The Pentagram map in higher dimensions and KdV flows

TL;DR

This paper generalizes the pentagram map to higher dimensions, explores its integrability via Lax form, and links its continuous limit to higher-dimensional KdV flows, extending known 2D results to broader geometric and integrable systems contexts.

Contribution

It introduces a higher-dimensional version of the pentagram map, establishes its integrability through Lax pairs, and connects its continuous limit to the (2,d+1)-flow of the KdV hierarchy.

Findings

Higher-dimensional pentagram map defined and analyzed.

Integrability shown via Lax form for closed and twisted polygons.

Continuous limit corresponds to higher-dimensional KdV flows.

Abstract

We extend the definition of the pentagram map from 2D to higher dimensions and describe its integrability properties for both closed and twisted polygons by presenting its Lax form. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-flow of the KdV hierarchy, generalizing the Boussinesq equation in 2D.