2018/03/31 by Andrew P. Kels
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Hypergeometric distribution #Hypergeometric function #Integrable system #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Path integral formulation #Physics #Pure mathematics #Quantum #Quantum mechanics #Star (game theory) #Yang–Baxter equation #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.1007/s11005-020-01255-3
published as Lett. Math. Phys. 110, 1477-1557 (2020) · 64 pages, 11 figures, v2: typos corrected; v3: improvements to text
arxiv created 2018/08/02 · openalex publication_date 2020/01/25 · arxiv updated 2020/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.