Wonder of sine-Gordon Y-systems

TL;DR

This paper proves the conjectured periodicity and dilogarithm identities of sine-Gordon Y-systems using polygon realizations of cluster algebras, revealing deep connections among various mathematical structures.

Contribution

It formulates sine-Gordon Y-systems via polygon realizations of cluster algebras and proves key conjectures in full generality.

Findings

Proved periodicity of sine-Gordon Y-systems.

Established dilogarithm identities for these Y-systems.

Revealed interplay among continued fractions, polygon triangulations, cluster algebras, and Y-systems.

Abstract

The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were introduced by Tateo in the 90's in the study of the integrable deformation of conformal field theory by the thermodynamic Bethe ansatz method. The periodicity property and the dilogarithm identities concerning these Y-systems were conjectured by Tateo, and only a part of them have been proved so far. In this paper we formulate these Y-systems by the polygon realization of cluster algebras of types A and D, and prove the conjectured periodicity and dilogarithm identities in full generality. As it turns out, there is a wonderful interplay among continued fractions, triangulations of polygons, cluster algebras, and Y-systems.