A Geometry for Multidimensional Integrable Systems

TL;DR

This paper develops a geometric framework using deformed differential calculus to describe multidimensional integrable systems, including the KP hierarchy, linking algebraic operators with geometric structures.

Contribution

It introduces a novel deformed calculus based on star-products that models pseudo-differential operators geometrically for multidimensional integrable systems.

Findings

Provides a geometric description of the KP hierarchy

Connects deformation limits to dispersionless hierarchies

Models algebra of pseudo-differential operators geometrically

Abstract

A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one obtains a geometric description of the operators. A dual theory is also possible, based on a deformation of differential forms. This calculus is applied to a number of multidimensional integrable systems, such as the KP hierarchy, thus obtaining a geometrical description of these systems. The limit in which the deformation disappears corresponds to taking the dispersionless limit in these hierarchies.