Anomalies and index theory

  • Michael Atiyah
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 208)


This has been a rather rapid survey of index theory and its relevance to Dirac operators and determinants. In particular we have seen that certain anomalies have a topological interpretation. However, I should emphasize that anomalies as understood by physicists are more precise algebraic objects than their topological counter parts. The precise relation between the algebraic and the topology is still a subject of investigation and for some new results in this direction I refer to 171.


Gauge Theory Vector Bundle Dirac Operator Gauge Field Index Theorem 
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  1. 1.
    M.F. Atiyah. The signature of fibre bundles, Collected Mathematical Papers in honour of K. Kodaira, Tokyo Univ. Press (1969).Google Scholar
  2. 2.
    M.F. Atiyah and I.M. Singer. The index of elliptic operators I, III, Ann. of Math. 87 (1968) 485–530, 546–604.Google Scholar
  3. 3.
    M.F. Atiyah and I.M. Singer. The index of elliptic operators IV, Ann. of Math. 93 (1971) 119–139.Google Scholar
  4. 4.
    M.F. Atiyah and I.M. Singer. The index of elliptic operators V, Ann. of Math. 93 (1971), 139–149.Google Scholar
  5. 5.
    M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry III, Math. Proc. Camb. Phil. Soc. 79 (1976), 71–99.Google Scholar
  6. 6.
    M.F. Atiyah and I.M. Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. USA 81 (1984).Google Scholar
  7. 7.
    O. Alvarez, I.M. Singer and B. Zumino. Gravitational anomalies and the family's index theorem, Berkeley Preprints (1984).Google Scholar
  8. 8.
    L. Alvarez-Gaumé and E. Witten, Gravitational Anomalies, Harvard Preprint, 1983 (to appear in Nuclear Physics B).Google Scholar
  9. 9.
    L. Alvarez-Gaumé and P. Ginsparg, Harvard preprint, 1983.Google Scholar
  10. 10.
    L. Baulieu, Algebraic construction of gauge invariant theories, Paris preprint, 1984 (to appear in Nuclear Physics).Google Scholar
  11. 11.
    L. Baulieu, Anomalies and gauge symmetry, Paris preprint, 1984.Google Scholar
  12. 12.
    T. Sumitani, Chiral anomalies and the generalized index theorem, Tokyo preprint, 1984.Google Scholar
  13. 13.
    C. Vafa and E. Witten, Eigenvalue inequalities for fermions in gauge theories, Princeton preprint, 1984.Google Scholar
  14. 14.
    E. Witten, A mod 2 anomaly, Princeton preprint.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Michael Atiyah
    • 1
  1. 1.Mathematical InstituteOxford

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