Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities

TL;DR

This paper develops a Hamiltonian and Lagrangian framework for mutations in cluster algebras, linking dilogarithm functions to these formalisms and deriving related identities.

Contribution

It introduces a novel Hamiltonian formalism using the Euler dilogarithm and connects it to the Rogers dilogarithm within cluster algebra mutations.

Findings

Hamiltonian formalism based on Euler dilogarithm

Lagrangian coincides with Rogers dilogarithm on a subspace

Derivation of dilogarithm identities from the formalism

Abstract

We introduce and study a Hamiltonian formalism of mutations in cluster algebras using canonical variables, where the Hamiltonian is given by the Euler dilogarithm. The corresponding Lagrangian, restricted to a certain subspace of the phase space, coincides with the Rogers dilogarithm. As an application, we show how the dilogarithm identity associated with a period of mutations in a cluster algebra arises from the Hamiltonian/Lagrangian point of view.