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


SIGMA 7 (2011), 102, 29 pages      arXiv:1104.4630      http://dx.doi.org/10.3842/SIGMA.2011.102

Classical and Quantum Dilogarithm Identities

Rinat M. Kashaev a and Tomoki Nakanishi b
a) Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland
b) Graduate School of Mathematics, Nagoya University, Nagoya, 464-8604, Japan

Received May 03, 2011, in final form October 26, 2011; Published online November 01, 2011

Abstract
Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method.

Key words: dilogarithm; quantum dilogarithm; cluster algebra.

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