The geometry of dented pentagram maps

Abstract

We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension $d$ there are $d-1$ such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps with those for the higher pentagram maps constructed with the help of short diagonal hyperplanes. When restricted to corrugated polygons, the dented pentagram maps coincide between themselves and with the corresponding corrugated pentagram map. Finally, we prove integrability for a variety of pentagram maps for generic and partially corrugated polygons in higher dimensions.