fields, 2018) and de Tilière (Electron J Probab 26:1–86, 2021); we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of Kenyon (Invent Math 150(2):409–439, 2002) and Kenyon and Okounkov (Duke Math J 131(3):499–524, 2006) on isoradial critical models. We give an explicit local expression for a two-parameter family of inverses of the Kasteleyn operator with no periodicity assumption on the underlying graph. When the minimal graph satisfies a natural condition, we construct a family of dimer Gibbs measures from these inverses, and describe the phase diagram of the model by deriving asymptotics of correlations in each phase. In the $$\mathbb {Z}^2$$ -periodic case, this gives an alternative description of the full set of ergodic Gibbs measures constructed in Kenyon et al. (Ann Math 163(3):1019–1056, 2006). We also establish a correspondence between elliptic dimer models on periodic minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite dimer model is invariant under the shrinking/expanding of 2-valent vertices and spider moves if and only if the associated Kasteleyn coefficients are antisymmetric and satisfy Fay’s trisecant identity.

The generalized theory of the ℛ-structure on affine operator Lie algebras is used to construct a complete theory of Lax integrable nonlinear dynamical systems in multidimensions. The operator bi-Hamiltonian structures and their functional reductions are discussed in great detail in the examples of operator Korteweg–de Vries and Benney–Kaup dynamical systems. As an important by-product of the developed algebraic theory, the Dirac canonical quantization problem is solved almost completely for the Neumann–Bogoliubov-type oscillatory dynamical system on spheres, associated via Moser with the spectral moment map on an affine associative metrized Lie coalgebra with a one-parameter gauge two-cocycle. Some remarks are given on the problem of extending the developed algebraic theory to the case of Lax integrable dynamical systems on discrete manifolds.

We present a survey of recent results concerning a remarkable class of unitary matrices, the CMV matrices. We are particularly interested in the role they play in the theory of random matrices and integrable systems. Throughout the paper we also emphasize the analogies and connections to Jacobi matrices.

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

This article describes an overview of completely integrable systems based on the self-dual Yang-Mills equations and its twistor correspondence. It also gives a brief survey of results obtained in the area. The Ward construction is briefly described. A classification of integrable systems in 2-dimensions as reductions of the self-dual Yang-Mills equations in 4-dimensions with two translational symmetries is briefly reviewed. Applications of the twistor constructions to the theory of such integrable systems are summarized and further problems are discussed.

This study investigates communication strategies in managing university-based conflicts and introduces the Integrated Layered Crisis Communication Model (ILCCM) as an empirically grounded contribution to the field of communication and conflict management. Conflicts in university unions often escalate due to poor communication, lack of transparency, and misalignment between management and union expectations. Utilizing a mixed methods approach, quantitative surveys and qualitative interviews were conducted among University of Eldoret management, staff, and union representatives. Findings revealed multi-layered communication challenges, inadequate message adaptation, systemic weaknesses, and mismatched crisis responses. The ILCCM, integrating Systems Theory, Communication Accommodation Theory (CAT), and Situational Crisis Communication Theory (SCCT) through the lens of Onion Theory of Conflict, provides a comprehensive framework comprising four interrelated phases: layered communication, strategic adaptation, systemic response and crisis-type strategies for layered, strategic, and systemic communication. By introducing ILCCM, the study offers a proactive, context-sensitive approach to conflict management as well as an innovative framework for scholars and practitioners in communication and conflict management, emphasizing tailored messaging, coordination across stakeholders and adaptive crisis communication training, transparent feedback mechanisms mediation structures, cultural sensitivity programs and employee support systems while enhancing institutional trust, conflict resolution, and organizational productivity.

Haantjes algebras and diagonalization

A general scheme of construction of Drinfeldians and Yangians from quantum non-twisted affine Kac-Moody algebras is presented. Explicit description of Drinfeldians and Yangians for all Lie algebras of the classical series A, B, C, D is given in terms of a Chevalley basis. Rational and trigonometric deformations of a universal enveloping algebra U(g(u)) (g is a finite-dimensional complex simple Lie algebra) - Yangian Yη(g) and a quantum (q-deformed) affine algebraUq(g(u)) - are playing increasingly a role in the theory of integrable systems and the quantum field theory. For applications it is useful to know various realizations (or bases) of these deformations. Among various realizations the Chevalley basis plays a special role due to its compactness and simplicity. In the case of the quantum affine algebras the Chevalley basis was originally introduced in the defining relations while the Yangians were initially de- fined in terms of a finite Cartesian basis (the first Drinfeld realization (2)) as well as in terms of a special infinite basis (the second Drinfeld realization (3)). It was not clear whether does exist a Chevalley basis for the Yangians. Recently in (12, 13) it was shown that the Yangian can be realized in terms of Chevalley basis and, moreover, there was found a simple connection of this realization with the corresponding quantum affine algebra (also see (14, 15)). This connection allows to introduce a new Hopf algebra called Drinfeldian which depends on two deformation parameters q and η. The Drinfeldian Dqη(g) can be considered as quantization of U(g(u)) in the direction of a classical r-matrix which is a sum of simplest rational and trigonometric r-matrices. When the parameter q goes to 1 we obtain the Yangian Yη(g) in terms of the Chevalley basis. In this paper we remind once more the procedure of construction of the Drin- feldian Dqη(g) and the Yangian Yη(g) in terms of the Chevalley basis starting from

At present there can be no doubt to say that the quantum group theory has been one of the most important, modern and rapidly developing directions of mathematics and mathematical physics in the end of the twentieth century. Although initially (in the early 1980’s) the quantum group theory was formulated for solving problems in the theory of integrable systems and statistical physics, later the surprising connection of this theory with many branches of mathematics, and the theoretical and mathematical physics was discovered. Today the quantum group theory is connected with such mathematical fields as special functions (especially, with (q-orthogonal polynomials and basic hypergeometric series), the theory of difference and differential equations, combinatorial analysis and representation theory, matrix and operator algebras, noncommutative geometry, knot theory, topology, category theory and so on. From point of view of the mathematical physics there exists interconnection of the quantum group theory with the quantum inverse scattering method, conformai and quantum fields theory and so on. It is expected that the quantum groups will provide deeper understanding of concept of symmetry in physics.KeywordsQuantum GroupSimple RootBasic Hypergeometric SeriesReduction ChainGeneral Projection OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.