Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry

TL;DR

This paper reviews the integrable structures of the box-ball system, a cellular automaton with deep connections to quantum and classical integrable systems, crystal bases, Bethe ansatz, and tropical geometry, highlighting two decades of developments.

Contribution

It provides a comprehensive overview of the integrable aspects and mathematical frameworks underlying the box-ball system and its generalizations.

Findings

Connections to Yang-Baxter models and crystal bases

Relations to soliton theory and tau functions

Insights into tropical geometry and spectral curves

Abstract

The box-ball system is an integrable cellular automaton on one dimensional lattice. It arises from either quantum or classical integrable systems by the procedures called crystallization and ultradiscretization, respectively. The double origin of the integrability has endowed the box-ball system with a variety of aspects related to Yang-Baxter integrable models in statistical mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz, geometric crystals, classical theory of solitons, tau functions, inverse scattering method, action-angle variables and invariant tori in completely integrable systems, spectral curves, tropical geometry and so forth. In this review article, we demonstrate these integrable structures of the box-ball system and its generalizations based on the developments in the last two decades.