Bi-differential calculi and integrable models

TL;DR

This paper explores how bi-differential calculus provides a unified framework to understand the infinite conserved currents in various integrable classical models, linking algebraic structures to physical integrability.

Contribution

It introduces the concept of gauged bi-differential calculus as a unifying approach to derive conserved currents in integrable models.

Findings

Infinite conserved currents are linked to closed forms in bi-differential calculus.

Gauged bi-differential calculus extends to flat covariant derivatives.

Framework applies to models like chiral, Toda, KP, and self-dual Yang-Mills.

Abstract

The existence of an infinite set of conserved currents in completely integrable classical models, including chiral and Toda models as well as the KP and self-dual Yang-Mills equations, is traced back to a simple construction of an infinite chain of closed (respectively, covariantly constant) 1-forms in a (gauged) bi-differential calculus. The latter consists of a differential algebra on which two differential maps act. In a gauged bi-differential calculus these maps are extended to flat covariant derivatives.