Desargues maps and the Hirota-Miwa equation

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.

💡 Research Summary

The paper introduces a novel discrete geometric object called the Desargues map, a mapping φ : ℤᴺ → ℙᴹ whose points and all of their forward nearest neighbours lie on a common projective line. This simple collinearity condition is shown to be equivalent to the classical Desargues theorem and its higher‑dimensional extensions when imposed on an N‑dimensional integer lattice. Consequently, the multidimensional compatibility of the map—i.e., the ability to extend the collinearity condition consistently throughout the whole lattice—is guaranteed by the same incidence geometry that underlies Desargues’ theorem.

From the projective coordinates of φ the authors construct non‑commutative variables τₙ and derive the algebraic relations between τ‑functions at neighboring lattice sites. The resulting system is precisely the non‑commutative Hirota–Miwa equation: for any pair of distinct directions i ≠ j, \