Dilogarithm identities in cluster scattering diagrams

Tomoki Nakanishi

Published 2021-11-18, updated 2021-12-05Version 2

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.