Discrete Connections and Difference Linear Equations

Following earlier works, we develop here a nonstandard discrete analogue of the theory of differential-geometric $GL_{n}$-connections on triangulated manifolds. This theory is based on the interpretation of a connection as a first-order linear difference equation—the “triangle equation”—for scalar functions of vertices in simplicial complexes. This theory appeared as a byproduct of the discretization of famous completely integrable systems such as the 2D Toda lattice. A nonstandard discretization of complex analysis based on these ideas was developed earlier. Here, a complete classification theory based on the mixture of abelian and nonabelian features is given for connections on triangulated manifolds.