New formulae for solutions of quantum Knizhnik-Zamolodchikov equations on level -4

TL;DR

This paper introduces a new, simplified integral form of solutions to the quantum Knizhnik-Zamolodchikov equations at level -4, linking them to hyper-elliptic integrals and the Riemann zeta-function, with implications for XXX model correlations.

Contribution

It presents a reduced product-of-integrals solution form for qKZ equations at level -4, connecting cohomological methods and hyper-elliptic integrals, and supports conjectures relating solutions to correlation functions.

Findings

Solution form is equivalent to Jimbo and Miwa's integral representation.

Reduced to products of single integrals, simplifying analysis.

Supports conjecture linking correlation functions to Riemann zeta-values.

Abstract

We present a new form of solution to the quantum Knizhnik-Zamolodchikov equation on level -4 in a special case corresponding to the Heisenberg XXX spin chain. Our form is equivalent to the integral representation obtained by Jimbo and Miwa in 1996 . An advantage of our form is that it is reduced to the product of single integrals. This fact is deeply related to a cohomological nature of our formulae. Our approach is also based on the deformation of hyper-elliptic integrals and their main property -- deformed Riemann bilinear relation. Jimbo and Miwa also suggested a nice conjecture which relates solution of the qKZ on level -4 to any correlation function of the XXX model. This conjecture together with our form of solution to the qKZ makes it possible to prove a conjecture that any correlation function of the XXX model can be expressed in terms of the Riemann zeta-function at odd arguments and rational coefficients suggested in our previous papers. This issue will be discussed in our next publication.