Skip to main content
SpringerLink
Log in

Monstrous Moonshine from orbifolds

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the Monster Module of Frenkel, Lepowsky, and Meurman as aZ 2 orbifold of a bosonic string compactified by the Leech lattice. We show that the main Conway and Norton Monstrous Moonshine properties, stating that the Thompson series for each Monster group conjugacy class has a modular invariance group of genus zero, follow from an orbifold construction based on an orbifold group composed of Monster group elements. it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each Thompson series. It is also shown that the Power Map formula of Conway and Norton follows from the same vacuum structure. Finally, we demonstrate the validity of the vacuum conjectures for sectors twisted by Leech lattice automorphisms in many cases.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Frenkel, I., Lepowsky, J., Meurman, A.: A moonshine module for the monster. In: Lepowsky, J. et al. (eds.). Vertex operators in mathematics and physics, Berlin, Heidelberg, New York: Springer 1985; Vertex operator algebras and the Monster. New York: Academic Press 1988

    Google Scholar 

  2. Griess, R.: The friendly giant. Invent. Math.69, 1 (1982)

    Article  Google Scholar 

  3. Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc.11, 308 (1979)

    Google Scholar 

  4. Borcherds, R.E.: University of Cambridge DPMMS preprint, 1989

  5. Borcherds, R.E.: Generlized Kac-Moody algebras. J. Algebra115, 501 (1988)

    Article  Google Scholar 

  6. Dixon, L., Harvey, J.A., Vafa, C., Witten, E.: Strings on orbifolds. Nucl. Phys. B261, 678 (1985)

    Article  Google Scholar 

  7. Dixon, L., Harvey, J.A., Vafa, C., Witten, E.: Strings on orbifolds II. Nucl. Phys. B274, 285 (1986)

    Article  Google Scholar 

  8. Dixon, L., Ginsparg, P., Harvey, J.: Beauty and the beast: Superconformal symmetry in a Monster Module. Commun. Math. Phys.119, 285 (1988)

    Article  Google Scholar 

  9. Narain, K.S.: New heterotic string theories in uncompactified dimensions <10. Phys. Lett.169 B, 41 (1986)

    Article  Google Scholar 

  10. Conway, J.H., Sloane, N.J.A.: Sphere packings, lattices and groups. Berlin, Heidelberg, New York: Springer 1988

    Google Scholar 

  11. Goddard, P., Olive, D.: Algebras, lattices and strings. In: Lepowsky, J. et al. (eds.). Vertex operators in mathematics and physics. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  12. Serre, J-P.: A course in arithmetic. Berlin, Heidelberg, New York: Springer 1970

    Google Scholar 

  13. Goddard, P.: Meromorphic conformal field theory. In: Proceedings of the CIRM-Luminy Conference, 1988. Singapore: World Scientific 1989

    Google Scholar 

  14. Vafa, C.: Modular invariance and discrete torsion on orbifolds. Nucl. Phys.B273, 592 (1986)

    Article  Google Scholar 

  15. Narain, K.S., Sarmadi, M.H., Vafa, C.: Asymmetric orbifolds. Nucl. Phys.B288, 551 (1987)

    Google Scholar 

  16. Thompson, J.G.: Finite groups and modular functions. Bull. Lond. Math. Soc.11, 347 (1979)

    Google Scholar 

  17. Gunning, R.C.: Lectures on modular forms. Princeton, NJ: Princeton Univ. Press 1962

    Google Scholar 

  18. Moore, G.: Atkin-Lehner symmetry. Nucl. Phys.B293, 139 (1987); Erratum. Nucl. Phys.B299, 847 (1988)

    Article  Google Scholar 

  19. Balog, J., Tuite, M.P.: The failure of Atkin-Lehner symmetry for lattice compactified strings. Nucl. Phys.B 319, 387 (1989)

    Google Scholar 

  20. Mason, G. (with an appendix by Norton, S.P.): Finite groups and modular functions. Proc. Symp. Pure Math.47, 181 (1987)

    Google Scholar 

  21. Apostol, T.M.: Modular functions and Dirichlet series in number theory. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  22. Ogg, A.P.: Hyperelliptic modular curves. Bull. Soc. Math. France102, 449 (1974)

    Google Scholar 

  23. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: An atlas of finite groups. Oxford: Clarendon Press 1985

    Google Scholar 

  24. Tits, J.: Normalisateurs de tores I. Groups de Coxeter étendus. J. Algebra4, 16 (1966)

    Article  Google Scholar 

  25. Dolan, L., Goddard, P., Montague, P.: Conformal field theory, triality and the Monster group. Preprint RU89/b1/45, DAMTP 89-29 (1989)

  26. Sorba, P., Torresani, B.: Twisted vertex operators and string theory. Int. J. Mod. Phys.3, 1451 (1988)

    Article  Google Scholar 

  27. Corrigan, E., Holowood, T.H.: Comments on the algebra of straight, twisted and intertwining vertex operators. Nucl. Phys.B304, 77 (1988)

    Article  Google Scholar 

  28. Tuite, M.P.: To appear. DIAS preprint, 1990

  29. Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Princeton, NJ: Princeton Univ. Press 1971

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland

    Michael P. Tuite

Authors
  1. Michael P. Tuite
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by N. Yu. Reshetikhin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tuite, M.P. Monstrous Moonshine from orbifolds. Commun.Math. Phys. 146, 277–309 (1992). https://doi.org/10.1007/BF02102629

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02102629

Keywords

Search

Navigation