Skip to main content
SpringerLink
Log in
Book cover

Number Theory and Physics pp 2–19Cite as

Z/NZ Conformal Field Theories

  • Conference paper

  • 424 Accesses

Part of the book series: Springer Proceedings in Physics ((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 (1) N WZW theories and the rational gaussian model.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Friedan, S. Shenker: The analytic formulation of 2D CFT. Nucl. Phys. B. 281, 509 1987.

    Article  MathSciNet  ADS  Google Scholar 

  2. G. Moore, N. Seiberg: Polynomial equations for Rational Conformai Field Theories. Preprint IASSNS-IIEP 88/18.

    Google Scholar 

  3. G. Moore, N. Seiberg: Naturality in Conformai Field Theory. Preprint IASSNS- HEP 88/31.

    Google Scholar 

  4. G. Moore, N. Seiberg. Classical and quantum Conformai Field Theory. Preprint IASSNS-HEP 88/35.

    Google Scholar 

  5. C. Vafa: Towards classification of Conformai Field Theories. Phys. Lett. B. 206, 421 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  6. J. Birman: Braids, Links and Mapping class groups. Ann. Math. Studies 82 (Princeton University Press 1974 )

    Google Scholar 

  7. E. Witten: Quantum Field Theory and the Jones polynomial. Preprint IQSSNS 88/33 to appear in Comm. Math. Phys.

    Google Scholar 

  8. G. Segal: The definition of CFT. Oxford University Preprint.

    Google Scholar 

  9. E. Verlinde: Fusion rules and modular transformations in 2D Conformai Field Theory. Nucl. Phys. B. 300 360 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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. K. Gawedzki: Theories Conformes Séminaire Bourbaki Novembre 1988.

    Google Scholar 

  12. C. Vafa: Conformai field theories and punctured surfaces Phys. Lett. B., 199, 195 1987.

    MathSciNet  ADS  Google Scholar 

  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. J.M. Drouffe, C. Itzykson: Statistical Field Theory to appear.

    Google Scholar 

  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. A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov: Infinite Conformai Sym¬metry in 2D-field theory. Nucl. Phys. B. 241, 333 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. P. Degiovanni: Z/NZ Conformai Field Theories. Preprint LPTENS 89/03.

    Google Scholar 

  18. J.P. Serre: Representation linéaire des groupes finis. ( Hermann, Paris 1968 )

    Google Scholar 

  19. A. Cappelli, C. Itzykson, J.B. Zuber: The ADE classification of and minimal conformai field theories. Comm. Math. Phys. 113, 1 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. D. Gepner, Z. Qiu: Modular invariant partition functions for parafermionic theories. Nucl. Phys. B.(FS 19 ) 285, 423 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  21. A. Kato: Classification of modular invariant partition functions in 2D CFT. Modern Phys. Lett. A, Vol 2, 585 (1987).

    Google Scholar 

  22. M. Bauer, C. Itzykson: Affine characters and modular transformations. See in this volume.

    Google Scholar 

  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. P. Cartier: Modular forms and elliptic functions. In the other volume of this school.

    Google Scholar 

  25. S. Lang: Algebraic number theory. ( Addison-Wesley, Reading 1970 )

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  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-Sud, 24 rue Lhomond, F-75005, Paris, France

    P. Degiovanni

Authors
  1. P. Degiovanni
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Service de Physique Théorique, C.E.N Saclay, F-91191, Gif-sur-Yvette Cedex, France

    Jean-Marc Luck  & Pierre Moussa  & 

  2. Institut Henri-Poincaré, Problèmes Diophantiens, Université de Paris 6, F-75231, Paris Cedex 05, France

    Michel Waldschmidt

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Degiovanni, P. (1990). Z/NZ Conformal Field Theories. In: Luck, JM., Moussa, P., Waldschmidt, M. (eds) Number Theory and Physics. Springer Proceedings in Physics, vol 47. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75405-0_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-75405-0_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-75407-4

  • Online ISBN: 978-3-642-75405-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation