Wonder of sine-gordon Y -systems

The sine-Gordon Y -systems and the reduced sine-Gordon Y - systems were introduced by Tateo in the 1990’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.