CLUSTER ALGEBRAS AND DILOGARITHM IDENTITIES

This is a reasonably self-contained exposition of the fascinating interplay between cluster algebras and the dilogarithm in the past two decades. The dilogarithm has a long and rich history since it was studied by Euler. The most intriguing property of the function is that it satisfies various functional relations, which we call dilogarithm identities (DIs). In the 1990s, various DIs were conjectured in the study of integrable models, but most of them were left unsolved. On the other hand, cluster algebras are a class of commutative algebras introduced by Fomin and Zelevinsky around 2000. In this text, we explain how the above DIs are proved using the techniques and results of cluster algebras. Also, we employ the DI associated with each period in a cluster pattern of cluster algebra as the leitmotif and present several proofs, variations, and generalizations of them with various methods and techniques. The quantum DIs are also treated from a unified point of view compared to the classical ones.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets