Skip to main content
SpringerLink
Log in

Sporadic Groups and String Theory

  • Chapter
First European Congress of Mathematics

Part of the book series: Progress in Mathematics ((PM,volume 3))

Abstract

I will start by giving some well known product identities. The first is

$$ {\sum\limits_{n \in {\mathbf{Z}}} {{{\left( { - 1} \right)}^n}q} ^{3{{\left( {n + 1/6} \right)}^2}/2}} = \,{q^{1/24}}\prod\limits_{n > 0} {\left( {1\, - \,{q^n}} \right)} $$

.

This is an expanded version of my talk at the ECM.

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

Access this chapter

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the monster, Proc. Natl Acad. Sci. USA 83 (1986), 3068–3071.

    Article  MathSciNet  Google Scholar 

  2. R. E. Borcherds, Generalized Kac-Moody algebras, J. Algebra 115 (1988), 501–512.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. E. Borcherds, The monster Lie algebra, Adv. Math. 83 (1) (1990).

    Google Scholar 

  4. R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent Math. 109 (1992), 405–444.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. H. Conway, S. Norton, Monstrous moonshine, Bull. Lond. Math. Soc. 11 (1979), 308–339.

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635–652.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Eichler, D. Zagier, The theory of Jacobi forms, Birkhäuser, Basel, 1985.

    MATH  Google Scholar 

  8. I. B. Frenkel, J. Lepowsky, A. Meurman, Vertex operator algebras and the monster, Academic Press, New York, 1988.

    MATH  Google Scholar 

  9. I. B. Frenkel, H. Garland, G. Zuckerman, Semi-infinite cohomology and string theory, Proc. Natl. Acad. Set. USA 83 (1986), 8442–8446.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Garland and J. Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), 37–76.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Goddard, C. B. Thorn, Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model, Phys. Lett. B 40 (2), (1972), 235–238.

    Google Scholar 

  12. V. G. Kac, Infinite dimensional Lie algebras, third edition, Cambridge University Press, New York, 1990.

    Book  MATH  Google Scholar 

  13. M. Koike, On Replication Formula and Hecke Operators. Nagoya University, preprint.

    Google Scholar 

  14. B. Kost ant, Lie algebra cohomology and the generalized Borel-Weil theorem, Annals of Math. 74 (1961), 329–387.

    Article  MathSciNet  Google Scholar 

  15. D. LĂĽst, S. Theisen, Lectures on string theory, Lecture Notes in Physics 346 (1989), Springer-Verlag, Heidelberg.

    Google Scholar 

  16. S. P. Norton, More on moonshine, in: Computational group theory, Academic Press, New York, 1984, 185–193.

    Google Scholar 

  17. S. P. Norton, Generalized Moonshine, Proc. Symp. Pure Math. 47 (1987), 208–209.

    MathSciNet  Google Scholar 

  18. J-P. Serre, Sur la lacunarite des puissances de η, Glasgow Math. Jour. 27 (1985), 203 - 221.

    Article  MATH  Google Scholar 

  19. D. Zagier, Eisenstein series and the Riemann zeta function, in: Automorphic forms, representation theory and arithmetic, Springer-Verlag, Heidelberg, 1981.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DPMMS, 16 Mill Lane, Cambridge, CB2 1SB, England

    Richard E. Borcherds

Authors
  1. Richard E. Borcherds
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques Fondamentales, Université Pierre et Marie Curie, 4, place Jussieu, 75252, Paris Cedex 05, France

    Anthony Joseph  & Rudolf Rentschler  & 

  2. The Weizmann Institute of Science, 76100, Rehovot, Israel

    Anthony Joseph

  3. Mathématiques Bâtiment 425, Université de Paris-Sud, 91405, Orsay Cedex, France

    Fulbert Mignot

  4. Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, 4, place Jussieu, 75252, Paris Cedex, France

    François Murat

  5. Laboratoire de Statistique Médicale, Université Paris V, 45, rue des Saints Pères, 75270, Paris Cedex, France

    Bernard Prum

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Birkhäuser Verlag

About this chapter

Cite this chapter

Borcherds, R.E. (1994). Sporadic Groups and String Theory. In: Joseph, A., Mignot, F., Murat, F., Prum, B., Rentschler, R. (eds) First European Congress of Mathematics . Progress in Mathematics, vol 3. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9110-3_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9110-3_13

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9911-6

  • Online ISBN: 978-3-0348-9110-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation