A geometric realization of the periodic discrete Toda lattice and its tropicalization

TL;DR

This paper provides a geometric interpretation of the periodic discrete Toda lattice using hyperelliptic curves and extends this to tropical geometry, offering new insights into integrable systems and their combinatorial counterparts.

Contribution

It introduces an explicit geometric realization of the Toda lattice's evolution through curve intersections and extends it to tropical geometry, connecting integrable systems with tropical combinatorics.

Findings

Explicit formula for curve intersections related to Toda lattice evolution

Realization of Toda dynamics as point addition on hyperelliptic curves

Tropical geometric construction of the periodic box-ball system

Abstract

An explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete Toda lattice is presented. First, the time evolution is realized as a point addition on a hyperelliptic curve, which is the spectral curve of the periodic discrete Toda lattice, then the point addition is translated into curve intersections. Next, it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the periodic discrete Toda lattice. Finally, the formulation is lifted to the framework of tropical geometry, and a tropical geometric realization of the periodic box-ball system is constructed via tropical curve intersections.