Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

Solutions of the Bethe ansatz equations associated to the XXX model of a simple Lie algebra g come in families called the populations. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra g . The proof is based on the correspondence between the solutions of the Bethe ansatz equations and special difference operators which we call the discrete Miura opers. The notion of a discrete Miura oper is one of the main results of the paper. For a discrete Miura oper D, associated to a point of a population, we show that all solutions of the difference equation DY = 0 are rational functions, and the solutions can be written explicitly in terms of points composing the population. ∗ Department of Mathematical Sciences, Indiana University Purdue University Indianapolis, 402 North Blackford St., Indianapolis, IN 46202-3216, USA Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA