Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 085, 34 pages      arXiv:1005.4199      https://doi.org/10.3842/SIGMA.2010.085

Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems

Tomoki Nakanishi a and Roberto Tateo b
a) Graduate School of Mathematics, Nagoya University, Nagoya, 464-8604, Japan
b) Dipartimento di Fisica Teorica and INFN, Universitè di Torino, Via P. Giuria 1, 10125 Torino, Italy

Received May 29, 2010, in final form October 16, 2010; Published online October 19, 2010

Abstract
We study the family of Y-systems and T-systems associated with the sine-Gordon models and the reduced sine-Gordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and the associated dilogarithm identities which have been conjectured earlier. In particular, this provides new examples of periodicities of seeds.

Key words: cluster algebras; quantum groups; integrable models.

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