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

Michael Gekhtman, Tomoki Nakanishi, Dylan Rupel

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.

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

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.

Paper Structure

This paper contains 31 sections, 31 theorems, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Outline and main results
  4. Preliminaries
  5. Mutations in cluster algebras
  6. Euler and Rogers dilogarithm functions
  7. Hamiltonian formalism of mutations
  8. Canonical and log-canonical variables
  9. Hamiltonian for infinitesimal nontropical mutation
  10. Small phase space and $x$-variables
  11. Tropical transformation
  12. Signed mutations
  13. Canonical quantization
  14. Quantization of $x$-variables through Dirac bracket
  15. Lagrangian formalism and Rogers dilogarithm
  16. ...and 16 more sections

Key Result

Lemma 3.1

We have the following formulas:

Figures (1)

  • Figure 1: Schematic diagram of a Hamiltonian flow for $T=2$, where for simplicity we use the same abbreviation as in \ref{['eq:flow1']}.

Theorems & Definitions (67)

  • Remark 1.1
  • Lemma 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • Proposition 3.6
  • proof
  • ...and 57 more