Dilogarithm identities in cluster scattering diagrams

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.

v1: 20 pp; v2: 20pp, minor changes in Sec 2.1; v3: 22pp, Thm. 3.4, modified and proof revised; Thm. 4.6 (former 4.7), proof revised; v4: 22pp, final version to appear in Nagoya Math. J

References in corpus (4)

• The quantum dilogarithm and representations quantum cluster varieties
• Scattering diagrams, theta functions, and refined tropical curve counts
• Comments on Exchange Graphs in Cluster Algebras
• Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities