Integrable quad equations derived from the quantum Yang-Baxter equation

TL;DR

This paper establishes a direct link between quantum Yang-Baxter equations and classical 3D-consistent quad equations, showing how the former can be used to derive the latter through quasi-classical expansion and hypergeometric integral interpretations.

Contribution

It provides the first explicit derivation of ABS quad equations from star-triangle relations, revealing their hypergeometric structure and new limiting relations.

Findings

Derived ABS equations from star-triangle relations.

Linked hypergeometric integrals to integrable systems.

Proved new limiting relations between integrable models.

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.