Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 032, 55 pages      arXiv:1001.4553
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

Quantum Integrable Model of an Arrangement of Hyperplanes

Alexander Varchenko
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Received July 19, 2010, in final form March 19, 2011; Published online March 28, 2011

The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe algebra of the corresponding Gaudin model. It is expected that this correspondence holds in general (without the assumptions). As a byproduct of constructions we show that in a Gaudin model (associated to an arbitrary simple Lie algebra), the Bethe vector, corresponding to an isolated critical point of the master function, is nonzero.

Key words: Gaudin model; arrangement of hyperplanes.

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