Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems

TL;DR

This paper explores integrable flows on Grassmannians defined by closed differential forms, linking them to well-known N-dimensional integrable systems and revealing dualities and connections to classical equations like Maxwell's.

Contribution

It introduces a geometric framework for N-dimensional integrable systems via closed forms on Grassmannians, unifying various equations and revealing duality structures.

Findings

Connections between Grassmannian flows and classical integrable equations

Identification of Maxwell equations as a special case at N=3

Dual systems related to projectively dual Grassmannians

Abstract

Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms $\Omega_{N-1}$ of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deformations of (N-1)-dimensional linear subspaces. For the class of solutions which are Laurent polynomials in one variable these systems coincide with N-dimensional integrable systems such as Liouville equation (N=2), dispersionless Kadomtsev-Petviashvili equation (N=3), dispersionless Toda equation (N=3), Plebanski second heavenly equation (N=4) and others. Gauge invariant part of the forms $\Omega_{N-1}$ provides us with the compact form of the corresponding hierarchies. Dual quasi-linear systems associated with the projectively dual Grassmannians Gr(2,N+1) are defined via the requirement of the closedness of the dual forms $\Omega_{N-1}^{\star}$. It is shown that at N=3 the self-dual quasi-linear system, which is associated with the harmonic (closed and co-closed) form $\Omega_{2}$, coincides with the Maxwell equations for orthogonal electric and magnetic fields.