Quasi-polynomials and the Bethe Ansatz

Abstract

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 sl_2.

This is the version published by Geometry & Topology Monographs on 19 March 2008

References in corpus (3)

• Quasiclassical asymptotics of solutions to the KZ equations
• Discrete Miura Opers and Solutions of the Bethe Ansatz Equations
• Populations of solutions of the XXX Bethe equations associated to Kac-Moody algebras

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• The $A_m^{(1)}$ Q-system
• Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians
• Feigin-Frenkel-Hernandez Opers and the QQ-system
• XXZ-type Bethe ansatz equations and quasi-polynomials
• On the Gaudin model of type G${}_2$
• On Wronskians and $qq$-systems
• Populations of Solutions to Cyclotomic Bethe Equations
• Opers on the projective line, Wronskian relations, and the Bethe Ansatz
• Superopers revisited