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Discretization Program of the famous Completely Integrable Systems andassociated Linear Operators was developed in 1990s. In particular, specificproperties of the second order difference operators on the triangulatedmanifolds and equilateral triangle lattices were studied in the works ofS.Novikov and I.Dynnikov since 1996. They involve factorization of operators,the so-called Laplace Transformations, new discretization of Complex Analysisand new discretization of $GL_n$ connections on the triangulated $n$-manifolds.The general theory of the new type discrete $GL_n$ connections was developed.However, the special case of $SL_n$-connections (and unimodular $SL_n^{\pm}$connections such that $\det A=\pm 1$) was not selected properly. As we prove inthis work, it plays fundamental role (similar to magnetic field in thecontinuous case) in the theory of self-adjoint discrete Schrodinger operatorsfor the equilateral triangle lattice in $\RR^2$. In Appendix~1 we present acomplete characterization of rank 1 unimodular $SL_n^{\pm}$ connections.Therefore we correct a mistake made in the previous versions of our paper (wewrongly claimed that for $n>2$ every unimodular $SL_n^{\pm}$ Connection isequivalent to the standard Canonical Connection). Using communications ofKorepanov we completely clarify connection of classical theory of electricchains and star-triangle with discrete Laplace transformation on the trianglelattices

Discrete SL2 Connections and Self-Adjoint Difference Operators on the Triangulated 2-manifold