Z/NZ Conformal Field Theories

  • P. Degiovanni
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 47)


In this paper, we give a brief introduction to Conformal Field Theory (CFT) following the presentation of G. Segal. We explain how to reconstruct part of a CFT from its fusion rules. The possible choices of S matrices are indexed by some automorphisms of the fusion algebra. We illustrate this procedure by computing the modular properties of the possible genus-one characters when the fusion algebra is the representation algebra of a finite group. We also classify the modular invariant partition functions of these theories. We recover as special cases the A N (1) WZW theories and the rational gaussian model.


Partition Function Riemann Surface Conformal Block Conformal Field Theory Fusion Rule 
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]
    D. Friedan, S. Shenker: The analytic formulation of 2D CFT. Nucl. Phys. B. 281, 509 1987.MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    G. Moore, N. Seiberg: Polynomial equations for Rational Conformai Field Theories. Preprint IASSNS-IIEP 88/18.Google Scholar
  3. [3]
    G. Moore, N. Seiberg: Naturality in Conformai Field Theory. Preprint IASSNS- HEP 88/31.Google Scholar
  4. [4]
    G. Moore, N. Seiberg. Classical and quantum Conformai Field Theory. Preprint IASSNS-HEP 88/35.Google Scholar
  5. [5]
    C. Vafa: Towards classification of Conformai Field Theories. Phys. Lett. B. 206, 421 (1988)MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    J. Birman: Braids, Links and Mapping class groups. Ann. Math. Studies 82 (Princeton University Press 1974 )Google Scholar
  7. [7]
    E. Witten: Quantum Field Theory and the Jones polynomial. Preprint IQSSNS 88/33 to appear in Comm. Math. Phys.Google Scholar
  8. [8]
    G. Segal: The definition of CFT. Oxford University Preprint.Google Scholar
  9. [9]
    E. Verlinde: Fusion rules and modular transformations in 2D Conformai Field Theory. Nucl. Phys. B. 300 360 (1988)MathSciNetADSMATHCrossRefGoogle Scholar
  10. [10]
    C. Itzykson: Level-one Kac-Moody characters and modular invariance. In P. Binetruy, P. Sorba, R. Stora, editors, Conformai Field Theories and related topics. Nucl. Phys. B. (Proc. Suppl.) 5, 150 ( North Holland, Amsterdam 1988 )Google Scholar
  11. [11]
    K. Gawedzki: Theories Conformes Séminaire Bourbaki Novembre 1988.Google Scholar
  12. [12]
    C. Vafa: Conformai field theories and punctured surfaces Phys. Lett. B., 199, 195 1987.MathSciNetADSGoogle Scholar
  13. [13]
    R. Dijkgraaf: Recent progress in rational conformai field theory. To appear in the proceedings of Les Ilouches XLIX-th summer school Strings, Fields and Critical phenomena E. Brézin, J. Zinn-Justin editors.Google Scholar
  14. [14]
    J.M. Drouffe, C. Itzykson: Statistical Field Theory to appear.Google Scholar
  15. [15]
    P. Ginsparg: Applied Conformai Field Theory. To appear in the proceedings of Les Houches XLIX-th summer school Strings, Fields and Critical phenomena E. Brézin, J. Zinn-Justin editors.Google Scholar
  16. [16]
    A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov: Infinite Conformai Sym¬metry in 2D-field theory. Nucl. Phys. B. 241, 333 (1984)MathSciNetADSMATHCrossRefGoogle Scholar
  17. [17]
    P. Degiovanni: Z/NZ Conformai Field Theories. Preprint LPTENS 89/03.Google Scholar
  18. [18]
    J.P. Serre: Representation linéaire des groupes finis. ( Hermann, Paris 1968 )Google Scholar
  19. [19]
    A. Cappelli, C. Itzykson, J.B. Zuber: The ADE classification of and minimal conformai field theories. Comm. Math. Phys. 113, 1 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  20. [20]
    D. Gepner, Z. Qiu: Modular invariant partition functions for parafermionic theories. Nucl. Phys. B.(FS 19 ) 285, 423 (1987)MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    A. Kato: Classification of modular invariant partition functions in 2D CFT. Modern Phys. Lett. A, Vol 2, 585 (1987).Google Scholar
  22. [22]
    M. Bauer, C. Itzykson: Affine characters and modular transformations. See in this volume.Google Scholar
  23. [23]
    R. Dijkgraaf, E. Verlinde: Modular invariance and the fusion algebra. In P. Bi-netruy, P. Sorba, R. Stora, editors, Conformai Field Theories and related topics. Nucl. Phys. B. (Proc. Suppl.) 5, 87 (North Holland, 1988 )Google Scholar
  24. [24]
    P. Cartier: Modular forms and elliptic functions. In the other volume of this school.Google Scholar
  25. [25]
    S. Lang: Algebraic number theory. ( Addison-Wesley, Reading 1970 )MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • P. Degiovanni
    • 1
  1. 1.LPTENS, Unité propre de Recherche du Centre National de la Recherche Scientifique, associée à l’Ecole Normale Supérieure et à l’Université de Paris-SudParisFrance

Personalised recommendations