Alexander Varchenko

The flag variety structure for solutions of the Bethe ansatz equations

The Bethe ansatz is a method in the theory of quantum integrable models to calculate eigenvectors for a certain family of commutative linear operators (hamiltonians of the model). One assigns the Bethe ansatz equations to a model. Then a solution of the equations gives an eigenvector of the commuting hamiltonians of the model. The simplest and interesting example of an integrable model is the Gaudin model associated with a complex simple Lie algebra $g$. It turns out that in this case solutions to the Bethe ansatz equations come in families called the populations. It also turns out that each population is isomorphic to the flag variety of the Langlands dual Lie algebra $g^t$. These facts are based on the correspondence between solutions of the Bethe ansatz equations and differential operators called the Miura opers.