Integrable quad equations derived from the quantum Yang-Baxter equation

Abstract

This paper presents an explicit correspondence between two different types of integrable equations; the quantum Yang-Baxter equation in its star-triangle relation form, and the classical 3D-consistent quad equations in the Adler-Bobenko-Suris (ABS) classification. Each of the 3D-consistent ABS quad equations of $H$-type, are respectively derived from the quasi-classical expansion of a counterpart star-triangle relation. Through these derivations it is seen that the star-triangle relation provides a natural path integral quantization of an ABS equation. The interpretation of the different star-triangle relations is also given in terms of (hyperbolic/rational/classical) hypergeometric integrals, revealing the hypergeometric structure that links the two different types of integrable systems. Many new limiting relations that exist between the star-triangle relations/hypergeometric integrals are proven for each case.

arXiv comment64 pages, 11 figures, v2: typos corrected; v3: improvements to text

References in corpus (16)

• Classification of integrable equations on quad-graphs. The consistency approach
• Quantum Dilogarithm
• Bäcklund transformation for the Krichever-Novikov equation
• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Lagrangian multiforms and multidimensional consistency
• Quivers, YBE and 3-manifolds
• Exact solution of the Faddeev-Volkov model
• New solutions of the star-triangle relation with discrete and continuous spin variables
• The star-triangle relation, lens partition function, and hypergeometric sum/integrals
• Yang-Baxter Maps, Discrete Integrable Equations and Quantum Groups
• On the Lagrangian structure of integrable quad-equations
• Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs
• Elliptic beta integrals and solvable models of statistical mechanics
• Lens elliptic gamma function solution of the Yang-Baxter equation at roots of unity
• On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations
• Exactly solved models on planar graphs with vertices in ${\mathbb{Z}}^3$

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• Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps
• Quasi-classical expansion of a hyperbolic solution to the star-star relation and multicomponent 5-point difference equations