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Norm of a Bethe vector and the Hessian of the master function

Published online by Cambridge University Press:  21 June 2005

Evgeny Mukhin  and
Alexander Varchenko
Evgeny Mukhin
Affiliation:
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 North Blackford St., Indianapolis, IN 46202-3216, USAmukhin@math.iupui.edu
Alexander Varchenko
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USAanv@email.unc.edu
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Abstract

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We show that the norm of a Bethe vector in the $sl_{r+1}$ Gaudin model is equal to the Hessian of the corresponding master function at the corresponding critical point. In particular the Bethe vectors corresponding to non-degenerate critical points are non-zero vectors. This result is a byproduct of functorial properties of Bethe vectors studied in this paper. As another byproduct of functoriality we show that the Bethe vectors form a basis in the tensor product of several copies of first and last fundamental $sl_{r+1}$-modules.

Keywords

Bethe ansatzmaster functionscritical points82B23 (primary)17B6714M15 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 141 , Issue 4 , July 2005 , pp. 1012 - 1028
Copyright
Foundation Compositio Mathematica 2005