Solutions to the Quantized Knizhnik-Zamolodchikov Equation and the Bethe Ansatz

Abstract

We give an integral representation for solutions to the quantized Knizhnik- Zamolodchikov equation (qKZ) associated with the Lie algebra $gl_{N+1}$. Asymptotic solutions to qKZ are constructed. The leading term of an asymptotic solution is the Bethe vector -- an eigenvector of the transfer-matrix of a quantum spin chain model. We show that the norm of the Bethe vector is equal to the product of the Hessian of a suitable function and an explicitly written rational function. This formula is a generalization of the Gaudin-Korepin formula for a norm of the Bethe vector. We show that, generically, the Bethe vectors form a base for the $gl_2$ case.