Wonder of sine-Gordon $Y$-systems
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- by Tomoki Nakanishi and Salvatore Stella PDF
- Trans. Amer. Math. Soc. 368 (2016), 6835-6886 Request permission
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.References
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Additional Information
- Tomoki Nakanishi
- Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8604, Japan
- Email: nakanisi@math.nagoya-u.ac.jp
- Salvatore Stella
- Affiliation: Department of Mathematics, North Carolina State University, Box 8205, Raleigh, North Carolina 27695-8205
- MR Author ID: 1025220
- ORCID: 0000-0001-5390-2081
- Email: sstella@ncsu.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6835-6886
- MSC (2010): Primary 13F60, 17B37
- DOI: https://doi.org/10.1090/tran/6505
- MathSciNet review: 3471079