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Mosc. Math. J., 2006, Volume 6, Number 1, Pages 195–210 (Mi mmj243)  

This article is cited in 9 scientific papers (total in 9 papers)

Bethe ansatz for arrangements of hyperplanes and the Gaudin model

A. N. Varchenko

Department of Mathematics, University of North Carolina at Chapel Hill

Abstract: We show that the Shapovalov norm of a Bethe vector in the Gaudin model is equal to the Hessian of the logarithm of the corresponding master function at the corresponding isolated critical point. We show that different Bethe vectors are orthogonal. These facts are corollaries of a general Bethe ansatz type construction, suggested in this paper and associated with an arbitrary arrangement of hyperplanes.

Key words and phrases: Gaudin model, Bethe vectors, arrangements of hyperplanes, Orlik–Solomon algebra, flag complex.

DOI: https://doi.org/10.17323/1609-4514-2006-6-1-195-210

Full text: http://www.ams.org/.../abst6-1-2006.html
References: PDF file   HTML file

Bibliographic databases:

MSC: Primary 82C20; Secondary 17B, 81R12, 82C23
Received: September 24, 2005
Language:

Citation: A. N. Varchenko, “Bethe ansatz for arrangements of hyperplanes and the Gaudin model”, Mosc. Math. J., 6:1 (2006), 195–210

Citation in format AMSBIB
\Bibitem{Var06}
\by A.~N.~Varchenko
\paper Bethe ansatz for arrangements of hyperplanes and the Gaudin model
\jour Mosc. Math.~J.
\yr 2006
\vol 6
\issue 1
\pages 195--210
\mathnet{http://mi.mathnet.ru/mmj243}
\crossref{https://doi.org/10.17323/1609-4514-2006-6-1-195-210}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2265955}
\zmath{https://zbmath.org/?q=an:05184506}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208595700012}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Mukhin E., Tarasov V., Varchenko A., “The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz”, Ann. of Math. (2), 170:2 (2009), 863–881  crossref  mathscinet  zmath  isi  scopus
    2. Alexander Varchenko, “Quantum Integrable Model of an Arrangement of Hyperplanes”, SIGMA, 7 (2011), 032, 55 pp.  mathnet  crossref  mathscinet
    3. Budur N., “Complements and higher resonance varieties of hyperplane arrangements”, Math. Res. Lett., 18:5 (2011), 859–873  crossref  mathscinet  zmath  isi
    4. Cohen D., Denham G., Falk M., Varchenko A., “Critical points and resonance of hyperplane arrangements”, Canad. J. Math., 63:5 (2011), 1038–1057  crossref  mathscinet  zmath  isi  scopus
    5. Mukhin E. Tarasov V. Varchenko A., “Three Sides of the Geometric Langlands Correspondence for Gl(N) Gaudin Model and Bethe Vector Averaging Maps”, Arrangements of Hyperplanes - Sapporo 2009, Advanced Studies in Pure Mathematics, 62, ed. Terao H. Yuzvinsky S., Math Soc Japan, 2012, 475–511  mathscinet  zmath  isi
    6. Jensen E.J., Varchenko A.N., “Norms of Eigenfunctions of Trigonometric Kzb Operators”, Int. Math. Res. Notices, 2013, no. 6, 1230–1267  crossref  mathscinet  zmath  isi  elib  scopus
    7. Varchenko A. Wright D., “Critical Points of Master Functions and Integrable Hierarchies”, Adv. Math., 263 (2014), 178–229  crossref  mathscinet  zmath  isi  elib  scopus
    8. Alexander Varchenko, Charles A. S. Young, “Populations of Solutions to Cyclotomic Bethe Equations”, SIGMA, 11 (2015), 091, 41 pp.  mathnet  crossref
    9. Varchenko A., “Solutions Modulo P of Gauss-Manin Differential Equations For Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz”, 5, no. 4, 2017, 52  crossref  zmath  isi  scopus
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