Periodicity and integrability for the cube recurrence

TL;DR

This paper proves periodicity and integrability for discrete dynamical systems based on the cube recurrence, extending Zamolodchikov's concepts from $T$- and $Y$-systems to a new setting.

Contribution

It establishes the periodicity and integrability of systems derived from the cube recurrence, confirming a conjecture by Henriques from 2007.

Findings

Proved periodicity for the cube recurrence system.

Established integrability properties of the system.

Extended Zamolodchikov's concepts to new discrete systems.

Abstract

Zamolodchikov periodicity is a property of $T$- and $Y$-systems, arising in the thermodynamic Bethe ansatz. Zamolodchikov integrability was recently considered as its affine analog in our joint work with P. Pylyavskyy. Here we prove periodicity and integrability for similar discrete dynamical systems based on the cube recurrence, also known as the discrete BKP equation. The periodicity part was conjectured by Henriques in 2007.