Classical and Quantum Dilogarithm Identities

TL;DR

This paper explores quantum dilogarithm identities within quantum cluster algebras, showing how classical identities emerge in the semiclassical limit through the saddle point method.

Contribution

It introduces various forms of quantum dilogarithm identities linked to quantum cluster algebra periodicities and demonstrates their classical limits.

Findings

Multiple forms of quantum dilogarithm identities are presented.

Classical dilogarithm identities are derived from quantum identities in the semiclassical limit.

The saddle point method connects quantum and classical identities.

Abstract

Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method.