Bethe Ansatz for Arrangements of Hyperplanes and the Gaudin Model

Abstract

We show that the Shapovalov norm of a Bethe vector in the Gaudin model is equal to the Hessian of the logarithm of the corresponding master function at the corresponding isolated critical point. We show that different Bethe vectors are orthogonal. These facts are corollaries of a general Bethe ansatz type construction, suggested in this paper and associated with an arbitrary arrangement of hyperplanes.