Integrable systems on quad-graphs

Abstract

Discrete (lattice) systems constitute a well-established part of the theory of integrable systems. They came up already in the early days of the theory (see, e.g. [11, 12]), and took gradually more and more important place in it (cf. a review in [18]). Nowadays many experts in the field agree that discrete integrable systems are in many respects even more fundamental than the continuous ones. They play a prominent role in various applications of integrable systems such as discrete differential geometry (see, e.g., a review in [9]). Traditionally, independent variables of discrete integrable systems are considered as belonging to a regular square lattice Z (or its multidimensional analogs Z). Only very recently, there appeared first hints on the existence of a rich and meaningful theory of integrable systems on nonsquare lattices and, more generally, on arbitrary graphs. The relevant publications are almost exhausted by [2, 3, 5, 6, 16, 20, 21, 22]. We define integrable systems on graphs as flat connections with the values in loop groups. This is very natural definition, and experts in discrete integrable systems will not only immediately accept it, but might even consider it trivial. Nevertheless, it crystallized only very recently, and seems not to appear in the literature before [3, 5, 6]. (It should be noted that a different framework for integrable systems on graphs is being developed by Novikov with collaborators [16, 20, 21].) We were led to considering such systems by our (with Hoffmann) investigations of circle patterns as objects of discrete complex analysis: in [5, 6] we demonstrated that certain classes of circle patterns with the combinatorics of regular hexagonal lattice