Cluster Algebras and Dilogarithm Identities

Abstract: This is a reasonably self-contained exposition of the fascinating interplay between cluster algebras and the dilogarithm in the recent two decades. The dilogarithm has a long and rich history since Euler. The most intriguing property of the function is that it satisfies various functional relations, which we call dilogarithm identities (DIs). In the 1990s Gliozzi and Tateo conjectured a family of DIs based on root systems, which vastly generalize the classic DIs such as Euler's and Abel's identities. On the other hand, the cluster algebras are a class of commutative algebras introduced by Fomin and Zelevinsky around 2000. It turned out that a DI is associated with each period in a cluster pattern of cluster algebra theory. In particular, they solve the conjecture on the DIs by Gliozzi and Tateo. We employ these DIs as the leitmotif and present several proofs, variations, and generalizations of them with various methods and techniques. The quantum DIs are also treated from the unified point of view with the classical ones.