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.

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We study the ground state eigenvalues of Baxter's Q-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the ∆ = − 1 2 six-vertex model. We show that these eigenvalues satisfy a non-stationary Schrodinger equation with the time-dependent potential given by the Weierstrass elliptic ℘ -function where the modular parameter τ plays the role of (imaginary) time. In the scaling limit the equation transforms into a "non-stationary Mathieu equation" for the vacuum eigenvalues of the Q-operators in the finite-volume massive sine-Gordon model at the super-symmetric point, which is closely related to the theory of dilute polymers on a cylinder and the Painlevé III equation. 1

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 Uq(sl2). 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 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.

In this paper we formulate an integrable model on the simple cubic lattice. The N -valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex type Tetrahedron Equation. In the thermodynamic limit our model is equivalent to the Bazhanov -Baxter Model. In the case when N = 2 we reproduce the Korepanov's and Hietarinta's solutions of the Tetrahedron equation as some special cases.

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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 ( sl (2)). 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. and the general scheme of quantum SoV has been developed [29][30][31][32]. The integral Q-operator for the case of the XXX chain was first calculated in [33]. It is worth noting that taking the limit N → ∞ [34] in the Bazhanov and Stroganov construction [22] one can recover the results of [27] and [33].The main difference of the above approach from the original Baxter method is that the "quantum" representation space is infinite-dimensional. It has the structure of a tensor product of Verma modules with the basis chosen as multi-variable polynomials p(x 1 , . . . , x M ), where M is the size of the system. The Q-operators appear as integral operators with an explicit action on such a polynomial basis. A detailed construction can be found in [35,36] for the XXX case and its generalization to the XXZ case in [37,38]. The non-compact case and applications of the Q-operators to Liouville theory are discussed in [39][40][41]. It is worth mentioning that the representation of the Q-operator by an integral operator is known only for the XXX case [33]. The proper deformation of such integral operator for the case of the six-vertex model is still a challenging problem.Another problem arises when spins take (half-) integer values. In this case the quantum space becomes reducible and the action of the Q-operator on the polynomial basis becomes singular. This difficulty can be overcome by expanding near the limit 2s → Z + as shown in [35,36]. However, a removal of such a regularization is technically challenging and it is desirable to have an alternative approach which is free from this difficulty.In 1997 Bazhanov, Lukyanov and Zamolodchikov (BLZ) suggested another method to derive the Q-operators related to the affine algebra U q ( sl(2)) [42,43]. Based on the universal R-matrix theory [44] they showed that the Q-operators can be constructed as special monodromy operators with the auxiliary space being an infinite-dimensional representation of the q-oscillator algebra. Although their original approach was developed in the context of quantum field theory, the results of [42,43] can be easily adjusted to the spin s = 1/2 XXZ chain [45]. However, the derivation of the local Q-operators from...

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