Asymptotic Solutions to the Knizhnik-Zamolodchikov Equation and Crystal Base

TL;DR

This paper explores the asymptotic solutions of the Knizhnik-Zamolodchikov equation for $s ext{l}_2$, linking them to crystal bases in quantum group modules and relating Bethe vectors to the crystal basis structure.

Contribution

It establishes a novel connection between asymptotic solutions of the KZ equation and crystal bases in quantum group representations, including a correspondence with Bethe vectors.

Findings

Describes transition functions between asymptotic solutions.

Links asymptotic solutions to crystal bases in $U_qs ext{l}_2$ modules.

Identifies a correspondence between Bethe vectors and the crystal basis.

Abstract

The Knizhnik-Zamolodchikov equation associated with $s\ell_2$ is considered. The transition functions between asymptotic solutions to the Knizhnik-Zamolodchikov equation are described. A connection between asymptotic solutions and the crystal base in the tensor product of modules over the quantum group $U_qs\ell_2$ is established, in particular, a correspondence between the Bethe vectors of the Gaudin model of an inhomogenious magnetic chain and the $\Bbb Q-$basis of the crystal base.