Wonder of sine-gordon Y -systems

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 deforma- tion 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 won- derful interplay among continued fractions, triangulations of polygons, cluster algebras, and Y -systems.