The Abel Prize pp 117-152 | Cite as

The Atiyah–Singer Index Theorem

  • Nigel HitchinEmail author
Part of the The Abel Prize book series (AP)


The Abel Prize citation for Michael Atiyah and Isadore Singer reads: “The Atiyah–Singer index theorem is one of the great landmarks of twentieth-century mathematics, influencing profoundly many of the most important later developments in topology, differential geometry and quantum field theory”. This article is an attempt to describe the theorem, where it came from, its different manifestations and a collection of applications. It is clear from the citation that the theorem spans many areas. I have attempted to define in the text the most important concepts but inevitably a certain level of sophistication is needed to appreciate all of them. In the applications I have tried to indicate how one can use the theorem as a tool in a concrete fashion without necessarily retreating into the details of proof. This reflects my own appreciation of the theorem in its various forms as part of the user community. The vision and intuition that went into its proof is still a remarkable achievement and the Abel Prize is a true recognition of that fact.

Mathematics Subject Classification (2000)

00-02 00A15 01A70 


  1. 1.
    Alvarez-Gaumé, L.: Supersymmetry and the Atiyah–Singer index theorem. Commun. Math. Phys. 90, 161–173 (1983) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atiyah, M.: Mathematician,
  3. 3.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators I. Ann. Math. 87, 484–530 (1968) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators III. Ann. Math. 87, 546–604 (1968) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators IV. Ann. Math. 93, 119–138 (1971) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators V. Ann. Math. 93, 139–149 (1971) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Atiyah, M.F., Singer, I.M., Segal, G.B.: The index of elliptic operators II. Ann. Math. 87, 531–545 (1968) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes I. Ann. Math. 86, 374–407 (1967) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes II Applications. Ann. Math. 88, 451–491 (1968) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Atiyah, M.F., Hirzebruch, F.: Spin-manifolds and group actions. In: Haefliger, A., Narasimhan, R. (eds.) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), pp. 18–28. Springer, New York (1970) CrossRefGoogle Scholar
  11. 11.
    Atiyah, M.F.: Riemann surfaces and spin structures. Ann. Sci. École Norm. Sup. 4, 47–62 (1971) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Deformations of instantons. Proc. Nat. Acad. Sci. U.S.A. 74, 2662–2663 (1977) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Atiyah, M.F., Hitchin, N.J., Drinfeld, V.G., Manin, Yu.I.: Construction of instantons. Phys. Lett. A 65, 185–187 (1978) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Springer, Berlin (1992) CrossRefGoogle Scholar
  15. 15.
    Bismut, J.-M.: A local index theorem for non-Kähler manifolds. Math. Ann. 284, 681–699 (1989) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bott, R., Taubes, C.: On the rigidity theorems of Witten. J. Am. Math. Soc. 2, 137–186 (1989) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford Univ. Press, Oxford (1990) zbMATHGoogle Scholar
  18. 18.
    Friedan, D., Windey, P.: Supersymmetric derivation of the Atiyah–Singer index and the chiral anomaly. Nucl. Phys. B 235, 395–416 (1984) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gelfand, I.M.: On elliptic equations. (Russ.) Usp. Mat. Nauk 15, 121–132 (1960) Google Scholar
  20. 20.
    Gelfand, I.M.: On elliptic equations. Russ. Math. Surv. 15, 113–123 (1960) CrossRefGoogle Scholar
  21. 21.
    Getzler, E.: A short proof of the local Atiyah–Singer index theorem. Topology 25, 111–117 (1986) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gray, J.J.: The Riemann–Roch theorem and geometry, 1854–1914. In: Proceedings of the International Congress of Mathematicians, vol. III, Berlin (1998). Doc. Math. 1998, Extra vol. III, 811–822 (electronic) Google Scholar
  23. 23.
    Gromov, M., Lawson, H.B. Jr.: The classification of simply connected manifolds of positive scalar curvature. Ann. Math. 111, 423–434 (1980) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hirzebruch, F.: Neue topologische Methoden in der algebraischen Geometrie. Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 9. Springer, Berlin (1956) zbMATHGoogle Scholar
  25. 25.
    Hirzebruch, F.: The signature theorem: reminiscences and recreation. In: Prospects in Mathematics. Ann. Math. Stud., vol. 70, pp. 3–31. Princeton Univ. Press, Princeton (1971) Google Scholar
  26. 26.
    Lichnerowicz, A.: Spineurs harmoniques. C. R. Acad. Sci. Paris 257, 7–9 (1963) MathSciNetzbMATHGoogle Scholar
  27. 27.
    Palais, R.S.: Seminar on the Atiyah–Singer Index Theorem. Ann. Math. Stud., vol. 57. Princeton Univ. Press, Princeton (1965) zbMATHGoogle Scholar
  28. 28.
    Singer, I.M.: Letter to Michael. In: Yau, S.-T. (ed.) The Founders of Index Theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer, pp. 296–297. International Press, Somerville (2003) Google Scholar
  29. 29.
    Stolz, S.: Simply connected manifolds of positive scalar curvature. Bull. Am. Math. Soc. 23, 427–432 (1990) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Mathematical InstituteOxford UniversityOxfordUK

Personalised recommendations