Gaudin model, Bethe Ansatz and critical level

Feigin, Boris; Frenkel, Edward; Reshetikhin, Nicolai

doi:10.1007/bf02099300

Cited by 276 publications

(396 citation statements)

References 41 publications

Supporting

Mentioning

378

Contrasting

Unclassified

Order By: Relevance

“…The Miura opers play an essential role in the Drinfeld-Sokolov reduction and geometric Langlands correspondence, see [DS,FFR,F2,FRS]. It would be interesting to see if our discrete opers may play a similar role in discrete versions of the Drinfeld-Sokolov reduction and geometric Langlands correspondence.…”

Section: Introductionmentioning

confidence: 99%

Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

Mukhin

Varchenko

2005

Commun. Math. Phys.

View full text Add to dashboard Cite

Abstract. Solutions of the Bethe ansatz equations associated to the XXX model of a simple Lie algebra g come in families called the populations. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra t g . The proof is based on the correspondence between the solutions of the Bethe ansatz equations and special difference operators which we call the discrete Miura opers. The notion of a discrete Miura oper is one of the main results of the paper.For a discrete Miura oper D, associated to a point of a population, we show that all solutions of the difference equation DY = 0 are rational functions, and the solutions can be written explicitly in terms of points composing the population.

show abstract

Section: Introductionmentioning

confidence: 99%

Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

Mukhin

Varchenko

2005

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…The Hamiltonians commute. The problem is to diagonalize simultaneously the Hamiltonians; see [B,BF,FFR,G,MV1,RV,ScV,V2].…”

Section: Application To the Bethe Ansatz In The Gaudin Model We Letmentioning

confidence: 99%

Multiple orthogonal polynomials and a counterexample to the Gaudin Bethe Ansatz Conjecture

Mukhin

Varchenko

2007

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two sl 2 irreducible modules. We study sequences of r polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two highest weight sl r+1 irreducible modules, with the restriction that the highest weight of one of the modules is a multiple of the first fundamental weight.We describe the recursion which can be used to compute these polynomials. Moreover, we show that the first polynomial in the sequence coincides with the Jacobi-Piñeiro multiple orthogonal polynomial and others are given by Wronskian-type determinants of Jacobi-Piñeiro polynomials.As a byproduct we describe a counterexample to the Bethe Ansatz Conjecture for the Gaudin model.

show abstract

“…The bibliography on this problem is enormous (cf. [Fr1,Fr2,FFR,MV]). It follows from the [FFR] construction that all elements of A(z 1 , .…”

Section: Introductionmentioning

confidence: 99%

The argument shift method and the Gaudin model

Rybnikov

2006

Funct Anal Its Appl

View full text Add to dashboard Cite

Abstract. We construct a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U (g) of a semisimple Lie algebra g . This family is parameterized by collections µ; z1, . . . , zn , where µ ∈ g * , and z1, . . . , zn are pairwise distinct complex numbers. The construction presented here generalizes the famous construction of the higher Gaudin hamiltonians due to Feigin, Frenkel, and Reshetikhin. For n = 1 , our construction gives a quantization of the family of maximal Poissoncommutative subalgebras of S(g) obtained by the argument shift method. Next, we describe natural representations of commutative algebras of our family in tensor products of finite-dimensional g -modules as certain degenerations of the Gaudin model. In the case of g = slr we prove that our commutative subalgebras have simple spectrum in tensor products of finite-dimensional g -modules for generic µ and zi . This implies simplicity of spectrum in the "generic" slr Gaudin model.

show abstract

Gaudin model, Bethe Ansatz and critical level

Cited by 276 publications

References 41 publications

Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

Multiple orthogonal polynomials and a counterexample to the Gaudin Bethe Ansatz Conjecture

The argument shift method and the Gaudin model

Contact Info

Product

Resources

About