About the Rationality of the Knizhnik-Zamolodchikov Equation Solution

TL;DR

This paper investigates the rationality of solutions to the Knizhnik-Zamolodchikov system, explicitly constructing solutions for specific cases and proving rationality conditions based on parameter values.

Contribution

It provides explicit hypergeometric solutions for the KZ system when n=4, m=2, and proves rationality for integer parameters, also showing non-rationality in a different case.

Findings

Solution is rational for integer  when n=4, m=2.

Explicit hypergeometric form of the solution is constructed.

The system lacks rational solutions for (n=4, m=5,  integer).

Abstract

In the paper the solution of KZ system (n=4, m=2) is constructed in the explicit form in terms of the hypergeometric functions. We proved that the corresponding solution is rational when the parameter $\rho$ is integer. We show that in the case (n=4, m=5, $\rho$ is integer) the corresponding KZ system hasn't got a rational solution.