nLab
Atiyah-Singer index theorem

Contents

Contents

Statement

Michael Atiyah and Isadore Singer associated two differently defined numbers to an elliptic operator on a manifold: the topological index and the analytical index. The index theorem asserts that the two are equal.

The index theorem generalizes earlier results such as the Riemann-Roch theorem.

References

General

The original articles are

  • Michael Atiyah, Isadore Singer, The index of elliptic operators I, Ann. of Math. (2) 87 (1968) pp. 484–530; III, Ann. of Math. (2) 87 (1968) pp. 546–604; IV, Ann. of Math. (2) 93 (1971) pp. 119–138; V, Ann. of Math. (2) 93 (1971) pp. 139–149

  • Michael Atiyah, Graeme Segal, The index of elliptic operators II, Ann. of Math. (2) 87 (1968) pp. 531–545

A proof of the Atiyah-Singer index theorem in terms of KK-theory/E-theory has been given by Nigel Higson, an account is in

A lightning review of the proof is on the last pages of

  • Introduction to KK-theory and E-theory, Lecture notes (Lisbon 2009) (pdf slides)

An introductory talk is given in

  • Rafe Mazzeo, The Atiyah-Singer Index theorem: what it is and why you should care, (slides)

See also

From the point of view of physics

The index theorem has an interpretation in terms of the quantum field theory of the superparticle on the given space.

Traditional physics arguments along these lines include for instance

Last revised on January 12, 2019 at 07:20:17. See the history of this page for a list of all contributions to it.