Bethe Ansatz for Arrangements of Hyperplanes and the Gaudin Model

2004/07/30 by Alexander Varchenko, Varchenko, Alexander · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.QA

paper · pdf · doi:10.48550/arxiv.math/0408001

Latex, 17 pages, an extended version, misprints corrected

openalex publication_date 2004/07/30 · arxiv created 2004/10/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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