Discretization Program of the famous Completely Integrable Systems and associated Linear Operators was developed in 1990s. In particular, specific properties of the second order difference operators on the triangulated manifolds were studied in the works of S.Novikov and I.Dynnikov since 1996. They involve factorization of operators, the so-called Laplace Transformations, New Discretization of Complex Analysis and 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 was not selected properly. Indeed, it appears in the theory of important self-adjoint operators. In the present work we construct a Theory of SL_2 discrete connections on the triangulated 2-manifolds. They are deeply associated with real self-adjoint difference operators similar to complex line bundles (magnetic fields) in the 2nd order Schrodinger operators on the plane in the standard continuous quantum mechanics.