Discrete SLn-connections and self-adjoint difference operators on two-dimensional manifolds

作者：Grinevich P G^*; Novikov S P

来源：Russian Mathematical Surveys, 2013, 68(5): 861-887.

DOI：10.1070/RM2013v068n05ABEH004859

• triangulated manifolds with black and white colouring
• discrete connections
• discrete complex structures
• factorization of self-adjoint operators
• Darboux and Laplace transformations
• discrete integrable systems

The programme of discretization of famous completely integrable systems and associated linear operators was launched in the 1990s. In particular, the properties of second-order difference operators on triangulated manifolds and equilateral triangular lattices have been studied by Novikov and Dynnikov since 1996. This study included Laplace transformations, new discretizations of complex analysis, and new discretizations of GL(n)-connections on triangulated n-dimensional manifolds. A general theory of discrete GL(n)-connections 'of rank one' has been developed (see the Introduction for definitions). The problem of distinguishing the subclass of SLn-connections (and unimodular SLn +/--connections, which satisfy det A - +/- 1) has not been solved. In the present paper it is shown that these connections play an important role (which is similar to the role of magnetic fields in the continuous case) in the theory of self-adjoint Schrodinger difference operators on equilateral triangular lattices in R-2. In Appendix 1 a complete characterization is given of unimodular SLn +/--connections of rank 1 for all n > 1, thus correcting a mistake (it was wrongly claimed that they reduce to a canonical connection for n > 2). With the help of a communication from Korepanov, a complete clarification is provided of how the classical theory of electrical circuits and star-triangle transformations is connected with the discrete Laplace transformations on triangular lattices.(1)

• 出版日期2013