Communications in Mathematical Physics

, Volume 344, Issue 3, pp 719–750 | Cite as

Bethe Ansatz and the Spectral Theory of Affine Lie Algebra-Valued Connections I. The simply-laced Case



We study the ODE/IM correspondence for ODE associated to \({\widehat{\mathfrak{g}}}\)-valued connections, for a simply-laced Lie algebra \({\mathfrak{g}}\). We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called \({\Psi}\)-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.


Dynkin Diagram Evaluation Representation Ordinary Differential Equa Representation Versus Monodromy Matrix 
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.
    Adamopoulou P., Dunning C.: Bethe Ansatz equations for the classical \({A_{n}^{(1)}}\) affine Toda field theories. J. Phys. A 47, 205205 (2014)ADSMathSciNetCrossRefMATHGoogle 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. Nucl. Phys. B 622(3), 475–547 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    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
  5. 5.
    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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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
  7. 7.
    Collingwood, D., McGovern, W.: Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Mathematical Series. Van Nostrand Reinhold Co., New-York (1993)Google Scholar
  8. 8.
    Destri C., de Vega H.J.: New thermodynamic Bethe ansatz equations without strings. Phys. Rev. Lett. 69, 2313–2317 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dorey P., Dunning C., Masoero D., Suzuki J., Tateo R.: Pseudo-differential equations, and the Bethe ansatz for the classical Lie algebras. Nucl. Phys. B 772(3), 249–289 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Dorey P., Faldella S., Negro S., Tateo R.: The Bethe Ansatz and the Tzitzeica-Bullough-Dodd equation. Phil. Trans. R. Soc. Lond. A 371, 20120052 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dorey P., Negro S., Tateo R.: Affine toda field theories and the bethe ansatz (2015, in preparation)Google 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.
    Dorey P., Tateo R.: On the relation between Stokes multipliers and the T-Q systems of conformal field theory. Nucl. Phys. B. 563(3), 573–602 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Drinfeld V.G., Sokolov V.V.: Lie algebras and equations of kdv type. Sov. J. Math. 30, 1975–2036 (1985)CrossRefMATHGoogle Scholar
  16. 16.
    Fedoryuk M.: Asymptotic analysis. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  17. 17.
    Feigin, B., Frenkel, E.: Quantization of soliton systems and Langlands duality. In: Exploring new structures and natural constructions in mathematical physics, Adv. Stud. Pure Math., vol. 61, pp. 185–274. Math. Soc. Japan, Tokyo (2011)Google Scholar
  18. 18.
    Fring A., Liao H., Olive D.: The Mass spectrum and coupling in affine Toda theories. Phys. Lett. B 266, 82–86 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fulton, W., Harris, J.: Representation theory, Graduate Texts in Mathematics, vol. 129. Springer-Verlag, New York, 1991. A first course, Readings in Mathematics (1991)Google Scholar
  20. 20.
    Gaiotto D., Moore G., Neitzke A.: Wall-crossing, hitchin systems, and the wkb approximation. Adv. Math. 234, 239–403 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    The GAP Group.: GAP—groups, algorithms, and programming, Version 4.7.6 (2014)Google Scholar
  22. 22.
    Ilyashenko, Y., Yakovenko, S.: Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86. American Mathematical Society, Providence (2008)Google Scholar
  23. 23.
    Kac, V., Scwarz, A.: Geometric interpretation of the partition function of 2D gravity. Phys. Lett. B 3–4(257), 329–334 (1991)Google Scholar
  24. 24.
    Kac V.G.: Infinite-dimensional Lie algebras, 3rd edn. Cambridge University Press, Cambridge (1990)Google Scholar
  25. 25.
    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
  26. 26.
    Levinson N.: The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Lukyanov S.L., Zamolodchikov A.B.: Quantum sine(h)-Gordon model and classical integrable equations. JHEP 1007, 008 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Masoero D.: Y-System and deformed thermodynamic bethe ansatz. Lett. Math. Phys. 94(2), 151–164 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Masoero D., Raimondo A., Valeri D.: Bethe Ansatz and the Spectral Theory of affine Lie algebra–valued connections II. The non simply–laced case. arXiv:1511.00895
  30. 30.
    Negro, S.: ODE/IM correspondence in Toda field theories and fermionic basis in sin(h)-Gordon model. PhD thesis, Università degli Studi di Torino (2014)Google Scholar
  31. 31.
    Reshetikhin N.Yu., Wiegmann P.B.: Towards the Classification of completely integrable quantum field theories. Phys. Lett. B 189, 125–131 (1987)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Sun, J.: Polynomial relations for q-characters via the ODE/IM correspondence. SIGMA Symmetry Integrability Geom. Methods Appl. 8: Paper 028, 34 (2012)Google Scholar
  33. 33.
    Suzuki, J.: Private communicationGoogle Scholar
  34. 34.
    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
  35. 35.
    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
    • 2
  • Daniele Valeri
    • 3
  1. 1.Grupo de Fisíca Matemática da Universidade de LisboaUniversidade de LisboaLisboaPortugal
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano-BicoccaMilanItaly
  3. 3.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

Personalised recommendations