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

TL;DR

This paper explores the Faddeev-Volkov solution to the Yang-Baxter equation, connecting it with quantum groups and discrete conformal geometry, and provides exact calculations of the model's free energy.

Contribution

It introduces a new integrable lattice model based on the Faddeev-Volkov solution, linking quantum algebra with discrete conformal geometry and calculating its free energy explicitly.

Findings

Exact free energy of the model in the thermodynamic limit

Connection between the model and quantum fluctuations of circle patterns

Description of discrete conformal transformations via the model

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 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.