分野別セミナー

The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were introduced by Tateo in 90's in the study of the integrable deformation of the minimal models in 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 the conjectures have been proved so far. 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. There is the wonderful interplay among continued fractions, triangulations of polygons, cluster algebras, and Y-systems.

This is a joint work with Salvatore Stella.