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

TL;DR

This paper explores the role of discrete SL2 connections in self-adjoint difference operators on triangulated 2-manifolds, correcting previous misconceptions and linking to classical electric chain theory.

Contribution

It develops the theory of discrete SLn connections, clarifies their importance in self-adjoint operators, and corrects earlier assumptions about unimodular SLn connections.

Findings

Characterization of rank 1 unimodular SLn± connections

Correction of previous claims about equivalence of unimodular SLn± connections

Connection established between classical electric chain theory and discrete Laplace transformations

Abstract

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 and equilateral triangle lattices 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 (and unimodular $SL_n^{\pm}$ connections such that $\det A=\pm 1$) was not selected properly. As we prove in this work, it plays fundamental role (similar to magnetic field in the continuous case) in the theory of self-adjoint discrete Schrodinger operators for the equilateral triangle lattice in $\RR^2$. In Appendix~1 we present a complete characterization of rank 1 unimodular $SL_n^{\pm}$ connections. Therefore we correct a mistake made in the previous versions of our paper (we wrongly claimed that for $n>2$ every unimodular $SL_n^{\pm}$ Connection is equivalent to the standard Canonical Connection). Using communications of Korepanov we completely clarify connection of classical theory of electric chains and star-triangle with discrete Laplace transformation on the triangle lattices