Smirnov's Integrals and Quantum Knizhnik-Zamolodchikov Equation of Level $0$

TL;DR

This paper derives an integral formula for solutions to the quantum Knizhnik-Zamolodchikov equation at level 0 for spin 1/2 representations, enabling the generation of multiple solutions through different integral kernels.

Contribution

It introduces a novel integral formula for solutions of the quantum KZ equation at level 0, applicable to arbitrary total spin and fixed integration cycles.

Findings

Integral formula for solutions with arbitrary total spin

Multiple solutions generated by different integral kernels

Applicable for |q|<1 in the quantum group context

Abstract

We study the quantum Knizhnik-Zamolodchikov equation of level $0$ associated with the spin $1/2$ representation of $U_q \bigl(\widehat{\frak s \frak l _{2}}\bigr)$. We find an integral formula for solutions in the case of an arbitrary total spin and $|q|<1$. In the formula, different solutions can be obtained by taking different integral kernels with the cycle of integration being fixed.