Holomorphic bundles and Scalar Difference Operators: One-Point Constructions

TL;DR

This paper develops new analytical methods for constructing commutative rings of higher-rank one-dimensional difference operators using Tyurin parameters, revealing richer structures and integrable systems, especially for even ranks, with implications for algebraic geometry and integrable systems.

Contribution

It introduces one-point constructions for higher-rank difference operators using Tyurin parameters, expanding the understanding beyond the rank one case and linking to integrable systems.

Findings

Constructed commutative rings of difference operators of rank > 1.

Discovered new integrable systems for rank 2, genus 1 case.

Showed that one-point constructions depend on functional parameters for even ranks.

Abstract

Commutative rings of one-dimensional difference operators of rank l>1 and their deformations are effectively constructed. Our analytical constructions are based on the so-called ''Tyurin parameters'' for the stable framed holomorphic vector bundles over algebraic curves of the genus equal to g and Chern number equal to lg. These parameters were heavily used by the present authors already in 1978-80 for the differential operators. Their deformations in the discrete case are governed by the 2D Toda Lattice hierarhy instead of KP. New integrable systems appear here in the case l=2,g=1. The theory of higher rank difference operators is much more rich than the rank one case where only 2-point constructions on the spectral curve were used in the previous literature (i.e. number of 'infinite points'' is equal to 2). One-point constructions appear in this problem for every even rank l=2k. Only in this case commutative rings depend on the functional parameters. Two-point constructions will be studied in the next work: even for higher rank l>1 this case can be solved in Theta-functions. It is not so for one-point constructions with rank l>1.