Integrability of higher pentagram maps

TL;DR

This paper introduces higher-dimensional generalizations of the pentagram map, proves their integrability via Lax representations, and explores their continuous limits related to the KdV hierarchy, with detailed analysis in 3D.

Contribution

It extends the pentagram map to higher dimensions, establishes integrability through Lax pairs, and connects discrete maps to continuous integrable systems in multiple dimensions.

Findings

Higher pentagram maps are integrable in any dimension.

The continuous limit relates to the $(2,d+1)$-equation of the KdV hierarchy.

Detailed integrability analysis for 3D case, including spectral curve and invariants.

Abstract

We define higher pentagram maps on polygons in $P^d$ for any dimension $d$, which extend R.Schwartz's definition of the 2D pentagram map. We prove their integrability by presenting Lax representations with a spectral parameter for scale invariant maps. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D. We also study in detail the 3D case, where we prove integrability for both closed and twisted polygons and describe the spectral curve, first integrals, the corresponding tori and the motion along them, as well as an invariant symplectic structure.