Communications in Mathematical Physics

, Volume 177, Issue 2, pp 381–398 | Cite as

Integrable structure of conformal field theory, quantum KdV theory and Thermodynamic Bethe Ansatz

  • Vladimir V. Bazhanov
  • Sergei L. Lukyanov
  • Alexander B. Zamolodchikov


We construct the quantum versions of the monodromy matrices of KdV theory. The traces of these quantum monodromy matrices, which will be called as “T-operators,” act in highest weight Virasoro modules. TheT-operators depend on the spectral parameter λ and their expansion around λ=∞ generates an infinite set of commuting Hamiltonians of the quantum KdV system. TheT-operators can be viewed as the continuous field theory versions of the commuting transfermatrices of integrable lattice theory. In particular, we show that for the values\(c = 1 - 3\frac{{3(2n + 1)^2 }}{{2n + 3}}\),n=1,2,3 .... of the Virasoro central charge the eigenvalues of theT-operators satisfy a closed system of functional equations sufficient for determining the spectrum. For the ground-state eigenvalue these functional equations are equivalent to those of the massless Thermodynamic Bethe Ansatz for the minimal conformal field theoryM2,2n+3; in general they provide a way to generalize the technique of the Thermodynamic Bethe Ansatz to the excited states. We discuss a generalization of our approach to the cases of massive field theories obtained by perturbing these Conformal Field Theories with the operator Φ1,3. The relation of theseT-operators to the boundary states is also briefly described.


Functional Equation Central Charge Closed System Conformal Field Theory Theory Version 
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.
    Itzykson, E., Saleur, H., Zuber, J.B. (eds.): Conformal Invariance and Applications to Statistical Mechanics. Singapore: World Scientific, 1988Google Scholar
  2. 2.
    Faddeev, L.D., Sklyanin, E.K., Takhtajan, L.A.: Quantum inverse scattering method. I. Theor. Math. Phys.40, 194–219 (1979) (in Russian)Google Scholar
  3. 3.
    Bogoliubov, N.M., Izergin, A.G., Korepin, V.E.: Correlation functions in integrable systems and the Quantum Inverse Scattering Method. Moscow: Nauka, 1992 (in Russian)Google Scholar
  4. 4.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. London: Academic Press, 1982Google Scholar
  5. 5.
    Zamolodchikov, Al.B.: From Tricritical Ising to Critical Ising by Thermodynamic Bethe Ansatz. Nucl. Phys.B358, 524–546 (1991)Google Scholar
  6. 6.
    Zamolodchikov, A.B., Zamolodchikov, Al.B.: Massless factorized scattering and sigma models with topological terms. Nucl. Phys.B379, 602–623 (1992)Google Scholar
  7. 7.
    Fendley, P., Saleur, H.: Massless integrable quantum field theories and massless scattering in 1+1 dimensions. Preprint USC-93-022, #hepth 9310058 (1993)Google Scholar
  8. 8.
    McCoy, B.M.: The connection between statistical mechanics and Quantum Field Theory. Preprint ITP-SB-94-07, #hepth 940384 (1994); to appear. In: Bazhanov, V.V., Burden, C.J. (eds.) Field Theory and Statistical Mechanics. Proceedings 7-th Physics Summer School at the Australian National University. Canberra. January 1994, Singapore: World Scientific, 1995Google Scholar
  9. 9.
    Yang, C.N., Yang, C.P.: Thermodynamics of one-dimensional system of bosons with repulsive delta-function potential. J. Math. Phys.10, 1115–1123 (1969)Google Scholar
  10. 10.
    Zamolodchikov, Al.B.: Thermodynamic Bethe ansatz in relativistic models: Scaling 3-state Potts and Lee-Yang models. Nucl. Phys.B342, 695–720 (1990)Google Scholar
  11. 11.
    Fendley, P.: Exited state thermodynamics. Nucl. Phys.B374, 667–691 (1992)Google Scholar
  12. 12.
    Sasaki, R., Yamanaka, I.: Virasoro algebra, vertex operators, quantum Sine-Gordon and solvable Quantum Field theories. Adv. Stud. in Pure Math.16, 271–296 (1988)Google Scholar
  13. 13.
    Eguchi, T., Yang, S.K.: Deformation of conformal field theories and soliton equations. Phys. Lett.B224, 373–378 (1989)Google Scholar
  14. 14.
    Kupershmidt, B.A., Mathieu, P.: Quantum KdV like equations and perturbed Conformal Field theories. Phys. Lett.B227, 245–250 (1989)Google Scholar
  15. 15.
    Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math.21, 467–490 (1968)Google Scholar
  16. 16.
    Miura, R.M.: Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. Phys. Rev. Lett.19, 1202–1204 (1968)Google Scholar
  17. 17.
    Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Method in the Theory of Solitons. New York: Springer, 1987Google Scholar
  18. 18.
    Feigin, B.L., Fuchs, D.B.: Representations of the Virasoro algebra. In: Faddeev, L.D., Mal'cev, A.A. (eds.) Topology. Proceedings, Leningrad 1982. Lect. Notes in Math.1060. Berlin, Heidelberg, New York: Springer, 1984Google Scholar
  19. 19.
    Dotsenko, Vl.S., Fateev, V.A.: Conformal algebra and multipoint correlation functions in 2d statistical models. Nucl. Phys.B240 [FS12], 312–348 (1984); Dotsenko, Vl.S., Fateev, V.A.: Four-point correlation functions and the operator algebra in 2d conformal invariant theories with central chargec≦1. Nucl. Phys.B251 [FS13] 691–734 (1985)Google Scholar
  20. 20.
    Fateev, V.A., Lukyanov, S.L.: Poisson-Lie group and classical W-algebras. Int. J. Mod. Phys.A7, 853–876 (1992); Fateev, V.A., Lukyanov, S.L.: Vertex operators and representations of quantum universal enveloping algebras. Int. J. Mod. Phys.A7, 1325–1359 (1992)Google Scholar
  21. 21.
    Kulish, P.P., Reshetikhin, N.Yu., Sklyanin, E.K.: Yang-Baxter equation and representation theory. Lett. Math. Phys.5, 393–403 (1981)Google Scholar
  22. 22.
    Fendley, P., Lesage, F., Saleur, H.: Solving 1d plasmas and 2d boundary problems using Jack polynomial and functional relations. Preprint USC-94-16, SPhT-94/107, #hepth 9409176 (1994)Google Scholar
  23. 23.
    de Vega, H.J., Destri, C.: Unified approach to thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories. Preprint IFUM 477/FT, LPTHE 94-28, #hepth 9407117 (1994)Google Scholar
  24. 24.
    Zamolodchikov, Al.B.: Private communicationGoogle Scholar
  25. 25.
    Kirilov, A.N., Reshetikhin, N.Yu.: Exact solution of the integrableXXZ Heisenberg model with arbitrary spin. J. Phys.A20, 1565–1585 (1987)Google Scholar
  26. 26.
    Bazhanov, V.V., Reshetikhin, N.Yu.: Critical RSOS models and conformal theory. Int. J. Mod. Phys.A4, 115–142 (1989)Google Scholar
  27. 27.
    Klümper, A., Pearce, P.A.: Conformal weights of RSOS lattice models and their fusion hierarchies. J. Phys.A183, 304–350 (1992)Google Scholar
  28. 28.
    Andrews, G., Baxter, R., Forrester, J.: Eight-vertex SOS-model and generalized Rogers-Ramanujan identities. J. Stat. Phys.35, 193–266 (1984)Google Scholar
  29. 29.
    Baxter, R.J., Pearce, P.A.: Hard hexagons: Interfacial tension and correlation length. J. Phys.A15, 897–910 (1982)Google Scholar
  30. 30.
    Kac, V.G.: Contravariant form for infinite-dimensional Lie algebras and superalgebras. Lect. Notes in Phys.94, Berlin, Heidelberg, New York: Springer, 1979, pp. 441–445Google Scholar
  31. 31.
    Zamolodchikov, Al.B.: On the TBA Equations for Reflectionless ADE Scattering Theories. Phys. Lett.B253, 391 (1991)Google Scholar
  32. 32.
    Klassen, T.R., Melzer, E.: Spectral flow between conformal field theories in 1+1 dimensions. Nucl. Phys.B370, 511–570 (1992)Google Scholar
  33. 33.
    Freund, P.G.O., Klassen, T.R., Melzer, E.:S-matrices for perturbations of certain conformal field theories. Phys. Lett.B229, 243–247 (1989)Google Scholar
  34. 34.
    Smirnov, F.A.: Reductions of Quantum Sine-Gordon Model as Perturbations of Minimal Models of Conformal Field Theory. Nucl. Phys.B337, 156–180 (1990)Google Scholar
  35. 35.
    Bazhanov, V.V., Lukyanov, S.L., Zamolodchikov, A.B.: In preparationGoogle Scholar
  36. 36.
    Kedem, R., Klassen, T.R., McCoy, B.M., Melzer, E.: Fermionic sum representations for conformal field theory characters. Phys. Lett.B307, 68–76 (1993)Google Scholar
  37. 37.
    Zamolodchikov, A.B.: Integrable field theory from conformal field theory. Adv. Stud. in Pure Math.19, 641–674 (1989)Google Scholar
  38. 38.
    LeClair, A.: Restricted Sine-Gordon theory and the minimal conformal series. Phys. Lett.B230, 103–107 (1989)Google Scholar
  39. 39.
    Yurov, V.P., Zamolodchikov, Al.B.: Truncated conformal space approach to scaling Lee-Yang model. Int. J. Mod. Phys.A5, 3221–3245 (1990)Google Scholar
  40. 40.
    Cardy, J.: Boundary conditions, fusion rules and the Verlinde formula. Nucl. Phys.B324, 581–596 (1989)Google Scholar
  41. 41.
    Ghosal, S., Zamolodchikov, A.B.: Boundary S-matrix and boundary state in two-dimensional integrable Quantum Field theory. Int. J. Mod. Phys.A9, 3841–3885 (1994)Google Scholar
  42. 42.
    Verlinde, E.: Fusion rules and modular transformations in 2d Conformal Field Theory. Nucl. Phys.B300, 360–376 (1988)Google Scholar
  43. 43.
    Zamolodchikov, Al.B.: Thermodynamic Bethe ansatz for RSOS scattering theories. Nucl. Phys.B358, 497–523 (1991)Google Scholar
  44. 44.
    Drinfel'd, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg de Vries type. J. Sov. Math.30, 1975–1980 (1985)Google Scholar
  45. 45.
    Feigin, B., Frenkel, E.: Integrals of motion and quantum groups. To appear in: Lect. Notes in Math.1620: Springer, 1995, #hepth 9310022Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Vladimir V. Bazhanov
    • 1
  • Sergei L. Lukyanov
    • 2
  • Alexander B. Zamolodchikov
    • 3
    • 4
  1. 1.Department of Theoretical Physics and Center of Mathematics and its ApplicationsIAS Australian National UniversityCanberraAustralia
  2. 2.Newman LaboratoryCornell UniversityIthacaUSA
  3. 3.Department of Physics and AstronomyRutgers UniversityPiscatawayUSA
  4. 4.L.D. Landau Institute for Theoretical PhysicsChernogolovkaRussia

Personalised recommendations