Communications in Mathematical Physics

, Volume 256, Issue 3, pp 565–588 | Cite as

Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

  • Evgeny Mukhin
  • Alexander Varchenko


Solutions of the Bethe ansatz equations associated to the XXX model of a simple Lie algebra Open image in new window come in families called the populations. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra Open image in new window 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.


Neural Network Statistical Physic Complex System Rational Function Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bogoliubov, N.M., Izergin, A.G., Korepin, V.E.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge: Cambridge University Press, 1993Google Scholar
  2. 2.
    Bernshtein, I.N., Gel’fand, I.M., Gel’fand, S.I.: Structure of representations generated by vectors of highest weight. Funct. Anal. Appl. 5, 1–8 (1971)CrossRefGoogle Scholar
  3. 3.
    Borisov, L., Mukhin, E.: Self-self-dual spaces of polynomials. QA/0308128, 2003
  4. 4.
    Drinfeld, V., Sokolov, V.: Lie algebras and KdV type equations. J. Sov. Math. 30, 1975–2036 (1985)Google Scholar
  5. 5.
    Faddeev, L.D.: Lectures on Quantum Inverse Scattering Method. In: Integrable Systems, X.-C. Song (ed.), Nankai Lectures Math Phys., Singapore: World Scientific, 1990, pp. 23–70Google Scholar
  6. 6.
    Faddeev, L.D., Takhtajan, L.A.: Quantum Inverse Problem Method and the Heisenberg XYZ-model. Russ. Math. Surveys 34, 11–68 (1979)Google Scholar
  7. 7.
    Feigin, B., Frenkel, E., Reshetikhin, N.: Gaudin model, Bethe Ansatz and Critical Level. Commun. Math. Phys. 166, 29–62 (1994)Google Scholar
  8. 8.
    Frenkel, E.: Opers on the projective line, flag manifolds and Bethe ansatz. http://arxiv. org/abs/math.QA/0308269, 2003
  9. 9.
    Frenkel, E.: Affine Algebras, Langlands Duality and Bethe Ansatz., 1995
  10. 10.
    Frenkel, E., Reshetikhin, N., Semenov-Tian-Shansky,: Drinfeld-Sokolov Reduction for Difference Operators and Deformations of W-algebras I. The Case of Virasoro Algebra. http://xxx.lanl/gov/abs/g-alg/9704011, 1997
  11. 11.
    Kac, V.: Infinite-dimensional Lie algebras. Cambridge: Cambridge University Press, 1990Google Scholar
  12. 12.
    Mukhin, E., Varchenko, A.: Critical Points of Master Functions and Flag Varieties. http://arxiv. org/abs/ math.QA/0209017, 2002
  13. 13.
    Mukhin, E., Varchenko, A.: Populations of solutions of the XXX Bethe equations associated to Kac-Moody algebras., 2002
  14. 14.
    Mukhin, E., Varchenko, A.: Solutions to the XXX type Bethe Ansatz equations and flag varieties. Cent. Eur. J. Math. 1(2), 238–271 (2003)Google Scholar
  15. 15.
    Mukhin, E., Varchenko, A.: Miura Opers and Critical Points of Master Functions., 2003
  16. 16.
    Ogievetsky, E., Wiegman, P.: Factorized S-matrix and the Bethe Ansatz for simple Lie groups. Phys. Lett. 168B(4), 360–366 (1986)Google Scholar
  17. 17.
    Sevostyanov, A.: Towards Drinfeld-Sokolov reduction for quantum groups. J. Geom. Phys. 33(3–4), 235–256 (2000)Google Scholar
  18. 18.
    Semenov-Tian-Shansky, M., Sevostyanov, A.: Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras. II. The general semisimple case. Commun. Math. Phys. 192(3), 631–647 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Evgeny Mukhin
    • 1
  • Alexander Varchenko
    • 2
  1. 1.Department of Mathematical SciencesIndiana University Purdue University IndianapolisIndianapolisUSA
  2. 2.Department of MathematicsUniversity of North Carolina at Chapel HillChapel HillUSA

Personalised recommendations