Integrable discrete nets in Grassmannians

TL;DR

This paper introduces a new class of discrete nets in Grassmannians that generalize known nets, proves their integrability through geometric methods, and links them to the noncommutative discrete Darboux system.

Contribution

It defines integrable discrete nets in Grassmannians, provides a geometric proof of their multidimensional consistency, and connects them to the noncommutative discrete Darboux system.

Findings

Proved integrability of discrete nets in Grassmannians.

Established geometric proof of multidimensional consistency.

Linked these nets to the noncommutative discrete Darboux system.

Abstract

We consider discrete nets in Grassmannians $\mathbb{G}^d_r$ which generalize Q-nets (maps $\mathbb{Z}^N\to\mathbb{P}^d$ with planar elementary quadrilaterals) and Darboux nets ($\mathbb{P}^d$-valued maps defined on the edges of $\mathbb{Z}^N$ such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.