Noncommutative geometry and a class of completely integrable models

TL;DR

This paper extends classical integrability of 2D harmonic maps into groups to a noncommutative setting using a newly introduced Hodge operator in noncommutative geometry.

Contribution

It introduces a Hodge operator in noncommutative geometry and demonstrates the integrability of noncommutative harmonic maps into matrix algebras.

Findings

Established a noncommutative Hodge operator.

Extended classical harmonic map integrability to noncommutative models.

Demonstrated integrability for maps into matrix algebras.

Abstract

We introduce a Hodge operator in a framework of noncommutative geometry. The complete integrability of 2-dimensional classical harmonic maps into groups (sigma-models or principal chiral models) is then extended to a class of 'noncommutative' harmonic maps into matrix algebras.