Dilogarithm identities in cluster scattering diagrams

TL;DR

This paper extends the concept of y-variables in cluster algebras to scattering diagrams and generalizes dilogarithm identities to loops in these diagrams, revealing their construction from trivial identities via pentagon identities.

Contribution

It introduces a new framework linking cluster scattering diagrams with dilogarithm identities and demonstrates their derivation through repeated application of the pentagon identity.

Findings

Dilogarithm identities correspond to loops in cluster scattering diagrams.

These identities can be reduced to trivial ones using pentagon identities.

The extension provides a unified view of identities in cluster algebra theory.

Abstract

We extend the notion of $y$-variables (coefficients) in cluster algebras to cluster scattering diagrams. Accordingly, we extend the dilogarithm identity associated with a period in a cluster pattern to the one associated with a loop in a cluster scattering diagram. We show that these identities are constructed from and reduced to a trivial one by applying the pentagon identity possibly infinitely many times.