Populations of Solutions to Cyclotomic Bethe Equations

TL;DR

This paper explores solutions to cyclotomic Bethe equations, linking them to critical points of master functions and populations of solutions, revealing geometric structures and symmetries in the cyclotomic Gaudin model.

Contribution

It introduces a new interpretation of cyclotomic Bethe solutions as critical points with symmetry and defines populations of solutions with geometric and algebraic structures.

Findings

Solutions correspond to critical points of a cyclotomic master function.

Populations of solutions are isomorphic to varieties of isotropic flags.

Associates vector spaces with bilinear forms to populations in type A.

Abstract

We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two interpretations of such solutions: as critical points of a cyclotomic master function, and as critical points with cyclotomic symmetry of a certain "extended" master function. In finite types, this yields a correspondence between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model and those of an "extended" non-cyclotomic Gaudin model. We proceed to define populations of solutions to the cyclotomic Bethe equations, in the sense of [Mukhin E., Varchenko A., Commun. Contemp. Math. 6 (2004), 111-163, math.QA/0209017], for diagram automorphisms of Kac-Moody Lie algebras. In the case of type A with the diagram automorphism, we associate to each population a vector space of quasi-polynomials with specified ramification conditions. This vector space is equipped with a ${\mathbb Z}_2$-gradation and a non-degenerate bilinear form which is (skew-)symmetric on the even (resp. odd) graded subspace. We show that the population of cyclotomic critical points is isomorphic to the variety of isotropic full flags in this space.