The Pentagram map in higher dimensions and KdV flows

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.

💡 Research Summary

The paper presents a comprehensive extension of the classical pentagram map—originally defined for planar polygons—to polygons in arbitrary projective spaces $\mathbb{RP}^d$. The authors begin by recalling that the 2‑dimensional pentagram map, which replaces each vertex of a polygon by the intersection of two “short” diagonals, is a discrete integrable system whose dynamics can be linearized on the Jacobian of an algebraic curve. Motivated by recent interest in higher‑dimensional discrete geometry, they construct a natural analogue for a “twisted” polygon ${V_i}{i\in\mathbb Z}$ in $\mathbb{RP}^d$, where $V_i$ and $V{i+d}$ are identified up to a projective monodromy. For each index $i$ they consider the $d$‑dimensional projective subspace spanned by $V_i,\dots,V_{i+d-1}$ and define the new vertex $V_i’$ as the intersection of two consecutive subspaces. This yields a well‑defined map $T:{V_i}\mapsto{V_i’}$ that is invariant under the action of $\operatorname{PGL}(d+1)$.

A central achievement of the work is the derivation of a Lax representation for $T$. Introducing a spectral parameter $\lambda$, the authors associate to each vertex a $(d+1)\times(d+1)$ matrix $L_i(\lambda)$ and a transition matrix $P_i(\lambda)$ such that \