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 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.

26 pages, 18 color figures, minor corrections

References in corpus (3)

• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant
• Exact solution of the Faddeev-Volkov model

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• A New Integrable Ising-type Model from 2d $\mathcal{N}$=(2,2) Dualities
• Lens partition function, pentagon identity and star-triangle relation
• Integrability as a consequence of discrete holomorphicity: loop models
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• Algebraic entropy for face-centered quad equations
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• Lax matrices for lattice equations which satisfy consistency-around-a-face-centered-cube
• Matrix factorization for solutions of the Yang-Baxter equation
• Quasi-classical expansion of a hyperbolic solution to the star-star relation and multicomponent 5-point difference equations