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

TL;DR

This paper introduces a new explicit integral form of solutions to the quantum Knizhnik-Zamolodchikov equations at level -4 for the XXX spin chain, facilitating the calculation of correlation functions and revealing deep mathematical structures.

Contribution

It presents a novel explicit integral solution to qKZ equations at level -4, linking it to cohomologies of deformed Jacobi varieties and applying it to correlation functions via the Jimbo-Miwa conjecture.

Findings

Solution reducible to one-dimensional integrals

Correlation functions agree with previous ansatz

Connection to Riemann zeta functions at odd arguments

Abstract

This paper is continuation of our previous papers hep-th/0209246 and hep-th/0304077 . We discuss in more detail a new form of solution to the quantum Knizhnik-Zamolodchikov equation [qKZ] on level -4 obtained in the paper hep-th/0304077 for the Heisenberg XXX spin chain. The main advantage of this form is it's explicit reducibility to one-dimensional integrals. We argue that the deep mathematical reason for this is some special cohomologies of deformed Jacobi varieties. We apply this new form of solution to the correlation functions using the Jimbo-Miwa conjecture. A formula (46) for the correlation functions obtained in this way is in a good agreement with the ansatz for the emptiness formation probability from the paper hep-th/0209246. Our previous conjecture on a structure of correlation functions of the XXX model in the homogeneous limit through the Riemann zeta functions at odd arguments is a corollary of the formula (46).