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Sergey M. Sergeev

Publications and source records attributed to Sergey M. Sergeev.

At least 19 recordsLinked to original sources
arXiv2026

Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions

In this paper we introduce a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices. Algebraically, this commutativity is based on a very special construction of local Boltzmann weights in terms of quantum dilogarithms satisfying the inversion and pentagon identities. We give three examples of such quantum dilogarithms, leading to integrable 3D lattice models. The partition function per site in these models can be exactly calculated in the limit of an infinite lattice by using the functional relations, symmetry and factorization properties of the transfer matrix. The results of such calculations for 3D models associated with the Faddeev modular quantum dilogarithm are briefly presented.

math-ph↗
arXiv2023

A distant descendant of the six-vertex model

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local "spin variables" can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent "dual" formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra $U_q(\widehat{sl}(2))$ without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.

hep-th↗
arXiv2022

An Ising-type formulation of the six-vertex model

We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the coordinate Bethe ansatz for the eigenvectors for all higher spin generalizations of the six-vertex model. The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. As a mathematical identity the emerging star-triangle relation is equivalent to the Pfaff-Saalschuetz-Jackson summation formula, originally discovered by J.~F.~Pfaff in 1797. Plausibly, all vertex models associated with quantized affine Lie algebras and superalgebras can be reformulated as Ising-type models with only two-spin interactions.

math-ph↗
arXiv2017

Spectral equations for the modular oscillator

Motivated by applications for non-perturbative topological strings in toric Calabi--Yau manifolds, we discuss the spectral problem for a pair of commuting modular conjugate (in the sense of Faddeev) Harper type operators, corresponding to a special case of the quantized mirror curve of local $\mathbb{P}^1\times\mathbb{P}^1$ and complex values of Planck's constant. We illustrate our analytical results by numerical calculations.

math-ph↗
arXiv2016

Yang-Baxter Maps, Discrete Integrable Equations and Quantum Groups

For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang-Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper we present detailed considerations of the above scheme on the example of the algebra $U_q(sl(2))$ leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra.

math-ph↗
arXiv2014

The pentagon relation and incidence geometry

We define a map S: D^2 x D^2 --> D^2 x D^2, where D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev-Petviashvili equation. Finally we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure - the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.

nlin.SI↗
arXiv2013

An integrable 3D lattice model with positive Boltzmann weights

In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0<q<1 and three continuous field variables. The layer-to-layer transfer matrices of the model form a two-parameter commutative family. This is the first example of a solvable 3D lattice model with non-negative Boltzmann weights.

math-ph↗
arXiv2013

An Elliptic Parameterisation of the Zamolodchikov Model

The Zamolodchikov model describes an exact relativistic factorized scattering theory of straight strings in (2+1)-dimensional space-time. It also defines an integrable 3D lattice model of statistical mechanics and quantum field theory. The three-string S-matrix satisfies the tetrahedron equation which is a 3D analog of the Yang-Baxter equation. Each S-matrix depends on three dihedral angles formed by three intersecting planes, whereas the tetrahedron equation contains five independent spectral parameters, associated with angles of an Euclidean tetrahedron. The vertex weights are given by rather complicated expressions involving square roots of trigonometric function of the spectral parameters, which is quite unusual from the point of view of 2D solvable lattice models. In this paper we consider a particular four-parameter specialization of the tetrahedron equation when one of its vertices goes to infinity and the tetrahedron itself degenerates into an infinite prism. We show that in this limit all the vertex weights in the tetrahedron equation can be represented as meromorphic functions on an elliptic curve. Moreover we show that a special reduction of the tetrahedron equation in this case leads precisely to an example of the tetrahedral Zamolodchikov algebra, previously constructed by Korepanov. This algebra plays important role for a "layered" construction of the Shastry's R-matrix and the 2D S-matrix appearing in the problem of the ADS/CFT correspondence for N=4 SUSY Yang-Mills theory in four dimensions. Possible applications of our results in this field are briefly discussed.

math-ph↗
arXiv2011

Elliptic gamma-function and multi-spin solutions of the Yang-Baxter equation

We present a generalization of the master solution to the quantum Yang-Baxter equation (obtained recently in arXiv:1006.0651) to the case of multi-component continuous spin variables taking values on a circle. The Boltzmann weights are expressed in terms of the elliptic gamma-function. The associated solvable lattice model admits various equivalent descriptions, including an interaction-round-a-face formulation with positive Boltzmann weights. In the quasi-classical limit the model leads to a new series of classical discrete integrable equations on planar graphs.

math-ph↗
arXiv2010

A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations

We obtain a new solution of the star-triangle relation with positive Boltzmann weights which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known. This new master solution defines an exactly solvable 2D lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters. If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on the circle. An absolute orientation of these positions on the circle slowly changes between lattice sites by overall rotations. Allowed configurations of these rotations are described by classical discrete integrable equations, closely related to the famous $Q_4$-equations by Adler Bobenko and Suris. Fluctuations between degenerate ground states in the vicinity of zero temperature are described by a rather general integrable lattice model with discrete spin variables. In some simple special cases the latter reduces to the Kashiwara-Miwa and chiral Potts models.

math-ph↗
arXiv2009

Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation

We study geometric consistency relations between angles of 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable "ultra-local" Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure allowed us to obtain new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation) as well as reproduce all those that were previously known. These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry.

math-ph↗
arXiv2009

Classical integrable field theories in discrete 2+1 dimensional space-time

We study "circular net" (discrete orthogonal net) equations for angular data generalized by external spectral parameters. A criterion defining physical regimes of these Hamiltonian equations is the reality of Lagrangian density. There are four distinct regimes for fields and spectral parameters classified by four types of spherical or hyperbolic triangles. Non-zero external spectral parameters provide the existence of field-theoretical ground states and soliton excitations. Spectral parameters of a spherical triangle correspond to a statistical mechanics; spectral parameters of hyperbolic triangles correspond to three different field theories with massless anisotropic dispersion relations.

nlin.SI↗
arXiv2009

Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations

In this paper we consider three-dimensional quantum q-oscillator field theory without spectral parameters. We construct an essentially big set of eigenstates of evolution with unity eigenvalue of discrete time evolution operator. All these eigenstates belong to a subspace of total Hilbert space where an action of evolution operator can be identified with quantized discrete BKP equations (synonym Miwa equations). The key ingredients of our construction are specific eigenstates of a single three-dimensional R-matrix. These eigenstates are boundary states for hidden three-dimensional structures of U_q(B_n^1) and U_q(D_n^1)$.

nlin.SI↗
arXiv2008

Geometry of quadrilateral nets: second Hamiltonian form

Discrete Darboux-Manakov-Zakharov systems possess two distinct Hamiltonian forms. In the framework of discrete-differential geometry one Hamiltonian form appears in a geometry of circular net. In this paper a geometry of second form is identified.

nlin.SI↗
arXiv2008

Quantum geometry of 3-dimensional lattices

We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable ``ultra-local'' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit.

hep-th↗
arXiv2007

Exact solution of the Faddeev-Volkov model

The Faddeev-Volkov model is an Ising-type lattice model with positive Boltzmann weights where the spin variables take continuous values on the real line. It serves as a lattice analog of the sinh-Gordon and Liouville models and intimately connected with the modular double of the quantum group U_q(sl_2). The free energy of the model is exactly calculated in the thermodynamic limit. In the quasi-classical limit c->infinity the model describes quantum fluctuations of discrete conformal transformations connected with the Thurston's discrete analogue of the Riemann mappings theorem. In the strongly-coupled limit c->1 the model turns into a discrete version of the D=2 Zamolodchikov's ``fishing-net'' model.

cond-mat.stat-mech↗
arXiv2007

Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry

The Faddeev-Volkov solution of the star-triangle relation is connected with the modular double of the quantum group U_q(sl_2). It defines an Ising-type lattice model with positive Boltzmann weights where the spin variables take continuous values on the real line. The free energy of the model is exactly calculated in the thermodynamic limit. The model describes quantum fluctuations of circle patterns and the associated discrete conformal transformations connected with the Thurston's discrete analogue of the Riemann mappings theorem. In particular, in the quasi-classical limit the model precisely describe the geometry of integrable circle patterns with prescribed intersection angles.

hep-th↗