Flat connection on four-dimensional lattice, related matrix difference equations and their solutions

TL;DR

This paper solves a special difference problem related to a four-dimensional zero-curvature condition, providing explicit solutions involving special functions, and extends previous work on reduced density matrices in integrable models.

Contribution

It presents a novel solution to a difference problem from quantum integrable models, connecting geometric flatness conditions with explicit special function solutions.

Findings

Solution expressed via Gamma-function and hypergeometric functions

Connected zero-curvature condition with geometric interpretation

Extended analysis of reduced density matrices for operator length 4

Abstract

In the paper [H.Boos, A.Hutsalyuk and Kh.Nirov, J.Phys.A:Math.Theor. 51 (2018) 445202] the reduced density matrix of the sl(3)-invariant fundamental exchange model was calculated for the operator length up to three by means of the reduced quantum Knizhnik-Zamolodchikov equation. In this paper we present the solution of some special difference problem originated from the study of the reduced density matrix for the operator length 4. This difference problem is related to a four-dimensional zero-curvature condition and has a clear geometrical meaning were we have a trivial fiber bundle CP^3 x C^4 with a vector function which takes value in C^4 and the base being the projective space CP^3. The local connection coefficients satisfy the above mentioned zero-curvature or flatness condition. The solution we discuss here is given in terms of the Gamma-function, its logarithmic derivative, hypergeometric functionand some other related functions defined via the functional relations of difference type.