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Symmetry, Integrability and Geometry: Methods and Applications, 2015, Volume 11, 091, 41 pp.
DOI: https://doi.org/10.3842/SIGMA.2015.091
(Mi sigma1072)
 

This article is cited in 1 scientific paper (total in 1 paper)

Populations of Solutions to Cyclotomic Bethe Equations

Alexander Varchenkoa, Charles A. S. Youngb

a Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
b School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK
Full-text PDF (681 kB) Citations (1)
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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.
Keywords: Bethe equations; cyclotomic symmetry.
Received: June 17, 2014; in final form November 5, 2015; Published online November 14, 2015
Bibliographic databases:
Document Type: Article
Language: English
Citation: Alexander Varchenko, Charles A. S. Young, “Populations of Solutions to Cyclotomic Bethe Equations”, SIGMA, 11 (2015), 091, 41 pp.
Citation in format AMSBIB
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\by Alexander~Varchenko, Charles~A.~S.~Young
\paper Populations of Solutions to Cyclotomic Bethe Equations
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Symmetry, Integrability and Geometry: Methods and Applications
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