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Wonder of sine-Gordon $ Y$-systems

Authors: Tomoki Nakanishi and Salvatore Stella
Journal: Trans. Amer. Math. Soc. 368 (2016), 6835-6886
MSC (2010): Primary 13F60, 17B37
Published electronically: January 22, 2016
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Abstract: 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.

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Additional Information

Tomoki Nakanishi
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8604, Japan

Salvatore Stella
Affiliation: Department of Mathematics, North Carolina State University, Box 8205, Raleigh, North Carolina 27695-8205

Received by editor(s): February 1, 2014
Received by editor(s) in revised form: June 24, 2014
Published electronically: January 22, 2016
Additional Notes: The second author was partially supported by A. Zelevinsky’s NSF grant DMS-1103813
Article copyright: © Copyright 2016 American Mathematical Society