nLab
Atiyah-Singer index theorem

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.

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 lighning review of the proof is on the last pages of

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

See also

From the point of view of physics

The index theorem is supposed to have 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

Revised on November 10, 2014 21:35:49 by Urs Schreiber (86.212.18.175)