S Matrices of the Tricritical Ising Model and Toda Systems

  • Philippe Christe
Part of the NATO ASI Series book series (NSSB, volume 245)


After the excellent discussion of the subject presented at this conference by E. Corrigan, it would be redundant to repeat in detail the different steps that make it possible to simplify the difficult problem of the determination of the S matrices in certain integrable off-critical systems. Instead, I will only briefly review the different tools, those developed by Zamolod-chikov [1], and the arguments that connect the subject to Toda field theories [2, 3, 4], in an effort to add new comments about each of them. The Tricritical Ising Model (TIM), that will be the standard example, is discussed in more detail in ref.[5], written with G. Mussardo. The interesting examples of the D n Toda systems are discussed in [6] and in the talk of E. Corrigan at this conference. In a second part, I will discuss in more detail certain selected features of Toda systems. The rich multipole structure and the connection of the exact solution for the S functions to the perturbative expansion is discussed. Finally I will comment on the relation between unitary and non-unitary off-critical integrable models that seems very closely related. The content of this talk represents the results obtained in collaboration with Giuseppe Mussardo.


Conformal Block Minimal Solution Conformal Field Theory Order Pole Tricritical Point 
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.


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  1. [1]
    A.B. Zamolodchikov, Integrable field theory form conformai field theory, Proc. of the Taniguchi Symposium, Kyoto 1988, to appear in Advanced Studies in Pure Mathematics. Google Scholar
  2. [2]
    T. Eguchi, S.K. Yang, Deformation of Conformai Field Theories and Soliton Equations, Kyoto preprint RIFP-797.Google Scholar
  3. [3]
    A. LeClair, Restricted Sine-Gordon Theory and the Minimal Conformai Series, Princeton preprint PUPT-1124.Google Scholar
  4. [4]
    T.J. Hollowood, P. Mansfield, Rational Conformai Field Theory at, and away from, Criticality as Toda Field Theories, Oxford preprint 89–17P.Google Scholar
  5. [5]
    P. Christe, G. Mussardo, Integrable Systems away from Criticality: the Toda Field Theory and S Matrix of the Tricritical Ising Model, Santa Barbara preprint UCSBTH-89–19.Google Scholar
  6. [6]
    H.W. Braden, E. Corrigan, P.E. Dorey, and R. Sasaki, Extended Toda Field Theory and Exact S-Matrices, Durham preprint UDCPT-89–23.Google Scholar
  7. [7]
    B.A. Kuperschmidt, P. Mathieu, Quantum Korteweg-de Vries like Equations and Perturbed Conformai Field Theories, Phys. Lett. B (to appear); P. Mathieu, Integrability of Perturbed Superconformai Minimal Models, Québec preprint.Google Scholar
  8. [8]
    John L. Cardy, private communication.Google Scholar
  9. [9]
    M. Blume, V.J. Emery, and R.B. Griffiths, Phys. Rev. A4(1971)1071.ADSGoogle Scholar
  10. [10]
    R.B. Griffiths, Phys. Rev. B7(1973)545.ADSGoogle Scholar
  11. [11]
    V. Rittenberg, private communication.Google Scholar
  12. [12]
    H. Saleur, M. Henkel, Remarks on the Mass Spectrum of non Critical Coset Models from Toda Theories, Saclay preprint.Google Scholar
  13. [13]
    S. Coleman and H.J. Thun, Comm.Math.Phys. 61(1978)31.MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    A.E. Arinshtein, V.A. Fateyev, and A.B. Zamolodchikov Phys.Lett. B87(1979)389.ADSGoogle Scholar
  15. [15]
    J.L. Cardy, G. Mussardo, S-matrix of the Yang-Lee Edge Singularity in Two Dimensions, Santa Barbara preprint UCSBTH-89–11, Phys.Lett. B (to appear).Google Scholar
  16. [16]
    P.G.O. Freund, T.R. Klassen, and E. Melzer, S-matrices for perturbations of certain conformai field theories, Chicago preprint EFI 89–29.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Philippe Christe
    • 1
  1. 1.Department of PhysicsUniversity of CaliforniaSanta BarbaraUSA

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