Communications in Mathematical Physics

, Volume 119, Issue 2, pp 221–241 | Cite as

Beauty and the beast: Superconformal symmetry in a monster module

  • L. Dixon
  • P. Ginsparg
  • J. Harvey


Frenkel, Lepowsky, and Meurman have constructed a representation of the largest sporadic simple finite group, the Fischer-Griess monster, as the automorphism group of the operator product algebra of a conformal field theory with central chargec=24. In string terminology, their construction corresponds to compactification on aZ2 asymmetric orbifold constructed from the torusR24/∧, where ∧ is the Leech lattice. In this note we point out that their construction naturally embodies as well a larger algebraic structure, namely a super-Virasoro algebra with central chargeĉ=16, with the supersymmetry generator constructed in terms of bosonic twist fields.


Automorphism Group Finite Group Quantum Computing Operator Product Algebraic Structure 
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.
    Frenkel, I. B., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess monster with the modular functionJ as character. Proc. Natl. Acad. Sci. U.S.A.81, 32566 (1984); Frenkel, I. B., Lepowsky, J., Meurman, A.: A moonshine module for the monster. In: Vertex operators in mathematics and physics. Lepowsky, J., Mandelstam, S., Singer, I (eds.). Publ. Math. Sciences Res. Inst. Vol. 3, p. 231. Berlin, Heidelberg, New York: Springer 1985; Frenkel, I. B., Lepowsky, J., Meurman, A.: Vertex operator algebras and the monster. New York: Academic Press (to appear)Google Scholar
  2. 2.
    Conway, J. H., Norton, S. P.: Monstrous moonshine. Bull. Lond. Math. Soc.11, 308 (1979)Google Scholar
  3. 3.
    Belavin, A. A., Polyakov, A. M., Zamolodchikov, A. B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys.B241, 333 (1984); Friedan, D.: Notes on string theory and two dimensional conformal field theory. In: Unified string theories. Green, M., Gross, D. (eds.). Singapore: World Scientific 1986; Peskin, M.: Introduction to string and superstring theory II. 1986 TASI lectures, SLAC-PUB-4251; Banks, T.: Lectures on conformal field theory. 1987 TASI lectures, SCIPP 87/111Google Scholar
  4. 4.
    Friedan, D., Qiu, Z., Shenker, S.: Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett.52, 1575 (1984)Google Scholar
  5. 5.
    Cardy, J.: Operator content of two-dimensional conformally invariant theories. Nucl. Phys.B270[FS16], 186 (1986)Google Scholar
  6. 6.
    Moore, G.: private communicationGoogle Scholar
  7. 7.
    Gliozzi, F., Scherk, J., Olive, D.: Supersymmetry, supergravity theories and the dual spinor model. Nucl. Phys.B122, 253 (1977); Friedan, D., Qiu, Z., Shenker, S.: Superconformal invariance in two dimensions and the tricritical Ising model. Phys. Lett.151B, 37 (1985)Google Scholar
  8. 8.
    Witten, E.: Constraints on supersymmetry breaking. Nucl. Phys.B202, 253 (1982)Google Scholar
  9. 9.
    Kastor, D.: Modular invariance in superconformal models. Nucl. Phys.B280[FS18] 304 (1987)Google Scholar
  10. 10.
    Dixon, L., Ginsparg, P., Harvey, J.:ĉ=1 superconformal field theory. Nucl. Phys.B306, 470 (1988)Google Scholar
  11. 11.
    Narain, K. S., Sarmadi, M. H., Witten, E.: A note on toroidal compactification of heterotic string theory. Nucl. Phys.B279 369 (1986); Ginsparg, P.: On toroidal compactification of heterotic superstrings. Phys. Rev.D35, 648 (1987)Google Scholar
  12. 12.
    Frenkel, I. B., Kac, V. G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math.62, 23 (1980); Segal, G.: Unitary representations of some infinite dimensional groups. Commun. Math. Phys.81, 301 (1981)Google Scholar
  13. 13.
    Dixon, L., Harvey, J., Vafa, C., Witten, E.: Strings on orbifolds I, II. Nucl. Phys.B261, 678 (1985); Nucl. Phys.B274 285 (1986)Google Scholar
  14. 14.
    Vafa, C.: Modular invariance and discrete torsion on orbifolds. Nucl. Phys.B273, 592 (1986)Google Scholar
  15. 15.
    Lepowsky, J.: Calculus of twisted vertex operators. Proc. Natl. Acad. Sci. USA82, 8295 (1985)Google Scholar
  16. 16.
    Narain, K. S., Sarmadi, M. H., Vafa, C.: Asymmetric orbifolds. Nucl. Phys.B288, 551 (1987)Google Scholar
  17. 17.
    Corrigan, E., Hollowood, T.: Comments on the algebra of straight, twisted, and intertwining vertex operators. Nucl. Phys.B304, 77 (1988)Google Scholar
  18. 18.
    Tits, J.: Normalisateurs de tores 1. Groups de Coxeter étendus. J. Alg.4, 96 (1966)Google Scholar
  19. 19.
    Griess, R. L., Jr.,: The friendly giant. Invent. Math.69, 1 (1982); Tits, J.: On R. Griess' ‘Friendly Giant’. Invent. Math.78, 491 (1984)Google Scholar
  20. 20.
    Conway, J. H., Sloane, N. J. A.: Twenty-three constructions for the Leech lattice. Proc. Roy. Soc. Lond. Ser.A381, 275 (1982)Google Scholar
  21. 21.
    Dixon, L., Friedan, D., Martinec, E., Shenker, S.: The conformal field theory of orbifolds. Nucl. Phys.B282, 13 (1987); Hamidi, S., Vafa, C.: Interactions on orbifolds. Nucl. Phys.B279, 465 (1987)Google Scholar
  22. 22.
    Mason G. (with an appendix by Norton, S.P.),: Finite groups and modular functions. Proc. Symposia in Pure Math.47, 181 (1987)Google Scholar
  23. 23.
    Moore, G.: Atkin-Lehner symmetry. Nucl. Phys.B293 139 (1987). Erratum ibid.B299, 847 (1988)Google Scholar
  24. 24.
    Ogg, A.: Hyperelliptic modular curves. Bull. Soc. Math. France102, 449 (1974)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • L. Dixon
    • 1
  • P. Ginsparg
    • 2
  • J. Harvey
    • 1
  1. 1.Physics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.Lyman Laboratory of PhysicsHarvard UniversityCambridgeUSA

Personalised recommendations