Wonder of sine-Gordon 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.
Keywords
Cite
@article{arxiv.1212.6853,
title = {Wonder of sine-Gordon Y-systems},
author = {Tomoki Nakanishi and Salvatore Stella},
journal= {arXiv preprint arXiv:1212.6853},
year = {2021}
}
Comments
v1: 66 pages; v2: 53 pages, the version to appear in Trans. Amer. Math. Soc. (in the journal version, the proofs of Props. 5.29-5.31 and Sect. 5.8 will be omitted due to the limitation of space); v3: 53 pages, minor improvement of figures; v4 (no text changes): Sage (v7.0 and higher) has built-in functions to plot the triangulations associated with sine-Gordon and reduced sine-Gordon Y-systems