Mutations of the cluster algebra of type $\boldsymbol{A^{(1)}_1}$ and the periodic discrete Toda lattice

TL;DR

This paper establishes a geometric link between the cluster algebra of type A^{(1)}_1 and the periodic discrete Toda lattice, interpreting seed mutations as point additions on an elliptic curve.

Contribution

It explicitly connects seed mutations in the cluster algebra with the time evolution of the Toda lattice via elliptic curve geometry.

Findings

Seed mutations correspond to point additions on an elliptic curve.

The Toda lattice evolution is realized as an orbit of a QRT map.

Specializations relate the algebraic and integrable systems.

Abstract

A direct connection between two sequences of points, one of which is generated by seed mutations in the cluster algebra of type $A^{(1)}_1$ and the other by time evolutions of the periodic discrete Toda lattice, is explicitly given. In this construction, each of them is realized as an orbit of a QRT map and specialization of the parameters in the maps and appropriate choices of the initial points relate them. The connection with the periodic discrete Toda lattice enables us a geometric interpretation of the seed mutations in the cluster algebra of type $A^{(1)}_1$ as addition of points on an elliptic curve arising as the spectral curve of the Toda lattice.