Discrete holomorphic geometry I. Darboux transformations and spectral curves

Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic line bundles over "discrete surfaces", by which we mean the vertex sets of triangulated surfaces with bi-colored set of faces. The resulting theory of discrete conformality is simultaneously Mobius invariant and based on linear equations. In the special case of maps into the 2-sphere we obtain a reinterpretation of the theory of complex holomorphic functions on discrete surfaces introduced by Dynnikov and Novikov. As an application of our theory we introduce a Darboux transformation for discrete surfaces in the conformal 4-sphere. This Darboux transformation can be interpreted as the space- and time-discrete Davey-Stewartson flow of Konopelchenko and Schief. For a generic map of a discrete torus with regular combinatorics, the space of all Darboux transforms has the structure of a compact Riemann surface, the spectral curve. This makes contact to the theory of algebraically completely integrable systems and is the starting point of a soliton theory for triangulated tori in 3- and 4-space devoid of special assumptions on the geometry of the surface.