Z/NZ Conformal Field Theories

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

Abstract

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.

Keywords

Manifold Covariance Kato Ustin 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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