The Darboux-KP system as an integrable Chern-Simons multiform theory in infinite dimensional space

TL;DR

This paper develops a hierarchy of Lagrangian multiforms for the Darboux-KP system, framing it as a universal 3D integrable system within an infinite-dimensional space of Miwa variables, connecting Chern-Simons theory and integrability.

Contribution

It introduces a hierarchy of Lagrangian multiforms for the Darboux-KP system as a Chern-Simons action in infinite dimensions, unifying integrable systems and variational structures.

Findings

Established a hierarchy of Lagrangian multiforms for the Darboux-KP system.

Connected the system to a universal 3D integrable structure in infinite-dimensional space.

Linked Chern-Simons theory with integrable hierarchies in a novel geometric framework.

Abstract

In a previous paper by one of the authors, a Lagrangian 3-form structure was established for a generalised Darboux system, originally describing orthogonal curvilinear coordinate systems, which encodes the Kadomtsev-Petviashvili (KP) hierarchy. Here a hierarchy of Lagrangian multiforms is established for the same system, viewed as a hierarchy of Chern-Simons actions in an infinite-dimensional space of Miwa variables, constituting the variational form of a universal 3D integrable system embedded in this infinite-dimensional space.