Vladimir V. Mangazeev

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

Abstract 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 Thurstons 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.

Q-operator and factorised separation chain for Jack polynomials

Abstract Applying Baxters method of the Q -operator to the set of Sekiguchis commuting partial differential operators we show that Jack polynomials P λ (1/g) ( χ 1 , …, χ n ) …, χ n ) are eigenfunctions of a one-parameter family of integral operators Q z . The operators Q z are expressed in terms of the Dirichlet-Liouville n -dimensional beta integral. From a composition of n operators Q zk we construct an integral operator S n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S n admits a factorisation described in terms of restricted Jack polynomials P λ (1/g) ( x 1 , …, x k , 1, … 1). Using the operator Q z for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.

On the Yang-Baxter equation for the six-vertex model

Abstract In this paper we review the theory of the Yang–Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R -matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R -matrix associated with the affine quantum algebra U q ( s l ( 2 ) ˆ ) . In the simplest case of the spin s = 1 / 2 this R -matrix naturally reduces to the R -matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q -operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s . We also give all functional equations satisfied by the transfer-matrices and Q -operators.

The eight-vertex model and Painlevé VI

In this paper we establish a connection of Picard-type elliptic solutions of the Painleve VI equation with the special solutions of the non-stationary Lame equation. The latter appeared in the study of the ground-state properties of Baxters solvable eight-vertex lattice model at a particular point, η = π/3, of the disordered regime.

The eight-vertex model and Painleve VI equation II: eigenvector results

We study a special anisotropic -model on a periodic chain of an odd length and conjecture exact expressions for certain components of the ground state eigenvectors. The results are written in terms of tau-functions associated with Picards elliptic solutions of the Painleve VI equation. Connections with other problems related to the eight-vertex model are briefly discussed.

Exact solution of the Faddeev-Volkov model

Abstract 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 → + ∞ the model describes quantum fluctuations of discrete conformal transformations connected with the Thurstons 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 Zamolodchikovs “fishing-net” model.

Analytic Theory of the Eight-Vertex Model

Abstract We observe that the exactly solved eight-vertex solid-on-solid model contains an hitherto unnoticed arbitrary field parameter, similar to the horizontal field in the six-vertex model. The parameter is required to describe a continuous spectrum of the unrestricted solid-on-solid model, which has an infinite-dimensional space of states even for a finite lattice. The introduction of the continuous field parameter allows us to completely review the theory of functional relations in the eight-vertex/SOS-model from a uniform analytic point of view. We also present a number of analytic and numerical techniques for the analysis of the Bethe ansatz equations. It turns out that different solutions of these equations can be obtained from each other by analytic continuation. In particular, for small lattices we explicitly demonstrate that the largest and smallest eigenvalues of the transfer matrix of the eight-vertex model are just different branches of the same multivalued function of the field parameter.

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 generalization 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 non-trivial solvable 3D lattice model with non-negative Boltzmann weights.

Q-operators in the six-vertex model

Abstract In this paper we continue the study of Q-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin R-matrix associated with the affine quantum algebra U q ( sl ( 2 ) ˆ ) . Taking a special limit in this R-matrix we obtained new formulas for the Q-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct Q-operators as integral operators with factorized kernels based on the original Baxters method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-)integer spins.

Stochastic R matrix for Uq(An(1))

We show that the quantum R matrix for symmetric tensor representations of Uq(An(1)) satisfies the sum rule required for its stochastic interpretation under a suitable gauge. Its matrix elements at a special point of the spectral parameter are found to factorize into the form that naturally extends Povolotskys local transition rate in the q-Hahn process for n=1. Based on these results we formulate new discrete and continuous time integrable Markov processes on a one-dimensional chain in terms of n species of particles obeying asymmetric stochastic dynamics. Bethe ansatz eigenvalues of the Markov matrices are also given.