Holomorphic bundles and commuting difference operators. Two-point constructions

Abstract

This paper is closely related to our previous paper [1]. As is well known, in current mathematical physics the theory of commuting one-dimensional operators appears as an auxiliary algebraic aspect of the theory of integration of non-linear soliton systems and the spectral theory of periodic finite-zone operators [2]–[5]. As a purely algebraic problem, the problem of classifying ordinary scalar differential operators had already been posed in the 1920s by Burchnall and Chaundy [6], who advanced a long way towards a solution of the problem in the case of operators of mutually prime orders (in which the rank is always equal to 1), completed in [3]. However, they remarked that the general problem for rank r > 1 seemed extraordinarily difficult. The first steps were taken in [7] and [8]. A method of effective classification of commuting differential operators of rank r > 1 in general position was created by the authors in [9] and [10]. Commuting pairs of rank r > 1 depend on (r − 1) arbitrary functions of one variable, a smooth algebraic curve Γ with one distinguished point P and a set of Tyurin parameters (characterizing a framed stable holomorphic bundle). We call these one-point constructions. For difference operators the whole theory, which has already become classical, of pairs of commuting operators of rank r = 1 was based only on two-point constructions [11], [12]. Rings of such operators turned out to be isomorphic to the rings A(Γ, P±) of meromorphic functions on an algebraic curve Γ with poles at a pair of distinguished points P±. In our previous paper [1] we showed that for rank 2l ≥ 2 a broad class of commuting difference operators can be obtained from the one-point construction. As in the continuous case, these operators depend on arbitrary functions of one variable n ∈ Z. In the present paper we have obtained a description of a broad class of commuting difference operators constructed starting from two-point constructions. In contrast to one-point constructions, there are no arbitrary functions here; the coefficients of the operators can be calculated by means of the Riemann theta function. As in the rank 1 case, these operators lead to solutions of the equations of the 2D Toda lattice and the whole hierarchy connected with them. We consider a smooth algebraic curve Γ of genus g with two distinguished points P±. Let (γ) be a collection of rg points γs, s = 1, . . . , rg, on Γ, and (α) be a collection of (r − 1)-dimensional vectors: αs = (αsj), j = 1, . . . , r − 1. According to [9], [10], these parameters (γ,α) are called Tyurin parameters. In the general case they determine a stable framed bundle E over Γ of rank r and degree c1(detE) = rg.