Communications in Mathematical Physics

, Volume 349, Issue 3, pp 1063–1105 | Cite as

Bethe Ansatz and the Spectral Theory of Affine Lie algebra–Valued Connections II: The Non Simply–Laced Case



We assess the ODE/IM correspondence for the quantum \({\mathfrak{g}}\)-KdV model, for a non-simply laced Lie algebra \({\mathfrak{g}}\). This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra \({{\mathfrak{g}}^{(1)}}\), and constructing the relevant \({\Psi}\)-system among subdominant solutions. We then use the \({\Psi}\)-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum \({\mathfrak{g}}\)-KdV model. We also consider generalized Airy functions for twisted Kac–Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adamopoulou P., Dunning C.: Bethe Ansatz equations for the classical \({A_n^{(1)}}\) affine Toda field theories. J. Phys. A 47, 205205 (2014)ADSCrossRefMATHGoogle Scholar
  2. 2.
    Bazhanov V.V., Hibberd A., Khoroshkin S.: Integrable structure of W3 conformal field theory, quantum Boussinesq theory and boundary affine Toda theory. Nuclear Phys. B 622(3), 475–547 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bazhanov V.V., Lukyanov S.: Integrable structure of quantum field theory: classical flat connections versus quantum stationary states. J. High Energy Phys. 2014(9), 1–69 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B.: Spectral determinants for Schrodinger equation and Q operators of conformal field theory. J. Stat. Phys. 102, 567–576 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B.: Integrable structure of conformal field theory. II. Q-operator and DDV equation. Commun. Math. Phys. 190(2), 247–278 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B.: Higher-level eigenvalues of Q-operators and Schroedinger equation. Adv. Theor. Math. Phys. 7, 711 (2004)CrossRefMATHGoogle Scholar
  7. 7.
    Berman S., Lee T., Moody R.: The spectrum of a coxeter transformation, affine coxeter transformations, and the defect map. J. Algebra 121, 339–357 (1987)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bertola, M., Dubrovin, B., Yang, D.: Simple Lie algebras and topological ODEs. arXiv:1508.03750
  9. 9.
    Collingwood, D., McGovern, W.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Mathematical Series. Van Nostrand Reinhold Co., New-York (1993)Google Scholar
  10. 10.
    Dorey P., Dunning C., Masoero D., Suzuki J., Tateo R.: Pseudo-differential equations, and the Bethe ansatz for the classical Lie algebras. Nuclear Phys. B 772(3), 249–289 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dorey P., Dunning C., Tateo R.: Differential equations for general SU(n) Bethe ansatz systems. J. Phys. A 33(47), 8427–8441 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dorey P., Faldella S., Negro S., Tateo R.: The Bethe Ansatz and the Tzitzeica–Bullough–Dodd equation. Philos. Trans. Roy. Soc. Lond. A 371, 20120052 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dorey P., Tateo R.: Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations. J. Phys. A 32, L419–L425 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Drinfeld V.G., Sokolov V.V.: Lie algebras and equations of KdV type. Soviet J. Math. 30, 1975–2036 (1985)CrossRefMATHGoogle Scholar
  15. 15.
    Eastham, M.S.P.: The Asymptotic Solution of Linear Differential Systems, Volume 4 of London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, New York (1989). (Applications of the Levinson theorem, Oxford Science Publications)Google Scholar
  16. 16.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Higher Transcendental Functions. Vols. I, II. McGraw-Hill Book Company, Inc., New York (1953)Google Scholar
  17. 17.
    Fedoryuk M.: Asymptotic Analysis. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  18. 18.
    Feigin, B., Frenkel, E.: Integrals of motion and quantum groups. In: Integrable Systems and Quantum Groups, pp. 349–418. Springer, BerlinGoogle Scholar
  19. 19.
    Feigin, B., Frenkel, E.: Quantization of soliton systems and Langlands duality. In: Exploring New Structures and Natural Constructions in Mathematical Physics. Advanced Studies in Pure Mathematics, vol. 61, pp. 185–274. Mathematical Society of Japan, Tokyo (2011)Google Scholar
  20. 20.
    Frenkel, E., Hernandez, D.: Spectra of quantum KdV Hamiltonians, Langlands duality, and affine opers. arXiv:1606.05301
  21. 21.
    Frenkel, E., Hernandez, D.: Baxter’s relations and spectra of quantum integrable models. Duke Math. J. 164(12), 2407–2460 (2015)Google Scholar
  22. 22.
    Fuchs J., Schellekens B., Schweigert C.: From Dynkin diagram symmetries to fixed point structures. Commun. Math. Phys. 180(1), 39–97 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Fulton, W., Harris, J.: Representation Theory, Volume 129 of Graduate Texts in Mathematics. Springer, New York (1991). (A first course, Readings in Mathematics).Google Scholar
  24. 24.
    Gaiotto D., Moore G., Neitzke A.: Wall-crossing, hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Hernandez, D., Jimbo, M. Asymptotic representations and Drinfeld rational fractions. Compositio Mathematica 148(5), 1593–623 (2012)Google Scholar
  26. 26.
    Howlett R., Rylands L., Taylor D.: Matrix generators for exceptional groups of Lie type. J. Symbolic Comput. 31(4), 429–445 (2001)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups, Volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)Google Scholar
  28. 28.
    Kac V.G.: Infinite-Dimensional Lie Algebras. 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  29. 29.
    Kojima, T.: Baxter’s q-operator for the W-algebra WN. J. Phys. A Math. Theor. 41(35), 355206 (2008)Google Scholar
  30. 30.
    Kostant B.: The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Am. J. Math. 81, 973–1032 (1959)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lukyanov S.L., Zamolodchikov A.B. Quantum sine(h)-Gordon model and classical integrable equations. JHEP. 1007:008 (2010)Google Scholar
  32. 32.
    Masoero D.: Y-system and deformed thermodynamic Bethe Ansatz. Lett. Math. Phys. 94(2), 151–164 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Masoero D., Raimondo A., Valeri D.: Bethe Ansatz and the spectral theory of affine lie algebra-valued connections I. The simply-laced case. Commun. Math. Phys. 344(3), 719–750 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Miller, P.: Applied Asymptotic Analysis, Volume 75 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2006)Google Scholar
  35. 35.
    Mukhin, E., Varchenko, A.: Populations of solutions of the XXX Bethe equations associated to Kac–Moody algebras. Contemp. Math. 392, 95–102 (2005)Google Scholar
  36. 36.
    Mukhin E., Varchenko A.: Quasi-polynomials and the bethe ansatz. Geom. Topol. Monogr. 13, 385–420 (2008)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Eswara R.S.: On representations of loop algebras. Commun. Algebra. 21(6), 2131–2153 (1993)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Reshetikhin N.Y., Wiegmann P.B.: Towards the Classification of completely integrable quantum field theories. Phys. Lett. B 189, 125–131 (1987)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Sun, J.: Polynomial relations for q-characters via the ODE/IM correspondence. SIGMA Symmetry Integr. Geom. Methods Appl. 8, 028–34 (2012)Google Scholar
  40. 40.
    Suzuki, J.: Stokes multipliers, spectral determinants and T-Q relations. Sūrikaisekikenkyūsho Kōkyūroku (1221), 21–37 (2001). [Development in discrete integrable systems—ultra-discretization, quantization (Japanese) (Kyoto, 2000)]Google Scholar
  41. 41.
    Suzuki, J.: Elementary functions in thermodynamic Bethe ansatz. J. Phys. A Math. Theor. 48(20), 205204 (2015)Google Scholar
  42. 42.
    Zamolodchikov A.B.: On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories. Phys. Lett. B 253(3–4), 391–394 (1991)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Davide Masoero
    • 1
  • Andrea Raimondo
    • 1
    • 2
  • Daniele Valeri
    • 3
  1. 1.Grupo de Física Matemática da Universidade de LisboaLisbonPortugal
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano-BicoccaMilanItaly
  3. 3.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

Personalised recommendations