# Cluster algebras, dilogarithm, and Y-systems

Cluster algebras were introduced by Fomin and Zelevinsky around 2000 as an underlying combinatorial structure in Lie theory. They also (often quite unexpectedly) appear in several branches of mathematics besides representation theory, e.g., hyperbolic geometry and Teichm\"uller theory, Poisson geometry, discrete dynamical systems, exact WKB analysis, etc. In this talk I review the application of cluster algebras to the dilogarithm and Y-systems, based on joint works over the recent years with R. Inoue, O. Iyama, R. Kashaev, B. Keller, A. Kuniba, R. Tateo, J. Suzuki, and S. Stella.

The talk consists of two parts. In the first part, after reviewing some basic properties of cluster algebras, I present the dilogarithm identity associated with any period of seeds in a cluster algebra. In the second part, I explain that this identity is related to the longstanding conjectures on the periodicities of Y-systems and the associated dilogarithm identities in conformal field theory, which arose through the thermodynamic Bethe ansatz approach in 90's. Then, I show how efficiently cluster algebra theory proves these conjectures.