Remarks on critical points of phase functions and norms of Bethe vectors

Abstract

We consider a tensor product of a Verma module and the basic linear representation of $sl(n + 1)$. We prove that the corresponding phase function, which is used in the solutions of the KZ equation with values in the tensor product, has a unique critical point and show that the Hessian of the logarithm of the phase function at this critical point equals the Shapovalov norm of the corresponding Bethe vector in the tensor product.