Cluster Algebras and Dilogarithm Identities

TL;DR

The paper elucidates a deep connection between cluster algebras and dilogarithm identities, showing that a DI naturally arises for each period in a cluster pattern and that this framework recovers and generalizes classical identities such as Euler’s and Abel’s pentagon relation. It develops both algebraic (constancy-based) and classical mechanical (Hamiltonian/Lagrangian) approaches to prove these DIs, and extends the analysis to Y-systems, their tropical dynamics, and cluster-scattering perspectives. A key insight is that Y-pattern periodicities, governed by Coxeter data, yield systematic dilogarithm sums that remain invariant under mutation, with tropicalization revealing the underlying combinatorial structure. The work further unifies classical DIs with quantum dilogarithm identities and provides multiple proofs and generalizations, highlighting the central role of cluster algebras in understanding dilogarithm relations and their physical and geometric applications.

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.

Figures (19)

• Figure 1.1: The graph of the Euler dilogarithm $\mathrm{Li}_2(x)$.
• Figure 1.2: The graph of the Rogers dilogarithm $L(x)$.
• Figure 1.3: The graph of the modified Rogers dilogarithm $\tilde{L}(x)$.
• Figure 1.4: Simply-laced Dynkin diagrams of finite type.
• Figure 2.1: The $n$-regular trees $\mathbb{T}_n$ for $n=1$, $2$, $3$.

Theorems & Definitions (65)

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