Cluster Algebras and Dilogarithm Identities

TL;DR

This paper explores the deep connections between cluster algebras and dilogarithm identities, providing proofs, generalizations, and a unified perspective on classical and quantum dilogarithm relations over the past two decades.

Contribution

It offers a comprehensive exposition of how cluster algebra techniques are used to prove and generalize dilogarithm identities, including quantum versions.

Findings

Proofs of dilogarithm identities using cluster algebra methods

Generalizations and variations of known identities

Unified treatment of classical and quantum dilogarithm identities

Abstract

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.