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## "Dilogarithm identities for conformal field theories and cluster algebras: simply laced case", Tomoki Nakanishi, 2011

*Reviewed October 13, 2023*

*Citation:* Nakanishi, Tomoki. "Dilogarithm identities for conformal field theories and cluster algebras: simply laced case." Nagoya Mathematical Journal 202 (2011): 23-43.

*Web:* https://arxiv.org/abs/0909.5480

*Tags:* Mathematical, Conformal-field-theory

This paper proves a special class of dilogarithm identities whenever the underlying Dynkin
diagram is simply laced. The interesting fact is that this is done from a cluster-theoretic perspective.
Cluster algebras seem to appear a lot in this general area of mathematical physics. For instance, they appear in the string-theoretic
study of (1+1)D conformal field theories:

>Cecotti, Sergio, Andrew Neitzke, and Cumrun Vafa. "R-twisting and 4d/2d correspondences." arXiv preprint arXiv:1006.3435 (2010).

They also appear in the TQFT-theoretic study of (2+1)D quantum gravity:

>Boldis, Bercel, and Péter Lévay. "Cluster algebraic description of entanglement patterns for the BTZ black hole." Physical Review D 105.4 (2022): 046020.

as well as in the theory of the modular group:

> Fock, Vladimir V., and Alexander B. Goncharov. "Cluster ensembles, quantization and the dilogarithm." Annales scientifiques de l'École normale supérieure. Vol. 42. No. 6. 2009.