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Quasi-polynomials and the Bethe Ansatz

E Mukhin and A Varchenko

Geometry & Topology Monographs 13 (2008) 385–420

DOI: 10.2140/gtm.2008.13.385

arXiv: math.QA/0604048


We study solutions of the Bethe Ansatz equation related to the trigonometric Gaudin model associated to a simple Lie algebra g and a tensor product of irreducible finite-dimensional representations. Having one solution, we describe a construction of new solutions. The collection of all solutions obtained from a given one is called a population. We show that the Weyl group of g acts on the points of a population freely and transitively (under certain conditions).

To a solution of the Bethe Ansatz equation, one assigns a common eigenvector (called the Bethe vector) of the trigonometric Gaudin operators. The dynamical Weyl group projectively acts on the common eigenvectors of the trigonometric Gaudin operators. We conjecture that this action preserves the set of Bethe vectors and coincides with the action induced by the action on points of populations. We prove the conjecture for sl2.


Bethe Ansatz, trigonometric Gaudin model, XXX model

Mathematical Subject Classification

Primary: 82B23

Secondary: 17B67


Received: 18 February 2006
Revised: 4 September 2006
Accepted: 11 October 2006
Published: 19 March 2008

E Mukhin
Department of Mathematical Sciences
Indiana University – Purdue University Indianapolis
402 North Blackford St
Indianapolis, IN 46202-3216
A Varchenko
Department of Mathematics
University of North Carolina at Chapel Hill
Chapel Hill, NC 27599-3250