Desargues maps and the Hirota-Miwa equation

TL;DR

This paper explores the geometric properties of Desargues maps and their relation to the Hirota-Miwa equation, using the nonlocal arar-dressing method to construct solutions and linking these maps to quadrilateral lattices.

Contribution

It establishes a connection between Desargues maps, the Hirota-Miwa system, and quadrilateral lattices, providing a geometric and analytical framework for their study.

Findings

Desargues maps generate collinear lattices linked to the Hirota-Miwa equation.

The nonlocal arar-dressing method constructs solutions and identifies the arar-dressing problem's Fredholm determinant with the ta-function.

Desargues maps are equivalent to quadrilateral lattices when considering their Laplace transforms.

Abstract

We study the Desargues maps $\phi:\ZZ^N\to\PP^M$, which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multidimensional compatibility of the map is equivalent to the Desargues theorem and its higher-dimensional generalizations. The nonlinear counterpart of the map is the non-commutative (in general) Hirota--Miwa system. In the commutative case of the complex field we apply the nonlocal $\bar\partial$-dressing method to construct Desargues maps and the corresponding solutions of the equation. In particular, we identify the Fredholm determinant of the integral equation inverting the nonlocal $\bar\partial$-dressing problem with the $\tau$-function. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.