Letters in Mathematical Physics

, Volume 83, Issue 2, pp 163–179 | Cite as

The Orbifold Transform and its Applications



We discuss the notion of the orbifold transform, and illustrate it on simple examples. The basic properties of the transform are presented, including transitivity and the exponential formula for symmetric products. The connection with the theory of permutation orbifolds is addressed, and the general results illustrated on the example of torus partition functions.

Mathematics Subject Classification (2000)



finitely generated groups permutation orbifolds wreath products 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bakalov B. and Kirillov A.A. (2001). Lectures on Tensor Categories and Modular Functors. University Lecture Series, vol. 21. American mathematical society, Providence Google Scholar
  2. 2.
    Bantay P. (1998). Characters and modular properties of permutation orbifolds. Phys. Lett. B 419: 175–178 MATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Bantay P. (2000). Frobenius–Schur indicators, the Klein-bottle amplitude and the principle of orbifold covariance. Phys. Lett. B 488: 207–210 MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Bantay P. (2001). Orbifoldization, covering surfaces and uniformization theory. Lett. Math. Phys. 57: 1–5 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bantay P. (2002). Permutation orbifolds. Nucl. Phys. B 633: 365–378 MATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Bantay P. (2003). The kernel of the modular representation and the Galois action in RCFT. Commun. Math. Phys. 233: 423–438 MATHADSMathSciNetGoogle Scholar
  7. 7.
    Bantay P. (2003). Symmetric products, permutation orbifolds and discrete torsion. Lett. Math. Phys. 63: 209–218 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Beardon A.F. (1983). The Geometry of Discrete Groups, GTM, vol. 91. Springer, New York Google Scholar
  9. 9.
    Belavin A.A., Polyakov A.M. and Zamolodchikov A.B. (1984). Infinite conformal symmetry in two-dimensional Quantum Field Theory. Nucl. Phys. B 241: 333 MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Borisov L., Halpern M.B. and Schweigert C. (1998). Int. J. Mod. Phys. A 13: 125 MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Cohen C. (1998). A Course in Computational Algebraic Number Theory, GTM, vol. 138. Springer, New York Google Scholar
  12. 12.
    Dijkgraaf, R.: Discrete torsion and symmetric products. hep-th/9912101 (1999)Google Scholar
  13. 13.
    Dijkgraaf R., Moore G., Verlinde E. and Verlinde H. (1997). Commun. Math. Phys. 185: 197 MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Dixon J.D. and Mortimer B. (1996). Permutation Groups, GTM, vol. 163. Springer, New York Google Scholar
  15. 15.
    Dixon L.J., Friedan D., Martinec E.J. and Shenker S.H. (1987). The conformal field theory of orbifolds. Nucl. Phys. B 282: 13–73 CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Dixon L.J., Harvey J.A., Vafa C. and Witten E. (1985). Strings on orbifolds. Nucl. Phys. B 261: 678–686 CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Dixon L.J., Harvey J.A., Vafa C. and Witten E. (1986). Strings on orbifolds 2. Nucl. Phys. B 274: 285–314 CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Farkas H.M. and Kra I. (1980). Riemann Surfaces, GTM, vol. 71. Springer, New York Google Scholar
  19. 19.
    Forster O. (1981). Lectures on Riemann Surfaces, GTM, vol. 81. Springer, New York Google Scholar
  20. 20.
    Di Francesco P., Mathieu P. and Sénéchal D. (1997). Conformal Field Theory. Springer, New York MATHGoogle Scholar
  21. 21.
    Frenkel I., Lepowsky J. and Meurman A. (1988). Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, vol. 134. Academic Press, New York Google Scholar
  22. 22.
    Green M., Schwarz J. and Witten E. (1987). Superstring Theory, vol. 1–2. Cambridge University Press, Cambridge Google Scholar
  23. 23.
    Kac V.G. (1997). Vertex Operators for Beginners. University Lecture Series, vol. 10. American Mathematical Society, Providence Google Scholar
  24. 24.
    Kerber A. (1971). Representations of Permutation Groups. Springer, Berlin MATHGoogle Scholar
  25. 25.
    Klemm A. and Schmidt M.G. (1990). Phys. Lett. B 245: 53 CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Lunin O. and Mathur S.D. (2001). Correlation functions for m(n)/s(n) orbifolds. Commun. Math. Phys. 219: 399–442 MATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Magnus W., Karras A. and Solitar D. (1966). Combinatorial Group Theory. Interscience Publishers. Wiley, New York Google Scholar
  28. 28.
    Polchinski J. (1998). String Theory, vol. 1–2. Cambridge University Press, Cambridge Google Scholar
  29. 29.
    Robinson D.J.S. (1982). A Course in the Theory of Groups, GTM, vol. 80. Springer, Berlin Google Scholar
  30. 30.
    Stanley R. (1999). Enumerative combinatorics. Cambridge University Press, Cambridge Google Scholar
  31. 31.
    Turaev V.G. (1994). Quantum Invariants of Knots and 3-Manifolds, Studies in Mathematics, vol. 18. de Gruyter, Berlin Google Scholar
  32. 32.
    Xu F. (2006). Some computations in the cyclic permutations of completely rational nets. Commun. Math. Phys. 267: 757 MATHCrossRefADSGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsEötvös Loránd UniversityBudapestHungary

Personalised recommendations