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.