Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems

TL;DR

This paper establishes new dilogarithm identities and periodicities for Y-systems and T-systems related to sine-Gordon models, using cluster algebra frameworks to prove conjectured properties.

Contribution

It formulates sine-Gordon Y-systems as finite type cluster algebras and proves their periodicities and dilogarithm identities, confirming previous conjectures.

Findings

Proved periodicities of sine-Gordon Y-systems

Established dilogarithm identities for these systems

Provided new examples of seed periodicities in cluster algebras

Abstract

We study the family of Y-systems and T-systems associated with the sine-Gordon models and the reduced sine-Gordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and the associated dilogarithm identities which have been conjectured earlier. In particular, this provides new examples of periodicities of seeds.