Desargues Maps and the Hirota–miwa Equation

We study the Desargues maps φ : Z → P , which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multidimensional consistency 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 ∂̄-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 ∂̄-dressing problem with the τ -function. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.