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

We present a new form of solution to the quantum Knizhnik-Zamolodchikov equation [qKZ] 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 [7]. 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 hyperelliptic 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 ζ -function at odd arguments and rational coefficients suggested in [8, 9]. This issue will be discussed in our next publication.