Populations of Solutions to Cyclotomic Bethe Equations

Symmetry Integrability and Geometry-Methods and Applications

article

Alexander Varchenko¹  Charles A.S. Young² 

[1] Department of Mathematics, University of North Carolina at Chapel Hill;School of Physics, University of Hertfordshire, College Lane

关键词: Bethe equations;    cyclotomic symmetry;   

DOI  :  10.3842/SIGMA.2015.091

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown