Communications in Mathematical Physics

, Volume 109, Issue 4, pp 525–536 | Cite as

Elliptic genera and quantum field theory

  • Edward Witten


It is shown that in elliptic cohomology — as recently formulated in the mathematical literature — the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the Dirac operator inK-theory. This leads to several insights concerning both elliptic cohomology and string theory. Some of the relevant calculations have been done previously by Schellekens and Warner in a different context.


Neural Network Statistical Physic Field Theory Complex System Quantum Field Theory 
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.
    Landweber, P.S., Stong, R.: Circle actions on spin manifolds and characteristic numbers, Rutgers University preprint 1985Google Scholar
  2. 2.
    Atiyah, M.F., Hirzebruch, F.: In: Essays on topology and related subjects, pp. 18–28. Berlin, Heidelberg, New York: Springer 1970Google Scholar
  3. 3.
    Atiyah, M.F., Singer, I.M.: Ann. Math.87, 484, 586 (1968)Google Scholar
  4. 3a.
    Atiyah, M.F., Segal, G.B.: Ann. Math.87, 531 (1968)Google Scholar
  5. 3b.
    Atiyah, M.F., Bott, R.: Ann. Math.87, 456 (1968)Google Scholar
  6. 4.
    Witten, E.: Fermion quantum numbers in Kaluza-Klein theory. In: Shelter Island II: Proceedings of the 1983 Shelter Island conference on quantum field theory and the fundamental problems of physics. Khuri, N. et al. (eds.). Cambridge, MA: MIT Press 1985Google Scholar
  7. 5.
    Borsari, L.: Bordism of semi-free circle actions on spin manifolds. Trans. Am. Math. Soc. (to appear)Google Scholar
  8. 6.
    Ochanine, S.: Sur les genres multiplicatifs définis par des intégrales elliptiques. Topology (to appear)Google Scholar
  9. 7.
    Chudnovsky, D.V., Chudnovsky, G.V.: Elliptic modular functions and elliptic genera. Columbia University preprint (1985)Google Scholar
  10. 8.
    Ochanine, S.: Elliptic genera forS 1 manifolds. Lecture at conference on elliptic curves and modular forms in algebraic topology. IAS (September 1986)Google Scholar
  11. 9.
    Landweber, P.S., Ravenel, D., Stong, R.: Periodic cohomology theories defined by elliptic curves. Preprint (to appear)Google Scholar
  12. 9a.
    Landweber, P.S.: Elliptic cohomology and modular forms. To appear in the proceedings of the conference on elliptic curves and modular forms in algebraic topology. IAS (September 1986)Google Scholar
  13. 10.
    Hopkins, M., Kuhn, N., Ravenel, D.: Preprint (to appear)Google Scholar
  14. 10a.
    Hopkins, M.: Lecture at the conference on elliptic curves and modular forms in algebraic topology. IAS (September 1986)Google Scholar
  15. 11.
    Dixon, L., Harvey, J.A., Vafa, C., Witten, E.: Strings on orbifolds. Nucl. Phys. B261, 678 (1985)Google Scholar
  16. 12.
    Schellekens, A., Warner, N.: Anomalies and modular invariance in string theory, Anomaly cancellation and self-dual lattices (MIT preprints 1986). Anomalies, characters and strings (CERN preprint TH 4529/86)Google Scholar
  17. 12a.
    Pilch, K., Schellekens, A., Warner, N.: Preprint, 1986Google Scholar
  18. 13.
    Witten, E.: J. Differ. Geom.17, 661 (1982), Sect. IV. In: Anomalies, geometry, and topology. Bardeen, W., White, A. (eds.). New York: World Scientific, 1985, pp. 61–99, especially pp. 91–95Google Scholar
  19. 14.
    Atiyah, M.F., Singer, I.M.: Ann. Math.93, 119 (1971)Google Scholar
  20. 15.
    Zagier, D.: A note on the Landweber-Stong elliptic genus (October 1986)Google Scholar
  21. 16.
    Asorey, M., Mitter, P.K.: Regularized, continuum Yang-Mills process and Feynman-Kac functional integral. Commun. Math. Phys.80, 43 (1981)Google Scholar
  22. 17.
    Bern, Z., Halpern, M.B., Sadun, L., Taubes, C.: Continuum regularization of QCD. Phys. Lett.165 B, 151 (1985)Google Scholar
  23. 18.
    Eichler, M., Zagier, D.: The theory of Jacobi forms. Boston: Birkhäuser 1985Google Scholar
  24. 19.
    Witten, E.: Non-abelian bosonization in two dimensions. Commun. Math. Phys.92, 455 (1984)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Edward Witten
    • 1
  1. 1.Joseph Henry LaboratoriesPrinceton UniversityPrincetonUSA

Personalised recommendations